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Experimental investigation of nonlinear coupled vibrations of bars and plates Schneider , Bernd C. 1969

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EXPERIMENTAL INVESTIGATION OF NONLINEAR COUPLED VIBRATIONS OF BARS AND PLATES  by  Bernd C. Schneider B.A.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1967 0  A Thesis Submitted i n P a r t i a l F u l f i l l m e n t of the Requirements f o r the Degree o f Master of Applied Science I n the Department of Mechanical Engineering  We accept t h i s t h e s i s as conforming t o the required standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1969  In p r e s e n t i n g  this thesis i n p a r t i a l f u l f i l m e n t of the requirements f o r  an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , the  L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e  I further agree that permission  f o rextensive  I agree  that  and Study.  copying of this  thesis  f o r s c h o l a r l y p u r p o s e s may b e g r a n t e d b y t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s .  It i s understood that copying o r p u b l i c a t i o n  o f t h i s t h e s . i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my wr i t t e n p e r m i s s i o n .  Department o f  Mechanical  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a Date  Engineering Columbia  A p r i l 30. 1969  I  TABLE OF CONTENTS Page  ABSTRACT  i  ACKNOWLEDGEMENT  iii  LIST OF FIGURES  iv  LIST OF TABLES  •  LIST OF APPENDICES  CHAPTER I  ...  NOMENCLATURE  vii  DEFINITION OF THE PROBLEM  1  Introduction Statement o f the Problem L i t e r a t u r e Survey L i m i t a t i o n s o f the Study Definitions CHAPTER II  vi  *  •  THEORETICAL CONSIDERATIONS  6  D e r i v a t i o n o f the D i f f e r e n t i a l E q u a t i o n s o f M o t i o n f o r a Beam Under Plane Motion D i s c u s s i o n o f the D e r i v e d E q u a t i o n s CHAPTER I I I  1 3 3 ^ 5  6 11  APPARATUS AND INSTRUMENTATION S i g n a l Flow I n d i v i d u a l Instruments T e s t Bed and Frame Suspension P l a t e Support Bar Clamp Models and Boundary C o n d i t i o n s Circular Plate C a n t i l e v e r Bar S t r a i n Gauges and B r i d g e s Circular Plate C a n t i l e v e r Bar Bridges  1^  '  ^ 18 21 22 23 3 2.h 2^ 25 25 25 26 26 2  Page  CHAPTER IV  TESTING.  27 C a l i b r a t i o n and Check Out T e s t Procedure • • Constant Power P l o t D i s c u s s i o n . I n t e r p r e t a t i o n o f Frequency S p e c t r a . . Chladni Figures P l o t t i n g and Photography  CHAPTER V  CHAPTER VI  CHAPTER VII  EXPERIMENTAL  RESULTS  27 28 30 33 3k J>k 35  Circular Plate  35  C a n t i l e v e r Bar  37  DISCUSSION OF EXPERIMENTAL  kl  RESULTS  Circular Plate  *H  C a n t i l e v e r Bar  kj>  SUMMARY AND CONCLUSION  kj  S u g g e s t i o n s f o r F u t u r e Research Summary Conclusion  •  •  k? k8 50  BIBLIOGRAPHY  52  APPENDIX  53  i  ABSTRACT  The theory presented describes the p h y s i c a l phenomenon of nonlinear coupling of l o n g i t u d i n a l and f l e x u r a l v i b r a t i o n s when a beam i s excited transversely at high frequencies.  Equations aire derived based on the  B e r n o u l l i - E u l e r theory of f l e x u r e , by energy methods, t o describe the transverse and the l o n g i t u d i n a l v i b r a t i o n of a beam of constant cross-section plane motion.  under  The i n i t i a l crookedness of the beam and the l o n g i t u d i n a l  i n e r t i a , accounted f o r i n the theory, give r i s e t o the coupled v i b r a t i o n s . No closed form s o l u t i o n i s presented.  However, a simple a n a l y s i s of some of  the coupling terms suggests the existence of several coupled v i b r a t i o n s .  By  the method proposed h e r e i n , the frequencies of these v i b r a t i o n s can be established.  I n p a r t i c u l a r , the theory p r e d i c t s two l o n g i t u d i n a l coupled  v i b r a t i o n s with the frequency r a t i o 1:2. The i s good.  agreement between the theory and the experimental r e s u l t s  The v i b r a t i o n s predicted  e x i s t and the frequency r a t i o f o r the pre-  d i c t e d l o n g i t u d i n a l v i b r a t i o n s was 1:2. Further, the experimental r e s u l t s i n d i c a t e that there are more l o n g i t u d i n a l v i b r a t i o n s than i n d i c a t e d by the theory.  A l o n g i t u d i n a l coupled v i b r a t i o n at three times the frequency of  transverse e x c i t a t i o n was recorded.  There are i n d i c a t i o n s i n the data that  coupled f l e x u r a l v i b r a t i o n s at twice the frequency of transverse e x c i t a t i o n exist. A c i r c u l a r plate c e n t r a l l y supported and transversely was also tested. The  excited  Two pronounced resonant r a d i a l v i b r a t i o n s were recorded.  frequency r a t i o was 1:2. Coupled f l e x u r a l v i b r a t i o n s were not i d e n t i f i e d .  ii The i n f l u e n c e of the l o n g i t u d i n a l v i b r a t i o n on the f l e x u r a l v i b r a t i o n of the beam i s examined.  The l i m i t a t i o n s of the theory, of the  experiment, and the s i g n i f i c a n c e of the resonant s t r a i n s i s discussed.  Iii  ACKNOWLEDGEMENT  I wish t o express my gratitude t o my advisors, Dr. C.R. H a z e l l and Dr. H. Ramsey, f o r g i v i n g me the opportunity t o work on t h i s challenging project and f o r granting me so much l a t i t u d e i n my work. Further, I take t h i s opportunity t o thank a l l the technicians and s e c r e t a r i e s i n the Department f o r c o n t r i b u t i n g d i r e c t l y , or i n d i r e c t l y , t o the research. This study was made p o s s i b l e through research grant No. 9513-07 provided by the Defense Research Board of Canada.  a  iv  LIST OF FIGURES  Figure  p  age  1.  Extension and Rotation of Central Plane F i b e r .  7  2.  Reference Axis f o r Displacement Measurements  9  3.  S i g n a l Flowchart  15  k.  O v e r a l l View of the Instrumentation  18  5.  Test Bed  21  6.  Bar Suspension  7.  Constant A c c e l e r a t i o n and Approximated Constant Power Frequency Spectra  31  Chladni Figures f o r the Transverse V i b r a t i o n of the Circular Plate  35  8. 9.  22  •  Chladni Figures f o r the Transverse V i b r a t i o n of the 38  C a n t i l e v e r Bar 10.  Waveforms at Resonance f o r C i r c u l a r P l a t e  39  11.  Waveforms at Resonance f o r C a n t i l e v e r Bar  ko 2....  A-12.  Graphical A d d i t i o n of Sinusoids Frequency Ratio  A-13.  Graphical A d d i t i o n of Sinusoids Frequency Ratio 3 . . . .  C-l*f.  Tangential S t r a i n Frequency Spectra f o r C i r c u l a r Plate  C-15.  R a d i a l S t r a i n Frequency Spectra f o r C i r c u l a r P l a t e . . .  C-l6.  R a d i a l plus Bending S t r a i n Frequency Spectra f o r Circular Plate Bending plus A x i a l Frequency Spectra f o r C a n t i l e v e r Bar A x i a l S t r a i n Frequency Spectra f o r C a n t i l e v e r Bar....  C-17. C-l8.  56  60 6l 62 63  6k  LIST OF TABLES  Page  Table 1  Allowable A c c e l e r a t i o n Level  Table 2  Non-Dimensionalized Nodal R a d i i f o r P l a t e . .  36  Table 3  Non-Dimensionalized Nodal Distances f o r Bar  38  Table k  T h e o r e t i c a l Resonant Frequencies f o r the Transverse V i b r a t i o n of the C i r c u l a r P l a t e . . .  *fl  Non-Dimensionalized T h e o r e t i c a l Nodal Distances f o r a C a n t i l e v e r Beam.  kj>  Table 5  Table 6  •  T h e o r e t i c a l Resonant Frequencies f o r F l e x u r a l -Vibrations of the Beam.......  30  kk  vi  LIST OF APPENDICES  Page  APPENDIX A  Addition of Two Sinusoids  53  APPENDIX B  Linear Equations f o r Plate and Beam  57  APPENDIX C  Approximate Constant Power Frequency Spectra  60  vii  NOMENCLATURE  Symbol  a  = diameter of p l a t e , i n  f  = frequency, cycles per second 2  g  = a c c e l e r a t i o n of g r a v i t y , in/sec  h  = bar and p l a t e thickness, i n  r  = r a d i a l distance, i n  t  = time, sec  W W  = d e f l e c t i o n from s t a t i c e q u i l i b r i u m p o s i t i o n , i n = s t a t i c d e f l e c t i o n of bar, or beam, i n (includes d e f l e c t i o n due t o i n i t i a l crookedness)  W ,U  = p a r t i a l d e r i v a t i v e of a displacement with respect t o the variable x 2  A  = cross-section o f beam, i n  D  = plate  stiffness 2  E  = modulus of e l a s t i c i t y , l b / i n  I  = moment of i n e r t i a  T  = period, sec -3  V  = volume, i n  0  = angle, radians  v  = Poisson's r a t i o  <P  = mass/volume  co  = frequency, radians per second  ^  = increment of some v a r i a b l e  viii  Abbreviations  BAM  = bridge a m p l i f i e r and meter  CRO  = cathode ray o s c i l l o s c o p e  Hg  = cycles per second  RMS  = root-mean-square value of a function  J  Q  = Bessel function of the f i r s t k i n d , i n t e g r a l order  Y  Q  = Bessel function of the second k i n d , i n t e g r a l order  I  Q  = Modified Bessel function of the f i r s t k i n d , i n t e g r a l order  K  o o  = Modified Bessel function of the second k i n d , i n t e g r a l order o  o  1 CHAPTER I  DEFINITION OF THE PROBLEM  Introduction The application of the simple linear thin-plate theory at frequencies of order 10 KHz encountered in this experimental investigation i s not appropriate.  Additional phenomena arise i n the mechanism of acoustic  energy transmission under these circumstances to those encountered in the audible frequency range usually considered i n architectural acoustics.  An  analysis by means of the simple linear theory indicates for the upper end of the acoustic frequency range that i n plates the wavelengths of longitudinal and flexural waves are of the same order of magnitude; at frequencies much below the above frequency range the wavelength of the two types of waves differ by an order of magnitude.  Consequently, there i s the likelihood of  coupling between the two types of waves at the upper end of the frequency range swept out i n the experiment, whereas the coupling seems to be insignificant at frequencies much below 10 KHz. A simple model was set up to explain the physical phenomenon of acoustic energy transmission at the upper frequency range encountered in the experiment.  Following the approach taken by Mettler (5), that i s , energy  methods and the Bernoulli-Euler theory of flexure, the coupled differential equations of motion for a beam of constant, prismatic cross section under plane motion were derived.  The formulation of the problem differs from the  usual linear theory essentially in one respect - the assumed strain expression. The equations account for the i n i t i a l crookedness, or i n i t i a l displacement, as * barred numbers i n parenthesis designate references i n the Bibliography.  2 w e l l as f o r ap. u s u a l l y omitted  i n e r t i a term.  The i n i t i a l crookedness and  the l o n g i t u d i n a l r i n e r t i a term account f o r the coupled v i b r a t i o n s recorded i n the experiment.  The l i m i t a t i o n s of the two t h e o r i e s , the l i n e a r uncoupled  theory and the non-linear coupled theory, are e s s e n t i a l l y the same, namely those of the B e r n o u l l i - E u l e r theory of f l e x u r e . assumed t o be d i r e c t l y proportional  The bending s t r a i n i s  t o the distance from the c e n t r a l plane.  Plane cross sections are assumed t o remain plane, that i s , the shear deformat i o n i s ignored.  Rotary . i n e r t i a i s omitted and damping i s ignored.  Summariz-  i n g , the proposed theory d i f f e r s from the l i n e a r theory i n that s t r a i n s i n the c e n t r a l plane are assumed t o e x i s t , whereas they are ignored i n the l i n e a r theory. No closed form s o l u t i o n i s presented f o r the equations of motion derived.  However, an examination of some of the second order coupling  terms i n d i c a t e s the existence of coupled v i b r a t i o n s .  Thus, f o r a beam excited i  at the supports, transversely, the a n a l y s i s of the f o l l o w i n g terms, WW X  ,W W X2C  X  , from the coupled equations i n d i c a t e s the existence of two l o n g i -  X3C  t u d i n a l v i b r a t i o n s with the frequency r a t i o of 1:2. Using the fact that there are two coupled l o n g i t u d i n a l v i b r a t i o n s , one can i n t e r p r e t the mixed second order coupling terms i n the complimentary equation i n the same manner.  Effect-  i n g the necessary s u b s t i t u t i o n s , two coupled transverse v i b r a t i o n s with the frequency r a t i o 1:2 should occur.  These represent the influence of the l o n g i -  t u d i n a l v i b r a t i o n on the transverse v i b r a t i o n .  The method adopted i n the  i n t e r p r e t a t i o n i n d i c a t e s only the existence of the coupled v i b r a t i o n . I t does not give any i n d i c a t i o n of t h e i r r e l a t i v e magnitude; i t does not throw any l i g h t on the influence of the l o n g i t u d i n a l v i b r a t i o n on the f l e x u r a l vibration. vibrations.  