Analytical And Experimental Studies Of Forced Vibration Of Impact-Damped Mechanical Systems

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Content INFORMATION TO USERS This material was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1. The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. You will find a good image of the page in the adjacent frame. 3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again — beginning below the first row and continuing on until complete. 4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. Silver prints of "photographs" may be ordered at additional charge by writing the Order Department, giving the catalog number, title, author and specific pages you wish reproduced. 5. PLEASE NOTE: Some pages may have indistinct print. Filmed as received. Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 74- 17,353 KAHYAI, Keykhosro, 1940- ANALYTICAL A N D EXPERIM ENTAL STUDIES O F F O R C E D VIBRATION O F IM PACT D A M P E D M ECHANICAL SYSTEMS. University of Southern C alifornia, Ph.D., 1974 Engineering, c iv il University Microfilms, A X E R O X Company, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. A N A LY TICA L AND E X P E R IM E N T A L STUDIES OF FO R C ED VIBRATION OF IM PA C T DAM PED M ECHANICAL SYSTEMS by K eykhosro Kahyai A D is s e rta tio n P r e s e n te d to the FA C U L T Y OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In P a r t i a l F u lfillm e n t of the R e q u ire m e n ts for the D eg ree DOCTOR OF PHILOSOPHY (Civil Engineering) June 1974 UNIVERSITY O F SOUTHERN CALIFORNIA TH E GRADUATE SC H O O L U N IV ERSITY PARK LO S A N G ELE S. CALI FO R N IA 9 0 0 0 7 This dissertation, written by Ke^y_khp_sroM Kahy_ajL.......................................... under the direction of h.is.... Dissertation Com mittee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillment of requirements of the degree of D O C T O R O F P H I L O S O P H Y Dean DISSERTATION COMMITTEE Chairman A BSTRA CT A study is m ade of the d y n am ical re s p o n s e of lin e a r m e c h a n ical im p a c t d am ped s y ste m s to h a rm o n ic excitation for two c la s s e s of p ro b le m s 1. M u lti-d e g re e -o f-fre e d o m s y s te m s 2. Continuous s y ste m s In the f ir s t case an analog c o m p u ter is used to sim u late a sev en s to ry building by using the dynam ic c h a r a c te r i s t ic s of the building obtained fro m e x p e rim e n ta l r e s u lts . The s y ste m , in addition of having viscous and s tr u c tu ra l dam ping, is a lso d am ped by an im pact d a m p e r a ttach e d to the top floor, and the building is subjected to h a rm o n ic ground excitation. The r e s u lt s a r e c o m p a re d to those obtained fro m an an aly tic al solution for a v a rie ty of c a s e s . Good a g r e e m e n t is found for the ran g e of in te r e s t and the effectiv en e ss of the im p act d a m p e r is d e m o n stra te d . In the second case an exact solution is p re s e n te d for the c la ss of lin e a r continuous m e c h a n ic a l s y s te m s p o sse sin g n o r m a l m o d es of vibration, that a r e im p act dam ped and subjected to a r b i t r a r y h a rm o n ic excitation. In p a rtic u la r, n u m e ric a l r e s u lt s a r e obtained for an e la stic plate and the validity of the technique a s w ell a s the ran g e of a g r e e m e n t is estab lish ed through the use of a m e c h a n ic a l m odel. A CKN O W LED G EM EN TS I w i s h to e x p re s s m y thanks to the m e m b e r s of the d is s e rta tio n co m m itte e , P r o f e s s o r s Sam i F a iz M a s ri, P a u l Seide, and John P i e r c e for th e ir advice and co m m en ts in w ritin g this d is s e rta tio n . I a m p a rtic u la rly indebted to m y c h a irm a n , P r o f e s s o r M a s ri, for his c o n siste n t guidance and continuous a s s is ta n c e throughout the c o u rs e of this r e s e a r c h . My love and a d m ira tio n goes to m y wife F e r y a for h e r patience and understanding. Special thanks a r e due M r. G eorge S choenherz, la b o ra to ry in s tru c to r at the U niversity of Southern C alifornia, whose e x p e rt advice and ingenuity w e re so e s s e n tia l during the e x p e rim e n ta l p a rts of this r e s e a r c h , and M rs. G race Jung for h e r e x p e rt typing. I a m fu rth e r indebted to the N ational Science F oundation for s u p p o rt of this w o rk under c o n tra c t No. GI-39417, and the U n iv e r sity of Southern C alifornia Computing C enter for the use of its f a c i lit ie s . iii TA B L E OF CONTENTS P age A B S T R A C T ...................................................................................................... ii A CK N O W LED G EM EN TS........................................................................ ill TA B L E OF C O N T EN TS.......................................................................... Lv LIST OF T A B L E S .................................................................................. vii LIST OF FIG U R E S ................................................................................ viii N O M E N C L A T U R E ................................................................................ xiv C H A PTE R 1. IN T R O D U C T IO N ....................................................... 1 1.1 H is to ric a l B a c k g ro u n d ............................................. 1 1.2 Scope of this R e s e a r c h ............................................. 3 C H A PT E R 2. FO R C ED VIBRATION OF A M U L T I D E G R E E SYSTEM WITH IM PA C T D A M P E R ................................................................... 4 2. 1 D e s c rip tio n of the P r o b l e m ................................ 4 2. 2 E quation of M otion...................................... 4 2. 3 N o rm a l M odes Solution of Undamped S y s te m .............................................................................. 7 2. 4 F o r c e d V ibration of the D am ped S y s te m .............................................................................. 8 2. 5 Solution of the P r o b l e m ......................... 12 2. 6 A n aly sis of D iscontinuous P o rtio n of the M otion...................................................................... 16 C H A PT E R 3. E X P E R IM E N T A L STUDIES............................. 19 3. 1 In tro d u ctio n .................................................................... 19 3. 2 D ynam ic C h a r a c te r is tic s of the m odel. . . . 20 iv P age 3. 3 Steady State Motion of B ase E xcited D am ped S y s te m ........................................................... 22 3. 4 E x p e rim e n ts with an E le c tric A n alo g 23 3.5 D igital C om puter S tu d ie s........................................ 35 3. 6 C o m p a riso n of A nalytical and E x p e r i m e n tal R e s u l t s ............................................................ 38 3. 7 Effect of D ifferent P a r a m e t e r s ......................... 54 C H A PT E R 4. FO R C E D VIBRATION OF TWO DIMENSIONAL IM PA C T DAM PED M ECHANICAL SYSTEM S............................. 62 4. 1 In tro d u ctio n ..................................................................... 62 4 .2 P r o b le m F o r m u la tio n .............................................. 63 C H A P T E R 5. A PP L IC A T IO N AND NUM ERICAL R E S U L T S ................................................................. 78 5.1 In tro d u ctio n ..................................................................... 78 5 .2 P la te s with U niform T h ic k n e s s ......................... 78 5. 3 B ase E xcitation of P l a t e s ...................................... 79 5. 4 E x p e rim e n ts with a M ech an ical Model. . . . 79 5 .5 D igital C om puter P r o g r a m ................................... 81 5 .6 N u m e ric a l R e s u lt s ...................................................... 86 5. 7 C onvergence of A nalytical Solution.. 99 C H A P T E R 6. SUMMARY AND CONCLUSIONS................ 102 B IB LIO G RA PH Y ..................................................................................... 105 A P P E N D IX A. VARIOUS M ATRICES AND VECTORS OF C H A PTE R 2 ............................................... 109 v P age A PP E N D IX B. D EFINITIO N OF SYMBOLS USED IN C H A PTE R 4 ................................................. 113 A PPE N D IX C ............................................................................................ 116 C l. D ata Input........................................................................ 116 C2. P r o g r a m L istin g ........................................................... 122 C3. A Sam ple O utput............................................................. 134 vi LIST OF TA BLES Page Table 3. 1 N a tu ra l F re q u e n c ie s of the Undamped 7 - D e g r e e - o f - F r e e d o m S y s te m ............................. 20 Table 3 .2 Mode Shapes of the Undamped S y s te m 22 Table 5. 1 N atu ral F re q u e n c ie s of the Square C an ti le v e r P late with a C oncentrated M ass a t the T ip ............................................................................................. 87 Table 5 .2 N atu ral F re q u e n c ie s of the Square C anti le v e r P late with a C oncentrated M ass at the C enter........................................................................... 88 Table 5. 3 N atu ral F re q u e n c ie s of a S im ply-S upported Square P late with a C oncentrated M ass at the C e n t e r ........................................................................... 97 Table 5 .4 N a tu ra l F re q u e n c ie s of a Sim ply-S upported Square P la te with a C oncentrated M ass at the C enter of a Q u a rte r of the P l a t e ................ 98 vii LIST OF FIGURES Page F ig u re 2. 1 M odel of the Im pact D am ped n -D e g re e - o f - F r e e d o m S y s te m ...................................................... 5 F ig u re 3. 1 M ech an ical M odel of the 7 Story Building. . 21 F ig u re 3 .2 F re q u e n c y R esp o n se of the B ase E xcited Building in the F i r s t Mode. (No Im pact D a m p e r, £ = 0. 01)................................................... 24 F ig u re 3. 3 F re q u e n c y R esp o n se of the B ase E xcited Building in the Second Mode. (No Im pact D a m p e r, = 0. 0 1 ) .................................................... 25 F ig u re 3 .4 F re q u e n c y R esp o n se of the B ase E xcited Building in the F i r s t Mode. (No Im pact D a m p e r, £ = 0. 05)...................................................... 26 F ig u r e 3 .5 F re q u e n c y R esp o n se of the B ase E xcited Building in the Second Mode. (No Im pact D a m p e r, £ = 0. 05) ...................................................... 27 F ig u re 3. 6 Modified M odel of the Im p act D a m p e r and its F u n c tio n s ..................................................................... 29 F ig u r e 3. 7 C irc u it for G enerating g(y)........................................ 30 F ig u r e 3 .8 C irc u it for G enerating h(y, y )................................... 31 F ig u re 3. 9 P h o to g rap h of E le c tro n ic Analog C om puter S etu p .......................................................................................... 33 viii P age F ig u r e 3. 10 Analog C om puter D ia g ra m for Equations ( 3 . 5 ) ..................................................................................... 34 F ig u r e 3. 11 E ffect of Im pact D am ping on the A m plitude R esp o n se of the Top F lo o r in the F i r s t Mode for Lightly D am ped S tru ctu re. (Q = U)j, Cj = - 01 ’ H = - 01)..................................... 39 F ig u re 3. 12 E ffect of Im pact D am ping on the A m plitude R esp o n se of the Top F lo o r in the F i r s t Mode for Lightly D am ped S tru ctu re, (fi = uur Q1 = 0.01, 0 .0 5 )........................................... 40 F ig u r e 3. 13 Effect of Im p act D am ping on the A m plitude R e sp o n se of the Top F lo o r in the Second Mode for Lightly D am ped S tru ctu re. (Q = w2 , q = 0.01, ^ = 0 . 0 1 ) ................................ 41 F ig u r e 3. 14 E ffect of Im p act D am ping on the A m plitude R esp o n se of the Top F lo o r in the Second Mode for Lightly D am ped S tru ctu re, (Q = U J 2, Cj = 0.01, ^ = 0 . 0 5 ) ................................ 42 F ig u re 3. 15 Effect of Im p act D am ping on the A m plitude R esp o n se of the Top F lo o r in the F i r s t Mode for Highly D am ped S tru ctu re. (Q = U , , Q = 0. 05, p = 0. 0 1 ) ................................ 44 ix Page F ig u r e F ig u re F ig u re F ig u re F ig u re F ig u re F ig u re . 16 E ffect of Im pact D am ping on the A m plitude R esp o n se of the Top F lo o r in the F i r s t Mode for Highly D am ped S tru c tu re . (Q = U ,lf Gj = 0.05, p = 0 .0 5 ) ........................................... 45 . 17 E ffect of Im p ac t D am ping on the A m plitude R esp o n se of the Top F lo o r in the Second Mode for Highly D am ped S tru ctu re. (Q = W 2 , Cx = ° - 05’ (4 = ° - 01) ........................................... 46 . 18 E ffect of Im pact D am ping on the A m plitude R e sp o n se of the Top F lo o r in the Second Mode for Highly D am ped S tru c tu re . (Q = 0 U 2 , Gi = ° - 0 5 > H = ° . ° 5 ) .......................................... 47 . 19 F re q u e n c y R e sp o n se of the Im pact D am ped B ase E xcited Building in the F i r s t Mode. (fi = ouj, = ° - 01» U = 0.01, e = 0 .2 5 )------- 49 .20 F re q u e n c y R e sp o n se of the Im pact D am ped B ase E xcited Building in the F i r s t Mode. (fl= U O j, Q 1 = ° . ° 1 . li= 0 .0 1 , e = 0 . 7 5 ) . . . . 50 . 21 F re q u e n c y R e sp o n se of the Im pact D am ped B ase E xcited Building in the F i r s t Mode. (fi = UJj. £]_ = ° - 01« p = 0. 05, e = 0. 25) . . . . 51 .22 F re q u e n c y R esp o n se of the Im pact D am ped x P age F ig u re F ig u re F igure F ig u re F ig u re F ig u re F ig u re B ase E xcited Building in the F i r s t Mode. (Q= ujj, Cj = ° - 01 « V J i = 0 .0 5 , e = .7 5 )........... 52 . 23 F re q u e n c y R esp o n se of the Im p act D am ped B ase E xcited Building in the Second Mode. (C l = i»2 . = °- 01, \ X = 0. 05, e = 0. 75) . . . . 53 . 24 F re q u e n c y R esp o n se of the Im pact D am ped B ase E xcited Building in the F i r s t Mode. (C l = UJj. Cj = ° - 05 > H = ° * 01’ e = 0 . 2 5 ) . . . . 55 . 25 F re q u e n c y R esp o n se of the Im p act D am ped B ase E xcited Building in the F i r s t Mode. (Q = m l , = O’ 0 5 , |J= 0 .0 1 , e = 0 .7 5 )------ 56 . 26 F re q u e n c y R esp o n se of the Im pact D am ped B ase E xcited Building in the F i r s t Mode. (Q = uoj. Cj = ° - 05> |i= ° - 05 > e = 0.25) . . . . 57 . 27 F re q u e n c y R esp o n se of the Im pact D am ped B ase E xcited Building in the F i r s t Mode. (Q = Cx = ° - 05 » L i = ° - 05 > e = 0. 75) . . . . 58 . 28 F re q u e n c y R e sp o n se of the Im pact D am ped B ase E xcited Building in the Second Mode. (0 = u)2 , Cj = °- 05» U = 0-05, e = 0. 75). . . . 59 . 1 Im p a c t D am p ed P la te . Subjected to Applied D ynam ic L o a d s ............................................................... 54 xi F ig u re 5. 1 F ig u re 5. 2 F ig u re 5. 3 F ig u re 5. 4 F ig u re 5. 5 F ig u r e 5. 6 F ig u re 5. 7 F ig u r e 5. 8 F ig u re 5. 9 F ig u r e 5. 1 F ig u r e 5. 1 Page Schem atic D ia g ra m of the M echanical M odel of Im p ac t D am ped C antilever P l a t e . .................................................................... 80 Schem atic D ia g ra m of the E x p e rim e n ta l S e t u p . ..................................................................... 82 Top View of the A ctual M echanical Model. . . 83 Side View of the A ctual M echanical Model. . . 84 G e n e ra l View of the A ctual E x p e rim e n ta l S etup..................................................................................... 85 C o m p a ris o n of T h e o re tic a l and E x p e r i m e n tal R e s u lt s ........................................................................ 89 F re q u e n c y -R e s p o n s e of B a se -E x c ite d C anti le v e r P late; x , / L = 0. 95 ...................................... 90 d F re q u e n c y -R e s p o n s e of B a s e -E x c ite d C anti le v e r P late; x , / L = 0. 5 ........................................... 91 d E ffect of V iscous D am ping. F i r s t Mode of C an tilev er P l a t e ........................................................... 92 0 Effect of V iscous D am ping. F i r s t Mode of C an tilev er P l a t e ............................................................ 93 1 Effect of D a m p e r M ass Ratio and Coefficient of R estitution. F i r s t Mode of C antilever P l a t e ...................................................................................... 94 xii P age F ig u re 5. 12 E ffect of D a m p e r Location and Mode Shapes. D is c re te F o r c e E x citatio n of a S im ply-S upported P la te ; (a) F i r s t Mode; (b) Second M o d e ............................................ 96 F ig u re 5. 13 E ffect of N um ber of M odes F o r D ifferent C le a ra n c e Ratio. F i r s t Mode of C anti le v e r P l a t e ....................................................................... 100 F ig u re 5. 14 E ffect of N um ber of M odes on the F r e quency R esp o n se of B ase Excited C anti le v e r P l a t e ....................................................................... 101 xiii N OM EN CLA TU RE Coefficient of viscous dam ping for the m o d e l of im pact d a m p e r A lin e a r d iffe re n tia l o p e ra to r D am ping m a trix G e n e ra liz e d dam ping for the i th m ode F le x u r a l rig id ity of the plate C le a ra n c e C oefficient of re s titu tio n A m plitude of applied load v ecto r A m plitude of applied load functions Spring constant for the m odel of im p act d a m p e r Stiffness m a tr ix G e n eralized stiffness coefficient for the i th mode A lin e a r d iffe re n tia l o p e ra to r M ass of the p a rtic le M a s s m a tr ix G e n e ra liz e d m a s s for the i th m ode N um ber of d e g re e s of fre e d o m A m plitude of base m otion T im e W(x, y ,t) D is p la c e m e n t function of the plate W (x ,y ,t) V elocity function of the plate W(x, y, t) A c c e le ra tio n function of the plate W (x, y) D is p la c e m e n t function of the plate a t the tim e of im pact o W (x, y) V elocity function of the plate a fte r an im pact a W (x, y) V elocity function of the plate before an im pact b (x(t)} D is p la c e m e n t vector fx(t)] V elocity vector [x (t)} A c c e le ra tio n vector {X}Q D isp lace v ecto r at the tim e of im p act [x ] V elocity vector a fte r an im pact 3. f x l V elocity v ector before an im pact L J b y(t) R elative d is p la c e m e n t of p a rtic le with r e s p e c t to the point w hich d a m p e r is attached to Z(t) A bsolute d is p la c e m e n t of the p a rtic le A bsolute d is p la c e m e n t of the p a rtic le at the tim e of im p act Z(t) A bsolute velocity of the p a rtic le Z A bsolute velocity of the p a rtic le before a n im pact Z^ A bsolute velocity of the p a rtic le a fte r a n im pact a P h a s e angle of excitation force ^o 5 xv a, p Coefficient of p ro p o rtio n a l dam ping 5 D i r a c 's function y j. M a s s d en sity of the uniform plate |_ l M a ss ra tio V4 L a p la c e 's o p e ra to r Q . C oefficient of c ritic a l dam ping for the i th mode ID . I th n a tu ra l frequency Q F r e q u e n c y of the excitation force [cp] E ig e n v e c to rs m a tr ix cp^(x, y) I th m ode shape of the plate M atrix Notations C olum n vector M a trix T ra n s p o s e d m a tr ix In v erte d m a trix Identity m a tr ix D iagonal m a tr ix T -1 xvi C H A P T E R 1 INTRODUCTION 1. 