University of Southern California Dissertations and Theses

Medium and high-frequency vibration analysis of flexible distributed parameter systems

(USC Thesis Other)

Medium and high-frequency vibration analysis of flexible distributed parameter systems

PDF

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content MEDIUM AND HIGH-FREQUENCY VIBRATION ANALYSIS OF FLEXIBLE
DISTRIBUTED PARAMETER SYSTEMS
by
Yichi Zhang
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)
August 2023
Copyright 2023 Yichi Zhang
ii

Table of Contents
List of Tables ............................................................................................................................................... v
List of Figures ............................................................................................................................................. vi
Abstract ....................................................................................................................................................... ix
Chapter 1 Introduction ............................................................................................................................... 1
Chapter 2 Augmented Distributed Transfer Function Method.............................................................. 7
2.1 Introduction ......................................................................................................................................... 7
2.2 Traditional Distributed Transfer Function Method ............................................................................. 7
2.3 Augmented Formulation for a Multi-Body Structure ....................................................................... 11
2.4 Mid- and High-frequency vibration analysis by the augmented DTFM ........................................... 14
2.4.1 The 𝑠 -domain Solution ............................................................................................................... 14
2.4.2 Eigenvalue Problem ................................................................................................................... 15
2.4.3 Frequency Response .................................................................................................................. 17
2.4.4 Transient Response .................................................................................................................... 17
2.4.5 Dynamic Stress and Strain ......................................................................................................... 18
Chapter 3 Vibration Analysis of Beam Structures at Mid- and High-Frequencies ............................ 20
3.1 Introduction ....................................................................................................................................... 20
3.2 Augmented Formulation for Beam Structures .................................................................................. 21
3.2.1 Euler-Bernoulli Beam Theory .................................................................................................... 21
3.2.2 Timoshenko Beam Theory ......................................................................................................... 23
3.3 Internal Forces of Beam Elements .................................................................................................... 25
3.3.1 Euler-Bernoulli Beam Elements ................................................................................................ 25
3.3.2 Timoshenko Beam Elements ..................................................................................................... 26
3.4 Boundary Conditions in Beam Structures ......................................................................................... 27
3.5 A demonstrative Example: Double-beam Structure ......................................................................... 30
3.6 Numerical Examples ......................................................................................................................... 33
iii

3.6.1 A Two-Beam Structure .............................................................................................................. 34
3.6.2 A Fifteen-Beam Structure .......................................................................................................... 35
3.6.3 A Thirty-Beam Structure ........................................................................................................... 37
Chapter 4 Transient Vibration Analysis of Beam Structures at Mid- and High-Frequencies .......... 54
4.1 Introduction ....................................................................................................................................... 54
4.2 Transient Analysis Tools .................................................................................................................. 54
4.2.1 Transient Analysis Tool 1: The Method of Inverse Laplace Transform .................................... 54
4.2.2 Transient Analysis Tool 2: The Generalized Modal Expansion ................................................ 57
4.2.3 Similarity and Difference Between Two Analysis Tools .......................................................... 59
4.3 Modal Reduction for Computational Efficiency ............................................................................... 60
4.4 Numerical Example .......................................................................................................................... 63
Chapter 5 Vibration Analysis of Thin Plates by Distributed Transfer Function Method ................. 72
5.1 Introduction ....................................................................................................................................... 72
5.2 Problem Statement ............................................................................................................................ 74
5.2.1 A Single Lévy Plate Resting on A Viscoelastic Foundation ...................................................... 74
5.2.2 Sandwich Plates with Viscoelastic Layers ................................................................................. 82
5.3 Models of Viscoelastic Layers .......................................................................................................... 87
5.4 Vibration Analysis by the Distributed Transfer Function Method ................................................... 89
5.4.1 Eigenvalue Problem ................................................................................................................... 89
5.4.2 Frequency Response .................................................................................................................. 90
5.5 Numerical Examples ......................................................................................................................... 91
Chapter 6 Vibration Analysis of Cylindrical Shell Structures ........................................................... 110
6.1 Introduction ..................................................................................................................................... 110
6.2 Distributed Transfer Function Formulation for Cylindrical Shells ................................................. 110
6.2.1 Distributed Transfer Function Formulation for A Single Cylindrical Shell ............................ 110
6.2.2 Distributed Transfer Function Formulation for Cylindrical Shells Connected by Viscoelastic
Layers ................................................................................................................................................ 113
6.3 A Simply Supported Shell Example ............................................................................................... 115
iv

Chapter 7 Conclusions ............................................................................................................................ 120
References ................................................................................................................................................ 123

v

List of Tables
Table 3.1 Structure parameters of the double-beam structure .................................................................... 51
Table 3.2 The natural frequencies fkof the elastically connected double-beam structure with fixed
boundary conditions (Hz) ........................................................................................................................... 52
Table 3.3 The natural frequencies fk of the elastically connected double-beam structure with simply
supported boundary conditions (Hz) ........................................................................................................... 53
Table 5.1 Natural frequencies 𝑓𝑘 of the elastically connected double-plate structure with SSSS
boundary conditions (Hz) .......................................................................................................................... 108
Table 5.2 Natural frequencies 𝑓𝑘 of the three-layer plate with partially connected elastic layers with
SSFF boundary conditions (Hz) ................................................................................................................ 109
Table 6.1 The natural frequencies 𝑓 0,𝑛 of the thin shell (Hz) ................................................................. 119

vi

List of Figures
Figure 2.1 A two-beam frame ..................................................................................................................... 19
Figure 3.1 A three-span beam carrying lamped masses .............................................................................. 41
Figure 3.2 An elastically connected double beam structure ....................................................................... 41
Figure 3.3 A two-span beam structure with a point-wise load ................................................................... 42
Figure 3.4 Nondimensionalized displacement at the middle point of the beam element 2 ......................... 42
Figure 3.5 Nondimensional displacement along the beam element 1 ......................................................... 43
Figure 3.6 Nondimensional energy flows at the joint point of two beams ................................................. 43
Figure 3.7 A beam frame with 15 elements ................................................................................................ 44
Figure 3.8 Nondimensional energy density along beam element 1 ............................................................ 44
Figure 3.9 Nondimensional energy density at the middle point of beam element 2 ................................... 45
Figure 3.10 A two-dimensional beam frame: simplified car frame ............................................................ 45
Figure 3.11 The spatial distribution of the time-averaged transverse displacement of beam
components (10) and (27) ........................................................................................................................... 46
Figure 3.12 The spatial distribution of the time-averaged shear force of beam components (8) and
(11) .............................................................................................................................................................. 47
Figure 3.13 Upper plot: The time-averaged angle of bending and angle of rotation. Lower plot: The
ratio of the shear displacement divided by the angle of rotation ................................................................ 48
Figure 3.14 The spatial distribution of the time-averaged shear force of beam components (8) and
(11) with the excitation frequency 𝑓 =5×103𝐻𝑧 ................................................................................... 49
Figure 3.15 The spatial distribution of the time-averaged shear force of beam components (8) and
(11) with the excitation frequency 𝑓 =5×104𝐻𝑧 ................................................................................... 50
Table 3.1 Structure parameters of the double-beam structure .................................................................... 51
Table 3.3 The natural frequencies 𝑓𝑘 of the elastically connected double-beam structure with
simply supported boundary conditions (Hz) ............................................................................................... 53
vii

Figure 4.1 Flowchart of the augmented DTFM .......................................................................................... 66
Figure 4.2 A five-beam frame ..................................................................................................................... 67
Figure 4.3 The transverse displacement at point G by DTFM .................................................................... 68
Figure 4.4 The transverse displacement at point G by DTFM and FEM .................................................... 69
Figure 4.5 The transverse displacement at point G by reduced model ....................................................... 70
Figure 4.6 The bending moment and shear force of at point G .................................................................. 71
Figure 5.1 A rectangular plate modeled by the Kirchhoff-Love plate theory resting on a viscoelastic
foundation ................................................................................................................................................... 97
Figure 5.2 A two-layer sandwich plate with viscoelastic layer, side view ................................................. 98
Figure 5.3 Mechanical schematic of Kelvin-Voigt viscoelastic model ...................................................... 99
Figure 5.4 Mechanical schematic of Maxwell viscoelastic model ........................................................... 100
Figure 5.5 (a) Mechanical schematic of Kelvin representation of standard linear solid model; (b)
Mechanical schematic of Maxwell representation of standard linear solid model ................................... 101
Figure 5.6 A three-layer sandwich plate with partially connected viscoelastic layers, side view ............ 102
Figure 5.7 The spatial distribution of the magnitude of transverse displacement along the line 𝑦 =
0.9 m on the upper plate and the line 𝑦 =0.6 m on the lower plate of the elastically connected two-
layer sandwich plate (SSFF boundary condition), with excitation at point 𝑥 =0.5 m, 𝑦 =0.9 m on
the upper plate and the excitation frequency 𝑓 =200 Hz: solid line – DTFM with 50 modes; dashed
line – FEM with 200 elements; dotted line – FEM with 3,200 element ................................................... 103
Figure 5.8 The magnitude of transverse displacement and shear force 𝑄𝑦 at point 𝑥 =0.5 m, 𝑦 =
0.9 m on the upper plate (SSSS boundary condition, Kelvin-Voigt model), with excitation at point
𝑥 =0.5 m, 𝑦 =0.9 m on the upper plate and the excitation frequency from 𝑓 =9,500 Hz to 𝑓 =
10,500 Hz: solid line – DTFM with 500 modes; dashed line – Analytical solution with 90,000
terms .......................................................................................................................................................... 104
viii

Figure 5.9 The spatial distribution of the magnitude of bending moment 𝑀𝑦𝑦 and shear force 𝑄𝑦
at point 𝑥 =0.5 m, 𝑦 =0.6 m on the lower plate (SSSS boundary condition, Kelvin-Voigt model),
with excitation at point 𝑥 =0.5 m, 𝑦 =0.9 m on the upper plate and the excitation frequency 𝑓 =
5×106 Hz: solid line – DTFM with 500 modes; dotted line – Analytical solution with 250,000
terms .......................................................................................................................................................... 105
Figure 5.10 The spatial distribution of the magnitude of transverse displacement, bending moment
𝑀𝑦𝑦 and shear force 𝑄𝑦 at point 𝑥 =0.55 m, 𝑦 =1.2 m on the middle plate (SSFF boundary
condition, standard linear solid model by Kelvin representation), with excitation at point 𝑥 =
0.55 m, 𝑦 =1.2 m on the top plate and the excitation frequency 𝑓 =2×105 Hz: solid line –
DTFM with 200 modes; dotted line – DTFM with 500 modes ................................................................ 106
Figure 5.11 The spatial distribution of the magnitude of transverse displacement, bending moment
𝑀𝑦𝑦 and shear force 𝑄𝑦 at point 𝑥 =0.55 m, 𝑦 =1.2 m on the middle plate (SSFF boundary
condition, standard linear solid model by Maxwell representation), with excitation at point 𝑥 =
0.55 m, 𝑦 =1.2 m on the top plate and the excitation frequency 𝑓 =2×105 Hz: solid line –
DTFM with 200 modes; dotted line – DTFM with 500 modes ................................................................ 107
Figure 6.1 Two cylindrical shells connected by the viscoelastic layer ..................................................... 117
Figure 6.2 Spatial distribution of the cylindrical shell at 𝜃 =0 under the excitation frequency 𝑓 =
5,000 Hz by 50-mode DTFM ................................................................................................................... 118

ix

Abstract
Vibration analysis of complex flexible distributed parameter systems is important to the
research and product design in automobile, aerospace, civil, machinery, and ship industries.
Depending on the frequency spectrum of interest, vibration analyses of flexible structures are
usually categorized into three groups: low-frequency analysis, mid-frequency analysis, and high-
frequency analysis. Through the years, numerical methods have been developed for vibration
analysis of complex structures. Low-frequency analysis tools, such as finite element analysis
(FEA), do not work well for mid- and high-frequency problems because they require a large
number of degrees-of-freedom and consequently costly computation, and sensitive to material
properties and boundary conditions. High-frequency analysis tools, such as the statistical energy
analysis (SEA) and energy flow analysis (EFA), cannot work properly for the mid-frequency
vibration analysis because the high-modal-density assumption needed by these methods may not
hold anymore. Existing mid-frequency analysis tools, such as the hybrid FEA/EFA method, rely
on the energy-based formulation, which can only provide energy densities and energy flows in the
frequency-domain. Local information about displacements and internal forces cannot be obtained
from them easily and much more additional work is required to obtain the transient response. Thus,
existing methods are not suitable for many mid-frequency problems.
An analytical method is developed for modeling and analysis of flexible distributed parameter
structures in this work. In this method, the vibration of the flexible beam, plate and shell structures
is modeled by the augmented formulation of the distributed transfer function method (DTFM).
The formulation does not rely on spatial discretization of structures and treats different types of
connection, and general boundary conditions in a unified manner. A highlight of the new method
x

is that it can deliver both frequency response and transient response with detailed information on
local displacement, slope, bending moment, and shear force for different frequency regions, from
low to high, which otherwise might be difficult to achieve by conventional analyses. For the
transient response, the proposed method does not require numerical integration and provides a
platform for model reduction in mid- and high-frequency analyses. The proposed method is
demonstrated in several examples. The accuracy of the method is validated with popular
conventional methods, such as modal analysis, FEA, and EFA. The efficiency is seen from the
comparison. As a result, the proposed method is a useful tool for design and optimization of
complex flexible structures.

1

Chapter 1 Introduction
Vibration analysis is essentially important to the research and development of complex
flexible structures for various applications in automobile, aerospace, civil, machinery, and ship
industries. Depending on the frequency spectrum of interest, vibration problems of flexible
structures are usually investigated in three frequency regions: low-frequency region, medium-
frequency (mid-frequency) region, and high-frequency region (Nefske, 1982; Lyon, 1995;
Rabbiolo, 2004; Vlahopoulos, 1999). When all vibrating substructures are short compared with
the wavelength, the problem is defined as the low-frequency vibrational problem, and the
frequency region is regarded as the low-frequency region. In contrast, the high-frequency region
is defined as a region where all vibrating substructures are long compared with the wavelength
(Vlahopoulos, 1999). Vibration analyses in these regions are subsequently known as low-, mid-
and high-frequency analyses. Through the years, many methods have been developed for vibration
analysis of complex structures, including the finite element analysis (FEA) (Cook, 2001; Bathe,
2006), the wave propagation methods (Von Flotow, 1986; Yong, 1992), and modal analysis
(Meirovitch, 1980).
Vibration analysis of structures is usually undertaken in two ways: frequency-domain analysis
and time-domain analysis (transient vibration analysis). For frequency-domain analysis, several
methods are available. The finite element analysis (FEA) is the most popular tool for low-
frequency vibration analysis (Cook, 2001; Bathe, 2006). In the FEA, a structure is discretized into
many small elements, and the required number of elements increases as the frequencies of interest
become high. The FEA may not be valid for mid- and high-frequency problems due to the
requirement for an incredible small mesh size and incurred errors. The statistical energy analysis
(SEA), on the other hand, is a suitable approach to solution of high-frequency vibro-acoustic
2

problems (Lyon, 1995; Woodhouse, 1980, 1981). It has been shown that both the SEA and
classical modal analysis can yield the same numerical results in a very high-frequency region
(Dowell, 1985). In the SEA, each subsystem of the structure is treated as one “element”, and the
governing equation is obtained according to the energy balance in the entire structure. The
interconnections among “elements” are modeled with coupling loss factors. Instead of its utility in
mid- to high-frequency regions, the SEA cannot provide detailed information inside subsystems
(Nefske, 1989).
Inspired by the SEA and FEA, the energy flow analysis (EFA) was developed for the high-
frequency vibration analyses to obtain the local information in subsystems (Nefske, 1989;
Wohlever, 1992; Bouthier, 1992; Cho, 1993, 1998). In the EFA, the energy flow behavior in a
subsystem is governed by the local energy equilibrium equation. Assumptions are made to derive
local energy equilibrium equations. The interconnections among subsystems are formulated via
energy transmission coefficients. The EFA governing equations are then solved by a finite element
method to obtain the local information. Compared with the FEA, the EFA-based computation
requires fewer elements.
Solution of mid-frequency vibration problems has been difficult. This is because the
definitions of low-, mid- and high-frequency regions are ambiguous, and dividing thresholds are
hard to determine (Rabbiolo, 2004). Hence, low-frequency methods and high-frequency methods
may not be sufficient to solve mid-frequency problem directly (Rabbiolo, 2004; Vlahopoulos,
1999). To deal with this issue, an improved EFA model was proposed for the mid-frequency region
specifically (Lase, 1996). The method is limited to one-dimensional two-span beam structures.
Later, a hybrid FEA-EFA approach was developed for the mid-frequency vibration of beam
structures (Zhao, 2004). The approach, however, is not valid for a vibration problem in which all
3

subsystems are under mid-frequency excitations. More recently, another hybrid finite
element/wave and finite element method (hybrid FE/WFE) was developed (Mace, 2005; Renno,
2013). The hybrid FE/WFE method derives the governing equation from wave element and finite
element method and formulates interconnection joints by the FEA.
Compared to frequency-domain analysis, transient vibration analysis of flexible structures at
medium and high frequencies is much more difficult because temporal coordinates must be
precisely estimated. Methods for transient solutions include the FEA, the transient statistical
energy analysis (TSEA) (Pinnington, 1996; Langley, 2019), the wave propagation methods (Von
Flotwo, 1986; Yong, 1992), the VTCR (Ladevèze, 2005), the method of reverberation-ray matrix
(MRRM) (Howard, 1998; Pao, 1999; Guo, 2008), and the transport model method (Le Guennec,
2011). The utility of these methods depends on the frequency region of interest.
In the transient analysis by the FEA, a structure is discretized into a number of small elements.
To obtain transient vibration solutions by the FEA, numerical integration schemes, such as the
Wilson-Theta method, the Newmark-Beta method, and the Runge-Kutta method, are usually
applied. In the FEA of a structure at medium or high frequencies, a huge number of degrees of
freedom are required to represent small-wavelength phenomena in the structure, which is
translated into tremendous computational effort. Moreover, the FEA solutions can be very
sensitive to material properties and boundary conditions at medium and high frequencies.
Therefore, the conventional FEA, which is versatile and accurate for low-frequency problems,
generally does not work well for mid-frequency and high-frequency problems.
The transient statistical energy analysis (TSEA), on the other hand, is a well-established
approach to solution of high-frequency vibro-acoustic problems (Langley, 2019). In the TSEA,
each element in the structure is treated as a subsystem that carries the vibrational energy; the
4

interconnections among the subsystems are described with coupling loss factors; and the governing
equations are derived through consideration of the energy balance in the entire multi-body
structure. The TSEA requires numerical integration of the governing equations, for which the time
step must be small enough to warrant accuracy at high frequencies. Because the SEA describes the
vibrational behaviors of a structure in a global manner, it cannot provide the details about the local
displacements and internal forces of the structure. Hence, the SEA and its variations are unsuitable
for mid-frequency vibration problems.
For transient vibration problems of beam structures, some specific methods have been
developed. The wave propagation methods use wave modes to describe the dynamics of each beam
element in the structure and adopt the wave scattering matrix to model the interconnections of
beam elements. With a wave method, the system response is normally presented in the frequency
domain, and because of this, the inverse Fast Fourier transform (FFT) is required to obtain transient
solutions. The method of reverberation-ray matrix (MRRM) is another wave approach, in which
the eigensolutions of a beam structure are first determined, and the transient vibration solutions
are then obtained by inverse FFT or by modal analysis. The MRRM has been applied to beam
structures subject to impulsive and step loadings. The above-mentioned wave methods are
applicable to periodic beam frames, and they may become inefficient if the structure in
consideration is complicated and non-periodic.
In summary of the previous efforts, there are several issues regarding mid- and high-
frequency vibration analyses of flexible structures. Although the available numerical tools can
provide mid-frequency vibration solutions to beam structures, they have certain limitations that
researchers need to pay attention to. First, the SEA, EFA, and hybrid FEA-EFA methods are based
on energy formulations of a structure. By the relevant solution process, only energy quantities,
5

such as energy density and energy flow can be obtained. Local details on structural response, such
as displacement, slope, axial force, bending moment and shear force, are unavailable. Second, the
EFA, hybrid FEA-EFA, and hybrid FE/WFE methods utilize the FEA to discretize the system. A
huge number of elements is still a must for a very high-frequency region and the mesh should be
updated in different frequency regions. These issues are translated into a demand for computer
storage and CPU time in the simulation, especially for a complex beam structure. Third, for
transient analyses, different numerical methods are needed in different frequency regions, and as
such, the transition of these methods, from low to high frequencies, may not be consistent.
Therefore, a vibration analysis method that can smoothly cover a wide frequency region is
desirable.
In this work, a new method for modeling and analysis of flexible beam and plate structures
in mid-frequency vibration is proposed. The new method is developed through the use of the
Distributed Transfer Function Method (DTFM) (Yang, 1994, 1994, 2005, 2010; Noh, 2014). In
the development, the vibration of a multi-body beam structure is described by an augmented state
formulation of the DTFM (augmented DTFM), with which, no discretization of beam elements is
required, and different types of interconnection joints and general boundary conditions are treated
in a systematic way. Also, the vibration of the sandwich plate and cylindrical shell is described by
a series of state equations. These series of state equations are cast into the DTFM formulation and
can be solved in the same manner.
The DTFM formulation remain the same for different frequency regions, from very low to
very high. The highlight of the proposed method is that exact, closed-form solutions about local
displacement, slope, moment and shear force can be obtained efficiently, with much less
computational effort. Moreover, the new method does not require a pre-calculation of coupling
6

loss factors or energy transmission coefficients. With the local information obtained, energy
density and energy flow at any point in the beam structure can be easily obtained. As shall be seen
in the numerical examples, the proposed method can deliver highly accurate solutions in mid-
frequency regions.
In Chapter 2, the augmented state-space formulation of the DTFM is presented, which
introduces how a multi-body system with one-dimensional continua can be formulated with all
boundary conditions described in a consistent way. Mid- and high-frequency vibration analysis be
the augmented DTFM is also proposed by utilizing the traditional DTFM solution method. In
Chapter 3, the augmented DTFM is applied to the vibration analysis of beam structures at mid-
and high frequencies. The solution procedure is presented step by step and demonstrated by
numerical examples. In Chapter 4, mid-frequency transient vibration analysis tools for beam
structures are developed based on the augmented DTFM, in which model reduction can be
integrated to save the computational effort without loss of accuracy. In Chapter 5, the DTFM is
used for modeling and analysis of the sandwich plate structures with viscoelastic layers. The
DTFM state equation for plates has a similar format as the traditional DTFM and augmented
DTFM, which makes it possible to provide good analysis results for plate vibration problems at
any frequency. Furthermore, the DTFM is applied to cylindrical shell structures in Chapter 7. It is
seen that the solution method for shell vibration analysis by the DTFM is accurate and efficient
compared with other analytical solutions. Finally, the main results from this research are
summarized and the future work of this work is discussed in Chapter 7.

7

Chapter 2 Augmented Distributed Transfer Function Method
2.1 Introduction
In this Chapter, the augmented Distributed Transfer Function Method (augmented DTFM) is
presented, which will be used for the modeling and analysis of flexible beam structures in mid-
and high-frequency regions. This chapter is organized as follows. First, the traditional Distributed
Transfer Function Method (traditional DTFM) is briefly reviewed in Chapter 2.2. Next, the
augmented formulation of the Distributed Transfer Function Method is introduced in Chapter 2.3.
Then, the mid- and high-frequency vibration analysis by the augmented DTFM is presented in
Chapter 2.4, which includes the eigenvalue problem, frequency response, transient response, and
dynamic stress and strain.

2.2 Traditional Distributed Transfer Function Method
The traditional Distributed Transfer Function Method (traditional DTFM) is a systematic
method that can provide the exact analytical solutions for various one-dimensional distributed
systems. With the traditional DTFM, both frequency response (Yang, 1992, 1994) and transient
response can be obtained (Yang, 2010).
Consider a general one-dimensional vibrating continuum, its dynamic response 𝑤 (𝑥 ,𝑡 ) is
governed by the linear partial differential equation
∑(𝑎 𝑘 𝜕 2
𝜕 𝑡 2
+𝑏 𝑘 𝜕 𝜕𝑡
+𝑐 𝑘 )
𝜕 𝑘 𝜕 𝑥 𝑘 𝑤 (𝑥 ,𝑡 )
𝑛 𝑘 =0
=𝑓 (𝑥 ,𝑡 ) (2.1)
for 𝑡 >0 and 0<𝑥 <𝐿 , with the inhomogeneous boundary conditions
8

𝑀 𝑗 𝑤 (𝑥 ,𝑡 )|
𝑥 =0
+ 𝑁 𝑗 𝑤 (𝑥 ,𝑡 )|
𝑥 =1
=𝛾 𝑗 (𝑡 ) (2.2)

for 𝑗 =0,1,2,…,𝑛 . In above equations, 𝑥 is the spatial coordinate; 𝑡 is the temporal parameter; n
is the highest order of spatial differentiation involved; 𝐿 is the length of the continuum; 𝑎 𝑘 , 𝑏 𝑘 and
𝑐 𝑘 are constants describing the physical properties of the continuum, such as inertia, damping,
stiffness, and axial force; 𝑓 (𝑥 ,𝑡 ) is the external disturbance; 𝑀 𝑗 and 𝑁 𝑗 are linear differential
operators; and 𝛾 𝑗 (𝑡 ) represents boundary disturbances. In this work, without loss of generality, a
one-dimensional continuum consisting of uniform distributed components is considered. As such,
the coefficients 𝑎 𝑘 , 𝑏 𝑘 and 𝑐 𝑘 in Eq. (2.1) are constants.
The distributed transfer function formulation is performed in the 𝑠 -domain (Yang, 1992, 1994,
2005). Taking Laplace transform of Eq. (2.1) with respect to time and formulating the resulting
equation in terms of state variables, eventually leads to the following spatial state-space equation
𝜕 𝜕𝑥
𝜼 (𝑥 ,𝑠 )=𝐅 (𝑠 )𝜼 (𝑥 ,𝑠 )+𝒒 (𝑥 ,𝑠 ) (2.3)
for 0<𝑥 <𝐿 , where 𝑠 is the Laplace transform parameter; 𝜼 (𝑥 ,𝑠 ) is an 𝑛 -by-1 state vector
defined by
𝜼 (𝑥 ,𝑠 )={
𝑤̅(𝑥 ,𝑠 )
𝜕 𝑤̅(𝑥 ,𝑠 )
𝜕𝑥
⋯
𝜕 𝑛 −1
𝑤̅(𝑥 ,𝑠 )
𝜕 𝑥 𝑛 −1
}
T
(2.4)
In the previous equations, 𝑤̅(𝑥 ,𝑠 ) is the Laplace transform of 𝑤 (𝑥 ,𝑡 ); 𝐅 (𝑠 ) is an 𝑛 -by-𝑛
state matrix with elements consisting of parameter 𝑠 and coefficients 𝑎 𝑘 , 𝑏 𝑘 and 𝑐 𝑘 ; and 𝒒 (𝑥 ,𝑠 ) is
an 𝑛 -by-1 vector containing external force 𝑓 (𝑥 ,𝑡 ) and initial disturbances. The boundary
conditions in Eq. (2.2) can be written in terms of the state vector as follows
𝐌 𝑏 (𝑠 )𝜼 (0,𝑠 )+𝐍 𝑏 (𝑠 )𝜼 (𝐿 ,𝑠 )=𝜸 𝑏 (𝑠 ) (2.5)
9

where 𝜼 (0,𝑠 ) and 𝜼 (𝐿 ,𝑠 ) are the values of the state vector at the two ends (𝑥 =0 and 𝑥 =𝐿 ) of
the continuum; 𝐌 𝑏 (𝑠 ) and 𝐍 𝑏 (𝑠 ) are two 𝑛 -by-𝑛 boundary matrices; and 𝜸 𝑏 (𝑠 ) is an 𝑛 -by-1
vector consisting of Laplace transforms of boundary disturbances at boundaries.
The state-space formulation described by Eqs. (2.4) and (2.5) is applicable to general one-
dimensional vibrating continua, including bars, shafts, strings, and beams. As shall be seen in the
next chapter, for a single beam modeled by Euler-Bernoulli beam theory or Timoshenko beam
theory under transverse vibration, the state matrix 𝐅 (𝑠 ) and boundary matrices 𝐌 𝑏 (𝑠 ) , 𝐍 𝑏 (𝑠 ) are
four-by-four complex matrices.
The 𝑠 -domain response of the continuum takes the form (Yang, 1992)
𝜼 (𝑥 ,𝑠 )=∫ 𝐆 (𝑥 ,𝜉 ,𝑠 )𝒒 (𝜉 ,𝑠 )𝑑𝜉 +𝐇 (𝑥 ,𝑠 )𝜸 𝑏 (𝑠 )
𝐿 0
(2.6)
where 𝐆 (𝑥 ,𝜉 ,𝑠 ) and 𝐇 (𝑥 ,𝑠 ) are the distributed transfer functions of the uniform one-dimensional
vibrating continuum, and they are given in the exact and analytical form
𝐆 (𝑥 ,𝜉 ,𝑠 )={
𝐇 (𝑥 ,𝑠 )𝐌 𝑏 (𝑠 )𝚽 (0,𝜉 ,𝑠 ), 𝜉 ≤𝑥 −𝐇 (𝑥 ,𝑠 )𝐍 𝑏 (𝑠 )𝚽 (𝐿 ,𝜉 ,𝑠 ),𝜉 >𝑥 𝐇 (𝑥 ,𝑠 )=𝚽 (𝑥 ,0,𝑠 )[𝐌 𝑏 +𝐍 𝑏 𝚽 (𝐿 ,0,𝑠 )]
−1
(2.7)
where 𝚽 (𝑥 ,𝜉 ,𝑠 ) is the transition matrix of the distributed system described by Eq. (2.3), and the
transition matrix that can be written in terms of an exponential matrix when the distributed system
is uniform
𝚽 (𝑥 ,𝜉 ,𝑠 )=exp[𝐅 (𝑠 )(𝑥 −𝜉 )] (2.8)
The traditional DTFM, with the solution format given by Eqs. (7) and (8), has been applied
to multi-body beam structures (Yang, 2005). For a multi-body dynamic system, the traditional
10

DTFM combines the closed form of distributed transfer functions for each subsystem and the
multi-body assembling capability of the finite element method. Considering the equations of force
balance and displacement continuity at all nodes of the multi-body structure, a global dynamic
stiffness matrix can be assembled from the distributed transfer functions of the subsystems which
are computed by Eq. (2.7). And the global governing equation takes the form
𝐊 (𝑠 )𝑼 (𝑠 )=𝑸 (𝑠 ) (2.9)
where 𝐊 (𝑠 ) is the global dynamic stiffness matrix; 𝑼 (𝑠 ) is the global nodal displacement vector;
𝑸 (𝑠 ) is the global nodal force vector. Equation (2.9) represents the dynamics of the multi-body
system by using the traditional DTFM.
When the traditional DTFM is applied to complex branched multi-body structures, its utility
in mid- and high-frequency vibration analyses is limited. Because the global dynamic stiffness
matrix 𝐊 (𝑠 ) carries the singularities of subsystem transfer functions, it becomes problematic in
computation, in which numerical instability may occur at higher frequencies. For this reason, the
traditional DTFM is not suitable for mid- and high-frequency vibration analyses.
Although the traditional DTFM is not viable for mid- and high-frequency analysis, the format
of the 𝑠 -domain solution, given by Eqs. (2.7) and (2.8), is still useful if the singularities of
subsystem transfer functions can be avoided.