Simply, the method just i n d i c a t e s the frequencies of the coupled  3  Statement of the Problem The purpose of t h i s study was: (1)  t o set up and check out the v i b r a t i o n system,  (2)  t o determine a n a l y t i c a l l y and t o i n t e r p r e t the coupled d i f f e r e n t i a l equations of motion f o r a beam of prismatic cross-section, under plane motion,  (3)  t o v e r i f y , experimentally, the occurrence of the coupled v i b r a t i o n s , i n the l o n g i t u d i n a l d i r e c t i o n ,  (4)  t o l o c a t e , experimentally, along the frequency spectrum l i n e , where the predicted resonant v i b r a t i o n s occur,  (5)  t o determine the importance of the associated resonant v i b r a t i o n s , and  (6)  t o determine f o r a c i r c u l a r plate supported and excited transversely i n the center the same experimental information as f o r the bar.  L i t e r a t u r e Survey To the best of the author's knowledge, the l i t e r a t u r e contains no record of the observance of acoustic coupling i n bars, or p l a t e s . t i c a l l y , t h i s coupled v i b r a t i o n has not been treated before.  Theore-  Up t o the  present time the transverse v i b r a t i o n of the bar, or the p l a t e , has been treated d i s s o c i a t e d from the l o n g i t u d i n a l v i b r a t i o n s .  Moreover, the l o n g i t u -  d i n a l v i b r a t i o n has been considered without regard t o the transverse v i b r a tions. The existence of s t r a i n s i n the c e n t r a l plane was postulated by mathematicians and s t r e s s analysts a long time ago. Pearson (6) and (7) showed that a beam under uniform transverse load does have s t r a i n s i n the  c e n t r a l plane; from p h y s i c a l considerations i t i s known that a non-developable surface has s t r a i n s i n the c e n t r a l plane.  I n the f i e l d of s t a t i c s , Timoshenko  (10) used t h i s f a c t t o derive the d e f l e c t i o n f o r a c i r c u l a r p l a t e .  To the  knowledge of the author s t r a i n s i n the c e n t r a l plane have not been p r e v i o u s l y considered f o r the dynamic case.  M e t t l e r (5) perhaps came c l o s e s t t o the  s o l u t i o n f o r the case of a beam i n plane motion.  I n h i s d e r i v a t i o n , he  retained the most s i g n i f i c a n t terms of the s t r a i n expression, thereby o b t a i n i n g coupling terms of the desired k i n d . There i s no i n d i c a t i o n i n h i s paper that he recognized the s i g n i f i c a n c e of these terms. L i m i t a t i o n s of the Study The prime reason f o r undertaking t h i s study was t o v e r i f y experimentally the existence of the coupled v i b r a t i o n s i n the p l a t e and i n the bar.  The l i m i t a t i o n s of the experimental i n v e s t i g a t i o n are as f o l l o w s .  Because of the a v a i l a b l e system, the experiment was r e s t r i c t e d t o 'approximate constant power' frequency spectra. Since the s t r a i n l e v e l was very s m a l l , the desired s i g n a l often vanished i n the e l e c t r o n i c noise and induced electromagnetism.  The experimental i n v e s t i g a t i o n was r e s t r i c t e d t o 10 KHz. The assumed  boundary conditions could not be r e a l i z e d i n p r a c t i c e f o r the c a n t i l e v e r bar. A s o l i d p l a t e could not be simulated.  I t was impossible t o i d e n t i f y , independ-  e n t l y from the s p e c t r a l s t r a i n record, the nonlinear coupled f l e x u r a l v i b r a tions. The a n a l y t i c a l expressions derived have l i m i t a t i o n s too. F i r s t , the study i s l i m i t e d t o n e a r l y f l a t surfaces and those s t r u c t u r e s able to develop bending stresses; second, the derived equations are not exact - the l i m i t a t i o n s of the B e r n o u l l i - E u l e r theory of f l e x u r e are present.  The method  adopted i n i n t e r p r e t i n g the coupled d i f f e r e n t i a l equations i n d i c a t e s only the frequency of the coupled v i b r a t i o n s .  The method adopted does not i n d i c a t e  5  the absolute amplitudes involved. Definitions Approximated Constant Power P l o t - the p l o t t i n g procedure used i n the s p e c t r a l analysis.  B r i e f l y , i t i s the break-up of the frequency spec-  trum i n t o short, constant a c c e l e r a t i o n segments making use of the maximum a v a i l a b l e power of the system.  For a f u l l  discus-  s i o n of t h i s method, see s e c t i o n Constant Power P l o t Discussion. Node - a l i n e , a point, or surface i n a standing wave where some c h a r a c t e r i s t i c of the wave f i e l d has zero amplitude. F i r s t Coupled Resonant Longitudinal  (Radial) V i b r a t i o n - l o n g i t u d i n a l ( r a d i a l )  v i b r a t i o n at the same frequency as the e x c i t a t i o n frequency i n the transverse d i r e c t i o n f o r the beam ( p l a t e ) . Second Coupled Resonant Longitudinal  (Radial) V i b r a t i o n - l o n g i t u d i n a l ( r a d i a l )  v i b r a t i o n at twice the frequency o f the transverse v i b r a t i o n of the beam ( p l a t e ) .  I t occurs at h a l f the frequency of the  e x c i t a t i o n as f o r the f i r s t l o n g i t u d i n a l ( r a d i a l ) v i b r a t i o n . Third Coupled Resonant Longitudinal  V i b r a t i o n - l o n g i t u d i n a l v i b r a t i o n at t h r i c e  the frequency of the e x c i t a t i o n i n the transverse d i r e c t i o n . I t occurs a t one t h i r d the frequency of the e x c i t a t i o n f o r the f i r s t l o n g i t u d i n a l v i b r a t i o n .  6  CHAPTER I I  THEORETICAL CONSIDERATIONS  D e r i v a t i o n of the D i f f e r e n t i a l Equations of Motion f o r a Beam Under Plane Motion As the f i r s t step towards the deduction of the equations of motion f o r a beam under plane motion, the s t r a i n expression has been derived. For the development of the s t r a i n expression f o r s t r a i n s i n the n e u t r a l plane f o r an i n i t i a l l y d e f l e c t e d l i n e segment, a geometric approach i s taken. Love (3) presents a mathematical d e r i v a t i o n .  The d i f f e r e n c e between the  equation deduced h e r e i n and the one i n the above c i t e d work l i e s i n the f a c t that i n the d e r i v a t i o n presented an i n i t i a l d e f l e c t i o n curve i s assumed and that the dynamic d e f l e c t i o n s are measured r e l a t i v e t o i t .  Love's deduction  assumes a f i x e d coordinate system i n which the i n i t i a l d e f l e c t i o n of any point of the n e u t r a l plane i s zero. The i n i t i a l d e f l e c t i o n curve assumed includes the s t a t i c displacement of the beam from a p e r f e c t l y f l a t surface and any e f f e c t s due t o the non-homogeneity of the m a t e r i a l .  A l s o , since the  l o n g i t u d i n a l i n e r t i a i s an i n t e g r a l part i n the proposed theory, the second  2 order term  has been retained i n the s t r a i n expression. Since an e l a s t i c  system i s assumed, the Lagrangian d e f i n i t i o n of s t r a i n i s used throughout. The barred l e t t e r s i n F i g . 1 r e f e r t o the s t a t i o n a r y system, that i s , these coordinates account f o r the i n i t i a l crookedness, s t a t i c d i s placement, and any e x i s t i n g non-homogeneity. i s shown.  A f i b e r of the c e n t r a l plane  w  F i g . 1. Extension and Rotation of C e n t r a l Plane Fiber Before deformation, a l i n e element of the a x i s of the beam has the length  ds  ~ ?p (1 + W dx x  =  when the beam has been deformed, the length i s ds  \L  =  (  (1 + U ) + (W + W ) x x x J 2  2  1 2  dx  The elongation of the a x i s of the beam i s given by 1  f (1 + U  ds - ds =  L  ) + (W  + W )  2  X  X  2  1  1 2  dx - (1 + W )  J  X  2  2  dx  X  Expanding each r a d i c a l by the binomial theorem, keeping second order terms of the same type, the t o t a l s t r a i n becomes . U +WW + J (U + W ) = ", :x xx 2 x x_ 2  g '  k  =  i  2  h 2 x 2  +  Since small transverse i n i t i a l displacements are considered, one i s e n t i t l e d  8  to w r i t e £  = U + W W + i (U + W ) x x x 2 x x 2  x  (1)  2  This i s the most general s t r a i n expression f o r the s t r a i n s i n the c e n t r a l plane of a beam f o r s t a t i c , or dynamic d e f l e c t i o n s i n two orthogonal d i r e c t i o n s . The above expression i s complete as i t stands.  However, an  2 estimate based on the experimental f i n d i n g s i n d i c a t e d that the term U i s X 2 approximately two orders of magnitude smaller, than  . For t h i s reason, the  2 term U  x  i n the s t r a i n expression i s omitted i n f u r t h e r d e r i v a t i o n s . The coupled equations of motion f o r the beam were derived by  Mettler (5).  I n the d e r i v a t i o n of the equations of motion, the basic assump-  t i o n s of the B e r n o u l l i - E u l e r theory of f l e x u r e are adopted.  Accordingly, i t  i s assumed that the bending s t r a i n s vary l i n e a r l y with the distance from the middle surface. Plane cross-sections are assumed t o remain plane, that i s , the shear deformation i s neglected.  Rotary ' i n e r t i a e f f e c t s are ignored.  Further, i n t e r n a l and e x t e r n a l damping and d i s t r i b u t e d transverse loads are not considered. Let W (x) be the i n i t i a l d e f l e c t i o n of the beam and Wv(x,t) be the dynamic d e f l e c t i o n of the beam measured r e l a t i v e t o the i n i t i a l d e f l e c t i o n curve as shown i n F i g . 2. The f i r s t assumption made i s that there are s t r a i n s i n the c e n t r a l plane of the beam. These can a r i s e from the non-developability of the c e n t r a l surface, from d i s t r i b u t e d transverse loads, or from l o n g i t u d i n a l inertia..  How the r e s u l t i n g s t r a i n i s made up, that i s , whether s t r a i n s  induced by ' i n e r t i a e f f e c t s predominate over s t r a i n s due t o the non-<  9 d e v e l o p a b i l i t y of the surface, i s not discussed i n t h i s t h e s i s .  INITIAL DEFLECTION CURVE  DYNAMIC DEFLECTION CURVE  F i g . 2.  Reference Axis f o r Displacement Measurements  The a d d i t i o n a l extension of a f i b e r at a distance  z  from  the a x i s i s F . &  =  - zW  1  XX  This follows from the usual beam bending theory, or from the assumption that plane cross-sections do remain plane.  £  =  C  x  The t o t a l s t r a i n i s therefore,  +  1  £  The next major assumption i s , as i s u s u a l l y done i n the beam bending theory, that stresses occur only i n the x - d i r e c t i o n . The e l a s t i c s t r a i n energy becomes U  =  •I.  £  2  dV  and c a r r y i n g out the p a r t i a l i n t e g r a t i o n over the cross s e c t i o n of a prismatic beam (U  x  + W W x x  +  i W ) dx + Et 2 x 2  2  W dx xx 2  10 The k i n e t i c energy i s given by  Making use of Hamilton's P r i n c i p l e  (TJ  - T) dt  0  =  (2)  t.  E f f e c t i n g the necessary s u b s t i t u t i o n s , using Calculus of V a r i a t i o n s , one obtains  -EA(U x + i w + WxW ) + tpA T7 2 x x x ^ tt  =  2  EIW  xxxx  I (U  - EA  + i W  \,x2x  + W W ) (W  2  xx  x  + W )  x  j  0  (3)  + Y<pAW. .  tt  x  =  ik)  0  These equations are the coupled nonlinear d i f f e r e n t i a l equations of motion f o r a beam under plane motion. s i m p l i f i e d s t r a i n expression. U  x  These equations are based on the  I f ' t h e f u l l s t r a i n expression i s used, or i f  i s of the same order of magnitude as W , one obtains the f o l l o w i n g set of x  equations - E A J |U  + fl W  + i ( U  V . L x x x 2 x  EI W - EA xxxx  U  TJ  2  + W  2  x J  + WW + ^ ( U + W )\(W xx 2 x X J 2  x  + U)V  ) ( l  x J  2  + ipAU..  tt  =  0  + W ) > +tpA W.. = 0 X x j ^ t t  (5)  (6)  x  The above equations, (5) and (6), d i f f e r from (3) and (h) r e s p e c t i v e l y , only  2 by the i n c l u s i o n of the term U  i n the s t r a i n expression.  This term might  be s i g n i f i c a n t f o r explaining coupling frequencies higher than twice the frequency of the e x c i t a t i o n , and i t might become important f o r e x c i t a t i o n s  11  much higher than those used i n t h i s research. To check the deduction, the equations are l i n e a r i z e d  EA U xx  = A cp U ^ tt  . \ '  U xx  = i2 c  U.. tt  (7)  and EI W + A cp V/ xxxx tt  =  0  (8)  The f i r s t of these equations corresponds to the c l a s s i c a l longitudinal v i b r a t i o n of rods; the second of these equations corresponds to the l a t e r a l v i b r a t i o n of a beam. Discussion of the Derived Equations In the paragraphs which follow the interpretation and the s i g nificance of p a r t i c u l a r coupling terms of the equations of motion f o r the beam are discussed.  Even though the method assumed i s q u a l i t a t i v e i n nature,  some concrete predictions can be made.  We consider now the equation of motion (3) f o r the beam. D i f f e r e n t i a t i n g and rearranging the equation, one obtains  - U + itr U. . = W W + W W + W W xx Q2 t t xx x x xx x xx  (9)  Assuming the f l e x u r a l displacement to be of the form V/ = so that  WW  =  H(x) c o s  cot  (10)  W W xx x  =  H. (x) cos cot 1  (ll)  WW x xx  =  H.