1 H ISTORICA L BACKGROUND An im p a c t d a m p e r is a sim p le v ib ra tio n a b s o r b e r which con s is ts of a m a s s allow ed to im p a c t betw een two stops so that e n e rg y is d iss ip a te d by c o n v e rs io n into noise and h eat, in addition to m o m e n tu m being t r a n s f e r r e d fro m the v ibrating s y s te m to the im p a c t ing m a s s and vice v e r s a , leading to atten u atio n of the re s p o n s e of the p r i m a r y m a s s . ( 1)* P a g e t p io n e e re d the study of the im p a c t d a m p e r. H is w o rk ( 2 ) w as followed by L e ib e r and J e n s e n who investigated the m o tio n of a single d e g re e of fre e d o m undam ped o s c illa to r coupled with im p act d a m p e r , by a s s u m in g a sim p le h a rm o n ic m otion, and co m p letely (3) p la stic im p a c ts (i.e . no rebound). Grubin, in h is investigation of this m odel, a s s u m e d a h a rm o n ic m otion c h a r a c te r iz e d by e x i s t ence of two s y m m e tr ic im p acts p er cycle, and he d e te rm in e d the v ehavior of the v isco u sly d am ped p r i m a r y s y s te m , by adding the (4) effects of m any im p a c ts. A rnold in v estig ated e x p e rim e n ta lly and th e o re tic a lly a s im ila r s y ste m , without viscous dam ping, by r e p r e senting the fo rce that a c ts during im p a c t by a F o u r i e r s e r ie s . (5) Sadek used a s im i la r technique to obtain a solution for the steady N u m b e rs in p a re n th e s is desig n ate r e f e r e n c e s a t the end of th e sis . 1 2 state m otion in w hich the two im p acts p e r cycle do not n e c e s s a r ily o ccu r a t equal tim e in te rv a ls (i. e. n o n s y m m e tric motion). His solution indicated that the am plitude vs freq u en cy curve of the im p act d a m p e r would exhibit a jum p phenom enon that is c h a r a c te r is tic of a nonlinear o s c illa to r with a h ard en in g spring. A c o n sid e ra b ly s im p le r m ethod w as suggested by W a r b u r t o n ^ which re q u ire d only (7) the c o n s id e ra tio n of two su c c e s s iv e im p a c ts. M a s r i used W ar- b u rto n 's technique to d e te rm in e the stability of two im pacts per cycle m otion. L a te r he g e n e ra liz e d the technique for m u lti- d e g re e - / g ) o f-fre e d o m s y s te m s by idealizing the im p act between the two m a s s e s as a discontinuous p r o c e s s d u rin g which the d is p la c e m e n ts a r e co n stan t w hile the velocity of only that m a s s which is subjected (9) to im p acts changes ab ruptly. T o k u m a ru and K o tera used a n in te g ra l fo rm u la tio n to study the m otion of an im p a c t-d a m p e r for a c o n c e n tr a te d - m a s s - c o n tin u u m s y s te m w ith infinitely m any d e g r e e s - o f-fre e d o m . R ecen tly M a s r i ^ ^ used the discontinuous p r o c e s s a p p ro ac h d e s c r ib e d in Ref. (8) to obtain the solution for the im pact d am ped E u le r b eam v ib ra tio n p ro b le m w h e re the b eam w as idealized by lum ped p a r a m e t e r s . Although the a fo re m e n tio n e d w o rk s d ealt p r im a r ily with the th e o re tic a l a s p e c ts of the p ro b le m , m any in v e stig a to rs studied the p ro b le m e x p e rim e n ta lly . A m ong th em a r e M c G o ld ric k .^ ^ L ieb er , . (12) (13) (14) . . (15) and Tripp, Sankey, Duckwald, Ibrahim , etc. 1. 2 SC O PE OF THIS RESEA RCH The objective of the p r e s e n t study is to extend and c o m p lem e n t the w ork of o th e r in v e stig a to rs in this field, to a s s e s s the p ra c tic a l feasib ility of the dev ice, and in p a rtic u la r to develop an an a ly tic a l a p p ro a c h based on a s s u m p tio n of two s y m m e tr ic im p acts p e r cycle to be used in a sy s te m a tic w ay for obtaining the solution for any lin e a r im p a c t d am p ed continuous s y ste m . To this end, the th e o r e tic a l solution for m u lti- d e g r e e - o f - f r e e d o m im p act dam ped s y s te m is stated and fully explored in C hapter 2, and its validity and p r a c tic a lity is checked by e le c tro n ic analog c o m p u ter sim u latio n of a m u ltis to r y building in C hapter 3. An exact solution is p re s e n te d for lin e a r im p a c t d am p ed continuous m e c h a n ic a l s y ste m s in C hapter 4, and a m e c h a n ic a l m o d e l is used to verify the n u m e ric a l re s u lts obtained by applying the m ethod of C hapter 4 to the v ib ratio n of an im p a c t dam ped c a n tile v er plate subjected to base excitation in Chap te r 5. C onclusions d ra w n fro m this r e s e a r c h , and re c o m m e n d a tio n s for future w o rk , a r e stated in C hapter 6. C H A PT E R 2 FO R C E D VIBRATION OF A M U L T ID E G R E E SYSTEM WITH AN IM PA C T D A M PER 2. 1 D E SC R IPT IO N OF THE P R O B L E M The lum ped p a r a m e t e r m odel of the n - d e g r e e - o f - f r e e d o m s y s te m being c o n sid e re d is shown in F ig. (2. 1). The sin u so id al fo rce vector is applied to the s y ste m and the im p act d a m p e r is attached to the j th m a s s . In the dev elo p m en t of this an aly tic al solution for stead y state m otion of the s y s te m it is a s s u m e d that during a period of the forcing function, two s y m m e tr ic im pacts o ccu r at equal tim e in terv als and a t opposite ends of the d a m p e r co n tain er. This a s sum ed m otion is c o n s is te n t with that which has been found to p r e dom inate in the e x p e rim e n ta l studies of im pact d a m p e rs as o b serv ed by m o s t in v e s tig a to rs in this field. 2. 2 EQUATION OF MOTION F ig . (2. lc) shows the fre e body d ia g ra m for a single m a s s of the s y ste m . D u rin g the tim e period fro m im m ed iately a fte r the j th im p act to im m e d ia te ly p rece ed in g the (j + l)th im pact (i. e. , t.< t < t. ) the m o tio n of the s y s te m in the ab se n c e of im pact d am p - J J+1 er is governed by the following se t of d iffe re n tia l equations obtained by applying N ew ton's law to any single m a s s . 4 - c l x l ^ i(x i-i x i) ^i+*i(x i-n x i) F , s ' i n f lt hnL ---- : ----► U x . (a) In ; I ► x j (C ) ( b ) F ig u re 2. 1 M odel of the Im pact D am ped n - D e g r e e - o f - F r e e d o m System . 6 m l * l + (cl +ci +C2) i l ' C2i 2 + (kl +k2)xr V 2 = F l ainnt m .x ,-c .x . +(c. +c! + c )x.-c x +(k.+k )x.-k. x i i l i - 1 1 1 1+1 1 1+1 1+1 1 1+1 1 1+1 1+1 = F. sin Q t l m x -c x , +(c +c' )x -k x , +k x = F s in Q t n n n m -1 n n m n m - 1 n n n T hese equations can be w ritte n In m a tr ix fo rm as follows [ m ] {x } + [ c ] {x} + [k ] jx} = {F} sin Qt (2.1) w h ere [ m ] Is a diagonal m a trix with positive e le m e n ts, and [k] and [ c ] a r e s y m m e tric . S im ila rly for the m otion of the p a rtic le between two im pacts in the a b se n c e of any e x te rn a l fo rce we can w rite m Z = 0 (2. 2) d w h ere Z denotes the absolute d is p la c e m e n t of the p article . C o n sid er the s te a d y -s ta te m otion of the s y ste m shown in F ig. 1 with the orig in of the tim e axis being shifted to coincide with the tim e im m e d ia te ly a fte r o c c u re n c e of an im pact at The r e s u lt of this shift is to m odify the excitation fo rce, giving in the new tim e scale. ( F ( t ) } = { F ] sin (Q t + a o ) (2.3) w h ere a Q = is a phase angle to be d e te rm in e d fro m the steady state m otion. 2. 3 NORMAL MODES SOLUTION OF UNDAMPED SYSTEM The undam ped n a tu ra l fre q u e n c ie s and m ode shapes a r e found fro m the hom ogeneous equation [ m ] {x} + [k ] {x} = {0} (2.4) Let [x ] = [m ]""5 {u} (2. 5) Then equation 2 .4 becom es [m ] [m ]"fe j-u } + [k] [ m ] " ^ {u} = 0 (2.6) -L P re m u ltip ly in g by [ m ] ^w e obtain [ m ] E [ m ] [ m ] ^ {u } + [ m ] s [ k ] r m ] s {u} = {0} (2.7) or [ I ] { u } + [ k ] { u ] = 1 0 } (2 . 8 ) [k ] is s y m m e tr ic so [k] is also s y m m e tr ic since [ k ] T = ( [ m ] " S [ k ] [ m ] " ^ ) T =j;m3 % k ] T [ m ] ' ^ = [ m ] " ^ = [k] A w ell known th e o re m in m a tr ix a lg e b ra (see P e r i l s , (17) H ildebrand ), s ta te s that a s y m m e tr ic m a tr ix can be diagona- lized by an orthogonal tra n s fo rm a tio n . L et [y ] be such an ortho- T gonal tr a n s f o r m a tio n m a tr ix i. e. , [y ] [y] = £MQ is a diagonal m a tr ix and let the tra n s fo rm a tio n be 8 {a} = [ y] {q } (2.9) Using the tr a n s f o r m a tio n given by Eq. (2. 9) and prem u ltip ly in g T by [y ] , equation (2. 8) becom es [ y ] T [y ] f q } + [ y ] T [k ] [ Y] {q } = {<>} (2 . 10) or {9 } + } = {0} (2. 11) The colum ns of [y ] a r e the e ig e n v e c to rs of the s y s te m d e s c rib e d by Eq. (2. 8). Let [cp] = [ m - f * [ y ] (2.12) Then [ C p]T [m][cpl = [ y ] T [ m ] " ^ [ m ] [ m 3 " i [ y ] = [ y ] T [Y]=rM3 (2. 13) and l > ] T [k ] [cp] = [ y ] T [ m ] ^ [ k ] [ m ] ^ [ y ] = [ Yl T [ k ] [ y ] = [K ] (2. 14) T h e re fo re colum ns of [cp] a r e eig e n v e c to rs of s y s te m d e s c rib e d by equation (2.4) and they can be obtained fro m e ig e n v e c to rs of [k ] by Eq. (2. 12). E igenvalues of two s y s te m s a r e the sam e. T h e re fo re the tra n s fo rm a tio n {x} = [ m ] “^[u} = [ m ] S [ Y] £ q }=[cp] { 9 } would uncouple undam ped m e c h a n ic a l s y s te m re p r e s e n te d by eq u a tion (2. 4). 2. 4 F O R C E D VIBRATION OF THE D A M PED SYSTEM Including now the dam ping m a tr ix and fo rce v e c to r, Eq. (2. 4) takes the fo rm of Eq. (2. 2) [ m ] [x } + [ c ] [x ] + [k ] [x ] = { F } s in Q t (2.2) As before let { x } = [cp] { q } T A fter p rem u ltip ly in g by [cp] Eq. (2. 2) b eco m es [cp ]T [ m ] [ cp] { q } + [c p ] T [ c ] [ c p ] { 4 } + [cp]T [ k ] [ c p ] { q } = [cp]T { F } or {q} + [cp]T [ c ] [cp] { 4 } + fK ]{ q ] = {Q} (2 .1 5 ) (18) R ayleigh showed that if the dam ping m a trix is a lin e a r com bination of the m a s s m a tr ix and stiffness m a trix , the dam ped s y s te m w ill p o s s e s s c l a s s ic a l n o rm a l m o d e s. The s a m e t r a n s f o r m a tio n that led to uncoupled equations in the undam ped case would a lso lead to uncoupled equations in the dam ped case. Let [ c ] = a [ m ] + (3 [k ] (2. 16) Then [c p ]T [ c ] [ c p ] = [ c p ] T (a [ m ] + (3 [k ]) [cp] = a [ c p ] T [ m ] [ c p ] + B [cp]T [ k ] [ c p ] = a f M ] + p £K] = [C ] (2.17) A ssu m in g dam ping m a tr ix is of the fo rm d e s c rib e d by Eq. (2. 16) and using Eq. (2. 17), Eq. (2. 15) would be uncoupled and the i th equation b eco m es M .q .+ C .q . + K.q. = Q.(t) = f . sin (Q t + a ) i 1 11 11 1 1 o 10 (2. 18) (2. 19) or in te r m of p e rc e n ta g e of c r itic a l dam ping ^i C c r . C. l (2 . 2 0 ) i 2 J K. M. v i i w h ere —— is i th n a tu ra l freq u en cy of the undam ped s y s- i tern. R ay leig h 's r e s u lt is very useful since it a tta c h e s a p h y sical d e s c rip tio n to the dam ping m a trix . If < x = 0 the dam ping m a tr ix is p ro p o rtio n al to the stiffness m a tr ix and the dam ping m e c h a n is m can be r e p r e s e n te d as in te r-flo o r dashpot, i .e . , re la tiv e dam ping. F r o m Eq. (2. 20) it follows that the p e rc e n ta g e of c r itic a l dam ping in this c a se is p ro p o rtio n al to the n a tu ra l frequency of the s y ste m . If p = 0 the dam ping m a trix is p ro p o rtio n a l to the m a s s m a tr ix and the dam ping m e c h a n is m can be r e p r e s e n te d as dashpots connecting the m a s s e s to the base of the s tr u c tu re , i.e . , absolute dam ping. In this c a se it follows fro m Eq. (2.20) that the p e rc e n ta g e of c r i tical dam ping is in v e rsely p ro p o rtio n a l to the n a tu ra l frequency. As shown above, the r e q u ir e m e n t that the dam ping m a tr ix be a lin e a r com bination of the stiffness m a tr ix and m a s s m a tr ix will lead to uncoupled equation of the type e x p re s s e d in Eq. (2. 18). 11 F u r t h e r m o r e , the sa m e tr a n s f o r m a tio n that uncouples the s y s te m w ithout dam ping w ill a lso uncouple the d am ped s y ste m , i. e. , the m ode shapes a r e the s a m e in the two c a s e s . While R ay leig h 's a ss u m p tio n is a sufficient condition, it is (19) not a n e c e s s a r y condition. This w as pointed out by Caughy who went on to show that the n e c e s s a r y and sufficient condition for the ex isten ce of c l a s s ic a l n o rm a l m odes is that the s a m e t r a n s f o r m a tion that d iagonalizes the dam ping m a tr ix a lso uncouples the un dam ped s y ste m . In an undam ped lin e a r s y ste m the m a s s e s pass through th e ir m a x im u m and m in im u m positions a t the s a m e instant of tim e. If c la s s ic a l n o rm a l m odes a r e to ex ist in a dam ped s y s te m the m ode shapes m u s t be the s a m e as for the undam ped s y ste m and the m a s s e s m u s t p ass through th e ir m a x im u m and m in im u m positions at the s a m e instant of tim e. L a te r, C aughy^*^ a lso showed that for lin e a r s y s te m s with s y m m e tr ic m a s s and stiffn ess m a tr ic e s and for d istin c t eigenvalues a n e c e s s a r y and sufficient condition for the ex isten ce of c la s s ic a l n o rm a l m odes could be e x p re s s e d in the following form : -L _ i M - l ^ [ m ] 2r C ] [ m ] D a ([m ] ^ k l f m ] s ) * (2.21) i = 0 “ In this equation n is the n u m b er of d e g r e e s of fre e d o m of the s y s tem . Eq. (2.21) e x p r e s s e s a m uch w id er c la s s than R ay leig h 's a ssu m p tio n . It can be s e e n that Eq. (2.21) leads to R ay leig h 's 12 a ss u m p tio n by letting a = a, a , = q and a_ = a„ = . . . a , = 0. o 1 2 3 m - 1 2. 5 SOLUTION OF THE P R O B L E M Eq. (2. 18) can be w ritte n as ” 2 i qL + 2 Ci ^i q i + id i q L - sin (Qt + a Q) (2. 22) i whe re C. n id . = 1 I — — , r. = ---------------- and f. = 7/ C D • F M U ---------- 1 t-j ‘V A The g e n e ra l solution of Eq. (2. 22) can be obtained by com bining the g e n e ra l solution of h o m o g en eo u s equation and the p a rtic u la r so lu tion of the n o n-hom ogeneous equation as follows: "£ iU Jit q.(t) = e (G cosn-uu-t + C sinr|.uu.t)+A.sin(nt+T.) (2.23) X 1 1 i ^ 1 1 1 1 w h e re f /K 1 1 Q ’» , - V r 7 ^ T • v y (1. r^ r ; 7 L 1 t- = a - Y • a n d Y - = tan i 2 C-r - . _ u,o x . — X . _ — 2 1 - r . i C and C a r e co n stan ts of in teg ratio n w hich could be d e te rm in e d X hi fro m initial condition. U sing so m e specified initial condition Eq. (2.23) becom es Ci 13 T U n . cos — Qt) sin T. i a . . n. i i - — r . ( s i n — Qt)cosT.} + A . s i n ( Q t + T.) , Ti. i r . i i i 11 i i = 1 , 2 , . . . n (2. 24) w h ere q 0 = q ^ 0) and qa = q (°+) i i and the + s u b s c r ip t indicates conditions im m e d ia te ly a fte r the specified tim e. D ifferen tiatio n of Eq. (2.24) with r e s p e c t to tim e yields Qt i 'Hi *H i q i (t) = 6 1 C ~ Ot + Ti-cos — Qt) qa u i i i uii -H i ^ -(— s i n —-Qt)q + (A (jj sin — Qt) sin T n. r . o . 11 r . l ' i l l i A.uj.r. n- ^ • 1 1 1 1 1 + ------------ (r sin — Qt - rl. cos — Qt) cos t. 1 n . i r. l r . l J 11 i i + A sin (Q t+ T .) , i = 1, 2, . . . , n (2. 25) L e t the initial d is p la c e m e n t and velocity a t t = 0^ be {x(°)} = {xo } {x(0+)} = {xa } (2.26) then {q(0)} = { q Q] = [cp ]'1 { * , } , (q<0+>) = { i a } = [ * ] ' * { i j (2. 27) S y stem s (2. 24) and (2. 25) can be w ritte n co n cisely as 14 {q(t)] = [ E ( t ) ] | [ B 2 (t)] {qa } + [ B ^ t ) ] { q j + [ B ^ t ) ^ } + [ B ^ t) ] { S 2 }| + {S3 (t)} (2. 28) and t4(t)} = [ E ( t n [ i a ] + [ B 13(t)] {qo } + [ B 14(t)] [ S ^ + [ B 15<t>] ( s 2 } ) + £S4 <«} (2-29) w h e re the m a tr ic e s a r e a s defined in the appendix. M ultiplying both sid es of equations (2. 28) and (2. 29) by [ C P] and m aking use of Eq. (2. 27) yields (x(t)} = [ B 21(t)] { x j + [ B 22(t)] {XoJ + [ B 2 3 (t(] {S l} + [ B 24( t n [S2 } + [®] tS 3 (*)} (2.30) (x (t)} = [ B 31(t)] { ia ) + [ B 32(t)] {xo } + [ B 33(t)] {S3 }+ [ B 34(t)] {S2 } + [ , ] [S4(t>] (2.31) 2. 6 ANALYSIS OF DISCONTINUOUS PO R TIO N OF THE MOTION D uring an im p act (idealized to be a discontinuous p ro c e s s ) the condition of the s y s te m r e m a in unchanged except fo r the v elo c itie s of im pacting bodies m , and m .. T h e re fo re the velocity vec- d j to r s before and a fte r im p act can be re la te d by C*b ] = TB63 { x j (2.32) w h e re is a co n stan t diagonal m a tr ix w hose all e le m e n ts a r e equal to unity, except the j th e le m e n t w hich is denoted by G, and can be calcu lated using the m o m e n tu m equation and the coefficient of re stitu tio n . Using th e se two definitions, velocities before and a fte r im p a c t of two im pacting bodies can be re la te d by V • * • x. m . + z m , = x. , m . + z . m , (2.33) J- J - d J+ J + d • • x - X " Z X e = - —^ ^ (2. 34) x . - z 3- Solving equations (2. 33) and (2. 34) for x. and x leads to J- J+ w h ere * j + = N z - + h 1 0 z + h = h = 7 1 + e 8 1 + e h = .g i 1 + >4 h = 9 1 + e 10 1 + e (2. 35) (2.36) and m . J In teg ra tio n of equation (2. 2) yields that Z is constant betw een two im p a c ts, th e re fo re in steady state m otion Z + = - Z (2. 37) L et y(t) be the re la tiv e d is p la c e m e n t of p a rtic le with r e s p e c t to j th m a s s which the im p a c t d a m p e r is attach ed to. Then y (t) = Z(t) - x ^ (t) Since the tim e o rig in conincides with the tim e of o c c u re n c e of an im p act, th e re fo re 16 y(0) = y = Z(0) - x (0) = + f (2.38) O J L * w h e re d is d a m p e r c le a ra n c e . L et T = be the period of v ib ratio n in stead y state m otion then ^ \ _ z (°) 2fi , , x Z ' 0 J ~ ~ ^ F 7 T = ~ — + Y J (2.39) + 1 / 4 tt OJ O Using (2. 39) and (2. 37) in (2. 35) we obtain x .(0 ) = x = -2 G (x . + y ) J - bj 2 oj o x .(0 ) = x . = -2 G (x . + y ) J + aj 1 oj o (2.40) w h e re G i = “ < h i n - h Q} * G? = “ < W < 2 - 41 > 1 tt 10 9 2 rr 8 7 E quations (2. 40) lead to G 2 x = —— x . bj G x aj or G G = G i T h e re fo re , m a tr ix ££5^ is co m p lete ly defined. In steady state m otion with two s y m m e tr ic im p acts p e r cycle of ex citatio n on opposite ends of co n tain er we have [ X < t ) } l o t = Tt = ' ( x ( 0 ) } = ■ [ x o 5 ( 2 ' 4 2 ) {x(t)5 Qt = t t ' -£i<0_) = -C ^b T = - CB6 3C*a T <2-43) By evaluating (x(t)} and {x(t)j a t tim e Qt=TT fro m equations (2. 