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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, University 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 for the Degree of Master of Applied Science In the Department of Mechanical Engineering We accept t h i s thesis as conforming to 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 t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s 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 , I a g r e e t h a t t h e 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 and S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be 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 . I t i s u n d e r s t o o d t h a t c o p y i n g 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 be 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 on. D e p a r t m e n t o f Mechanical Engineering The 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 V a n c o u v e r 8, Canada D a t e A p r i l 30. 1969  I TABLE OF CONTENTS Page ABSTRACT i ACKNOWLEDGEMENT i i i LIST OF FIGURES i v LIST OF TABLES • LIST OF APPENDICES ... v i NOMENCLATURE v i i CHAPTER I DEFINITION OF THE PROBLEM 1 In t roduct ion 1 Statement of the Problem 3 L i t e r a tu re Survey • 3 L im i ta t ions of the Study * ^ D e f i n i t i o n s 5 CHAPTER II THEORETICAL CONSIDERATIONS 6 Der i va t ion 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 6 D iscuss ion of the Derived Equations 11 CHAPTER II I APPARATUS AND INSTRUMENTATION 1^  S igna l Flow ^ Ind i v idua l Instruments 18 Test Bed and Frame - 21 Suspension 22 P la te Support 23 Bar Clamp ' 2 3 Models and Boundary Condit ions 2.h C i r c u l a r P la te 2^ Cant i l eve r Bar 25 S t r a i n Gauges and Bridges 25 C i r c u l a r P la te 25 Cant i l eve r Bar 26 Bridges 26 Page CHAPTER IV TESTING. 27 C a l i b r a t i o n and Check Out 27 Test Procedure • • 28 Constant Power P lo t D i s cuss ion . 30 In te rpre ta t ion of Frequency Spec t r a . . 33 Chladni F igures 3k P l o t t i n g and Photography J>k CHAPTER V EXPERIMENTAL RESULTS 35 C i r c u l a r P la te 35 Cant i l eve r Bar 37 CHAPTER VI DISCUSSION OF EXPERIMENTAL RESULTS kl C i r c u l a r P la te *H Cant i l eve r Bar kj> CHAPTER VII SUMMARY AND CONCLUSION kj Suggestions f o r Future Research k? Summary k8 Conclusion • • 50 BIBLIOGRAPHY 52 APPENDIX 53 i ABSTRACT The theory presented describes the physical phenomenon of nonlinear coupling of longitudinal and f l e x u r a l vibrations when a beam i s excited transversely at high frequencies. Equations aire derived based on the Bernoulli-Euler theory of flexure, by energy methods, to describe the trans-verse and the longitudinal vibration of a beam of constant cross-section under plane motion. The i n i t i a l crookedness of the beam and the longitudinal i n e r t i a , accounted for i n the theory, give r i s e to the coupled vibrations. No closed form solution i s presented. However, a simple analysis of some of the coupling terms suggests the existence of several coupled vibrations. By the method proposed herein, the frequencies of these vibrations can be established. I n p a r t i c u l a r , the theory predicts two longitudinal coupled vibrations with the frequency r a t i o 1:2. The agreement between the theory and the experimental results i s good. The vibrations predicted exist and the frequency r a t i o for the pre-dicted longitudinal vibrations was 1:2. Further, the experimental results indicate that there are more longitudinal vibrations than indicated by the theory. A longitudinal coupled vibration at three times the frequency of transverse excitation was recorded. There are indications i n the data that coupled f l e x u r a l vibrations at twice the frequency of transverse excitation e x i s t . A c i r c u l a r plate c e n t r a l l y supported and transversely excited was also tested. Two pronounced resonant r a d i a l vibrations were recorded. The frequency r a t i o was 1:2. Coupled f l e x u r a l vibrations were not i d e n t i f i e d . i i The influence of the longitudinal 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 significance of the resonant strains i s discussed. I i i ACKNOWLEDGEMENT I wish to express my gratitude to my advisors, Dr. C.R. Hazell and Dr. H. Ramsey, for giving me the opportunity to work on t h i s challenging project and for granting me so much la t i t u d e i n my work. Further, I take t h i s opportunity to thank a l l the technicians and secretaries i n the Department for contributing d i r e c t l y , or i n d i r e c t l y , to the research. This study was made possible through research grant No. 9513-07 provided by the Defense Research Board of Canada. a i v LIST OF FIGURES Figure page 1. Extension and Rotation of Central Plane Fiber. 7 2 . Reference Axis for Displacement Measurements 9 3 . Signal Flowchart 15 k. Overall View of the Instrumentation 18 5 . Test Bed 21 6. Bar Suspension • 22 7. Constant Acceleration and Approximated Constant Power Frequency Spectra 31 8 . Chladni Figures for the Transverse Vibration of the Ci r c u l a r Plate 35 9 . Chladni Figures for the Transverse Vibration of the Cantilever Bar 38 1 0 . Waveforms at Resonance for C i r c u l a r Plate 39 11. Waveforms at Resonance for Cantilever Bar ko A-12. Graphical Addition of Sinusoids Frequency Ratio 2 . . . . A-13. Graphical Addition of Sinusoids Frequency Ratio 3 . . . . 56 C-l*f. Tangential S t r a i n Frequency Spectra for C i r c u l a r Plate 60 C-15. Radial S t r a i n Frequency Spectra for C i r c u l a r Plate... 6 l C - l 6 . Radial plus Bending S t r a i n Frequency Spectra for Circular Plate 62 C-17. Bending plus A x i a l Frequency Spectra for Cantilever Bar 63 C - l 8 . A x i a l S t r a i n Frequency Spectra for Cantilever Bar.... 6k LIST OF TABLES Page Table 1 Allowable Acceleration Level • 30 Table 2 Non-Dimensionalized Nodal Radii for Plate.. 36 Table 3 Non-Dimensionalized Nodal Distances for Bar 38 Table k Theoretical Resonant Frequencies for the Transverse Vibration of the Circular Plate... * f l Table 5 Non-Dimensionalized Theoretical Nodal Distances for a Cantilever Beam. kj> Table 6 Theoretical Resonant Frequencies for Flexural -Vibrations of the Beam....... kk v i LIST OF APPENDICES Page APPENDIX A Addition of Two Sinusoids 53 APPENDIX B Linear Equations for Plate and Beam 57 APPENDIX C Approximate Constant Power Frequency Spectra 60 v i i NOMENCLATURE Symbol a = diameter of plate, i n f = frequency, cycles per second 2 g = acceleration of gravity, in/sec h = bar and plate thickness, i n r = r a d i a l distance, i n t = time, sec W = deflection from s t a t i c equilibrium position, i n W = s t a t i c deflection of bar, or beam, i n (includes deflection due to i n i t i a l crookedness) W ,U = p a r t i a l derivative of a displacement with respect to the variable x 2 A = cross-section of beam, i n D = plate s t i f f n e s s 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 variable v i i i Abbreviations BAM = bridge amplifier and meter CRO = cathode ray oscilloscope Hg = cycles per second RMS = root-mean-square value of a function J Q = Bessel function of the f i r s t kind, i n t e g r a l order o Y Q = Bessel function of the second kind, i n t e g r a l order o I Q = Modified Bessel function of the f i r s t kind, i n t e g r a l order o K = Modified Bessel function of the second kind, i n t e g r a l order 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 is not appropriate. Additional phenomena arise in the mechanism of acoustic energy transmission under these circumstances to those encountered in the audible frequency range usually considered in architectural acoustics. An analysis by means of the simple linear theory indicates for the upper end of the acoustic frequency range that in 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 is the likelihood of coupling between the two types of waves at the upper end of the frequency range swept out in the experiment, whereas the coupling seems to be insigni-ficant 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 in parenthesis designate references in the Bibliography. 