11

2.3 Augmented Formulation for a Multi-Body Structure
An augmented formulation is utilized to resolve the above-mentioned singularity issue with
the traditional DTFM (Noh, 2014). Based on the augmented formulation, a new DTFM for mid-
frequency vibration analysis for multi-body structures is developed.
In the proposed augmented formulation for a multi-body structure, instead of using the
transfer functions that are determined from the local state equations of the subsystems, a global
state equation of the multi-body structure is directly constructed in terms of global state variables.
Solution of the global state equation gives the dynamic response of the entire structure in a straight
way. This process of the augmented formulation and solution is named the augmented Distributed
Transfer Function Method (augmented DTFM).
Consider a structure of 𝑁 subsystems (beam components). Modeling of the multi-body
structure by the augmented DTFM takes the following three steps.
Step 1. Formation of Nondimensional Subsystem State Equations
In this step, the state equations of all the subsystems shown in Eq. (2.3) are converted to
nondimensional state equations in a global frame. For the 𝑖 th
subsystem, its 𝑠 -domain response
can be expressed by the following nondimensional state equation
𝜕 𝜕𝑥
𝜼̂
𝑖 (𝑥 ,𝑠 )=𝐅̂
𝑖 (𝑠 )𝜼̂
𝑖 (𝑥 ,𝑠 )+𝒒̂
𝑖 (𝑥 ,𝑠 ) (2.10)
for 0<𝑥 <1, where x is a global nondimensional spatial coordinate; 𝜼̂
𝑖 (𝑥 ,𝑠 ) , 𝐅̂
𝑖 (𝑠 ) , and 𝒒̂
𝑖 (𝑥 ,𝑠 )
are the state vector, state matrix, and external force vector of the 𝑖 th
subsystem after the
nondimensionalization, respectively. Equation (2.10) can be easily obtained from Eq. (2.3) by
12

proper scaling, as shall be seen in Chapter 3. The key in this step is to use the unified spatial domain
0<𝑥 <1 for all subsystems.
Step 2. Formation of an Augmented State Equation for the Multi-Body Structure
The state equations of the 𝑁 subsystems obtained in Step 1 are cast into the following
augmented form
𝜕 𝜕𝑥
𝜼̂(𝑥 ,𝑠 )=𝐅̂
(𝑠 )𝜼̂(𝑥 ,𝑠 )+𝒒̂(𝑥 ,𝑠 ), 0<𝑥 <1 (2.11)
where 𝜼̂(𝑥 ,𝑠 ) , 𝐅̂
(𝑥 ,𝑠 ) and 𝒒̂(𝑥 ,𝑠 ) are global state vector, global state matrix, and global force
vector, respectively; and they are given as follows
𝜼̂(𝑥 ,𝑠 )={𝜼̂
1
T
(𝑥 ,𝑠 ), 𝜼̂
2
T
(𝑥 ,𝑠 ), ⋯ 𝜼̂
𝑁 T
(𝑥 ,𝑠 )}
T
𝐅̂
(𝑥 ,𝑠 )=diag{𝐅̂
𝑖 (𝑥 ,𝑠 )}
𝒒̂(𝑥 ,𝑠 )={𝒒̂
1
T
(𝑥 ,𝑠 ), 𝒒̂
2
T
(𝑥 ,𝑠 ), ⋯ 𝒒̂
𝑁 T
(𝑥 ,𝑠 )}
T
(2.12)
with 𝜼̂
𝑖 (𝑥 ,𝑠 ) , 𝐅̂
𝑖 (𝑠 ) , and 𝒒̂
𝑖 (𝑥 ,𝑠 ) , which have been constructed in Step 1.
Step 3. Description of the Dynamic Interactions among Subsystems
The global state equation (2.11) only governs the dynamic response of each subsystem at its
interior domain (0<𝑥 <1), and it does not describe the dynamic interactions among the multiple
subsystems. In structural dynamics, the dynamic interactions of the subsystems of a structure are
described by the matching conditions from force balance and displacement continuity at the nodes
where subsystems are interconnected. In addition, the boundary conditions of the multi-body
structure need to be specified at those boundary nodes. Details about how to describe the matching
conditions and boundary conditions are introduced as follows.
13

Denote a node of the 𝑖 th
subsystem (beam element) of the structure by 𝜁 𝑖 ∗
, which can be
either 0 or 1, depending on the local coordinate system. If 𝜁 𝑖 ∗
is a boundary node of the structure
(see point A in Figure 2.1 for example), the boundary conditions of the subsystem at this node can
be written as
𝚪 𝑖 (𝑠 )𝜼̂
𝑖 (𝜁 𝑖 ∗
,𝑠 )=𝜸̂
𝑖 (𝑠 ) (2.13)
where 𝚪 𝑖 (𝑠 ) is a matrix consisting of the coefficients that can be extracted from some boundary
matrices, like 𝐌 𝑏 (𝑠 ) and 𝐍 𝑏 (𝑠 ) in Eq. (2.5); and 𝜸̂
𝑖 (𝑠 ) is a vector of boundary disturbances at the
node.
If 𝜁 𝑖 ∗
is an interior node (joint) of the structure, where 𝑚 beam elements 𝑒 1
,𝑒 2
,…,𝑒 𝑚 are
interconnected (see point C in Figure 2.1 for example), the matching conditions at the node, which
describe displacement continuity and force balance, are of the form
∑𝐂 𝑖 ,𝑗 (𝑠 )𝜼̂
𝑗 (𝜁 𝑒𝑗
∗
,𝑠 )
𝑒 𝑚 𝑗 =𝑒 1
=𝜸̂
𝑖 (𝑠 ) (2.14)
where 𝜁 𝑒𝑗
∗
refers to the same node, but it is in the local coordinate of element 𝑒 𝑗 ; 𝐂 𝑖 ,𝑗 (𝑠 ) are
appropriate coupling matrices; and 𝜸̂
𝑖 (𝑠 ) is a vector of external loading at this joint.
Because the value of 𝜁 𝑒 𝑗 ∗
in Eq. (2.14) is either 0 or 1, the matching conditions can be viewed
as the boundary conditions at the interior nodes. Hence, in the augmented formulation, boundary
conditions are specified at all the nodes (boundary nodes and interior nodes) of the structure. In
this sense, assembly of Eqs. (2.13, 2.14) for all the nodes and use of the global state vector 𝜼̂(𝑥 ,𝑠 ) ,
eventually yields the following global boundary condition for the multi-body structure
𝐌 𝑔 (𝑠 )𝜼̂(0,𝑠 )+𝐍 𝑔 𝜼̂(1,𝑠 )=𝜸̂
𝑔 (𝑠 ) (2.15)

14

where 𝜼̂(0,𝑠 ) and 𝜼̂(1,𝑠 ) are the values of the global state vector at two ends: 𝑥 =0 and 𝑥 =1,
respectively; 𝐌 𝑔 (𝑠 ) and 𝐍 𝑔 (𝑠 ) are global boundary matrices composed of the coefficients of
matrices 𝚪 𝑖 (𝑠 ) and 𝐂 𝑖 ,𝑗 (𝑠 ) ; and 𝜸̂
𝑔 (𝑠 ) is a global boundary disturbance vector consisting of the
components of 𝜸̂
𝑖 (𝑠 ) . Equation (2.15) contains all matching conditions and boundary conditions
of the 𝑁 -body structure in the proposed augmented formulation.
In the augmented formulation, given by Eqs. (2.11) and (2.15), the global state matrix 𝐅̂
(𝑥 ,𝑠 )
and the global boundary matrices 𝐌 𝑔 (𝑠 ) and 𝐍 𝑔 (𝑠 ) are formed by the governing equations of
subsystems and dynamic interaction among them, without usage of subsystem transfer functions.
As such, the numerical instability issue due to the local singularities of subsystem transfer
functions in the traditional DTFM is completely avoided in the augmented DTFM.

2.4 Mid- and High-frequency vibration analysis by the augmented DTFM
Chapter 2.3 shows the modeling of a multi-body structure by an augmented formulation
(augmented DTFM). Once the global state-space formulation is established, the mid- and high-
frequency vibration analysis of the multi-body structure can be carried out.
2.4.1 The 𝑠 -domain Solution
Note that the global state equation (2.11) and the global boundary condition (2.15) for a multi-
body structure have the same form as Eqs. (2.3) and (2.5) for a single body. Because of the
consistent format, the dynamic response of the entire structure, which is the solution of Eqs. (2.11)
and (2.15), can be obtained by following the single body solution, which is provided by Eqs. (2.6-
2.8). Hence, the 𝑠 -domain response of the multi-body structure is given by
15

𝜼̂(𝑥 ,𝑠 )=∫ 𝐆̂
(𝑥 ,𝜉 ,𝑠 )𝒒̂(𝜉 ,𝑠 )𝑑𝜉 1
0
+𝐇̂
(𝑥 ,𝑠 )𝜸̂
𝑔 (𝑠 ) (2.16)
where 𝐆̂
(𝑥 ,𝜉 ,𝑠 ) and 𝐇̂
(𝑥 ,𝑠 ) are the global distributed transfer functions of the structure and they
are given in the exact and analytical form
𝐆̂
(𝑥 ,𝜉 ,𝑠 )={
𝐇̂
(𝑥 ,𝑠 )𝐌 𝑔 (𝑠 )𝚽̂
𝑔 (0,𝜉 ,𝑠 ),𝜉 ≤𝑥 −𝐇̂
(𝑥 ,𝑠 )𝐍 𝑔 (𝑠 )𝚽̂
𝑔 (1,𝜉 ,𝑠 ),𝜉 >𝑥 𝐇̂
(𝑥 ,𝑠 )=𝚽̂
𝑔 (𝑥 ,0,𝑠 )[𝐌 𝑔 +𝐍 𝑔 𝚽̂
𝑔 (1,0,𝑠 )]
−1
(2.17)
where 𝚽̂
𝑔 (𝑥 ,𝜉 ,𝑠 ) is the transition matrix of the global state equation given by
𝚽̂
𝑔 (𝑥 ,𝜉 ,𝑠 )=exp[𝐅̂
(𝑠 )(𝑥 −𝜉 )] (2.18)
With the augmented formulation and the global distributed transfer function formulas,
dynamic problems of multi-body structures consist of one-dimensional continua can be
conveniently solved.

2.4.2 Eigenvalue Problem
For an undamped structure, its natural frequencies can be solved from the eigenvalues of its
governing equation. With the augmented state-space formulation, the eigenvalue problem of a
beam structure is described by the following homogeneous state equation (Yang, 1992)
𝜕 𝜕𝑥
𝝍̂
(𝑥 ,𝑠 )=𝐅̂
(𝑠 )𝝍̂
(𝑥 ,𝑠 ) (2.19)
for 0<𝑥 <1, subject to the boundary condition
𝐌 𝑔 (𝑠 )𝝍̂
(0,𝑠 )+𝐍 𝑔 (𝑠 )𝝍̂
(1,𝑠 )=𝟎 (2.20)
16

where 𝑠 is an eigenvalue of the system, 𝝍̂
(𝑥 ,𝑠 ) is the associate eigenfunction in the vector form,
and 𝐅̂
(𝑥 ,𝑠 ) , 𝐌 𝑔 (𝑠 ) and 𝐍 𝑔 (𝑠 ) are matrices from Eqs. (2.11, 2.15).
A nontrivial solution of Eq. (2.19) is given by
𝝍̂
(𝑥 ,𝑠 )=𝚽̂
𝑔 (𝑥 ,0,𝑠 )𝒂 (2.21)
where 𝚽̂
𝑔 (𝑥 ,0,𝑠 ) can be computed from Eq. (2.18), and 𝒂 is a non-zero constant vector to be
determined. Substituting Eq. (2.21) into the boundary condition (2.20) leads to an eigenequation
of state-space form as follows
[𝐌 𝑔 (𝑠 )+𝐍 𝑔 (𝑠 )𝚽̂
𝑔 (1,0,𝑠 )]𝒂 =𝟎 (2.22)
It follows that the eigenvalues of the structure are the roots of the characteristic equation
det[𝐌 𝑔 (𝑠 )+𝐍 𝑔 (𝑠 )𝚽̂
𝑔 (1,0,𝑠 )]=0 (2.23)
Once a root 𝑠 is determined by Eq. (2.23), one natural frequency 𝜔 of the structure can be
obtained from 𝑠 =𝑗𝜔 , where 𝑗 =√−1. Meanwhile, a nonzero 𝒂 can be obtained by solving Eq.
(2.23), which gives the associate eigenfunction by Eq. (2.21). After the vector-form eigenfunction
is determined, the eigenfunctions in the conventional scalar form can be obtained by taking
appropriate component(s) of 𝝍̂
(𝑥 ,𝑠 ) . Also, from 𝝍̂
(𝑥 ,𝑠 ) , the modal information about all state
variables can be determined.
In the above derivation, no discretization or approximation has been made; the eigensolutions
determined are of exact and analytical form. Because there is no singularity issue with the
augmented formulation, the natural frequencies of the structure in any frequency region (from low
to very high frequencies) can be efficiently computed without numerical instability with good
17

accuracy. This unique feature facilitates the development of the two proposed transient analysis
tools in mid- and high-frequency regions, as shall be presented later.

2.4.3 Frequency Response
The steady-state response of a multi-body structure subject to a sinusoidal force of excitation
frequency 𝛺 can be obtained from Eq. (2.16) with 𝑠 =𝑗𝛺 , where 𝑗 =√−1. By Eq. (2.17), the
exact frequency response transfer functions 𝐆̂
(𝑥 ,𝜉 ,𝑗𝛺 ) and 𝐇̂
(𝑥 ,𝑗𝛺 ) can be obtained directly. The
frequency response can also be determined by using the generalized modal expansion with the
eigensolutions solved in the eigenvalue problem. However, in mid- and high-frequency regions,
the modal expansion must require a very large number of modes for convergent and accurate
results, which needs extensive computational effort for a complex multi-body structure with many
components. For this reason, Eq. (2.16) is more suitable to determine the frequency response of a
multi-body structure in mid- to high-frequency regions. Again, because the augmented formulation
does not contain the singularities from the subsystem transfer functions, it can be used to accurately
predict the frequency response of the structure in any frequency region, which will be shown in
Chapter 3.

2.4.4 Transient Response
With the 𝑠 -domain solution and the eigenvalue problem of the multi-body structure governed
by Eq. (2.11) and Eq. (2.15), two approaches can be proposed to solve the transient response. The
transient response of a multi-body structure can be determined through inverse Laplace transform
of Eq. (2.16) via a residue theorem (Yang, 2010; Morse, 1953). The transient response can also be
18

determined by using the generalized modal expansion after solving the eigenvalue problem. One
advantage of the augmented DTFM is that the transient response can be solved from low to high
frequencies. The details about the transient vibration analysis will be discussed in Chapter 4.

2.4.5 Dynamic Stress and Strain
The state vector 𝜼̂(𝑥 ,𝑠 ) in Eq. (2.16) contains not only displacements of the subsystems, but
also the special derivatives of the displacements, as shown in Eq. (2.4). Because of this, the
augmented DTFM can provide the local information about displacement, strain, and internal forces
of a structure, from low to high frequencies and without the need for approximations. This special
feature differentiates the proposed method from most existing techniques for mid-frequency
vibration analysis.
Note that the global distributed transfer functions given in Eq. (2.17) have fixed dimensions.
For example, in a two-dimensional beam structure containing 𝑁 beam components (with either
Euler-Bernoulli beam theory or Timoshenko beam theory), the number of state variables is always
6𝑁 . This property implies that the level of computational effort in the augmented DTFM-based
vibration analysis is consistent in any frequency region. On the other hand, most existing methods,
such as FEM and hybrid WFE/FE, require a huge and ever-increasing number of degrees of
freedom to capture the small-wavelength phenomena and high-frequency dynamics in the structure,
which is translated into substantial computational effort.

19

Figure 2.1 A two-beam frame

20

Chapter 3 Vibration Analysis of Beam Structures at Mid- and High-
Frequencies
3.1 Introduction
In this Chapter, the augmented DTFM presented in Chapter 2 is applied to beam structures
composed of uniform components. As shall be shown, exact analytical solutions for eigenvalue
problems and frequency-response problems in any frequency region can be obtained by the
proposed method. And the augmented DTFM can be applied to beam structures, modeled by
different beam theories, such as two popular theories, the Euler-Bernoulli beam theory and the
Timoshenko beam theory. In Chapter 3.2, the augmented formulation for beam structures is
presented, which shows how the global state equation of the entire is constructed step by step. The
calculation of local information, such as internal force, is demonstrated in Chapter 3.3. The shear
deformation in the beam structures at higher frequencies is also discussed. The boundary
conditions subjected to the global state equation are discussed in Chapter 3.4, which presents how
general complex boundary conditions and matching conditions in the structure can be formulated
in a consistent way. In Chapter 3.4, a double-beam structure is formulated by the augmented
DTFM as a demonstrative example. Finally, numerical examples are presented in Chapter 3.5,
from which the accuracy, efficiency and other advantages of the proposed method can be seen.

21

3.2 Augmented Formulation for Beam Structures
The augmented state-space formulation (augmented DTFM) presented in Chapter 2 is valid
for structure elements of various beam theories. In this work, the augmented DTFM is applied to
two-dimensional Euler-Bernoulli and Timoshenko beam structures.
3.2.1 Euler-Bernoulli Beam Theory
Consider a two-dimensional multi-body structure composed of 𝑁 uniform Euler-Bernoulli
beam components. For the 𝑖 th
uniform beam element, its longitudinal displacement 𝑢 𝑖 (𝑥 ,𝑡 ) and
transverse displacement 𝑤 𝑖 (𝑥 ,𝑡 ) are governed by
𝜌 𝑖 𝐴 𝑖 𝜕 2
𝜕𝑡
2
𝑢 𝑖 (𝑥 ,𝑡 )−
𝐸 𝑖 𝐴 𝑖 𝐿 𝑖 2
𝜕 2
𝜕𝑥
2
𝑢 𝑖 (𝑥 ,𝑡 )=𝑓 𝐿 ,𝑖 (𝑥 ,𝑡 ) (3.1𝑎 )
𝜌 𝑖 𝐴 𝑖 𝜕 2
𝜕𝑡
2
𝑤 𝑖 (𝑥 ,𝑡 )+
𝐸 𝑖 𝐼 𝑖 𝐿 𝑖 4
𝜕 4
𝜕𝑥
4
𝑤 𝑖 (𝑥 ,𝑡 )=𝑓 𝑇 ,𝑖 (𝑥 ,𝑡 ) (3.1𝑏 )
for 0<𝑥 <1, and 𝑡 >0, where 𝐿 𝑖 , 𝜌 𝑖 , 𝐸 𝑖 , 𝐴 𝑖 , 𝐼 𝑖 are the length, density, Young’s modulus, cross -
section area and area moment of inertia of the beam element, respectively; 𝑥 is a non-dimensional
(scaled) local coordinate; 𝑓 𝐿 ,𝑖 (𝑥 ,𝑡 ) and 𝑓 𝑇 ,𝑖 (𝑥 ,𝑡 ) are external longitudinal and transverse forces of
the 𝑖 th
beam element. Assume zero initial disturbances for convenience of presentation. Take
Laplace transform of Eq. (3.1) with respect to time to obtain
𝜌 𝑖 𝐴 𝑖 𝑠 2
𝑢̅
𝑖 (𝑥 ,𝑠 )−
𝐸 𝑖 𝐴 𝑖 𝐿 𝑖 2
𝜕 2
𝜕𝑥
2
𝑢̅
𝑖 (𝑥 ,𝑠 )=𝑓 ̅
𝐿 ,𝑖 (𝑥 ,𝑠 ) (3.2𝑎 )
𝜌 𝑖 𝐴 𝑖 𝑠 2
𝑤̅
𝑖 (𝑥 ,𝑠 )+
𝐸 𝑖 𝐼 𝑖 𝐿 𝑖 4
𝜕 4
𝜕𝑥
4
𝑤̅
𝑖 (𝑥 ,𝑠 )=𝑓 ̅
𝑇 ,𝑖 (𝑥 ,𝑠 ) (3.2𝑏 )
where the overbar stands for Laplace transformation.
22

In the 𝑠 -domain, the state vector 𝜼̂
𝑖 (𝑥 ,𝑠 ) for the 𝑖 th
beam component is defined as
𝜼̂
𝑖 (𝑥 ,𝑠 )={𝒖̂
𝑖 T
(𝑥 ,𝑠 ), 𝒘̂
𝑖 T
(𝑥 ,𝑠 )}
T
(3.3)
with
𝒖̂
𝑖 (𝑥 ,𝑠 )={𝑢̅
𝑖 (𝑥 ,𝑠 ) 𝜕 𝑢̅
𝑖 (𝑥 ,𝑠 ) 𝜕𝑥 ⁄ }
T
𝒘̂
𝑖 (𝑥 ,𝑠 )={𝑤̅
𝑖 (𝑥 ,𝑠 ) 𝜕 𝑤̅
𝑖 (𝑥 ,𝑠 ) 𝜕𝑥 ⁄ 𝜕 𝑤̅
𝑖 2
(𝑥 ,𝑠 ) 𝜕 𝑥 2
⁄ 𝜕 𝑤̅
𝑖 3
(𝑥 ,𝑠 ) 𝜕 𝑥 3
⁄ }
T
(3.4)
With the above-defined state vector, the s-domain governing equations (3.2) are converted to
the state equation (2.10), with
𝐅̂
𝑖 (𝑠 )=
[

0 1 0 0 0 0
𝜌 𝑖 𝐴 𝑖 𝑠 2
𝐸 𝑖 𝐴 𝑖 𝐿 𝑖 2
0 0 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 0
−𝜌 𝑖 𝐴 𝑖 𝑠 2
𝐸 𝑖 𝐼 𝑖 𝐿 𝑖 4
0 0 0
]

(3.5𝑎 )
𝒒̂
𝑖 (𝑥 ,𝑠 )={0 −
𝑓 ̅
𝐿 ,𝑖 (𝑥 ,𝑠 )
𝐸 𝑖 𝐴 𝑖 𝐿 𝑖 2
0
𝑓̅
𝑇 ,𝑖 (𝑥 ,𝑠 )
𝐸 𝑖 𝐼 𝑖 𝐿 𝑖 4
0 0}
T
(3.5𝑏 )
As a special case, if only the transverse displacement 𝑤 𝑖 (𝑥 ,𝑡 ) is considered, the state vector
of the element becomes 𝜼̂
𝑖 (𝑥 ,𝑠 )=𝒘̂
𝑖 (𝑥 ,𝑠 ) and the state matrix is given by
𝐅̂
𝑖 (𝑠 )=
[

0 1 0 0
0 0 1 0
0 0 0 1
−𝜌 𝑖 𝐴 𝑖 𝑠 2
𝐸 𝑖 𝐼 𝑖 𝐿 𝑖 4
0 0 0
]

(3.5𝑐 )
With 𝐅̂
𝑖 (𝑠 ) and 𝒒̂
𝑖 (𝑥 ,𝑠 ) for all beam components, the global state equation (2.11) of the entire
Euler-Bernoulli beam structure can be constructed by using Eq. (2.11). Also, the boundary
23

conditions and the matching conditions of the beam structure can be specified by the formation of
matrices 𝐌 𝑔 (𝑠 ) and 𝐍 𝑔 (𝑠 ) in the global boundary condition (2.15).
As can be seen in this formulation procedure, no discretization or approximation has been
made and arbitrary boundary conditions and external loads are treated in a systematic manner.
3.2.2 Timoshenko Beam Theory
According to the literature, the Timoshenko beam theory is probably a more suitable beam
theory for high-frequency vibrations (Mei, 2005; Renno, 2013). For the 𝑖 th
beam element of the
two-dimensional beam structure modeled by the Timoshenko beam theory, its longitudinal
displacement 𝑢 𝑖 (𝑥 ,𝑡 ), transverse displacement 𝑤 𝑖 (𝑥 ,𝑡 ) and angle 𝜑 𝑖 (𝑥 ,𝑡 ) of rotation are
governed by
𝜌 𝑖 𝐴 𝑖 𝜕 2
𝜕𝑡
2
𝑢 𝑖 (𝑥 ,𝑡 )−
𝐸 𝑖 𝐴 𝑖 𝐿 𝑖 2
𝜕 2
𝜕𝑥
2
𝑢 𝑖 (𝑥 ,𝑡 )=𝑓 𝐿 ,𝑖 (𝑥 ,𝑡 ) (3.6𝑎 )
𝜌 𝑖 𝐴 𝑖 𝜕 2
𝜕𝑡
2
𝑤 𝑖 (𝑥 ,𝑡 )−𝑓 𝑇 ,𝑖 (𝑥 ,𝑡 )=𝜅 𝑖 𝐴 𝑖 𝐺 𝑖 [
1
𝐿 𝑖 2
𝜕 2
𝜕 𝑥 2
𝑤 𝑖 (𝑥 ,𝑡 )−
1
𝐿 𝑖 𝜕 𝜕𝑥
𝜑 𝑖 (𝑥 ,𝑡 )] (3.6𝑏 )
𝜌 𝑖 𝐼 𝑖 𝜕 2
𝜕𝑡
2
𝜑 𝑖 (𝑥 ,𝑡 )=
𝐸 𝑖 𝐼 𝑖 𝐿 𝑖 2
𝜕 2
𝜕 𝑥 2
𝜑 𝑖 (𝑥 ,𝑡 )+𝜅 𝑖 𝐴 𝑖 𝐺 𝑖 [
1
𝐿 𝑖 𝜕 𝜕𝑥
𝑤 𝑖 (𝑥 ,𝑡 )−𝜑 𝑖 (𝑥 ,𝑡 )] (3.6𝑐 )
for 0<𝑥 <1, and 𝑡 >0, where 𝐿 𝑖 , 𝜌 𝑖 , 𝐸 𝑖 , 𝐴 𝑖 , 𝐼 𝑖 are the same as those defined for Eq. (3.1); 𝜅 𝑖 is
the Timoshenko shear coefficient; and 𝐺 𝑖 is the shear modulus of the element; 𝑥 is a non-
dimensional (scaled) local coordinate; 𝑓 𝐿 ,𝑖 (𝑥 ,𝑡 ) and 𝑓 𝑇 ,𝑖 (𝑥 ,𝑡 ) are external longitudinal and
transverse forces.
Following the Euler-Bernoulli beam element, take Laplace transform of Eq. (3.6) with respect
to time and assume zero initial disturbances, to obtain
24

𝜌 𝑖 𝐴 𝑖 𝑠 2
𝑢̅
𝑖 (𝑥 ,𝑠 )−
𝐸 𝑖 𝐴 𝑖 𝐿 𝑖 2
𝜕 2
𝜕𝑥
2
𝑢 𝑖 (𝑥 ,𝑡 )=𝑓 ̅
𝐿 ,𝑖 (𝑥 ,𝑠 ) (3.7𝑎 )