30) and (2. 31) and m aking substitution defined by equations (2. 42) and (2. 43) and noting that ts 3<s>} - -ts ,} TT, CS4 ( ^ } = - n t s 2 } we obtain the following equations -(tri+C b 22(S )])[x o } = [ B 21§ ] { i a } H [ B 23(a)3- [ r f )|;s1] + r B 2 4§ ] { S 2 ] (2 . 4 4) - ( [ B6 ] + [ B 31§ ] [ i a } = [ B 32(3 )][x o 3 4 [B 33(n)] [ S l3 + ([ B34(S)] - n [cp]) Cs 2 } <2 - 4 5 > E quations (2.44) and (2.45) can be e x p r e s s e d as ^ = ^ + ^ ^ + ^ 2 8 ^ Cs 2 } < 2 ‘ 46> = ^ ^ + CB 37^ ^ + CB38] CS2 } (2 >47) E lim in atin g [x } fro m Eq. (2.46) using Eq. (2.47) yields c L (x 0 } = [ B44 ] {s i} + [ B 4 5 ] {S2 } (2.48) Substituting {XQ} *n Eq. (2.47) we obtain t x a 5 = £B46] { S ^ + C B ^ ] {S2 } (2.49) In equations (2.48) and (2.49), {S ] and {S } a r e functions of sin 1 u O _ o and cos q, , th e re fo re they can be e x p r e s s e d as Cs ! } = {S5 1 s i n cc0 - {S6 } cos a Q (2.50) {S2 } = {S5 } cos ct0 + {S6 } sin a Q Substituting th e se values for {S ] and {S ] back in equations (2.48) j L m and (2. 49) r e s u lt s in 18 C XQ} = C s 7 } sin ao + [Sg] cos a Q (2.51) = ( s 9 ] s i n % + ( s 10} cos a Q < 2 - 52) The j th equations of sy ste m s (2.51) and (2.52) are x . = S sin a + S cos a (2.53) oj 7j o 8j o *aj = S9j sin a o + S10j cos a o (2’ 54) But from Eq. (2. 40) we know that x . = - 2 G (x . + y ) (2. 55) aj 1 oj / o Eliminating x and x ^ from equations (2.53), (2.54), and (2.55) leads to h J sin a Q + h2 cos a Q = h 3 (2.56) which has the following solution ' 2 . . . . . *3 a = tan ' - 2 — (2.57) h 2h 3 ; N V h? + h 2 ' h 3 - i h i h 3 - h z ' ) l h i + h 2 ' h : 1 " i i i ■ ■ I ■ m — . ... ■ - w h ere h 's a r e functions of S ^ - S ^ . With a d e te rm in e d fro m Eq. (2.57), the r e s t of the unknowns can be found by back substitution. C H A P T E R 3 E X P E R IM E N T A L STUDIES 3. 1 INTRODUCTION The e x p e rim e n ta l stu d ie s w e re p e rfo rm e d to fulfill p r im a r ily the following objectives. a) To v erify the validity of the a n aly tic al m ethod d e s c rib e d in C hapter 2. b) To e s ta b lis h the ran g e of a g re e m e n t between the analy tic al and e x p e rim e n ta l r e s u lt s . c) To investigate fe a sib ility of using of im p act v ib ratio n d a m p e r s to s u p p re s s e a rth q u ak e-in d u c ed or w ind-induced o s c il lations of ta ll buildings. d) Investigation of the p o ssib le design p ro b le m s that m a y be en co u n tered in the a c tu a l co n stru c tio n of such a sy ste m . e) E ffect of viscous and s tr u c t u r a l dam ping on the effectiv e n e s s of im p act d a m p e r a s applied to the tall buildings. In o r d e r to obtain so m e in fo rm atio n re le v a n t to th e se m a tte r s , the dynam ic c h a r a c t e r i s t i c s of a r e a l building which w e re obtained (22 ) through e x p e rim e n ta l stu d ies, by using n a tu ra l m o d es and fre - (23) q u en cies of the building, w e re used to c o n s tru c t a d y n a m ic a l m o d e l for the building. B a se excitation w as applied to this m o d e l while an im p a c t d a m p e r w as attach ed to the top floor and the follow- 19 20 ing e x p e rim e n ts and studies w e re conducted as d e s c rib e d below: 1- e x p e rim e n ts witb an e le c tr ic analog 2- n u m e ric a l studies involving a d ig ital c o m p u te r. 3. 2 DYNAMIC CH ARACTERISTICS OF THE M O D E L A sev en s to ry building w as m o d eled by a sev en d e g r e e -o f - fre e d o m m e c h a n ic a l s y s te m co n sistin g of lum ped m a s s e s , sp rin g s and dash p o ts. F ig . 3. 1 shows the lum ped p a r a m e t e r m e c h a n ic a l m o d e l w hose n a tu ra l fre q u e n c ie s and m ode sh ap es ap p ro x im a te those obtained e x p e rim e n ta lly fro m full sc a le d y n am ic te s ts of a m o d e rn building. The n a tu ra l fre q u e n c ie s and m ode shapes of the undam ped s y s te m a r e given in Table 3. 1 and Table 3 .2 . M odel N um ber N a tu ra l F re q u e n c e Rad. per sec. Cyc. per sec. 1 15. 9 2. 5 2 41. 6 6. 6 3 63. 5 10. 1 4 83. 3 13. 3 5 104. 0 16.6 6 122. 8 19. 5 7 143. 1 22. 8 Table 3. 1 - N a tu ra l F r e q u e n c ie s of the U ndam ped 7 -D e g re e -o f- F r e e d o m S ystem . 21 k L (i^ L E L .Io "4 ) m L ( K i j a . s g c 2) ZA5 5.0 5.0 5.0 5.0 5.0 6.0 6.5 i = @ ' ki F ig u re 3. 1 M ech an ical M odel of the 7 Story Building 22 F lo o r No. Mode N um ber 1 2 3 4 5 6 7 1 . 10 -. 33 . 56 - 1. 11 1. 41 -1 1 .5 473. 0 2 .22 - .6 3 . 89 -1. 30 . 86 0. 07 372. 0 3 . 42 -. 86 .49 . 77 -2. 53 10. 4 215.0 4 . 58 -. 79 - .3 3 1. 75 - . 34 - 20. 8 - 96. 0 5 . 77 - .3 1 -1. 05 - .29 3. 68 11. 8 28. 0 6 . 89 . 31 - .6 1 -1. 75 -3. 24 -5. 04 - 7. 15 7 1.0 1. 0 1. 0 1.0 1.0 1. 0 1. 0 Table 3 .2 - Mode Shapes of the U ndam ped System . A bsolute d am p in g w as a s s u m e d to be z e ro (< 3, = 0) and two d iffere n t d am p ed s y s te m s w e re c o n s id e re d with r e g a r d to re la tiv e dam ping: a) Lightly d am p ed s tr u c t u r e (£ = 0.01 leading to (3 = 0.001255) b) Highly d a m p e d s tr u c t u r e (£ = 0.05 leading to p = 0.006275) 3. 3 STEADY S T A T E MOTION OF BASE EX CITED D A M PED SYSTEM E quations (2. 1) w e re given in c h ap ter 2 as the governing eq u a tions of m otion for the fo rced v ibration of a dam ped m u l ti- d e g re e - o f-fre e d o m s y s te m . If the s y s te m is excited through s u p p o rt m o tion r a t h e r than d ire c tly applied load, then letting S(t) = So s in Q t be the rigid body tr a n s la tio n of the b ase and x^(t) be the d is p la c e m e n t of the ith floor m e a s u r e d re la tiv e to rigid body m otion, the g o v e rn ing equation of m o tio n (2. 1) r e m a in s the s a m e except that the applied load is re p la c e d by the in e rtia load 23 2 [ F ] sin Q t = {m} S (t) = - Q Sq [ m ] sin Q t (3. 1) w h ere f m ] = col. fm , m . . . m 1 (3.2) c - 1 L 1 2 n J T h e re fo re , we can conclude that in the case of base excitation F . = Q2 S m. 1 o 1 F ig u r e s 3. 2 through 3. 5 show the am plitude re s p o n s e vs frequency c u rv e s in the f i r s t and second m o d e s for the base excitation of the s y s te m shown in F ig . (3. 1), for two d iffere n t dam ping coefficients. 3. 4 E X P E R IM E N T S WITH AN E L E C T R IC ANALOG An e x p e rim e n ta l study w as needed to co m p le m e n t the an a ly sis in those p a r ts w hich a n a ly tic a l ap p ro a c h w ill not produce sufficient in fo rm atio n as d e s c r ib e d below: a) R e a liz a tio n of stab le solution. b) D e te rm in a tio n of the ra n g e of validity for the a s s u m p tio n of two s y m m e tr ic im p acts p er cycle. F o r this p u rp o se , the e le c tric analog w as chosen a s an e x p e r im e n tal setup b ecau se of its following d e s ira b le p ro p e rtie s : a) e a s e of v arying the p a r a m e t e r s of the m o d e l ov er a wide ra n g e . b) sim p lic ity of m o n ito rin g , m e a su rin g , and re c o rd in g the v a ria b le s of the s y s te m . To ac c o m o d a te the lim ita tio n im posed by using r e a l com ponents in 24 90. 0 90. 0 60. 0 60. 0 FLOOR 7 THV 5 THA 3 RdA 1 st\ 30. 0 30. 0 1. 08 0. 96 1. 00 0. 92 f t F ig u re 3 .2 F re q u e n c y R esp o n se of the B ase Excited Building in the F i r s t M ode. (No Im pact D a m p e r, = 0.01). 25 15. 0 x : I FLOOR 7 THn 5 TH-s, 3 RDn 1 ST s s o 1. 00 1. 06 0. 88 f t F ig u r e 3. 3 F re q u e n c y R esp o n se of the B ase E xcited Building in the Second M ode. (No Im pact D a m p e r, £ = 0.01) 26 14. 0 14. 0 X L FLOOR 7 THX 5 TH\ 3 R D \ S„ 1. 05 1. 09 1. 00 0. 95 f t . F ig u re 3 .4 F r e q u e n c y R esp o n se of the B ase E xcited Building in the F i r s t Mode. (No Im pact D a m p e r, = 0. 05). col X 27 3. 1. 0. 1. 05 0. 35 1. 00 0. 83 t t F ig u r e 3.5 F re q u e n c y R esp o n se of the B ase Excited Building in the Second M ode. (No Im pact D a m p e r, £ = 0. 05) 28 (24) a n e le c tro n ic analog com puter, the following m odifications had to be m ade in the e le c tr ic a l m odel: a) The infinitely rigid m a th e m a tic a l stops w e re re p la c e d by v e ry stiff s p rin g s , a s c o m p a re d to other in te r-flo o r sp rin g s. b) The sim plified concept of coefficient of restitu tio n , i.e . , re la tin g the discontinuous values of the velocities im m e d iately p re c e e d in g and succeeding an im pact, w as re p la c e d by a continuous p r o c e s s with the s a m e end r e s u lts . Ju stific a tio n for th e se two step s a r e w ell docum ented. F ig . 3. 6 shows a s c h e m a tic d ia g ra m of the m odified im pact d a m p e r m e c h a n ism , attach ed to rm and the functions re p re s e n tin g its p ie c e -w is e lin e a r behavior. By a p r o p e r choice of k^, the p ie c e -w is e lin ear sp rin g s can sim u late a rigid b a r r i e r to any d e s ir e d d e g re e of a c c u ra c y . c_ d in conjunction w ith h(y, y) p ro v id es m e a n s for sim u latin g inelastic im p a c ts , ranging fro m the co m p lete ly pla stic up to the ela stic ones. The c irc u its that w e re used to g e n e ra te the nonlinear functions g(y) and h(y, y) a r e shown in F ig . 3. 7 and 3. 8. In the p r e s e n t study, the im p act d a m p e r w as attached to the ( 18 ) top floor exclu siv ely , b eca u se it h as been shown that due to te le sc o p ic effect, the top floor is the m o s t effective location for the im p a c t d a m p e r. But other s y s te m p a r a m e t e r s w e re v arie d to study 29 b " 7 " H y=Z-x (a) (b) (c) F ig u r e 3. 6 M odified M odel of the Im pact D a m p e r and its F u n ctio n s. 150K IN914A g(y) 120K 50K 100K 50K IN914A IN914A g(y) 5 OK IN914A IN914A 100K ■VWSr 100K 50K 900K i r IN914A +E 150K 00 o F ig u re 3. 7 C irc u it for G enerating g(y) 31 IN914A 1M I 1M U a a a a - 1 1M O — — W W -1 W 1M p V W W IN914A 0 .5 M -M- /W W — - W W 50K IN628 50K x o 50K + V 0— vw v IN628 100K -V AAAA— 100K 100K 100K IN914A +V=2. 7 IN 914A 100K 100K - V=2. 7 100K 100K x. A F ig u re 3. 8 C irc u it for G enerating h(y, y) 32 the effect of each one s e p a ra te ly and in com bination with o th e rs . The a c tu a l analog c o m p u ter setup is shown in Fig. 3. 9. U sing the concept of m odified im pact d a m p e r d e s c rib e d before, and its nonlinear functions, the governing d iffe re n tia l equations for the m otion of the base excited im p a c t dam ped seven d e g r e e s - o f - fre e d o m s y s te m becom e m l * l + (° l +C2 ) x l ' C2x 2 + < V k2> * 1- V 2 = "“ I S m 2x 2 - c2* 1 +(c2+C3) V C3 V k2Xl +(k2+k3)x2‘ k3X3= ' m 2® m 3i ’3 - C3*2+(c3+04)5‘3 ' C4;; - k3x2+( V k4)x3-k4X4 = ‘ m 3 ® • • • * • • • m .x -c .x., + (c + c _ )x - c _ x - k .x„+(k .+k_ ) x -k_x_. = - m S 4 4 4 3 4 5 4 5 4 3 4 5 4 5 5 4 • * • ■ * * • m _x _-c_ x . + (c_+c , ) x _ - c .x - k _ x .+(k_+k. )x - k . x . = - m _ S (3.4) 5 5 5 4 5 6 5 6 54 5 6 5 6 6 5 m 6x 6 " C6X5 + (c6+c7)x6 ' c 7x 7 -k 6x 5 +(k6+k7)x6 - k 7x 7 = - m 6® m ?x 7- c 7x 6+ c7x 7- k 7x 6+k7x 7- c d h(y, y) - kd g(y) = - m ? S m d Z + cd h (y , y) + kd g(y) = " ^ S w h ere x. is the re la tiv e d is p la c e m e n t of the i th m a s s with r e s p e c t to base, Z is the re la tiv e d is p la c e m e n t of p a rtic le w ith r e s p e c t to base and y is the re la tiv e d is p la c e m e n t of p a rtic le with r e s p e c t to top f lo o r . E quations (3.4) can be w ritte n in the following fo rm w hich is m o r e s u ita b le fo r a n a lo g im plem entation. * 1 = - D ! - M , V ! + M 1 D2 - S x 2 = -D2 - M2 V2 + M 2 D3 + VX - S 00 F ig u re 3.9 Photograph of E le c tro n ic Analog C o m p u ter Setup. \ • ( , v v \ MM,) S -------- V < M4 > [ -(M4) • s l > -X, 2, -X. ■•to?) •pc3 .,» Q • « » 4 > ^ •pc4 .4» > - ^ - ( D > — *— f • ( 2 C 4 i o 4 > | • (« 5 ) • < 2 C 5 « u 5 ) > -4— ( e 0 j---------- \ ' ' “ s1 \ •|M5) s ------- © - 0 0 - ! -‘M D N '<M 6> ) -S ----- r i > -x . -XS •<4> ■«:***» •<4> f •<2V6> I % I •<4> f ' U :616l V •<4> ■uv ? i © > ^ - 0 © H 0 — 0 — -s — g(y> F ig u re 3. 10 Analog C om puter D ia g ra m for Equations (3.5) U J The co m p u ter d ia g ra m re p re s e n tin g Eq. (3.5) is given in F ig . 3. 10. C a se s under study co n siste d of two d iffere n t m a s s ra tio s (|j = 0. 01, jj, = 0. 05) and two d iffere n t coefficients of re stitu tio n (e = 0 .2 5 , e = 0. 75) in the f i r s t two m odes for a total of eight c a s e s for each of the two d iffere n tly dam ped s y s te m s . 3. 5 D IG ITA L C O M PU T E R STUDIES B ased on fo rm u la tio n p re s e n te d in C hapter 2, a d igital c o m p u te r p r o g r a m w as developed to calculate the value of a, and other unknowns of the p ro b lem . The value of a, can be d e te rm in e d fro m Eq. (2.57) re s ta te d h e re . -1 , h lh3 ± h2 V * j + ^ - h ‘i R 2 ' 3 6 a = ta n ” * ( —-— ---------- - -■ ■ ------ ) (2.57) h 2h 3 * hl R R ' hl This equation and consequently the im pact dam ped p ro b le m h as a solution if A = h i + h 2 " h 3 ^ 0 (3‘ 6) Since a is a function of d, (-1, e, Q and other c h a r a c te r i s t ic s of the m e c h a n ic a l s y s te m , th e re fo re , for so m e com bination of th e se p a r a m e t e r s we have th re e d iffere n t c a s e s . 1) A > 0 In this c a se llh3 ± R 2 ± u 2 x 2 + h2 ' h 3 h h + V h i + 2 3 ' 1 2 3 h a s two r e a l solutions y^ and y^. C o rresp o n d in g to any of these so lutions, a Q has two a n s w e rs b e tw een o and 2yr. Let a^j» cc2 j> an^ ^ 1 2 ’ a 22 be tbe so ^ution 0:E equations tan ccQ = yj and tan a = y ^ re sp e c tiv e ly , then as can e a s ily be shown a 21 = a l l + n (3.7) Solution of equations (2.51) and (2.52) c o rre sp o n d in g to th e se two values of n a r e o { * > ! ! » = t s 7 } s i n a u + l s 8 ) 0 0 3 a n t * a <an >} = Cs 9 ) l i n o u + f s 1 0 3 003 a u 37 and or t x o ( a 2i>} = - t s ?5 ! i “ « n - f s s ’ c o s a n = ' t s 9^ S i n a ll ' t S 10> COSaU t x o (al l , 5 = ' t x o (a2 1 ) 5 t xa^a l l ^ = - t* a <a2 1 )} T h e re fo re , solutions c o rre sp o n d in g to & and a , , a r e not d istin c t L i 1 X X and they a r e d iffe re n t b ecau se they a r e re la te d to opposite ends of c o n ta in e r. The sam e a r g u m e n t is also valid for and & 2 Z ' H ence we can conclude that th e re e x ists two d istin c t solutions c o rre s p o n d in g to two d is tin c t values of y^ and y^. 2) A = 0 In this c a se + h 2 - h 3 ’ ° or h l + *2 - G l ^ = ° w hich can be used to calcu la te the th e o re tic a l m a x im u m c le a ra n c e value. 3) A < 0 In this c a s e which c o rre s p o n d s to d > d no solution c o r- r m ax 38 responding to a s s u m e d conditions (steady state with two s y m m e tr ic im pacts per cycle) ex ists. The digital co m p u ter p r o g r a m c a lcu la tes the value of h and h based on s y ste m c h a r a c te r i s t ic s and then X L i using the value of h^ obtained for d iffere n t c le a ra n c e such that d < d m ax, d e te rm in e s the two d istin ct values of n denoted by a o o and a • o Each one of these values for a, , would be used in subsequent o calculation of other unknowns. T h e re fo re , for any se t of p a r a m e te r s two d iffere n t b ran ch e s of solutions can be obtained by m e a n s of digital p ro g ra m . The p r o g r a m is w ritte n ex clu siv ely in FO R TR A N IV language, and w as executed by m e an s of an IBM 360 c o m p u ter in the U n iv ersity C om puter C enter of the U n iv ersity of Southern C alifornia. 3. 6 COMPARISON OF A N A LY TICA L AND E X P E R IM E N T A L RESULTS It w as shown prev io u sly that for the case of [ \ > 0, the a n a lytical m ethod p re se n te d in C hapter 2 would produce two d istin ct solutions c o rre s p o n d in g to two d ifferen t values of phase angle. But only one of these solutions is stable. Stability an a ly sis has been used su ccessfu lly to d e te rm in e the stable range of the solution for the re sp o n se of a single d e g re e of fre e d o m im p act d am ped (25) s y ste m . But for the m u lti- d e g re e - o f - f r e e d o m s y s te m s , the a n a ly sis is tedious and m o re com plicated. The other a lte rn a tiv e x7/sf t x7/s, 39 D/XP-, M = 0. 01 e =0. 25 60. 0 30. 0 01+ ------ - 01- ................. E X P . ° 0 0 360. 0 270. 0 180. 0 D/Sn 90. 0 1. 0 0. 5 a. X D/XP? A=0. 01 e =0. 75 60. 0 30. 0 0!+ ------- - 0|- ......... E X P. 0 0 0 720. 0 510. 0 360. 0 D/S_ 180. 0 F ig u r e 3. 11 E ffect of Im p act D am ping on the A m plitude R e- ponse of the Top F lo o r in the F i r s t Mode for Lightly D am ped S tru c tu re . (Q = (* ) ,£ =. 01, H = . 01) 40 D / X P - , 60. 0 (X+ - CX- •• EXP. o 165. 0 220. 0 110. 0 o/sn 55. 0 1. 0 0. 5 0. 0 c L ~ x d/ xp7 A=0. 05 e =0.75 60. 0 Oi-f ------ - (X- ......... EXP.® o • 195. 0 260. 0 130. 0 D/S0 65. 0 F ig u re 3. 12 E ffect of Im p ac t D am ping on the A m plitude R e sponse of the Top F lo o r in the F i r s t Mode for Lightly D am ped S tru c tu re . (Q = 0J, » Ci=0- 01> H = 0.05) 41 D/XP, o co N . N - X G. § A=0. 01. e =0.25 C X + - CX- • EXP.* 0 0 68 . 0 17. 0 1. 0 Q _ X D/XP-, 12. co — x~ 6. 0. 16. 0 A=0. 01- e =0. 75 cx+ cx- EXP, 0 200. 0 150. 0 50. 0 1. 0 O L X 0. 5 F ig u re 3. 13 E ffect of Im pact D am ping on the A m plitude R e sponse of the Top F lo o r in the Second Mode for Lightly D am ped S tru c tu re . (Q = & , £ = 0. 01, J J L = 0.01) 42 D/XP-7 o m h- x 6. 0. § A=0. 05 . e =0. 25 01 + EXP, 0 52. 0 33. 0 13. 0 26. 0 0 /S n 1. 0 0. 0 D/XP? § M ~ 0. 05 . e =0.75 0 0 0 84. 0 63. 0 42. 0 D/Sn F ig u r e 3. 14 E ffect of Im pact D am ping on the A m plitude R esp o n se of the Top F lo o r in the Second M ode for Lightly D am ped S tru c tu re . (Q = uu2 , Ci = 0- °1. |i = 0. 05) •dX/lX :d X /!X 43 to stability a n a ly sis is the c o m p a ris o n of a n a ly tic a l r e s u lt s with e x p e rim e n ta l ones to d e te r m in e the stable b ranch of solution. In addition, e x p e rim e n ta l d ata can be used to find the range of steady state m o tio n for two s y m m e tr ic im p acts p e r cycle. F ig s . 3. 11 through 3 .2 8 show the e x p e rim e n ta l and n u m e ric a l r e sults for d iffere n t com binations of p a r a m e t e r s . In a ll c a s e s the a n aly tic al solution is r e p r e s e n t e d by two d iffere n t b ra n c h e s of c u rv e s identified by a^_ and a a n d e x p e rim e n ta l r e s u lts a r e given at d is c r e te points. Two fa m ilie s of curve-s a r e given in this section. 1) R e sp o n se vs. c le a ra n c e c u rv e s for a fixed ex citatio n f r e quency. 2) R esp o n se vs. freq u en cy c u rv e s for a fixed c le a ra n c e . In each of th e se two c a te g o rie s , in so m e re g io n s, d is c r e p a n c ie s can be s e e n betw een the a n a ly tic a l and the e x p e rim e n ta l re s u lts w hich a r e due to the fa ilu re of the o rig in a l a s s u m p tio n of two s y m m e tr ic im p acts p er cycle, a s d is c u s s e d below. F ig s . 