2 well as for ap. usually omitted i n e r t i a term. The i n i t i a l crookedness and the longitudinal r i n e r t i a term account for the coupled vibrations recorded i n the experiment. The l i m i t a t i o n s of the two theories, the l i n e a r uncoupled theory and the non-linear coupled theory, are es s e n t i a l l y the same, namely those of the Bernoulli-Euler theory of flexure. The bending s t r a i n i s assumed to be d i r e c t l y proportional to the distance from the central plane. Plane cross sections are assumed to remain plane, that i s , the shear deforma-t i o n i s ignored. Rotary . i n e r t i a i s omitted and damping i s ignored. Summariz-ing, the proposed theory d i f f e r s from the l i n e a r theory i n that strains i n the central plane are assumed to e x i s t , whereas they are ignored i n the l i n e a r theory. No closed form solution i s presented for the equations of motion derived. However, an examination of some of the second order coupling terms indicates the existence of coupled vibrations. Thus, for a beam excited at the supports, transversely, the analysis of the following terms, i WW , W W , from the coupled equations indicates the existence of two l o n g i -X X2C X X3C tudinal vibrations with the frequency r a t i o of 1:2. Using the fact that there are two coupled longitudinal vibrations, one can interpret the mixed second order coupling terms i n the complimentary equation i n the same manner. Effect-ing the necessary substitutions, two coupled transverse vibrations with the frequency r a t i o 1:2 should occur. These represent the influence of the l o n g i -tudinal 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 interpretation indicates only the existence of the coupled vibration. I t does not give any indication 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 longitudinal vibration on the f l e x u r a l v i b r a t i o n . Simply, the method just indicates the frequencies of the coupled vibrations. 3 Statement of the Problem The purpose of t h i s study was: (1) to set up and check out the vibration system, (2) to determine a n a l y t i c a l l y and to interpret the coupled d i f f e r e n t i a l equations of motion for a beam of prismatic cross-section, under plane motion, (3) to v e r i f y , experimentally, the occurrence of the coupled vibrations, i n the longitudinal d i r e c t i o n , (4) to locate, experimentally, along the frequency spectrum l i n e , where the predicted resonant vibrations occur, (5) to determine the importance of the associated resonant vibrations, and (6) to determine for a c i r c u l a r plate supported and excited transversely i n the center the same experimental information as for the bar. Literature 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 plates. Theore-t i c a l l y , t h i s coupled vibrat i o n has not been treated before. Up to the present time the transverse vibration of the bar, or the plate, has been treated dissociated from the longitudinal vibrations. Moreover, the longitu-d i n a l vibration has been considered without regard to the transverse v i b r a -t i o n s . The existence of strains i n the central plane was postulated by mathematicians and stress analysts a long time ago. Pearson (6) and (7) showed that a beam under uniform transverse load does have strains i n the central plane; from physical considerations i t i s known that a non-developable surface has strains i n the central plane. In the f i e l d of s t a t i c s , Timoshenko (10) used t h i s fact to derive the deflection for a c i r c u l a r plate. To the knowledge of the author strains i n the central plane have not been previously considered for the dynamic case. Mettler (5) perhaps came closest to the solution for the case of a beam i n plane motion. In h i s derivation, 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 obtain-ing coupling terms of the desired kind. There i s no indication i n his paper that he recognized the significance of these terms. Limitations of the Study The prime reason for undertaking t h i s study was to v e r i f y experimentally the existence of the coupled vibrations i n the plate and i n the bar. The l i m i t a t i o n s of the experimental investigation are as follows. Because of the available system, the experiment was r e s t r i c t e d to 'approximate constant power' frequency spectra. Since the s t r a i n l e v e l was very small, the desired signal often vanished i n the electronic noise and induced electromag-netism. The experimental investigation was r e s t r i c t e d to 10 KHz. The assumed boundary conditions could not be realized i n practice for the cantilever bar. A s o l i d plate could not be simulated. I t was impossible to i d e n t i f y , independ-ently from the spectral 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 li m i t a t i o n s too. F i r s t , the study i s li m i t e d to nearly f l a t surfaces and those structures able to develop bending stresses; second, the derived equations are not exact - the li m i t a t i o n s of the Bernoulli-Euler theory of flexure are present. The method adopted i n interpreting the coupled d i f f e r e n t i a l equations indicates only the frequency of the coupled vibrations. The method adopted does not indicate 5 the absolute amplitudes involved. Definitions Approximated Constant Power Plot - the p l o t t i n g procedure used i n the spectral analysis. B r i e f l y , i t i s the break-up of the frequency spec-trum into short, constant acceleration segments making use of the maximum available power of the system. For a f u l l discus-sion of t h i s method, see section Constant Power Plot Discussion. Node - a l i n e , a point, or surface i n a standing wave where some characteristic of the wave f i e l d has zero amplitude. F i r s t Coupled Resonant Longitudinal (Radial) Vibration - longitudinal (radial) v i b r a t i o n at the same frequency as the excitation frequency i n the transverse dir e c t i o n for the beam (plate). Second Coupled Resonant Longitudinal (Radial) Vibration - longitudinal (radial) v i b r a t i o n at twice the frequency of the transverse vibration of the beam (plate). I t occurs at h a l f the frequency of the excitation as for the f i r s t longitudinal (radial) vibration. Third Coupled Resonant Longitudinal Vibration - longitudinal vibration at thrice the frequency of the excitation i n the transverse d i r e c t i o n . I t occurs at one t h i r d the frequency of the excitation for the f i r s t longitudinal vibration. 6 CHAPTER I I THEORETICAL CONSIDERATIONS Derivation of the D i f f e r e n t i a l Equations of Motion for a Beam Under Plane  Motion As the f i r s t step towards the deduction of the equations of motion for 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 for strains i n the neutral plane for an i n i t i a l l y deflected l i n e segment, a geometric approach i s taken. Love (3) presents a mathematical derivation. The difference between the equation deduced herein and the one i n the above c i t e d work l i e s i n the fact that i n the derivation presented an i n i t i a l deflection curve i s assumed and that the dynamic deflections are measured r e l a t i v e to i t . Love's deduction assumes a fixed coordinate system i n which the i n i t i a l deflection of any point of the neutral plane i s zero. The i n i t i a l deflection curve assumed includes the s t a t i c displacement of the beam from a perfectly f l a t surface and any effects due to the non-homogeneity of the material. Also, since the longitudinal i n e r t i a i s an in 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 refer to the stationary system, that i s , these coordinates account for the i n i t i a l crookedness, s t a t i c d i s -placement, and any exi s t i n g non-homogeneity. A f i b e r of the central plane i s shown. w F i g . 1. Extension and Rotation of Central Plane Fiber Before deformation, a l i n e element of the axis of the beam has the length ~ ? p ds = (1 + W dx x when the beam has been deformed, the length i s 1 ds = \ (1 + U ) 2 + (W + W ) 2 ( 2 dx L x x x J The elongation of the axis of the beam i s given by 1 1 ds - ds = f (1 + U ) 2 + (W + W ) 2 1 2 dx - (1 + W 2 ) 2 dx L X X X J 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 2 + W 2 ) g = ", :x x x 2 x x_ 'k = i + h 2 2 x 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 write £ = U + W W + i (U 2 + W 2) (1) x x x x 2 x x This i s the most general s t r a i n expression for the strains i n the central plane of a beam for s t a t i c , or dynamic deflections i n two orthogonal directions. The above expression i s complete as i t stands. However, an 2 estimate based on the experimental findings indicated 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 further derivations. The coupled equations of motion for the beam were derived by Mettler ( 5 ) . In the derivation of the equations of motion, the basic assump-tions of the Bernoulli-Euler theory of flexure are adopted. Accordingly, i t i s assumed that the bending strains vary l i n e a r l y with the distance from the middle surface. Plane cross-sections are assumed to remain plane, that i s , the shear deformation i s neglected. Rotary ' i n e r t i a effects are ignored. Further, i n t e r n a l and external damping and distributed transverse loads are not considered. Let W (x) be the i n i t i a l deflection of the beam and Wv(x,t) be the dynamic deflection of the beam measured r e l a t i v e to the i n i t i a l deflection curve as shown i n F i g . 2. The f i r s t assumption made i s that there are strains i n the central plane of the beam. These can arise from the non-developability of the central surface, from distributed transverse loads, or from longitudinal i n e r t i a . . How the re s u l t i n g s t r a i n i s made up, that i s , whether strains induced by ' i n e r t i a effects predominate over strains due to the non-< 9 developability of the surface, i s not discussed i n t h i s thesis. the axis i s INITIAL DEFLECTION CURVE DYNAMIC DEFLECTION CURVE F i g . 