𝜌 𝑖 𝐴 𝑖 𝑠 2
𝑤̅
𝑖 (𝑥 ,𝑠 )−𝑓 ̅
𝑇 ,𝑖 (𝑥 ,𝑠 )=𝜅 𝑖 𝐴 𝑖 𝐺 𝑖 [
1
𝐿 𝑖 2
𝜕 2
𝜕 𝑥 2
𝑤̅
𝑖 (𝑥 ,𝑠 )−
1
𝐿 𝑖 𝜕 𝜕𝑥
𝜑̅
𝑖 (𝑥 ,𝑠 )] (3.7𝑏 )
𝜌 𝑖 𝐼 𝑖 𝑠 2
𝜑̅
𝑖 (𝑥 ,𝑠 )=
𝐸 𝑖 𝐼 𝑖 𝐿 𝑖 2
𝜕 2
𝜕 𝑥 2
𝜑̅
𝑖 (𝑥 ,𝑠 )+𝜅 𝑖 𝐴 𝑖 𝐺 𝑖 [
1
𝐿 𝑖 𝜕 𝜕𝑥
𝑤̅
𝑖 (𝑥 ,𝑠 )−𝜑̅
𝑖 (𝑥 ,𝑠 )] (3.7𝑐 )

where the overbar stands for Laplace transformation, and the state vector is defined as
𝜼̂
𝑖 (𝑥 ,𝑠 )={𝒖̂
𝑖 T
(𝑥 ,𝑠 ), 𝒘̂
𝑖 T
(𝑥 ,𝑠 )}
T
(3.8)
with
𝒖̂
𝑖 (𝑥 ,𝑠 )={𝑢̅
𝑖 (𝑥 ,𝑠 ) 𝜕 𝑢̅
𝑖 (𝑥 ,𝑠 ) 𝜕𝑥 ⁄ }
T
𝒘̂
𝑖 (𝑥 ,𝑠 )={𝑤̅
𝑖 (𝑥 ,𝑠 ) 𝜕 𝑤̅
𝑖 (𝑥 ,𝑠 ) 𝜕𝑥 ⁄ 𝜑̅
𝑖 (𝑥 ,𝑠 ) 𝜕 𝜑̅
𝑖 (𝑥 ,𝑠 ) 𝜕𝑥 ⁄ }
T
(3.9)
With the above-defined state vector for Timoshenko beam elements, the s-domain governing
equations (3.7) are converted to the state equation (2.10), with
𝐅̂
𝑖 (𝑠 )=
[

0 1 0 0 0 0
𝜌 𝑖 𝐴 𝑖 𝑠 2
𝐸 𝑖 𝐴 𝑖 𝐿 𝑖 2
0 0 0 0 0
0 0 0 1 0 0
0 0
𝜌 𝑖 𝐴 𝑖 𝑠 2
𝜅 𝑖 𝐴 𝑖 𝐺 𝑖 𝐿 𝑖 2
0 0 𝐿 𝑖 0 0 0 0 0 1
0 0 0
−𝜅 𝑖 𝐴 𝑖 𝐺 𝑖 𝐸 𝑖 𝐼 𝑖 𝐿 𝑖 (𝜅 𝑖 𝐴 𝑖 𝐺 𝑖 +𝜌 𝑖 𝐼 𝑖 𝑠 2
)
𝐸 𝑖 𝐼 𝑖 𝐿 𝑖 2
0
]

(3.10𝑎 )
𝒒̂
𝑖 (𝑥 ,𝑠 )={0 −
𝑓 ̅
𝐿 ,𝑖 (𝑥 ,𝑠 )
𝐸 𝑖 𝐴 𝑖 𝐿 𝑖 2
0 −
𝑓 ̅
𝑇 ,𝑖 (𝑥 ,𝑠 )
𝜅 𝑖 𝐴 𝑖 𝐺 𝑖 𝐿 𝑖 2
0 0}
T
(3.10𝑏 )
Again, the global state equation (2.11) of the entire Timoshenko beam structure can be
constructed by using Eqs. (3.8-3.10). The boundary conditions of beam elements can be converted
to the global boundary condition (2.15).
25

As seen from the previous derivations, for different beam theories and general boundary
conditions, the augmented formulation described by Eqs. (2.11, 2.15) remains the same although
the state matrices in these equations have different elements. This feature of symbolic
manipulation makes the augmented formulation highly efficient in vibration analysis of beam
structures.
3.3 Internal Forces of Beam Elements
The local information about the internal forces of a beam element, besides the displacements,
can be conveniently obtained by the state-space formulation, and it is done without the need to
differentiate the beam displacement functions. This useful feature, which is important in the mid-
frequency analysis of beam structures, is shown as follows.
3.3.1 Euler-Bernoulli Beam Elements
For an Euler-Bernoulli beam element described by Eq. (3.1), its normal force (axial force)
𝑇̅
𝑖 (𝑥 ,𝑠 ) , bending moment 𝑀̅
𝑖 (𝑥 ,𝑠 ) and shear force 𝑄̅
𝑖 (𝑥 ,𝑠 ) at any internal point (0<𝑥 <1) can
be written as
𝑇̅
𝑖 (𝑥 ,𝑠 )=
𝐸 𝑖 𝐴 𝑖 𝐿 𝑖 𝜕 𝜕𝑥
𝑢̅
𝑖 (𝑥 ,𝑠 )
𝑀̅
𝑖 (𝑥 ,𝑠 )=
𝐸 𝐼 𝑖 𝐿 𝑖 2
𝜕 2
𝜕 𝑥 2
𝑤̅
𝑖 (𝑥 ,𝑠 )
𝑄̅
𝑖 (𝑥 ,𝑠 )=
−𝐸 𝐼 𝑖 𝐿 𝑖 3
𝜕 3
𝜕 𝑥 3
𝑤̅
𝑖 (𝑥 ,𝑠 )
(3.11)
It follows from Eqs. (3.3, 3.4) that the internal forces of the beam element at any interior point
are expressed by the state vector as follows
(𝑇̅
𝑖 (𝑥 ,𝑠 ) 𝑀̅
𝑖 (𝑥 ,𝑠 ) 𝑄̅
𝑖 (𝑥 ,𝑠 ))
T
=𝐃 𝐸𝐵 ,𝑖 𝜼̂
𝑖 (𝑥 ,𝑠 ) (3.12)
26

where 𝐃 𝐸𝐵 ,𝑖 is a three-by-six constant matrix, which is given by
𝐃 𝐸𝐵 ,𝑖 =[
0 𝐸 𝑖 𝐴 𝑖 /𝐿 𝑖 0 0 0 0
0 0 0 0 𝐸 𝐼 𝑖 /𝐿 𝑖 2
0
0 0 0 0 0 −𝐸 𝐼 𝑖 /𝐿 𝑖 3
] (3.13)
Once the state vector 𝜼̂
𝑖 (𝑥 ,𝑠 ) is determined by the 𝑠 -domain solution (2.16), the
displacement, rotation, shear deformation and internal forces of the beam component at any point
can be easily computed by Eq. (3.12) in any frequency region.

3.3.2 Timoshenko Beam Elements
The internal forces of the Timoshenko beam element described by Eq. (3.6) can be similarly
expressed by the state vector as defined by Eqs. (3.8, 3.9). The transverse displacement 𝑤̅
𝑇 ,𝑖 (𝑥 ,𝑠 ) ,
slope (angle of rotation) 𝜃 ̅
𝑇 ,𝑖 (𝑥 ,𝑠 ) , bending moment 𝑀̅
𝑇 ,𝑖 (𝑥 ,𝑠 ) , and shear force 𝑄̅
𝑇 ,𝑖 (𝑥 ,𝑠 ) at any
interior point of the beam component (0<𝑥 <1) are
{

𝑤̅
𝑇 ,𝑖 (𝑥 ,𝑠 )=𝑤̅
𝑖 (𝑥 ,𝑠 )
𝜃 ̅
𝑇 ,𝑖 (𝑥 ,𝑠 )=𝜑̅
𝑖 (𝑥 ,𝑠 )
𝑀̅
𝑇 ,𝑖 (𝑥 ,𝑠 )=−𝐸 𝑖 𝐼 𝑖 1
𝐿 𝑖 𝜕 𝜑̅
𝑖 (𝑥 ,𝑠 )
𝜕𝑥
𝑄̅
𝑇 ,𝑖 (𝑥 ,𝑠 )=𝜅 𝑖 𝐺 𝑖 𝐴 𝑖 [
1
𝐿 𝑖 𝜕 𝜕𝑥
𝑤̅
𝑖 (𝑥 ,𝑠 )−𝜑̅
𝑖 (𝑥 ,𝑠 )]
(3.14)
As pointed out by researchers, when a beam structure vibrates at higher frequencies, the shear
deformation of the beam components cannot be neglected (Mei, 2005). Thus, in mid-frequency
analysis, it is necessary to investigate the effect of shear deformation in Timoshenko beam
components on the dynamic response of a structure. In the 𝑠 -domain, the shear deformation of the
𝑖 th
beam component is expressed as
27

𝛽 ̅
𝑖 (𝑥 ,𝑠 )=
1
𝐿 𝑖 𝜕 𝜕𝑥
𝑤̅
𝑖 (𝑥 ,𝑠 )−𝜑̅
𝑖 (𝑥 ,𝑠 ), 0<𝑥 <1 (3.15)
By Eqs. (3.14, 3.15), the displacements, internal forces and shear deformation of the beam
component can be expressed in terms of the state vector as follows
{𝑤̅
𝑇 ,𝑖 (𝑥 ,𝑠 ) 𝜃 ̅
𝑇 ,𝑖 (𝑥 ,𝑠 ) 𝛽 ̅
𝑖 (𝑥 ,𝑠 ) 𝑀̅
𝑇 ,𝑖 (𝑥 ,𝑠 ) 𝑄̅
𝑇 ,𝑖 (𝑥 ,𝑠 )}
T
=𝐃 𝑇𝑀 ,𝑖 𝜼̂
𝑖 (𝑥 ,𝑠 ) (3.16)
where 𝐃 𝑇𝑀 ,𝑖 is a five-by-six constant matrix given by
𝐃 𝑇𝑀 ,𝑖 =
[

0 0 1 0 0 0
0 0 0 0 1 0
0 0 0
1
𝐿 𝑖 −1 0
0 0 0 0 0 −
𝐸 𝑖 𝐼 𝑖 𝐿 𝑖 0 0 0
𝜅 𝑖 𝐺 𝑖 𝐴 𝑖 𝐿 𝑖 −𝜅 𝑖 𝐺 𝑖 𝐴 𝑖 0
]

(3.17)
As shall be seen in the subsequent numerical examples, the internal forces and shear
deformation of a Timoshenko beam structure in a mid- or high-frequency vibration analysis can
be computed by Eqs. (3.16, 3.17) once the state vectors of the beam elements are known. With the
accurate internal forces and shear deformation at mid- and high frequencies, the difference
between Euler-Bernoulli beam theory and Timoshenko beam theory can be studied.

3.4 Boundary Conditions in Beam Structures
As introduced in Chapter 2, the augmented formulation for a multi-body structure contains
three main steps. In Chapter 3.2, the first two steps in the augmented DTFM, formulation of
nondimensional subsystem state equations and formation of an augmented state equation for the
multi-body structure, have been discussed. And the global state equation (2.11) of the multi-body
structure modeled by Euler-Bernoulli beam theory or Timoshenko beam theory has been derived.
28

It has been that the augmented DTFM can provide local information, such as displacement,
moment, and shear force, conveniently. The next step is to obtain the description of the dynamic
interactions among subsystems to get the boundary condition equation (2.15). A detailed example
is presented to demonstrate how to treat boundary conditions in the multi-body beam structures
systematically.
Consider a multi-span beam structure modeled by the Timoshenko beam theory in Figure 3.1,
which carries a lumped mass at point A (a boundary node). The boundary conditions at point A
are
𝐽 𝐴 𝜕 2
𝜕 𝑡 2
𝜑 1
(0,𝑡 )−𝐸 1
𝐼 1
1
𝐿 1
𝜕 𝜑 1
(0,𝑡 )
𝜕𝑥
=0
𝑚 𝐴 𝜕 2
𝜕 𝑡 2
𝑤 1
(0,𝑡 )+𝑐 𝐴 𝜕 𝜕𝑡
𝑤 1
(0,𝑡 )+𝑘 𝐴 𝑤 1
(0,𝑡 )−𝜅 1
𝐺 1
𝐴 1
[
1
𝐿 1
𝜕 𝜕𝑥
𝑤 1
(0,𝑡 )−𝜑 1
(0,𝑡 )]=0
(3.18)
where 𝑤 1
(𝑥 ,𝑡 ) and 𝜑 1
(𝑥 ,𝑡 ) are the transverse displacement and angle of rotation of the beam
component 1; 𝐸 1
, 𝐼 1
, 𝐿 1
are the same as those defined in Eq. (3.6); 𝑚 𝐴 and 𝐽 𝐴 are the inertia and
moment of inertia of the mass, respectively; 𝑐 𝐴 is the coefficient of the damper; and 𝑘 𝐴 is the
coefficient of the spring. Equations (3.18), after Laplace transform, are written in the form
𝐌 𝐴 (𝑠 )𝒘̂
1
(0,𝑠 )=[
0
0
] (3.19)
where 𝒘̂
1
(0,𝑠 ) is the boundary value of 𝒘̂
1
(𝑥 ,𝑠 ) defined by Eq. (3.9) at point A (𝑥 =0) ; and
𝐌 𝐴 (𝑠 ) is a two-by-four matrix given by
𝐌 𝐴 (𝑠 )=
[

0 0 𝐽 𝐴 𝑠 2
−𝐸 1
𝐼 1
1
𝐿 1
𝑚 𝐴 𝑠 2
+𝑐 𝐴 𝑠 +𝑘 𝐴 −𝜅 1
𝐺 1
𝐴 1
1
𝐿 1
𝜅 1
𝐺 1
𝐴 1
0
]

(3.20)
29

In the modeling of the structure, Eq. (3.19) will be integrated into the global boundary condition
(2.15).
Now consider point C (an interior node) in Figure 3.1, where beam components 2 and 3 are
interconnected, and a lumped mass is attached. There are four matching conditions at point C
𝑤 2
(1,𝑡 )=𝑤 3
(0,𝑡 )
𝜑 2
(1,𝑡 )=𝜑 3
(1,𝑡 )
𝑚 𝐶 𝜕 2
𝜕 𝑡 2
𝑤 2
(1,𝑡 )+𝜅 2
𝐺 2
𝐴 2
[
𝜕 𝐿 2
𝜕𝑥
𝑤 2
(1,𝑡 )−𝜑 2
(1,𝑡 )]
−𝜅 3
𝐺 3
𝐴 3
[
𝜕 𝐿 3
𝜕𝑥
𝑤 3
(0,𝑡 )−𝜑 3
(0,𝑡 )]=0
𝐽 𝐶 𝜕 2
𝜕 𝑡 2
𝜑 2
(1,𝑡 )+𝐸 2
𝐼 2
1
𝐿 2
𝜕 𝜑 2
(1,𝑡 )
𝜕𝑥
−𝐸 3
𝐼 3
1
𝐿 3
𝜕 𝜑 3
(0,𝑡 )
𝜕𝑥
=0
(3.21)
where 𝑤 𝑖 (𝑥 ,𝑡 ) and 𝜑 𝑖 (𝑥 ,𝑡 ) are the transverse displacement and angle of rotation of the beam
component 𝑖 ; 𝐸 2
, 𝐼 2
, 𝐿 2
, 𝐸 3
, 𝐼 3
, 𝐿 3
are the same as those defined in Eq. (3.6); and 𝑚 𝐶 and 𝐽 𝐶 are
the inertia and moment of inertia of the lamped mass C, respectively. Equations (3.21) can be cast
into the following form
𝐌 𝐶 (𝑠 )𝒘̂
3
(0,𝑠 )+𝐍 𝐶 (𝑠 )𝒘̂
2
(1,𝑠 )=[𝟎 ]
4×1
(3.22)
where 𝒘̂
2
(1,𝑠 ) and 𝒘̂
3
(0,𝑠 ) are the boundary values of 𝒘̂
2
(𝑥 ,𝑠 ) and 𝒘̂
3
(𝑥 ,𝑠 ) at point C; and
𝐌 𝐶 (𝑠 ) and 𝐍 𝐶 (𝑠 ) are given by
30

𝐌 𝐶 (𝑠 )=
[

−1 0 0 0
0 0 −1 0
0 −𝜅 3
𝐺 3
𝐴 3
1
𝐿 3
𝜅 3
𝐺 3
𝐴 3
0
0 0 0 −𝐸 3
𝐼 3
1
𝐿 3
]

𝐍 𝐶 (𝑠 )=
[

1 0 0 0
0 0 1 0
𝑚 𝐶 𝑠 2
𝜅 2
𝐺 2
𝐴 2
1
𝐿 2
−𝜅 2
𝐺 2
𝐴 2
0
0 0 𝐽 𝐶 𝑠 2
𝐸 2
𝐼 2
1
𝐿 2
]

(3.23)
Again, in modeling, Eq. (3.22) will be integrated into the global boundary condition (2.15).
Structures carrying lumped masses are widely seen in various engineering applications. In the
augmented DTFM, lumped masses are treated as parts of the boundary or matching conditions at
the nodes of the structure, which is included in boundary matrices 𝐌 𝑔 (𝑠 ) and 𝐍 𝑔 (𝑠 ) . However,
for other energy-based methods, such as, EFA and SEA, coupling loss factors or energy
transmission coefficients have to be determined before the structure analysis, which requires more
experimental work or numerical tests.

3.5 A demonstrative Example: Double-beam Structure
The augmented DTFM is illustrated on a double-beam structure as shown in Figure 3.2, where
two uniform Timoshenko beam components are elastically coupled by a uniformly distributed
spring of coefficient 𝑘 𝑠 . In modeling, the upper beam component is labeled as beam 1, and the
lower beam component beam 2. The transverse displacements of the beam components are
governed by the coupled partial differential equations as follows
31

𝜌 1
𝐴 1
𝜕 2
𝜕𝑡
2
𝑤 1
(𝑥 ,𝑡 )−𝑓 𝑇 ,𝑖 (𝑥 ,𝑡 )=
𝜅 1
𝐴 1
𝐺 1
[
1
𝐿 1
2
𝜕 2
𝜕 𝑥 2
𝑤 1
(𝑥 ,𝑡 )−
1
𝐿 1
𝜕 𝜕𝑥
𝜑 1
(𝑥 ,𝑡 )]−𝑘 𝑠 (𝑤 1
(𝑥 ,𝑡 )−𝑤 2
(𝑥 ,𝑡 )) (3.24𝑎 )
𝜌 1
𝐼 1
𝜕 2
𝜕𝑡
2
𝜑 1
(𝑥 ,𝑡 )=
𝐸 1
𝐼 1
𝐿 1
2
𝜕 2
𝜕 𝑥 2
𝜑 1
(𝑥 ,𝑡 )+𝜅 1
𝐴 1
𝐺 1
[
1
𝐿 1
𝜕 𝜕𝑥
𝑤 1
(𝑥 ,𝑡 )−𝜑 1
(𝑥 ,𝑡 )] (3.24𝑏 )

𝜌 2
𝐴 1
𝜕 2
𝜕𝑡
2
𝑤 2
(𝑥 ,𝑡 )−𝑓 𝑇 ,2
(𝑥 ,𝑡 )=
𝜅 2
𝐴 2
𝐺 2
[
1
𝐿 2
2
𝜕 2
𝜕 𝑥 2
𝑤 2
(𝑥 ,𝑡 )−
1
𝐿 2
𝜕 𝜕𝑥
𝜑 2
(𝑥 ,𝑡 )]−𝑘 𝑠 (𝑤 2
(𝑥 ,𝑡 )−𝑤 1
(𝑥 ,𝑡 )) (3.24𝑐 )
𝜌 2
𝐼 2
𝜕 2
𝜕𝑡
2
𝜑 2
(𝑥 ,𝑡 )=
𝐸 2
𝐼 2
𝐿 2
2
𝜕 2
𝜕 𝑥 2
𝜑 2
(𝑥 ,𝑡 )+𝜅 2
𝐴 2
𝐺 2
[
1
𝐿 2
𝜕 𝜕𝑥
𝑤 2
(𝑥 ,𝑡 )−𝜑 2
(𝑥 ,𝑡 )] (3.24𝑑 )

for 𝑡 >0, 0<𝑥 <1, where the equations have been scaled with the nondimensional spatial
coordinate 𝑥 , and the definition of the beam parameters is the same as that for Eq. (3.6).
Because longitudinal deformation is not considered in the beam components, each beam
component is described by two partial differential equations. Thus, the global state vector 𝜼̂(𝑥 ,𝑠 )
has eight elements. Equations (3.24a, 3.24b) are converted to the augmented form as Eq. (2.11),
for which the global quantities are given by
𝜼̂(𝑥 ,𝑠 )={𝒘̂
1
T
(𝑥 ,𝑠 ), 𝒘̂
2
T
(𝑥 ,𝑠 )}
T
𝐅̂
(𝑠 )=
[

0 1 0 0 0 0 0 0
(𝜌 1
𝐴 1
𝑠 2
+𝑘 𝑠 )
𝜅 1
𝐴 1
𝐺 1
𝐿 1
2
0 0 𝐿 1
−𝑘 𝑠 𝜅 1
𝐴 1
𝐺 1
𝐿 1
2
0 0 0
0 0 0 1 0 0 0 0
0
−𝜅 1
𝐴 1
𝐺 1
𝐸 1
𝐼 1
𝐿 1
(𝜅 1
𝐴 1
𝐺 1
+𝜌 1
𝐼 1
𝑠 2
)
𝐸 1
𝐼 1
𝐿 1
2
0 0 0 0 0
0 0 0 0 0 1 0 0
−𝑘 𝑠 𝜅 2
𝐴 2
𝐺 2
𝐿 2
2
0 0 0
(𝜌 2
𝐴 2
𝑠 2
+𝑘 𝑠 )
𝜅 2
𝐴 2
𝐺 2
𝐿 2
2
0 0 𝐿 2
0 0 0 0 0 0 0 1
0 0 0 0 0
−𝜅 2
𝐴 2
𝐺 2
𝐸 2
𝐼 2
𝐿 2
(𝜅 2
𝐴 2
𝐺 2
+𝜌 2
𝐼 2
𝑠 2
)
𝐸 2
𝐼 2
𝐿 2
2
0
]

𝒒̂(𝑥 ,𝑠 )={0 −
𝑓 ̅
𝑇 ,1
(𝑥 ,𝑠 )
𝜅 1
𝐴 1
𝐺 1
𝐿 1
2
0 0 0 −
𝑓 ̅
𝑇 ,2
(𝑥 ,𝑠 )
𝜅 2
𝐴 2
𝐺 2
𝐿 2
2
0 0}
T
(3.25)
32

with
𝒘̂
𝑖 (𝑥 ,𝑠 )={𝑤̅
𝑖 (𝑥 ,𝑠 ) 𝜕 𝑤̅
𝑖 (𝑥 ,𝑠 ) 𝜕𝑥 ⁄ 𝜑̅
𝑖 (𝑥 ,𝑠 ) 𝜕 𝜑̅
𝑖 (𝑥 ,𝑠 ) 𝜕𝑥 ⁄ }
T
(3.26)
For demonstration purpose, natural frequencies of the double-beam structure are computed
by Eq. (2.23) with beam parameters shown in Table 3.1. The results (exact solutions) obtained by
the augmented DTFM are compared with those by the FEM and those by an analytical solution
method given in reference (Chen 1994). Assume that all the boundary nodes (points A, B, C and
D) of the structure are fixed (clamped). Listed in Table 3.2 are the first five and some higher-mode
natural frequencies of the beam structure. For the first five natural frequencies, the results from the
reference can match with the exact solutions given by the augmented DTFM, and the 10-element
FEM provides reasonably good results. For the natural frequencies of the 51
th
to 55
th
modes, the
FEM requires 1,000 elements to get convergent results, compared to the results given by the
augmented DTFM.
If all the boundary nodes are simply supported, for the first five natural frequencies, the results
by the augmented DTFM and those from reference (Chen, 1994) match well; see Table 3.3. Also,
as shown in the table, the proposed method has no difficulty determining the natural frequencies
of higher modes. Further simulations show that the augmented DTFM can determine the natural
frequencies of the structure in a very high frequency region, with almost the same computational
effort as in a low-frequency region. Additionally, comparison of Tables 3.2 and 3.3 reveals that
the boundary conditions of the beam structure have fewer effects on the natural frequencies of
higher modes. This is physically expected – as the wave number increases, the effects of the
boundary conditions decrease.
33

As seen in the above demonstrative example, the augmented DTFM can deal with complex
beam structures in a systematic way. With Eqs. (3.25, 3.26), the natural frequencies of the double-
beam structure can be accurately and efficiently computed. Besides, distributed viscous damping
can be conveniently added through proper modification of the state matrix 𝐅̂
(𝑠 ) in Eq. (3.25). This
shows the versatility of the proposed method in modeling, analysis, and computation. More results
about the frequency response of various complex beam structures will be presented in following
numerical examples.

3.6 Numerical Examples
The DTFM-cased vibration analysis, which is presented in this Chapter, is illustrated on three
examples: a two-span beam, a fifteen-beam structure, and a 30-beam frame as a simplified
structure model of a car. As shall be seen from the numerical simulation, the proposed method is
accurate and efficient in determination of mid-frequency responses for various types of
Timoshenko beam structures. For validation, the numerical predictions by the augmented DTFM
are compared with the results by the finite element method (FEM) (Friedman, 1993).
In simulations, the time-averaged variables are used to represent the frequency response of a
structure predicted by different methods. The time-averaged displacements, internal forces and
shear deformation of the beam component 𝑖 at point 𝑥 𝑝 are given by
〈𝑣 ̅
𝑖 (𝑥 𝑝 ,𝑠 )〉=[
1
2
(𝑣 ̅
𝑖 (𝑥 𝑝 ,𝑠 ))(𝑣 ̅
𝑖 (𝑥 𝑝 ,𝑠 ))
∗
]
1 2 ⁄
(3.27)

34

where 〈∙〉 represents a time-averaged variable; (∙)
∗
represents complex conjugate; and 𝑠 = 𝑗𝛺
with 𝑗 =√−1 and 𝛺 being an excitation frequency. Here, 𝑣 ̅
𝑖 (𝑥 𝑝 ,𝑠 ) is the local information of the
beam structure, which can be obtained by Eqs. (3.12, 3.16).

3.6.1 A Two-Beam Structure
Shown in Figure 3.3 is a simple structure of two collinearly coupled beams. This example has
been used in Reference (Nefske, 1989). These two beam elements of the structure are identical.
The beam structure is simply supported at nodes 𝐴 , 𝐵 , 𝐶 , and a pointwise harmonically varying
force of magnitude 𝐹 is acting at point 𝐷 (the midpoint of beam element 1). In simulation, the
following non-dimensional values of the frame parameters are assigned:
𝜌 1
=𝜌 2
=27, 𝐸 𝐼 1
=𝐸 𝐼 2
=200, 𝐿 1
=𝐿 2
=1
The first natural frequency of each beam element is 𝑓 1
=7.998 Hz, and the damping ratio of the
complex modulus of elasticity for the structure is 𝜁 =0.1.
In Figure 3.4, the displacement magnitude curves at the middle point of beam element 2 is
plotted against the excitation frequency, from low- to high-frequency regions, which are obtained
by several methods. The methods in comparison are: the augmented DTFM, which delivers exam
solutions of frequency response; the modal expansion with the first 200 eigenfunctions obtained
by the DTF, the SEA, and the EFA. At lower frequencies (10 to 3.5×10
3
Hz), the augmented
DTFM and the modal expansion obtain the same results. In the region of 3.5×10
3
to 10
5
Hz, the
augmented DTFM gives more accurate predictions of the system response than the EFA. The
modal expansion solution becomes erroneous beyond 10
4
Hz, due to the insufficient number of
modes used in the computation. Further computation indicates that the modal expansion requires
35

2,000 modes to get good results. In this case, the augmented DTFM only needs eight state variables
in Eq. (2.12), and it provides highly accurate results from low- to high frequencies.
The spatial distribution of the displacement of beam element 1 at the excitation frequency of
1,000 Hz is plotted in Figure 3.5, in which the curves are obtained by the augmented DTFM, the
FEA with 200 elements, and the EFA. As can be seen from the figure, the prediction by the
augmented DTFM and that of the FEA are in good agreement. With the EFA, however, the near-
field effects are neglected, and the displacements are smoothed, which leads to the significant
difference between the augmented DTFM and the EFA. Indeed, the EFA cannot give a spatial
distribution of the beam displacement, but the augmented DTFM can provide such local
information, with accuracy and efficiency.
Figure 3.6 plots the energy flows at the joint point 𝐵 in the frequency region 10
1
to 10
5
Hz,
which are determined by the augmented DTFM and the EFA. The augmented DTFM presents the
correct trend of the energy flow change as the excitation frequency increases. As can be seen, the
energy flow by the EFA is smaller than that by the augmented DTFM at higher frequencies. This
is because certain neglections are made in the EFA formulation. A similar phenomenon is also
seen in the displacement curves in Figure 3.4. Additionally, as shown in Figures 3.4 and 3.6, the
augmented DTFM exhibits vibration resonance at different frequencies, but the EFA fails to do
that.