3. 11 through 3 .1 8 show the r e s p o n s e vs c le a ra n c e c u rv e for the v a rie ty of p a r a m e t e r s in the f i r s t and second m o d e s. It is shown that the b ran ch of solutions c o rre sp o n d in g to which h a s a low er value than the o th er one, is stable. If d = d , then A = 0 m ax and the two solutions would coincide and we have ju s t one solution that, b ecause of its lim itin g n a tu re , is unstable. In addition, e x p e rim e n ta l r e s u lt s show that for values of d close to d , in m o s t m a x 44 d /x p 7 12. 0 o X . 0 1+ -------- CX- ............ EXP. ° • 0 6 3 . 0 * ♦2 . 0 D/ Sn D/XP? 1 6 . 0 0 A = 0 . 0 1 e = 0 . 7 5 12. 0 0 . C X + ------- ot~ ........ EXP.® ° 0 0 0 210 . 0 2 8 0 . 0 1 4 0 . 0 D/S- 7 0 . 0 F ig u re 3. 15 E ffect of Im p act D am ping on the A m plitude R e sponse of the Top F lo o r in the F i r s t Mode for Highly D am p ed S tru c tu re . (Q = (*),. C = 0 - 05 » M - = 0 . 01 ) •dX/'X ' d X / ' X D/XP? A=0. 05 e =0.25 12. 0 o X . 0 1+ - 0! - • EXP.° 1?. 0 3H. 0 D/S„ 68. 0 51. 0 D/XP-, 12. 0 o X 105. 0 35. 0 70. 0 a. X D/SL F ig u re 3. 16 E ffect of Im p act D am ping on the A m plitude R e sp o n se of the Top F lo o r in the F i r s t Mode for H ighly D am ped S tru c tu re . (Q = uk > Ci = H = 0.05) 46 C O v. N - X D/XP7 § A=0. 01 * = 0 . 25 Oi+ EXP, 0 0 1G. 0 12. 0 D/S, 1. 0 Q_ X 0. 5 (0 N , 0. d/ xp7 16. 0 § A=0. 01 e =0. 75 01+ ot- EXP, 0 60. 0 30. 0 D/Sft 15. 0 1. 0 Q _ X F ig u r e 3 .1 7 E ffect of Im pact D am ping on the A m plitude R e sp onse of the Top F lo o r in the Second Mode for Highly D am ped S tru c tu re . (Q = (jj , £ = 0. 05, n = 0.01) 47 D/XP, o to i x 0. § M - 0 . 05 e =0.25 01 + 01- EXP, 0 16. 0 12. 0 C L X X D/XP-, o co v . 1 ^ - X 1. 0. 16. 0 § M = 0 . 05 e =0. 75 0 HO. 0 30. 0 20. 0 D /Sn 10. 0 1. 0 C L X F ig u re 3. 18 E ffect of Im p act D am ping on the A m plitude R e sponse of the Top F lo o r in the Second Mode for Highly D am ped S tru c tu re , (fi = ^ , £ = 0.05, |J = 0 . 0 5 ) 48 c a s e s , the a s s u m p tio n of two s y m m e tr ic im p acts p er cycle is over o p tim istic b ecau se if we s t a r t the s y s te m fro m z e ro initial condition no im p a c t would o c c u r, o r in so m e c a s e s a fte r a few im p a c ts x . would be re d u c e d and would c au se few er im p acts p er cycle which r e s u lt s in in c re a s in g value for x^. and consequently m o r e im p acts. As this cycle is re p e a te d , beat phenom enon would a p p e a r in re s p o n s e of the s y s te m . H ow ever, r e s p o n s e m o d e s c o rre s p o n d in g to la rg e d can be e s ta b lis h e d e x p e rim e n ta lly if initial conditions other than z e r o a r e im posed (startin g position of m ^ close to w all, initial v elo city for m ,, etc. ). d In the o th e r lim iting c a s e , if d = 0, the h^ = 0 and ag ain we ob tain only one solution fro m the a n a ly tic a l m ethod c o rre sp o n d in g to C X = tan * (- h 2 /h l). P h y s ic a lly this solution c o rre s p o n d s to the r e s p o n s e of the s y s te m w ithout im p act d a m p e r , provided that m is in c re a s e d by m . T h e re fo re , the r e s p o n s e for this c a s e should be independent of coefficient of re s titu tio n . But as can be s een in fig u re s 3. 11, 3. 12, 3. 14 and 3. 16, this is not the case in the a n a ly tical r e s u lt s . The r e a s o n is, as we a p p ro a c h this lim it, one can see that if d is s m a ll c o m p a re d to m agnitude of x ., the condition of two °J s y m m e tr ic im p a c ts p e r cycle cannot be e sta b lish e d . In re a lity , th e re a r e m o r e than two im p a c ts p er cycle and they a r e not n e c e s s a r ily s y m m e tr ic , a s h as been verified by e x p e rim e n ts . The p r e - 49 30. 0 SO. 0 0. 01 UNSTEADY RRNSE EXPERIMENT o • N O D A M PER ----- . 60. 0 30. 0 30. 0 1. 08 1. 00 0. 92 0. 96 f t F ig u r e 3. 19 F re q u e n c y R esp o n se of the Im p ac t D am ped B ase E xcited Building in the F i r s t Mode. (Q = U ), » G. = 0. 01, |i = 0. 01, e = 0. 25) 50 90. 0 90. 0 0. 01 UNSTEADY RRNSE EXPERIMENT o « N O DAM PER 60. 0 60. 0 30. 0 cr. 1. 08 1. 00 0. 32 0. 96 £ L O), F ig u r e 3. 20 F re q u e n c y R esp o n se of the Im p act D am ped B ase E xcited Building in the F i r s t Mode. (Q = U J1 > = 0. 01, = 0. 01, e = 0. 75) 51 30. 0 90. 0 A = 0. 05 e = 0. 25 UNSTEADY RAN GE EXPERIMENT o • N O D A M PER ....... 30. 1. 08 0. 9G 1. 00 1. 01 0. 92 CO F ig u re 3.21 F re q u e n c y R esp o n se of the Im p act D am ped B ase E xcited Building in the F i r s t Mode. (Q = uo, » Ci = 0. 01, |i = 0. 05, e = 0. 25) 52 30. 0 90. 0 A= 0. 05 e = 0. ?5 UNSTEADY RANGE EXPERIMENT o o N O D A M PER ------ . GO. 0 60. 0 30. 0 30. 0 . 1. 08 1. 00 1. O il 0 . 96 0. 92 f t O), F ig u r e 3.22 F re q u e n c y R e sp o n se of the Im p act D am ped Base E xcited Building in the F i r s t Mode. (Q = uk , £ = 0. 01, |i = 0. 05, e = . 75) 53 15. 0 10. 0 5. 0 0. 0 0. 88 I'X I I o • o l “ “ * UNSTEADY RAN GE t EXPERIMENTo • « ’ N O D A M PER ............ M= 0. 05 e = 0. 75 D/S0 = 50. / o'Y \ / • / \ / / \° \ \ ° c / : \ i / 0 '■ \ » ° \ 0 1 - 1 --------------- 1 0. 94 1. 00 J L 1. 06 15. 0 10 . 0 5. 0 0. 0 1. 12 F ig u re 3.23 F re q u e n c y R esp o n se of the Im p act D am ped B ase Excited Building in the Second Mode. (Q = U)_, Q, = 0. 01, |i = 0. 05, e = 0. 75) 54 sen t an a ly sis can be m odified to acco u n t for m o r e than two im p acts p e r cycle for e ith e r s y m m e tr ic or u n s y m m e tric c a s e , as h as been (21 ) done for s in g le - d e g r e e - o f - f r e e d o m . O ther than th e se two r e gions m entioned above, good a g r e e m e n t is found betw een an aly tic a l and e x p e rim e n ta l re s u lts e lse w h e re . F ig s . 3. 19 through 3 .2 8 show the r e s p o n s e vs frequency c u rv e s with and without the d a m p e r for d iffere n t c a s e s . The c l e a r ance ra tio for each c a s e is fixed and is the one which would provide significant attenuation if the s y s te m w as o p erated a t the n a tu ra l fre q u e n c ie s. A gain, it is shown that the b ran ch of solution c o r responding to ct+ is the stable one. Good a g r e e m e n t is also found betw een e x p e rim e n ta l and an aly tic a l r e s u lt s a lm o s t e v e ry w h e re , sp ecifically in the neighborhood of n a tu ra l fre q u e n c ie s . In g e n e ra l, by studying th e se c u rv e s , we can conclude that, that p o rtio n of the stable cu rv e which lies inside of the stead y state r e s p o n s e curve without im pact d a m p e r is a p p ro x im a te ly the re g io n for w hich the a ss u m p tio n of two s y m m e tr ic im p acts p er cycle holds. 3. 7 E F F E C T OF D IF F E R E N T P A R A M E T E R S The effect of d iffere n t p a r a m e t e r s h as been investigated in d e ta il by m any in v e stig a to rs. A lm o s t the s a m e c h a r a c te r i s t ic s and behavior is o b se rv e d in the base ex citatio n of im p a c t d am p ed s e v e n - d e g r e e - o f- f r e e d o m s y ste m . Som e of th e se o b se rv a tio n s a s obtained by the p r e s e n t study a r e s u m m a r iz e d below. 55 UNSTEADY RANGE { EXPERIMENT o o o N O D A M PER ........... e = 0. 25 D/SQ= G O 14. 0 S . o 1. 05 1. 09 1. 00 S I F ig u r e 3 .2 4 F re q u e n c y R esp o n se of the Im pact D am ped B ase E xcited Building in the F i r s t Mode. (Q = cu , £ = 0. 05, jjl = 0 .0 1 , e = 0. 25) 56 UNSTEADY RANGE J A= 0. 01 e = 0. 75 EXPERIMENT « o o N O DRM PER ............ m . 0 14. 0 1. 09 1. 05 0. 95 1. 00 F ig u r e 3. 25 F re q u e n c y R esp o n se of the Im p ac t D am ped B ase Excited Building in the F i r s t Mode. (Q = uk > Ci = 0.0 5 , ^ = 0. 01, e = 0. 75) 57 0 A= 0. 05 e = 0. 25 UNSTEADY RA N G E EXPERIMENT » o o N O DAM PER .......... D/S = 40 14. 0 0 0 0 0 1. 09 1. 05 1. 00 0. 95 J L O), F ig u r e 3 .2 6 F re q u e n c y R esp o n se of the Im pact D am ped B ase Excited Building in the F i r s t Mode. (Q = m u , Q = 0. 05, jj = 0. 05, e = 0. 25) 58 UNSTEADY RANGE f EXPERIMENT o o o NO DAMPER .............. //= 0. 05 e = 0. 75 D/S = 8 0 14.0 m. o 1. 05 1. 09 0. 95 1. 00 C O , F ig u r e 3 .2 7 F re q u e n c y R esp o n se of the Im p act D am ped B ase E xcited Building in the F i r s t Mode. (Q = yj , r = 0 .0 5 , ^ = 0. 05, e = 0. 75) 59 4. 5 ?! = 0 . 05 UNSTEADY RANGE J EXPERIMENT o • . NO DAMPER ............. D/S0= 25. A= 0. 05 e = 0. 75 3. 0 1. 5 . 0. 0 0 . 8 9 -L. 0 . 9 5 1. 00 JGL a_ 'J. 5 3. 0 1. 5 1. 0 5 0. 0 1. 11 F ig u re 3. 28 F re q u e n c y R e sp o n se of the Im p a c t D am ped Base E xcited Building in the Second Mode. (Q = m , £ 0. 05, ^ = 0. 05, e = 0. 75) 60 a) With a given d a m p e r m a s s ra tio , the m a x im u m p e rc e n ta g e red u c tio n in the r e s p o n s e of the s tr u c tu re is achieved for the building with the le a s t a m o u n t of viscous dam ping. b) F o r any d am ped s tr u c t u r e the m a x im u m red u c tio n d e c r e a s e s with m ode n u m b e r. F o r instance, in the case of lightly d am ped s tr u c tu re , for ^ = . 05 and e = 0. 75 the m a x im u m red u c tio n d e c r e a s e s fro m 87 p e rc e n t in the firs t, to 75 p e r c e n t in the second (F ig u re s 3. 12 and 3. 14). One re a s o n for this re d u c tio n in the efficiency of the d a m p e r is that, w hen only re la tiv e viscous dam ping is p re s e n t, £ , the ra tio of c r itic a l dam ping in the i th m ode is p r o p o rtio n a l to i th n a tu ra l freq u en cy dj. (Eq. 2. 20) and higher dam ping p e rc e n ta g e would be re s u lte d for hig h er m odes. T h e re fo re , based on the finding stated in p a r t (a), le s s re d u c tio n is expected, c) E ven with m a s s ra tio on the o r d e r of one p e rc e n t, con sid e ra b le am o u n t of red u c tio n can be achieved. F ig u re s 3.11 and 3. 15 show that in the f ir s t m ode for jj. = 0.01 m a x im u m red u ctio n s of 60 and 25 p e rc e n t is achieved for two s tr u c t u r e s with d iffere n t am o u n t of dam ping. d) With the r e s t of p a r a m e t e r s the s a m e , a h ig h e r coefficient of re s titu tio n r e s u lt s in m o r e reduction. But the d is a d v a n tage of a im p a c t d a m p e r m e c h a n is m of this type is that it 61 is v ery s e n sitiv e to c le a ra n c e v a ria tio n (Figs. 3. 11, 3. 12, 3. 14 and 3.16) and freq u en cy changes. This c h a r a c te r i s t ic is m o r e visible in the c a s e of lightly dam p ed s tr u c t u r e w hich le ad s to v ery lim ited ra n g e of use. (F igs. 3 .2 0 , 3 .2 2 , 3.23 and 3.28). e) Although in som e c a s e s , for one s e t of p a r a m e t e r s , n o tic e able re d u c tio n cannot be obtained, s till by choosing the rig h t co m b in atio n of m a s s r a tio and coefficient of r e s t i t u tion, d e s ir a b le re s u lts can be achieved. F ig . 3 .1 7 show s that for the highly d am ped s tr u c t u r e , with one p e r c e n t m a s s r a tio , the effect of im p act d am ping in the second m ode is z e ro . But with ^ = 0. 05 and e = 0. 75, 50 p e rc e n t red u ctio n , would be the r e s u lt for the o s c il la tion of the s a m e s tr u c t u r e in the s a m e m ode. This r e s u l t is shown in F ig u re 3. 18. C H A P T E R 4 F O R C E D VIBRATION OF TWO DIMENSIONAL IM PA C T D A M PED M ECH AN ICA L SYSTEMS 4. 1 INTRODUCTION The idea of attaching a discontinuous m a s s to an o scillating plate with the a im of re d u c in g the fo rced v ib ra tio n of the plate w as ( 2 ) investigated as far back a s 1943 by L e ib e r and J e n s e n who con ducted a n an aly tic a l and e x p e rim e n ta l study of an a irp la n e c o n tro l s u rfa c e which w as provided with discontinuous p a r tic le s that w e re fre e to o sc illa te with a given c le a ra n c e re la tiv e to the wing. L u n d g ren et a l m a d e a n a p p ro x im a te a n a ly sis of the effect of an im p a c t d a m p e r attach e d to the c e n te r of a sq u a re plate. They concluded on the basis of th e ir lim ited a n a ly sis that the device is not a v e ry effective d a m p e r. This se c tio n p r e s e n ts an "ex act" solution for the stead y - state m o tio n of a visco u sly d am ped plate of a r b i t r a r y shape and with a r b i t r a r y boundary conditions, that is subjected to sinusoidally varying d is trib u te d a s w ell a s c o n ce n trate d loads, and provided at so m e point w ithin its d o m a in with a discontinuous m a s s that is fre e to o scillate fric tio n le ss ly and u n id irectio n ally with a given c le a ra n c e . It is shown, both an aly tic ally and e x p e rim e n ta lly (C hapter 5), that a p r o p e rly designed d evice of the type under d is c u s s io n can 62 s e rv e a s a n efficient d a m p e r of plate v ibration, even with a s m a ll m a s s ratio . 4. 2 P R O B L E M FO R M U LA TIO N The m o d e l of the s y s te m under c o n s id e ra tio n is shown in F ig . 4. 1. In the a b se n c e of the discontinuous m a s s , the m o tio n of the plate is governed by the p a rtia l d iffe re n tia l equation over the re g io n R in x, y plane occupied by the plate, w h ere L [W (x ,y , t)] + “ T C [W (x ,y ,t)] + M ( x ,y ) n t a t (4. 1) L - is a lin e a r hom ogeneous se lf-a d jo in t d iffe re n tia l o p e ra to r of o r d e r 2p with r e s p e c t to s p atial c o o rd i nates x and y, that sp ecifies the stiffn ess d is tr i b u tion of the plate C - is a n o p e ra to r that is a lin e a r com bination of o p e ra to r L and function M (4.2) w h e re & and g a r e co n stan t coefficients M (x,y) - is a function that sp ecifies the m a s s d is trib u tio n of the plate f(x, y, t) - is a sin u soidally varying load equal to f(x ,y )sin Q t with JL f(x, y) = p(x,y) + S F k 6(x-xk> y -y fc) k=l (4. 3) p(x, y) -d is trib u te d excitation force 64 X F ig u r e 4. 1 Im p a c t D am p ed P la te . Subjected to Applied D y n am ic Loads. 65 F -c o n c e n tra te d excitation fo rc e actin g a t point x = x K K and y = y k 6(x-x , - s p a tia l D i r a c 's d elta functions defined by K y' yk ) 6(x-xk> y - y k ) = 0 if X + x k o r y 4 y k (4.4) J 6(x-xk , y - y k ) d x d y = 1 R A t the boundary c u rv e th e re a r e p boundary conditions of the type B^[W(x, y, t)] = 0 i = 1, 2, . . . p (4. 5) which should be satisfied a t each boundary point, w h ere B. a r e lin e a r hom ogeneous d iffe re n tia l o p e r a to r s of o r d e r 2 p - l . L et cp.(x, y) be the i th eigenfunction a s s o c ia te d with the h o m o geneous equation of the undam ped s y s te m given by the d iffe re n tia l equation L[W (x, y)] = u)2 M(x, y) W(x, y) (4.6) and boundary conditions Bi [ w ( x ,y ) ] = 0 i = 1, 2, . . . p (4.7) and a s s u m e that eigenfunctions satisfy the orthogonality condition* ^U sually cpj.(x, y) is uncoupled in t e r m s of x and y and can be w ritte n a s cpi < x > y) = fn (x) g m (y ) (4.6a) But we can define a byjective m apping S:PxP-»P such that S(n, m)=i w h e re P is the s e t of positive in te g e rs. T h e re fo re having any one of the fo rm of eigenfunction in equation 4. 6 the o th e r one a lso would be co m p lete ly defined. To sim plify the notation and avoid use of double su m m a tio n and in teg ratio n in fo rth co m in g d is c u s s io n , single function of two v aria b le with single s u b s c rip t is used in this ch a p te r a lm o s t exclusively. H ow ever, w h en ev er use of coupled fo rm would c o m p lica te p h y sical m eaning of the p ro b le m , uncoupled fo rm is used instead. 66 r cp.(x, y) M(x, y) ^.{x, y) d x d y = 6; - M. (4.8) 1 J lJ 1 R f cp; (x, y) L[cp.(x, y)] d x d y = 6-.K (4.9) Ti J J -J i Using g e n e ra l ex p an sio n th e o re m the solution of the p ro b le m c a n be expanded in t e r m s of n o r m a l m o d e s (eigenfunctions) as 00 w ( x , y , t ) = 23 cp.(x, y) q.(t) (4.10) i = l Substituting (4. 10) fo r W (x ,y ,t) in (4. 1) and m ak in g use of o r th o gonality conditions (4. 8) and (4. 9) lead s to following s e t of equations M. q .(t) + C. q.(t) + K. q.(t) = Q.(t) (4.11) 1 1 11 11 l w h ere = & M., + (3 K. (4. 12) and Q.(t) = f f(x, y, t) cp.(x, y) d x d y (4.13) 1 R 1 Follow ing the s a m e a p p ro a c h used in C hapter 2 we shift the o rig in of the tim e ax is to coincide with tim e of o c c u re n c e of an im p a c t a t t = t . The net r e s u l t of this shift is to m odify the e x c i tation fo rc e to r e p r e s e n t the sa m e function in the new tim e scale . f(x> yj t) = f(x, y) sin(fit + o,o ) (4.14) w h e re a Q = (1* is a p h ase angle to be d e te rm in e d . T h e re fo re fro m Eq. 4. 13 we obtain Q . ( t ) = f f(x, y) s i n ( n t + a ) cp. (x, y)dxdy=f. s i n ( Q t+ a ) 1 u _ 0^1 1 o X V w h e re f = J f(x,y) cp.(x,y) d x d y (4.15) R 67 U sing this re la tio n s h ip Eq. (4. 11) becom es M i (t) + C., qj(t) + kjCjj(t) = E sin (Qt + a Q)» i=1, 2. . . o o (4. 16) T hese equations have the s a m e fo rm a s Eq, (2. 18) in ch a p te r 2 and have the following solutions ^i l “ Hi “ Hi 1 “ Hi . q. (t) = exp(-— Qt)[ — (C.sin— Qt+p.cos— fit)q + (s in — Qt)q i r. p . x r. 'i r. o. m.p. r. a. X 1 1 1 X X X 'X : X X A i ^i ■ H i A i ^i - — (C-s i n — fit + p.cos — fit) sin T. -— r . (sin — Qt)cos T.] p. x r. 'x r. x p . x r. x J 1 x x x 'x x + A. sin (Qt + t\ ) (4, 17) and r i ^i ^i . " i ^i q (t) = exp(- — Qt)L — (- C;s i n — Qt+n; cos — Qt)q - (— s i n — fit)q x r. p. x r. i r. a. p. r. o. x 'x x i x 'x x x T 1 . A - ( J U - r - T |. r\. l i l L i l + (A.uj.sin— Qt)sin T. +------------( r .s i n — Q t-p .c o s— fit)cos T.l x x r. x p . n r. 'x r. xJ x 1 x x x + Q A . c o s (Qt + T.) (4,18) E q s ' 4. 17 and 4. 18 can be w ritte n in concise fo rm as q .(t> = E . ( t ) [ B2 .(t)4a .+ B3 i( t,q o .+B4 .(t)S l .+B51(t)S2 .1+S3 .(t) (4.19) and q .(t) = E . ( t ) [ B 12.(t)4a l+ B 13.(t)q o .+B 14.(t)S1.+B 15.(t)S2 .1 + S4 .(t) (4. 19a) w h ere a ll the coefficients a r e defined in Appendix B and v = % (0> and % i = F r o m Eq. (4. 10) we can s e e that M ultiplying both sides of equation (4. 20) by cp^(x, y)dxdy, integrating over the reg io n , and m aking use of orthogonality condition yields 0 0 J * cp.(x> y)M(x, y) W(x, y, t)dxdy = f cp.(x, y)M(x, y ) [ £ cp.(x, y)q.(t)]dxdy R 1 j=l 3 J or f cp.(x, y) M(x, y) W(x, y, t) dxdy = q.(t) M. (4.21) R 1 1 1 S im ila rly r cp.(x, y) M(x, y) W(x, y, t) dxdy = q.(t)M . (4.22) R 1 1 1 Using the initial condition we obtain q oi = J* cp^x, y) M(x, y) W q ( x , y) dxdy (4. 23) R and w h ere q . M. = f cp.(x, y) M(x, y) W (x, y)dxdy (4.24) 3.1 1 « — 1 3 -K W (x, y) = W (x, y ,0 ) o and W (x, y) = W(x, y, 0 ) 3 * Substituting the value for q and q fro m E q s. (4. 23) and (4. 24) into E q s. (4. 19) yields J cp. (x, y)M(x, y)W (x, y)dxdy q.(t) = E l(t ) [ B2 . ( t ) - ^ - i -------- 5----------------- + B 3 .(t) i J cp. (x, y)M(x, y)W (x, y)dxdy + V t)sn +B5i(t)s2i} + s 3i(t) 1 69 J RcPi(x, y) M (x , y) " ^ ( x , y) dxdy q .lt) . E .