2. Reference Axis for Displacement Measurements The additional extension of a f i b e r at a distance z from F . = - z W & 1 XX This follows from the usual beam bending theory, or from the assumption that plane cross-sections do remain plane. The t o t a l s t r a i n i s therefore, £ = C x + £ 1 The next major assumption i s , as i s usually done i n the beam bending theory, that stresses occur only i n the x-direction. The e l a s t i c s t r a i n energy becomes U = • I . £ 2 dV and carrying out the p a r t i a l integration over the cross section of a prismatic beam (U + W W + i W 2 ) 2 dx + x x x 2 x Et W 2 dx xx 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 ( T J - T) dt = 0 (2) t. E f f e c t i n g the necessary substitutions, using Calculus of Variations, one obtains -EA(U + i w 2 + WW) + tpA T7 = 0 (3) x 2 x x x x ^ t t EIW - EA I (U + i W 2 + W W ) (W + W ) + <pAW. . = 0 ik) xxxx \ , x 2 x x x x x j x Y tt These equations are the coupled nonlinear d i f f e r e n t i a l equations of motion for a beam under plane motion. These equations are based on the si m p l i f i e d s t r a i n expression. If'the f u l l s t r a i n expression i s used, or i f U x i s of the same order of magnitude as Wx, one obtains the following set of equations - E A J |U + fl W + i ( U 2 + W 2 ) ( l + U ) V + ipAU.. = 0 (5) V . L x x x 2 x x J x J t t EI W - EA xxxx UTJ + WW + ^ ( U 2 + W 2)\(W + W)> +tpA W.. = 0 (6) x x x 2 x X J X x j x ^ t t The above equations, (5) and (6), d i f f e r from (3) and (h) respectively, only 2 by the inclusion 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 for explaining coupling frequencies higher than twice the frequency of the exci t a t i o n , and i t might become important for excitations 11 much higher than those used i n this research. To check the deduction, the equations are linearized EA U = A cp U . \ U = i- U.. (7) and xx ^ t t ' xx c 2 t t EI W + A cp V/ = 0 (8) xxxx t t The f i r s t of these equations corresponds to the classical longitudinal vibration of rods; the second of these equations corresponds to the lateral vibration of a beam. Discussion of the Derived Equations In the paragraphs which follow the interpretation and the sig-nificance of particular coupling terms of the equations of motion for the beam are discussed. Even though the method assumed i s qualitative i n nature, some concrete predictions can be made. We consider now the equation of motion (3) for the beam. Differentiating and rearranging the equation, one obtains - U + itr U. . = W W + W W + W W (9) xx Q2 t t xx x x xx x xx Assuming the flexural displacement to be of the form V/ = F(x) cos cot so that WW = H(x) cos 2 cot (10) x xx and W W = H. (x) cos cot ( l l ) xx x 1 WW = H.(x) cos cot (12) x xx 2 12 where W and W are just functions of x . x xx From a n a l y t i c a l geometry 2 1 cos cot = ^ ( l + cos 2 cot) therefore, WW = H_(x) f 1 + cos 2 cot (13) X X X j L J Next, i t i s assumed that the terras on the right-hand side of (9) are forcing functions. Substituting expressions ( l l ) , (12), and (13) for the terms on the right hand side of (9)« one obtains - TJ^ + ^ U t t = | H^x) + H 2(x) | cos cot + H^x) |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) i n the following form, EIW - E A l U W + U W 1+ AcpW. . = EA I U W + TJ W (15) XXXX [ XX X x xx J t t [ XX X X XX J Having established, a n a l y t i c a l l y and experimentally, the fact that there are at least two longitudinal coupled vibrations, one might sub-s t i t u t e t h e i r expressions, guided by the approach taken for the longitudinal vibrations, for the right-hand side cross products of (15)• As a consequence one should expect also two transverse, or f l e x u r a l coupled vibrations, with the frequency r a t i o of 1:2. These would represent the influence of longitu-dinal vibrations on the transverse vibration. In examining the expression (1*0 and (15) one can draw the following conclusion: Expression (1*0: At any one instant of time the longitudinal vibration of the beam i s the result of two d i f f e r e n t , superimposed, sinusoidal excitations, 13 the frequencies of which are i n the r a t i o of 1:2. Moreover, at least two different resonant frequencies are to be expected - one fundamental at the frequency of f l e x u r a l vibration; the second at twice the frequency of trans-verse vibration. Expression (13): Making use of the fact that the beam i s excited l o n g i t u d i n a l l y by two forcing functions of frequency co and 2co, one can show that the r i g h t -hand side of (15) consists of the following functions G^(x) cos cot + G,, (x) cos 2 cot Two attendant resonant frequencies are to be expected. 14 CHAPTER III APPARATUS AND INSTRUMENTATION Signal Flow The signal flowchart i s shown i n Fig. 3. Essentially, the instrumentation consisted of a vibration control feedback system and ancillary instruments to record diverse signals. Thus, a sound level meter was used for dynamic sound pressure waveform recording and frequency reference. The output was displayed either on the CRO, or on the level recorder. An optical non-contacting displacement transducer, called a Fotonic Sensor (MTT Instru-ments Division, Hatham, N.Y.), was used as an independent frequency reference. The output was displayed on the CRO. Finally, a BAM and strain gauges were used for frequency spectra recording and waveform analysis. The vibration control loop consisted of the generator, amplifiers, electromagnetic exciter, and accelerometer. The generator (henceforth called the control unit) sent out a pre-programmed signal, corres-ponding to the vibration level, to the 50 lbs. force vector exciter; the actual vibration level was sensed by the accelerometer mounted r i g i d l y on the adapter for the specimen. The vibration level as recorded by the accelero-meter was compared with the set point vibration level. Any error between the set point control and the feedback signal was corrected by the vibration control unit. For synchronization of the generator and the frequency c a l i -brated paper on the level recorder an electrical connection was provided between these two units. Finally, a signal proportional to the acceleration A . C C K L B f t O U K T B a a a u a i T i v i T v QEWERATOH., F E E D B A C K . t SYNCHRONIZATION 1-BEA.U 'SCOPE • 8 0 U V J 0 L 1 V K L U L T S a tfOTOUIC A C C K L C R O U E T C n A C T I V C S T I U I M Q A A U -ELECTRO-MAGNETIC EXCITER />/////////////> D U U U Y « T C M P C O U P C , * ^ * • FOO. VfNCHflOI4!lA.TtOU LEVEL RECORDER S I Q K J A t L F L O W C H A R T Fig. 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. All three were useful and practical. They are considered in 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 break-down 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 fifty 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 its reflections in 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 first and second longitudinal vibration of the bar and the resonant first 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 in appropriate positions. These gauges were hooked up in suitable networks with, or without temperature compensating gauges in external half, or fu l l bridges to the bridge amplifying meter. Since, in 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 net-work. 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 lock on which the dummy guages were mounted, Ind iv idua l Instruments F i g . k. Ove ra l l View of the Instrumentation 1 Power Amp l i f i e r 2 Amp l i f i e r 3 CRO k Po lar Height Gauge 5 Electromagnetic Exc i t e r 6 Test Bed 7 Frame 8 Frequency Counter 9 Contro l Uni t 10 Leve l Recorder 11 BAM 12 Roots Blower An o v e r a l l view of the instrumentat ion set up i s shown i n F i g . *f. Not shown i n the f igure i s the Fotonic Sensor, the storage o s c i l l o s -cope, and the vacuum tube vol tmeter . With automatic c o n t r o l , the system can be operated from 10 Hz to 10,200 Hz. The maximum force ava i l ab le i s 50 l b s ; the maximum displacement of the plunger of the electromagnet i s r e s t r i c t e d to l e s s than 0.5 inches peak-to-peak. The system can be run backward or forward, or 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 control unit can be synchronized with the f r e -quency calibrated 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 control unit; an average frequency reading can be obtained from a d i g i t a l counter. Without automatic control, the frequency range can be extended from 10,200 Hz to 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 control, the control unit and the l e v e l recorder are taken out of the c i r c u i t . The output of an external function generator replaces the control unit. Under these conditions the controls revert to manual - t h i s extension requires a l o