3.6.2 A Fifteen-Beam Structure
Consider a beam frame with 15 elements as shown in Figure 3.7. A similar structure has been
considered as a simplified light truck chassis frame in Reference (Cho, 1993). A pointwise
36

harmonically varying force is applied at the middle point of beam element 2. All the beam elements
have the same material properties, with Young’s modulus being 200 GPa and linear density
7860 Kg/m
3
.
First, compare the proposed method with the FEA. The displacement at the midpoint of beam
element 15 in the frequency region of 10
3
to 10
5
Hz is computed by the augmented DTFM and
the FEA with 500 and 1500 elements, and the results are plotted in Figure 3.8. The damping ratio
of the complex modulus of elasticity for the simulation is chosen as 0.05. As can be seen from the
figure, at lower frequencies, the FEA results and the DTFM predictions are in good agreement.
However, in the mid-frequency range (10
4
to 10
5
Hz), the FEA with 600 elements may not be
accurate enough. In this case, more elements (say 1500 elements) are needed to obtain results
matching those by the DTFM. Further simulations show that the DTFM is applicable to vibration
analyses at higher frequencies. Thus, the augmented DTFM can perform vibration analyses in both
low and mid- frequency regions.
Second, compare the proposed method with the EFA. To this end, the energy density at the
midpoint of beam element 15 is computed by the augmented DTFM and the EFA with 300 and
1200 elements. In the EFA, beam elements are assumed to be semi-infinite to derive the energy
transmission coefficients at interconnection joints. With the assumption, fixed boundary nodes are
simplified as free nodes. For comparison purposes, in the DTFM-based analysis, all the boundary
nodes are set to be free ones. The DTFM, however, is applicable to beam structures with arbitrary
boundary conditions.
The computed results on the energy density are plotted in Figure 3.9, for a frequency region
of 10
5
to 10
8
Hz. The damping ratio of the complex modulus of elasticity for the simulation is
37

chosen as 0.025. As seen from the figure, the EFA model with 300 elements is not good enough
to cover the entire frequency region. The EFA model of 1200 elements, while being able to follow
the trend predicted by the DTFM, misses the local variations of the energy density. Additional
simulations show that the DTFM can obtain accurate results for the excitation frequency way
higher than 10
8
Hz. This indicates that the augmented DTFM can produce highly accurate results
in mid- to high-frequency analyses of multi-body beam structures.

3.6.3 A Thirty-Beam Structure
Shown in Figure 3.10 is a thirty-beam frame, which can be viewed as a simplified two-
dimensional car frame. In Figure 3.10, number 𝑖 designates the 𝑖 th
beam component, and circles
represent the nodes where the beam components are rigidly connected. In this example, all beam
components undergo both longitudinal vibration and transverse vibration. Thus, with 30 beams,
the global state vector where 𝜼̂(𝑥 ,𝑠 ) has 180 state variables.
All the beam components have the same parameters as follows: density 𝜌 =7,800 kg/m
3
,
Young’s modulus 𝐸 =210 Gpa, Poisson’s ratio 𝜈 =0.30, shear modulus 𝐺 =80.77 Gpa, beam
width 𝑏 =0.01 m, beam thickness ℎ=0.01 m; and the Timoshenko shear coefficient of the
cross-section 𝜅 =10(1+𝜈 ) (12+11𝜈 )=0.8497 ⁄ . Assume that the beam frame has modal
damping ratio 𝜉 =0.05, for all modes. Also, a lumped mass of inertia 40 kg is attached to the
node that connects beam components 10 and 11. Note that all the 30 components of the frame can
be viewed as thin beams because the length-to-height ratio (𝐿 /ℎ) is larger than 10. As shall be
seen, even with these thin beam components, the effect of shear deformation at medium and high
38

frequencies is quite noticeable. In other words, the Timoshenko beam theory is useful in mid-
frequency analysis of structures with thin beam components.
Consider the forced vibration of the structure. A point-wise sinusoidal force with magnitude
of 10 N is applied at the lumped mass vertically. Let the excitation frequency of the force be 𝑓 =
5×10
3
Hz, which is near the 108
th
natural frequency of the beam structure. Figure 3.11 plots the
spatial distributions of the time-averaged transverse displacement of beam components 10 and 27,
which are obtained by the augmented DTFM and the FEA with 3,000 elements. (To match the
augmented DTFM predictions, at least 3,000 finite elements in the DTFM are required.) The
distribution curves by two methods are in good agreement. Compared to the FEM, the proposed
method gives accurate analytical results with much less computational effort. In addition, the shear
force distributions of beam components 8 and 11 are presented in Figure 3.12, and good agreement
between the proposed method and the FEA is seen.
As observed from Figures 3.11 and 3.12, beam components 10 and 11 exhibit lower-mode
vibration with a longer wavelength while beam components 8 and 27 present higher-mode
vibration with a shorter wavelength. This interesting phenomenon can be explained as follows. As
mentioned before, beam components 10 and 11 are very close (connected) to the lumped mass,
and beam components 8 and 27 are a bit far away from the mass. From the kinetic energy viewpoint,
in the neighborhood of the mass, higher-frequency local modes of vibration can hardly be excited,
and as such, lower-frequency local modes become dominant. On the other hand, for a component
that is relatively far from the mass, higher-frequency modes can be excited.
Next, the usefulness of the Timoshenko beam theory in the mid-frequency vibration analysis
of beam structures is demonstrated. As mentioned before, all beam components in this example
are thin beams in the conventional sense in low-frequency analysis. However, the shear
39

deformation of the beam structure can be significant as the excitation frequency increases. Again,
a point-wise sinusoidal force with magnitude of 10 N is applied at the lumped mass vertically. In
this simulation case, a wide excitation frequency region, from a relatively low frequency 𝑓 =
1,000 Hz to a very high frequency 𝑓 =1×10
6
Hz, is considered. Near the high end of this
frequency region, the FEM analysis requires too much computational effort in terms of the number
of degrees of freedom and computation time and it still can give inaccurate results. For this reason,
only the proposed method is used here.
By the augmented DTFM, the angle of bending and the total angle of ratio at the midpoint of
the beam component 11 are plotted against the excitation frequency in the upper part of Figure
3.13. Also plotted in the lower part of Figure 3.13 is the ratio of the shear deformation to the total
angle of rotation
Ratio=(
𝛽 ̅
𝑖 𝜑̅
𝑖 )
𝑥 =𝑥 ∗
(3.28)

where 𝑖 =11 (beam component number), and 𝑥 ∗
represents the midpoint of component 11. As
observed in Figure 3.13, when the excitation frequency is below 1×10
4
Hz, the difference
between the angle of bending and the total angle of rotation is small, and the shear deformation is
less than 10% of the total angle of rotation. This means that the shear deformation of the structure
is neglectable for the excitation frequency less than 1×10
4
Hz. At a higher excitation frequency,
say 5×10
4
Hz, the shear deformation is about 100% of the total angle of rotation or more. Thus,
the excitation frequency 𝑓 =1×10
4
Hz can be regarded as a cut-off frequency. Below this cut-
off frequency, the shear deformation is less than 10% of the total angle of ratio. Beyond the cut-
off frequency, the shear deformation becomes obvious, and its effect on the beam vibration must
be considered when the excitation frequency is higher than 5×10
4
Hz.
40

To further investigate the difference between the Euler-Bernoulli beam theory and the
Timoshenko beam theory, the shear forces computed by two beam theories are compared. Plotted
in Figure 3.14 are the spatial distributions of the time-averaged shear force of beam components 8
and 11, at excitation frequency 𝑓 =5×10
3
Hz. Although the current excitation frequency is
below the aforementioned cut-off frequency (1×10
4
), the difference between the distributions
of shear force obtained by two beam theories is noticeable. If the excitation frequency is above the
cut-off frequency, say frequency 𝑓 =5×10
4
Hz, the discrepancy in the shear force predictions
by two beam theories is obvious. This is shown in the spatial distribution curves in Figure 3.15 for
the same beam components.
In this example, the augmented DTFM is shown to be able to deliver accurate solutions, for
displacement, strain and internal forces and at both low and high frequencies. Also, the proposed
method is computationally efficient because the global state vector contains a fixed number of
elements at any high frequency, without the need for discretization.

41

Figure 3.1 A three-span beam carrying lamped masses

Figure 3.2 An elastically connected double beam structure

42

Figure 3.3 A two-span beam structure with a point-wise load

Figure 3.4 Nondimensionalized displacement at the middle point of the beam element 2

43

Figure 3.5 Nondimensional displacement along the beam element 1

Figure 3.6 Nondimensional energy flows at the joint point of two beams

44

Figure 3.7 A beam frame with 15 elements

Figure 3.8 Nondimensional energy density along beam element 1

45

Figure 3.9 Nondimensional energy density at the middle point of beam element 2

Figure 3.10 A two-dimensional beam frame: simplified car frame

46

Figure 3.11 The spatial distribution of the time-averaged transverse displacement of beam
components (10) and (27)

47

Figure 3.12 The spatial distribution of the time-averaged shear force of beam components (8)
and (11)

48

Figure 3.13 Upper plot: The time-averaged angle of bending and angle of rotation. Lower plot:
The ratio of the shear displacement divided by the angle of rotation

49

Figure 3.14 The spatial distribution of the time-averaged shear force of beam components (8)
and (11) with the excitation frequency 𝑓 =5×10
3
𝐻𝑧
50

Figure 3.15 The spatial distribution of the time-averaged shear force of beam components (8)
and (11) with the excitation frequency 𝑓 =5×10
4
𝐻𝑧

51

Table 3.1 Structure parameters of the double-beam structure

Parameters Data
Young’s modulus GPa 200
Shear modulus (GPa) 76.92
Poisson’s ratio 0.3
Timoshenko shear coefficient 0.87
Density (kg/m
3
) 7,600
Spring stiffness (N/m) 8,000
Width of the upper beam (m) 0.01
Height of the upper beam (m) 0.01
Width of the lower beam (m) 0.01
Height of the lower beam (m) 0.005
Beam length of two beam elements (m) 1
52

Table 3.2 The natural frequencies f
k
of the elastically connected double-beam structure with fixed
boundary conditions (Hz)

k
Reference (Chen,
1994)
FEM (Friedman, 1993) (elements)
DTFM
10 1000
1 33.96 33.5624 33.5485 33.9554
2 55.85 56.1461 55.1168 55.8451
3 76.17 76.4485 76.1690 76.1711
4 144.2 144.8454 144.2319 144.2320
5 146.1 146.1937 144.2427 146.0777
51 / / 10,419.3875 10,418.5434
52 / / 10,883.2369 10,882.3152
53 / / 11,087.9893 11,086.9728
54 / / 11,775.0436 11,773.8271
55 / / 11,798.3554 11,797.1746
53

Table 3.3 The natural frequencies f
k
of the elastically connected double-beam structure with
simply supported boundary conditions (Hz)

𝒌 Reference (Chen, 1994) DTFM
1 18.83 18.8312
2 33.49 33.4879
3 51.71 51.7119
4 94.53 94.5280
5 107.1 107.1351
51 / 10,106.0278
52 / 10,477.4175
53 / 10,766.5607
54 / 11,382.6474
55 / 11,445.7218
54

Chapter 4 Transient Vibration Analysis of Beam Structures at Mid-
and High-Frequencies
4.1 Introduction
With the augmented DTFM proposed in Chapters 2 and 3, a new method for transient analyses
of two-dimensional beam structures in mid- and high-frequency regions is developed. Based on
the augmented DTFM, two transient analysis tools are proposed: the method of inverse Laplace
transform and the generalized modal expansion. As shall be seen, these tools do not rely on
discretization in space and numerical integration in time, and they give transient solutions in exact
analytical form. In Chapter 4.2, two transient analysis tools are introduced; and similarity and
difference between the two analysis tools are discussed. A model reduction is then proposed for
the augmented DTFM transient analysis tools in Chapter 4.3, which can improve computational
efficiency without loss of accuracy. In Chapter 4.4, a numerical example is presented, such that
the accuracy of the proposed transient analysis method is validated, and the efficiency of the
proposed model reduction is discussed.

4.2 Transient Analysis Tools
4.2.1 Transient Analysis Tool 1: The Method of Inverse Laplace Transform
The 𝑠 -domain solution and the eigenvalue problem of the multibody-structure have been
introduced in Chapter 2, which can be utilized to develop the method of inverse Laplace transform.
According to Eq. (2.17) and Eq. (2.23), the poles of the global distributed transfer functions are
55

the roots of the characteristic equation in the eigenvalue problem. So, transfer function poles and
the system eigenvalues are the same.
By inverse Laplace transform of Eq. (2.16), the transient response of the multi-body structure
is expressed by the following Green’s function integral
𝜼 (𝑥 ,𝑡 )=∫ ∫ 𝐆 (𝑥 ,𝜉 ,𝑡 −𝜏 )𝒑 (𝜉 ,𝜏 )𝑑𝜉 1
0
𝑑𝜏 𝑡 0
+∫𝐇 (𝑥 ,𝑡 −𝜏 )𝜸 𝑏 (𝜏 )𝑑𝜏 𝑡 0
(4.1)
for 0<𝑥 <1, 𝑡 >0, where the time domain solution 𝜼 (𝑥 ,𝑡 ) is the inverse Laplace transform of
the 𝑠 -domain solution 𝜼̂(𝑥 ,𝑠 ); the global Green’s functions 𝐆 (𝑥 ,𝜉 ,𝑡 ) and 𝐇 (𝑥 ,𝑡 ) are the inverse
Laplace transforms of global distributed transfer functions 𝐆̂
(𝑥 ,𝜉 ,𝑠 ) and 𝐇̂
(𝑥 ,𝑠 ) , respectively;
and 𝒑 (𝜉 ,𝜏 ) and 𝜸 𝑏 (𝜏 ) are related to external, boundary and initial disturbances of the structure.
Let the poles of 𝐆̂
(𝑥 ,𝜉 ,𝑠 ) and 𝐇̂
(𝑥 ,𝑠 ) be 𝑠 ±𝑘 , 𝑘 =1,2,…. The transfer function poles of an
undamped structure are of the form 𝑠 ±𝑘 =±𝑗 𝜔 𝑘 , with 𝑗 =√−1 and 𝜔 𝑘 being the 𝑘 th
natural
frequency of the structure. The transfer function poles of a damped structure generally are of the
form 𝑠 ±𝑘 =− 
𝑘 ±𝑗 𝜔 𝑘 . The poles are obtained by solving the characteristic equation (2.23).
By the theorem of residues, the Green’s functions in Eq. 4.1) are given by
𝐆 (𝑥 ,𝜉 ,𝑡 )=∑𝐆 𝒌 (𝑥 ,𝜉 ,𝑡 )
∞
𝑘 =1
𝐇 (𝑥 ,𝑡 )=∑𝐇 𝒌 (𝑥 ,𝑡 )
∞
𝑘 =1
(4.2)
where 𝐆 𝒌 (𝑥 ,𝜉 ,𝑡 ) and 𝐇 𝒌 (𝑥 ,𝑡 ) are the Green’s functions of the structure in the 𝑘 th
mode of
vibration and they are given by (Yang, 2010)
56

𝐆 𝒌 (𝑥 ,𝜉 ,𝑡 )=𝚽̂
𝑔 (𝑥 ,0,𝑠 𝑘 )𝐑 𝑘 𝐃 (𝑥 ,𝜉 ,𝑠 𝑘 )𝑒 𝑠 𝑘 𝑡 +𝚽̂
𝑔 (𝑥 ,0,−𝑠 𝑘 )𝐑 −𝑘 𝐃 (𝑥 ,𝜉 ,−𝑠 𝑘 )𝑒 −𝑠 𝑘 𝑡 (4.3𝑎 )

𝐇 𝒌 (𝑥 ,𝑡 )=𝚽̂
𝑔 (𝑥 ,0,𝑠 𝑘 )𝐑 𝑘 𝑒 𝑠 𝑘 𝑡 +𝚽̂
𝑔 (𝑥 ,0,−𝑠 𝑘 )𝐑 −𝑘 𝑒 −𝑠 𝑘 𝑡 (4.3𝑏 )
with
𝐃 (𝑥 ,𝜉 ,𝑠 𝑘 )={
𝐌 𝑔 (𝑠 𝑘 )𝚽̂
𝑔 (𝑥 ,𝜉 ,𝑠 𝑘 ), 𝜉 ≤𝑥 −𝐍 𝑔 (𝑠 𝑘 )𝚽̂
𝑔 (1,𝜉 ,𝑠 𝑘 ), 𝜉 >𝑥 (4.4)
and
𝐑 ±𝑘 =Res(𝐙 (±𝑠 𝑘 )
−1
) (4.5𝑎 )
𝐙 (𝑠 )=𝐌 𝑔 (𝑠 )+𝐍 𝑔 (𝑠 )𝚽̂
𝑔 (1,0,𝑠 ) (4.5𝑏 )
In the previous equations, matrices 𝐑 𝑘 and 𝐑 −𝑘 are the residues of the distributed transfer
functions at the poles 𝑠 𝑘 and 𝑠 −𝑘 , respectively. These residues are given in the exact and closed
form (Yang, 2010; Morse, 1953)
𝐑 ±𝑘 =
adj𝐙 (±𝑠 𝑘 )
𝑑 𝑑𝑠 {det𝐙 (𝑠 )}
𝑠 =±𝑠 𝑘 (4.6)

where adj𝐙 (𝑠 ) is the adjoint of 𝐙 (𝑠 ) .
With the exact transfer function residues given by Eq. (4.6), the transient response of the
structure subject to external, boundary and initial disturbances can be determined analytically by
the Green’s function integral in Eq. 4.1). Hence, the computation of 𝜼 (𝑥 ,𝑡 ) does not depend on
any numerical time integration algorithm, which is commonly seen in approximate methods, such
as the inverse Fast Fourier Transform used by FEA, energy methods and wave methods.
Furthermore, because the poles and residues of 𝐆̂
(𝑥 ,𝜉 ,𝑠 ) and 𝐇̂
(𝑥 ,𝑠 ) can be precisely estimated
57

at high frequencies as shown in Chapter 3, the method of inverse Laplace transform is applicable
to mid- and high-frequency problems of bean structures.
One special property of this tool is that the spatial derivatives of beam displacements at any
point can be conveniently obtained from the state vector 𝜼 (𝑥 ,𝑡 ) . This means that the new method
can provide the local information about the transient displacements and transient internal forces
even at very high frequencies, which is similar to the way shown in Eqs. (3.12, 3.16). Otherwise,
the determination of local information would be difficult by most existing methods for mid- and
high-frequency analyses.

4.2.2 Transient Analysis Tool 2: The Generalized Modal Expansion
In this analysis tool, the eigensolutions of a multi-body structure, which are obtained by the
augmented formulation in Chapter 3, are used to construct transient vibration solutions. To this
end, express the transient response of the structure by
𝒘 (𝑥 ,𝑡 )=∑𝑼 𝑘 (𝑥 )𝑞 𝑘 (𝑡 )
∞
𝑘 =1
(4.7)
where 𝒘 (𝑥 ,𝑡 ) is a vector of the displacements of the structure; 𝑼 𝑘 (𝑥 ) is a vector of the normalized
eigenfunctions of all subsystems in the 𝑘 th
mode of vibration, and 𝑞 𝑘 (𝑡 ) is the corresponding
modal coordinate. For instance, for a structure with 𝑁 Euler-Bernoulli beam elements, 𝒘 (𝑥 ,𝑡 ) and
𝑼 𝑘 (𝑥 ) can be written as
𝒘 (𝑥 ,𝑡 )={
𝑦 1
T
(𝑥 ,𝑡 ) 𝑦 2
T
(𝑥 ,𝑡 ) ⋯ 𝑦 𝑁 T
(𝑥 ,𝑡 )}
T
𝑼 𝑘 (𝑥 )={𝑉 𝑘 ,1
T
(𝑥 ) 𝑉 𝑘 ,2
T
(𝑥 ) ⋯ 𝑉 𝑘 ,𝑁 T
(𝑥 )}
T
(4.8)
58

where 𝑦 𝑚 (𝑥 ,𝑡 ) is the displacement vector of the 𝑚 th
element given by
𝑦 𝑚 (𝑥 ,𝑡 )={𝑢 𝑚 (𝑥 ,𝑡 ) 𝑤 𝑚 (𝑥 ,𝑡 )}
T
(4.9𝑎 )
and 𝑉 𝑘 ,𝑚 (𝑥 ) is the modal displacement vector of the 𝑚 th
element in the 𝑘 th
mode of vibration
given by
𝑉 𝑘 ,𝑚 (𝑥 )={𝑢̃
𝑘 ,𝑚 (𝑥 ) 𝑤̃
𝑘 ,𝑚 (𝑥 )}
T
(4.9𝑏 )
where 𝑢̃
𝑘 ,𝑚 (𝑥 ) is the normalized eigenfunction of the 𝑚 th
element in the 𝑘 th
mode of longitudinal
vibration, and 𝑤̃
𝑘 ,𝑚 (𝑥 ) is the normalized eigenfunction of the 𝑚 th
element in the 𝑘 th
mode of
transverse vibration.
If Timoshenko beam elements are considered, 𝒘 (𝑥 ,𝑡 ) and 𝑼 𝑘 (𝑥 ) have the same format as
shown in Eq. (4.8), with their components given by
𝑦 𝑚 (𝑥 ,𝑡 )={𝑢 𝑚 (𝑥 ,𝑡 ) 𝑤 𝑚 (𝑥 ,𝑡 ) 𝜑 𝑚 (𝑥 ,𝑡 )}
T
(4.10𝑎 )
𝑉 𝑘 ,𝑚 (𝑥 )={𝑢̃
𝑘 ,𝑚 (𝑥 ) 𝑤̃
𝑘 ,𝑚 (𝑥 ) 𝜑̃
𝑘 ,𝑚 (𝑥 )}
T
(4.10𝑏 )
where 𝑢̃
𝑘 ,𝑚 (𝑥 ) , 𝑤̃
𝑘 ,𝑚 (𝑥 ) , and 𝜑̃
𝑘 ,𝑚 (𝑥 ) are the normalized eigenfunctions of the 𝑚 th
element in
the 𝑘 th
mode of longitudinal, transverse and rotational vibrations, respectively.
Substituting the modal series, Eq. (4.7), into the governing equations, Eq. (3.1) or Eq. (3.6),
and using the orthonormal conditions of the eigensolutions, eventually yields an infinite set of
decoupled differential equations about the modal coordinates
𝑞 ̈𝑘 (𝑡 )+𝜔 𝑘 2
𝑞 𝑘 (𝑡 )=𝑄 𝑘 (𝑡 ), 𝑘 =1,2,3,… (4.11)
where 𝑄 𝑘 is the modal force corresponding to the 𝑘 th
mode and it is given by
59

𝑄 𝑘 (𝑡 )=∫
{𝑼 𝑘 (𝜉 )}
T
{𝒇̅
(𝜉 ,𝑡 )}𝑑𝜉 1
0
(4.12)
with {𝒇̅
(𝜉 ,𝑡 )} being the vector of external forces that are from Eq. (3.1) or Eq. (3.6). Thus, the
solution to Eq. (4.11) is
𝑞 𝑘 (𝑡 )=𝑞 𝑘 (0)cos𝜔 𝑘 𝑡 +𝑞 ̇𝑘 (0)
1
𝜔 𝑘 sin𝜔 𝑘 𝑡 +∫
1
𝜔 𝑘 sin[𝜔 𝑘 (𝑡 −𝜏 )]𝑄 𝑘 (𝜏 )𝑑𝜏 𝑡 0
(4.13)
for 𝑘 =1,2,3,…, where the initial values 𝑞 𝑘 (0) and 𝑞 ̇𝑘 (0) of the modal coordinates are obtained
through use of the initial conditions of the beam elements.
If modal damping is considered, the modal equation (4.11) will be modified as follows
𝑞 ̈𝑘 (𝑡 )+2𝜁 𝑘 𝜔 𝑘 𝑞 ̇𝑘 (𝑡 )+𝜔 𝑘 2
𝑞 𝑘 (𝑡 )=𝑄 𝑘 (𝑡 ) (4.14)
where 𝜁 𝑘 is the damping ratio of the 𝑘 th
vibration mode. In this case, the damped modal equations
can be easily solved by a conventional method. After the modal coordinates are determined, the
transient response of the structure is given by the modal series (4.7).

4.2.3 Similarity and Difference Between Two Analysis Tools
The method of inverse Laplace transform and the generalized modal expansion bear the same
advantages, including determination of analytical transient responses by infinite series, no use of
spatial discretization and numerical integration, presentation of local information about beam
displacement and internal forces, and consistent and accurate solutions from low to high
frequencies.
One important key to the utility of these two tools in mid- and high-frequency problems is
the capability to obtain accurate system eigensolutions at high frequencies, which many existing
60

methods lack. This capability is made possible by the augmented state-space formulation as
described in Chapters 2 and 3.
For transient analysis of a damped structure, however, the two tools differ in the way of
damping specification. Tool 1 (the method of inverse Laplace transform) is convenient for
describing damping physically, such as partially distributed viscous damping layers and dampers
(dashpots) at nodes. Tool 2 (the generalized modal expansion), on the other hand, is good for
assigning modal damping, which treats damping in a global manner.

4.3 Modal Reduction for Computational Efficiency
The two tools of the augmented DTFM developed in Chapter 4.2 give transient solutions by
the infinite series, Eq. (4.2) and Eq. (4.7). This implies that many terms in these series may be
involved in transient analysis of a beam structure at medium and high frequencies. In this section,
to improve computational efficiency, model reduction by truncated series is investigated. The key
in such model reduction is the balance between accuracy and efficiency in computation.
Assume that a beam structure is subject to excitations in a mid- or high-frequency region. The
idea of the model reduction here is to truncate the infinite series (4.2) and (4.7) according to the
following two requirements:
(i) The dominant low-frequency modes of the structure are included in vibration; and
(ii) The excitation spectrum in the mid- or high-frequency region is properly covered.
61

The dominant low-frequency modes of the structure are necessary to describe the vibration
of the structure subject to initial disturbances, quasi-static loads, and loads with constant
components or components that converge to constants as time goes by.
The coverage of the excitation spectrum may adopt a 𝜇 −𝜎 rule, which is stated as follows.
Let the bandwidth of the excitation spectrum be 𝜎 . For the excitation at a single frequency, 𝜎 is
determined such that several neighboring natural frequencies of the structure are included in the
spectrum. Now, select a frequency region of bandwidth 𝜇 ×𝜎 , with 𝜇 >1, such that the excitation
spectrum is in the middle of the region. Apparently, the larger the 𝜇 , the more accurate results are
expected, which is at the cost of more computational effort.
According to the previous discussion, as the first step in transient analysis, the transfer
function poles or system eigenvalues are to be determined in the following two frequency regions:
(i) Low-frequency region: Ω
𝐿 ={𝜔 |0≤𝜔 ≤𝜔 𝐿 } , which includes the dominant low-frequency
modes of the structure; and
(ii) High-frequency region: Ω
𝐻 ={𝜔 | 𝜔 𝐻 1
≤𝜔 ≤𝜔 𝐻 2
, 𝜔 𝐻 2
−𝜔 𝐻 1
=𝑛𝜎 } , which properly
covers the excitation spectrum,
where 𝜔 𝐿 , 𝜔 𝐻 1
, and 𝜔 𝐻 2
are the pre-selected bounds of the frequency regions. The transfer
function poles or system eigensolutions in Ω
𝐿 and Ω
𝐻 can be precisely computed by Eq. (2.23).
With the poles or eigensolutions determined in the selected frequency regions, model
reduction proceeds. With the method of inverse Laplace transform, the Green’s functions in Eq.
(4.2) are approximated by the truncated series
62

𝐆 (𝑥 ,𝜉 ,𝑡 )≈∑𝐆 𝑗 (𝑥 ,𝜉 ,𝑡 )
𝑁 0
𝑗 =1
+ ∑ 𝐆 𝑘 (𝑥 ,𝜉 ,𝑡 )
𝑁 2
𝑘 =𝑁 1

𝐇 (𝑥 ,𝑡 )≈∑𝐇 𝑗 (𝑥 ,𝑡 )
𝑁 0
𝑗 =1
+ ∑ 𝐇 𝑘 (𝑥 ,𝑡 )
𝑁 2
𝑘 =𝑁 1
(4.15)
where 𝑁 0
is the mode number corresponding to the upper bound 𝜔 𝐿 of the low-frequency region
Ω
𝐿 ; and 𝑁 1
and 𝑁 2
are the mode numbers corresponding to the bounds (𝜔 𝐻 1
,𝜔 𝐻 2
) of the high-
frequency region Ω
𝐻 . Each of the two series in Eq. (4.15) has two sums, with the first sum related
to low-frequency vibration and the second sum about high-frequency vibration.
Similarly, with the generalized modal expansion, the transient solution given in Eq. (4.7) is
approximated by the following truncated series
𝒘 (𝑥 ,𝑡 )≈∑𝑼 𝑗 (𝑥 )𝑞 𝑗 (𝑡 )
𝑁 0
𝑗 =1
+ ∑ 𝑼 𝑘 (𝑥 )𝑞 𝑘 (𝑡 )
𝑁 2
𝑘 =𝑁 1
(4.16)
where the mode numbers 𝑁 0
, 𝑁 1
and 𝑁 2
are the same as those defined in Eq. (4.15).
The truncated series Eq. (4.15) and (4.16) give two reduced models for mid- and high-
frequency analyses of beam structures. The number of terms in each of the truncated series is 𝑁 𝑅 =
𝑁 0
+𝑁 2
−𝑁 1
+1. With properly selected numbers 𝑁 0
, 𝑁 1
and 𝑁 2
, significant computational
effort can be saved, and at the same time, accurate transient solutions still can be obtained. This
shall be shown in the numerical examples.
It should be pointed out that the truncated series (4.15) and (4.16) are obtained without the
need to deal with large matrices, which is often the case in other numerical methods. Regardless
of how many terms or modes are selected for computation, the dimensions of the Green’s functions
63

𝐆 𝑘 (𝑥 ,𝜉 ,𝑡 ) and 𝐇 𝑘 (𝑥 ,𝑡 ) in the series (4.2), and the eigenfunctions 𝑼 𝑘 (𝑥 ) in series (4.7) remain the
same. In fact, the computational effort for a low-frequency term and that for a high-frequency term
in the truncated series (4.15) and (4.16) are almost the same. Therefore, the truncated models given
in this section can be used to improve computational efficiency in mid- and high-frequency
analysis of beam structures.
In summary, mid- and high-frequency transient vibration analyses of beam frames by the
augmented DTFM take the following four steps:
(i) Formulate each beam element in the frame by Eqs. (2.10);
(ii) Assemble all beam elements into the augmented DTFM by Eqs. (2.11, 2.12, 2.15);
(iii) Compute the transient response by the inverse Laplace transform, Eqs. (4.1, 4.2); or
compute the transient response by the modal expansion, Eqs. (4.7, 4.14);
(iv) Use model reduction to improve computational efficiency, Eq. (4.15) or Eq. (4.16).
A flowchart of the proposed transient analysis is shown in Figure 4.1.