(x) cos cot 2  (12)  x and  F(x) cos cot  2  xx  12 where W  x  and W  are just functions of  xx  x .  From a n a l y t i c a l geometry cos therefore,  2  WW  cot =  X  1 ^ ( l + cos 2 cot)  =  H_(x) j  XX  f 1 + cos 2 cot  (13)  L  J  Next, i t i s assumed that the terras on the right-hand side of (9) are functions.  S u b s t i t u t i n g expressions ( l l ) , (12), and  the r i g h t hand side of (9)« - TJ^ + ^ U  =|  t t  H^x)  one  forcing  (13) f o r the terms on  obtains  + H (x) | cos cot + H^x) 2  | l + cos 2 cot ^  (1*0  By analogy, expanding, d i f f e r e n t i a t i n g , and rearranging as was  done above, one can write (k)  EIW  XXXX  - E A l U W + U W [  XX  X  i n the f o l l o w i n g form, 1+  x xx J  AcpW. . = EA I U tt  [ XX  W  X  + TJ W  X XX  J  (15)  Having established, a n a l y t i c a l l y and experimentally, the f a c t that there are at l e a s t two l o n g i t u d i n a l coupled v i b r a t i o n s , one might subs t i t u t e t h e i r expressions, guided by the approach taken for the v i b r a t i o n s , f o r the right-hand side cross products of (15)•  longitudinal  As a consequence  one should expect also two transverse, or f l e x u r a l coupled v i b r a t i o n s , with the frequency r a t i o of 1:2.  These would represent the influence of l o n g i t u -  d i n a l v i b r a t i o n s on the transverse v i b r a t i o n . I n examining the expression (1*0 following Expression  and (15) one can draw the  conclusion: (1*0: At any one i n s t a n t of time the l o n g i t u d i n a l v i b r a t i o n of the  beam i s the r e s u l t of two d i f f e r e n t , superimposed, s i n u s o i d a l e x c i t a t i o n s ,  13  the frequencies of which are i n the r a t i o of 1:2.  Moreover, at l e a s t two  d i f f e r e n t resonant frequencies are to be expected - one fundamental at the frequency of f l e x u r a l v i b r a t i o n ; the second at twice the frequency of t r a n s verse v i b r a t i o n . Expression (13): Making use of the f a c t that the beam i s excited l o n g i t u d i n a l l y by two f o r c i n g functions of frequency  co and 2co, one can show that the r i g h t -  hand side of (15) c o n s i s t s of the f o l l o w i n g functions G^(x) cos cot + G,, (x) cos 2 cot Two attendant resonant frequencies are to be expected.  14  CHAPTER I I I  APPARATUS AND INSTRUMENTATION  Signal Flow The  signal flowchart i s shown i n F i g . 3.  E s s e n t i a l l y , the  instrumentation consisted of a v i b r a t i o n control feedback system and a n c i l l a r y instruments to record diverse s i g n a l s .  Thus, a sound l e v e l meter was used  for dynamic sound pressure waveform recording and frequency reference. output was displayed  either on the CRO, or on the l e v e l recorder.  The  An o p t i c a l  non-contacting displacement transducer, c a l l e d a Fotonic Sensor (MTT Instruments D i v i s i o n , Hatham, N.Y.), was used as an independent frequency The  output was displayed  on the CRO.  reference.  F i n a l l y , a BAM and s t r a i n gauges were  used f o r frequency spectra recording and waveform analysis. The v i b r a t i o n control loop consisted of the generator, amplifiers, electromagnetic e x c i t e r , and accelerometer.  The generator  (henceforth c a l l e d the control unit) sent out a pre-programmed s i g n a l , corresponding to the v i b r a t i o n l e v e l , to the 50 l b s . force vector exciter; the actual v i b r a t i o n l e v e l was sensed by the accelerometer mounted r i g i d l y on the adapter for the specimen.  The v i b r a t i o n l e v e l as recorded by the accelero-  meter was compared with the set point vibration l e v e l .  Any error between  the set point control and the feedback signal was corrected by the v i b r a t i o n control unit.  For synchronization  of the generator and the frequency c a l i -  brated paper on the l e v e l recorder an e l e c t r i c a l connection was provided between these two units.  F i n a l l y , a signal proportional  to the acceleration  1-BEA.U 'SCOPE  •  80UVJ0 L1VKL ULTSa  tfOTOUIC A.CCKLBftOUKTBa aauaiTiviTv  ACCKLCROUETCn ACTIVC  S T I U I M  Q A A U  -  LEVEL RECORDER  QEWERATOH., FEEDBACK.  t  SYNCHRONIZATION  ELECTROMAGNETIC EXCITER  DUUUY « T C M P COUP C , * ^ *  />/////////////> • FOO. VfNCHflOI4!lA.TtOU  SIQKJAtL  FLOWCHART  F i g . 3. Signal Flowchart  16  was displayed as a frequency reference on the CRO. Essentially, three electronically independent instruments were used for signal recording. These were: displacement transducer, sound level transducer, and strain gauge transducer. practical.  A l l three were useful and  They are considered i n turn. Displacement Measurements - a fiber optic, non-contacting  displacement transducer,the Fotonic Sensor, was used at the beginning of the experiment. Its use had to be discontinued later on because of the breakdown of the instrument.  Moreover, this particular instrument was unsuitable  for the amplitude measurements around 10 KHz. The amplitudes were less than one angstrom in magnitude for the vibration levels involved In the experiment. The available instrument was simply not sensitive enough for amplitudes of vibration of that order. Sound Level Measurements - these were recorded by means of a General Radio Sound level meter, type I565-A.  This instrument was used  essentially for waveform recording and frequency reference. The sound level for the specimens fluctuated approximately twenty decibels, between the limits of f i f t y and one hundred and ten decibels; the instrument emitted a signal proportional to the sound level fluctuation over a range of ten decibels. In recording the sound pressure level, the signal was displayed either directly on the CRO, or on the level recorder. The sound pressure level frequency spectrum data turned out to be useless due to the background noise compounded with the noise emitted by the specimen and i t s reflections i n the best room. No trend could be established. However, with the addition of an anechoic chamber, sealing off of the background noise and perhaps attenuating the reflections, such records might prove useful.  17  To establish whether the characteristic strain waveform at the resonant f i r s t and second longitudinal vibration of the bar and the resonant f i r s t and second radial vibration of the plate could be recorded independently, the sound level meter output was displayed simultaneously on the CRO. The microphone was suspended approximately four feet above the plate, and off center.  For the bar, the microphone was suspended approximately six  inches above the bar and over one cantilever.  With this arrangement, the  sound level meter recorded the net result at that point of a l l sound waves emanated and reflected from the specimen and surroundings. Strain Gauge Measurements - on both the bar and the plate, strain gauges were applied i n appropriate positions.  These gauges were hooked  up in suitable networks with, or without temperature compensating gauges i n external half, or f u l l bridges to the bridge amplifying meter. Since, i n general, the strains measured were extremely small, less than 20 uin/in, high resistance wire strain gauges and 120 ohm film strain gauges temperature compensated for steel were used.  In addition, dry cells were used in series  with the internal excitation of the bridge amplifying meter to make the bridges as sensitive as was possible without overloading the electronic network. From the BAM, the signal was taken either directly to the oscilloscope, for qualitative and quantitative measurements, or the signal was amplified in another stage of amplification and then recorded on the level recorder. The last amplification mentioned was a necessity since the level recorder had a minimum signal level of 10 mv  RMS, and the maximum signal recorded at the  output of the BAM was less than hO mv  peak-to-peak.  Most instruments were hooked up for floating ground - grounding was done through the oscilloscope.  The components grounded were:  the power  amplifier, the sound level meter, the Fotonic Sensor, the specimen, the  18 o s c i l l o s c o p e , and the b l o c k on which the dummy guages were mounted, Individual  Instruments  F i g . k. 1 2  3 k 5 6  O v e r a l l View o f the  Instrumentation  Power A m p l i f i e r Amplifier CRO P o l a r Height Gauge Electromagnetic E x c i t e r T e s t Bed  7 8 9 10  11 12  Frame Frequency Counter Control Unit L e v e l Recorder BAM Roots Blower  An o v e r a l l view o f the i n s t r u m e n t a t i o n s e t up i s shown i n Fig.  *f.  Not shown i n the f i g u r e i s the F o t o n i c S e n s o r , the s t o r a g e o s c i l l o s -  c o p e , and the vacuum tube v o l t m e t e r .  With automatic c o n t r o l , the system can be operated from 10 Hz to  10,200  Hz.  The maximum f o r c e a v a i l a b l e i s  50 l b s ;  the maximum d i s p l a c e m e n t  o f the p l u n g e r o f the electromagnet i s r e s t r i c t e d t o l e s s than 0.5 peak-to-peak.  inches  The system can be run backward o r f o r w a r d , o r backward and  19  forward continuously over the whole frequency range, or over segments of i t s range.  The system can be run continuously at maximum power.  The frequency  sweep of the generator of the c o n t r o l u n i t can be synchronized with the f r e quency c a l i b r a t e d paper feed of the l e v e l recorder.  The recording can be  done over the f u l l frequency range of the c o n t r o l u n i t ; an average frequency reading can be obtained from a d i g i t a l counter.  Without automatic c o n t r o l ,  the frequency range can be extended from 10,200 Hz t o 20,000 Hz, but at a reduced force.  The manufacturer suggests an upper l i m i t of 25 l b s . under these  circumstances.  Without automatic c o n t r o l , the c o n t r o l u n i t and the l e v e l  recorder are taken out of the c i r c u i t . generator replaces the c o n t r o l u n i t .  The output of an e x t e r n a l f u n c t i o n Under these conditions the c o n t r o l s  r e v e r t t o manual - t h i s extension requires a l o t of patience and a t t e n t i v e ness on the part of the operator. The Bruel and Kjaer Automatic V i b r a t i o n E x c i t e r Control Unit Type 1025 provides a constant displacement c o n t r o l up t o 2 KHz and a maximum amplitude of 0.1 i n . peak-to-peak; constant v e l o c i t y c o n t r o l i s a v a i l a b l e up t o 2 KHz and up t o 100 in/sec peak value; constant a c c e l e r a t i o n c o n t r o l i s f e a s i b l e up t o 10 KHz and 1000 g peak value ( l g = a c c e l e r a t i o n of g r a v i t y ) . Since the frequency d i a l on the instrument was not properly c a l i b r a t e d , and even i f i t had been, i t i s impossible t o read frequencies c l o s e r than 50 Hz at the upper end of the frequency s c a l e , an e x t e r n a l General Radio Company D i g i t a l Time and Frequency Meter, Type 1151-A, S e r i a l 3^7 was used f o r f r e quency measurements.  At low scanning speeds of the c o n t r o l u n i t , t h i s u n i t  provides not an instantaneous, but an average frequency.  At steady.state, i t  d i s p l a y s the true frequency of the e x c i t a t i o n (provided the s i g n a l i s s t a b l e and the - 1 count inaccuracy has been taken i n t o account).  The c o n t r o l u n i t  has s i x f i x e d scanning speeds i f used with the synchronized l e v e l recorder,  20 or 132 scanning speeds otherwise.  In terms of frequencies being swept out  per u n i t of time, the concept of a f i x e d scanning speed has no meaning.  This  a r i s e s from the f a c t that the generator produces the frequencies logarithmic-;. a l l y with time, whereas the scanning device i s synchronous, or constant with time.  The c o n t r o l u n i t has f i x e d or pre-programmed compressor speeds. The B r u e l and Kjaer Level Recorder, Type 2305, was  i z e d e l e c t r o n i c a l l y with the c o n t r o l u n i t f o r most t e s t s . medium 50 mm  synchron-  As p l o t t i n g  wax paper, frequency c a l i b r a t e d , was used i n conjunction w i t h  a sapphire pen and every p l o t was c a l i b r a t e d separately. This l e v e l recorder does not reproduce the instantaneous value of the f u n c t i o n , but p l o t s e i t h e r the RMS, DC, or peak value.  The RMS f u n c t i o n was chosen f o r t h i s experiment -  the i n f l u e n c e of sudden extraneous s i g n a l s i s minimized thereby.  Different  paper and w r i t i n g speeds are a v a i l a b l e . The E l l i s Associates BAM 1 Bridge A m p l i f i e r and  Meter has a  DC a m p l i f i e r whose frequency response i s not f l a t over the used frequency range.  At 10 KHz the attenuations i s approximately 3%»  A Hewlett Packard DC  Vacuum Tube Voltmeter was used to measure the e f f e c t i v e e x c i t a t i o n voltage of the d i f f e r e n t bridges.  The a m p l i f i e r s e c t i o n of the Bruel and Kjaer f r e -  quency analyzer Type 2107 was used as cascading a m p l i f i e r a f t e r the BAM  unit.  