t of patience and attentive-ness on the part of the operator. The Bruel and Kjaer Automatic Vibration Exciter Control Unit Type 1025 provides a constant displacement control up to 2 KHz and a maximum amplitude of 0 .1 i n . peak-to-peak; constant v e l o c i t y control i s available up to 2 KHz and up to 100 in/sec peak value; constant acceleration control i s feasible up to 10 KHz and 1000 g peak value ( l g = acceleration of gravity). Since the frequency d i a l on the instrument was not properly calibrated, and even i f i t had been, i t i s impossible to read frequencies closer than 50 Hz at the upper end of the frequency scale, an external 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 for f r e -quency measurements. At low scanning speeds of the control u n i t , t h i s unit provides not an instantaneous, but an average frequency. At steady.state, i t displays the true frequency of the excitation (provided the signal i s stable and the - 1 count inaccuracy has been taken into account). The control unit has s i x fixed 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 unit of time, the concept of a fixed scanning speed has no meaning. This arises from the fact 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 control unit has fixed or pre-programmed compressor speeds. The Bruel and Kjaer Level Recorder, Type 2305, was synchron-ized e l e c t r o n i c a l l y with the control unit for most te s t s . As p l o t t i n g medium 50 mm wax paper, frequency calibrated, was used i n conjunction with a sapphire pen and every plot was calibrated separately. This l e v e l recorder does not reproduce the instantaneous value of the function, but plots either the RMS, DC, or peak value. The RMS function was chosen for t h i s experiment -the influence of sudden extraneous signals i s minimized thereby. Different paper and writing speeds are available. The E l l i s Associates BAM 1 Bridge Amplifier and Meter has a DC amplifier 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 effective excitation voltage of the different bridges. The amplifier section of the Bruel and Kjaer f r e -quency analyzer Type 2107 was used as cascading amplifier after the BAM unit. The MB Electronics Model 2250 Power Amplifier i s capable of delivering up to kOO watts of power to the exciter. The Bruel and Kjaer dual channel accelerometer preamplifier Type 2622 has a b u i l t - i n s e n s i t i v i t y attenuator for setting of the accelerometer s e n s i t i v i t y . For correct setting of the accelerometer s e n s i t i v i t y , the channel output i s scaled to 10 mv/g. The waveforms of interest i n the experiment were displayed on the Tektronix 565 Dual Beam (or four beams with chopper) oscilloscope and photographed 21 therefrom. F i n a l l y , the po la r height gauge and Fotonic Sensor probe holder was designed and b u i l t i n the Department. These devices al low displacement measurements i n three coordinates - 6, r, h. Test Bed and Frame The tes t bed on which the v i b r a t i o n e x c i t e r was mounted con -s i s t e d of a b lock of wood, a s t e e l cas ing , and a two foot by two foot b lock of concrete . The whole assembly was embedded i n the ground beneath the f l o o r * of the l abora tory . See F i g . 5* F i g . 5. Test Bed The concrete b lock was s t r u c t u r a l l y i s o l a t e d from the b u i l d i n g . S ince the t e s t s were ca r r i ed out dur ing the n i g h tj the in f luence of Rayle igh Surface waves (caused by t rucks and cars pass ing the nearby road, unbalanced machinery i n s i de the bu i ld ing ) was minimized. The s t e e l cas ing shown i n the f igure cons is ted of two heavy T-sect ions jo ined on the top by 1/2 inch s t e e l p l a t e , braced by two gussets , welded together , and bo l ted to the concrete b lock . A wooden b lock was screwed to the s t e e l p l a t e . Th i s was used as a damper as 22 wel l as a support fo r the frame to which the suspension and a u x i l i a r y f i x -tures were attached. The frame running the f u l l length of the working s u r -face cons i s t s of a 3/16 x 1% inch angle i r o n , welded together and screwed to the wood. A hole 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 center of the v i b r a t i o n exc i t e r fo r attachment of the specimen suspension. Suspension For the p la te as wel l as fo r the bar , an external suspension had to be provided to take the weight of the moving element, specimen, and any necessary appendages o f f the electromagnet. In the experiment, t h i s was a necess i ty i n order to obta in the l e v e l of a cce l e ra t ion used throughout the t e s t s . sp r ing systems i n p a r a l l e l . F i r s t , there was the sp r ing ac t ion provided by an annular rubber diaphragm incorporated i n the v i b r a t i o n e x c i t e r . Second, there was provided an externa l spr ing system cons i s t i ng of c o i l sp r ings . For The suspension fo r the ba r , shown i n F i g . 6, cons is ted of two F i g . 6. Bar Suspension 23 the plate t h i s system comprised four springs i n p a r a l l e l ; for 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 for the choise of springs was to keep the natural frequency of the complete spring-mass system as f a r away as possible from the lowest operating frequency. Without power flow to the ex c i t e r , a natural frequency between four and f i v e cycles per second for both systems was attained with the available springs. The suspension was attached to 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 large nut to keep the rod from s l i d i n g down and to support the weight to be suspended. This nut and a locking nut were threaded on the far end of the rod to an appropriate height. The hemi-spherical nut, resting on the hole i n the angle i r o n , allowed for small misalignments of the moving parts i n two orthogonal directions. Plate Support The plate was supported e s s e n t i a l l y , by a threaded shaft, one end of which had a wide flange which was bolted to the moving element of the electromagnetic exciter. The length of the shaft consisted of four types of thread - three external, one i n t e r n a l . The central external thread was a tapered pipe thread to match the pipe thread i n the central hole of the plate. The i n t e r n a l one was used to fasten the accelerometer to the top of the shaft. The upper machine thread on the shaft was used to fasten the external spring system to the shaft. 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 into the center of the upper bross beam; the lower wide flanged section of the shaft was bolted to the plunger of the exciter; the springs were attached to the upper cross beam. The double Zk can t i l e ve r bar arrangement was used to provide a balanced system on the shaker t a b l e . Models and Boundary Condi t ions  C i r c u l a r P l a te For the experiment a c i r c u l a r p la te of mi ld s t e e l , l/k x 16" d i a . (nominal) was used. Th i s diameter was chosen fo r the p la te so as to b r i ng the breath ing mode frequency i n t o the 8 KHz range. In the center of the p la te a 1/4" pipe thread hole was machined. The surfaces were ground on a surface gr inder and hand sanded to remove rust spots . Due to the g r ind ing opera t ion , the p la te had assumed a curvature - i n s i g n i f i c a n t to be considered a ' curved ' su r face . The boundary condi t ions aimed f o r were: a c i r c u l a r p la te without transverse l o a d , f ree edges, and supported and exci ted i n the center . As was mentioned under the heading ' P l a te Suppor t ' , the p la te was supported by a threaded sha f t . The sec t i on of the shaf t with the pipe thread was screwed hand t i gh t i n t o the hole of the p l a t e . The i n t en t i on was to simulate a continuous p l a t e , without a cen t r a l ho le . The Chladni f igures obtained were to be an i n d i c a t i o n of the symmetry of the v i b r a t i o n . Two other support ing condi t ions were considered. One, a s o l i d p la te clamped, or s imply supported along i t s per iphery ; two, a clamped p la te i n the center . The f i r s t model was dismissed as i m p r a c t i c a l , cons ider ing the ava i l ab l e exc i t a t i on and the d i f f i c u l t y i n producing the exact boundary cond i t i ons . The second one was t r i e d . The p la te was clamped between two nuts run up on the shaf t and against the p l a t e . In sp i t e of the care taken i n machining the surfaces i nvo l ved , pecu l i a r Chladni f igures were obta ined. There fore , t h i s boundary cond i t ion was abandoned. 