4.4 Numerical Example
The augmented DTFM transient analysis tools and the model reduction are illustrated on one
numerical example: a branched five-beam structure. In the example, the mid- and high-frequency
transient responses of the structures are computed. Because modal damping is considered, the
generalized modal expansion is used. Also, for validation purposes, the numerical results obtained
by the proposed method are compared with those by the FEA.
64

Consider a five-beam structure in Figure 4.2, where the boundary points A, C, E, and F are
all fixed, and symbols , , …, are the beam element numbers. A sinusoidal external force,
𝐹 (𝑡 )=𝐹 0
∗sin (2𝜋𝑓𝑡 ) , is applied to the midpoint H of element (5). Assume that every beam
element undergoes both longitudinal and transverse displacements, as described by Eq. (3.1). In
the state-space formulation, the state vector and the state matrix are given by Eqs. (2.12).
In the numerical simulation, all the parameters are assigned non-dimensional values. Let all
the beam elements have the same linear density, bending stiffness and longitudinal rigidity: 𝜌𝐴 =
27, 𝐸𝐼 =700, and 𝐸𝐴 =7×10
6
. The geometric parameters of the frame are specified as: 𝐿 1
=
2, 𝐿 2
=1.25, 𝐿 3
=1.5, 𝐿 4
=1, 𝐿 5
=√2, 𝛼 =37°. Also, the excitation amplitude and frequency
are assigned as 𝐹 0
=1 and 𝑓 =2,000 Hz. The beam structure is initially at rest.
The transient response of element (1) at its midpoint G is computed. A convergence test shows
that the augmented DTFM needs 500 modes to get convergent results. This is shown in Figure 4.3,
where the transverse displacement of element (1) at point G is computed by the proposed method,
with 200, 400 and 500 modes. Figure 4.4 compares the transverse displacement curves of the beam
element (1) at point G that are computed by the augmented DTFM with 500 modes and the FEA
with 2000 elements. As seen from the figure and relevant computation, the FEA requires at least
2000 elements to obtain convergent solutions for this five-beam frame. The computing time for
the FEA with 2000 elements, however, is about 30 times that by the augmented DTFM with 500
modes. This indicates that the augmented DTFM is much more computationally efficient than the
FEA in mid- and high-frequency transient analyses.
Again, for further improvement of computational efficiency, Eq. (4.16) is used to obtain the
following two reduced models:
65

(i) A 14-mode model: 8 modes are chosen from the low-frequency region and 6 modes are
chosen from the high-frequency region of 1,940 to 2,060 Hz.
(ii) A 26-mode model: 16 modes are chosen from the low-frequency region and 10 modes
are chosen from the high-frequency region of 1,910 Hz to 2,090 Hz.
Note that the high-frequency region of each reduced model properly covers the excitation
frequency 𝑓 =2,000 Hz.
The transverse displacement of element (1) at its midpoint G is computed by the reduced
models and the results are plotted in Figure 4.5. For comparison, the prediction by the augmented
DTFM with 500 modes serves as a reference solution, which is also plotted in the figure. It is seen
from Figure 4.5 that the 26-mode curve matches the reference solution well, and apparently it is
better than the 14-mode result. However, the 14-mode result does not deviate from the reference
solution significantly, and it correctly predicts the trend of the transient response. The ratio of the
time used in the simulation by the 500-mode model to that by the 26-mode model is about 20. This
implies that with acceptable simulation results, about 95% of computation time is saved by the
reduced model. Of course, if high accuracy is critically important, the “full model” of modes
can be used, which is still computationally more efficient than many existing methods.
One advantage of the augmented DTFM is that the transient internal forces (e.g., bending
moment and shear force) of a beam structure at any point can be conveniently obtained from the
state vector 𝜼 (𝑥 ,𝑡 ). The transient bending moment and shear force of the element (1) at the
midpoint G are plotted in Figure 4.6 through use of the augmented DTFM with 500 modes. The
local information about internal forces at high frequencies, as given in Figure 4.6, should be useful
for optimal design and health monitoring of flexible structures in various engineering applications.
66

Figure 4.1 Flowchart of the augmented DTFM

67

Figure 4.2 A five-beam frame

68

Figure 4.3 The transverse displacement at point G by DTFM

69

Figure 4.4 The transverse displacement at point G by DTFM and FEM

70

Figure 4.5 The transverse displacement at point G by reduced model

71

Figure 4.6 The bending moment and shear force of at point G

72

Chapter 5 Vibration Analysis of Thin Plates by Distributed
Transfer Function Method
5.1 Introduction
Sandwich plates are found in numerous engineering applications, including aerospace,
mechanical and civil engineering (Mouritz, 2001; Ning, 2007). A typical sandwich plate is
configured of two parallel plates that are connected by an elastic or viscoelastic layer. In
engineering practices, the plate elements have same or different material properties and boundary
conditions. In broad applications, sandwich plate structures have multiple parallel plate elements,
and the multi-layer sandwich plate can fulfill more practical interests (Liaw, 1967). Viscoelastic
damping can be used for vibration control (Mead, 1998). The adjacent two parallel plates in
sandwich plates hereby are fully or partially connected by one or multiple types of viscoelastic
materials. Thus, the vibration analysis of multi-layer sandwich plates with viscoelastic layers is
essential for design and optimization of complex structures.
Several analytical solution methods have been proposed for vibration analysis of sandwich
plates (Alam, 1984; Rao, 2004; Foraboschi, 2013). Because these analytical solutions were derived
by using the series approximation based on the Navier solution, they can only be applied to plates
with simply supported boundary conditions. Also, it has been pointed out that for the high
frequency vibration analysis, hundreds of thousands of terms are required by the series solution to
obtain the converged result (Alaimo, 2019). There are some alternative methods for the vibration
analysis of sandwich plates with viscoelastic layers, such as the model strain energy method (MSE
Method) (He, 1988) and the wave spectral finite element method (WSFEM) (Mejdi, 2015). The
MSE Method was developed when the damping of the sandwich layer was proportional and the
73

FEA was used as a numerical solver, which means it is suitable for low frequency vibration
analysis. The WSFEM, on the other hand, can only provide accurate results at high frequencies,
when the flexural wavelength is small and near-field effect can be ignored. In summary of existing
methods, there are several issues regarding vibration analysis of sandwich plates with viscoelastic
layers. First, almost all existing methods can only work one frequency region at a time. High-
frequency vibration analysis tools cannot be applied to low-frequency problems due to the
assumptions made in the modeling, low-frequency vibration methods are not suitable for high-
frequency problems because of the intensive computation, and mid-frequency vibration problem
is difficult because hybrid formulation and solution procedure is required. Second, at mid and high
frequencies, for methods relying on the spatial discretization and approximation in series, a lot of
computing effort is required to obtain accurate results. Third, many existing methods cannot
provide the local information at any location in the structure, such as bending moment and shear
force. Therefore, a method that can conduct vibration analysis for sandwich plates with viscoelastic
layers in different frequency regions efficiently and is capable of providing detailed local
information accurately is in demand.
In this chapter, a new analytical approach for vibration analysis of sandwich plate with
viscoelastic layers is presented. The approach is based on the distributed transfer function method
(DTFM) (Yang, 1992) and its augmented formulation (Noh, 2014). The DTFM is applied to
multilayer sandwich plates with Lévy type boundary conditions that are fully or partially connected
by different types of viscoelastic layers. The systematic way of formulating different sandwich
plates shows the flexibility of the new method. Also, the DTFM delivers the analytical solution to
the eigenvalue problem and frequency response of sandwich plates. One highlight of the proposed
method is that regardless of the frequency region, the consistent formulation and solution
74

procedure can always provide the detailed local information of the sandwich plate, such as bending
moment and shear force at any location, without difficulty.
The remainder of the chapter is organized as follows. Chapter 5.2 states the vibration problem
a sandwich Lévy plate structure with viscoelastic layers in consideration. The formulations of
several viscoelastic models are introduced in Chapter 5.3. Chapter 5.4 presents the analytical
solution to the eigenvalue problem and frequency response of the plate structure by the distributed
transfer function method (DTFM). In Chapter 5.5, three numerical examples are presented to
illustrate the accuracy and efficiency of the proposed method, from low-frequency region to high-
frequency region. The results obtained in this work are summarized in Chapter 5.6.

5.2 Problem Statement
In this section, the vibration problem of Lévy plate structures with viscoelastic layers is stated.
The DTFM formulation for a single Lévy plate resting on a viscoelastic foundation is first
introduced in Chapter 5.2.1. The DTFM formulation for sandwich Lévy plates with viscoelastic
layers is given in Chapter 5.2.2.

5.2.1 A Single Lévy Plate Resting on A Viscoelastic Foundation
Consider a single Lévy plate rests on a viscoelastic foundation as shown in Figure 5.1. The
homogeneous rectangular thin plate is modeled by the Kirchhoff-Love plate theory and the
viscoelastic foundation is described by the Kelvin-Voigt Model. When the in-plane deformation
75

of the plate is ignored, the displacement of the out-plane transverse vibration 𝑤 (𝑥 ,𝑦 ,𝑡 ) is governed
by the linear partial differential equation
𝐷 [
𝜕 4
𝑤 (𝑥 ,𝑦 ,𝑡 )
𝜕 𝑥 4
+
𝜕 4
𝑤 (𝑥 ,𝑦 ,𝑡 )
𝜕 𝑥 2
𝜕 𝑦 2
+
𝜕 4
𝑤 (𝑥 ,𝑦 ,𝑡 )
𝜕 𝑦 4
]+𝑘 𝑓 𝑤 (𝑥 ,𝑦 ,𝑡 )+𝑐 𝑓 𝑤 ̇(𝑥 ,𝑦 ,𝑡 )
=𝑞 (𝑥 ,𝑦 ,𝑡 )−2𝜌 ℎ
𝜕 2
𝑤 (𝑥 ,𝑦 ,𝑡 )
𝜕 𝑡 2
(5.1)
for 0<𝑥 <𝑎 , 0<𝑦 <𝑏 , where 𝑥 and 𝑦 are spatial coordinates in two directions; 𝑡 is the
temporal parameter; 𝑎 is the length of the plate in the 𝑥 direction; 𝑏 is the length of the plate in the
𝑦 direction; 2ℎ is the thickness of the plate; 𝜌 , 𝐸 and 𝜈 are the density, Young’s modulus and
Poisson’s ratio of the plate material, respectively; 𝐷 is bending stiffness of the plate where 𝐷 =
2ℎ
3
𝐸 3(1−𝜈 2
)
; 𝑞 (𝑥 ,𝑦 ,𝑡 ) is the transverse external force acting on the plate; 𝑘 𝑓 describes the elasticity of
the viscoelastic foundation and 𝑐 𝑓 describes the viscosity of the foundation.
The thin rectangular plate described by Eq. (5.1) is subject to Lévy-type boundary conditions,
which means two opposite edges of the plate are simply supported and the other two opposite
edges have arbitrary boundary conditions, such as, free, clamped and simply supported boundary
conditions. Without loss of generality, as shown in Figure 1, let two edges 𝑥 =0 and 𝑥 =𝑎 be
simply supported; and two edges 𝑦 =0 and 𝑦 =𝑏 have arbitrary boundary conditions. In this case,
the transverse displacement 𝑤 (𝑥 ,𝑦 ,𝑡 ) of the Lévy plate can be expressed in terms of single Fourier
series
𝑤 (𝑥 ,𝑦 ,𝑡 )= ∑sin(
𝑚𝜋𝑥 𝑎 )𝑌 𝑚 (𝑦 ,𝑡 )
∞
𝑚 =1
(5.2)
76

where 𝑚 =1,2,3,…. The simply supported boundary conditions on two edges 𝑥 =0 and 𝑥 =𝑎
are satisfied by the term sin(
𝑚𝜋𝑥 𝑎 ), and arbitrary boundary conditions on two edges 𝑦 =0 and
𝑦 =𝑏 are fulfilled by the term 𝑌 𝑚 (𝑦 ,𝑡 ) , which is an unknown function to be determined, and
𝑌 𝑚 (𝑦 ,𝑡 ) is a function of location 𝑦 and time 𝑡 .
For consistency, the loading function 𝑞 (𝑥 ,𝑦 ,𝑡 ) is also written in terms of the series by
𝑞 (𝑥 ,𝑦 ,𝑡 )= ∑𝑞 𝑚 (𝑦 ,𝑡 )sin(
𝑚𝜋𝑥 𝑎 )
∞
𝑚 =1
(5.3𝑎 )
with
𝑞 𝑚 (𝑦 ,𝑡 )=
𝑎 2
∫ 𝑞 (𝑥 ,𝑦 ,𝑡 )sin(
𝑚𝜋𝑥 𝑎 )𝑑𝑥 𝑎 0
(5.3𝑏 )
To cast the governing equation of the single Lévy plate into the spatial state-space equation,
take Laplace transform of Eqs. (5.1-5.3) with respect to time and assume zero initial conditions to
obtain the 𝑠 -domain equations
𝐷 [
𝜕 4
𝑤̅(𝑥 ,𝑦 ,𝑠 )
𝜕 𝑥 4
+
𝜕 4
𝑤̅(𝑥 ,𝑦 ,𝑠 )
𝜕 𝑥 2
𝜕 𝑦 2
+
𝜕 4
𝑤̅(𝑥 ,𝑦 ,𝑠 )
𝜕 𝑦 4
]+𝑘 𝑓 𝑤̅(𝑥 ,𝑦 ,𝑠 )+𝑐 𝑓 𝑠 𝑤̅(𝑥 ,𝑦 ,𝑠 )
=𝑞̅(𝑥 ,𝑦 ,𝑠 )−2𝜌 ℎ𝑠 2
𝑤̅(𝑥 ,𝑦 ,𝑠 )
(5.4)
𝑤̅(𝑥 ,𝑦 ,𝑠 )= ∑𝑌̅
𝑚 (𝑦 ,𝑠 )sin(
𝑚𝜋𝑥 𝑎 )
∞
𝑚 =1
(5.5)
𝑞̅(𝑥 ,𝑦 ,𝑠 )= ∑𝑞̅
𝑚 (𝑦 ,𝑠 )sin(
𝑚𝜋𝑥 𝑎 )
∞
𝑚 =1
(5.6𝑎 )
𝑞̅
𝑚 (𝑦 ,𝑠 )=
2
𝑎 ∫ 𝑞̅(𝑥 ,𝑦 ,𝑠 )sin(
𝑚𝜋𝑥 𝑎 )𝑑𝑥 𝑎 0
(5.6𝑏 )
77

where 𝑠 is the complex transform parameter, and the overbar stands for the 𝑠 -domain variables.
Plugging Eqs (5.5) and (5.6) into Eq. (5.4) yields
∑{
𝐷 [(
𝑚𝜋
𝑎 )
4
𝑌̅
𝑚 (𝑦 ,𝑠 )−2(
𝑚𝜋
𝑎 )
2
𝜕 2
𝑌̅
𝑚 (𝑦 ,𝑠 )
𝜕 𝑦 2
+
𝜕 4
𝑌̅
𝑚 (𝑦 ,𝑠 )
𝜕 𝑦 4
]
+𝑘 𝑓 𝑌̅
𝑚 (𝑦 ,𝑠 )+𝑐 𝑓 𝑠 𝑌̅
𝑚 (𝑦 ,𝑠 )−𝑞̅
𝑚 (𝑦 ,𝑠 )+2𝜌 ℎ𝑠 2
𝑌̅
𝑚 (𝑦 ,𝑠 )
}sin(
𝑚𝜋𝑥 𝑎 )
∞
𝑚 =1
=0 (5.7)
for 0<𝑦 <𝑏 , where 𝑚 =1,2,3,… . Equation (5.7) contains an infinite set of independent
(decoupled) equations. For the 𝑚 th
term in the summation, its vibration mode is corresponding to
an 𝑠 -domain governing equation
𝐷 [(
𝑚𝜋
𝑎 )
4
𝑌̅
𝑚 (𝑦 ,𝑠 )−2(
𝑚𝜋
𝑎 )
2
𝜕 2
𝑌̅
𝑚 (𝑦 ,𝑠 )
𝜕 𝑦 2
+
𝜕 4
𝑌̅
𝑚 (𝑦 ,𝑠 )
𝜕 𝑦 4
]+𝑘 𝑓 𝑌̅
𝑚 (𝑦 ,𝑠 )+𝑐 𝑓 𝑠 𝑌̅
𝑚 (𝑦 ,𝑠 )
=𝑞̅
𝑚 (𝑦 ,𝑠 )−2𝜌 ℎ𝑠 2
𝑌̅
𝑚 (𝑦 ,𝑠 )
(5.8)
By solving the ordinary differential equation (5.8), the unknown function 𝑌̅
𝑚 (𝑦 ,𝑠 ) that
represents the vibration mode in the 𝑦 direction can be obtained. In the distributed transfer
function method (DTFM), Eq. (5.8) is cast into a spatial state-space equation. For the 𝑚 th
mode,
define the state vector 𝜼̂
𝑚 (𝑦 ,𝑠 ) as follows
𝜼̂
𝑚 (𝑦 ,𝑠 )={𝑌̅
𝑚 (𝑦 ,𝑠 )
𝜕 𝑌̅
𝑚 (𝑦 ,𝑠 )
𝜕𝑦
𝜕 2
𝑌̅
𝑚 (𝑦 ,𝑠 )
𝜕 𝑦 2
𝜕 3
𝑌̅
𝑚 (𝑦 ,𝑠 )
𝜕 𝑦 3
}
T
(5.9)
By using the 4-by-1 state vector 𝜼̂
𝑚 (𝑦 ,𝑠 ) that contains 𝑌̅
𝑚 (𝑦 ,𝑠 ) and its spatial derivatives,
Eq. (5.8) can be written in a state-space form
𝜕 𝜕𝑦
𝜼̂
𝑚 (𝑦 ,𝑠 )=𝐅̂
𝑚 (𝑦 ,𝑠 )+𝒑̂
𝑚 (𝑦 ,𝑠 ) (5.10)
for 0<𝑦 <𝑏 , where 𝐅̂
𝑚 (𝑦 ,𝑠 ) is a 4-by-4 state matrix and 𝒑̂
𝑚 (𝑦 ,𝑠 ) is a 4-by-1 loading vector,
and they are given by
78

𝐅̂
𝑚 (𝑦 ,𝑠 )=
[

0 1 0 0
0 0 1 0
0 0 0 1
[
−2𝜌 ℎ𝑠 2
−𝑘 𝑓 −𝑐 𝑓 𝑠 𝐷 −(
𝑚𝜋
𝑎 )
4
] 0 2(
𝑚𝜋
𝑎 )
2
0
]

(5.11)
𝒑̂
𝑚 (𝑦 ,𝑠 )={
0 0 0
𝑞̅
𝑚 (𝑦 ,𝑠 )
𝐷 }
T
(5.12)
When the initial conditions are nonzero, the Laplace transform of the initial disturbances can
be added to the loading vector 𝒑̂
𝑚 (𝑦 ,𝑠 ). It has been shown that for a single Lévy plate, its
transverse vibration can be formulated by infinite independent (decoupled) 𝑠 -domain governing
equations. Each governing equation corresponds to one vibration mode sin(
𝑚𝜋𝑥 𝑎 ) and is expressed
by the state equation (5.10).
Since the state vector 𝜼̂
𝑚 (𝑦 ,𝑠 ) defined by Eq. (5.9) contains the transverse displacement of
the plate and its spatial derivatives, the DTFM can provide the local information about
displacement, slope, moment, and shear force of the thin plate easily. In the DTFM, slopes
𝜃 ̅
𝑥 (𝑥 ,𝑦 ,𝑠 ) and 𝜃 ̅
𝑦 (𝑥 ,𝑦 ,𝑠 ), bending moments 𝑀̅
𝑥𝑥
(𝑥 ,𝑦 ,𝑠 ) and 𝑀̅
𝑦𝑦
(𝑥 ,𝑦 ,𝑠 ), twisting moment
𝑀̅
𝑥𝑦
(𝑥 ,𝑦 ,𝑠 ), and shear forces 𝑄̅
𝑥𝑥
(𝑥 ,𝑦 ,𝑠 ) and 𝑄̅
𝑦𝑦
(𝑥 ,𝑦 ,𝑠 ) at any point of the plate can be
obtained in form of series as follows
𝜃 ̅
𝑥 (𝑥 ,𝑦 ,𝑠 )= ∑(
𝑚𝜋
𝑎 )𝑌̅
𝑚 (𝑦 ,𝑠 )cos(
𝑚𝜋𝑥 𝑎 )
∞
𝑚 =1
(5.13𝑎 )
𝜃 ̅
𝑦 (𝑥 ,𝑦 ,𝑠 )= ∑
𝜕 𝑌̅
𝑚 (𝑦 ,𝑠 )
𝜕𝑦
sin(
𝑚𝜋𝑥 𝑎 )
∞
𝑚 =1
(5.13𝑏 )
𝑀̅
𝑥𝑥
(𝑥 ,𝑦 ,𝑠 )= ∑{−𝐷 [−(
𝑚𝜋
𝑎 )
2
𝑌̅
𝑚 (𝑦 ,𝑠 )+ 𝜈 𝜕 2
𝑌̅
𝑚 (𝑦 ,𝑠 )
𝜕 𝑦 2
]}
∞
𝑚 =1
sin(
𝑚𝜋𝑥 𝑎 ) (5.13𝑐 )
79

𝑀̅
𝑦𝑦
(𝑥 ,𝑦 ,𝑠 )= ∑{−𝐷 [−𝜈 (
𝑚𝜋
𝑎 )
2
𝑌̅
𝑚 (𝑦 ,𝑠 )+
𝜕 2
𝑌̅
𝑚 (𝑦 ,𝑠 )
𝜕 𝑦 2
]}
∞
𝑚 =1
sin(
𝑚𝜋𝑥 𝑎 ) (5.13𝑑 )
𝑀̅
𝑥𝑦
(𝑥 ,𝑦 ,𝑠 )= ∑{−𝐷 (1−𝜈 )[(
𝑚𝜋
𝑎 )
𝜕 𝑌̅
𝑚 (𝑦 ,𝑠 )
𝜕𝑦
]}
∞
𝑚 =1
cos(
𝑚𝜋𝑥 𝑎 ) (5.13𝑒 )
𝑄̅
𝑥𝑥
(𝑥 ,𝑦 ,𝑠 )= ∑{−𝐷 (
𝑚𝜋
𝑎 )[−(
𝑚𝜋
𝑎 )
2
𝑌̅
𝑚 (𝑦 ,𝑠 )+
𝜕 2
𝑌̅
𝑚 (𝑦 ,𝑠 )
𝜕 𝑦 2
]}
∞
𝑚 =1
cos(
𝑚𝜋𝑥 𝑎 ) (5.13𝑓 )
𝑄̅
𝑦𝑦
(𝑥 ,𝑦 ,𝑠 )= ∑{−𝐷 [−(
𝑚𝜋
𝑎 )
2
𝜕 𝑌̅
𝑚 (𝑦 ,𝑠 )
𝜕𝑦
+
𝜕 3
𝑌̅
𝑚 (𝑦 ,𝑠 )
𝜕 𝑦 3
]}
∞
𝑚 =1
sin(
𝑚𝜋𝑥 𝑎 ) (5.13𝑔 )
In a Lévy plate, arbitrary boundary conditions are assigned to edges 𝑦 =0 and 𝑦 =𝑏 . These
boundary conditions can be expanded and written in terms of series as
∑{𝐌 𝑚 (𝑠 )𝜼̂
𝑚 (0,𝑠 )+𝐍 𝑚 (𝑠 )𝜼̂
𝑚 (𝑏 ,𝑠 )−𝜸̂
𝑚 (𝑠 )}sin(
𝑚𝜋𝑥 𝑎 )
∞
𝑚 =1
=0 (5.14)
where 𝑚 =1,2,3,…; 𝜼̂
𝑚 (0,𝑠 ) and 𝜼̂
𝑚 (𝑏 ,𝑠 ) are values of the 4-by-1 state vector on edges 𝑦 =0
and 𝑦 =𝑏 , respectively; 𝐌 𝑚 (𝑠 ) and 𝐍 𝑚 (𝑠 ) are two 4-by-4 boundary matrices containing
coefficients of the state variables; and 𝜸̂
𝑚 (𝑠 ) is a 4-by-1 loading vector consisting of Laplace
transforms of boundary disturbances. Hence, for the 𝑚 th
vibration mode sin(
𝑚𝜋𝑥 𝑎 ), its 𝑠 -domain
governing equation (5.10) is subject to the following boundary conditions
𝐌 𝑚 (𝑠 )𝜼̂
𝑚 (0,𝑠 )+𝐍 𝑚 (𝑠 )𝜼̂
𝑚 (𝑏 ,𝑠 )=𝜸̂
𝑚 (𝑠 ) (5.15)
For illustration purpose, several common boundary conditions, such as, simply supported (hinged),
clamped (fixed), and free edges are listed below:
80

Simply supported: 𝑤 (𝑥 ,0,𝑡 )=𝑤 (𝑥 ,𝑏 ,𝑡 )=0, 𝑀 𝑦𝑦
(𝑥 ,0,𝑡 )=𝑀 𝑦𝑦
(𝑥 ,𝑏 ,𝑡 )=0 (5.16𝑎 )
Clamped: 𝑤 (𝑥 ,0,𝑡 )=𝑤 (𝑥 ,𝑏 ,𝑡 )=0, 𝑄 𝑦𝑦
(𝑥 ,0,𝑡 )=𝑄 𝑦𝑦
(𝑥 ,𝑏 ,𝑡 )=0 (5.16𝑏 )
Free: 𝑀 𝑦𝑦
(𝑥 ,0,𝑡 )=𝑀 𝑦𝑦
(𝑥 ,𝑏 ,𝑡 )=0, 𝑉 𝑦𝑦
(𝑥 ,0,𝑡 )=𝑉 𝑦𝑦
(𝑥 ,𝑏 ,𝑡 )=0 (5.16𝑐 )
where 𝑉 𝑦𝑦
(𝑥 ,𝑦 ,𝑡 ) is the Kirchhoff free edge condition, and it is given by [Reddy, 2006;Ugural,
2009]
𝑉 𝑦𝑦
(𝑥 ,𝑦 ,𝑡 )=𝑄 𝑦𝑦
(𝑥 ,𝑦 ,𝑡 )+
𝜕 𝑀 𝑥𝑦
(𝑥 ,𝑏 ,𝑡 )
𝜕𝑥
(5.17)

Taking Laplace transform of Eq. (5.16) and expanding resulting equations by sine and cosine
functions gives boundary conditions as follows
Simply supported: 𝑌̅
𝑚 (0,𝑠 )=𝑌̅
𝑚 (𝑏 ,𝑠 )=0,
−𝐷 [−𝜈 (
𝑚𝜋
𝑎 )
2
𝑌̅
𝑚 (0,𝑠 )+
𝜕 2
𝑌̅
𝑚 (0,𝑠 )
𝜕 𝑦 2
]=0
−𝐷 [−𝜈 (
𝑚𝜋
𝑎 )
2
𝑌̅
𝑚 (𝑏 ,𝑠 )+
𝜕 2
𝑌̅
𝑚 (𝑏 ,𝑠 )
𝜕 𝑦 2
]=0
(5.18𝑎 )
Clamped: 𝑌̅
𝑚 (0,𝑠 )=𝑌̅
𝑚 (𝑏 ,𝑠 )=0,
−𝐷 [−(
𝑚𝜋
𝑎 )
2
𝜕 𝑌̅
𝑚 (0,𝑠 )
𝜕𝑦
+
𝜕 3
𝑌̅
𝑚 (0,𝑠 )
𝜕 𝑦 3
]=0
−𝐷 [−(
𝑚𝜋
𝑎 )
2
𝜕 𝑌̅
𝑚 (𝑏 ,𝑠 )
𝜕𝑦
+
𝜕 3
𝑌̅
𝑚 (𝑏 ,𝑠 )
𝜕 𝑦 3
]=0
(5.18𝑏 )
Free:
−𝐷 [−𝜈 (
𝑚𝜋
𝑎 )
2
𝑌̅
𝑚 (0,𝑠 )+
𝜕 2
𝑌̅
𝑚 (0,𝑠 )
𝜕 𝑦 2
]=0 (5.18𝑐 )
81

−𝐷 [−𝜈 (
𝑚𝜋
𝑎 )
2
𝑌̅
𝑚 (𝑏 ,𝑠 )+
𝜕 2
𝑌̅
𝑚 (𝑏 ,𝑠 )
𝜕 𝑦 2
]=0
−𝐷 [(𝜈 −2)(
𝑚𝜋
𝑎 )
2
𝜕 𝑌̅
𝑚 (0,𝑠 )
𝜕𝑦
+
𝜕 3
𝑌̅
𝑚 (0,𝑠 )
𝜕 𝑦 3
]=0
−𝐷 [(𝜈 −2)(
𝑚𝜋
𝑎 )
2
𝜕 𝑌̅
𝑚 (𝑏 ,𝑠 )
𝜕𝑦
+
𝜕 3
𝑌̅
𝑚 (𝑏 ,𝑠 )
𝜕 𝑦 3
]=0
By using the state vector 𝜼̂
𝑚 (𝑦 ,𝑠 ) , boundary conditions in Eq. (5.18) can be cast into the 𝑠 -domain
spatial state matrix form, Eq. (5.15).
For instance, boundary condition matrices 𝐌 𝑚 (𝑠 ) and 𝐍 𝑚 (𝑠 ) and loading vector 𝜸̂
𝑚 (𝑠 ) for
free boundary conditions on edges 𝑦 =0 and 𝑦 =𝑏 are given by
𝐌 𝑚 (𝑠 )=
[

𝐷𝜈 (
𝑚𝜋
𝑎 )
2
0 −𝐷 0
0 −𝐷 (𝜈 −2)(
𝑚𝜋
𝑎 )
2
0 −𝐷 0 0 0 0
0 0 0 0
]

(5.19𝑎 )
𝐍 𝑚 (𝑠 )=
[

0 0 0 0
0 0 0 0
𝐷𝜈 (
𝑚𝜋
𝑎 )
2
0 −𝐷 0
0 −𝐷 (𝜈 −2)(
𝑚𝜋
𝑎 )
2
0 −𝐷 ]

(5.19𝑏 )
𝜸̂
𝑚 (𝑠 )=𝟎 (5.19𝑐 )
For a single thin Lévy plate resting on a viscoelastic foundation, when its transverse vibration
is governed by Eq. (5.1), state equation (5.10) and boundary condition (5.15) formulate the 𝑚 th

vibration mode in the 𝑠 -domain. By solving the ordinary differential equation, the unknown
vibration mode in the 𝑦 direction, 𝑌̅
𝑚 (𝑦 ,𝑠 ), is obtained. Meanwhile, with state variables in
82

𝜼̂
𝑚 (𝑦 ,𝑠 ) , local information about slope, moment and shear force can be computed by Eq. (5.13)
directly without further post-processing work.