(t) [ B 121<t) ----------------j j - -----------=------------------ + B 13.(t) 1 J R cp.(x, y) M (x, y) W (x, y) dxdy -----------------ST --------------------------------- +B14i<t)Sn +B15i<t) S 2 i l +S4L(t) 1 (4 .2 6 ) Substituting the values for q^(t) and q^(t) fro m E qs. (4.21) and (4 .2 2 ) in E qs. ( 4. 25) and (4. 26) leads to following equations f cp.(x, y)M (x, y) W (x, y, t) dxdy = R 1 f c p .(x ,y )M (x ,y )W (x ,y )d x d y [ B2 i<‘> — S T 2----------------- 1 + B ,.( t) J R<Pi(x,y)M(x,y)Wo(x,y) dxdy 3i M . i 4i li 5i 2 iJ i 3i and J cp^x, y)M( x, y)W(x, y, t)dxdy = R M .E .(t) [ £ > . ( ! ) J RCPi(x, y)M(x, y) W ^ x, y) dxdy i i' L 12i ' M . i + B 10.(t) J*R cPi^x » y)M (x » y)WQ(x, y) dxdy 13 i ' M . L + B14i(t>Sli + B15i(t)S2 i l +Mi V t) (4' 28) E q s. (4.27) and (4.28) can be w ritte n in co n cise fo rm as follows: ( * cp. (x, y)M(x, y)W(x, y, t) dxdy = R 1 + B 2 l i ^ J W i^x * y) M^ x * y) Wa^x ’ y^ dxdy R B (t:) f cp. (x, y)M (x, y) W (x, y) dxdy j i i v i a R + B (t) r cp.(x, y) M(x, y) W (x, y) dxdy D C, i d ° K . + B331( t ) S lL + B34i(t)S 2i + Mi S4L(t) < 4 ' 301 Follow ing the s a m e p ro c e d u re for idealizing the im p act as a discontinuous p r o c e s s used in C hapter 2, we ag ain a s s u m e that du rin g an im p a c t the d is p la c e m e n t of the s y s te m r e m a in s the sam e W ( x ,y ,o ) + = W (x,y,o)_ (4.31) But the velocity of the s y s te m changes d iscontinuously only a t point of im p act d esig n ated by x^ and y^. T h e re fo re the velocity of point (xd , y d ) before and a fte r im p act can be re la te d by W(xd- V o)b = yd' o)a (4- 32) By using D i r a c 's function and Eq. (4. 32) the re la tio n betw een the velocity functions before and a fte r an im p a c t can be w ritte n as: I W(x, y, o)b = [ 1+ ( G - l) 6(xd - x ) 6(yd -y)]W (x, y, o)a (4. 33) G can be calculated by using m o m e n tu m equation and the coefficient of restitu tio n . U sing th e se concepts, the v elocities before and a fte r im p a c t of the p a rtic le and the point of im p act can be re la te d by 71 W (Xd,yd ) M (xd> yd ) + Z _m d = W(xd> yd V M ^ Xd ’ yd^ + ^ + m d ^ 4 * 3 4 ^ and W(x y ) - Z e = - — ■ — - — ------------— (4 35) W(xd , y d ) _ - Z _ ( 4 > 3 ) w h e re Z and Z + a r e the ab so lu te v elo cities of p a rtic le before and a fte r im p act re s p e c tiv e ly . Solving E q s. (4. 34)and (4. 35) for W (x,, y , ) and W(x ,, y ,) . d d - d d + leads to W(xd ,y d )_ = h 7 Z _ + h g i + * N i - + h 10 ^ + w h e re h = e- LJ4 , h = -1 + ^ 7 1 + e 8 1 + e h = e_(l + H') h = 1 ~ U 1 e 9 1 + e ’ 10 1 + e (4. 36) (4. 37) and u ' = “ W L et y(t) = Z(t) - W (x ^ ,y ^ ,t) be the re la tiv e d is p la c e m e n t of p a rtic le with r e s p e c t to point of a tta c h m e n t of im p act d a m p e r , then a s w as shown before for the fre e m otion of the p a rtic le in c o n tain er the following re la tio n s h ip holds. N = K y(°) = Z(o) - W(xd , y d>o) = + j (4.38) a n d • 2 0 Z +(°) = - ^ [W(xd , y d ,o) + y(o)] (4.39) Substituting those values in E q s. (4. 36) we obtain, W(xd , y d »°)_ = - 2 G2 [W(xd> yd> o) + y(o)] ™(x d , y d ’ ° h = - 2 G1 C w (xd > yd * °) + y(°)] (4. 40) w h ere G i - £ <hi o - V (4.41) and G = — (h - h ) 2 T T o I F r o m equations (4.40) we obtain g , W(xd . y d ,c.). = W (xd ,y d ,o ) + or G2 G = 1 5 T ~ l In stead y sta te m otion w ith two s y m m e tr ic im p acts p e r cycle of ex citatio n on opposite ends of c o n tain er we have W ( x ,y ,t) j = - W ( x , y , o ) = -W (x, y) (4.42) |U t - T T O W(x, y, t) ^ = - W ( x , y , o ) _ = - W ( x ,y ,o ) b = -[1 + ( G - l ) 6(xd - x ) 6(yd ~y)] W(x, y, o)a (4.43) By evaluating both sid e s of equations (4.29), and (4. 30) a t tim e Qt = t t and m aking the su b stitu tio n defined by E q s. (4. 42) and (4. 43) and noting that 73 w e o b ta in the follow ing e q u a tio n s - J cp.(x, y) M(x, y) W^(x, y) dxdy = R B21i<o> I cp^x, y)M (x, y) Wa (x, y) dxdy R + B22i^n ^ J * cpi^X’ y ^M ^x ’ y^ Wo (x, y) dxdy R + B ( —) S .. + B „. ( —) S0 . - M. S ,. 23i n l i 24 i n 2i i li " ^ 1 + B2 2 i§ ^ I ®i(x’ y)M(x’ y)w 0(x > y) dxdy = R B2 1 i ^ J cpi^x » y)M (x » y)W&(x, y) dxdy R + CB2 3 l € >- M l’ S U + B2 4 l ® SZi <4- 44) n ii 24i*n and - J cp^x, y) M(x, y) W b(x, y) dxdy = R B 3 1 i < ^ ) J C pi(x ,y ) M ( x ,y )W a (x,y) dxdy Q - R + B3 2 i ^ J ® P i^ x * y) M^ x » y) w 0(x * y) dxdy R + B 33i<H) S n + B 34i<S>S2 i - ^ i S2 . (4.45) U sing Eq. (4. 43) left hand side of Eq. (4. 45) can be w ritte n as: - J cp.(x, y) M(x, y) W ^ x , y) dxdy = R - J* cp^x, y)M(x, y )[l+ ( G - l)6(x-xd )6(y-yd )]W (x;y)dxdy = “J cp^x, y)M(x, y)Wa (x, y)dxdy-(G-l)cp.(xd,yd )M(xd,yd)W a(xd , yd ) R (4. 46) M aking use of Eq. (4.46), then Eq. (4.45) b e c o m e s <s> [ 1+B31i(^)] J cp.(x,y)M(x,y)Wa (x,y) dxdy = R B 32i(^) J q > . ( * . y)M(x, y)WQ(x, y) dxdy R + B33i<H) S li + CB3 4 i< S » -° M i l S2i + ( G - l ) v .(xd ,y d )M(xd , y d )Wa (Xd,y d ) (4.47) Equations (4. 44) and (4. 47) can be ex p ressed as J cpJx, y) M(x, y) W q (x, y) dxdy = R B26i J ®i(*.y)M (*.y)W a (x ,y )d Xdy + B ^ . S j . + B ^ . S.,. R (4. 48) and J cp. (x, y) M(x, y) W (x, y) dxdy = R a B36i I ^ i^ ’ M^ X > y)WG (x’ y^ dxdy + B 37i S l i R + B 3 8 i S2i + D n * a < V V <4 - 4 ’ > Substituting the value of right hand side of Eq. (4. 49) for the left hand side in Eq. (4. 48) yields J ^ (x, y) M(x, y) W q ( x , y) dxdy = R B26i^- B 36iJ* cPi^x ’ y ^ M ^ x> y)w 0(x > y) dxdy R 75 or B4 1 i J > i <X' y)M<X>y)Wo<X,y)dxciy = B4 2 iSli + B43iS2i+D2il!fa (xd - yd ) R or J C.IX, y)M(x, y)Wo (x, y) dxdy - B ^ . S u + B ^ . + D j . W J x ^ yd ) R (4. 50) Substituting the rig h t hand side of Eq. (4. 50) for its left hand side, in Eq. (4. 49) yields j a .I x .y lM l x .y lW tx .y l d x d y = [ B ^ . + B ^ . S ^ + D ^ W ^ ) R + B S +B S +D W (x , y ) 37i li 38i Z i 1l a d d or J- tp.(x, y)M(x, y)Wa(x, y)dxdy = B ^ . S j . + D 4 .W (xd , yd > R (4.51) E quations (4. 50) and (4. 51) can be w ritte n as Mi*l(0> = B44iSli + B45iS2 i + V a (xd^d> (4 - 5 2 > M. q. (0 ) = B . .S . + B . _.S + D^.W ( x , , y j i M i +' 46i l i 47i 2i 4i a d ’ yd or qi(0> = B5 1 i Sli + B5 2 i S2 i + D 5 i W a (V yd> < 4' 53> •»l‘V * B61i Sl i + B62i S2L + D 6i \ < X d' yd> < 4 ' 54> w h e re i = 1, 2 , . . . o o M ultiplying each equation of the type (4. 53) and (4. 54) by m .( x ,,y .) and s u m m a tio n o v er i le ad s to d d 76 S pj(xd ,y d )q.(0) = T - ®i(*d .y d ) [ B 5 1 . S u + B5 2 . S.,.] 1=1 1=1 and C O + W a ,Xd ' yd ) S 'Pi(xd ’ yd)D5i <4’ 55) 1=1 £ 'Pi (xd ' y«i,<4i <0+) = r , ' V V V r B61i Sl i + B62i S2i ] 1=1 1=1 <4' 56) 1=1 U sing Eq. (4. 10) we obtain Wo (xd ’ y'd> = ^ f i <X d ’ S,d )[ B 5 H Sl i + B52LS2 i l +Wa (xd''r d>S 1'l!i(xd-yd)D 5i 1=1 1 = 1 (4.57) and Wa (xd ’ srd )= 5'f>i(x d ’ 'rd )CB61iS n +B62iS2 i l +Wa <xd ’ yd ) S ® i <xd-ird) D 6i 1=1 1 = 1 (4. 58) (4.59) But S , . and S_. ca n be w ritte n as l i 2i S = S sin a, - S .. cos a l i 5i o 6i o S = S cos a + s /- sin a 2i 5i o 6i o T h e re fo re E q s. (4.57) and (4.58) becom e Wo (xd ' yd> = S ln % .^ tPi<X d ' yd)CB51iS5L+B52iS6il 1=1 + c o a a 0 S c o ^ . yd , t - B5US6 i+B52iS5 i ] +Wa (xd ' yd> 1=1 00 2 D 5i (4- 60) 1=1 77 Wa (V yd )= sin “ o .S tPi<*d . yd ) CB6 u S5i + B62LS6i3 i=l + = °» a 0 E 'P i< xd -yd )C-B6HS6i+B62iS5i] + W a <xd ' yd> 1=1 Z| <Pi <*d . yd > ° 6i <*•«> 1=1 In stead y state m o tio n a ll s e r i e s should be finite. T h e re fo re , E q s. (4. 60) and (4. 61) can be w ritte n as V V V = S7 s i n °o + S8 COSao + \ (xd - yd) D 7 (4' 62) * a (xd ' yd ’ = S9 s l n a o + S1 0 COSao + * a (xd ’ yd) D 8 (4’ 63) But fro m Eq. (4. 40) we know that K <X d ’ V = - 2 G1 [ W0 (Xd- yd> + ^ (4- 64) E lim in atin g W ( x , , y , ) and W (x , y ) fro m E q s . (4.62), (4.63), and o d d a d d (4. 64) le ad s to h sin a "t h cos a = (4 - 65) 1 o 2 o i w hich h a s the following solution . 2 l a = tan ‘ ^ ----;------------— (4. 66) 1 ~ + h 2 - h 2 2 3 w h e re h 's a r e functions of S and D ^ ,D g . W ith a ^ d e te rm in e d fro m Eq. (4. 66) the r e s t of the unknowns ca n be found by back substitution. -1 V 3 ± h2 V h? +h2 - h 1 . . I - - - - - ,lh3 4 h lV ^ l C H A P T E R 5 A P P L IC A T IO N AND N U M ERICA L R ESU LTS 5. 1 INTRODUCTION The a n a ly tic a l m ethod p re s e n te d in C hapter 4 ca n be applied to any two d im e n s io n a l p ro b le m , provided the eigenvalue p ro b le m r e p r e s e n te d by equation (4. 6) can be solved to obtain n a tu ra l f r e quencies and m ode sh ap es of the s y s te m . B ased on the a s s u m p tio n that the n a tu ra l fre q u e n c ie s and m ode sh ap es a r e av ailab le for input, a dig ital c o m p u te r p r o g r a m w as developed to p e r f o r m the r e s t of a n a ly sis and in p a r tic u la r this p r o g r a m w as used to obtain detailed in fo rm atio n re le v a n t to follow ing c a s e s : 1. B ase ex citatio n of a c a n tile v e r s q u a re plate with uniform th ic k n e ss 2. D is c r e te fo rc e ex citatio n of a s im p ly -s u p p o rte d plate with u n ifo rm th ick n ess In addition, e x p e rim e n ts w e re p e rfo rm e d with a m e c h a n ic a l m o d e l of the c a n tile v e r sq u a re plate to v erify the n u m e ric a l r e s u lt s . 5 - 2 p l a t e s WITH UNIFORM t h i c k n e s s F o r a u n ifo rm thin plate satisfying a s s u m p tio n s of the K irchfoff's h y p o th e sis, the p re v io u s ly m entioned d iffe re n tia l o p e r a t o r s L and M becom e 4 L = D v , M(x, y) = u e o 4 w h ere is the fle x u ra l rig id ity of the plate, v is the b ih a rm o n ic o p e ra to r and is the m a s s d ensity. 5. 3 BASE EX CITA TIO N OF P L A T E S If a plate is excited through s u p p o rt m otion r a th e r than d i r e c t ly applied load, then letting SQ(t) be the rigid body tr a n s la tio n and W (x ,y ,t) be the e la stic d e fo rm a tio n m e a s u r e d re la tiv e to rigid body m otion, the governing equation of m o tio n (4. 1) r e m a in s the s a m e except that the applied load f(x, y, t) is re p la c e d by the in e rtia load M(x, y) S Q(t). 5. 4 E X P E R IM E N T S WITH A M EC H A N IC A L M O D EL A sc h e m a tic d ia g r a m of the m e c h a n ic a l m o d e l and p la te 's p r o p e r tie s is given in F ig . 5 .1 . The co n tain er is a half cy lin d er and a m ovable d is k inside of this cy lin d er p ro v id es the d e s ir e d c le a ra n c e . The p a rtic le is r e p r e s e n t e d by a s te e l ball w hose d ia m e te r is 0 . 001 in. le s s than in te rio r d ia m e te r of the cy lin d er to avoid side m otion. But the ball can m ove along the length of the cy lin d er fre e ly with little frictio n . The coefficient of re s titu tio n , for the im p act of the ball to e ith e r side of the c o n tain er w as found to be a p p ro x im a te ly 0. 4. The c o n ta in e r could be bolted to the plate a t d iffere n t locations. The plate w as m a d e of alu m in u m . V iscous d am ping w as p r o vided by attaching v ic o e la stic m a t e r i a l to the plate. F r e e v ib ratio n 80 16" • 16' 1 S s in f it o T 1 T M a s s D e n sity of the P la te M a ss of the C on tain er M a ss of th e P a r t ic le 0.000042 0.001685 Lb. S ec In3 Lb. Sec" 0.00038 In' Lb. Sec" In F ig u r e 5. 1 S ch em atic D ia g ra m of the M ech an ical M odel of Im p ac t D am ped C a n tile v e r P late. 81 and the concept of lo g a rith m ic d e c r e m e n t w as used to obtain the dam ping coefficient of plate in the f i r s t m ode to be £ = 0.007. D is p la c e m e n t of the plate a t d e s ir e d point w as m e a s u r e d by use of an a c c e le r o m e te r . The output fro m a c c e le r o m e te r w as d i r e c te d through a v ib ratio n m e te r with double in teg ratio n capability and c a lib ra tio n w as m a d e such that the re a d in g fro m v ib ratio n m e te r would c o rre s p o n d to re la tiv e m otion of the plate with r e s p e c t to the base. F ig . 5. 2 show s the s c h e m a tic d ia g ra m of the e x p e rim e n ta l setup. P h o to g rap h s of the a c tu a l m o d e l and the n e c e s s a r y in s tr u m e n ta tio n a r e given in F ig u r e s 5. 3 through 5 .5 . 5. 5 D IGITA L C O M PU T E R PROGRAM A digital c o m p u te r p r o g r a m , based on the fo rm u la tio n given in C h ap ter 4, w as developed to calculate the values of p h ase angle a, and two b ra n c h e s of solutions. The dynam ic p ro p e rtie s of the m e c h a n ic a l s y s te m would be defined by a finite n u m b e r of its n a tu ra l fre q u e n c ie s and c o rre s p o n d in g m ode shapes a s input to the p r o g r a m . T h e re fo re , the dig ital p r o g r a m can be used for d is c r e t e s y s te m s as w ell a s continuous ones. In the c a s e of continuous s y s te m s , the m ode sh ap es a r e given at d is c r e t e , n u m b e re d points in the re g io n and the solution a t any or all of these points can be d e te rm in e d . O ther p a r a m e t e r s , re le v a n t to the im p a c t d a m p e r itse lf or con tro llin g the output of the p r o g r a m , should be a lso input. The 82 Accelerometer Particle Container Cantilever Plate Oci1lating Base Strain Gages Strain Indicator Vibration Meter Recorder Oci1loscope F ig u r e 5 .2 S chem atic D ia g ra m of the E x p e rim e n ta l Setup. F ig u r e 5. 3 T op V iew of the A c tu a l M e c h a n ic a l M odel. F ig u re 5. 4 Side View of the A ctual M echanical M odel. F ig u re 5. 5 G en eral View of the A ctual E x p e rim e n ta l Setup. 86 listing of the c o m p u te r p r o g r a m , d e ta il of its input p re p a ra tio n , and the output in te rp re ta tio n a r e given in Appendix C. 5. 6 N U M ER IC A L R ESU LTS F ig u r e 5 .6 show s the r e s p o n s e of a ty p ical uniform sq u a re c a n tile v e r plate provided w ith a discontinuous m a s s and subjected to sin u so id al base excitation. Since no ex ac t an a ly tic a l solution is av ailab le in a clo sed fo rm fo r a ll m o d e s of a plate with a co n cen tra te d m a s s a t so m e point (m a s s of the c o n tain er), the n a tu ra l f r e q uencies and m ode sh ap es used in the p r e s e n t a n a ly sis w e re obtained (27) fro m a finite e le m e n t solution using s ta n d a rd techniques . The f i r s t ten n a tu ra l fre q u e n c ie s of the c a n tile v e r plate with a co n ce n tra te d m a s s for two d iffe re n t locations of the co n tain er as obtained by using finite e le m e n t technique a r e given in T ables 5. 1 and 5. 2. The left hand side o rd in a te of F ig. 5. 6 is the ra tio of the m a x im u m d is p la c e m e n t a t fre e end, W(L, 0) , for the plate using the d a m p e r , divided by the peak d is p la c e m e n t, W(L, 0) , of the sa m e point with no d a m p e r being used. In the a b s e n c e of the d a m p e r, the solution c u rv e is a s tr a ig h t line a t W(L, 0) /W (L , 0) = 1. m a x p The low er a b s c i s s a in F ig . 5. 6 is the c le a ra n c e ra tio , d /S , in o w hich is the am p litu d e of the s in u so id a l base m otion. E x p e r im e n ta l r e s u l t s obtained fro m the m e c h a n ic a l m o d e l w e re used to ch ec k the th e o re tic a l r e s u l t s and to d e te rm in e which b ran ch of the solution, if any, w as stable. As indicated by the 87 Mode F req u en c y R ad ian / s e c . cps. 1 89. 53 14. 25 2 280. 7 44. 68 3 528. 1 84. 05 4 841. 3 133. 9 5 1010.0 160. 8 6 1419.0 225. 9 7 1958.0 311. 7 8 1995.0 317. 5 9 2289.0 364. 4 10 2576. 0 410. 0 Table 5. 1 N a tu ra l F r e q u e n c ie s of the Square C an tilev er P la te with a C o n cen trated M ass a t the Tip Mode F req u en c y R adian cps. 1 111. 1 17. 68 2 280. 7 44. 68 3 621. 0 98. 84 4 791. 9 126. 0 5 1010.0 160. 8 6 1554.0 247. 3 7 1978.0 314. 9 8 1995.0 317. 5 9 2 289.0 364. 4 10 2921. 0 464. 9 Table 5 .2 N a tu ra l F r e q u e n c ie s of a Square C an ti le v e r P la te -with a C o n cen trated M ass a t the C enter 89 n / u | = i fj. - 0.03 C , = 0.007 e = 0.40 t —| S Q sin&t| W(L,0) 1.0 .8 D 5 - 6 X o E O * ■ _j .4 .2 . - ........... 1 1 ------ Theory o o o Experiment J Range of unsteady motion _ \ y - x oA = o-5 • v d A = 0.0 ' O S. 1 N x d /L = 0.95 y dA=o.o i - - - 1 0 80 60 o co N O E 40 9. £ 20 40 « /s 0 80 120 F ig u re 5. 6 C o m p a ris o n of T h e o re tic a l and E x p e r i m e n ta l R e su lts. £=0.007 fjL =0.03 e = .40 0 . 0 0.03 Theory _ — Experiment o + ' u i S0sin,Qt# W(L,0) 1,^-N o Damper (u=0.0) d / S = 30 X d/L=0.95 [Yd/ L = 0.0 0.95 1.00 1.05 Q s / tcu, F ig u r e 5 .7 F r e q u e n c y - R e s p o n s e of B a s e - E x c ite d C an ti le v e r P late; x / L = 0. 95. d ^=0.007 /x = 0.03 e = .40 0.0 0.03 Theory --------- Experiment o + S0sini2t^ W(L,0) 120 No Damper (^.=0.0) 80 o ^ 40 20 0.95 1 .0 0 1.05 1.10 F ig u re 5 .8 F r e q u e n c y -R e s p o n s e of B a s e - E x c ite d C a n ti le v e r P la te ; x,,/L = 0. 5. a 92 First mode (,ft/u^=l) fjL = 0.0 5 e = 0 .2 5 X „/L = I Y d =0 S0sinXlt I W (L,0) 20 £,= 0.01 o i n X o O _J £ £= 0. 0 £, = 0 .0 5 50 0 10 20 30 4 0 F ig u r e 5. 9 E ffec t of V iscous D am ping. F i r s t Mode of C a n tile v e r P la te . 93 First mode [ £ l / u \ = I) fj. = 0.05 e = 0.25 X„/L = I Y „ =0 W (L,0) 0.6 o 0 ) Q. o . J \ 0.3 X o £ ,= 0 . 0 1 Q . - J 50 . 40 30 20 F ig u re 5. 10 E ffec t of V iscous D am ping. F i r s t Mode of C a n tile v e r P la te . ^W{L,0)m o x /w(L,0)p e a I ( /xW(L,0)I [ ,ox/W(L,0) 94 ^ ,=o .o i £ l / d , = I Xd/L s .95 \ A =o S0 sin,ftt W(L,0) 50 75 25 100 125 4 > a. 0 .0 1 1 .0 a. {JL - 0 . 0 1 0.005 ,25 5 .75 1 . 0 1.25 o • i d /S , 20 25 0.01 0.2 fj. - 0.05 a. e = 0.25 0.005 e = 0.5 .25 .75 1 .0 1.25 F ig u re 5. 11 E ffec t of D a m p e r M a s s R atio and Coefficient of R estiu tio n . F i r s t Mode of C a n ile v e r P la te . 95 d is c r e te points shown in F ig. 5 .6 , the solution cu rv e a s s o c ia te d with a + (the upper choice of signs in Eq. 4. 66) w as found to be stable throughout m o s t of its ran g e. C o m p a ris o n of the e x p e rim e n ta l and an a ly tic a l r e s u lts for a typical c a s e is a lso shown in F ig u r e s 5. 7 and 5. 8. It is c le a r that the e x p e rim e n ta l m e a s u r e m e n ts co m p letely c o r r o b o r a te the th e o r e tic a l p re d ic tio n s. F ig u re 5. 9 shows the influence of visco u s d a m p ing on the r e s p o n s e of a can tile v e r plate provided with a d a m p e r. Although, F ig. 5. 9 shows that the low est am p litu d e of re s p o n s e c o rre s p o n d s to the plate with h ig h e st viscous dam ping, one should note that still the net red u c tio n in re s p o n s e am plitude due to im p act dam ping is h ig h e st for the plate with le a s t am o u n t of dam ping as shown in F ig. 5. 10. The effects of d a m p e r m a s s ra tio , |j, and coefficient of r e s ti tu ti o n , e, on the r e s p o n s e of a b a s e -e x c ite d c a n tile v e r plate that is o s c il la t in g with a fo rcin g freq u en cy c o r r e s ponding to the fu n d am en tal frequency, is shown in F ig. 5. 