25 Cantilever Bar The bar was fabricated from 1/4 x 1" (nominal) cold rolled 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 longitudinal resonant vibration occurred in the 8 KHz range. The cross beam width was 17/32". The surfaces were ground so as to remove surface pits. The boundary condition to be simulated was that of two canti-lever bars, obtained by clamping a single bar at the center with the cross beam. Obviously, the arrangement assumed w i l l not yield true fixed end conditions. For bending, the boundary conditions were exact; for longitu-dinal wave propagation, the width of the cross beam has to be taken into account. Strain Gauges and Bridges  Circular Plate Two kinds of gauges were placed on the plate - for the tangent-i a l strains BUDD C 6 - l 4 l B film gauges of 120 ohm resistance were used; for the bending plus radial strain and the radial strain records 500 ohm gauges SR-4 CD-8 Baldwin Lima wound wire gauges were used. Identical gauges were used wherever dummy, or temperature compensating gauges were needed. The gauges were cemented to the surfaces with GA-1 and SR-4 strain gauge cement, res-pectively. The location and hook-up of the gauges was as follows: Tangential - three gauges placed at random along the periphery and hooked up in series to form one arm of a 360 ohm external half bridge. The gauges were located as symmetrically as was possible about the central plane. Bending plus radial and radial strain - 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 plate surface i n corresponding positions. 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 plate were hooked up i n series to form one arm of a half 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 active gauges, one on each side of the-plate, connected so as to eliminate the bending s t r a i n . The location 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 arb i t r a r y since an optimum position for a l l frequencies could not be determined. Cantilever Bar Four BUDD C6-141-B gauges were located i n symmetrical loca-tions r e l a t i v e to the cross beam on both cantilevers. One gauge was located on the top surface and one on the lower surface i n corresponding positions on each cantilever. The gauges were located 3%" away from the end and equi-distant from the sides. GA-1 cement was used for t h e i r adhesion. Because of the higher s t r a i n l e v e l involved, the gauges were hooked up into 120 ohm h a l f bridge for the bending plus r a d i a l s t r a i n record and into 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 tolerable, additional external excitation of the bridge was provided. The c r i t e r i o n for the choice of magnitude of the external excitation was dictated by the available sources and the dissipative capacity of the s t r a i n gauges. As suggested by t h e i r respective manufacturers, 25 ma. was taken as the l i m i t i n g current for the SR-h gauges and 5 watts per square inch of grid area was taken as the l i m i t i n g power for the C-6 gauges. 27 CHAPTER IV TESTING C a l i b r a t i o n and Check Out The accelerometer used i n the experiment was c a l i b r a t e d at the factory. No c a l i b r a t i o n check was made p r i o r to the experiment. The only t e s t done was to compare i t s behaviour against a s i m i l a r u n i t . Their outputs agreed reasonably well over the f u l l frequency range. The influence of a cooling medium on the output of the accelerometer was examined. No appreciable differences were recorded. 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 set on the control u n i t , and the a c c e l e r a t i o n recorded by the accelerometer agreed exactly. The set point a c c e l e r a t i o n was not always constant throughout the d i f f e r e n t f r e -quency ranges. The a c c e l e r a t i o n meter on the co n t r o l unit i n d i c a t e d small deviations from the set point. In general, these deviations were unpredict-able and i n s i g n i f i c a n t . The os c i l l o s c o p e ' s c a l i b r a t i o n was checked p r i o r to the experiment. The frequency response of the BAM was checked. The frequency response complies with the manufacturer's s p e c i f i c a t i o n . The frequency res-ponse of the cascaded a m p l i f i e r ( a m p l i f i e r of the frequency analyzer) was checked. I t s response i s f l a t over the frequency range involved. For the p a r t i c u l a r a p p l i c a t i o n , an external frequency counter i n conjunction with the c o n t r o l unit was a necessity. Frequencies cannot be read from the co n t r o l u n i t frequency scale c l o s e r than approximately 250 Hz at 10 KHz. In ad d i t i o n , 28 the frequency sca le was not proper ly ad justed, or c a l i b r a t e d . The required a i r f l ow through the electromagnetic exc 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 ow rate and the power input to the compressor were inadequately s p e c i f i e d by t h e i r respect ive manufacturers. A doubl ing of the f i r s t quant i ty and t r i p l i n g of the second quant i ty was necessary. The coo l ing system as now designed i s adequate f o r continuous serv i ce at f u l l power d i s s i p a t i o n . The synchronizat ion between the l e v e l recorder and the con t ro l un i t was good. The natura l frequency of the complete suspension of the system was es tab l i shed . To determine t h i s frequency, a storage osc i l l o scope and the Foton ic Sensor p lus the height gauge were used. The probe of the Fotonic Sensor was suspended over the surface of the p l a t e , and the dynamic output of the instrument was d isp layed on the storage o s c i l l o s c o p e . F inger pressure was used to d r i ve the system (specimen, adapters, p lunger , accelerometer) at i t s na tura l frequency. Th is procedure was repeated severa l t imes, and the average of the recorded frequencies was taken. The dev ia t ions from the mean were very sma l l . Noise problems a r i s i n g from ground loops were e l im ina ted . A l l cables and wires used were sh i e lded . Electromagnetic pick-up was m i n i -mized. Test Procedure Since the t es t procedure fo r the p la te i s s i m i l a r to that of the ba r , only one i s descr ibed below. The p la te with s t r a i n gauges and lead wires attached was bo l ted to the plunger of the electromagnet, making use of the appropriate adapter. The connector fo r the suspension and the a c ce l e ro -meter was at tached. By means of a center ing nut , a s t r i n g , and a plumb, 29 the exciter was centered d i r e c t l y under the hole i n the frame. The suspen-sion was attached to the frame, as outlined 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 least excitation. This required a t r i a l and error procedure. Next, the stay r i n g was attached to the frame, and the lead wires to i t . By the procedure indicated e a r l i e r , the natural frequency of the system (specimen, adapters, accelerometer, and plunger of electromagnetic exciter) was determined. Next, the active gauges and the temperature com-pensating gauges were checked for t h e i r resistance; the gauges were combined in',the".proper order to form the desired c i r c u i t s and then connected to the appropriate posts of the BAM. The shields of the lead wires and the ground wire of the specimen and the temperature compensation gauge block were con-nected to the ground of the oscilloscope. The external excitation was added to the BAM. A l l instruments were properly interconnected, and connected to the power source. Then the instruments were turned on and l e f t i n the stand-by position for at least two hours. The next phase of the test procedure was to determine the specimen characteristics - what acceleration to use over which frequency range. This required a t r i a l and error procedure. The figures i n Table 1 were found to be satisfactory for the plate and the bar, respectively. This tabulation i s useful i n l a t e r discussions. The actual testing got under way af t e r : (a) s e t t i n g the set point acceleration, (b) synchronizing the l e v e l recorder with the control 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 writing speed, and the type of response of the l e v e l recorder, (f) cooling the exciter with forced a i r , and (g) c a l i b r a t i n g the BAM and measuring the 30 P la te Cha rac t e r i s t i c s Bar Cha rac t e r i s t i c s Frequency Maximum Frequency Maximum . ./Range,.; Acce le ra t ion Range Acce le ra t ion Hz g Hz g 150 200 2 400 1000 40 200 700 5.5 1000 1200 20 700 800 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 Table 1. Al lowable Acce le ra t ion Leve l exc i t a t i on vol tage of the b r idge . S ince each tes t extended over long per iods of t ime, the l a s t step had to be done p e r i o d i c a l l y dur ing the course of the experiment. The h igh externa l vol tage source ' s l i f e was l im i t ed - up to 50?o reduct ion i n exc i t a t i on was encountered. Constant Power P lo t D iscuss ion The i d e a l approach to the experiment would have been, of course, a constant displacement frequency sweep of the specimen. A p r a c t i c a l example expla ins best t h i s cho ice . A ro t a t i ng shaf t to which a c i r c u l a r d i s c i s r i g i d l y attached i s cons idered. I f the bear ing i n which the shaft i s r o t a t i ng i s misa l igned , so that the d i sc experiences a s i nuso ida l transverse pe r i od i c e x c i t a t i o n , according to the theory proposed h e r e i n , two r a d i a l v i b r a t i ons of the d i s c and the assoc ia ted r a d i a l resonant v ib ra t i ons must be expected. S ince the a x i a l displacement of the shaft can be considered constant , regard less of the frequency of 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 c r i t e r i o n , i s h i g h l y d e s i r a b l e . As was s t a t e d i n the d e s c r i p t i o n of the 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 next best t e s t procedure would have been a constant power p l o t , however, the system i s not designed f o r t h i s k i n d of c o n t r o l . The next a l t e r n a t i v e was a constant 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 of c o n t r o l was a l s o u n s u i t a b l e . Since the frequency sweep extends from 100 Hz t o 10 KHz, i t can be shown that the peak amplitudes i n the center of the p l a t e are d r a s t i c a l l y reduced, by a f a c t o r of 10 , and, consequently, the r e s u l t i n g s t r a i n s are very s m a l l , perhaps too s m a l l t o be d i s t i n g u i s h e d from the inherent e l e c t r o n i c n o i s e . This po i n t i s c l a r i f i e d by the two ch a r t s below, F i g . 