5.2.2 Sandwich Plates with Viscoelastic Layers
Consider a sandwich plate with a viscoelastic layer shown in Figure 5.2, in which two parallel
Lévy plates of the same dimensions are interconnected by a uniformly distributed viscoelastic
layer. Two plate elements are both modeled by the Kirchhoff-Love thin plate theory and they both
have Lévy type boundary conditions with two edges 𝑥 =0 and 𝑥 =𝑎 being simply supported and
the other two edges 𝑦 =0 and 𝑦 =𝑏 being arbitrary boundary conditions. Two plates can have
different material properties, thicknesses and boundary conditions at edges 𝑦 =0 and 𝑦 =𝑏 .
Without loss of generality, assume the viscoelastic layer between two plates is described by the
Kelvin-Voigt Model. When the in-plane deformation is ignored, the transverse vibration of the
sandwich plate structure is governed by two linear partial differential equations
𝐷 1
[
𝜕 4
𝑤 1
(𝑥 ,𝑦 ,𝑡 )
𝜕 𝑥 4
+
𝜕 4
𝑤 1
(𝑥 ,𝑦 ,𝑡 )
𝜕 𝑥 2
𝜕 𝑦 2
+
𝜕 4
𝑤 1
(𝑥 ,𝑦 ,𝑡 )
𝜕 𝑦 4
]+𝑘 𝑙 [𝑤 1
(𝑥 ,𝑦 ,𝑡 )−𝑤 2
(𝑥 ,𝑦 ,𝑡 )]
+𝑐 𝑙 [𝑤 ̇ 1
(𝑥 ,𝑦 ,𝑡 )−𝑤 ̇ 2
(𝑥 ,𝑦 ,𝑡 )]=𝑞 1
(𝑥 ,𝑦 ,𝑡 )−2𝜌 1
ℎ
1
𝜕 2
𝑤 1
(𝑥 ,𝑦 ,𝑡 )
𝜕 𝑡 2
(5.20𝑎 )
𝐷 2
[
𝜕 4
𝑤 2
(𝑥 ,𝑦 ,𝑡 )
𝜕 𝑥 4
+
𝜕 4
𝑤 2
(𝑥 ,𝑦 ,𝑡 )
𝜕 𝑥 2
𝜕 𝑦 2
+
𝜕 4
𝑤 2
(𝑥 ,𝑦 ,𝑡 )
𝜕 𝑦 4
]+𝑘 𝑙 [𝑤 2
(𝑥 ,𝑦 ,𝑡 )−𝑤 1
(𝑥 ,𝑦 ,𝑡 )]
+𝑐 𝑙 [𝑤 ̇ 2
(𝑥 ,𝑦 ,𝑡 )−𝑤 ̇ 1
(𝑥 ,𝑦 ,𝑡 )]=𝑞 2
(𝑥 ,𝑦 ,𝑡 )−2𝜌 2
ℎ
2
𝜕 2
𝑤 2
(𝑥 ,𝑦 ,𝑡 )
𝜕 𝑡 2
(5.20𝑏 )
for 0<𝑥 <𝑎 , 0<𝑦 <𝑏 , where 𝑥 and 𝑦 are spatial coordinates in two directions; 𝑡 is the
temporal parameter; 𝑎 is the length of the plate structure in the 𝑥 direction; 𝑏 is the length of the
plate structure in the 𝑦 direction; 𝑤 1
(𝑥 ,𝑦 ,𝑡 ) and 𝑤 2
(𝑥 ,𝑦 ,𝑡 ) are transverse displacements of two
83

plate elements, respectively. For the plate element 𝑖 , 𝑖 =1,2, 2ℎ
𝑖 , 𝜌 𝑖 , 𝐸 𝑖 , 𝜈 𝑖 , 𝐷 𝑖 and 𝑞 𝑖 (𝑥 ,𝑦 ,𝑡 ) are
the same as those defined in Eq. (5.1). Meanwhile, 𝑘 𝑙 describes the elasticity of the viscoelastic
layer and 𝑐 𝑙 describes the viscosity of the layer.
The transverse displacement 𝑤 𝑖 (𝑥 ,𝑦 ,𝑡 ) of Lévy plates can be expressed in terms of single
Fourier series
𝑤 𝑖 (𝑥 ,𝑦 ,𝑡 )= ∑sin(
𝑚𝜋𝑥 𝑎 )𝑌 𝑖 ,𝑚 (𝑦 ,𝑡 )
∞
𝑚 =1
(5.21)
Also, the loading functions 𝑞 𝑖 (𝑥 ,𝑦 ,𝑡 ) is also written in terms of the series by
𝑞 𝑖 (𝑥 ,𝑦 ,𝑡 )= ∑sin(
𝑚𝜋𝑥 𝑎 )𝑞 𝑖 ,𝑚 (𝑦 ,𝑡 )
∞
𝑚 =1
(5.22𝑎 )
with
𝑞 𝑖 ,𝑚 (𝑦 ,𝑡 )=
𝑎 2
∫ 𝑞 𝑖 (𝑥 ,𝑦 ,𝑡 )sin(
𝑚𝜋𝑥 𝑎 )𝑑𝑥 𝑎 0
(5.22𝑏 )
Again, the vibration problem described by Eq. (5.20) is solved by the DTFM. To this end,
take Laplace transform of Eqs. (5.20-5.22) with zero initial conditions to obtain
𝐷 1
[
𝜕 4
𝑤̅
1
(𝑥 ,𝑦 ,𝑠 )
𝜕 𝑥 4
+
𝜕 4
𝑤̅
1
(𝑥 ,𝑦 ,𝑠 )
𝜕 𝑥 2
𝜕 𝑦 2
+
𝜕 4
𝑤̅
1
(𝑥 ,𝑦 ,𝑠 )
𝜕 𝑦 4
]+𝑘 𝑙 [𝑤̅
1
(𝑥 ,𝑦 ,𝑠 )−𝑤̅
2
(𝑥 ,𝑦 ,𝑠 )]
+𝑐 𝑙 𝑠 [𝑤̅
1
(𝑥 ,𝑦 ,𝑠 )−𝑤̅
2
(𝑥 ,𝑦 ,𝑠 )]=𝑞̅
1
(𝑥 ,𝑦 ,𝑠 )−2𝜌 1
ℎ
1
𝑠 2
𝑤̅
1
(𝑥 ,𝑦 ,𝑠 )
(5.23𝑎 )
𝐷 2
[
𝜕 4
𝑤̅
2
(𝑥 ,𝑦 ,𝑠 )
𝜕 𝑥 4
+
𝜕 4
𝑤̅
2
(𝑥 ,𝑦 ,𝑠 )
𝜕 𝑥 2
𝜕 𝑦 2
+
𝜕 4
𝑤̅
2
(𝑥 ,𝑦 ,𝑠 )
𝜕 𝑦 4
]+𝑘 𝑙 [𝑤̅
2
(𝑥 ,𝑦 ,𝑠 )−𝑤̅
1
(𝑥 ,𝑦 ,𝑠 )]
+𝑐 𝑙 [𝑤̅
2
(𝑥 ,𝑦 ,𝑠 )−𝑤̅
1
(𝑥 ,𝑦 ,𝑠 )]=𝑞̅
2
(𝑥 ,𝑦 ,𝑡 )−2𝜌 2
ℎ
2
𝑠 2
𝑤̅
2
(𝑥 ,𝑦 ,𝑠 )
(5.23𝑏 )
𝑤̅
𝑖 (𝑥 ,𝑦 ,𝑠 )= ∑𝑌̅
𝑖 ,𝑚 (𝑦 ,𝑠 )sin(
𝑚𝜋𝑥 𝑎 )
∞
𝑚 =1
(5.24)
84

𝑞̅
𝑖 (𝑥 ,𝑦 ,𝑠 )= ∑𝑞̅
𝑖 ,𝑚 (𝑦 ,𝑠 )sin(
𝑚𝜋𝑥 𝑎 )
∞
𝑚 =1
(5.25𝑎 )
𝑞̅
𝑖 ,𝑚 (𝑦 ,𝑠 )=
2
𝑎 ∫ 𝑞̅
𝑖 (𝑥 ,𝑦 ,𝑠 )sin(
𝑚𝜋𝑥 𝑎 )𝑑𝑥 𝑎 0
(5.25𝑏 )
where the overbar stands for the 𝑠 -domain variables.
Similar to the single plate discussed in Chapter 5.2.1, plugging Eqs. (5.24) and (5.25) into Eq.
(5.23) yields an infinite set of independent equations. For the 𝑚 th
vibration mode, it is governed
by the 𝑠 -domain equations
𝐷 1
[(
𝑚𝜋
𝑎 )
4
𝑌̅
1,𝑚 (𝑦 ,𝑠 )−2(
𝑚𝜋
𝑎 )
2
𝜕 2
𝑌̅
1,𝑚 (𝑦 ,𝑠 )
𝜕 𝑦 2
+
𝜕 4
𝑌̅
1,𝑚 (𝑦 ,𝑠 )
𝜕 𝑦 4
]
+𝑘 𝑓 [𝑌̅
1,𝑚 (𝑦 ,𝑠 )−𝑌̅
2,𝑚 (𝑦 ,𝑠 )]+𝑐 𝑓 𝑠 [𝑌̅
1,𝑚 (𝑦 ,𝑠 )−𝑌̅
2,𝑚 (𝑦 ,𝑠 )]
=𝑞̅
1,𝑚 (𝑦 ,𝑠 )−2𝜌 1
ℎ
1
𝑠 2
𝑌̅
1,𝑚 (𝑦 ,𝑠 )
(5.26𝑎 )
𝐷 2
[(
𝑚𝜋
𝑎 )
4
𝑌̅
2,𝑚 (𝑦 ,𝑠 )−2(
𝑚𝜋
𝑎 )
2
𝜕 2
𝑌̅
2,𝑚 (𝑦 ,𝑠 )
𝜕 𝑦 2
+
𝜕 4
𝑌̅
2,𝑚 (𝑦 ,𝑠 )
𝜕 𝑦 4
]
+𝑘 𝑓 [𝑌̅
2,𝑚 (𝑦 ,𝑠 )−𝑌̅
1,𝑚 (𝑦 ,𝑠 )]+𝑐 𝑓 𝑠 [𝑌̅
2,𝑚 (𝑦 ,𝑠 )−𝑌̅
1,𝑚 (𝑦 ,𝑠 )]
=𝑞̅
2,𝑚 (𝑦 ,𝑠 )−2𝜌 2
ℎ
2
𝑠 2
𝑌̅
2,𝑚 (𝑦 ,𝑠 )
(5.26𝑏 )
Similar to Eq. (5.9), state vectors 𝜼̂
𝑖 ,𝑚 (𝑦 ,𝑠 ) for each plate element can be defined by
𝜼̂
𝑖 ,𝑚 (𝑦 ,𝑠 )={𝑌̅
𝑖 ,𝑚 (𝑦 ,𝑠 )
𝜕 𝑌̅
𝑖 ,𝑚 (𝑦 ,𝑠 )
𝜕𝑦
𝜕 2
𝑌̅
𝑖 ,𝑚 (𝑦 ,𝑠 )
𝜕 𝑦 2
𝜕 3
𝑌̅
𝑖 ,𝑚 (𝑦 ,𝑠 )
𝜕 𝑦 3
}
T
(5.27)
And the global state vector 𝜼̂
𝑔 ,𝑚 (𝑦 ,𝑠 ) of the sandwich plate structure with the viscoelastic layer
can be written as
𝜼̂
𝑔 ,𝑚 (𝑦 ,𝑠 )={
𝜼̂
T
1,𝑚 (𝑦 ,𝑠 ) 𝜼̂
T
2,𝑚 (𝑦 ,𝑠 )
}
T
(5.28)
85

In the DTFM, the governing equations of the sandwich plate is cast into the following state-
space form
𝜕 𝜕𝑦
𝜼̂
𝑔 ,𝑚 (𝑦 ,𝑠 )=𝐅̂
𝑔 ,𝑚 (𝑦 ,𝑠 )+𝒑̂
𝑔 ,𝑚 (𝑦 ,𝑠 ) (5.29)
for 𝑚 =1,2,3,…, 0<𝑦 <𝑏 , where 𝐅̂
𝑔 ,𝑚 (𝑦 ,𝑠 ) is an 8-by-8 global state matrix and 𝒑̂
𝑔 ,𝑚 (𝑦 ,𝑠 ) is
an 8-by-1 global loading vector, and they are given by
𝐅̂
𝑔 ,𝑚 (𝑦 ,𝑠 )=
[

0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
𝐅̂
𝑔 ,𝑚 ,4,1
0 𝐅̂
𝑔 ,𝑚 ,4,3
0 𝐅̂
𝑔 ,𝑚 ,4,5
0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
𝐅̂
𝑔 ,𝑚 ,8,1
0 0 0 𝐅̂
𝑔 ,𝑚 ,8,5
0 𝐅̂
𝑔 ,𝑚 ,8,7
0
]

(5.30𝑎 )
with
𝐅̂
𝑔 ,𝑚 ,4,1
=
−2𝜌 1
ℎ
1
𝑠 2
−𝐺 𝑙 (𝑠 )
𝐷 1
−(
𝑚𝜋
𝑎 )
4
(5.30𝑏 )
𝐅̂
𝑔 ,𝑚 ,8,5
=
−2𝜌 2
ℎ
2
𝑠 2
−𝐺 𝑙 (𝑠 )
𝐷 2
−(
𝑚𝜋
𝑎 )
4
(5.30𝑐 )
𝐅̂
𝑔 ,𝑚 ,4,3
=𝐅̂
𝑔 ,𝑚 ,8,7
=2(
𝑚𝜋
𝑎 )
4
(5.30𝑑 )
𝐅̂
𝑔 ,𝑚 ,4,5
=
𝐺 𝑙 (𝑠 )
𝐷 1
(5.30𝑒 )
𝐅̂
𝑔 ,𝑚 ,8,1
=
𝐺 𝑙 (𝑠 )
𝐷 2
(5.30𝑓 )
and
86

𝒑̂
𝑔 ,𝑚 (𝑦 ,𝑠 )={𝒑̂
1,𝑚 (𝑦 ,𝑠 ) 𝒑̂
2,𝑚 (𝑦 ,𝑠 )}
T
(5.31𝑎 )
with
𝒑̂
𝑖 ,𝑚 (𝑦 ,𝑠 )={0 0 0
𝑞̅
𝑖 ,𝑚 (𝑦 ,𝑠 )
𝐷 𝑖 }
T
(5.31𝑏 )
where
𝐺 𝑙 (𝑠 )=𝑘 𝑙 +𝑐 𝑙 𝑠 (5.32)
𝐺 𝑙 (𝑠 ) describes the viscoelasticity of the sandwich core. When the viscoelastic layer is modeled
by the Kelvin-Voigt model, it can be expressed by Eq. (5.32). For different viscoelastic models, a
general way of expressing 𝐺 𝑙 (𝑠 ) will be presented in Chapter 5.3.
By using the 8-by-1 global state vector 𝜼̂
𝑔 ,𝑚 (𝑦 ,𝑠 ) of the sandwich plate, the local information
about transverse displacement 𝑤̅
𝑖 (𝑥 ,𝑦 ,𝑠 ) , slopes 𝜃 ̅
𝑖 ,𝑥 (𝑥 ,𝑦 ,𝑠 ) and 𝜃 ̅
𝑖 ,𝑦 (𝑥 ,𝑦 ,𝑠 ) , bending moments
𝑀̅
𝑖 ,𝑥𝑥
(𝑥 ,𝑦 ,𝑠 ) and 𝑀̅
𝑖 ,𝑦𝑦
(𝑥 ,𝑦 ,𝑠 ), twisting moment 𝑀̅
𝑖 ,𝑥𝑦
(𝑥 ,𝑦 ,𝑠 ), and shear forces 𝑄̅
𝑖 ,𝑥𝑥
(𝑥 ,𝑦 ,𝑠 )
and 𝑄̅
𝑖 ,𝑦𝑦
(𝑥 ,𝑦 ,𝑠 ) at any point of two plate elements can be obtained by Eq. (5.13).
Since two plate elements both have Lévy boundary conditions, four edges at 𝑦 =0 and 𝑦 =
𝑏 can be arbitrary. They all can be expanded and written in the state formulation by using the
global state vector as
𝐌 𝑔 ,𝑚 (𝑠 )𝜼̂
𝑔 ,𝑚 (0,𝑠 )+𝐍 𝑔 ,𝑚 (𝑠 )𝜼̂
𝑔 ,𝑚 (𝑏 ,𝑠 )=𝜸̂
𝑔 ,𝑚 (𝑠 ) (5.33)

where 𝜼̂
𝑔 ,𝑚 (0,𝑠 ) and 𝜼̂
𝑔 ,𝑚 (𝑏 ,𝑠 ) are values of the global state vector on edges at 𝑦 =0 and 𝑦 =𝑏 ,
respectively; 𝐌 𝑔 ,𝑚 (𝑠 ) and 𝐍 𝑔 ,𝑚 (𝑠 ) are two 8-by-8 boundary matrices consisting of coefficients of
87

the state variables for two plate element; and 𝜸̂
𝑔 ,𝑚 (𝑠 ) is an 8-by-1 vector containing Laplace
transforms of boundary disturbances.
Therefore, for a sandwich plate with a viscoelastic layer shown in Figure 5.2 governed by Eq.
(5.20), its transverse vibration can be formulated by the state equation (5.29) and boundary
conditions (5.33). By solving Eqs. (5.29) and (5.33), the dynamic response of transverse
displacement, slope, moment and shear force at any location of the structure can be obtained.
The DTFM formulation can be applied to the sandwich plate structures with multiple plate
components that are partially connected with general viscoelastic layers. The vibration of sandwich
Lévy plates with viscoelastic layers and sandwich Lévy plates with partial viscoelastic layers is
formulated by the DTFM in the same systematic way. The format of the governing equations and
boundary conditions is consistent. Equations (5.29) and (5.33) can be solved by an analytical
solution method from low- to high-frequency regions accurately and efficiency that will be
introduced in Chapter 5.4. Local information in plate structures at any point about displacement,
slope, moment and shear force can be obtained easily by using the state variables. In the DTFM
formulation, the viscoelasticity of the sandwich core is represented by the transfer function 𝐺 (𝑠 )
in a neat way. More details about viscoelastic models of interconnection layers in sandwich plates
will be discussed in Chapter 5.3.

5.3 Models of Viscoelastic Layers
The sandwich plate structures introduced in Chapter 5.2 can have different types of
viscoelastic cores. In the DTFM formulation, the viscoelasticity of the sandwich core is
represented by an 𝑠 -domain transfer function 𝐺 (𝑠 ) . In this section, several prevalent viscoelastic
88

models are reviewed, such as Kelvin-Voigt model, Maxwell model and standard linear solid
models (Christensen, 2012; Zener, 1948). The viscoelasticity of different models is transformed to
the 𝑠 -domain transfer function 𝐺 (𝑠 ) , which is prepared for vibration analysis of plates.
Shown in Figure 5.3 is the mechanical model corresponding to the Kelvin-Voigt model of the
viscoelastic material. The Kelvin-Voigt model can be used to describe the creep deformation of
solid material. In the mechanical model, the spring and the dashpot are in the parallel connection.
The transfer function of the Kelvin-Voigt model 𝐺 𝑣 (𝑠 ) is given by
𝐺 𝑣 (𝑠 )=(𝑘 𝑣 +𝑐 𝑣 𝑠 ) (5.34)
Shown in Figure 5.4 is the mechanical model that corresponds to the Maxwell model of the
viscoelastic material. The Maxwell model can be used to describe relaxation in the soft solid
material. In the mechanical model, the spring and the dashpot are in the serial connection. the
transfer function of the Maxwell model can be written as
𝐺 𝑚 (𝑠 )=
𝑘 𝑣 ∗𝑐 𝑣 𝑠 𝑘 𝑣 +𝑐 𝑣 𝑠 (5.35)
Both the Kelvin-Voigt model and the Maxwell model cannot describe the creep and relaxation
at the same time. To predict the material property more precisely, standard linear solid models are
introduced (Zener, 1948). Shown in Figure 5.5 are mechanical models of two examples of standard
linear solid models, one is the Kelvin-Voigt representation and the other one is the Maxwell
representation. Similarly, transfer functions describing these two models can be expressed as
𝐺 𝑆𝐿𝑆 ,𝐾𝑉
(𝑠 )=
𝑘 2,𝑣 (𝑘 1,𝑣 +𝑐 𝑣 𝑠 )
𝑘 1,𝑣 +𝑘 2,𝑣 +𝑐 𝑣 𝑠 (5.36)
89

𝐺 𝑆𝐿𝑆 ,𝑀 (𝑠 )=
𝑘 1,𝑚 𝑘 2,𝑚 +𝑘 1,𝑚 𝑐 𝑚 𝑠 +𝑘 2,𝑚 𝑐 𝑚 𝑠 𝑘 2,𝑚 +𝑐 𝑚 𝑠 (5.37)
where 𝐺 𝑆𝐿𝑆 ,𝐾𝑉
(𝑠 ) and 𝐺 𝑆𝐿𝑆 ,𝑀 (𝑠 ) are transfer functions of the Kelvin-Voigt representation and
Maxwell representation of standard linear solid models, respectively; 𝑘 1,𝑣 , 𝑘 2,𝑣 , 𝑐 𝑣 , 𝑘 1,𝑚 , 𝑘 2,𝑚 , 𝑐 𝑚
are material properties.
For more general viscoelastic models, dynamic modulus of the material in the time domain
can be transformed in the 𝑠 -domain and their transfer functions can be obtained (Christensen,
2012). The 𝑠 -domain transfer functions that describe the viscoelasticity of sandwich cores in plate
structures can be inserted into the DTFM formulation directly. By using them, the vibration of
sandwich plates with viscoelastic layers can be modeled without difficulty.

5.4 Vibration Analysis by the Distributed Transfer Function Method
With the 𝑠 -domain state space equations formulated in Chapter 5.2, it has been shown that
the vibration problem of Lévy plate with viscoelastic layers is to solve the governing equation
(5.29) subject to boundary conditions and compatibility conditions (5.33). The solution method of
the vibration analysis by the DTFM is introduced in this Section.
5.4.1 Eigenvalue Problem
The exact eigensolutions of sandwich plates can be determined by the DTFM. The eigenvalue
problem of the plate structure formulated by Eqs. (5.29) and (5.33) is defined as
𝜕 𝜕𝑦
𝜓̂
𝑔 ,𝑚 (𝑦 ,𝑠 )=F
̂
𝑔 ,𝑚 (𝑠 )𝜓̂
𝑔 ,𝑚 (𝑦 ,𝑠 ) (5.38)
90

for 𝑚 =1,2,3,…, 0<𝑦 <𝑏 , subject to the homogeneous boundary condition
M
𝑔 ,𝑚 (𝑠 )𝜂 ̂
𝑔 ,𝑚 (0,𝑠 )+N
𝑔 ,𝑚 (𝑠 )𝜂 ̂
𝑔 ,𝑚 (𝑏 ,𝑠 )=0 (5.39)

where 𝑠 is the eigenvalue of the ordinary differential equation; 𝜓̂
𝑚 (𝑦 ,𝑠 ) is the eigenfunction in
the vector form that contains all state variables in global state vector 𝜂 ̂
𝑔 ,𝑚 (𝑦 ,𝑠 ) .
The eigenvalues are roots of the transcendental characteristic equation
det[M
𝑔 ,𝑚 (𝑠 )+N
𝑔 ,𝑚 (𝑠 )Φ
̂
𝑔 ,𝑚 (𝑏 ,0,𝑠 )]=0 (5.40)
where Φ
̂
𝑔 ,𝑚 (𝑥 ,𝜉 ,𝑠 ) is the transition matrix of the global state equation.
The characteristic equation (5.40) can be solved by using the bisection method to find the
natural frequencies of the undamped plates in any frequency region, from low to high. With
accurate eigensolutions, a generalized modal expansion can be implemented to predict the
frequency response and transient response of the structure.

5.4.2 Frequency Response
The 𝑠 -domain state space equations (5.29) and (5.33) have the same format as the traditional
DTFM (Yang, 1992). Hence, the 𝑠 -domain dynamic response corresponding to the 𝑚 th
mode of
the plate is given by
𝜂 ̂
𝑔 ,𝑚 (𝑦 ,𝑠 )=∫ G
̂
𝑔 ,𝑚 (𝑦 ,𝜉 ,𝑠 )𝑝 ̂
𝑔 ,𝑚 (𝜉 ,𝑠 )𝑑𝜉 +H
̂
𝑔 ,𝑚 (𝑦 ,𝑠 )𝛾̂
𝑔 ,𝑚 (𝑠 )
𝑏 0
(5.41)
for 0<𝑦 <𝑏 , where G
̂
𝑔 ,𝑚 (𝑦 ,𝜉 ,𝑠 ) and H
̂
𝑔 ,𝑚 (𝑦 ,𝑠 ) are the distributed transfer functions of the 𝑚 th

mode and they are given in the exact and closed form by
91

G
̂
𝑔 ,𝑚 (𝑦 ,𝜉 ,𝑠 )={
H
̂
𝑔 ,𝑚 (𝑦 ,𝑠 )M
𝑔 ,𝑚 (𝑠 )Φ
̂
𝑔 ,𝑚 (0,𝜉 ,𝑠 ),𝜉 ≤𝑦 −H
̂
𝑔 ,𝑚 (𝑦 ,𝑠 )N
𝑔 ,𝑚 (𝑠 )Φ
̂
𝑔 ,𝑚 (𝑏 ,𝜉 ,𝑠 ),𝜉 >𝑦 H
̂
𝑔 ,𝑚 (𝑦 ,𝑠 )=Φ
̂
𝑔 ,𝑚 (𝑦 ,0,𝑠 )[M
𝑔 ,𝑚 (𝑠 )+N
𝑔 ,𝑚 (𝑠 )Φ
̂
𝑔 ,𝑚 (𝑏 ,0,𝑠 )]
−1
(5.42)
The steady-state response (frequency response) of a plate structure subject to a harmonic force
of excitation frequency Ω can be obtained from Eq. (5.41) with 𝑠 =𝑗 Ω, where 𝑗 =√−1. Once the
global state vector 𝜂 ̂
𝑔 ,𝑚 (𝑦 ,𝑠 ) is solved, the values of 𝑌̅
𝑖 ,𝑚 (𝑦 ,𝑠 ) and its derivatives can be obtained.
The frequency response of transverse displacement, slope, moment and shear force at any point of
the plate can be computed by Eqs. (5.5) and (5.13).
Though the response is represented by the series solution, a truncated series is used in practice.
As can be seen from the DTFM formulation presented in Chapter 5.2, no spatial discretization has
been made in the solution procedure. Thus, the only approximation made in this solution is the
truncated series. When the excitation frequency Ω becomes higher, the number terms used in the
truncated series also increases. However, it will be shown that to get accurate results for mid- and
high-frequency vibration analysis, the number of terms required by the DTFM is much less than
the number of terms used by other analytical methods and the number of elements required by the
FEA in numerical examples.

5.5 Numerical Examples
The DTFM-based solution method introduced in Chapter 5.2 and 5.4, is illustrated on three
numerical examples: a double-plate with fully connected viscoelastic layer shown in Figure 5.2
and a three-layer sandwich plate with partially connected viscoelastic layers shown in Figure 5.6.
These three examples will show that the proposed method can provide accurate and efficient
92

vibration analysis for various sandwich plates with viscoelastic layers in different frequency
regions, from low to high. For validation purpose, the numerical results by the DTFM are
compared with results by the existing analytical solution and the finite element method which
utilizes the plate element proposed in the literature (Melosh, 1963).