The MB E l e c t r o n i c s Model 2250 Power A m p l i f i e r i s capable of d e l i v e r i n g up t o kOO watts of power t o the e x c i t e r .  The Bruel and Kjaer  dual channel accelerometer p r e a m p l i f i e r Type 2622 has a b u i l t - i n s e n s i t i v i t y attenuator f o r s e t t i n g of the accelerometer s e n s i t i v i t y .  For c o r r e c t s e t t i n g  of the accelerometer s e n s i t i v i t y , the channel output i s scaled t o 10  mv/g.  The waveforms of i n t e r e s t i n the experiment were displayed on the Tektronix 565 Dual Beam (or four beams with chopper) o s c i l l o s c o p e and photographed  21  therefrom.  Finally,  the p o l a r h e i g h t gauge and F o t o n i c Sensor probe h o l d e r  was designed and b u i l t i n the Department. measurements i n t h r e e c o o r d i n a t e s - 6, r,  These d e v i c e s a l l o w  displacement  h.  T e s t Bed and Frame The t e s t bed on which the v i b r a t i o n e x c i t e r was mounted c o n s i s t e d o f a b l o c k o f wood, a s t e e l c a s i n g , and a two f o o t by two f o o t b l o c k of concrete.  The whole assembly was embedded i n the ground beneath the  floor  * o f the l a b o r a t o r y .  See F i g .  5*  F i g . 5.  T e s t Bed  The c o n c r e t e b l o c k was s t r u c t u r a l l y  i s o l a t e d from the  S i n c e the t e s t s were c a r r i e d out d u r i n g the n i g h t j the i n f l u e n c e o f  building.  Rayleigh  S u r f a c e waves (caused by t r u c k s and c a r s p a s s i n g the nearby r o a d , unbalanced machinery i n s i d e the b u i l d i n g ) was m i n i m i z e d .  The s t e e l c a s i n g shown i n  f i g u r e c o n s i s t e d o f two heavy T - s e c t i o n s j o i n e d on the t o p by 1/2 i n c h p l a t e , braced by two g u s s e t s , welded t o g e t h e r , A wooden b l o c k was screwed t o the s t e e l p l a t e .  the  steel  and b o l t e d t o the c o n c r e t e b l o c k . T h i s was used as a damper as  22  w e l l as a support f o r the frame t o which the s u s p e n s i o n and a u x i l i a r y t u r e s were a t t a c h e d .  The frame r u n n i n g the f u l l l e n g t h o f the working s u r -  f a c e c o n s i s t s o f a 3/16 the wood.  fix-  x 1% i n c h angle i r o n , welded t o g e t h e r and screwed t o  A h o l e was d r i l l e d through the angle i r o n d i r e c t l y above the  c e n t e r of the v i b r a t i o n e x c i t e r f o r attachment o f the specimen s u s p e n s i o n .  Suspension F o r the p l a t e as w e l l as f o r the b a r , an e x t e r n a l s u s p e n s i o n had t o be p r o v i d e d t o take the weight of the moving element, specimen, and any n e c e s s a r y appendages o f f the e l e c t r o m a g n e t . a n e c e s s i t y i n o r d e r t o o b t a i n the l e v e l  In the experiment, t h i s  was  o f a c c e l e r a t i o n used throughout  the  tests.  The s u s p e n s i o n f o r the b a r , shown i n F i g . 6, c o n s i s t e d o f two  Fig.  s p r i n g systems i n p a r a l l e l .  6.  First,  Bar  Suspension  t h e r e was the s p r i n g a c t i o n p r o v i d e d by  an a n n u l a r rubber diaphragm i n c o r p o r a t e d i n the v i b r a t i o n e x c i t e r .  Second,  t h e r e was p r o v i d e d an e x t e r n a l s p r i n g system c o n s i s t i n g o f c o i l s p r i n g s .  For  23 the p l a t e t h i s system comprised four springs i n p a r a l l e l ; f o r the bar two springs were used i n p a r a l l e l .  The c r i t e r i o n f o r the choise of springs was  to keep the n a t u r a l frequency of the complete spring-mass system as f a r away as p o s s i b l e from the lowest operating frequency.  Without power flow to the  e x c i t e r , a n a t u r a l frequency between four and f i v e cycles per second f o r both systems was a t t a i n e d with the a v a i l a b l e s p r i n g s . The suspension was attached t o the frame v i a a long  1/4"  threaded rod running through the hole i n the upper angle of the frame.  A  hemispherical surface was machined on a l a r g e nut to keep the rod from s l i d i n g down and t o support the weight t o be suspended.  This nut and a l o c k i n g nut  were threaded on the f a r end of the rod to an appropriate height.  The hemi-  s p h e r i c a l nut, r e s t i n g on the hole i n the angle i r o n , allowed f o r small misalignments of the moving parts i n two orthogonal d i r e c t i o n s . P l a t e Support The p l a t e was supported e s s e n t i a l l y , by a threaded s h a f t , one end of which had a wide flange which was bolted t o the moving element of the electromagnetic e x c i t e r .  The length of the shaft consisted of four types of  thread - three e x t e r n a l , one i n t e r n a l .  The c e n t r a l external thread was a  tapered pipe thread to match the pipe thread i n the c e n t r a l hole of the p l a t e . The i n t e r n a l one was used to fasten the accelerometer t o the top of the s h a f t . The upper machine thread on the shaft was used to fasten the external spring system t o the s h a f t . Bar Clamp The bar was clamped between two short cross beams, one of which was an i n t e g r a l part of a shaft arrangement as i n the previous case. The accelerometer was threaded i n t o the center of the upper bross beam; the lower wide flanged s e c t i o n of the shaft was bolted to the plunger of the e x c i t e r ; the springs were attached t o the upper cross beam. The double  Zk  cantilever shaker  b a r arrangement was used t o p r o v i d e a b a l a n c e d system on the  table.  Models and Boundary C o n d i t i o n s Circular Plate F o r the experiment a c i r c u l a r p l a t e o f m i l d s t e e l , l/k dia.  (nominal) was u s e d .  T h i s diameter was chosen f o r the p l a t e so as t o  b r i n g the b r e a t h i n g mode f r e q u e n c y i n t o the 8 KHz r a n g e . the p l a t e a 1/4"  x 16"  p i p e t h r e a d h o l e was machined.  In the c e n t e r o f  The s u r f a c e s were ground on  a s u r f a c e g r i n d e r and hand sanded t o remove r u s t s p o t s .  Due t o the g r i n d i n g  o p e r a t i o n , the p l a t e had assumed a c u r v a t u r e - i n s i g n i f i c a n t t o be c o n s i d e r e d a  'curved'  surface. The boundary c o n d i t i o n s aimed f o r were:  without  a circular  plate  t r a n s v e r s e l o a d , f r e e edges, and supported and e x c i t e d i n the  center.  As was mentioned under the h e a d i n g ' P l a t e S u p p o r t ' , the p l a t e was supported by a threaded s h a f t .  The s e c t i o n o f the s h a f t w i t h the p i p e t h r e a d was  screwed hand t i g h t i n t o the h o l e o f the p l a t e . a c o n t i n u o u s p l a t e , without  a central hole.  The i n t e n t i o n was t o  The C h l a d n i f i g u r e s  were t o be an i n d i c a t i o n o f the symmetry o f the  simulate  obtained  vibration.  Two o t h e r s u p p o r t i n g c o n d i t i o n s were c o n s i d e r e d .  One, a s o l i d  p l a t e clamped, o r s i m p l y supported a l o n g i t s p e r i p h e r y ; two, a clamped p l a t e i n the c e n t e r . the a v a i l a b l e conditions.  The f i r s t  model was d i s m i s s e d as i m p r a c t i c a l , c o n s i d e r i n g  e x c i t a t i o n and the d i f f i c u l t y i n p r o d u c i n g the exact boundary The second one was t r i e d .  The p l a t e was clamped between two  n u t s run up on the s h a f t and a g a i n s t the p l a t e .  In s p i t e o f the c a r e  taken  i n machining the s u r f a c e s i n v o l v e d , p e c u l i a r C h l a d n i f i g u r e s were o b t a i n e d . T h e r e f o r e , t h i s boundary c o n d i t i o n was abandoned.  25 Cantilever Bar The bar was fabricated from 1/4 x 1" (nominal) cold r o l l e d steel.  The f u l l length of the double cantilever bar was  12.38". This length  was chosen so that the f i r s t coupled l o n g i t u d i n a l resonant v i b r a t i o n occurred i n the 8 KHz range.  The cross beam width was  17/32". The surfaces were  ground so as t o remove surface p i t s .  The boundary condition to be simulated was that of two c a n t i l e v e r bars, obtained by clamping a single bar at the center with the cross beam.  Obviously, the arrangement assumed w i l l not y i e l d true fixed end  conditions.  For bending, the boundary conditions were exact; f o r l o n g i t u -  d i n a l wave propagation, the width of the cross beam has to be taken into account.  S t r a i n Gauges and Bridges C i r c u l a r Plate Two kinds of gauges were placed on the plate - for the tangenti a l s t r a i n s BUDD C 6 - l 4 l B f i l m gauges of 120 ohm resistance were used; f o r the bending plus r a d i a l s t r a i n and the r a d i a l s t r a i n records 500 ohm gauges SR-4 CD-8 Baldwin Lima wound wire gauges were used.  I d e n t i c a l gauges were used  wherever dummy, or temperature compensating gauges were needed.  The gauges  were cemented to the surfaces with GA-1 and SR-4 s t r a i n gauge cement, r e s pectively.  The l o c a t i o n and hook-up of the gauges was as follows: Tangential - three gauges placed at random along the periphery and hooked up i n s e r i e s t o form one arm of a 360 ohm external h a l f bridge.  The gauges were  located as symmetrically as was possible about the c e n t r a l plane.  Bending  plus r a d i a l and r a d i a l s t r a i n - two gauges, of short filament length located within 1/64 of an inch from the outer edge.  One gauge was located on each  26 side of the p l a t e surface i n corresponding p o s i t i o n s . Two such gauge arrangements were placed at random along the periphery.  For bending plus  r a d i a l s t r a i n , two gauges on the same side of the p l a t e were hooked up i n s e r i e s t o form one arm of a h a l f bridge.  For r a d i a l s t r a i n records, a f u l l  500 ohm bridge was connected with the a c t i v e gauges, one on each side of thep l a t e , connected so as to eliminate the bending s t r a i n .  The l o c a t i o n of the  s t r a i n gauges i n the r a d i a l d i r e c t i o n was a r b i t r a r y since an optimum p o s i t i o n for  a l l frequencies could not be determined. C a n t i l e v e r Bar Four BUDD C6-141-B gauges were l o c a t e d i n symmetrical  t i o n s r e l a t i v e t o the cross beam on both c a n t i l e v e r s .  loca-  One gauge was l o c a t e d  on the top surface and one on the lower surface i n corresponding p o s i t i o n s on each c a n t i l e v e r .  The gauges were l o c a t e d 3%" away from the end and equi-  d i s t a n t from the s i d e s .  GA-1 cement was used f o r t h e i r adhesion.  Because of  the higher s t r a i n l e v e l i n v o l v e d , the gauges were hooked up i n t o 120 ohm h a l f bridge f o r the bending plus r a d i a l s t r a i n record and i n t o 120 ohm f u l l bridge for  the r a d i a l s t r a i n record. Bridges As f a r as i t was t o l e r a b l e , a d d i t i o n a l external e x c i t a t i o n of  the bridge was provided.  The c r i t e r i o n f o r the choice of magnitude of the  e x t e r n a l e x c i t a t i o n was d i c t a t e d by the a v a i l a b l e sources and the d i s s i p a t i v e capacity of the s t r a i n gauges. As suggested by t h e i r respective 25 ma.  manufacturers,  was taken as the l i m i t i n g current f o r the SR-h gauges and 5 watts per  square inch of g r i d area was taken as the l i m i t i n g power f o r the C-6 gauges.  27  CHAPTER IV  TESTING  C a l i b r a t i o n and Check Out The a c c e l e r o m e t e r used i n the experiment the f a c t o r y .  No c a l i b r a t i o n check was  o n l y t e s t done was  was  made p r i o r t o the experiment.  t o compare i t s b e h a v i o u r a g a i n s t  o u t p u t s agreed r e a s o n a b l y w e l l over the f u l l  a similar unit.  f r e q u e n c y range.  o f a c o o l i n g medium on the output o f the a c c e l e r o m e t e r was appreciable  c a l i b r a t e d at The Their  The  influence  examined.  No  d i f f e r e n c e s were r e c o r d e d .  The  system as a whole and i t s i n d i v i d u a l components were  checked as f a r as t h i s was  possible.  The a c c e l e r a t i o n as s e t on the  control  u n i t , and the a c c e l e r a t i o n r e c o r d e d by the a c c e l e r o m e t e r agreed e x a c t l y . set point  a c c e l e r a t i o n was  quency ranges. deviations  not always  a b l e and i n s i g n i f i c a n t . the experiment.  constant throughout the d i f f e r e n t f r e -  The a c c e l e r a t i o n meter on the c o n t r o l u n i t i n d i c a t e d  from the s e t p o i n t .  In general,  these d e v i a t i o n s  The o s c i l l o s c o p e ' s c a l i b r a t i o n was  The f r e q u e n c y response o f the BAM  was  c o n t r o l u n i t was  a necessity.  unpredict-  checked p r i o r t o The  frequency  a n a l y z e r ) was  over the f r e q u e n c y range i n v o l v e d .  p a r t i c u l a r a p p l i c a t i o n , an e x t e r n a l  small  The f r e q u e n c y r e s -  ponse o f the cascaded a m p l i f i e r ( a m p l i f i e r o f the f r e q u e n c y I t s response i s f l a t  were  checked.  