7. (a) (b) F i g . 7* Constant A c c e l e r a t i o n and Approximated Constant Power Frequency Spectra (a) Constant a c c e l e r a t i o n (b) Approximated constant power 32 In F i g . 7(a) the constant acceleration plot for the plate i s shown, f o r both r a d i a l and tangential s t r a i n . Only one resonant v i b r a -t i o n (tangential s t r a i n at 4405 Hz) shows up c l e a r l y . In Fig. 7(b) the approximated constant power frequency spectra for the plate i s shown. Two resonant vibrations, one at 4405 Hz and one at 8810 Hz, can be i d e n t i f i e d c l e a r l y . The additional information obtained from the approximated constant power plot dictated i t s choice over the constant acceleration control. 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 into a number of con-venient, small segments along which the acceleration i s kept constant; d i f -ferent acceleration l e v e l s are used f o r different segments such that at one point of each segment, the power input to the plate i s a maximum (i-70 watts approximately). The number of segments and t h e i r location i s an engineering decision - an optimization process. Table 1 l i s t s these frequency segments and the constant acceleration l e v e l . This procedure allows for the fact that the maximum allowable acceleration l e v e l of the specimen plus i t s adapters, i s lowest at the resonant frequencies. This procedure results i n an optimum te s t i n g , considering a l l the factors involved. This procedure allows the constant acceleration control f a c i l i t y of the system to be used. This proce-2 dure results i n a decrease i n amplitude of vibration by only a factor bf 10 if as compared to 10 for a constant acceleration frequency spectra between the frequency extremes. This procedure i s the f i r s t approximation to a constant, maximum power frequency spectra. The frequency response spectra, as w i l l be shown shortly, were plotted i n a pa r t i c u l a r manner to s a t i s f y two conditions. F i r s t , the different frequency segments were separated from each other. This was a necessity since transient voltages were set up whenever certain controls 33 were changed. For instance, i n changing the acceleration l e v e l , the scriber 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 acceleration l e v e l s resulted i n different locations of t h i s trace, different attenuation of the signal at the l e v e l recorder was used for d i f -ferent frequency segments. Second, since the damping i s small i n s t e e l , the resonant vibrations do show up as spikes i n the records. To make absolutely sure that none of these spikes were missed, s u f f i c i e n t frequency overlap was provided. This was achieved by running the whole system 'backwards1, approxi-mately 50 - 100 cycles and then forv/ard again, after having decided on a new attenuation l e v e l . ' Interpretation of Frequency Spectra A l l frequency spectra shown herein were obtained by magnifying the s t r a i n gauge signal i n the BAM 120 x (depends s l i g h t l y on the frequency) and 1000 x thereafter i n the cascaded amplifier. To correlate the trace on the record with the measured s t r a i n , the following must be known: (a) the bridge excitation used at that time ( t h i s variable i s a function of time), (b) the bridge arrangement and the gauge factor, (c) the acceleration 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 tr a n s l a t i o n of the voltage l e v e l into a true s t r a i n record i s a tedious one, and because the aim of t h i s thesis i s to show only the existence of the predicted vibrations, no such translation was carried out. The only s t r a i n calculations carried out were at the predicted longitudinal resonant vibrations of the bar and at s p e c i f i c frequencies close by. The purpose was to show the significance of t h i s resonant'vibra-t i o n . 34 Chladni Figures The Chladni figures reproduced herein were obtained by uniformly d i s t r i b u t i n g ordinary sugar on the plate. The exact transverse resonant vibrations were established i n the following manner. The f i r s t i n d i c a t i o n of a resonant transverse vibration was obtained from the sugar -the i n d i v i d u a l sugar grains moved to a nodal c i r c l e . An even closer approxi-mation could be obtained by observing the output voltage of the exciter control unit. As a narrow frequency range straddling a transverse natural frequency was swept through, the output voltage suddenly swung to a high value, and then f e l l off again. The method used to obtain the figures l i s t e d herein i s as follows. The probe of the Fotonic Sensor was suspended over the specimen, at a fixed 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 oscilloscope. A frequency response survey was made, at discrete frequencies, through the expected range. At the frequency where a natural 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 calibrated, 50 mm, logarithmic, wax paper. A sapphire stylus was used as scriber. This com-bination resulted i n a very high resolution of the signals. Ink on ordinary paper would have been easier to reproduce, however, the resolution 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 oscilloscope, i n the single sweep mode, with a Pentax camera and Tri-X f i l m . 35 CHAPTER V EXPERIMENTAL RESULTS In t h i s s ection the experimental findings are presented. An i n t e r p r e t a t i o n and a discussion of the same i s presented i n Chapter VI. To make the discussion coherent, a l l the findings of the plate are presented f i r s t , followed by those f o r the bar. C i r c u l a r Plate Figure 8 shows the Chladni figures f o r the transverse resonant v i b r a t i o n of the pl a t e , second to f i f t h natural frequency. 4th Mode 5th Mode Fi g . 8. Chladni Figures f o r the Transverse Vi b r a t i o n of the C i r c u l a r Plate 36 For photographic purposes, more sugar than was a c tua l l y needed was o s c i l l a t i n g on the p l a t e . The r e so lu t i on can be improved above that shown i n the f i g u r e . The natura l f requencies of t ransverse v i b r a t i o n of the p l a t e , and the non-dimensionalized nodal r a d i i , with t h e i r respect ive l a rges t de v i a -t i o n from the mean r ad i u s , are shown i n Table 2. The natura l f requencies serve as a tes t f o r the boundary cond i t ions ; the l a rges t dev ia t ion from the mean rad ius i s a measure of the symmetry of the v i b r a t i n g system. _ Non-Dimensionalized Largest Dev ia t ion Mode sequency M e a n N o d a l R a d i i f r o m M e a n Radius Hz xn 2 ?67 3 2204 0.87 0.51 0.02 4 4335 0.91 0.65 0.36 0.03 5 7130 0.93 0.73 0.50 0.28 0.04 Table 2. Non-Dimensionalized Nodal Rad i i f o r P la te Note: The average dev ia t ion i s much smal ler than the values quoted above. The non-dimen-s i o n a l i z e d nodal r a d i i are based on a f u l l radius (p late without a cen t r a l h o l e ) . The mean nodal r a d i i were determined by averaging three r a n -dom rad ius measurements. The frequencies quoted above were measured with the d i g i t a l counter dur ing steady-state v i b r a t i o n . The e r rors and i naccu ra -c i e s invo lved i n the frequencies quoted a re : (a) experimental e r ror a r i s i n g from inaccurate l o ca t i on of the resonant peak - l e s s than 1 Hz , (b) the round o f f e r ro r from the four th and f i f t h d i g i t , and 37 (c) inherent inaccurac ies of the counting dev ice . In Appendix C, F i g . C-l4 to F i g . C-l6 are shown three ' a p p r o x i -mated constant power* frequency spec t ra fo r the p l a t e . Two spikes i n the r a d i a l s t r a i n record and the tangent ia l s t r a i n r eco rd , F i g . C-15 and F i g . C-l4, are shown. The f i r s t occurs at kh05 Hz and the second at 8810 Hz. There i s one l i m i t a t i o n inherent i n the frequency spec t ra . Incorporated i n these records i s a con t r ibu t ion from the electromagnetic i n -duct ion i n the g a u g e . i n s t a l l a t i o n . In the experiment, t h i s extraneous s i g n a l was minimized. However, i t could not be e l iminated . The s i gna l impressed by t h i s electromagnetic induc t ion has always the frequency of the e x c i t a t i o n . For the resonant peaks, t h e i r in f luence i s smal l (approximately 5% of the amplitude f o r the fundamental l ong i t ud ina l f requency) ; at other exc i t a t i on frequencies t h i s s i gna l i s very s i g n i f i c a n t . Care must be exerc i sed , the re -f o r e , i n i n t e r p r e t i n g the spec t r a l records . F i g . 