Example 1. A Double-Plate with Fully Connected Viscoelastic Layer
This example is a benchmark for validation of the accuracy, efficiency and consistency of the
DTFM in vibration analysis of the sandwich plate from low to high frequencies. Shown in Figure
5.2 is a sandwich plate with fully connected viscoelastic layer described by the Kelvin-Voigt model.
As stated before, for each plate element in the sandwich structure, two edges at 𝑥 =0 and 𝑥 =𝑎
are simply supported and the other two edges at 𝑦 =0 and 𝑦 =𝑏 can be arbitrary. For numerical
simulation, the parameters of the double-plate structure are listed as follows.
𝑎 =1 m, 𝑏 =1.5 m, 𝑡 =0.025 m, 𝐸 =200 GPa
𝜈 =0.3, 𝜌 =7800 kg/m
3
, 𝑘 𝑣 =2.5×10
6
N/m, 𝑐 𝑣 =2.5×10
5
Ns/m
Consider a point-wise sinusoidal force acting at point 𝑥 =0.5 m, 𝑦 =0.9 m on the upper
plate, 𝐹 𝑓 (𝑡 )=𝐹 0
sin(2𝜋𝑓𝑡 ) , where 𝑓 is the excitation frequency in Hertz. Also, the unity force
amplitude is used, 𝐹 0
=1 N. In the subsequent simulations, sandwich plates are assumed to be
initially at rest.
First, for validation purposes, natural frequencies of the plate structure are computed by the
DTFM, and the FEM with 50, 200 and 3,200 elements. To make the comparison intuitive, the
sandwich core is set to be elastic such that all natural frequencies are real numbers. Listed in Table
93

5.1 are the first five and some higher-mode natural frequencies of the plate structure with all edges
being simply supported (SSSS boundary condition). Similarly, natural frequencies of the plate
structures with edges at 𝑥 =0 and 𝑥 =𝑎 being simply supported and edges at 𝑦 =0 and 𝑦 =𝑏
being clamped (SSCC boundary condition). It can be seen that for different types of boundary
conditions, FEM results are always converging to the DTFM results, which are the exact natural
frequencies. For the first five natural frequencies, 50-element FEM can provide good results while
for higher-mode natural frequencies, more elements are necessary for the FEM, such as 200 and
3,200 elements.
Second, compute the frequency response of the plate structure with viscoelastic layer.
Consider the excitation frequency as 𝑓 =200 Hz, which is between the fourth and the fifth natural
frequency of the undamped structure. The boundary conditions of the plate structure are set to be
edges at 𝑥 =0 and 𝑥 =𝑎 being simply supported and edges at 𝑦 =0 and 𝑦 =𝑏 being free (SSFF
boundary condition). The frequency response is computed by the DTFM with Eqs. (5.5) and (5.59),
and the FEM with 200 elements and 3,200 elements for comparison. The spatial distribution of the
magnitude of transverse displacement along two lines on two plate elements are plotted in Figure
5.7. As shown in the figure, the transverse displacement obtained by the FEM with 3,200 elements
is in good agreement with the DTFM with 50 modes. Meanwhile, the transverse displacement
obtained by the 200-element FEM cannot match with 50-mode DTFM and 3,200-element FEM
well at the location in the lower plate, though 200-element FEM can provide relatively accurate
natural frequencies in the low frequency region. The comparison validates the DTFM result and
indicates that the FEM requires many elements to provide accurate results at certain location of
the plate. When 3,200-element FEM is used, the dimension of the stiffness and mass matrices is
9,922×9,922. When DTFM is used, the dimension of the global state matrix is 8×8, and the
94

50-mode DTFM requires 50 terms in the series expansion. It can be seen that much more
computation effort is need for the FEM.
Since the FEM is not suitable for mid- and high-frequency analysis. To validate the accuracy
of the DTFM in the high-frequency region, the analytical solution (Navier solution) for plates with
SSSS boundary condition is used (Reddy, 2006; Ugural, 2009). The frequency response of the
sandwich plate with SSSS boundary condition is computed in the high-frequency region, from
9,500 Hz to 10,500 Hz, which includes the 347
th
to 380
th
natural frequencies of the undamped
structure. In Figure 5.8, transverse displacement and shear force 𝑄 𝑦 responses at the excitation
point are plotted. It shows that frequency responses by the 500-mode DTFM and 90,000-term
analytical solution (300×300) can match with each other very well. Increase the excitation
frequency and let it be 𝑓 =5×10
6
Hz. Plotted in Figure 5.9 are the spatial distribution curves for
the bending moment 𝑀 𝑦𝑦
and shear force 𝑄 𝑦 . Again, the result obtained by the DTFM with 500
modes and the analytical solution with 250,000 (500×500) terms are in good agreement. Hence,
the DTFM also is capable of providing accurate vibration responses in the high-frequency region.
In this benchmark example, the accuracy and efficiency of the DTFM has been illustrated by
comparison with the FEM and the analytical solution, from low-frequency region to mid- and high-
frequency regions. Compared with the existing numerical methods, the DTFM has one advantage
that is providing local displacement, moment and shear force. Compared with the existing
analytical solution (Navier solution), the DTFM can deal with different types of boundary
conditions easily. The flexibility of the DTFM will be seen in the following two numerical
examples which are sandwich plates with partially connected viscoelastic layers.

95

Example 3. A Three-Layer Sandwich Plate with Partially Connected Viscoelastic Layers
Now, consider a three-layer sandwich plate with partially connected viscoelastic layers, as
shown in Figure 5.6. In the sandwich plate structure, the top plate and the middle plate are partially
connected in the region 𝐵𝐶
̅̅̅̅
−𝐵 ′
𝐶 ′
̅̅̅̅̅̅
; the middle plate and the bottom plate are partially connected
in the region 𝐶 ′
𝐹 ′
̅̅̅̅̅̅
−𝐶 ′′
𝐹 ′′
̅̅̅̅̅̅̅
. For numerical simulation, some parameters of the double-plate
structure are listed as follows.
𝜈 =0.3, 𝜌 =7800 kg/m
3
, 𝑘 𝑣 =2.5×10
6
N/m, 𝑐 𝑣 =2.5×10
5
Ns/m, 𝑎 =1.1 m
And dimensions in the 𝑦 direction are shown in Figure 5.6. A point-wise sinusoidal force is acting
at point 𝐹 on the upper plate, 𝐹 𝑓 (𝑡 )=𝐹 0
sin(2𝜋𝑓𝑡 ) , where 𝑓 is the excitation frequency in Hertz,
and 𝐹 0
=1 N.
Natural frequencies of the undamped plate structure with SSFF boundary condition are
computed by the DTFM and the FEM with 300, 1,200 and 4,800 elements and results are presented
in Table 5.2. Good agreement is seen between DTFM and 4,800-element FEM. Again, the
accuracy of the DTFM is validated by the FEM.
As introduced in Chapter 5.3, complex viscoelastic models can be described by a simple way
in the DTFM. In this case, the standard linear solid models by Kelvin representation and Maxwell
representation are considered. First, let the viscoelastic layers be the standard linear solid model
by Kelvin representation and they have the parameters:𝑘 1,𝑣 =𝑘 2,𝑣 =3×10
6
N/m , 𝑐 𝑣 =
3×10
5
Ns/m . Set the excitation frequency as 𝑓 =2×10
5
Hz. In Figure 5.10, the spatial
distribution of displacement, moment 𝑀 𝑦𝑦
and shear force 𝑄 𝑦 at point 𝐶 on the middle plate are
plotted. The frequency responses are obtained by the 200-mode DTFM and 500-mode DTFM and
two DTFM solutions match well. Next, change the viscoelastic layers to the standard linear solid
96

model by Maxwell representation that has the parameters: 𝑘 1,𝑚 =𝑘 2,𝑚 =1.5×10
6
N/m, 𝑐 𝑣 =
3×10
5
Ns/m. Frequency responses of displacement, moment and shear force at the same point
are shown in Figure 5.11. Again, 200-mode DTFM results and 500-DTFM results are in good
agreement.
From this example, it has been shown that the DTFM can deal with the high-frequency
vibration analysis of sandwich structures with different types of viscoelastic layers without
difficulty.

97

Figure 5.1 A rectangular plate modeled by the Kirchhoff-Love plate
theory resting on a viscoelastic foundation

98

Figure 5.2 A two-layer sandwich plate with viscoelastic layer, side
view
99

Figure 5.3 Mechanical schematic of Kelvin-Voigt viscoelastic model

100

Figure 5.4 Mechanical schematic of Maxwell viscoelastic model

101

(a)

(b)
Figure 5.5 (a) Mechanical schematic of Kelvin representation of
standard linear solid model; (b) Mechanical schematic of Maxwell
representation of standard linear solid model

102

Figure 5.6 A three-layer sandwich plate with partially connected
viscoelastic layers, side view

103

Figure 5.7 The spatial distribution of the magnitude of transverse
displacement along the line 𝑦 =0.9 m on the upper plate and the line
𝑦 =0.6 m on the lower plate of the elastically connected two-layer
sandwich plate (SSFF boundary condition), with excitation at point
𝑥 =0.5 m, 𝑦 =0.9 m on the upper plate and the excitation frequency
𝑓 =200 Hz: solid line – DTFM with 50 modes; dashed line – FEM
with 200 elements; dotted line – FEM with 3,200 element

104

Figure 5.8 The magnitude of transverse displacement and shear force
𝑄 𝑦 at point 𝑥 =0.5 m, 𝑦 =0.9 m on the upper plate (SSSS boundary
condition, Kelvin-Voigt model), with excitation at point 𝑥 =0.5 m,
𝑦 =0.9 m on the upper plate and the excitation frequency from 𝑓 =
9,500 Hz to 𝑓 =10,500 Hz: solid line – DTFM with 500 modes;
dashed line – Analytical solution with 90,000 terms

105

Figure 5.9 The spatial distribution of the magnitude of bending
moment 𝑀 𝑦𝑦
and shear force 𝑄 𝑦 at point 𝑥 =0.5 m, 𝑦 =0.6 m on
the lower plate (SSSS boundary condition, Kelvin-Voigt model), with
excitation at point 𝑥 =0.5 m, 𝑦 =0.9 m on the upper plate and the
excitation frequency 𝑓 =5×10
6
Hz: solid line – DTFM with 500
modes; dotted line – Analytical solution with 250,000 terms

106

Figure 5.10 The spatial distribution of the magnitude of transverse
displacement, bending moment 𝑀 𝑦𝑦
and shear force 𝑄 𝑦 at point 𝑥 =
0.55 m, 𝑦 =1.2 m on the middle plate (SSFF boundary condition,
standard linear solid model by Kelvin representation), with excitation
at point 𝑥 =0.55 m, 𝑦 =1.2 m on the top plate and the excitation
frequency 𝑓 =2×10
5
Hz: solid line – DTFM with 200 modes; dotted
line – DTFM with 500 modes

107

Figure 5.11 The spatial distribution of the magnitude of transverse
displacement, bending moment 𝑀 𝑦𝑦
and shear force 𝑄 𝑦 at point 𝑥 =
0.55 m, 𝑦 =1.2 m on the middle plate (SSFF boundary condition,
standard linear solid model by Maxwell representation), with
excitation at point 𝑥 =0.55 m, 𝑦 =1.2 m on the top plate and the
excitation frequency 𝑓 =2×10
5
Hz: solid line – DTFM with 200
modes; dotted line – DTFM with 500 modes

Table 5.
108

Table 5.1 Natural frequencies 𝑓 𝑘 of the elastically connected double-plate structure with SSSS
boundary conditions (Hz)
𝑘
FEM (elements)
DTFM
50 200 3,200
1 85.0390 86.4263 86.8882 86.9195
2 88.7757 90.1055 90.5486 90.5786
3 161.1637 165.4778 167.0437 167.1528
4 163.1664 167.4287 168.9766 169.0845
5 260.1657 265.3827 267.3096 267.4445
10 323.0613 341.1225 348.1126 348.6107
20 599.0271 645.5216 667.3677 669.0969
30 897.0924 983.4272 989.3284 989.8729
40 1067.6546 1164.9535 1204.1859 1210.4549
50 1361.6047 1510.5935 1546.7057 1551.3877

109

Table 4.2 Natural frequencies 𝑓 𝑘 of the three-layer plate with partially connected elastic layers
with SSFF boundary conditions (Hz)
𝑘
FEM (elements)
DTFM
300 1,200 4,800
1 53.7829 53.7028 53.6821 53.6752
2 54.9444 54.8623 54.8412 54.8341
3 57.4559 57.3714 57.3497 57.3425
4 75.1837 75.1723 75.1687 75.1674
5 75.5086 75.4956 75.4916 75.4902
10 217.4733 216.8162 216.6319 216.5686
20 313.8678 316.1529 316.7388 316.9354
30 492.1189 490.0905 489.4404 492.6234
40 657.7398 661.2479 662.5154 662.9800
50 874.2904 872.5249 870.9846 870.5649

110

Chapter 6 Vibration Analysis of Cylindrical Shell Structures

6.1 Introduction
Shell structures have been widely seen in engineering applications, such as, pressure vessels,
ship hulls and fuselages of airplanes. This chapter is concerned with DTFM solution method used
for modeling and analysis of cylindrical shell structures. In Chapter 6.2, the DTFM formulation
for thin cylindrical shells is introduced. In Chapter 6.3, the DTFM solution method for vibration
analysis of cylindrical shells is presented. A benchmark example is presented in Chapter 6.4 to
illustrate the accuracy and efficiency of the proposed method.

6.2 Distributed Transfer Function Formulation for Cylindrical Shells
6.2.1 Distributed Transfer Function Formulation for A Single Cylindrical Shell
Consider a general elastic homogeneous rectangular thin plate modeled by the classical shell
theory (CST), the displacements of the in-plane vibration, 𝑢 (𝑥 ,𝜃 ,𝑡 ) and 𝑣 (𝑥 ,𝜃 ,𝑡 ) and the out-
plane transverse vibration, 𝑤 (𝑥 ,𝜃 ,𝑡 ) , is governed by the linear partial differential equations
𝜕 𝑁 𝑥𝑥
𝜕𝑥
+
1
𝑅 𝜕 𝜕𝜃
𝑁 𝑥𝜃
+𝑓 𝑥 =𝐼 0
𝜕 2
𝑢 𝜕 𝑡 2
𝑓 𝜃 +
𝜕 𝑁 𝑥𝜃
𝜕𝑥
+
1
𝑅 𝜕 𝜕𝜃
𝑁 𝜃𝜃
+
1
𝑅 (
𝜕 𝑀 𝑥𝜃
𝜕𝑥
+
1
𝑅 𝜕 𝑀 𝜃𝜃
𝜕𝜃
)=𝐼 0
𝜕 2
𝑣 𝜕 𝑡 2
−𝐼 2
𝜕 3
𝑤 𝑅𝜕𝑥𝜕 𝑡 2
𝜕 2
𝑀 𝑥𝑥
𝜕 𝑥 2
+
1
𝑅 2
𝜕 𝑀 𝜃𝜃
2
𝜕 𝜃 2
−
2
𝑅 𝜕 2
𝑀 𝑥𝜃
𝜕𝑥𝜕𝜃 −
𝑁 𝜃𝜃
𝑅 +𝑓 𝑧 =𝐼 0
𝜕 2
𝑤 𝜕 𝑡 2
−𝐼 2
𝜕 2
𝜕 𝑡 2
(
𝜕 2
𝑤 𝜕 𝑥 2
+
1
𝑅 2
𝜕 2
𝑤 𝜕 𝜃 2
)
(6.1)
where 𝑁 𝑥𝑥
, 𝑁 𝑥𝜃
and 𝑁 𝜃𝜃
are resultant membrane forces; 𝑀 𝑥𝑥
, 𝑀 𝑥𝜃
and 𝑀 𝜃𝜃
are resultant
moments; 𝑓 𝑥 , 𝑓 𝜃 and 𝑓 𝑧 are external forces acting on the middle surface of the cylindrical shell in
111

the axial, circumferential and radial directions; 𝐼 0
and 𝐼 2
are the mass inertia terms; and 𝑅 is the
radius of the shell.
The above governing equations can be written in terms of the displacements only as follows
(𝐴 11
𝜕 2
𝜕 𝑥 2
+
𝐴 66
𝑅 2
𝜕 2
𝜕 𝜃 2
−𝐼 0
𝜕 2
𝜕 𝑡 2
)𝑢 +((
𝐴 12
𝑅 +
𝐴 66
𝑅 )
𝜕 2
𝜕𝑥𝜕𝜃 )𝑣 +(𝐴 12
𝜕 𝜕𝑥
)𝑤 =−𝑓 𝑥 (
𝐴 12
+𝐴 66
𝑅 )
𝜕 2
𝜕𝑥𝜕𝜃 𝑢 +(𝐴 66
𝜕 2
𝜕 𝑥 2
+
𝐴 22
𝑅 2
𝜕 2
𝜕 𝜃 2
−
𝐼 0
𝜕 2
𝜕 𝑡 2
)𝑣 +
𝐴 22
𝑅 2
𝜕 𝜕𝜃
𝑤 =−𝑓 𝜃 𝐴 12
𝑅 𝜕 𝜕𝑥
𝑢 +
𝐴 22
𝑅 2
𝜕 𝜕𝜃
𝑣 +(𝐷 11
𝜕 4
𝜕 𝑥 4
+
2𝐷 11
𝑅 2
𝜕 4
𝜕 𝑥 2
𝜕 𝜃 2
+
𝐷 11
𝑅 4
𝜕 4
𝜕 𝑥 4
)𝑤 +
𝐴 22
𝑅 2
𝑤 +(
𝐼 0
𝜕 2
𝜕 𝑡 2
−
𝐼 2
𝜕 2
𝜕 𝑡 2
(
𝜕 2
𝜕 𝑥 2
+
1
𝑅 2
𝜕 2
𝜕 𝜃 2
))𝑤 =−𝑓 𝑧 (6.2)
where 𝐴 11
, 𝐴 12
, 𝐴 22
, 𝐴 66
, 𝐷 11
, 𝐷 12
, 𝐷 22
and 𝐷 66
are parameters of the thin cylindrical shell from
its constitutive equations and geometry.
Take Laplace transform of Eq. (6.2) with zero initial conditions and the 𝑠 -domain governing
equations can be obtained
(𝐴 11
𝜕 2
𝜕 𝑥 2
+
𝐴 66
𝑅 2
𝜕 2
𝜕 𝜃 2
−𝐼 0
𝑠 2
)𝑢̂+((
𝐴 12
𝑅 +
𝐴 66
𝑅 )
𝜕 2
𝜕𝑥𝜕𝜃 )𝑣̂+(𝐴 12
𝜕 𝜕𝑥
)𝑤̂ =−𝑓̂
𝑥 (
𝐴 12
+𝐴 66
𝑅 )
𝜕 2
𝜕𝑥𝜕𝜃 𝑢̂+(𝐴 66
𝜕 2
𝜕 𝑥 2
+
𝐴 22
𝑅 2
𝜕 2
𝜕 𝜃 2
−𝐼 0
𝑠 2
)𝑣̂+
𝐴 22
𝑅 2
𝜕 𝜕𝜃
𝑤̂ =−𝑓̂
𝜃 𝐴 12
𝑅 𝜕 𝜕𝑥
𝑢̂+
𝐴 22
𝑅 2
𝜕 𝜕𝜃
𝑣̂+(𝐷 11
𝜕 4
𝜕 𝑥 4
+
2𝐷 11
𝑅 2
𝜕 4
𝜕 𝑥 2
𝜕 𝜃 2
+
𝐷 11
𝑅 4
𝜕 4
𝜕 𝑥 4
+
𝐴 22
𝑅 2
)𝑤̂
(𝐼 0
𝑠 2
−𝐼 2
𝑠 2
(
𝜕 2
𝜕 𝑥 2
+
1
𝑅 2
𝜕 2
𝜕 𝜃 2
))𝑤̂ =𝑓̂
𝑧 (6.3)
where 𝑢̂=𝑢̂(𝑥 ,𝜃 ,𝑠 ) , 𝑣̂=𝑣̂(𝑥 ,𝜃 ,𝑠 ) and 𝑤̂ =𝑤̂(𝑥 ,𝜃 ,𝑠 ) are 𝑠 -domain variables of the displacements.
By using the Lévy solution, the displacements can be expressed by series as follows
112

[
𝑢̂(𝑥 ,𝜃 ,𝑠 )
𝑣̂(𝑥 ,𝜃 ,𝑠 )
𝑤̂(𝑥 ,𝜃 ,𝑠 )
]=∑[
𝑢̂
𝑚 (𝑥 ,𝑠 )cos(𝑚𝜃 )
𝑣̂
𝑚 (𝑥 ,𝑠 )sin(𝑚𝜃 )
𝑤̂
𝑚 (𝑥 ,𝑠 )cos(𝑚𝜃 )
]
∞
𝑚 =0
(6.4)
Plugging Eq. (6.4) into Eq. (6.3) gives three equations for the 𝑚 th
vibration mode
[
𝐴 66
𝑅 2
(−𝑚 2
)−𝐼 0
𝑠 2
]𝑢̂
𝑚 (𝑥 ,𝑠 )+𝐴 11
𝑢̂
𝑚 ′′
(𝑥 ,𝑠 )
+(
𝐴 12
+𝐴 66
𝑅 )(𝑚 )𝑣̂
𝑚 ′
(𝑥 ,𝑠 )+
𝐴 12
𝑅 𝑤̂
𝑚 ′
(𝑥 ,𝑠 )=−𝑓̂
𝑥 ,𝑚 (𝑥 ,𝑠 )
(6.5𝑎 )
𝐴 12
+𝐴 66
𝑅 (−𝑚 )𝑢̂
𝑚 ′
(𝑥 ,𝑠 )+[
𝐴 22
𝑅 (−𝑚 2
)−𝐼 0
𝑠 2
]𝑣̂(𝑥 ,𝑠 )+𝐴 66
𝑣̂
𝑚 ′′
(𝑥 ,𝑠 )
+
𝐴 22
𝑅 2
(−𝑚 )𝑤̂
𝑚 ′
(𝑥 ,𝑠 )=−𝑓̂
𝜃 ,𝑚 (𝑥 ,𝑠 )
(6.5𝑏 )
𝐴 12
𝑅 𝑢̂
𝑚 ′
(𝑥 ,𝑠 )+
𝐴 22
𝑅 2
(𝑚 )𝑣̂
𝑚 (𝑥 ,𝑠 )+ 𝐷 11
𝑤̂
𝑚 ′′′′
(𝑥 ,𝑠 )+[
−2𝐷 11
𝑚 2
𝑅 2
−𝐼 2
𝑠 2
]𝑤̂
𝑚 ′′
(𝑥 ,𝑠 )
+[
𝐷 11
𝑅 4
(𝑚 4
)+
𝐴 22
𝑅 2
+𝐼 0
𝑠 2
+
𝐼 2
𝑠 2
𝑚 2
𝑅 2
]𝑤̂
𝑚 (𝑥 ,𝑠 )=𝑓̂
𝑧 ,𝑚 (𝑥 ,𝑠 )
(6.5𝑐 )
In the 𝑠 -domain, define the state vector 𝜼̂
𝑚 (𝑥 ,𝑠 ) corresponding to the 𝑚 th
mode as
𝜼̂
𝑚 (𝑥 ,𝑠 )={
𝑢̂
𝑚 𝑢̂
𝑚 ′
𝑣̂
𝑚 𝑣̂
𝑚 ′
𝑤̂
𝑚 𝑤̂
𝑚 ′
𝑤̂
𝑚 ′′
𝑤̂
𝑚 ′′′
}
T
(6.6)
With the above-defined state vector, the 𝑠 -domain governing equation (6.5) of the 𝑚 th
mode can
be converted to the state equation
𝜕 𝜕𝑥
𝜼̂
𝑚 (𝑥 ,𝑠 )=𝐅̂
𝑚 (𝑠 )𝜼̂
𝑚 (𝑥 ,𝑠 )+𝒑̂
𝑚 (𝑥 ,𝑠 ) (6.7)
with an 8-by-8 state matrix 𝐅̂
𝑚 (𝑠 ) , whose elements are
𝐅̂
𝑚 (1,2)=𝐅̂
𝑚 (3,4)=𝐅̂
𝑚 (5,6)=𝐅̂
𝑚 (6,7)=𝐅̂
𝑚 (7,8)=1
𝐅̂
𝑚 (2,1)=
1
𝐴 11
(
𝐴 66
𝑚 2
𝑅 2
+𝐼 0
𝑠 2
) 𝐅̂
𝑚 (2,4)=−
(𝐴 12
+𝐴 66
)
𝑅 𝐴 11
𝑚 𝐅̂
𝑚 (2,6)=−
𝐴 12
𝑅 𝐴 11

113

𝐅̂
𝑚 (4,2)=
(𝐴 12
+𝐴 66
)
𝑅 𝐴 66
𝑚 𝐅̂
𝑚 (4,3)=
1
𝐴 66
(
𝐴 22
𝑚 2
𝑅 +𝐼 0
𝑠 2
) 𝐅̂
𝑚 (4,5)=𝑚 𝐴 22
𝑅 2
𝐴 66

𝐅̂
𝑚 (8,2)=
−𝐴 12
𝑅 𝐷 11
𝐅̂
𝑚 (8,3)=
−𝐴 22
𝑅 2
𝐷 11
𝑚 𝐅̂
𝑚 (8,5)=(−
1
𝐷 11
)(
𝐷 11
𝑅 4
𝑚 4
+
𝐴 22
𝑅 2
+𝐼 0
𝑠 2
+
𝐼 2
𝑠 2
𝑅 2
𝑚 2
) 𝐅̂
𝑚 (8,7)=(
1
𝐷 11
)(
2𝐷 11
𝑅 2
𝑚 2
+𝐼 2
𝑠 2
)
(6.8)
Meanwhile, the boundary conditions of the cylindrical shell can be written in terms of state
variables and cast into the matrix form as
𝐌 𝑚 (𝑠 )𝜼̂
𝑚 (0,𝑠 )+𝐍 𝑚 (𝑠 )𝜼̂
𝑚 (𝐿 ,𝑠 )=𝜸̂
𝑚 (𝑠 ) (6.9)

6.2.2 Distributed Transfer Function Formulation for Cylindrical Shells Connected by
Viscoelastic Layers
Consider two cylindrical shells connected by a viscoelastic layer as shown in Figure 6.1. The
𝑠 -domain governing equations can be written as follows
114

(𝐴 11
𝜕 2
𝜕 𝑥 2
+
𝐴 66
𝑅 2
𝜕 2
𝜕 𝜃 2
−𝐼 0
𝑠 2
)𝑢̂
1
+((
𝐴 12
𝑅 +
𝐴 66
𝑅 )
𝜕 2
𝜕𝑥𝜕𝜃 )𝑣̂
1
+(𝐴 12
𝜕 𝜕𝑥
)𝑤̂
1
=−𝑓̂
𝑥 ,1
(
𝐴 12
+𝐴 66
𝑅 )
𝜕 2
𝜕𝑥𝜕𝜃 𝑢̂
1
+(𝐴 66
𝜕 2
𝜕 𝑥 2
+
𝐴 22
𝑅 2
𝜕 2
𝜕 𝜃 2
−𝐼 0
𝑠 2
)𝑣̂
1
+
𝐴 22
𝑅 2
𝜕 𝜕𝜃
𝑤̂
1
=−𝑓̂
𝜃 ,1
𝐴 12
𝑅 𝜕 𝜕𝑥
𝑢̂
1
+
𝐴 22
𝑅 2
𝜕 𝜕𝜃
𝑣̂
1
+(𝐷 11
𝜕 4
𝜕 𝑥 4
+
2𝐷 11
𝑅 2
𝜕 4
𝜕 𝑥 2
𝜕 𝜃 2
+
𝐷 11
𝑅 4
𝜕 4
𝜕 𝑥 4
+
𝐴 22
𝑅 2
)𝑤̂
1
(𝐼 0
𝑠 2
−𝐼 2
𝑠 2
(
𝜕 2
𝜕 𝑥 2
+
1
𝑅 2
𝜕 2
𝜕 𝜃 2
))𝑤̂
1
+𝐺̂
(𝑠 )(𝑤̂
1
−𝑤̂
2
)=𝑓̂
𝑧 ,1
(𝐴 11
𝜕 2
𝜕 𝑥 2
+
𝐴 66
𝑅 2
𝜕 2
𝜕 𝜃 2
−𝐼 0
𝑠 2
)𝑢̂
2
+((
𝐴 12
𝑅 +
𝐴 66
𝑅 )
𝜕 2
𝜕𝑥𝜕𝜃 )𝑣̂
2
+(𝐴 12
𝜕 𝜕𝑥
)𝑤̂
2
=−𝑓̂
𝑥 ,2
(
𝐴 12
+𝐴 66
𝑅 )
𝜕 2
𝜕𝑥𝜕𝜃 𝑢̂
2
+(𝐴 66
𝜕 2
𝜕 𝑥 2
+
𝐴 22
𝑅 2
𝜕 2
𝜕 𝜃 2
−𝐼 0
𝑠 2
)𝑣̂
2
+
𝐴 22
𝑅 2
𝜕 𝜕𝜃
𝑤̂
1
=−𝑓̂
𝜃 ,2
𝐴 12
𝑅 𝜕 𝜕𝑥
𝑢̂
2
+
𝐴 22
𝑅 2
𝜕 𝜕𝜃
𝑣̂
2
+(𝐷 11
𝜕 4
𝜕 𝑥 4
+
2𝐷 11
𝑅 2
𝜕 4
𝜕 𝑥 2
𝜕 𝜃 2
+
𝐷 11
𝑅 4
𝜕 4
𝜕 𝑥 4
+
𝐴 22
𝑅 2
)𝑤̂
2
(𝐼 0
𝑠 2
−𝐼 2
𝑠 2
(
𝜕 2
𝜕 𝑥 2
+
1
𝑅 2
𝜕 2
𝜕 𝜃 2
))𝑤̂
2
+𝐺̂
(𝑠 )(𝑤̂
2
−𝑤̂
1
)=𝑓̂
𝑧 ,2
(6.10)
where 𝐺̂
(𝑠 ) is the transfer function describing the material property of the viscoelastic layer that
has been discussed in Chapter 5.3.
Similar to Eq. (6.6), define the state vector 𝜼̂
𝑚 (𝑥 ,𝑠 ) corresponding to the 𝑚 th
mode as
𝜼̂
𝑚 (𝑥 ,𝑠 )={
{𝑢̂
𝑚 ,1
𝑢̂
𝑚 ,1
′
𝑣̂
𝑚 ,1
𝑣̂
𝑚 ,1
′
𝑤̂
𝑚 ,1
𝑤̂
𝑚 ,1
′
𝑤̂
𝑚 ,1
′′
𝑤̂
𝑚 ,1
′′′
}
T
{𝑢̂
𝑚 ,2
𝑢̂
𝑚 ,2
′
𝑣̂
𝑚 ,2
𝑣̂
𝑚 ,2
′
𝑤̂
𝑚 ,2
𝑤̂
𝑚 ,2
′
𝑤̂
𝑚 ,2
′′
𝑤̂
𝑚 ,2
′′′
}
T
} (6.11)
Hence, the 𝑠 -domain governing equation (6.10) of the 𝑚 th
mode can be converted to the state
equation similar to Eq. (6.7) while the state matrix 𝐅̂
𝑚 (𝑠 ) is now 16-by-16. More importantly, the
boundary conditions can be expressed in the matrix form as Eq. (6.9), too.
It is seen that the state equations . and . have the same format as plates’ state equations
(5.29) and (5.33). Subsequently, the dynamic response of each vibration mode of the cylindrical
shell can be obtained by following the standard DTFM solution procedure. The eigenvalue
115

problem can be solved for the free vibration analysis and transient response. The closed-form
Green’s functions for the frequency response of the cylindrical shell can be obtained.