response complies w i t h the manufacturer's s p e c i f i c a t i o n .  checked.  The  F o r the  frequency counter i n conjunction  w i t h the  F r e q u e n c i e s cannot be r e a d from the c o n t r o l  u n i t f r e q u e n c y s c a l e c l o s e r than a p p r o x i m a t e l y 250  Hz a t 10 KHz.  In a d d i t i o n ,  28  the frequency s c a l e was not p r o p e r l y a d j u s t e d , o r c a l i b r a t e d .  The r e q u i r e d  a i r f l o w through the e l e c t r o m a g n e t i c e x c i t e r was checked at maximum, constant power d i s s i p a t i o n .  The a i r f l o w r a t e and the power i n p u t t o the compressor  were i n a d e q u a t e l y s p e c i f i e d by t h e i r r e s p e c t i v e m a n u f a c t u r e r s . the f i r s t  A doubling of  q u a n t i t y and t r i p l i n g o f the second q u a n t i t y was n e c e s s a r y .  c o o l i n g system as now d e s i g n e d i s adequate f o r continuous s e r v i c e at power d i s s i p a t i o n .  The full  The s y n c h r o n i z a t i o n between the l e v e l r e c o r d e r and the  c o n t r o l u n i t was good. The n a t u r a l f r e q u e n c y o f the complete s u s p e n s i o n o f the system was e s t a b l i s h e d .  To determine t h i s f r e q u e n c y , a s t o r a g e o s c i l l o s c o p e and t h e  F o t o n i c Sensor p l u s the h e i g h t gauge were u s e d .  The probe o f the F o t o n i c  Sensor was suspended over the s u r f a c e o f the p l a t e , and the dynamic output o f the i n s t r u m e n t was d i s p l a y e d on the s t o r a g e o s c i l l o s c o p e .  Finger pressure  was used t o d r i v e the system (specimen, a d a p t e r s , p l u n g e r , a c c e l e r o m e t e r ) i t s natural  frequency.  at  T h i s procedure was repeated s e v e r a l t i m e s , and the  average o f the r e c o r d e d f r e q u e n c i e s was t a k e n .  The d e v i a t i o n s from the mean  were v e r y s m a l l .  Noise problems a r i s i n g from ground l o o p s were e l i m i n a t e d . A l l c a b l e s and w i r e s used were s h i e l d e d .  E l e c t r o m a g n e t i c p i c k - u p was m i n i -  mized. Test  Procedure S i n c e the t e s t procedure f o r the p l a t e i s s i m i l a r t o t h a t o f  the b a r , o n l y one i s d e s c r i b e d below.  The p l a t e w i t h s t r a i n gauges and l e a d  w i r e s a t t a c h e d was b o l t e d t o the p l u n g e r o f the e l e c t r o m a g n e t , making use o f the a p p r o p r i a t e a d a p t e r . meter was a t t a c h e d .  The c o n n e c t o r f o r the s u s p e n s i o n and the  accelero-  By means o f a c e n t e r i n g n u t , a s t r i n g , and a plumb,  29  the  e x c i t e r was centered d i r e c t l y under the hole i n the frame.  The suspen-  s i o n was attached t o the frame, as o u t l i n e d e a r l i e r , and the whole assembly was l i f t e d by means of the /hemispherical nut u n t i l the specimen, adapters, etc.  vibrated with the l e a s t e x c i t a t i o n .  procedure. to i t .  This required a t r i a l and e r r o r  Next, the stay r i n g was attached t o the frame, and the lead wires  By the procedure i n d i c a t e d e a r l i e r , the n a t u r a l frequency of the  system (specimen, adapters, accelerometer, and plunger of electromagnetic e x c i t e r ) was determined.  Next, the a c t i v e gauges and the temperature com-  pensating gauges were checked f o r t h e i r r e s i s t a n c e ; the gauges were combined in',the".proper order t o form the desired c i r c u i t s and then connected t o the appropriate posts of the BAM.  The s h i e l d s of the lead wires and the ground  wire of the specimen and the temperature compensation gauge block were connected t o the ground of the o s c i l l o s c o p e . to the BAM. the  The e x t e r n a l e x c i t a t i o n was added  A l l instruments were properly interconnected, and connected t o  power source.  Then the instruments were turned on and l e f t i n the stand-  by p o s i t i o n f o r at l e a s t two hours. The next phase of the t e s t procedure was t o determine the specimen c h a r a c t e r i s t i c s - what a c c e l e r a t i o n t o use over which frequency range.  This required a t r i a l and e r r o r procedure.  The f i g u r e s i n Table 1  were found t o be s a t i s f a c t o r y f o r the p l a t e and the bar, r e s p e c t i v e l y .  This  t a b u l a t i o n i s u s e f u l i n l a t e r discussions. The a c t u a l t e s t i n g got under way a f t e r : (a)  s e t t i n g the set point a c c e l e r a t i o n , (b) synchronizing the l e v e l recorder  with the c o n t r o l u n i t , (c) choosing the scanning speed and d i r e c t i o n , (d)  determining the compressor speed, (e) choosing the attenuation, the  w r i t i n g speed, and the type of response of the l e v e l recorder, ( f ) the  cooling  e x c i t e r with forced a i r , and (g) c a l i b r a t i n g the BAM and measuring the  30 Plate Characteristics Frequency . ./Range,.; Hz  Bar  Maximum Acceleration g  150  200  2  200  700  700  Characteristics  Frequency Range Hz  Maximum Acceleration g  400  1000  40  5.5  1000  1200  20  800  0.5  1200  1500  1.4  800  1000  4.5  1500  3600  50  1000  1500  30  3600  4000  7  1500  2000  26  4000  7000  100  2000  2400  3  7000  7500  14  2400  7000  110  7500  9500  100  7000  7400  15  9500  10000  60  7400  10000  8o  T a b l e 1. e x c i t a t i o n voltage o f t i m e , the l a s t experiment.  Allowable A c c e l e r a t i o n Level  o f the b r i d g e .  S i n c e each t e s t extended over l o n g p e r i o d s  s t e p had t o be done p e r i o d i c a l l y d u r i n g the course o f  The h i g h e x t e r n a l v o l t a g e s o u r c e ' s l i f e  the  was l i m i t e d - up t o 50?o  r e d u c t i o n i n e x c i t a t i o n was e n c o u n t e r e d .  Constant Power P l o t D i s c u s s i o n The i d e a l approach t o the experiment would have b e e n , o f c o u r s e , a c o n s t a n t d i s p l a c e m e n t f r e q u e n c y sweep o f the specimen. example e x p l a i n s b e s t t h i s i s r i g i d l y attached i s  choice.  considered.  A rotating shaft If  A practical  t o which a c i r c u l a r  the b e a r i n g i n which the s h a f t  disc  is  r o t a t i n g i s m i s a l i g n e d , so t h a t the d i s c e x p e r i e n c e s a s i n u s o i d a l  transverse  p e r i o d i c e x c i t a t i o n , a c c o r d i n g t o the t h e o r y proposed h e r e i n , two  radial  vibrations  o f the d i s c and the a s s o c i a t e d r a d i a l r e s o n a n t v i b r a t i o n s must be  expected.  S i n c e the a x i a l d i s p l a c e m e n t o f the s h a f t  can be c o n s i d e r e d c o n s t a n t ,  r e g a r d l e s s o f the f r e q u e n c y o f r o t a t i o n , an i n v e s t i g a t i o n ,  assuming the  above  31 criterion, i s highly desirable.  As was s t a t e d i n the d e s c r i p t i o n o f t h e  o v e r a l l system, such an i n v e s t i g a t i o n was not p o s s i b l e .  The n e x t b e s t t e s t procedure would have been a c o n s t a n t power p l o t , however, t h e system i s not d e s i g n e d f o r t h i s k i n d o f c o n t r o l .  The next a l t e r n a t i v e was a c o n s t a n t a c c e l e r a t i o n c o n t r o l . T h i s k i n d o f c o n t r o l was a l s o u n s u i t a b l e .  S i n c e t h e f r e q u e n c y sweep extends  from 100 Hz t o 10 KHz, i t can be shown t h a t t h e peak a m p l i t u d e s i n t h e c e n t e r o f t h e p l a t e a r e d r a s t i c a l l y r e d u c e d , by a f a c t o r o f 10 , and, c o n s e q u e n t l y , the r e s u l t i n g s t r a i n s a r e v e r y s m a l l , perhaps t o o s m a l l t o be d i s t i n g u i s h e d from the i n h e r e n t e l e c t r o n i c n o i s e .  T h i s p o i n t i s c l a r i f i e d by t h e two  c h a r t s below, F i g . 7.  (a)  (b)  F i g . 7*  C o n s t a n t A c c e l e r a t i o n and Approximated Constant Power Frequency S p e c t r a (a)  Constant a c c e l e r a t i o n  (b)  Approximated c o n s t a n t power  32  In F i g . 7(a) the constant a c c e l e r a t i o n p l o t f o r the plate i s shown, f o r both r a d i a l and t a n g e n t i a l s t r a i n .  Only one resonant v i b r a -  t i o n ( t a n g e n t i a l s t r a i n at 4405 Hz) shows up c l e a r l y . I n F i g . 7(b) the approximated constant power frequency spectra f o r the plate i s shown.  Two  resonant v i b r a t i o n s , one at 4405 Hz and one at 8810 Hz, can be i d e n t i f i e d clearly.  The a d d i t i o n a l information obtained from the approximated constant  power p l o t d i c t a t e d i t s choice over the constant a c c e l e r a t i o n c o n t r o l . By d e f i n i t i o n , the approximated constant power frequency spectra i s the break-up of the f u l l frequency spectrum i n t o a number of convenient, small segments along which the a c c e l e r a t i o n i s kept constant; d i f ferent a c c e l e r a t i o n l e v e l s are used f o r d i f f e r e n t segments such that a t one point of each segment, the power input t o the plate i s a maximum (i-70 watts approximately).  The number of segments and t h e i r l o c a t i o n i s an engineering  d e c i s i o n - an optimization process.  Table 1 l i s t s these frequency segments  and the constant a c c e l e r a t i o n l e v e l .  This procedure allows f o r the f a c t that  the maximum allowable a c c e l e r a t i o n l e v e l of the specimen plus i t s adapters, i s lowest at the resonant frequencies.  This procedure r e s u l t s i n an optimum  t e s t i n g , considering a l l the f a c t o r s involved.  This procedure allows the  constant a c c e l e r a t i o n c o n t r o l f a c i l i t y of the system t o be used.  This proce2  dure r e s u l t s i n a decrease i n amplitude of v i b r a t i o n by only a f a c t o r bf 10 if as compared t o 10 frequency extremes.  f o r a constant a c c e l e r a t i o n frequency spectra between the This procedure i s the f i r s t approximation t o a constant,  maximum power frequency spectra. The frequency response spectra, as w i l l be shown s h o r t l y , were p l o t t e d i n a p a r t i c u l a r manner t o s a t i s f y two conditions. d i f f e r e n t frequency segments were separated from each other.  F i r s t , the  This was a  necessity since t r a n s i e n t voltages were set up whenever c e r t a i n controls  33  were changed.  For i n s t a n c e , i n changing the a c c e l e r a t i o n l e v e l , the s c r i b e r  of the l e v e l recorder had to be l i f t e d o f f the paper.  Further, since d i f -  ferent a c c e l e r a t i o n l e v e l s r e s u l t e d i n d i f f e r e n t l o c a t i o n s of t h i s t r a c e , d i f f e r e n t attenuation of the s i g n a l at the l e v e l recorder was used f o r d i f ferent frequency segments.  Second, since the damping i s small i n s t e e l , the  resonant v i b r a t i o n s do show up as spikes i n the records.  To make a b s o l u t e l y  sure that none of these spikes were missed, s u f f i c i e n t frequency overlap provided.  was  This was achieved by running the whole system 'backwards , approxi1  mately 50 - 100 c y c l e s and then forv/ard again, a f t e r having decided on a new attenuation l e v e l .  '  I n t e r p r e t a t i o n of Frequency Spectra A l l frequency spectra shown h e r e i n were obtained by  magnifying  the s t r a i n gauge s i g n a l i n the BAM 120 x (depends s l i g h t l y on the frequency) and 1000 x t h e r e a f t e r i n the cascaded a m p l i f i e r .  To c o r r e l a t e the trace on  the record with the measured s t r a i n , the f o l l o w i n g must be known: (a)  the bridge e x c i t a t i o n used at that time ( t h i s v a r i a b l e i s a f u n c t i o n of time),  (b)  the bridge arrangement and the gauge f a c t o r ,  (c)  the a c c e l e r a t i o n l e v e l i n t h i s p a r t i c u l a r frequency range, and  (d)  the attenuation of the l e v e l recorder. Because the t r a n s l a t i o n of the voltage l e v e l i n t o a true  s t r a i n record i s a tedious one, and because the aim of t h i s t h e s i s i s to show only the existence of the predicted v i b r a t i o n s , no such t r a n s l a t i o n was c a r r i e d out.  The only s t r a i n c a l c u l a t i o n s c a r r i e d out were at the predicted  l o n g i t u d i n a l resonant v i b r a t i o n s of the bar and at s p e c i f i c frequencies close by. tion.  The purpose was to show the s i g n i f i c a n c e of t h i s resonant'vibra-  34  Chladni Figures The Chladni f i g u r e s reproduced h e r e i n were obtained by uniformly d i s t r i b u t i n g ordinary sugar on the p l a t e .  The exact transverse  resonant v i b r a t i o n s were established i n the f o l l o w i n g manner.  The  first  i n d i c a t i o n of a resonant transverse v i b r a t i o n was obtained from the sugar the i n d i v i d u a l sugar grains moved t o a nodal c i r c l e .  An even c l o s e r approxi-  mation could be obtained by observing the output voltage of the e x c i t e r control unit.  