10 dep ic ts the waveforms at the two r a d i a l resonant v i b r a t i o n of the p l a t e . The i r i n t e rp r e t a t i on i s given i n Appendix A. Cant i l ever Bar F igure 9 shows the Chladni f igures f o r the transverse v i b r a -t i o n of the bar . Next, the na tura l f requencies fo r the transverse v i b r a t i o n and the non-dimensionalized nodal d is tances from the b u i l t - i n end are t abu -l a t e d . The frequencies ind i ca ted i n Table 3 have the same l i m i t a -t i ons as the measured frequencies fo r the p l a t e . The resonant frequencies were measured to check the assumed boundary cond i t i ons . 38 2nd Mode 3 r d Mode F i g . 9» Chladni F igures fo r the Transverse V ib ra t i on of the Cant i l eve r Bar Mode Frequency Hz Non-Dimensionalized Nodal Distances from the B u i l t - i n End 1 2 2 3 4 1367 3805 7364 O.78 0 .50 0.86 Table 3» Non-Dimensionalized Nodal Distances fo r the Bar Note: The nodal d istances fo r the fourth mode could not be measured. The nodal l i n e s are poor ly def ined at 7364. Fur ther , a l l non-dimensionalized nodal d istances were ca l cu la ted us ing the exposed length of the bar as reference l eng th . The 'approximated constant power' frequency spectra f o r the bar are shown i n Appendix C, F i g . C-17 and F i g . C - l 8 . The two resonant l ong i tud ina l v ib ra t ions 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 record , F i g . C - l 8 . Expected l oca t i ons of h igher v i b ra t i ons than twice the frequency 39 Breathing Mode (a) Frequency 88lO Hz Signa l Order (top to bottom) Radial S t r a i n Tangential S t r a i n Acceleration Displacement (b) Frequency kkok Hz Signal Order (top to bottom) Bending and Radial S t r a i n Sound Pressure Acceleration (c) Frequency Vf03 Hz Signal Order (top to bottom) Radial S t r a i n Sound Pressure Acceleration (d) Frequency kk05 Hz Signal Order (top to bottom) Tangential S t r a i n Sound Pressure Acceleration F i g . 10. Waveforms at Resonance f o r C i r c u l a r Plate ko of the e x c i t a t i o n are indicated as well as the expected transverse coupled v i b r a t i o n s . In F i g . 11 are shown t y p i c a l waveforms at the resonant trans-verse v i b r a t i o n . 2nd Longitudinal Vibration Frequency 4lJ5 Hz Order of Signals (top to bottom) Longitudinal S t r a i n Sound Pressure Acceleration 3rd Longitudinal V i b r a t i o n Frequency 2756 Hz Order of Signals (top to bottom) Longitudinal S t r a i n Acceleration F i g . 11. Waveforms at Resonance f o r Cantilever Bar kl CHAPTER VI DISCUSSION OF EXPERIMENTAL RESULTS C i r c u l a r P l a te Table k l i s t s the t heo re t i c a l resonant frequencies fo r the transverse v i b r a t i o n . Mode Frequency Hz 2 7 0 0 3 2 0 3 0 k kooo Table k. Theore t i ca l Resonant Frequencies f o r the Transverse V ib ra t i on of the C i r -cu la r P la te Comparing t h i s data with the data i n Table 2 , great d i f f e r e n -ces i n the frequencies are observed. The d i f f e rence i n frequencies suggests that the boundary condi t ions aimed fo r could not be r e a l i z e d . Fur ther , the t h e o r e t i c a l f requencies i n Table k are based on a p la te with a f ree edge and supported i n the center . The experimental f requencies l i s t e d i n Table 2 , however, are based on a p la te with a f ree edge and 'c lamped 1 i n the center over a smal l area. For transverse v i b r a t i o n , the smal l ' c lamping ' area r e s u l t s i n a s t i f f e r p la te than a p la te clamped s t r i c t l y i n the center . The higher frequencies i n Table 2 as compared to those i n Table k seem, there fore , to be j u s t i f i e d . 42 The calculation for the breathing mode (assuming a s o l i d plate) indicates a r a d i a l resonance at 8900 Hz. The measured resonant r a d i a l vibration frequency was 8810 Hz. For the r a d i a l v i b r a t i o n , a plate with a discontinuity i n the center, i n the form of a c i r c u l a r hole, i s a less s t i f f structure than a plate without a hole. Therefore, i t i s to be expected that the natural frequencies for r a d i a l v i b r a t i o n w i l l be lower. The largest deviation 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 small. Consequently, the vibration was nearly symmetrical with respect to the center, or no gross s i n g u l a r i t i e s existed i n the plate with exception of the center. The s t a t i c maximum deflection of the plate was determined to be 7»l8 x (10) inches. In addition to t h i s d e f l e c t i o n , there existed the deflection due to the i n i t i a l crookedness of the plate. The dynamic deflec-tions were, i n general, much smaller than the s t a t i c deflection. The tangential 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 also these two spikes i n addition to the resonant bending peaks. I t i s important to note the difference i n shape of the two peaks - the bending resonant strains do not buil d up as fast as the r a d i a l resonant s t r a i n s . Also, i t i s worthy to examine the lower r a d i a l resonant peak i n these figures. F i r s t , one sees from Table 2 that the fourth mode of f l e x u r a l vibration occurs at 4335 Hz; second, the r a d i a l vibration occurs at 4405 Hz. The spike i s just to the right 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 position of maximum bending s t r a i n , i t must be concluded 43 that the r a d i a l resonant s t r a i n i s smal ler than the bending s t r a i n at resonance. The r a d i a l s t r a i n , F i g . C-15, record corroborates these statements. Two peaks, at 4405 Hz and 8810 Hz r e spec t i v e l y , corresponding to the r a d i a l resonant v i b r a t i o n , are i nd i c a t ed . In a d d i t i o n , vest iges of a l l the bending resonant s t r a i ns are v i s i b l e . The explanat ion fo r t h i s might be, one, an incomplete compensation of the gauges and, two, the in f luence of the bending s t r a i n on the l ong i t ud i na l v i b r a t i o n . Next, the waveforms involved are examined. The r a d i a l and the tangent i a l s t r a i n waveforms, at 8810 Hz, are shown i n F i g . 10 . The waveforms are pe r iod i c and of the same frequency as the e x c i t a t i o n . There fore , at 8810 Hz a resonant r a d i a l v i b r a t i o n , the 'b rea th ing mode', e x i s t s . F igure 10 (b ) , ( c ) , and (d) represent , r e spec t i v e l y , the bending plus r a d i a l s t r a i n , the r a d i a l s t r a i n , and the tangent ia l s t r a i n . Appendix A has to be consulted f o r t h e i r i n t e r p r e t a t i o n . I t i s seen that the s i gna l i s e s s e n t i a l l y the super -p o s i t i o n of two harmonics - one of the frequency of the e x c i t a t i o n , and one twice that frequency. The waveform recorded by the sound l e v e l meter i s the same as that of the s t r a i n s i g n a l . Can t i l eve r Bar The t h e o r e t i c a l l y determined nodal d istances fo r a . .cant i lever beam, exc i ted at the b u i l t - i n end, are l i s t e d i n Table 5« Non-Dimensionalized Distances Mode from the B u i l t - i n End 1 2 3 2 O.783 3 0.504 0.868 4 0.358 0.644 0 .906 Table 5« Non-Dimensionalized Theore t i ca l Nodal Distances f o r a Cant i l ever Beam kk In Table 6 are l i s t e d the theoretical resonant frequencies for the transverse vibration of the beam as well as estimates of the maximum error (upper bound) for the calculated frequencies. The upper bound error estimates take into account the inaccuracies i n the measurements of the physical proper-t i e s and parameters of the bar. M , Theoretical Frequency Maximum Error o a e Hz Hz 2 ikOO ikS 3 3860 ko6 k 7550 795 Table 6. Theoretical Resonant Frequencies for Flexural Vibration of the Beam The predicted resonant frequencies for f l e x u r a l vibration l i s t e d i n Table 6 were derived from the cantilever beam frequency equation. Not l i s t e d are the predicted resonant frequencies for f l e x u r a l vibration for a cantilever plate. These frequencies are somewhat higher than those l i s t e d i n Table 6. 