11. It is se e n fro m F ig . 5 .1 1 (a ) that, even with a = 0.0 1 , a red u c tio n of « 80% can be achieved if the c le a ra n c e ra tio is adju sted to its o p tim u m value. The re s p o n s e of a sim p ly -su p p o rte d s q u a re plate subjected to a d is c r e t e sin u so id al fo rce is shown in F ig u r e s 5. 12 w h ere the effects of m ode shape, fo rce location, and d a m p e r location a r e exhibited. A gain, a finite e le m e n t p r o g r a m w as used to obtain 96 (a) (b) £ , ■ £ * ■ a .01 f t = 0.05 e = 0.25 T L 1 J Y ♦ Y d ------- t ► F ftsinXlt £ V l V l '0 .0 0.0 -1/4 -1 /4 - -1/4 1/4 Vl/4 1/4 F i r s t m ode fl/ujj = 1 w/ 2 5 d/w(0,0)s t a t lc — 2nd m ode n/u> 2 = 1 £ r — l —*i 1 5 10 V l V l '0.0 0.0 -1/4 1/4 -1/4 -1/4 ✓ 1/4 1/4 . / 6.25 d /W (0,0) 12.5 s t a t i c F ig u r e 5. 12 E ffect of D a m p e r L o catio n and Mode Shapes, D is c re te F o r c e E x c ita tio n of a S im p ly -S u p p o rted P late; (a) F i r s t Mode; (b) Second Mode. W(0,0)m Q /W (0 ,0 )stotic 97 Mode F requency R adian cps 1 512. 4 81. 56 2 1661.0 264. 3 3 1661.0 264. 3 4 2389.0 380. 2 5 2674.0 425. 6 6 3346.0 532. 5 7 43 75.0 696. 2 8 43 75.0 696. 2 9 5336.0 849. 2 10 5734. 0 912. 6 Table 5. 3 N a tu ra l F r e q u e n c ie s of a Sim ply-S upported S quare P late with a C oncentrated M ass at the C enter 98 Mode F req u en c y R adian cps. 1 605. 7 96. 39 2 1260.0 200. 5 3 1661.0 264. 3 4 2292.0 364. 7 5 3182.0 506. 4 6 3346.0 532. 5 7 4008.0 637. 9 8 4 375.0 696. 2 9 5734.0 912. 6 10 5734. 0 912. 6 Table 5 . 4 N a tu ra l F r e q u e n c ie s of a Sim ply-Supported S quare P la te with a C oncentrated M ass at the C en ter of a Q u a rte r of the P late 99 n a tu ra l fre q u e n c ie s and m ode shapes. The f ir s t ten n a tu ra l f r e q uencies of the s im p ly -su p p o rte d plate with a c o n c e n tra te d m a s s for two d iffe re n t locations of the container a r e given in T ables 5. 3 and 5 .4 . It w as found that a p ro p e rly designed d a m p e r is effective at any selec ted freq u en cy . In g e n e ra l, the o p tim u m location of the d a m p e r (re g a rd in g am plitude attenuation) coincides with the point of m a x im u m deflection. 5. 7 CO N V ERG EN CE OF A N A LY TIC A L SOLUTION It w as shown in C hapter 4, that the solution of the im pact d a m p e r p ro b le m w as a function of som e infinite s e r i e s , co n sistin g of su m m a tio n s o v er m ode shapes and other p a r a m e t e r s . F o r the steady state m otion, th e se s e r ie s should be finite. In o r d e r to ob tain n u m e ric a l r e s u lt s for the an aly tic a l fo rm u latio n , the infinite s e r ie s should be re p la c e d by a finite n u m b er of th e rm s . T h e re fo re , c o n v erg en ce is re q u ir e d for any of these s e r ie s . F ig u r e s 5. 14 and 5. 15 show the effect of n u m b er of m odes on the solution of the pro b lem . As can be seen , with m a x im u m of s ev en m o d e s used, good a c c u ra c y would be re s u lte d . W(L,0)m o x /w(L,0) 100 X l/tU,= I j j. = 0.03 x « A = i £ ; = 0.007 e = 0.40 Y d =0 u S0sinX2t| T W(L,0) o 0> a . 0.2 NUMBER OF MODES 0.1 > 7 20 40 0 60 80 F ig u re 5. 13 E ffec t of N u m b e r of M odes F o r D ifferent C le a ra n c e Ratio. F i r s t Mode of C a n ti le v e r P la te . 101 f j , = 0 . 0 3 X. A = i £ ; = 0.007 e = 0 . 4 0 Y d =0 iY - — L -* | A L U S0sin£2t< T W(L,0) < n X o ^_E Q . _ j 4 0 3 0 NUMBER OF MODES 20 7< 10 D / S = . 3 0 0.90 0.95 1.00 1.05 J 2 co F ig u r e 5. 14 E ffect of N u m b e r of M odes on the F r e quency R e sp o n se of B ase E x cited C a n ti le v e r P la te . C H A PT E R 6 SUMMARY AND CONCLUSIONS An an a ly tic a l m ethod w as p re s e n te d for the solution of the fo rced v ib ratio n of the im pact d am ped m u l ti- d e g re e - o f - f r e e d o m s y ste m . In g e n e ra l this a p p ro ac h yields two b ran ch e s of solutions. An analog co m p u te r w as used to v erify the a n aly tic al re s u lts and to d e te rm in e which bran ch of solution, if any, is stable. The f e a s i bility of using the im p a c t d a m p e r a s a p r a c t ic a l tool to s u p p re s s the o scillatio n of tall buildings due to wind or ea rth q u a k e s w as in v e sti gated by applying the a n aly tic al m ethod to a m o d e l of a m o d e rn sev en s to ry building and sim ulating the s a m e building on an analog co m p u te r . F o r m o s t of the c a s e s , good a g r e e m e n t w as found betw een an a ly tic a l re s u lts and e x p e rim e n ta l ones. The effects of d iffere n t p a r a m e t e r s w e re studied and detailed d ata, useful for d e sig n p u r p o s e s , w e re obtained for v a rie ty of c a s e s . In g e n e ra l, it w as found that im p act d a m p e r can be used su c c e ss fu lly to re d u ce the re s p o n s e am plitude of h ig h - r i s e buildings to h a rm o n ic base e x c i tation. ( 28 ) It has been found that the im p a c t d a m p e r can also be used effectively to re d u c e the am plitude r e s p o n s e of s in g le -d e g re e -o f- fre e d o m s y s te m s subjected to ra n d o m ex citatio n applied to the su p 102 103 p o rt. Since the r e s u l t s obtained in this study for h a rm o n ic base excited m u l ti- d e g r e e - o f - f r e e d o m s y s te m show the s a m e behavior as h a rm o n ic base excited s in g le - d e g r e e - o f - f r e e d o m s y s te m , one can s u s p e c t that the im p act d a m p e r can be used as an effective m e a n to atten u ate the m otion of ta ll buildings due to ra n d o m e x c i tation and in p a rtic u la r earth q u ak e excitation. H ow ever, m o re study is s till needed in this field. In the second p a r t of p r e s e n t w ork, an exact th eo ry w as p r e sented for the s te a d y -s ta te m otion of a v isco u sly d am p ed plate that is provided with a n in e lastic d iscontinuous m a s s attached to som e a r b i t r a r y point in the plate and w as su b sequently excited by a sin u so id al d is trib u te d load, or d is c r e t e fo rc e s , or both, applied to any a r b i t r a r y point. The m ethod w as developed in its m o s t g e n e ra l fo rm and it w as shown that it could be applied to any type of co ntin uous s y s te m such as sh e lls or b e a m s, provided that the n a tu ra l fre q u e n c ie s and m ode sh ap es of the s y s te m a r e available to r e p r e sen t the s y ste m . P r e d ic tio n of the th e o ry w e re c o r r o b o r a te d by e x p e rim e n ta l studies with a m e c h a n ic a l m odel. C onvergence of the a n aly tic al solution w as studied and it w as found that the r a te of convergence is v e ry fa s t and a few n u m b er of m o d e s would yield a good a c c u ra c y . N eed less to say, the p r e s e n t study does not exhaust the supply of in te re s tin g p ro b le m s re la te d to the behavior of the im p act d a m p e r 104 that aw ait solution. F o r in sta n ce, in the dev elo p m en t of the a n a ly tical solution for continuous s y s te m , it w as a s s u m e d that the effect of a n im p act, im m e d ia te ly a fte r the im pact, would be felt only by the point of im pact. This a s s u m p tio n is justified if the m a s s of c o n tain er e n te rin g the m o m e n tu m equation is significant. H ow ever, if the m a s s of c o n tain er is negligible, then the im p act would be felt through a re g io n which can be ap p ro x im a te d by a (29) c irc le c e n te re d a t point of im p a c t . T h e re fo re , a m o re s o p h is ticated th e o ry is needed to account for this p artic ip a tin g a r e a . O ther p r a c t ic a l a r e a s w hich should be fu rth e r exam ined a r e that of im p a c t d am p ed s y s te m s operating in m o r e than two i m p a c ts p e r cycle m o d e , n o n - s y m m e tr ic im p acts m ode, and m ultiple p a r tic le s . b i b l i o g r a p h y 1. P aget, A. L. , "M e ch an ic al D am ping by Im pact, " (1930). This r e p o r t is an appendix in "D am ping Effect of S teel B alls in T u r bine B lades, " by E. Jo h ansson, G e n e ra l E le c tric R e s e a rc h Lab. , M em o R e p o rt C-226 (1952) . 2. L ie b e r, P. and Je n se n , D. P. , "An A c c e le ra tio n D a m p e r: D evelopm ent, D esign, and Some A p p lic a tio n s ," T ra n s . ASME vol. 67 (1945), pp. 523-530. 3. Grubin, C. , "On the T heory of the A c c e le ra tio n D a m p e r, " J o u rn al of Applied M ech an ics, vol. 2 3 , T ra n s . A S M E , vol. 78 (1 9 5 6 ), p p . 3 7 3 - 3 7 8 . 4. A rnold, R. N. , "R esp o n se of an Im p act V ibration A b s o rb e r to F o r c e d V ib ra tio n ," Ninth In tern a tio n a l C o n g re ss of Applied M ech an ics (1956). 5. Sadek, M. M. , "T he B ehavior of Im p act D a m p e r, " P ro c e e d in g s of the Institution of M ech an ical E n g in e e rin g , vol. 180, P a r t I, 1965-1966. 6. W a rb u rta n , G. B. , D is c u s sio n of "On the T h eo ry of the A c c e l e r ation D a m p e r, " J o u rn a l of Applied M e c h a n ic s , vol. 24, T r a n s . ASM E, vol. 79 (1957), pp. 322-324. 7. M a s ri, S. F . , "A nalytical and E x p e rim e n ta l Studies of Im pact D a m p e rs , " P h. D. T h e sis, C alifornia Institute of Technology, 1965. 8. M a s ri, S. F . , "S teady-S tate R esp o n se of a M u ltid eg ree S ystem With an Im p act D a m p e r, " Jo u rn a l of Applied M e c h a n ic s , vol. 40 (1973), pp. 127-132. 9. T o k u m aru , H. and K o tera, T. "On Im p a c t-D a m p e r for C oncen tra te d -M a s s -C o n tin u u m Syst^j^i" B ulletin of JSM E vol. 13 No. 59, (1970). 10. M a s ri, S. F . " F o rc e d V ibration of C la ss of N onlinear D is s i p a tive B eam s," J o u rn a l of E n g in eerin g M ech an ics D iv is io n , P r o - . ceedings of ASCE, vol. 99, No. EM 4, (1973). 11. M cG oldrick, R. T. , " E x p e rim e n ts with an Im p a c t V ibration D a m p e r, " David T aylor M odel B asin R e p o rt No. 816, (1952). 12. L e ib e r, P. and Tripp, F . , " E x p e rim e n ta l R esu lts on the A c c e le ra tio n D a m p e r, " R e n s s e la e r P olytechnic Institute A e ro n a u ti cal L ab o ra to ry , R e p o rt No. TR A E 5401 (1954). 13. Sankey, G. O. , "Som e E x p e rim e n ts on a P a r tic le or 'shot' D a m p e r , " M em o ra n d u m , W estinghouse R e s e a r c h Labs (1954). 14. D uckw ald, C. S ., "Im p a c t D am ping for T urbine B u c k e ts ," G e n e ra l E n g in eerin g L a b o ra to ry , G e n e ra l E le c tr ic , R e p o rt No. R55GL108 (1955). 15. Ib ra h im , A .M . "Studies of Im p act D a m p e r s , " Ph. D. T h e s is, U n iv ersity of S outhern C alifornia (1971). 16. P e r i l s , S. , T h eo ry of M a t r i c e s , A d d iso n -W e sle y P ublishing Com pany, Inc. , (1958). 17. H ildebrand, F . G. , M ethods of Applied M a th e m a tic s , P r e n ti c e - H all (1952). 18. S trutt, J. W. , B aron R a y le ig h , The T h e o ry of Sound, D over P u b licatio n s (1945). 19. Caughey, T. K. , " C la s s ic a l N o rm a l M odes in D am ped L in ear D ynam ic S y stem s, " J o u rn a l of Applied M e c h a n ic s , 59-A -62 (1959). 20. Caughey, T. K. and O 'K elly, M. E, J. , " G e n e ra l T h eo ry of V ibration of D am ped L in ear D ynam ic S y ste m s, " D y n am ics 107 L a b o ra to ry , C alifornia Institute of Technology, P a sa d e n a , C alifornia (June, 1963). 21. M a s ri, S. F . " G e n e ra l M otion of Im p a c t D a m p e r s , " The Jo u rn a l of the A c o u stica l Society of A m e r ic a , vol. 47, No. 1, (1970). 22. K uroiw a, J. H. "V ib ratio n T e s t of a M u ltisto ry B uilding," E ng in eerin g D e g re e T h e s is , C alifornia Institute of Technology, 1967. 23. N ielsen, N. N. "D ynam ic R esp o n se of M u ltisto ry B u ild in g s," Ph. D. T h e s is, C alifornia Institute of Technology, 1964. 24. Jackson, A. S. , Analog C om putation, M c G r a w - H i l l Book C o m pany, In c ., New Y ork, N. Y. (I960). 25. M a s ri, S. F . , and Caughey, T. K. , "On the Stability of the Im p act D a m p e r, " J o u rn a l of A pplied M e c h a n ic s , vol. 33 T ra n s . ASM E. vol. 88, (1966), pp. 586-592. 26. Lundgren, T. S. , Chang, C. C. , and Whang, Y. C. , "D am ping of R ectan g u la r P late V ib ratio n s, " WADC T ech n ical R e p o rt 59-544, (I960). 27. Bathe, K. J. , W ilson, E. L. and P e te r s o n , F . W. , SAP IV A S tru c tu ra l A n alysis P r o g r a m F o r Static and D ynam ic R esponse of L in e a r S y ste m s, R e p o rt No. E E R C 73-11. U n iv ersity of C alifornia. B erk eley , C alifornia. June 1973. 28. M a s r i, S. F . and Ib ra h im , A. M. , " R e sp o n se of the Im pact D a m p e r to Stationary R andom E x citatio n , " The Jo u rn a l of the A c o u stic a l Society of A m e r i c a , vol. 53 (1973), pp. 200-211. 2 9 . H a r r i s , C. M . and C re d e , C. E . " Shock and V ibration Handbook" M c G r a w - H i l l Book Com pany, Inc. , New Y ork, N. Y. (1 9 6 1 ), vol. 1, pp . 9. 2 - 9. 3. 108 30. L e is s a , A. W. V ibration of P l a t e s , NASA S P -160 (1969). 31. M e i r o v i t c h , L. A n aly tical M ethods in V ib ratio n s, The M a c m illa n Com pany, New York, N. Y. , (1967). 32. Young, D. and F e lg a r , R. P. Table of C h a r a c te r is tic F unctions R e p re s e n tin g N o rm a l M odes of V ibration of a B eam , The Uni v e rs ity of T exas P ublication, No. 4913 (1949). 33. R eed, W. H. , III, and D uncan, R. L. , " D a m p e rs to S up p ress W ind-Induced O scillations of T a ll F lex ib le S t r u c t u r e s , " D evelop m e n ts in M ech an ics, vol. 4, C e rm a k , J. E. , and Goodman, J. R. , eds. , Johnson P u b lishing C o ., (1968), pp. 881-897. 34. R eed, W. H. , III, "H anging-C hain Im p ac t D a m p e r s : A Simple M ethod for D am ping Tall F le x ib le S tru c tu re s , " Wind E ffects on Buildings and S tru c tu re s , vol. II, U n iv e rsity of Toronto P r e s s , (1968), pp. 283 - 321. 35. Z ienkiew icz, O. C. , The F in ite E le m e n t M ethod in E n g in eerin g S cie n ces, M c G ra w -H ill, (1971). A PP E N D IX A VARIOUS M ATRICES AND V ECTO RS OF C H A PT E R 2 [E (t)] = D iagonal m a tr ix such that its i th e le m e n t is Ci exp ( - ------- Q t). i [B ^(t)] = D iagonal m a tr ix such that its i th e le m e n t is 1 • 111 n* s i n — Qt. 1 [B (t)] = D iagonal m a tr ix such that its i th e le m e n t is I ^ i ^ i - ( r . sin — Q t + n . cos — Q t). n . i r. 1 i r. II l l [B (t)] = D iagonal m a tr ix such that its i th e le m e n t is 1 ^ i ^ i - — (C ■ sin — Q t + ri . cos — Q t). Tl. i r. 11 r . h i l [B ^(t)] = D iagonal m a t r i x such that its ith e le m e n t is 1 . ^ i - r . s m Qt. T l . 1 r. 1 i l [ B ^ ( t ) ] = D iagonal m a tr ix such that its ith e le m e n t is 1 ^ i ^ i — ( - r ■ sin — Qt + ri - cos — Qt). ^ i r . 11 r. ^ i i l [B ^^(t)] = D iagonal m a t r i x such that its ith e le m e n t is ^ _ Hi s m — Qt. i rij. [B ^^(t)] = D iagonal m a tr ix such that its ith e le m e n t is . ^ i (D . sin — Q t. i r . i [B (t)] s D iagonal m a t r i x such that its ith e le m e n t is 1 109 I V t v {S3 (t)} tv tv [ B 2 i(t)3 t B22< ‘>] [ B23<t>] CB24(t)t! t B31(t)tl [ B3 2 (t)] [ B 33W ] Cb 34< ‘>] t V ■"ir i ■H i ’U n i (r . sin — fit - ri. cos — Qt) Bi r. h r. 110 col. f A , sin t , •••» A. sin r., • • ■ » A ^ sin t } L 1 1 i i 1 1 n J col. (A , c o s t ,, . .. , A .cos A cos t 1 u 1 1 l i n n J col. [A j sin(Qt + r J , .... A ,sin(Qt + TJ , . . . , A sin(Qt + t )} n n J Cs 4(t)} s col. {Q A j cos (Qt + T i), . . . , Q A. cos (Qt + T.), . . . , Q A cos (Qt + T )} n n J col . fA , cos Y, , • • • , A. cos Y.« . . . , A cos Y 1 L 1 1 • i i n n J col. (A^ sin Yj , . . . , A., sin Y ^» . . , A sin Y } n n J [cp] [E(t) [cp] [E(t) [cp] [E(t) [cp] [E(t) [cp] [E(t) [cp] [E(t) [cp] [E(t) [cp] [E(t) CB-(t)] [cp] [ B 3(t)] [cp] [ B .(t)] [ B J t ) ] [ B 12(t)] [cp] [ B 13(t)] [cp] [ B 14(t)] t B 15(t)3 -1 -1 -1 -1 D iagonal m a tr in such that B 6.. li if i ? j tss) [s*g] + l9 sJ C^g] - 2 t8 s} 1 9 s3 [S f ,g] + tss) [**a] 2 t'-s) [8 £ g] + [s,g] [9 £ g] 2 t L* a [i£g] + [wg] [9 £ a] = [9 *a [£tg] 1_[Itg] 2 [s” g [wg] 2 C wa [8 zg] + [8 £ g] [9 Z g] 2 [£ ” g [izg ] + [i£g] [9 Z g] 2 [wg [9 £ g] C 9 Z g] - C i3 2 C "a Mu - [(g) *£ g ]> T _ C s£a3 2 [8 £ a [(g) £ £ g] t_[s£g] 2 [i£g [(g ) z£ a] t.[5 £ g] 2 [9 £ g [< g)1 £ a] + [9 a]) - 2 [s£a [(g ) ^ . [ ^ a ] 2 [8 zg ([*] - [(g ) £ Z g]) t.[szg] 2 [iza [(g) lza] 1.[szg] 2 [9 zg ([(g) zza] + [i3) - = [szg ff " 9 I I I 112 tS 9] s t s io} " £B46l tS55 + CB47] £S6] - [ b46] {S6} + [ b4?] £S5] S0 + 2 G, S„ 9. 1 7. J J S + 2G S 1 0 . 1 8 . J J -2 G y 1 yo A PP E N D IX B D EFIN ITIO N OF SYMBOLS USED IN C H A P T E R 4 C L E . (t) = exp ( - — Qt) i r . 1 1 ^ i B (t) = sin — Qt 2i uu.ri. r. I ^ i ^ i B (t) = — ( Q . sin — Qt + n . cos — Qt) 3i p . 1 - r . ' l r . II l i 1 T ' i ^ i B . .(t) = (C ■ sin —- Qt + ri . cos —- Qt) 4i p • 1 r • 1 r - 1 i i l 1 . ^ i Bc .(t) = — r. sin — Qt i J r* • \ * * t 1 , O J . l l 5i r) ^ i r . S, . = A. sin t . li i i S„. = A. cos t. 2i l i S3i (t) = A. sin (Q t + T|_ ) I ^ i ^ i B (t) = (- C . sin — Qt + p . cos —- Q t) 12i p . ^ i r . 11 r . II i l tl) ■ B l o : (t) = - — sin — Qt i ' 1 O * \ u f 13i n i r. ^ i B 14i(t) = ^ s i n — n t i V i 71 i i B 15i(t) = — — (Ci sin — Qt - p . cos — Qt) 1 i i i S4 .(t) = Q A. cos (Qt + T.) B2 u (t) = E . ( t ) B 2 .(t) , B22.(t) = E . ( t ) B 3 .(t) B23i(t) = M lE i (t) B4 i(,) ’ B24i(,) = M iE i (t| B5 i(t) 113 5341(E ,- n M i3 / B 35i (damper m a ss/co n ta in er m a ss) A PP E N D IX C The d e ta il of input p r e p a r a tio n for the c o m p u te r p r o g r a m used in a n a ly sis of the fo rced v ib ratio n of im p act dam ped continuous s y s te m s is given in se c tio n C l. P r o g r a m listing is given in sectio n C2 and the p r o g r a m 's output for a sam p le case is included in sectio n C3 C l. DATA IN PU T Input data to c o m p u te r p r o g r a m c o n s is ts of two p a rts: a) S y stem c h a r a c t e r i s t i c s data which includes m ode shapes and n a tu ra l fre q u e n c ie s of the continuous s y s te m under s tud y . b) C ase c o n tro l data, co n siste d of n u m b er of d iffere n t c a s e s and re le v a n t p a r a m e t e r s for each case to be analyzed in one execution tim e. F o r a continuous s y s te m , re p r e s e n te d by its m ode shapes and n a tu ra l fre q u e n c ie s defined in p a r t (a), a n u m b e r of d iffere n t c a s e s with r e g a r d to d iffe re n t choices of the im pact dam ping of the s y ste m can be a n aly se d in one execution tim e. In addition for each case, m any s u b c a s e s can be defined based on uniform v aria tio n of m a ss ratio , c le a ra n c e , coefficient of re s titu tio n and the freq u en cy of the excitation fo rc e for each su b case. The detail of each sectio n is given below. a) S y stem c h a r a c te r i s t ic data 116 C ard num ber F o r m a t Colum n V ariable Notes 1 (215) 1-5 NPO ( 1) 6-10 NM ( 2) 2- k ( 9 F 5 .0 ) 1-45 DXY(I) (3) k+i - a (19x, E 1 2 .5 ) 20-31 WCPS( I) (4) & + 1 - m ( 32x, E 1 2 . 