As a narrow frequency range s t r a d d l i n g a transverse n a t u r a l  frequency was swept through, the output voltage suddenly swung to a high value, and then f e l l o f f again. h e r e i n i s as f o l l o w s .  The method used to obtain the f i g u r e s l i s t e d  The probe of the Fotonic Sensor was suspended over  the specimen, at a f i x e d p o s i t i o n , and i t s output, at a constant s e t t i n g , was displayed on the storage o s c i l l o s c o p e .  A frequency response survey was  made, at d i s c r e t e frequencies, through the expected range.  At the frequency  where a n a t u r a l frequency occurs, a maximum response was recorded on the oscilloscope. P l o t t i n g and  Photography The p l o t t i n g was done on frequency c a l i b r a t e d , 50  l o g a r i t h m i c , wax paper.  A sapphire s t y l u s was used as s c r i b e r .  b i n a t i o n r e s u l t e d i n a very high r e s o l u t i o n of the s i g n a l s .  mm, This com-  Ink on ordinary  paper would have been e a s i e r to reproduce, however, the r e s o l u t i o n of t h i s combination i s not as good. The appended photographs of waveforms, F i g . 10 and F i g . 11, were taken d i r e c t l y from the screen of the Tektronix o s c i l l o s c o p e , i n the s i n g l e sweep mode, with a Pentax camera and T r i - X f i l m .  35 CHAPTER V  EXPERIMENTAL RESULTS  I n t h i s s e c t i o n the e x p e r i m e n t a l f i n d i n g s a r e p r e s e n t e d . i n t e r p r e t a t i o n and a d i s c u s s i o n o f the same i s p r e s e n t e d i n Chapter V I . make the d i s c u s s i o n c o h e r e n t , a l l the f i n d i n g s o f t h e p l a t e a r e p r e s e n t e d first,  followed  by those f o r the b a r .  Circular Plate F i g u r e 8 shows the C h l a d n i  figures f o r the transverse  resonant v i b r a t i o n o f the p l a t e , second t o f i f t h  4th Mode  Fig.  8.  natural  frequency.  5th Mode  Chladni Figures f o r the Transverse V i b r a t i o n o f the C i r c u l a r P l a t e  An To  36 F o r p h o t o g r a p h i c p u r p o s e s , more sugar than was needed was o s c i l l a t i n g on the p l a t e . t h a t shown i n the  actually  The r e s o l u t i o n can be improved above  figure.  The n a t u r a l f r e q u e n c i e s o f t r a n s v e r s e v i b r a t i o n o f the  plate,  and the n o n - d i m e n s i o n a l i z e d n o d a l r a d i i , w i t h t h e i r r e s p e c t i v e l a r g e s t  devia-  t i o n from the mean r a d i u s , are shown i n T a b l e 2. s e r v e as a t e s t  The n a t u r a l  frequencies  f o r the boundary c o n d i t i o n s ; the l a r g e s t d e v i a t i o n from the  mean r a d i u s i s a measure o f the symmetry o f the v i b r a t i n g system.  Mode  _ sequency Hz  Non-Dimensionalized M  e  a  n  N  o  d  a  l  R  a  d  i  Largest  i  f  r  o  m  M  e  a  Deviation Radius  n  xn  2  ?67  3  2204  0.87  0.51  4  4335  0.91  0.65  0.36  5  7130  0.93  0.73  0.50  T a b l e 2. Note:  0.02  0.03 0.28  0.04  Non-Dimensionalized Nodal R a d i i f o r P l a t e  The average d e v i a t i o n i s much s m a l l e r than the v a l u e s quoted above. The non-dimens i o n a l i z e d n o d a l r a d i i are based on a f u l l r a d i u s ( p l a t e without a c e n t r a l h o l e ) .  The mean n o d a l r a d i i were determined by a v e r a g i n g t h r e e dom r a d i u s measurements.  ran-  The f r e q u e n c i e s quoted above were measured w i t h  the d i g i t a l counter d u r i n g s t e a d y - s t a t e v i b r a t i o n .  The e r r o r s and i n a c c u r a -  c i e s i n v o l v e d i n the f r e q u e n c i e s quoted a r e : (a)  e x p e r i m e n t a l e r r o r a r i s i n g from i n a c c u r a t e l o c a t i o n o f the resonant peak - l e s s than 1 H z ,  (b)  the round o f f e r r o r from the f o u r t h and f i f t h d i g i t ,  and  37  (c)  i n h e r e n t i n a c c u r a c i e s o f the c o u n t i n g d e v i c e . In Appendix C , F i g . C-l4 t o F i g . C - l 6 are shown t h r e e  mated c o n s t a n t power*  f r e q u e n c y s p e c t r a f o r the p l a t e .  Two s p i k e s i n  'approxithe  r a d i a l s t r a i n r e c o r d and the t a n g e n t i a l s t r a i n r e c o r d , F i g . C-15 and F i g . C - l 4 , are shown.  The f i r s t  o c c u r s at  kh05  Hz and the second at  8810  Hz.  There i s one l i m i t a t i o n i n h e r e n t i n the frequency s p e c t r a . I n c o r p o r a t e d i n these r e c o r d s i s a c o n t r i b u t i o n from the e l e c t r o m a g n e t i c d u c t i o n i n the g a u g e . i n s t a l l a t i o n . was m i n i m i z e d .  However, i t  in-  I n the e x p e r i m e n t , t h i s extraneous s i g n a l  c o u l d not be e l i m i n a t e d .  The s i g n a l impressed by  t h i s e l e c t r o m a g n e t i c i n d u c t i o n has always the f r e q u e n c y o f the  excitation.  F o r the resonant p e a k s , t h e i r i n f l u e n c e i s s m a l l ( a p p r o x i m a t e l y 5% o f amplitude f o r the fundamental l o n g i t u d i n a l f r e q u e n c y ) ; frequencies t h i s signal i s very s i g n i f i c a n t .  at o t h e r  the  excitation  Care must be e x e r c i s e d ,  there-  f o r e , i n i n t e r p r e t i n g the s p e c t r a l r e c o r d s .  F i g . 10 d e p i c t s the waveforms at the two r a d i a l v i b r a t i o n o f the p l a t e . Cantilever  resonant  T h e i r i n t e r p r e t a t i o n i s g i v e n i n Appendix A.  Bar F i g u r e 9 shows the C h l a d n i f i g u r e s f o r the t r a n s v e r s e  t i o n o f the  vibra-  bar. N e x t , the n a t u r a l f r e q u e n c i e s f o r the t r a n s v e r s e  vibration  and the n o n - d i m e n s i o n a l i z e d n o d a l d i s t a n c e s from the b u i l t - i n end are t a b u lated. The f r e q u e n c i e s i n d i c a t e d i n T a b l e 3 have the same l i m i t a t i o n s as t h e measured f r e q u e n c i e s f o r the p l a t e .  The resonant  were measured t o check the assumed boundary c o n d i t i o n s .  frequencies  38  2nd Mode Fig.  9»  Mode  3 r d Mode  C h l a d n i F i g u r e s f o r the T r a n s v e r s e o f the C a n t i l e v e r Bar  Frequency Hz  Non-Dimensionalized Nodal D i s t a n c e s from the B u i l t - i n End  1  2  2  1367  O.78  3  3805  0.50  4  7364  T a b l e 3»  0.86  Non-Dimensionalized Nodal D i s t a n c e s f o r the  Note:  Vibration  Bar  The n o d a l d i s t a n c e s f o r the f o u r t h mode c o u l d not be measured. The n o d a l l i n e s are p o o r l y d e f i n e d at 7364. F u r t h e r , a l l non-dimensionalized nodal d i s t a n c e s were c a l c u l a t e d u s i n g the exposed l e n g t h o f the bar as r e f e r e n c e l e n g t h .  The 'approximated constant power' b a r are shown i n Appendix C, F i g . C-17  frequency s p e c t r a f o r  and F i g . C - l 8 .  the  The two resonant  l o n g i t u d i n a l v i b r a t i o n s are c l e a r l y l a b e l l e d i n the a x i a l s t r a i n r e c o r d , Fig.  C-l8.  Expected l o c a t i o n s o f h i g h e r v i b r a t i o n s than t w i c e the  frequency  39 B r e a t h i n g Mode (a)  Frequency  88lO Hz  S i g n a l Order ( t o p t o bottom) Radial Strain Tangential S t r a i n Acceleration Displacement  (b)  Frequency kkok Hz S i g n a l Order ( t o p t o bottom) Bending and R a d i a l S t r a i n Sound P r e s s u r e Acceleration  (c)  Frequency  Vf03 Hz  S i g n a l Order ( t o p t o bottom) Radial Strain Sound P r e s s u r e Acceleration  (d)  Frequency  kk05 Hz  S i g n a l Order ( t o p t o bottom) Tangential S t r a i n Sound P r e s s u r e Acceleration  F i g . 10.  Waveforms a t Resonance f o r C i r c u l a r P l a t e  ko  o f t h e e x c i t a t i o n are i n d i c a t e d as w e l l as the expected t r a n s v e r s e  coupled  vibrations.  In F i g . 11 a r e shown t y p i c a l waveforms at the resonant  trans-  verse v i b r a t i o n .  2nd L o n g i t u d i n a l Frequency  4lJ5  Vibration  Hz  Order o f S i g n a l s ( t o p t o bottom) Longitudinal Strain Sound P r e s s u r e Acceleration  3rd L o n g i t u d i n a l Frequency  2756  Vibration  Hz  Order o f S i g n a l s  ( t o p t o bottom)  Longitudinal Strain Acceleration  F i g . 11.  Waveforms a t Resonance f o r C a n t i l e v e r Bar  kl  CHAPTER  VI  DISCUSSION OF EXPERIMENTAL RESULTS  Circular Plate Table k l i s t s transverse  the t h e o r e t i c a l resonant f r e q u e n c i e s f o r  vibration.  Mode  T a b l e k.  Frequency Hz  2  700  3  2030  k  kooo  T h e o r e t i c a l Resonant F r e q u e n c i e s f o r the T r a n s v e r s e V i b r a t i o n o f the C i r cular Plate  Comparing t h i s d a t a w i t h the d a t a i n T a b l e 2 , g r e a t ces i n the f r e q u e n c i e s are o b s e r v e d .  Further,  the  f r e q u e n c i e s i n T a b l e k are based on a p l a t e w i t h a f r e e edge and  supported i n the c e n t e r . however,  differen-  The d i f f e r e n c e i n f r e q u e n c i e s suggests  t h a t the boundary c o n d i t i o n s aimed f o r c o u l d not be r e a l i z e d . theoretical  the  The e x p e r i m e n t a l f r e q u e n c i e s l i s t e d i n T a b l e 2 ,  are based on a p l a t e w i t h a f r e e edge and ' c l a m p e d  over a s m a l l a r e a .  1  i n the  center  F o r t r a n s v e r s e v i b r a t i o n , the s m a l l ' c l a m p i n g ' a r e a  r e s u l t s i n a s t i f f e r p l a t e than a p l a t e clamped s t r i c t l y  i n the c e n t e r .  The  h i g h e r f r e q u e n c i e s i n T a b l e 2 as compared t o those i n T a b l e k seem, t h e r e f o r e , t o be  justified.  42  The c a l c u l a t i o n f o r the breathing mode (assuming a s o l i d p l a t e ) i n d i c a t e s a r a d i a l resonance at 8900 Hz.  The measured resonant  r a d i a l v i b r a t i o n frequency was 8810 Hz. For the r a d i a l v i b r a t i o n , a p l a t e with a d i s c o n t i n u i t y i n the center, i n the form of a c i r c u l a r h o l e , i s a l e s s s t i f f s t r u c t u r e than a plate without a hole. Therefore, i t i s t o be expected that the n a t u r a l frequencies f o r r a d i a l v i b r a t i o n w i l l be lower. The l a r g e s t d e v i a t i o n from the mean radius of the nodal r a d i i , Table 2, are r e l a t i v e l y s m a l l .  Consequently, the v i b r a t i o n was nearly  symmetrical with respect t o the center, or no gross s i n g u l a r i t i e s existed i n the p l a t e with exception of the center. The s t a t i c maximum d e f l e c t i o n of the p l a t e was determined t o be 7»l8 x (10)  inches.  I n a d d i t i o n t o t h i s d e f l e c t i o n , there existed the  d e f l e c t i o n due t o the i n i t i a l crookedness of the p l a t e .  The dynamic d e f l e c -  t i o n s were, i n general, much smaller than the s t a t i c d e f l e c t i o n . The t a n g e n t i a l s t r a i n record, F i g . C-l4, shows two pronounced peaks - one at 4405 Hz and one at 8810 Hz. Another at 1160 Hz cannot be explained.  No other s i g n i f i c a n t peaks were found.  The r a d i a l plus bending  s t r a i n record, F i g . C-l6, shows a l s o these two spikes i n a d d i t i o n t o the resonant bending peaks.  I t i s important t o note the d i f f e r e n c e i n shape of  the two peaks - the bending resonant s t r a i n s do not b u i l d up as f a s t as the r a d i a l resonant s t r a i n s .  A l s o , i t i s worthy t o examine the lower r a d i a l  resonant peak i n these f i g u r e s .  F i r s t , one sees from Table 2 that the fourth  mode of f l e x u r a l v i b r a t i o n occurs at 4335 Hz; second, the r a d i a l v i b r a t i o n occurs at 4405 Hz. The spike i s just t o the r i g h t of the peak bending s t r a i n and i s indeed approximately of the same magnitude.  However, since the gauge  was not located at a p o s i t i o n of maximum bending s t r a i n , i t must be concluded  43  t h a t the r a d i a l resonant s t r a i n i s s m a l l e r than the bending s t r a i n resonance.  The r a d i a l s t r a i n , F i g . C - 1 5 ,  r e c o r d c o r r o b o r a t e s these  Two p e a k s , at 4405 Hz and 8810 Hz r e s p e c t i v e l y , resonant v i b r a t i o n ,  are i n d i c a t e d .  resonant s t r a i n s are v i s i b l e .  at statements.  c o r r e s p o n d i n g t o the  In a d d i t i o n , v e s t i g e s  of a l l  radial  the bending  The e x p l a n a t i o n f o r t h i s might b e , one, an  i n c o m p l e t e compensation o f the gauges a n d , two, the i n f l u e n c e o f the bending s t r a i n on the l o n g i t u d i n a l  vibration.  N e x t , the waveforms i n v o l v e d are examined. tangential  s t r a i n waveforms,  at 8810 H z , are shown i n F i g . 1 0 .  a r e p e r i o d i c and o f the same f r e q u e n c y as the e x c i t a t i o n . 8810 Hz a resonant r a d i a l v i b r a t i o n , 10 ( b ) ,  (c),  and (d)  represent,  the  It  The waveforms  Therefore,  ' b r e a t h i n g mode', e x i s t s .  respectively,  the r a d i a l s t r a i n , and the t a n g e n t i a l for their interpretation.  