6.3 A Simply Supported Shell Example
The DTFM solution for the cylindrical shell is illustrated on one simple benchmark example:
a thin shell modeled by the classical shell theory (CST) with two ends simply supported. In this
example, the natural frequencies of the plate are computed and compared with natural frequencies
from the analytical solution (2006, Reddy; 1992 Reddy) to validate the DTFM formulation. The
frequency response of the shell by the DTFM is also presented.
The parameters of the single cylindrical shell used in the simulation are listed below.
𝐿 =1 m, 𝑅 =1 m, 𝑡 =1/60 m, 𝐸 =200 GPa, 𝜈 =0.3, 𝜌 =7800 kg/m
3

First, for validation purposes, compute the natural frequencies of the plate. The results
obtained by the DTFM are compared with those by the analytical solution. Listed in Table 6.1 are
the first five and some higher-mode natural frequencies of the plate for 𝑚 =0. It can be seen that
results from both methods can match with each other exactly. This is because the DTFM also
provides the exact solution of the eigenvalue problem. Hence, the DTFM formulation for the thin
shell works well in the modeling of the cylindrical shell vibration.
Next, compute the frequency response of the plate subject to the line load, 𝐹 𝑓 (𝑥 ,𝜃 ,𝑡 )=
𝐹 0
𝛿 (𝑥 −
𝐿 2
)sin(2𝜋𝑓𝑡 ) . Let excitation frequency be 𝑓 =5×10
3
Hz. Plotted in Figure 6.2 is the
spatial distribution of the transverse displacement of the shell at 𝜃 =0 by using only 50 modes.
116

The simulation can be done on a PC within five seconds, which implies the DTFM solution method
is very efficient.

117

Figure 6.1 Two cylindrical shells connected by the viscoelastic layer

118

Figure 6.2 Spatial distribution of the cylindrical shell at 𝜃 =0 under
the excitation frequency 𝑓 =5,000 Hz by 50-mode DTFM

119

Table 5.1 The natural frequencies 𝑓 0,𝑛 of the thin shell (Hz)

Analytical Solution DTFM
𝒏
1 802.8001 802.8001
2 820.4007 820.4007
3 844.8257 844.8257
4 881.7346 881.7346
5 1028.1128 1028.1128
30 / 12,561.4859
31 / 12,565.8109
40 / 18,440.2110
41 / 18,580.4201

120

Chapter 7 Conclusions
In this work, a new method for modeling and analysis of flexible parameter structures in mid-
and high-frequency vibration. The proposed method is based on the distributed transfer function
method (DTFM) and it has been applied to beam networks, sandwich plates and cylindrical shells.
The main results obtained from the previous effort are summarized and the future work of this
research project is discussed.
The DTFM-based approach does not require spatial discretization or approximation in
modeling structures so it can provide closed-form analytical frequency-domain solutions. With the
augmented state formulation, the local singularities in the traditional DTFM have been avoided.
Different types of boundary conditions and matching conditions are treated systematically, with
no need for coupling loss factors or energy transmission coefficients. Furthermore, the DTFM
yields a global state equation, the solution of which yields the mid- to high-frequency vibration
response and energy density of a beam structure without any numerical instability at high
frequencies.
The DTFM has also been developed for the transient vibration analyses of beam structures in
mid- and high-frequency regions. By using the 𝑠 -domain global transfer function and the exact
eigensolutions provided by the augmented DTFM, two analysis tools, the method of inverse
Laplace transform and the generalized modal expansion, are presented. Moreover, a model
reduction procedure can be applied to the augmented DTFM transient tools, which yields the
reduced model by truncated series. The reduced models can save significant computational effort
and produce transient solutions of adequate accuracy.
121

In various numerical examples, good agreement is seen when the augmented DTFM is
compared with other methods, such as, the EFA and FEA. It is shown that the new method is
accurate and efficient in mid- to high-frequency regions for both frequency response and the
transient response. With this feature, the augmented DTFM has been used to study the shear
deformation in Timoshenko beam structures at high frequencies.
The augmented DTFM has been extended to the modeling and vibration analysis of sandwich
plates and cylindrical shells. By a series representation, the proposed method has a systematic
formulation for governing equations and boundary conditions. Because the state formulation of
sandwich plate structures and cylindrical shells by the augmented DTFM is the same as the
traditional DTFM, the frequency response and transient response of the plate can be obtained in
the same manner. Compared with existing methods, FEA-based methods require much effort to
get shear force and energy-based methods have difficulty providing local information; and the
analytical solutions are limited to simple boundary conditions and double-plate structures. In
numerical examples, the accuracy and efficiency of the DTFM is validated through the comparison
with the FEA and analytical solutions. It has been shown that the proposed method can work well
in all frequency regions, from low to high. Though the proposed method provides the frequency
response in the series expression, the required computational effort is much less than the existing
methods. By using the DTFM, engineers can select hundreds of terms in advance to do vibration
analysis at mid- and high frequencies accurately.
One advantage of proposed methods based on the augmented DTFM is that detailed local
information about flexible distributed structures, including displacement, slope, moment and shear
force, can be obtained from the state vector directly. Energy density and energy flow can be easily
computed. This functionality of having local information, which most conventional methods for
122

mid- or high- frequency analyses lack, makes the proposed method a versatile tool in design and
optimization of complex structures in various applications.
The highlight of proposal methods is that the formulation procedures and solution methods
do not change when the excitation frequency varies. The consistent formulation steps and solution
procedures make it possible to have one vibration analysis method for the same structure from the
low-frequency region to high-frequency region, which is suitable for mid-frequency vibration
analysis.
Current results can be extended in several ways which can be the future work of the research
project. First, the DTFM can be extended to more complex structures, such as, thick plate theories,
thick shell theories and the plate-beam structures. The shear effect in various types of structures at
mid and high frequencies can be studied. Second, the frequency region threshold can be discovered
by the DTFM in a quantitative way. As natural frequencies, the frequency region threshold is one
of the structure’s dynamic characteristics. If the frequency region threshold can be found, mid -
and high-frequency vibration analysis of complex structures will be easier.

123

References
Akoussan, K., Boudaoud, H., Daya, E. M., and Carrera, E. (2015). Vibration modeling of
multilayer composite structures with viscoelastic layers. Mechanics of Advanced Materials and
Structures, 22(1-2), 136-149.
Alaimo, A., Orlando, C., and Valvano, S. (2019). Analytical frequency response solution for
composite plates embedding viscoelastic layers. Aerospace Science and Technology, 92, 429-445.
Alam, N., and Asnani, N. T. (1984). Vibration and damping analysis of a multilayered cylindrical
shell. I-Theoretical analysis. AIAA Journal, 22(6), 803-810.
Alam, N., and Asnani, N. T. (1984). Vibration and damping analysis of a multilayered cylindrical
shell, part II: numerical results. AIAA Journal, 22(7), 975-981.
Alam, N., and Asnani, N. T. (1984). Vibration and damping analysis of multilayered rectangular
plates with constrained viscoelastic layers. Journal of Sound and Vibration, 97(4), 597-614.
Bagley, R. L., and Torvik, P. J. (1983). A theoretical basis for the application of fractional calculus
to viscoelasticity. Journal of Rheology, 27(3), 201-210.
Bagley, R. L., and Torvik, P. J. (1986). On the fractional calculus model of viscoelastic
behavior. Journal of Rheology, 30(1), 133-155.
Bathe, K. J. (2006). Finite Element Procedures. Klaus-Jurgen Bathe.
Bouthier, O. M. (1992). Energetics of Vibrating Systems. Purdue University.
Bouthier, O. M., and Bernhard, R. J. (1995). Simple models of the energetics of transversely
vibrating plates. Journal of Sound and Vibration, 182(1), 149-164.
Chen, Y. H., and Sheu, J. T. (1994). Dynamic characteristics of layered beam with flexible core.
Cho, P. E. H. (1993). Energy Flow Analysis of Coupled Structures. Purdue University.
Cho, P. E., and Bernhard, R. J. (1998). Energy flow analysis of coupled beams. Journal of Sound
and Vibration, 211(4), 593-605.
Christensen, R. (2012). Theory of Viscoelasticity: An Introduction. Elsevier.
Cook, R. D., Malkus, D. S., Plesha, M. E., and Witt, R. J., 2001, Concepts and Applications of
Finite Element Analysis, Fourth Edition, John Wiley and Sons, New York, NY.
Cremer, L., Heckl, M., and Ungar, E. E. (1974). Structure-borne sound. Journal of Applied
Mechanics, 41(3), 839.
124

Di Paola, M., Heuer, R., and Pirrotta, A. (2013). Fractional visco-elastic Euler–Bernoulli
beam. International Journal of Solids and Structures, 50(22-23), 3505-3510.
Donnell, L. H. (1935). Stability of thin-walled tubes under torsion (No. NACA-TR-479).
Dowell, E. H., and Kubota, Y. (1985). Asymptotic modal analysis and statistical energy analysis
of dynamical systems, 949-957, Journal of Applied Mechanics, 52(4), 949-957
Doyle, J. F., and Kamle, S. (1985). An experimental study of the reflection and transmission of
flexural waves at discontinuities, Journal of Applied Mechanics, 52(3), 669
Duhamel, D., Mace, B. R., and Brennan, M. J. (2006). Finite element analysis of the vibrations of
waveguides and periodic structures. Journal of sound and vibration, 294(1-2), 205-220.
Foraboschi, P. (2013). Three-layered sandwich plate: Exact mathematical model. Composites Part
B: Engineering, 45(1), 1601-1612.
Fredö, C.R., (1997). A SEA-like approach for the derivation of energy flow coefficients with a
finite element model. Journal of Sound and Vibration, 199(4), 645-666.
Friedman, Z., and Kosmatka, J. B. (1993). An improved two-node Timoshenko beam finite
element. Computers & Structures, 47(3), 473-481.
Guo, Y., Chen, W., and Pao, Y. H. (2008). Dynamic analysis of space frames: The method of
reverberation-ray matrix and the orthogonality of normal modes. Journal of Sound and
Vibration, 317(3-5), 716-738.
Hambric, S. A. (1990). Power flow and mechanical intensity calculations in structural finite
element analysis. Journal of Vibration and Acoustics, 112(4), 542-549
Han, F., Bernhard, R.J. and Mongeau, L.G., (1997). Energy flow analysis of vibrating beams and
plates for discrete random excitations. Journal of Sound and Vibration, 208(5), 841-859.
He, J. F., and Ma, B. A. (1988). Analysis of flexural vibration of viscoelastically damped sandwich
plates. Journal of Sound and Vibration, 126(1), 37-47.
Hong, S. B., Wang, A., and Vlahopoulos, N. (2006). A hybrid finite element formulation for a
beam-plate system. Journal of Sound and Vibration, 298(1-2), 233-256.
Hong, S. W., and Kim, J. W. (1999). Modal analysis of multi-span Timoshenko beams connected
or supported by resilient joints with damping. Journal of sound and vibration, 227(4), 787-806.
Howard, S. M., and Pao, Y. H. (1998). Analysis and experiments on stress waves in planar
trusses. Journal of Engineering Mechanics, 124(8), 884-891.
Howard, S. M., and Pao, Y. H. (1998). Analysis and experiments on stress waves in planar
trusses. Journal of Engineering Mechanics, 124(8), 884-891.
Huff Jr, J. E. (1997). Analysis of structural systems in the mid-frequency range, Purdue University
125

Hughes, O. F. (2010). Ship structural analysis and design. Published by: The Society of Naval
Architects and Marine Engineers, SNAMNE, New Jersey, ISBN: 978-0-939773-78-3.
Hwang, H. D., Maxit, L., Ege, K., Gerges, Y., and Guyader, J. L. (2017). SmEdA vibro-acoustic
modelling in the mid-frequency range including the effect of dissipative treatments. Journal of
Sound and Vibration, 393, 187-215.
Ladevèze, P., and Chevreuil, M. (2005). A new computational method for transient dynamics
including the low‐and the medium‐frequency ranges. International Journal for Numerical
Methods in Engineering, 64(4), 503-527.
Lam, K. Y., and Loy, C. T. (1995). Effects of boundary conditions on frequencies of a multi-
layered cylindrical shell. Journal of Sound and vibration, 188(3), 363-384.
Langley, R.S., (1995). On the vibrational conductivity approach to high frequency dynamics for
two-dimensional structural components. Journal of Sound and Vibration, 182(4), 637-657.
Langley, R. S., Hawes, D. H., Butlin, T., and Ishii, Y. (2019). A derivation of the Transient
Statistical Energy Analysis (TSEA) equations with benchmark applications to plate
systems. Journal of Sound and Vibration, 445, 88-102.
Lase, Y., Ichchou, M. N., and Jezequel, L. (1996). Energy flow analysis of bars and beams:
theoretical formulations. Journal of Sound and Vibration, 192(1), 281-305.
Le Guennec, Y., and Savin, É. (2011). A transport model and numerical simulation of the high-
frequency dynamics of three-dimensional beam trusses. The Journal of the Acoustical Society of
America, 130(6), 3706-3722.
Liaw, B. D., and Little, R. W. (1967). Theory of bending multi-layer sandwich plates. AIAA
Journal, 5(2), 301-304.
Liu, S., and Yang, B. (2019). A closed-form analytical solution method for vibration analysis of
elastically connected double-beam systems. Composite Structures, 212, 598-608.
Lu, Y. P., Killian, J. W., and Everstine, G. C. (1979). Vibrations of three layered damped sandwich
plate composites. Journal of Sound and Vibration, 64(1), 63-71.
Lueschen, G. G. G., Bergman, L. A., and McFarland, D. M. (1996). Green's functions for uniform
Timoshenko beams. Journal of sound and vibration, 194(1), 93-102.
Lyon, R. H., DeJong, R. G., & Heckl, M. (1995). Theory and Application of Statistical Energy
Analysis, Second Edition. Butterworth-Heinemann, Newton, MA.
Ma, X., & Vakakis, A. F. (1999). Karhunen-Loève Decomposition of the transient dynamics of a
multibay truss. AIAA journal, 37(8), 939-946.
126

Mace, B. R., Duhamel, D., Brennan, M. J., and Hinke, L. (2005). Finite element prediction of wave
motion in structural waveguides. The Journal of the Acoustical Society of America, 117(5), 2835-
2843.
Mead, D. J. (1998). Passive Vibration Control. John Wiley & Sons.
Mei, C., and Mace, B. R. (2005). Wave reflection and transmission in Timoshenko beams and
wave analysis of Timoshenko beam structures, Journal of Vibration and Acoustics, 127(4), 382-
394
Meirovitch, L. (1980). Computational methods in structural dynamics (Vol. 5). Springer Science
& Business Media.
Mejdi, A., Atalla, N., and Ghinet, S. (2015). Wave spectral finite element model for the prediction
of sound transmission loss and damping of sandwich panels. Computers & Structures, 158, 251-
258.
Melosh, R. J. (1963). Basis for derivation of matrices for the direct stiffness method. AIAA
Journal, 1(7), 1631-1637.
Morse, P. M., and Feshbach, H. (1954). Methods of theoretical physics. American Journal of
Physics, 22(6), 410-413.
Mouritz, A. P., Gellert, E., Burchill, P., and Challis, K. (2001). Review of advanced composite
structures for naval ships and submarines. Composite Structures, 53(1), 21-42.
Nefske, D. J., Wolf Jr, J. A., and Howell, L. J. (1982). Structural-acoustic finite element analysis
of the automobile passenger compartment: a review of current practice. Journal of sound and
vibration, 80(2), 247-266.
Nefske, D. J., and Sung, S. H. (1989). Power flow finite element analysis of dynamic systems:
basic theory and application to beams, Journal of Vibration, Acoustics, Stress, and Reliability in
Design, 111(1): 94-100.
Ning, H., Janowski, G. M., Vaidya, U. K., and Husman, G. (2007). Thermoplastic sandwich
structure design and manufacturing for the body panel of mass transit vehicle. Composite
Structures, 80(1), 82-91.
Noh, K., and Yang, B. (2014). An augmented state formulation for modeling and analysis of
multibody distributed dynamic systems. Journal of Applied Mechanics, 81(5): 051011.
Nosier, A., and Reddy, J. N. (1992). Vibration and stability analyses of cross-ply laminated circular
cylindrical shells. Journal of Sound and Vibration, 157(1), 139-159.
Pao, Y. H., Keh, D. C., and Howard, S. M. (1999). Dynamic response and wave propagation in
plane trusses and frames. AIAA Journal, 37(5), 594-603.
127

Pinnington, R. J., and Lednik, D. (1996). Transient energy flow between two coupled
beams. Journal of Sound and Vibration, 189(2), 265-287.
Pinnington, R. J., and Lednik, D. (1996). Transient energy flow between two coupled
beams. Journal of Sound and Vibration, 189(2), 265-287.
Pritz, T. (1996). Analysis of four-parameter fractional derivative model of real solid
materials. Journal of Sound and Vibration, 195(1), 103-115.
Qu, Y., Hua, H., and Meng, G. (2013). A domain decomposition approach for vibration analysis
of isotropic and composite cylindrical shells with arbitrary boundaries. Composite Structures, 95,
307-321.
Qu, Y., Long, X., Wu, S., and Meng, G. (2013). A unified formulation for vibration analysis of
composite laminated shells of revolution including shear deformation and rotary
inertia. Composite Structures, 98, 169-191.
Rabbiolo, G., Bernhard, R. J., and Milner, F. A. (2004). Definition of a high-frequency threshold
for plates and acoustical spaces. Journal of Sound and Vibration, 277(4-5), 647-667.
Rao, M. D. (2003). Recent applications of viscoelastic damping for noise control in automobiles
and commercial airplanes. Journal of Sound and Vibration, 262(3), 457-474.
Rao, M. K., & Desai, Y. M. (2004). Analytical solutions for vibrations of laminated and sandwich
plates using mixed theory. Composite Structures, 63(3-4), 361-373.
Reddy, J. N. (1984). Exact solutions of moderately thick laminated shells. Journal of Engineering
Mechanics, 110(5), 794-809.
Reddy, J. N. (2006). Theory and Analysis of Elastic Plates and Shells. CRC press.
Renno, J. M., & Mace, B. R. (2013). Calculation of reflection and transmission coefficients of
joints using a hybrid finite element/wave and finite element approach. Journal of Sound and
Vibration, 332(9), 2149-2164.
Renno, J. M., & Mace, B. R. (2014). Calculating the forced response of cylinders and cylindrical
shells using the wave and finite element method. Journal of Sound and Vibration, 333(21), 5340-
5355.
Shorter, P. J., & Langley, R. S. (2005). Vibro-acoustic analysis of complex systems. Journal of
Sound and Vibration, 288(3), 669-699.
Soize, C. (1982). Medium frequency linear vibrations of anisotropic. Recherche Aerospatiale
(English edition), 5, 65-87.
Soize, C., Desanti, A., & David, J. M. (1992). Numerical methods in elastoacoustic for low and
medium frequency ranges. La Recherche Aérospatiale (English Edition), 5, 25-44.
128

Timoshenko, S. P., 1921, On the correction for shear of the differential equation for transverse
vibrations of prismatic bars, Philos. Mag., 41, pp. 744–746.
Timoshenko, S. P. (1922). X. On the transverse vibrations of bars of uniform cross-section. The
London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 43(253), 125-
131.
Timoshenko, S., & Woinowsky-Krieger, S. (1959). Theory of Plates and Shells (Vol. 2, pp. 240-
246). New York: McGraw-hill.
Ugural, A. C. (2009). Stresses in Beams, Plates, and Shells. CRC press.
Vlahopoulos, N., and Zhao, X. (1999). Basic development of hybrid finite element method for
midfrequency structural vibrations. AIAA journal, 37(11), 1495-1505.
Von Flotow, A. H. (1986). Disturbance propagation in structural networks. Journal of Sound and
Vibration, 106(3), 433-450.
Wang, C., and Lai, J. C. S. (2000). Prediction of natural frequencies of finite length circular
cylindrical shells. Applied Acoustics, 59(4), 385-400.
Wang, T., Li, S., Rajaram, S., and Nutt, S. R. (2010). Predicting the sound transmission loss of
sandwich panels by statistical energy analysis approach, Journal of Vibration and
Acoustics, 132(1).
Warburton, G. B. (1965). Vibration of thin cylindrical shells, Journal of Mechanical Engineering
Science, 7(4), 399-407.
Wohlever, J. C., and Bernhard, R. J. (1992). Mechanical energy flow models of rods and
beams, Journal of Sound and Vibration, 153(1), 1-19.
Woodhouse, J. (1981). An approach to the theoretical background of statistical energy analysis
applied to structural vibration, The Journal of the Acoustical Society of America, 69(6), 1695-1709.
Woodhouse, J. (1981). An introduction to statistical energy analysis of structural
vibration, Applied Acoustics, 14(6), 455-469.
Yang, B., and Tan, C. A. (1992). Transfer Functions of One-Dimensional Distributed Parameter
Systems. Journal of Applied Mechanics, 59(4), 1009-1014.
Yang, B. (1994). Distributed Transfer Function Analysis of Complex Distributed Parameter
Systems. Journal of Applied Mechanics, 61(1) 84-92.
Yang, B. (2005). Stress, Strain, and Structural Dynamics: An Interactive Handbook of Formulas,
Solutions, and MATLAB Toolboxes. Elsevier
Yang, B. (2010). Exact transient vibration of stepped bars, shafts and strings carrying lumped
masses. Journal of Sound and Vibration, 329(8), 1191-1207.
129

Yang, B., and Liu, S. (2017). Closed-form analytical solutions of transient heat conduction in
hollow composite cylinders with any number of layers. International Journal of Heat and Mass
Transfer, 108, 907-917.
Yang, Y., Mace, B., & Kingan, M. (2018, December). Prediction of Noise Transmission through
Honeycomb Sandwich Panels Using a Wave and Finite Element Method. In INTER-NOISE and
NOISE-CON Congress and Conference Proceedings (Vol. 257, No. 1, pp. 122-132). Institute of
Noise Control Engineering.
Yong, Y., and Lin, Y. K. (1992). Dynamic response analysis of truss-type structural networks: a
wave propagation approach. Journal of Sound and Vibration, 156(1), 27-45.
Zener, C. M., & Siegel, S. (1949). Elasticity and Anelasticity of metals. The Journal of Physical
Chemistry, 53(9), 1468-1468.
Zhang, X. (2011). The Fourier Spectral Element Method for Vibration Analysis of General
Dynamic Structures. Wayne State University.
Zhang, X. M., Liu, G. R., and Lam, K. Y. (2001). Vibration analysis of thin cylindrical shells using
wave propagation approach. Journal of Sound and Vibration, 239(3), 397-403.
Zhao, X., and Vlahopoulos, N. (2004). A basic hybrid finite element formulation for mid-
frequency analysis of beams connected at an arbitrary angle, Journal of Sound and
Vibration, 269(1-2), 135-164.
Zhou, J., and Yang, B. (1995). Distributed transfer function method for analysis of cylindrical
shells. AIAA Journal, 33(9), 1698-1708.
Zhou, J., and Yang, B. (1996). Three-dimensional analysis of simply supported laminated
cylindrical shells with arbitrary thickness. AIAA Journal, 34(9), 1960-1964.
Zhou, R., & Crocker, M. J. (2010). Sound transmission loss of foam-filled honeycomb sandwich
panels using statistical energy analysis and theoretical and measured dynamic properties. Journal
of Sound and Vibration, 329(6), 673-686.

Abstract (if available)

Abstract Vibration analysis of complex flexible structures is important to the research and product design in automobile, aerospace, civil, machinery, and ship industries. Depending on the frequency spectrum of interest, vibration analyses of flexible structures are usually categorized into three groups: low-frequency analysis, mid-frequency analysis, and high-frequency analysis. An analytical method is developed for modeling and analysis of flexible distributed parameter structures in this work. In the proposed method, the vibration of the flexible beam, plate, and shell structures is modeled by the augmented formulation of the distributed transfer function method (DTFM). The formulation does not rely on discretization and treats different types of connection, and general boundary conditions in a unified manner. A highlight of the new method is that it can deliver both frequency response and transient response with detailed information on local displacement, slope, bending moment, and shear force for different frequency regions, from low to high, which otherwise might be difficult to achieve by conventional analyses. For the transient response, the proposed method does not require numerical integration and provides a platform for model reduction in mid- and high-frequency analyses. The proposed method is demonstrated in several examples. The accuracy of the method is validated with popular conventional methods, such as modal analysis, FEA, and EFA. The efficiency is seen from the comparison. As a result, the proposed method is a useful tool for design and optimization of complex flexible structures.

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

Dynamic analysis and control of one-dimensional distributed parameter systems

PDF

Modeling, analysis and experimental validation of flexible rotor systems with water-lubricated rubber bearings

PDF

An approach to dynamic modeling of percussive mechanisms

PDF

An approximation of frequency response of vibration system in mid-frequency area

PDF

Control of spacecraft with flexible structures using pulse-modulated thrusters

PDF

Periodic solutions of flexible systems under discontinuous control

PDF

Transient modeling, dynamic analysis, and feedback control of the Inductrack Maglev system

PDF

Modeling and dynamic analysis of coupled structure-moving subsystem problem

PDF

Active delay output feedback control for high performance flexible servo systems

PDF

Design, modeling and analysis of piezoelectric forceps actuator

PDF

Dynamic modeling and simulations of rigid-flexible coupled systems using quaternion dynamics

PDF

Modeling and vibration analysis of wheelchair-users

PDF

Optimal design, nonlinear analysis and shape control of deployable mesh reflectors

PDF

Numerical study of shock-wave/turbulent boundary layer interactions on flexible and rigid panels with wall-modeled large-eddy simulations

PDF

Data-driven image analysis, modeling, synthesis and anomaly localization techniques

PDF

Nonlinear control of flexible rotating system with varying velocity

PDF

Form finding and shape control of space deployable truss structures

PDF

Efficient machine learning techniques for low- and high-dimensional data sources

PDF

Development and applications of a body-force propulsor model for high-fidelity CFD

PDF

Single-cell analysis with high frequency ultrasound

Asset Metadata

Creator Zhang, Yichi (author)

Core Title Medium and high-frequency vibration analysis of flexible distributed parameter systems

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Degree Conferral Date 2023-08

Publication Date 07/10/2023

Defense Date 05/03/2023

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag beam structures,cylindrical shells,distributed transfer function method,mid- and high-frequency analysis,OAI-PMH Harvest,sandwich plates,vibrations

Format theses (aat)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Yang, Bingen (committee chair), Flashner, Henryk (committee member), Nakano, Aiichiro (committee member)

Creator Email echizhang@gmail.com,yichiz@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC113263143

Unique identifier UC113263143

Identifier etd-ZhangYichi-12053.pdf (filename)

Legacy Identifier etd-ZhangYichi-12053

Document Type Dissertation

Format theses (aat)

Rights Zhang, Yichi

Internet Media Type application/pdf

Type texts

Source 20230710-usctheses-batch-1065 (batch), University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright.

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA

Repository Email cisadmin@lib.usc.edu