5) 33-44 PH I 0(1) (5) C ase co n tro l data C ard num ber F o r m a t Colum n Var iable Notes 1 (15) 1-5 NCASE (6) 2 (215) 1-5 NDO (7) 6-10 NF (8) 3 (315) 1-5 INC (9) 6-10 KW RITE ( 10) 11-15 NDSP (H ) 4 (15) 1-5 NP (12) 5 (1615) 1-80 ISOL(I) (13) 6 (215) 1-5 ND (14) 6-10 NCOM P (15) 7 (5 F 1 0 .0 ) 1-10 SO (16) 11-20 CA LPHA (17) 21-30 CBETA (17) 31-40 CONT (18) 41-50 DENSTY (19) 118 (6F10. 0) (6F10. 0) Colum n Y aria ble Notes 1-10 UO (20) 11-20 D ELU 21-30 UMAX 31-40 EO (21) 41-50 D E L E 51-60 EMAX 1-10 OMGO (22) 11-20 D ELO M G 21-30 OMGMAX 31-40 DOO (23) 41-50 D ELD 51-60 DMAX Notes (1) N PO is the n u m b er of d is c r e te points used to r e p r e s e n t m ode shapes of the sy ste m . (2) NM is the n u m b e r of m odes to be used in an a ly sis. (3) DXY(I) is the a r e a (or length) assig n ed to point I, to be used in calculation of g e n e ra liz e d m a s s and g e n e ra liz e d fo rce. Each c a rd in this sectio n contains nine values of DXY(I) s ta rtin g fro m I = 1 up to I = NPO. T h e re a r e as m any c a r d s in this section as n e c e s s a r y to contain all values of DXY(I). (4) WCPS(I) is the Ith n a tu ra l freq u en cy of the s y s te m in cps. T h e re a r e NM c a rd s in this sectio n each containing one value of WCPS(I) for I = 1, 2, . . . NM. (5) PHIO(I, J) is the value of Ith m ode at point J. T h e re a r e N PO * NM c a rd s in this sectio n each containing one value of PHIO(I, J) s ta rtin g with I = 1 up to I = NM for each value of J fro m NPO to 1. (6) NCASE is to n u m b e r of c a s e s to be an aly se d for the defined sy ste m . (7) NDO is the n u m b e r of the point which im p act d a m p e r is attached to. (8) NF defines the forcing function. O value of NF indicates base excitation. O ther value of NF gives the n u m b e rs of the point, w hich c o n ce n trate d force is applied to. (9) INC is the n u m b e r of d ivision in a half period for the c a lc u la tion of the peaks. (10) KW RITE d e te r m in e s the am o u n t of output to be printed. If KW RITE = 1, then detailed output would be printed. Any other value of KW RITE re s u lts in le s s am o u n t of output. (11) NDSP is a flag which is defined a s follows: If NDSP . EQ. O, then the values of DOO, D ELD and DMAX would be se t equal to those specified on input card . If NDSP. GT. O, then the above p a r a m e t e r s a r e calcu lated by the p r o g r a m a c c o rd in g to follow ing fo rm u la s: 120 DOO = 0 . 0 DMAX = SQRT (Hl**2 + H2**2)/G1 D ELD = D M A X /F L O A T (NDSP-1) (12) N P is the red u ce d n u m b e r of points for solution set. (13) ISOL(I) is the n u m b e r of Ith point in solution set. Up to m a x i m u m of sixteen points can be specified and the point of im pact should a lso be included in this set. (14) ND is the n u m b e r of the point in red u ce d se t which d a m p e r is attach ed to. (15) N COM P is the n u m b e r of the point in red u ce d se t which is used for n o rm a liz a tio n of solution v e c to rs. (16) SO is the am p litu d e of forcing function or base excitation. (17) CALPHA and CBETA a r e the coefficients of p ro p o rtio n al dam ping. (18) CONT is the m a s s of the co n tain er. (19) DENSTY is the m a s s d ensity of the s y s te m . (20) UO, D ELU and UMAX a r e the initial, in c re m e n t and final value of m a s s ra tio re sp e c tiv e ly . (21) EO, D E L E and EM AX a r e the initial, in c re m e n t and final value of coefficient of re s titu tio n re sp e c tiv e ly . (22) OMGO, D ELO M G and OMGMAX a r e the initial, in c re m e n t and final value of the freq u en cy of the excitation force re s p e c tiv e ly . (23) DOO, D ELD and DMAX a r e the initial, in c re m e n t and final value of the clearance ratio respectively. C2. PROGRAM LISTING FORTRAN I V G L E V E L 21 OATE = 7 A 0 S 4 1 5 / 1 4 / 3 1 PAGE 0 0 0 1 0001 0 0 0 2 0 0 0 3 0 0 3 4 r oo5 0CC6 c C - c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c • c c I D ICO T H I S PROGRAM CALCULATES THE RESPONSE OF AN IMPACT OAMOED CONTINUOUS SYSTEM TO S IN U S O ID A L E X C IT A T I O N D E S C R IP T IO N OP PARAMETERS NPO s NIJMPfiP Oe POINTS THAT APE USED TO DEPINE THE MODI SHAPES = NUMBER OF MODES USED I N A N A LY S IS = AREA COR LENGTH) CO NTR IB UTIO N OP P OINT I FOR CALCULATIO N OF INTEGRALS REPRESENTING GEN ER ALIZED MASSES AND GENERALIZED FORCES = I TH NATURAL FREQUENCY I N CPS = VALUE OF I TH MODE AT P OINT J = NUMPSB OF D IFFE RE NT CASES = NUMp.ER OF THE P OINT IN NPO SET THAT THE DAMPER I S ATTACHED TO • E Q*0 BASE E X C IT A T I C N • GT«0 NUMBER OF THE P OINT IN NPO SET THAT CONCENTRATED FORCE IS A P P L IE D TO s NUMBER OF D I V I S I O N S I N HALF PERIOD FOR CALCULATIO N OF PEAKS •E Q * 1 D E T A IL E D OUTPUT « N E •1 REGULAR OUTPUT • CQ«0 0 0 . DELD AND DMAX W IL L BE SET EQUAL TO T HE IR CORRESPOND ING INPUT VALUES • GT«G DO .DE LD AND OM*X W IL L BE CALCULATED INTERNALLY AS FOLLOWS : 0 0 = 0 * 0 OMAX = S 0 R T ( H l * * 2 + H 2 * * 2 ) / Gl DFLD = DMAX / FLOATCNDSR - 1) = NUMBER OF P OINTS FOR THE SOLUTION SET s VECTOR CO N TA IN IN G O R I G IN A L NUMBERS OF THE SOLUTION P OINTS = NUMBER OF THE P OINT I N NP S rT REPRESENTING NDO = N'JMHLR OF THE P OINT IN NR SET USED FOP NOR M ALIZ A T I C N CP THE SOLUTION VECTORS = CONSTANTS FOP PROPORTIONAL TAMPING = BASE DISPLACEMENT AMPLITUD E OR CONCENTRATED FORCE AMPLITUD E = E X C IT A T I O N FREQUENCY = CLEARANCE = MASS RA TIO = C H F F F IC IE N T OF R E S T IT U T IO N sMASS DE NS ITY OF THE SYSTEM - MASS O F THE CONTAINER NM D X Y NCASE NDO NF INC KWRITE NDSP NP IFO L ND NOOMP CALPHA.CBETA SO OMFGA D u OHNSTY - C3NT C O M M O N /C R O /C A L B H A .C P E T A .O M F G A .D E N S T Y .C O N T ,IS 0 L (5 0 > C O M M O N / C B l / N P . N M , P I , C F N M ( 2 0 ) , Z E T A ( 2 0 ) . R ( 2 0 ) . E T A ( 2C > • W( 2 0 ) • A ( 2 0 ) • 1 P S K 2 0 ) . P S I P I 2 P ) . r ( 2 C ) . P H l < 2 0 . 5 G > . K W R I T E . P f e A K l 5 0 ) C O M M O N /C 3 2 / S I ( 2 3 ) . S 2 ( 2 0 ) . Q O ( 2 0 ) • ODOTA<2 3 ) C U M M 0 N / C B 3 / G . S T #S 3 . S 9 . S 1 0 . B S l ( 2 O ) • B 5 2 ( 2 0 ) • B 6 1 < 2 0 > . B 6 2 ( 2 0 1 > D 5 ( 2 C ) • 1 D 6 ( 2 G ) C 3 M M C N / C R 4 / I N C . X M A X ( S O ) * TX M A X (5 0 ) • XO( 5 0 ) , X D 0 T A ( 5 0 ) . X D O T B ( 5 0 ) COMMON /Cn5/fcU2( 2 0 ) . E D 3 ( 2 0 ) • £ 0 4 ( 2 0 ) * E B S ( 2 0 ) . E B 1 2 ( 2 0 ) . E B 1 3 ( 2 0 ) • 122 FDR TRAN I V G L E V E L 21 DA TE = 7 4 0 5 4 1 5 / 1 4 / 3 1 C C c 3C03 o o r o 1 0(M 0 0011 0 0 1 2 0 013 f 0 1 4 0 1 5 0 > t 6 0 01 ▼ 001 o 0 0 1 3 0 2 0 0 2 1 0 2 2 0 >23 0 0 2 4 0 02 5 0 >26 17 0 0 2 7 0 0 9 13 0 0 2 3 12 O J i n 00 31 C 032 00 33 2C P-V54 C 335 0 036 2 CO 37 0 0 33 O l j Q C-3 40 0041 0 0 4 2 C04 3 0 0 4 4 3 CO 45 0 0 4 6 0 04 7 0 0 4 8 4 0 0 4 0 0 0 5 0 5 0 0 5 1 . 0 3 5 2 I 5 « 1 4 < 2 P > . E « I 5 C 2 0 > D IM E N S IO N P H I O < 2 0 » i e C ) , D X Y C 1 0 0 ) , W C P S C 2 0 > READ DATA READC 5 * 1 ) NPP«NM FORMAT( I 61 5 I WRTTllC 6 * 6 1 ) M3 0 fN M 61 FORmaTC1 H I , / / , S X , • NUMBER OF POIN TS = ' , I 5 , / , 5 X , 1 'NUMBER OF MODES = ' . I 5 * / / 1 P I = 3 * 1 4 1 5 0 READ( 5 * 3 6 ) CDXYCI) . I = 1 , N P 0 ) © A rO =M A T<4F5,0> R E A 0 C 5 . F 3 ) < W C » S ( l) • I = 1 ,N M ) 93 F *1 R M A *M 1 0X *G 1? .5I DO a s I = 1,NW w(i> = 2 . o *p t * wcpsc i ) 9 5 C L N T lN U r DO 10 I = 1 • NPO I I = NPG + 1 - 1 P .E A n«5*B4) ( PH 1 D ( J , I 1 ) , J = 1 ,N M ) 10 CONTINUE 34 - O R M A T < 3 2 X , r i 2 * 5 > WR JT E< 6,1 '7 ) F Q R M A T C //5 X 'M A T R I X P H IO • ) DO 1« 1 = 1 .KM W R IT E ( 6 , 1 2 ) ( P H I O C l . J > * J = 1 * N P Q ) F O R M A T C E X I P l f £ 1 0 * 2 ) WRITCCfc.15) 1 5 FORMAT( / / • 5X , • AREAC OR L EN GTH ) ASSIGNE D TO EACH P O IN T FOR INTE G RAT I 1') N • » /1 W R IT E ( 6 , 1 1 } <DXV( I ) , f a t , N P O ) CONTINUE .\FA'3<5,1 ) NCAG.E W F I T S < 6 , 2 ) NCASz FORMAT C/ / / , 5 * , * NCASF ss ' . 1 5 ) DO lOi'O K K = 1 « NCASE R £ A r » C 5 . t I N DO .N* R r A D ( 5 # l ) I N C . K W R I T F , NDSP R ~ A D C 5 , 1 I NO RE. AO(5 « 1 ) C I S O L C I ) « I « i * N P ) R L A O C S . l ) ND.NCOMP RE ADC 5 . 3 ) SO,CALPHA*CBETA•CONTtDENSTV F P R M A T C P F 1 0 . C ) RE ADC 5 * 3 ) UQ»D E L U , U M A X , E 0 , DELE,EMAX RuADC 5 , 3 ) CMGO.OELOMG,3MGMAX, 0 0 0 • DELD*DMAX W i? I T 5 ( r, A ) KK, NCASF F r * 4 « t T ( l r i l . I X , ' T H I S I S CASE ' . I 3 . 2 X , ' OF ' , ! 4 , 2 X , ' CASE •» W R IT 1 : ( C . 5 ) N P , N D O , N O ,I N C ,K W R I T E , N C D M P , N F , N D S P F C R M A T C / / . 5 X , * I'D 1 00 I IMPACT DAMPED BEAM OR PLATE • • / / / / * 5 X * 1 • NP = • . 1 3 , 5 X , ' N D O = • , I 3 • 5 X • * ND = • • I 3 , 5 X • • INC = ' , ! 3 , S X . 2 * KWRITE = • , I 3 . 5 X , / , 5 X , • NCOMP = • • I 3 • 5 X , 3 ' N p = • , I 3 »5X , • ND S° = • » I 3 ) . W R IF E C 6 , 5 2 ) ( I S O L C I ) • I = 1 .N P) 5 2 F 0 R M A T C / / . 5 X . ' P O I N T S THAT SOLUTIONS AT THOSE ARE S O U G H T ', 1 / / , 1 6 1 7 , / ) 0 0 0 2 123 FO RTRAN I V G L E V E L 21 M A IN DA TE * 7 4 0 5 4 1 5 / 1 4 / 3 1 PAGE 0003 C O S ? 3 0 5 4 0 0 5 5 005* 03*7 003.3 00^9 0 05 3 C 1 6 1 0 V-2 0CS3 0 0 6 4 0 0 o 5 0 1 * 7 0C n3 0 0 6 9 C O 70 0071 00 7 2 C073 00?4 C 075 CO 76 0 0 7 7 0073 COT? 10SD r 0 3 l 0032 0 0 3 3 0 0 5 4 0 0 5 5 C o 36 0 1 3 7 OIF 4 0 0 3? 0043 - 009 1 0 3 « 2 CC 93 0094 0 1 > 5 C r o 6 W R IT F ( 6 . 6 ) S P . C A L P H A . C 3 E T A . D E N S T Y . C 0 N T FORMAT( / / ♦ 5 X » • SO . I P E 1 0 # 2 . 4 X . * CALPHA s * . 1 PE 1 2 * 4 . 4 X . • CBETA * • 1 * I P E 1 2 * 4 . 4 X . • DE NS ITY = • ♦ 1 P F 9 . 1 • 3 X * « CO NTA INF« = • • l P E 1 0 * 2 . / > W R I T E ! 6 * 7 ) U C . D S L U . U M A X .F O . D c L E .E M A X 7 F O R M A T < 5 X . » U 0 = * • 1 P E I 0 • 2 • 5 X • « P E L U = ' • 1 P E 1 0 . 2 • 5 X . • UMAX = « .1 P E 1 0 * 2 * 1 1 OX • • trn=» #1 PE1 0 #2 «5X« • O r L E = » . 1 P C 1 0 . 2 . 5 X . »F.MAX=» • I P F 1 0 * 2 . / > WKIT - ( 6. A > f'MG'Jf D" L0MG * -jMGM A X * DCO.DELD * DMAX 5 FORMAT CSX• ■OMGO=» « 1 P F 1 C * 2 * 5 X . 'D E L1M G= * . 1 P S 1 0 . 2 . 5 X . «OMGMAX=* . 1 1 P C 1 0 # 2 * 5 X .» P 0 0 = » , 1 P E 1 0 . 2 . 5 X , » O E L D = ' « IPS 1 0 • 2 . 5X . *DMAX=»• l P E 1 0 « 2 . / > CALCULATE GEN ER ALIZED MASS DO 22 I = l . N M G=NM( I 1 = 0 * 0 DO 21 J = 1 * NP 0 21 = G F N M ( I ) ♦ P H I 0 ( I • J > * * 2 * D X Y < J) C t'Z NM ( I ) = GENM(I )* P F N S T Y 4 P H I O C I , N 0 O ) * * 2 * C 0 N T 2 2 CONTINUE OMEGA = CMGO START OF THE LOOP FOR FREOUFNCY SWEEP 30 CONTINUE CALCULATE g e n e r a l i z e d FORCE I F ( N F • GT *0 I GO TO 62 FO = SO * C M EG A**2 * DENSTY DP 24 I = l . N M F ( I ) = 0 * 0 PC 2 3 J = l . K ’PO 2 3 - » F ( I ) ♦ P H I O < I » J ) * D X Y ( J ) F ( I ) s F ( I ) * * 0 ♦ PH 1 0 ( I* N D O )* F O * C O N T / D E N S T Y 2 4 CCNTINUF GO TO 63 6 2 DC 64 I = l . N M F( I ) = SO * P H 10 C I . N F ) 6 4 CONTINUE 5 3 CC NTINU F DO 51 I = l . N P I I = I S O L C I ) CO 51 J = l . N M P H I ( J . I ) = P H I O C J . I I ) 51 CONTINUE CALL L IN E A R U * UO 40 E = EO so o = oro IP S = c IF ( N D S P * GT * 0 ) D = C.O F K 7 = ( r - U ) / ( 1 . 4 E ) F K S s ( 1 . +!J ) / ( 1 ) p K9 = E * ( U + U ) / ( l « 4 E ) F K 10 = < l . - U * £ £ ) / ( l . * F > - ........................ G 1 = C M E G A * ( F K 1 0 - F K 9 ) / P I G 2 = 2 M F G A * < F K f l - F K 7 ) / P I 124 POSTMAN IV G LEVEL 21 MAIN DATc * 74054 1 5 / 1 4 / 3 1 PAGE 9 01? 4 125 I (V • • x c z V* 1 O 1 U) C D (A N < — V O s s « K « • 4 h Z » N • UI * tvj | a O « w* u incu to a u. > v X *10 u. A O • 0 z NO) 0 4 1 1 Z c * i n # h x — » # r* f U X < N a « . * 1 3 0 # < z < c 41 ts * tr * > a c IL 4 1 — # c - « o - J C O * * I *u. Ul •4 »l Z i N C v - X V u 0 — x : ^ w c a • K ' C i r x z i + i*;**' C Cr 4 C C IT • < l ; nj 0 N « Z 4 Li C - \ r a to P.* S i \ H C• »0) *SC.U1 u: S O j f t « K a d IL h ^; W Z l / ) W ( f t l l ' - N II Z u O - H i C C l! l< *-• cr II L J< J II 1 1 2 X I - J O K II K Z I I J - < H t ( j Z % Ift*-. V | < ^ * r u u r i r t ’ ^ C ) tii c o a o : 0 U X I * - 0 » l J 00 z > — * u «o 10 0 m IT <0 0 0 ru < v z * • « c n n r • I V ) <UC If -* * • CMO z z + < c « y • v # o * * • 0 1 2 # I I • • I’ jQ. 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X O C I I • X O O T A d I • XDOTfM I 1 , XMAX( I )»TX MAX{ 11 1 «P E A K ( I ) . 0 1 , 0 2 * 0 3 0 1 5 5 45 FORMAT ( l x , 1 2 , IPO c l 2 . 4 ) C 159 W R T T 2 ( 6 , 4 7 ) 2 0 , 7D0TA 0 1 6 0 42 eOiJMAT ( 6 X» TO** , 1 PE 1 6 • 4 , 5X 9 ZD0 TA = • , 1 PE 1 2 * 4 ) 0161 ALP” ALP HA 0 / ° I r i o a wPITc ( 0 , 3 3 1 3 I G N , U , L , C M E G A , 0 , A L P 01 53 3 3 FORMAT( 5 X* SIGNS* . F f . 1 , 4 X #U=» , F 7 « 4 , 4 X •£ =« , F 6 « 3 , AX*CMEGA=• , 1 P E 1 2 * 4 » 1 w X ' O —* ,1 Pc 1 3 * 2 , 4X* A L P H A O /P I = • , 1 PF. 1 2 • 4 / / / ) 01 54 130 CONTINUC 0 1 6 5 IFC'3 • GH • 0 MA X ) GO TO 190 0 1 6 6 0 = D + .3 ELD 0 1 6 7 GO TO 6C 0 1 6 5 1 90 IF CE *GF *E M A X > GO TO 191 C I 6 9 E = E * DLLS 0 17 0 GO TO 50 C 1 71 1 01 I T ( J . G 5 . T 1 A X ) GO TO 192 0 1 7 2 U = U ♦ 3FLU 01 73 GO TO 40 0 17 4 1*3? I p ( OMH GA• GF« OMGMA X 1 GO TO 1 9 3 0 1 75 .JMLGA = OMEGA + OELOMG 01 76 CO TO 3C 0 1 7 7 1 41 CC NT I4U2 0 1 7 3 1CC0 CONTINUE C l? o W R I T C ( 6 , 2 o ) 0 ! * ? 26 F P P M A T ( / / / , F X , • NATUPAL FND OF I D 1 9 0 * ) 0151 STOP 01* 2 fc ’ ND 126 FORTRAN oooi C 0 * 7 OOC4 0 0 0 5 OCf 6 C 9 0 7 no os n ?C 9 c 0 3 1 1 CO! 2 on 3 0 0 1 A CO 1 5 001f t C 01 7 ooia 0 0 1 R 0 0 ’ 3 O C ? l r.** 0 0 2 3 0 0 2 4 0C 2 S 00. ? a CP 27 002^ C 0 ?5 0 0 3 0 C 0 3 I 0 0 72 0033 P 0~ 4 C >33 0 0 3 6 0077 0 '20 0 0 3 5 0 0 AC 30-1 O C A £ I V G L r V E L 21 L IN E A R O A T £ a 7 4 0 5 4 1 5 / 1 4 / 3 1 S U B R O U T IN E L IN E A R C C T H I S S U B R O U T IN E C A L C U L A T E S S TEA DY S TAT E RESPCNSE OF THE SYSTEM C I N THF ABSENCE OF THE IM P A C T DAMPER AND ALSO GENERATES L IN E A R AND C C E N F R A L I 7 T 0 PARAM ETERS f q r SUBSEQUENT PAR TS C C O M M O N /C -l^ /c A L P H A , C B E T A . 0 M E G A |D E N S T Y ,C 0 N T , I SOL< 5 0 ) CRMm h m / c r i /N , M , o j * 3f N M ( 2 C ) , Z E T A ( 2 0 ) , 0 ( 2 3 ) .E T A C 2 0 ) , W ( 2 C ) • A ( 2 0 ) , 1 P S I ( 2 f > , P £ I P ( 2 C ) , F ( 2 0 ) . P H I < 2 0 , 5 0 ) • KWRIT£*PLA< ( 5 0 ) CCMMON/C = ? 4 /IN C . XMAX { 5C ) , TX»A <150 ) . X 0 C 5 G ) • X D O T A (S O ) ,X O G T R (S O > D IM E N S IO N X { 5 0 ) , 0 ( 2 0 ) W R I T E ( 6 . 2 R > 2 4 F O R M A T ! / 2 X » I * , 7 X » M « , f 3 X * C , , 9 X * K « t 9 X * W » , R X * R * , 7 X * Z E T A * , 7 X * E T A * , 1 9X * F * , 1 0 X • A * • S X • P S I / P I • ) n o 10 i = i , m SK = G F N '•* ( I ) * W ( T ) * * 2 s c = C A L ;»HA»C.ENM( I ) ♦ C R E T A * SK Z E T A ( I ) = SC / ( 2 . 0 * S Q R T ( C E N M ( I ) * S K ) ) R A ( I ) a F ( I ) / S K / S O R T C < 1 . 0 - P ( I > * * 2 ) * * 2 ♦ ( 2 * 0 4 7F T A * * 2 ) ) PS I P ( I ) a P S I C l ) / P I 1 0 W PIT»M * * 3 3 1 I , -3ENM( I ) , SC « SK , W ( I ) * R ( I ) « ZETAC I ) , c T A ( I ) , F ( I ) , A ( 1 ) , I P S I P ( I ) 3 0 F O R M A T ( 1 X , 1 2 , 2 X , 1 P 7 E 1 0 • 3 , 1 P 3 E .I1 * 3 ) C C D E T E R M IN S SS D IS P L A C E M E N T RESPONSE AND MAX D I S P C I F ( < W R I T E « N . i . l ) GO TO 93 W R IT ' £ ( 6 , 3 C > PO F O R M A T C / / / 5 X * THE F O LLO W IN G I S S * S * D I S P L A C E M E N T * ) 9 3 DO 4 0 1 = 1 ,N X M A X C I) = 3 • 4C T X M A X ( I ) = 0 • I N C 1= 1 NC♦ 1 DC Lw T = P I / F L O A T C I N C ) DO **2 1 = 1 • IN C 1 W T = F L D A T (1 - 1 ) FDELWT C C CO NSTRUCT 0 . 0 , X .X c PC 4 2 j = l . M 4 3 0 ( J ) = A ( J ) * S I N( WT - P S K J ) ) C A L L M V P R D C P H I , Q , X , M , N ) I F ( K . * f R I T 2 « N 2 * l ) GO TO 3 4 - W R T T t ( 6 , 4 7 ) I , W T , ( X C J ) » J = 1 » N ) 4 ? F O R M A T ( 1 X* ! = • • I 3 . 2 X * W T = * , F 7 * 4 • 2 X • X = • , C I P 1 0 E 1 0 . 2 ) ) 9 4 OP 4 4 J = 1 , N IF C A B S C X C J ) ) * L E « A Q S C X M A X C J )) ) GO TO 4 4 XM AX C J )= X { J ) T XM4 X C J )= W T 4 4 . C E N T IN U E 4 2 C O N T IN U E r.P o 0 I = I , N 9 0 P E A K { I )^ X M A X ( I ) FORTRAN I V G L E V E L c 1 L I N E A P DATE = 7 4 0 5 4 1 5 / 1 4 / 3 1 PAGE 0 0 0 2 0 0 4 3 WRI T £ { 6« 4 6 ) 0 0 4 4 4 6 F O R 4 A T ( / /* 5 X • S * S * D I S P * A M P .Q ? X«> ■>045 OH S 3 I = 1 • N C C 46 J = I S O L ( I > 0 0 4 ? 5 J W F I T r ( 6 , 4 5 ) J • XM A X { I > • TX M A X ( I ) 0 0 4 A 4 5 F O R M A T (£ X « I = * • 1 2 * 5 X « X M A X ( I > = » . 1 P E I 2 • 4 « 5 X • T X M A X < I ! = • • 1 P E 1 < « 4 » 0 0 4 0 IF < K * n I T E * N 'i« 1 ) GO TO 3 5 0050 « » lT t£ ( 6i5G ) 0 0 5 1 5 0 F OPMA T C / / / 5 X * THE FO LL O W IN G I S S * S * XDOT* I C C D E T E R M IN E or»OT*XDOT c 0 0 5 2 DO 4 3 1 = 1 • IN C t 0.053 V T * F L 0 A T ( t - 1 )« tt? L W T C-054 DO 4 0 J = 1*M 0 0 - 5 4 0 Q ( J ) = n MFn 4 * A ( J )* C O S (W T - PS I ( J ) ) 0 0 * 6 C A L L M V P P n (P H l COST 4 3 WP T T ~ ( 6 * 5 1 1 X « w T t ( X ( J > • J = 1 » N l 3 0 * 3 51 F O R M A T !1 X » ! = • • I 3 , 2 X » W T = * . F 7 * 4 , 2 X • X D O T = • . I 1 P 1 0 E 1 0 * 2 > ) 0 0 5 9 3 5 C O N T IN U E 0 0 6 0 RTTURN 5 0 6 1 .................. - END POPTPAM IV < • LEVEL 21 NLINR DATE = 74054 1 5 / 1 4 / 3 1 PAGE C C C l 129 z O o C J H ——in < o c o -J WIT • 3 U * * o _| X^CJ < • U j — u — a cj O *'i) a cj.u x o — —H • a C<»x — UIHO'O Q v L J J f\| 1" _J X * w a C ? • Ui (A — U I ——c c z * c m • HCJ • — t f l Z w O O U J u ffc jc j C»X«-w a • ••M C V J U J ► - T i n to i - c a c l/)W • • U J z ——— h lt>4CC C.HCJN z #MJ— W < N IL X a o x x i n u • * c to s —— • U J OC-OO H • f\JCJ — < < — —H J h i t • 3 w Z — O ' U a a tO H -I u o a • < • *XC) u *H x a — • U J a »C;N z -jy O ’H H o z ^ t r 3 N S l / S r: r- a fo r n c c xar. cr ^ t - u * u 3 1 N X - N rna Z Z C T ' j C w Clu, ' tn« 2 ■ ? —> < r i < ? XU c n x r .i HO u u u * « o < J H H -tiJ w — w U lZ X O C .C O O o * “■ inajo tr jU 3 1 0 Ui u. to c o o t to •- c a a * - * X — x < » - l rt II x ( - < l i < < < i I I I I i-tTH Q CL. <-»c- 1 - 2 . t o ~ a I v < — ?kj — * — — —— a — •- n n •r .j h x L C N • n cr « 1 I ~ i n ii o — < 'r' < « < < H f !- — » ■ • L 'Z S '•.♦-•a l. t n « r # c r 2 to ui « « c j r <t m tL a cr ct « * * ■ « to to C J — *** f (T r in cr U O IT tnc; r*fn c . + a ffi CM fO *t If* C J 1 h h m h OC \ O S \ a c r e • x ri • « *+ r — —/v ^ c m w H X M v r i i : cj — rof) UJUJlJlM I C —C | E C I I I I I I IH<} if* •n i r e UJUolli! I I I ) I I I I I I I I I I I I I I I I I I I I I I I I I I I I 5CXE * • I C ■ ’ a n * C J I - -• I t " » I c C l • - o — II .1 L'iON'i; *X cj cvjoj i*) i1 ) to f*) cj o. cj c » ; r»n n n »x *t a . c c c d c t d c c c c r r c ' u c a i i s c j i r : u u u u c c o o ic «c n t it r» ^ ' I r ' <j io >o c- <r c <*. x o - w «* m « < • c < r o» <*j r< «r m*c H <r c? x m p - <# in «r h w c* n ’f.i-ii. r i n r H i ' i ^ f i " ^ t ' o »t •»•» <t 4 4 a m 066 «*0 ' ' ^ c c ) c c c f n o c c o o c ' f i c c o y f j o c c c i o c o c c n o c c o c o c c r e o d o o c o o o c o c c t o o o c c c c i t c o o c c o o o o c o t t o o o o o o v o o c y o FOOTSAM IV G CEVEL El NLINR DATE = 74P5A 1 5 / 1 4 / 3 1 PAGE 0 0? ?. 