The r a d i a l and the  strain.  at  Figure  the bending p l u s r a d i a l  strain,  Appendix A has t o be c o n s u l t e d  i s seen t h a t the s i g n a l i s e s s e n t i a l l y  the  super-  p o s i t i o n o f two harmonics - one o f the frequency of the e x c i t a t i o n , and one twice that frequency.  The waveform r e c o r d e d by the sound l e v e l meter i s  same as t h a t o f t h e s t r a i n Cantilever  the  signal.  Bar The t h e o r e t i c a l l y  determined n o d a l d i s t a n c e s f o r a  ..cantilever  beam, e x c i t e d a t the b u i l t - i n end, are l i s t e d i n T a b l e 5«  Non-Dimensionalized D i s t a n c e s from the B u i l t - i n End  Mode 1  2  2  O.783  3  0.504  0.868  4  0.358  0.644  T a b l e 5«  3  0.906  Non-Dimensionalized T h e o r e t i c a l Nodal D i s t a n c e s f o r a C a n t i l e v e r Beam  kk  In Table 6 are l i s t e d the t h e o r e t i c a l resonant frequencies f o r the transverse v i b r a t i o n of the beam as w e l l as estimates of the maximum e r r o r (upper bound) f o r the c a l c u l a t e d frequencies. The upper bound e r r o r estimates take i n t o account the inaccuracies i n the measurements of the p h y s i c a l propert i e s and parameters of the bar.  ,  M  o a e  T h e o r e t i c a l Frequency Hz  Maximum E r r o r Hz  2  ikOO  ikS  3  3860  ko6  k  7550  795  Table 6.  T h e o r e t i c a l Resonant Frequencies f o r F l e x u r a l V i b r a t i o n of the Beam  The predicted resonant frequencies f o r f l e x u r a l v i b r a t i o n l i s t e d i n Table 6 were derived from the c a n t i l e v e r beam frequency equation. Not l i s t e d are the predicted resonant frequencies f o r f l e x u r a l v i b r a t i o n f o r a cantilever plate. i n Table 6.  These frequencies are somewhat higher than those l i s t e d  The actual resonant f l e x u r a l frequencies of the bar, Table 3,  are lower than t£iose i n Table 6.  However, comparing the frequencies i n  Table 3 with those i n Table 6, taking i n t o account the upper bound on the e r r o r , one must conclude that the agreement between the two sets of data i s reasonable.  I t seems that the bar behaved more l i k e a beam than a p l a t e . The agreement between the measured non-dimensionalized nodal  distances from the b u i l t - i n end, Table 3, and the t h e o r e t i c a l nodal distances from the b u i l t - e n d tabulated i n Table 5 i s good.  This corroborates the con-  c l u s i o n a r r i v e d at i n the previous paragraph. The t h e o r e t i c a l l y predicted f i r s t coupled resonant l o n g i t u d i n a l  45 v i b r a t i o n s h o u l d o c c u r at o f the bar i s i t s  8520  H z . i f X i t i s assumed t h a t the e f f e c t i v e  exposed l e n g t h .  i s c o n s i d e r e d , the f i r s t  If  length  the h a l f l e n g t h o f the f u l l double b a r  c o u p l e d l o n g i t u d i n a l resonant v i b r a t i o n ought t o  o c c u r at 8150 H z . Comparing t h i s d a t a w i t h the e x p e r i m e n t a l f i n d i n g s , a d i s c r e p a n c y i s observed i n the f i r s t frequency.  c o u p l e d l o n g i t u d i n a l resonant  vibration  T h i s s u g g e s t s t h a t the boundary c o n d i t i o n s aimed f o r c o u l d not  be r e a l i z e d .  In c o n s i d e r i n g the l o n g i t u d i n a l v i b r a t i o n , the e f f e c t i v e  of the b a r i s not i t s  exposed l e n g t h , but i t  i n c l u d e s an a d d i t i o n a l  length  length  under the clamp. Examining next the s t r a i n r e c o r d s , the f o l l o w i n g are made. s p i k e at  A x i a l S t r a i n - as can be seen from F i g . C - l 8 , t h e r e i s a pronounced  8270  H z , the f i r s t  c o u p l e d l o n g i t u d i n a l resonant v i b r a t i o n , and one  at 4155 Hz e x c i t a t i o n , the second coupled l o n g i t u d i n a l resonant Further,  t h e r e i s a v e r y , v e r y s m a l l s p i k e at  can be i d e n t i f i e d p o s i t i v e l y .  2755  excitation.  No o t h e r s  A l s o , as f o r the p l a t e , F i g . C - l 4 , one sees  p r e t e d i n the same l i g h t as those f o r the p l a t e .  C-17,  vibration.  Hz l a b e l l e d A^.  v e s t i g e s o f the f l e x u r a l s t r a i n resonant v i b r a t i o n s .  in Fig.  observations  t h e r e i s a pronounced s p i k e at  None can be seen at  2755  Hz.  These must be i n t e r -  Bending p l u s A x i a l S t r a i n -  8270  Hz and one at  4135  Hz  As b e f o r e , the resonant l o n g i t u -  d i n a l s t r a i n i s at most equal i n magnitude t o the resonant f l e x u r a l In the same s p e c t r a , one sees the f l e x u r a l resonant p e a k s .  strain.  A g a i n , one notes  t h a t the f l e x u r a l resonant v i b r a t i o n s b u i l d up much more s l o w l y than the l o n g i t u d i n a l resonant v i b r a t i o n s .  One o t h e r item i n these s p e c t r a i s worthy  o f note - s p i k e s l a b e l l e d A^ and A^.  The f i r s t  one o c c u r s at o n e - h a l f  the  f r e q u e n c y o f the e x c i t a t i o n o f the second bending resonant mode; the second one o c c u r s at o n e - h a l f the f r e q u e n c y o f the e x c i t a t i o n o f the f o u r t h resonant  k6  bending mode.  Both peaks are extremely s m a l l compared t o the s t r a i n s  p r i o r and j u s t a f t e r i t s  o c c u r r e n c e and both peaks show the  slow b u i l d - u p o f bending resonant s t r a i n s . s m a l l f o r an a c c u r a t e waveform a n a l y s i s .  just  characteristic  The s i g n a l s i n v o l v e d are t o o However, by analogy w i t h the more  pronounced l o n g i t u d i n a l c o u p l e d v i b r a t i o n f i n d i n g s , i t  can be assumed t h a t  t h e s e r e p r e s e n t f l e x u r a l v i b r a t i o n s o f the bar at t w i c e the f r e q u e n c y o f excitation.  In o t h e r words, these s m a l l peaks r e p r e s e n t the i n f l u e n c e o f  l o n g i t u d i n a l coupled v i b r a t i o n on the f l e x u r a l v i b r a t i o n as i n d i c a t e d i n  the the the  t h e o r y by the term (U W ) . X X X Examining next the waveforms at the t h r e e s i g n i f i c a n t for  spikes  l o n g i t u d i n a l v i b r a t i o n , F i g . 11, making use of Appendix A, one has t o  conclude the f o l l o w i n g : v i b r a t i o n of f r e q u e n c y  at  8270  8270  Hz e x c i t a t i o n , a l o n g i t u d i n a l resonant  Hz e x i s t s ; at  resonant v i b r a t i o n o f f r e q u e n c y  8270  4135  Hz e x c i t a t i o n a l o n g i t u d i n a l  Hz e x i s t s ; at  t u d i n a l resonant v i b r a t i o n o f f r e q u e n c y  8270  2755  Hz e x i s t s .  (  Hz e x c i t a t i o n a l o n g i -  47  CHAPTER  VII  SUMMARY AND CONCLUSION  Suggestions f o r Future  Research  B e s i d e s the obvious r e s e a r c h t o be done, such as the c a l s o l u t i o n o f the exact f o r m u l a t i o n , a number o f e x p e r i m e n t a l  analyti-  investigations  based on the p r e d i c t i o n s o f the s i m p l e , proposed f o r m u l a t i o n can and s h o u l d be done.  The r e l a t i o n s h i p between response and e x c i t a t i o n s h o u l d be examined.  T h i s i n v e s t i g a t i o n can be done w i t h the p r e s e n t s e t up.  Next, t h i s  research  s h o u l d be extended t o the upper end o f the a c o u s t i c frequency r a n g e .  The  damping o f the t r a n s v e r s e and l o n g i t u d i n a l v i b r a t i o n s h o u l d be i n v e s t i g a t e d . Quantitative  answers s h o u l d be found f o r the i n f l u e n c e e x e r t e d on the  and frequency response by v i s c o e l a s t i c and e l a s t i c - v i s c o e l a s t i c on the s u r f a c e s o f the b a r . elastic-viscoelastic  amplitudes  damping l a y e r s  O p t i m i z i n g parameters f o r the v i s c o e l a s t i c  damping l a y e r s s h o u l d be f o u n d .  The i s o l a t i o n  and  properties  o f t h e s e m a t e r i a l s at the f r e q u e n c y range encountered s h o u l d be i n v e s t i g a t e d . T h e i r t r a n s m i s s i o n - o f - e n e r g y p r o p e r t i e s are o f i n t e r e s t . e l a s t i c m a t e r i a l s s h o u l d be i n v e s t i g a t e d s h o u l d be e v a l u a t e d ;  'in situ  The s u r f a c e treatment  c u l a r damping m a t e r i a l must be d e t e r m i n e d .  1  Conventional v i s c o -  and o t h e r p r o m i s i n g m a t e r i a l s  o f the b a r , o r p l a t e , f o r a p a r t i The f a t i g u e and temperature depend-  ence o f v i s c o e l a s t i c l a y e r s must be i n v e s t i g a t e d . The i n s t r u m e n t a t i o n can be r e f i n e d .  If  possible, a control  u n i t f o r c o n s t a n t displacement c o n t r o l up t o the upper end o f the  acoustic  f r e q u e n c y range s h o u l d be a c q u i r e d - the c o n t r o l technique adopted f o r  this  k8  r e s e a r c h i s not the b e s t one. s h o u l d be a v a i l a b l e . investigations  F o r f u r t h e r i n v e s t i g a t i o n s , a F o t o n i c Sensor  T h i s i n s t r u m e n t i s the most p r o m i s i n g one f o r waveform  - n o i s e problems a s s o c i a t e d w i t h the o t h e r two s e t s of  measuring i n s t r u m e n t s are e l i m i n a t e d h e r e b y . future,  It  i s suggested t h a t i n  the  s t r a i n gauges d e s i g n e d e x p l i c i t l y f o r dynamic a p p l i c a t i o n be u s e d .  Time and c a p i t a l o u t l a y can be saved t h e r e b y .  The BAM 1 s h o u l d be  t o a BAM 2 through the a c q u i s i t i o n o f a s u i t a b l e E l l i s A s s o c i a t e s  altered amplifier.  The f r e q u e n c y response o f the p r e s e n t u n i t i s down a p p r o x i m a t e l y 3% at 10 KHz. An a m p l i f i e r o f s u i t a b l e g a i n and f r e q u e n c y response s h o u l d be d e s i g n e d t o r e p l a c e the a m p l i f i e r o f the frequency a n a l y z e r . for i t s  intended purpose.  T h i s u n i t s h o u l d be  An anechoic chamber s h o u l d be i n s t a l l e d and  a c o u s t i c c h a r a c t e r i s t i c s be s t u d i e d .  s h o u l d be i n c o r p o r a t e d i n t o the system.  t o the sweep c o n t r o l o f the g e n e r a t o r . be approximated more c l o s e l y .  its  The G e n e r a l Radio Sound l e v e l meter  s h o u l d be r e p l a c e d by an i n s t r u m e n t h a v i n g a wider l i n e a r r a n g e . filter  available  A band pass  The band s h o u l d be s y n c h r o n i z e d  The d e s i r e d boundary c o n d i t i o n s s h o u l d  The arrangement used f o r the c a n t i l e v e r  seems t o be more p r o m i s i n g than the support f o r the p l a t e .  bar  A technique  to  e v a l u a t e damping m a t e r i a l s s h o u l d be d e t e r m i n e d .  Summary The u n d e r l y i n g r e a s o n f o r d o i n g t h i s e x p e r i m e n t a l workjwas. t o verify  the s p e c u l a t i o n t h a t the t r a n s v e r s e  t u d i n a l v i b r a t i o n s o f the beam. up.  e x c i t a t i o n o f a beam l e a d s t o  To t h i s e f f e c t ,  a s i m p l i f i e d model was s e t  Assuming s t r a i n s i n the c e n t r a l p l a n e , coupled n o n l i n e a r p a r t i a l  ferential  longi-  e q u a t i o n s o f motion were d e r i v e d f o r a beam.  dif-  By s u i t a b l y m a n i p u l a -  t i n g the e q u a t i o n s and i n t e r p r e t i n g some o f the c o u p l i n g t e r m s , two coupled l o n g i t u d i n a l v i b r a t i o n s , w i t h a frequency r a t i o o f 1:2,  were p r e d i c t e d .  a d d i t i o n , two c o u p l e d f l e x u r a l v i b r a t i o n s , w i t h a f r e q u e n c y r a t i o o f  1:2,  In  49 were a n t i c i p a t e d . The e x p e r i m e n t a l i n v e s t i g a t i o n proved the v a l i d i t y above p r e d i c t i o n s .  of  the  Two resonant l o n g i t u d i n a l v i b r a t i o n s were determined f o r  a b a r and the f r e q u e n c y r a t i o was 1:2.  Further,  through the  investigation,  the l o c a t i o n o f the resonant peaks i n r e g a r d s t o the frequency o f the t i o n was e s t a b l i s h e d .  The f i r s t  excita-  l o n g i t u d i n a l resonant v i b r a t i o n o c c u r r e d at  the fundamental f r e q u e n c y f o r l o n g i t u d i n a l resonant v i b r a t i o n .  The second  l o n g i t u d i n a l resonant v i b r a t i o n o c c u r r e d at 1/2 times the f r e q u e n c y o f transverse  the  e x c i t a t i o n at the fundamental f r e q u e n c y f o r l o n g i t u d i n a l resonant  v i b r a t i o n and the frequency o f the v i b r a t i o n was t h a t o f the fundamental l o n g i t u d i n a l resonant  vibration. A l s o , one l o n g i t u d i n a l v i b r a t i o n at t h r e e times the  o f the e x c i t a t i o n i n the t r a n s v e r s e d i r e c t i o n was r e c o r d e d .  frequency  The f r e q u e n c y o f  the v i b r a t i o n seemed t o be t h a t o f the fundamental l o n g i t u d i n a l resonant vibration.  I n the s t r a i n r e c o r d s t h e r e i s evidence t h a t t h e r e are more  f l e x u r a l resonant v i b r a t i o n s than i n d i c a t e d by the l i n e a r t h e o r y . at o n e - h a l f the f r e q u e n c y f o r f l e x u r a l resonant s t r a i n s .  These o c c u r  A waveform  analysis  was not c a r r i e d out s i n c e the s i g n a l i n v o l v e d i s v e r y s m a l l . Some g e n e r a l c o n c l u s i o n s can be drawn i n r e g a r d s t o the  rela-  t i v e v a l u e o f the magnitudes at resonance between the f l e x u r a l v i b r a t i o n and the l o n g i t u d i n a l v i b r a t i o n .  At b e s t , the l o n g i t u d i n a l s t r a i n at  resonance  e q u a l s the bending p l u s l o n g i t u d i n a l s t r a i n f o r t r a n s v e r s e r e s o n a n c e ; general, i t  i s smaller.  Further,  it  seems t h a t t h e magnitude o f the  in strains  at resonance d e c r e a s e s as the o r d e r of the l o n g i t u d i n a l f r e q u e n c y i n c r e a s e s .  An e x p e r i m e n t a l i n v e s t i g a t i o n , analogous t o t h a t of the b a r , was c a r r i e d out f o r a c i r c u l a r p l a t e suspended i n the c e n t e r and e x c i t e d  50 transversely. recorded. 1/2  Two c o u p l e d resonant v i b r a t i o n s i n the r a d i a l d i r e c t i o n were  T h e i r f r e q u e n c y r a t i o was 1:2;  times the f r e q u e n c y o f the t r a n s v e r s e  quency f o r r a d i a l resonant v i b r a t i o n .  they o c c u r r e d r e s p e c t i v e l y  at 1 and  e x c i t a t i o n f o r the fundamental  No n o n l i n e a r coupled f l e x u r a l  at o n e - h a l f the f r e q u e n c y o f resonant f l e x u r a l  'linear*  fre-  vibration  v i b r a t i o n were i d e n t i -  fied.  Conclusion From the i n v e s t i g a t i o n , the f o l l o w i n g c o n c l u s i o n s were drawn: (1)  The a n t i c i p a t e d l o n g i t u d i n a l v i b r a t i o n s e x i s t , when a b a r i s  excited  transversely. (2)  There i s some evidence t h a t the a n t i c i p a t e d f l e x u r a l v i b r a t i o n s o f beam at t w i c e the f r e q u e n c y o f the e x c i t a t i o n e x i s t . however,  the  Their influence,  on the t r a n s v e r s e v i b r a t i o n , even at t h e i r resonant c o n d i t i o n ,  was extremely s m a l l . (3)  The d e r i v e d d i f f e r e n t i a l e q u a t i o n s o f motion f o r the beam are  correct  i n s o f a r as the p r e d i c t i o n s o f f r e q u e n c i e s o f coupled v i b r a t i o n a r e  con-  cerned. (k)  The method used i n i n t e r p r e t i n g the s p e c i a l c o u p l i n g terms i s  (5)  Experimentally,  it  was determined t h a t h i g h e r l o n g i t u d i n a l  than the second do o c c u r .  correct.  vibrations  A t h i r d resonant l o n g i t u d i n a l v i b r a t i o n  was  recorded. (6)  Resonant v i b r a t i o n s o c c u r at the i n v e r s e r a t i o o f t h e i r That i s ,  frequencies.  f o r the beam, the l o n g i t u d i n a l v i b r a t i o n frequency r a t i o i s  the resonant l o n g i t u d i n a l v i b r a t i o n s o c c u r r e s p e c t i v e l y t i m e s the t r a n s v e r s e  1:2,  at 1 and 1/2  e x c i t a t i o n at the fundamental l o n g i t u d i n a l resonant  vibration. (7)  It  seems t h a t ,  at b e s t , the amplitude f o r resonant l o n g i t u d i n a l  vibration  51 s t r a i n e q u a l s the amplitude f o r resonant a x i a l p l u s bending s t r a i n . general, i t  i s smaller.  Nevertheless,  the amplitude f o r resonant  In  longi-  t u d i n a l v i b r a t i o n s t r a i n i s l a r g e r than the amplitude f o r l o n g i t u d i n a l p l u s bending s t r a i n at near-by non-resonant c o n d i t i o n . The l o n g i t u d i n a l resonant s t r a i n s b u i l d up much f a s t e r than the resonant  flexural  strains.  F o r the p l a t e , two r a d i a l coupled v i b r a t i o n s were r e c o r d e d . quency r a t i o was 1:2;  t h e y o c c u r r e d at 1 and 1/2  f r e q u e n c y f o r the b r e a t h i n g mode.  times the  Their  fre-  transverse  52  BIBLIOGRAPHY  H a r r i s , C.M. and C r e d e , C . E . , "Shock and V i b r a t i o n Handbook", v o l . 1, McGraw-Hill Book C o . , N.Y. 1961. J a c o b s e n , L . S . and A y r e , R . S . , " E n g i n e e r i n g V i b r a t i o n s " , McGraw-Hill Book C o . , N.Y. 19587^ L o v e , A . E . H . , "A T r e a t i s e on the Mathematical Theory o f t i c i t y " , 3 r d e d . , Cambridge, U n i v e r s i t y P r e s s , 1920.  Elas-  McLeod, A . J . and B i s h o p , R . E . D . , "The F o r c e d V i b r a t i o n o f C i r c u l a r F l a t P l a t e s " , Mechanical Engineering Science, monograph n o . 1, The I n s t i t u t i o n o f M e c h a n i c a l E n g i n e e r s , London, March, 19^5. M e t t l e r , E . , Dynamic B u c k l i n g , "Handbook o f E n g i n e e r i n g M e c h a n i c s " , 1 s t e d . , F l u e g g e , W., e d i t o r , McGraw-Hill Book Company, I n c . 1962. P e a r s o n , K., "Memoir on the F l e x u r e o f Heavy Beams S u b j e c t e d t o Continuous Systems o f L o a d " , Q u a r t e r l y J o u r n a l o f Pure and A p p l i e d Mathematics, v o l . 2 4 , Longmans, Green, and C o . , London, 1 8 9 0 . P e a r s o n , K., and F i l o n , L . H . G . , "On the F l e x u r e o f Heavy Beams S u b j e c t e d t o Continuous Systems o f L o a d " , Q u a r t e r l y J o u r n a l o f Pure and A p p l i e d Mathematics, v o l . 3 1 , Longmans, Green and C o . , London, 1900. S t r u t t , J . W . , L o r d R a y l e i g h , " T h e o r y o f S o u n d " , v o l . 1 and 2 , Dover P u b l i c a t i o n s , N . Y . , 19k~5~. Thomson, W . T . , " V i b r a t i o n Theory and A p p l i c a t i o n s " , P r e n t i c e H a l l , I n c . , N.J., 1965. Timoshenko, S. and Woinowsky-Krieger, S . , " T h e o r y o f P l a t e s and S h e l l s " . McGraw-Hill Book Company, I n c . , N . Y . , 1959.  53  APPENDIX A  ADDITION OF TWO SINUSOIDS  In t h i s s e c t i o n , a n a l y t i c a l graphical interpretations amplitude, d i f f e r e n t  e x p r e s s i o n s are d e r i v e d and  shown f o r the a d d i t i o n of two s i n u s o i d s o f  f r e q u e n c i e s , and i n phase such t h a t both f u n c t i o n s c r o s s  the time a x i s at the same i n s t a n t  o f t i m e , and t h a t an i n s t a n t l a t e r  f u n c t i o n s are e i t h e r p o s i t i v e , o r  negative.  Part  different  both  1 Different  amplitude  One f r e q u e n c y b e i n g t w i c e the In We c o n s i d e r the  other  phase  function V  =  A s i n 9 + B s i n 26  =  (A * 2B cos G) s i n e  =  (Modulated A m p l i t u d e ) ( C i r c u l a r  (l6) Function)  E q u a t i o n ( l 6 ) r e p r e s e n t s a modulated amplitude r o t a t i n g at the frequency o f the fundamental f u n c t i o n , t h a t i s , r e g a r d l e s s how l a r g e the i n d i v i d u a l a m p l i tudes a r e , the r e s u l t i n g waveform has one p e r i o d e q u a l t o t h a t o f the f u n d a mental f u n c t i o n . The maxima and the minima o f the r e s u l t i n g f u n c t i o n  are  found from  -- k  <">  5^  ^  £2  =  _  s i n 29 - A s i n  As good as the above approach i s , i t e v a l u a t i o n o f the f u n c t i o n (17) and (18) above.  6  does not a f f o r d an easy-  F o r t h i s r e a s o n , the g r a p h i -  c a l e v a l u a t i o n o f the o r i g i n a l f u n c t i o n i s shown i n F i g . 12. has been d e r i v e d above, a n a l y t i c a l l y ,  Everything  can be v e r i f i e d i n these  AMPLITUDE  F i g . 12.  (l8)  which  figures.  RATIO |:|  Graphical A d d i t i o n bf Sinusoids - Frequency R a t i o 2  The frequency r a t i o i s d e f i n e d as  follows  Frequency o f 'Harmonic' Frequency o f Fundamental and the d e f i n i t i o n o f the amplitude r a t i o  is  Amplitude o f 'Harmonic' Amplitude o f Fundamental The important i t e m s which deserve a t t e n t i o n i n F i g . 12 first,  are,  the waveform i n v o l v e d , a n d , s e c o n d , the r e l a t i v e magnitude o f the  peaks i n v o l v e d .  As the amplitude o f the h a r m o n i c , t w i c e the f r e q u e n c y of  two the  55 fundamental, i s i n c r e a s e d , l e a v i n g the o t h e r unchanged, the r e l a t i v e a m p l i tude o f the second peak i n c r e a s e s .  As l o n g as the s i n g l e frequency i s  a r e l a t i v e i n c r e a s e o f the amplitude o f the second harmonic f u n c t i o n  finite,  will  cause the amplitude o f the second peak t o approach the amplitude o f the peak.  However,  allows  one t o estimate the r a t i o o f magnitudes i n v o l v e d , without  a Fourier  it  w i l l never equal i t .  first  T h i s i s an important d e d u c t i o n .  It  c a r r y i n g out  analysis. As a f i n a l a n a l y t i c a l  d e r i v a t i o n i n t h i s s e c t i o n , the RMS  v a l u e o f the r e s u l t i n g f u n c t i o n w i l l be d e t e r m i n e d . interpretation  T h i s i s of value i n  the  o f s t r a i n f r e q u e n c y - s p e c t r a s i n c e , f o r the sake of m i n i m i z a t i o n  o f e r r o r s , RMS v a l u e s were p l o t t e d t h e r e o n .  V  -  R M S  v (e)de  j l j  2  To make the s o l u t i o n as g e n e r a l as p o s s i b l e , the f o l l o w i n g f u n c t i o n i s  con-  sidered V  where  n  i s an i n t e g e r ,  n >  =  A s i n G + B s i n nG  1  Substituting  V  RMS  =  sJl  (A  "  +  B  2  )  C o n s e q u e n t l y , the RMS v a l u e o f a f u n c t i o n c o n s i s t i n g o f the s u p e r p o s i t i o n o f two s i n e f u n c t i o n s , the f r e q u e n c y o f one b e i n g an i n t e g r a l m u l t i p l e o f o t h e r , i n phase, i s always g i v e n by the above e x p r e s s i o n .  the  56  Part 2 Different  amplitude  One frequency b e i n g t h r i c e the In It  other  phase  can be shown t h a t V  =  A sin  =  £ A + B (3 c o s  =  Q+  B sin 3 £ 2  Q - sin  Q )]  2  sin Q  £ Modulated a m p l i t u d e J ( c i r c u l a r  (18)  function)  As b e f o r e , the c o n d i t i o n f o r a maxima, o r a minima i s by cos Q  |  A + B (3 c o s  2  Q - 9 sin  2  J  0. ) I  =  given  0  and  dfv dQ  -  A sin  Q +  B sin  Q (9  sin  2  Q - 29  cos  2  Q )  2  E v e r y t h i n g s a i d f o r P a r t 1 has a c o u n t e r p a r t i n P a r t 2.  For  s i m i l a r r e a s o n s , the g r a p h i c s o l u t i o n o f the p r o b l e m , P a r t 2, i s shown below i n F i g , 13» i n Part  The d e f i n i t i o n o f amplitude r a t i o and f r e q u e n c y r a t i o i s  1.  AMPLITUDE RATIO  F i g , 13,  1=3  Graphical Addition of Sinusoids - Frequency R a t i o 3  given  57 APPENDIX B  LINEAR EQUATIONS FOR PLATE AND BEAM  Circular Plate The  'linear'  e q u a t i o n o f motion f o r t r a n s v e r s e  s y m m e t r i c a l about the c e n t e r a c c o r d i n g t o (4) i s  vibration  written  whose s o l u t i o n , assuming a harmonic e x c i t a t i o n , i s ' W ( r , t ) = J" A J  '  I  o  (Kr)  + B Y  o  *4  w h e r e  and A , B,  (Kr)  + C I  o  (Kr)  + D K  o  (Kr)  J  cos cot  gut  C , and D a r e determined by the boundary c o n d i t i o n s . F o r a c i r c u l a r p l a t e clamped i n the c e n t e r , with f o r c e e x c i -  t a t i o n i n the c e n t e r and o u t e r edge f r e e , the n a t u r a l t r a n s v e r s e i s g i v e n by ( l )  vibration  as  a)  =  B  n  I  ,  \l ^  ^  .  (1 - ^ )  ^/SEC  2  and  B  1  = 4.35 The  B  2  = 24.26  B^ = 70.39  ' b r e a t h i n g mode', o r the f i r s t  deduced from the e q u a t i o n s l i s t e d i n  (3)  B^ = 138.85  r a d i a l v i b r a t i o n can be  5 8  dJ (Kr) =  -hsy  J  o  (  K  r  -  )  J  i  (  K  r  )  Ka = 4 . 1  By t r i a l and e r r o r but  K  therefore  cp  =  -  co  E  -  f  ( 1  1 3 . 9 4  a  ( 1 0 ) * *  H  s  C a n t i l e v e r Bar The l i n e a r equations of motion f o r the transverse v i b r a t i o n of a beam, v i b r a t i n g under i t s own weight, assuming a harmonic e x c i t a t i o n , i s given by ( 9 ) as  ^  - nS =  dx 4  where  n  _  0  2  cpco  EI  \4iose general s o l u t i o n i s y  =  A cosh nx + B s i n h nx + C cos n^x + D s i n nix  and A, B, C, and D are determined by the boundary conditions. The frequency equation and the respective constants according to  ( 2 )  is co  and  (^E)  (r e) k  =  2  2  n  =  ( r nI) g  / ET T— Acp  —r-r-  3 . 5 2  =120.8  ./ U  2  (r  2  T)  2  =  2 2 . 0  RAD,-™  /SEC '  (r^L)  2  =  6l ?5 0  59 The f i r s t  l o n g i t u d i n a l v i b r a t i o n a c c o r d i n g t o (2)  c u l a t e d by u s i n g  W  n  =  (2B  -  1)  2l  „. n  / E X*  is  cal-  60 APPENDIX C  APPROXIMATE CONSTANT POWER FREQUENCY SPECTRA  Fig.  C-14.  T a n g e n t i a l S t r a i n Frequency S p e c t r a for Circular Plate  F i g . C-15.  R a d i a l S t r a i n Frequency S p e c t r a for Circular Plate  Fig. C-l6.  R a d i a l p l u s Bending Strain FrequencySpectra f o r C i r c u l a r Plate  63  F i g . C-17.  Bending p l u s A x i a l Frequency S p e c t r a f o r C a n t i l e v e r Bar  Fig. C-l8.  A x i a l S t r a i n Frequency S p e c t r a f o r C a n t i l e v e r Bar  

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