130 (0 W n a v > + to # r > « ■ » < /> n o # * i / i * ■ * * # ** fw aj w C M C 03 + + <M ■* <* n a r - c t r F T O E 11' S V * # « ■o o (\j n ' W <\* O 4 <* ( r t t w r j f f c C £ tl II II II II I pi r3 <* in * «i n * j *t r C O O C I I . C N O ro n E d *■«</ LO <t n | # ■ e e a. rrar • « in < n s O v \ N + > 4 lO fn « j< * r Cj CT T .C r . in * N O C O r * + t/> C O r v H - ' O — O N S «t > » E C <# -10 c ► C O o r. c. r. ^ c: o — o C L 0 I 1 o N • N O O n f . # • W # C w t C' — « - s i - * - n r - o ii ii n u n « « * ■ . 1 5 1 1 0 II 1 1 X i ) a x e . || || *.>-• ►-*-« I ? rc 'C 0 : c o o s e * + I-*-* l!* i x r t u i n cc q a z w o itn z •-< ii a I I I I I - H I I I I 3 rr c i-o N B ' G P h M D ’ J Z ccuhknn'3'l; n *« < v n ^ in ■ < ) p- a) <» c »< cj i" in *r n t) o**» p i p i # in -o n cr» in it in in it ir m ir u i < *• o *o *n »o -n s.* *r -c n n n n n n p. n n o p ^ o c j o r . c c c c n t c c r . c '■‘ o n o r e o n e o 0 0 0 0 0 0 0 0 X 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 FORTRAN I V G L E V E L 21 1 5 / 1 4 / 3 1 ^or3 3 0 C * 0025 0 0 0 7 000.5 0 7 0 - ‘ > - 0 0 1 0 .ox 1 0 0 1 2 0 0 1 3 0*14 0 3 1 5 0 0 1 s 0 0 1 7 2 31 5 0 3 1 ? 0 0 2 0 0 0 2 1 C 0 2 2 0*2? 0 3 2 4 0 0 2 5 0 37 6 0 0 2 7 . 0 3 2 4 a J 2 4 0 0 3 3 0 3 3 1 0 0 7 2 0033 0 5 2 4 C 0 3 5 0 3 2 6 0037 0 0 3 3 0079 0 3 4 0 S U B R O U T IN E MAXMTN< A L P H A 0 ) T H I S SUE RQ U T IM E C A L C U L A T E S T h E MAXIMUM VALU E S OF X AND <DOT I N ONE H A L F OF THE P E R IO D C O » 'M O N /C 3 1 /N • « * P U G E N M ( 2 0 1 . Z E T A U O ) . M 2 Q ) « FT A ( 2 3 ) . W ( 2 0 ) . A ( 2 0 ) « 1 R S I ( 2 0 ) . " S t P ( 2 J ) . F ( 2C > . P H I < 2 3 . 5 0 ) .KWR I T E .P F A K ( 5 0 ) C C M « C N /C 3 7 / S I ( 2 0 ) , S 2 < 2 0 ) .0 C < 20 ) • Q D O T A (2 0 I C C f t v iD N /C n L / lN C . X V A X < 5 0 > . T X M A X ( S O ) * X O ( 5 C ) . X 0 ) T A ( 5 0 ) . X D O TP < 50 ) C '.W M C "J /C '< 5 /C r^ < 20 ) , £ R 3 ( 2 0 ) * £ 0 4 1 2 0 ) » E B 5 ( 2 0 ) . £ Ell 2 < 2 3 > • E D I 3 < 2 0 ) ♦ 1 F l3 l 4 0 0 1 1 = 1 . N 1 * U A X ( I ) = 9 . I N C 1 = I N C + 1 D“ L VT = P I / P L O A T ( I N C ) I c ( K W R I T E . Z O . l ) W P I T S ( 6 . 1 5 ) 1 5 FORMAT < / / f . X * IM P A C T DAMPED D IS P L A C E M E N T * ) DO 2 1=1 . I NCI W T = F L P A T < 1 - 1 ) 40E LW T C A L L M A T R IX < WT > o n < 3 J = 1 .M S " * ( J ) = 4 < J ) * S T N ( W T ♦ ALPHAO - P S I ( J ) ) 9 Q( J ) = r s ? ( J ) « O D O T A (J ) ♦ E B 3 ( J ) * 0 D ( J ) ♦ E B 4 ( J ) * S 1 ( J ) 1 ♦ p n c < J ) * S 2 ( i ) ♦ S 3 ( J ) C A L L M V P R D ( P H I . O . X . M . N ) I F ( K # « I T 2 . E 0 * 1 ) W R IT E ( 6 . 8 ) I . W T , ( X ( J ) . J = l « N ) D E T E R M IN E XM A X ( J ) DG 3 J = 1 * N 1 r < A H S ( X { J ) ) . L E * A R S (X M A X ( J ) ) ) GO TO 3 X M A X ( J ) = X ( J ) T X M A X ( J )= W T 3 C O N T IN U E 2 C O N T IN U E I F C < W R IT (-* N 3 * 1 ) GO TO 11 D E T E R M IN E X D O T (W T ) W R IT r < 6 . 7 ) 7 P D K M A T ( / / / / / 5 X * X D 0 T « ) DO 6 T=1 . I NCI WT = CL 0 A T ( 1 - 1 )*O E L W T C A L L M A T R J X (W T ) DO I f . J = l . M S 4 ( J ) = A ( J ) * C O S ( W T ♦ ALPHAO - P S I ( J ) ) 10 O D O T ( J ) = F > ) 1 2 ( J ) * QDOT A ( J ) ♦ EH 1 3 < J >* Q 0 < J > ♦ E P 1 4 ( J ) * S 1 ( J ) I + F rU 5 ( J ) * ? 2 ( J > ♦ S 4 ( J ) C A LL M V P R D ( P H I . Q D O T . X . M . N ) 6 WR I T T ( 6 . 8 ) I . WT ♦ ( X ( J ) . J = l . N ) B F O R V A T d X * I = * * I 3 . 2 X * W T = * . 1 P ~ 1 0 * 2 . ( 1 P 1 0 E 1 0 . 2 ) ) 1 1 RETURN END 0001 131 e ORTRAN IV G LEVEL 21 MATRIX DATE = 74054 1 5 / 1 4 / 3 1 PAGE 0 0 0 1 132 -> c -»c O W C N tn —ip — w t f w r j 011X03 w *»l iy Mo.0. • * O • — H w ii ic J 2 —acy a < o t r *•«'■» > o * a in • » » D M < K > ■ O * » t» — h Z »- a • • < U J U 1 0 — H a Z t f O O td S IU luHlflW u. 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X o*g-« o 3»s p. m -o i «n o o t> O —X 2 6 £ I —I n ( / 1 ' I I " </, t ' l t n / . t , ( < 'j (if ( lU 'i ('! I'. I' I' INFORMATION TO USERS This materia! was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1.T he sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. 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Silver prints of "photographs" may be ordered at additional charge by writing the Order Department, giving the catalog number, title, author and specific pages you wish reproduced. 5. PLEASE NOTE: Some pages may have indistinct print. Filmed as received. Xerox University Microfilms 300 North Z eeb Road Ann Arbor, Michigan 48106 74-17,354 KQRNFELD, Joyce Lynne, 1942- ASSERTIVE TRAINING WITH JUVENILE DELINQUENTS. University of Southern California, Ph.D., 1974 Education, psychology University Microfilms, A X E R O X Company , Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. ASSERTIVE TRAINING WITH JUVENILE DELINQUENTS by Joyce Lynne Kornfeld A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Educational Psychology) January 1974 UNIVERSITY O F SOUTHERN CALIFORNIA TH E GRADUATE SC H O O L U NIVERSITY PA RK LO S A N G ELES. CALI FO R N IA 9 0 0 0 7 This dissertation, written by jJ-Q Y -CE.-JL iYHNE KD.RNJF.EIiH............................. under the direction of h.ex... Dissertation Com mittee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillment of requirements of the degree of D O C T O R OF P H I L O S O P H Y Dean D ate . 1 . ^ 4 $ 7 DISSERTATION COM M ITTEE Chairman ACKNOWLEDGEMENTS This investigation represents the combined effort, thought, and cooperation of many associates of the writer, foremost of whom is Dr. Frank H. Fox, who in addition to his interest and encouragement, provided a consistent model of strength, wisdom, humor and intelligence. Appreciation is extended to Dr. Robert Smith, Dr. Eddie Williams, and Dr. Eugene Briere for their valuable suggestions and advice during the various stages of the research. The writer wishes to express deep appreciation to Dr. Rodney C. Hoeltzel for his exceptional performance as therapist-facilitator for the assertive training group. His special blend of self-assertiveness and empathy earned him the respect and cooperation of a marvelous group of eight young people. Appreciation is extended to the students at the Santa Monica Boys Club (CDC), who provided hours of in terest, humor, love, and frustration and to the staff of the school who provided understanding and cooperation. Finally, special thanks are extended to my mother and father who provided continual support, interest and encouragement through every phase of my life and my educa tion. ii TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ..................................... ii TABLE OF CONTENTS.................................. . iii LIST OF TABLES ............................................................................................................... V Chapter I THE PROBLEM..................................... 1 Introduction Purpose of the Study Delineation of the Problem Importance of the Study Definitions of Terms Used Delimitations of the Study Organization of the Remainder of the Study II REVIEW OF THE LITERATURE........................12 Self Concept Delinquency and Self Concept Delinquency as a Defense Delinquency Reinforced by the Environment The Delinquent Subculture The Learning of Appropriately Assertive Behaviors Summary III METHODOLOGY.......................................25 Organization of the Chapter Research Design and Statistical Analysis Research Sample Data Collection Procedures and Recording Methodological Assumptions Limitations of the Study Null Hypotheses IV ANALYSIS AND DISCUSSION OF FINDINGS ........... 35 iii TABLE OF CONTENTS (Continued) Chapter Page Analysis of Results Discussion of Results V SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS . . . 48 Summary Conclusions Recommendations BIBLIOGRAPHY ............................................ 55 ADDENDUM................................................. 60 iv LIST OF TABLES Table Page 1. DIRECTION OF CHANGE PRETEST TO POSTTEST .... 34 2. ANALYSIS OF FACTOR CHANGES, PRETEST TO POSTTEST.........................................36 3. RESULTS OF ANALYSIS OF FISHER'S EXACT PROBABILITY TEST FOR THE FACTOR, BEHAVIOR . . 37 4. RESULTS OF ANALYSIS OF FISHER’S EXACT PROBABILITY TEST FOR THE FACTOR, INTELLECTUAL AND SCHOOL STATUS ............... 38 5. RESULTS OF ANALYSIS OF FISHER'S EXACT PROBABILITY TEST FOR THE FACTOR, PHYSICAL APPEARANCE AND ATTRIBUTES..................... 39 6. RESULTS OF ANALYSIS OF FISHER'S EXACT PROBABILITY TEST FOR THE FACTOR, ANXIETY . . . 40 7. RESULTS OF ANALYSIS OF FISHER'S EXACT PROBABILITY TEST FOR THE FACTOR, POPULARITY.......................................41 8. RESULTS OF ANALYSIS OF FISHER'S EXACT PROBABILITY TEST FOR THE FACTOR, HAPPINESS AND SATISFACTION..................... 42 v CHAPTER I THE PROBLEM Introduction Self concept is a relatively neglected area of research in regard to adolescent aggression and juvenile delinquency. Studies such as Deitche (1959) and Jensen (1972) suggest that juvenile delinquents test lower (less positive) on self concept scales than non-delinquents. The research of Gold and Mann (1972) and Dimitz, et al. (1962) indicates that juvenile delinquency may be a de fense against a derogated self-image. There is much pres tige attached to delinquent acts within the deviant sub culture where aggression is required and subsequently rewarded. Data obtained by Bandura and Walters (195) and Sykes and Matza (1957) indicate that juvenile delinquency is a learned set of behaviors which are expected and sub sequently reinforced by the environment (subculture). An experiment by Videbeck (1960) supports the proposition that self concepts are learned and that the evaluative responses of others play a significant part in that learning process, indicating that self concept is 1 2 amenable to change through the processes of modeling, practice and reinforcement by peers. Assertion training offers the juvenile delinquent a supportive environment for change. It offers him the opportunity to broaden his repertoire of behaviors and gives him a choice of responses to any particular set of experiences. Assertion training involves a dynamic skills training procedure which allows the individual to realize a greater range of his behavioral and attitudinal poten tial. Background Bandura and Walters (1963) offer substantial ev idence that novel modes of aggressive (or deviant) behav ior are readily acquired through observation of aggressive (or deviant) models. Results of these investigations corroborates field studies which have demonstrated the role of modeling in the origin of anti-social aggressive (or deviant) behavior (McCord and McCord, 1958); and in the cultural transmission of these aggressive response patterns (Bateson, 1936; Whiting, 1941). Assertion is also a learned response. Just as an individual may learn inappropriate responses to specific situations by observing models (Walters and Brown, 1963; 3 Bandura and Walters, 1963; and Wheeler, 1966); the in dividual may also learn appropriately assertive behaviors through modeling (Chittenden, 1942). It has been suggested (Alberti and Emmons, 1970) that assertive training may serve to increase the in dividuals feelings about himself in a positive direction. Problem Situation— Area of Concern A number of studies, for example, (Deitche, 195 9; Teichman, 1972; Jensen, 1972) indicate that mean self con cept scores on scales of self concept of non-delinquent boys are significantly higher (more positive), than mean self concept scores of delinquent boys. It is possible, then, that in the process of learning appropriately assertive behaviors and as a result of the positive consequences resulting from the use of this new behavioral repertoire, self concept may be in creased in a positive direction. The relative superiority of a behaviorally orient ed approach probably stems from the fact that a basic change in behavior provides an objective and genuine basis by which one feels self-respect, self confidence, and dignity [Albert Bandura, 1969, p. 91]. 4 Purpose of the Study The present investigation was undertaken to ex plore the efficacy of the use of assertive training to increase (in a positive direction) the self concepts of a select group of juvenile delinquents. Results of studies suggest that self concept has pervasive and significant effects on the individual's behavior. Various investigations (Deutsch and Krauss, 1965; and Deutsch and Solomon, 1959) employing diverse tests, situations, and techniques provide consistent data regarding the behavior of individuals. Findings suggest that individuals high in self-esteem are more effective in meeting environmental demands and enjoy a greater sense of well-being than individuals with low self-esteem. The consequences of self-esteem are multifaceted in their expression and represent an integration of behavior pat terns, attitudes, and perceptions. Self concept is a learned behavior (Videbeck, 1960; Kinch, 1963) and therefore amenable to change through reinforcement and practice. Assertive training offers an atmosphere for elicitation and support (re inforcement) of selective behavior change. If delinquent behavior is a consequence of derogated self-image subject to reinforcement by the delinquent subculture, and if self 5 concept is in reality a dynamic system of learned char acteristics and behaviors, and therefore subject to change, then assertive training may well be an important addition to programs designed to alleviate stress and change behav ior. Assertive training consists of modeled assertive behaviors and behavioral rehearsal which includes the techniques of role-playing and role reversal. Behavioral rehearsal is based on the principle of successive approx imations. The individual begins with small assertions which are likely to be rewarded. When the individual begins to feel more comfortable with his assertive behav ior, the encounters presented become more and more threat ening as the individual can successfully handle each suc ceeding situation. The individual chooses his own situa tions to practice and creates his own assertion hierarchy. The purpose of the assertive training group is to provide a supportive atmosphere which offers the opportun ity for the individual to learn and practice appropriately assertive behaviors within a reference or peer group which support these changes in behavior. Questions to be Answered In terms of the hypothesized relationship between 6 assertion training and self concept, the objectives of the investigation was clarified in terms of the following question. 1. Will assertion training increase self concept, in a positive direction, in a group of juvenile delinquents? Delineation of the Problem The research problem was concerned with ascertain ing whether or not assertive training would increase self concept in a positive direction for a select group of juvenile delinquents: 1. A sample of eight 13-17 year old court ruled ju venile delinquents and a non-equivalent group of five 13-17 year old court ruled juvenile delin quents . Research Hypothesis Relative to the research problem the following hypothesis was formulated: 1. Does assertive training increase the self concept of identified juvenile delinquents? 7 Importance of the Study As a review of the literature in the next chapter indicates, there is a paucity of research that has attempt ed, by experimental means, to facilitate positive change in the self concepts of juvenile delinquents, even though low self-esteem and derogated self-image have been linked as factors relating to, or emanating from, juvenile delin quency. It appears that, assertive training, individually or in groups, has not previously been used experimentally specifically to increase self concept in a positive direc tion. This study attempted to explore the possibilities of employing assertive training procedures to facilitate behavior change and increase self concept in a positive direction. Definitions of Terms Used Assertive Behavior. "Behavior which enables a person to act in his own best interests, to stand up for himself without undue anxiety, to exercise his rights without denying the rights of others, is called assertive behavior!' [Alberti and Emmons, 1970, p. 7]. 8 Assertive Training. An action oriented therapy based on the principles of reciprocal inhibition and re inforcement theory and incorporating such methods and procedures as: Modeling. The therapist or another group member will sometimes play the trainee's role in order to model ways in which the subject might respond in a certain situation. Behavioral Rehearsal. Non-threatening role-play ing situations which are enacted until the trainee feels comfortable with his role. The encounters then become mildly threatening as the trainee acts out his more difficult situations until he feels comfortable with them. Behavioral rehearsal encompasses the var ious techniques of role-playing, role-reversal, and modeling. Role-Playing. Acting out common stressful situa tions, for practice, which will be enacted or have been enacted in real life. Role Reversal. In a given situation the therapist^ or another trainee, may reverse roles with the trainee and play the trainee while the trainee plays the part of his antagonist (mother, boss, salesman, teacher, 9 etc.). Successive Approximations. (Shaping). The train ee should begin with small assertions that are likely to be rewarded and from there proceed to more difficult situations. If the trainee does suffer a setback, the therapist must be ready to help him analyze the situa tion and regain his confidence. The trainee's initial attempts at assertion should be chosen for their high potential for success, so as to provide reinforcement. The more successfully one asserts himself, the more likely he is to do so from then on. Reciprocal Inhibition. When two responses are mutually incompatible, performing one response sup presses the other. Delimitations of the Study The present study was subject to the following delimitations that served to narrow its focus and general- izability. 1. A selected group of thirteen to seventeen year old court-ruled juvenile delinquents (on probation), from the Santa Monica Boys Club (California

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Creator Kahyai, Keykhosro (author)

Core Title Analytical And Experimental Studies Of Forced Vibration Of Impact-Damped Mechanical Systems

Contributor Digitized by ProQuest (provenance)

Degree Doctor of Philosophy

Degree Program Civil Engineering

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

Tag engineering, civil,OAI-PMH Harvest

Language English

Advisor Masri, Sami Faiz (committee chair), Pierce, John G. (committee member), Seide, Paul (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-686648

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