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 into account the upper bound on the error, 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 plate. The agreement between the measured non-dimensionalized nodal distances from the b u i l t - i n end, Table 3, and the theoretical nodal distances from the built-end tabulated i n Table 5 i s good. This corroborates the con-clusion arrived 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 longitudinal 45 v i b r a t i o n should occur at 8520 H z . i f X i t i s assumed that the e f f e c t i v e length of the bar i s i t s exposed l eng th . I f the h a l f length of the f u l l double bar i s cons idered, the f i r s t coupled l ong i t ud i na l resonant v i b r a t i on ought to occur at 8150 Hz. Comparing t h i s data with the experimental f i n d i n g s , a d i s -crepancy i s observed i n the f i r s t coupled l ong i t ud ina l resonant v i b r a t i o n frequency. Th i s suggests that the boundary condi t ions aimed fo r could not be r e a l i z e d . In cons ider ing the l ong i t ud i na l v i b r a t i o n , the e f f e c t i v e length of the bar i s not i t s exposed l eng th , but i t inc ludes an add i t i ona l length under the clamp. Examining next the s t r a i n records , the fo l low ing observat ions are made. A x i a l S t r a i n - as can be seen from F i g . C-l8, there i s a pronounced spike at 8270 Hz , the f i r s t coupled l ong i t ud ina 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 ong i t ud i na l resonant v i b r a t i o n . Fu r the r , there i s a very , very smal l spike at 2755 Hz l a b e l l e d A^. No others can be i d e n t i f i e d p o s i t i v e l y . A l s o , as f o r the p l a t e , F i g . C-l4, one sees vest iges of 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 . These must be i n t e r -preted i n the same l i g h t as those fo r the p l a t e . Bending plus A x i a l S t r a i n -i n F i g . C-17, there i s a pronounced spike at 8270 Hz and one at 4135 Hz e x c i t a t i o n . None can be seen at 2755 Hz. As be fore , 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 to the resonant f l e x u r a l s t r a i n . In the same spec t r a , one sees the f l e x u r a l resonant peaks. Aga in , one notes that the f l e x u r a l resonant v ib ra t i ons b u i l d up much more s lowly than the l ong i t ud i na l resonant v i b r a t i o n s . One other item i n these spect ra i s worthy of note - spikes l a b e l l e d A^ and A^. The f i r s t one occurs at one-half the frequency of the exc i t a t i on of the second bending resonant mode; the second one occurs at one-half the frequency of the exc i t a t i on of the fourth resonant k6 bending mode. Both peaks are extremely smal l compared to the s t r a i ns just p r i o r and just a f t e r i t s occurrence and both peaks show the cha r a c t e r i s t i c slow bui ld-up of bending resonant s t r a i n s . The s igna l s invo lved are too smal l f o r an accurate waveform ana l y s i s . However, by analogy with the more pronounced l ong i t ud i na l coupled v i b r a t i o n f i n d i n g s , i t can be assumed that these represent f l e x u r a l v ib ra t i ons of the bar at twice the frequency of the e x c i t a t i o n . In other words, these small peaks represent the in f luence of the l ong i t ud i na 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 on as ind ica ted i n the theory by the term (U W ) . X X X Examining next the waveforms at the three s i g n i f i c a n t spikes f o r l ong i t ud ina l v i b r a t i o n , F i g . 11, making use of Appendix A, one has to conclude the fo l l ow ing : at 8270 Hz e x c i t a t i o n , a l ong i t ud ina l resonant v i b r a t i o n of frequency 8270 Hz e x i s t s ; at 4135 Hz exc i t a t i on a l ong i t ud i na l resonant v i b r a t i o n of frequency 8270 Hz e x i s t s ; at 2755 Hz exc i t a t i on a l o n g i -t ud ina l resonant v i b r a t i o n of frequency 8270 Hz e x i s t s . ( 47 CHAPTER VII SUMMARY AND CONCLUSION Suggestions fo r Future Research Besides the obvious research to be done, such as the a n a l y t i -c a l s o l u t i o n of the exact formula t ion , a number of experimental i n ves t i ga t i ons based on the p red i c t i ons of the s imple , proposed formulat ion can and should be done. The r e l a t i o n s h i p between response and exc i t a t i on should be examined. Th i s i n ves t i ga t i on can be done with the present set up. Next, t h i s research should be extended to the upper end of the acoust ic frequency range. The damping of the transverse and l ong i t ud ina l v i b r a t i o n should be i nves t iga ted . Quant i ta t i ve answers should be found fo r the in f luence exerted on the amplitudes 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 damping l aye rs on the surfaces of the bar . Opt imiz ing parameters f o r the 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 damping l aye rs should be found. The i s o l a t i o n proper t ies of these mater ia ls at the frequency range encountered should be i nves t iga ted . The i r transmission-of-energy proper t ies are of i n t e r e s t . Conventional v i s c o -e l a s t i c mater ia ls should be inves t iga ted ' i n s i t u 1 and other promising mater ia ls should be eva luated; The surface treatment of the ba r , or p l a t e , f o r a p a r t i -cu l a r damping mater ia l must be determined. The fa t igue and temperature depend-ence of v i s c o e l a s t i c l aye rs must be i nves t i ga ted . The instrumentat ion can be r e f i n e d . I f pos s i b l e , a cont ro l un i t f o r constant displacement con t ro l up to the upper end of the acoust ic frequency range should be acquired - the con t ro l technique adopted fo r t h i s k8 research i s not the best one. For fu r ther i n v e s t i g a t i o n s , a Fotonic Sensor should be a v a i l a b l e . Th i s instrument i s the most promising one f o r waveform inves t i ga t i ons - noise problems assoc iated with the other two sets of measuring instruments are e l iminated hereby. I t i s suggested that i n the fu tu re , s t r a i n gauges designed e x p l i c i t l y f o r dynamic app l i c a t i on be used. Time and c a p i t a l out lay can be saved thereby. The BAM 1 should be a l t e red to a BAM 2 through the a c q u i s i t i o n of a su i t ab l e E l l i s Associates a m p l i f i e r . The frequency response of the present un i t i s down approximately 3% at 10 KHz. An amp l i f i e r of su i t ab le gain and frequency response should be designed to rep lace the amp l i f i e r of the frequency ana lyzer . Th is un i t should be ava i l ab l e f o r i t s intended purpose. An anechoic chamber should be i n s t a l l e d and i t s acoust ic c h a r a c t e r i s t i c s be s tud ied . The General Radio Sound l e v e l meter should be replaced by an instrument having a wider l i n e a r range. A band pass f i l t e r should be incorporated i n to the system. The band should be synchronized to the sweep con t ro l of the generator. The des i red boundary condi t ions should be approximated more c l o s e l y . The arrangement used fo r the can t i l e ve r bar seems to be more promising than the support fo r the p l a t e . A technique to evaluate damping mater ia ls should be determined. Summary The under ly ing reason f o r doing t h i s experimental workjwas. to v e r i f y the specu la t ion that the transverse exc i t a t i on of a beam leads to l o n g i -t ud ina l v ib ra t i ons of the beam. To th i s e f f e c t , a s i m p l i f i e d model was set up. Assuming s t r a i n s i n the cen t ra l p lane, coupled nonl inear p a r t i a l d i f -f e r e n t i a l equations of motion were der ived fo r a beam. By su i t ab l y manipula-t i n g the equations and i n t e r p r e t i n g some of the coupl ing terms, two coupled l ong i t ud i na l v i b r a t i o n s , with a frequency r a t i o of 1:2, were p red i c ted . In a d d i t i o n , two coupled f l e x u r a l v i b r a t i o n s , with a frequency r a t i o of 1:2, 49 were an t i c i pa ted . The experimental i n ves t i ga t i on proved the v a l i d i t y of the above p r e d i c t i o n s . Two resonant l ong i t ud ina l v ib ra t i ons were determined f o r a bar and the frequency r a t i o was 1:2. Fu r ther , through the i n v e s t i g a t i o n , the l o c a t i o n of the resonant peaks i n regards to the frequency of the e x c i t a -t i o n was es tab l i shed . The f i r s t l ong i t ud ina l resonant v i b r a t i o n occurred at the fundamental frequency f o r l ong i t ud i na l resonant v i b r a t i o n . The second l ong i t ud ina l resonant v i b r a t i o n occurred at 1/2 times the frequency of the transverse exc i t a t i on at the fundamental frequency f o r l ong i t ud ina l resonant v i b r a t i o n and the frequency of the v i b r a t i o n was that of the fundamental l o n g i -t ud ina l resonant v i b r a t i o n . A l s o , one l ong i t ud ina l v i b r a t i o n at three times the frequency of the exc i t a t i on i n the transverse d i r e c t i o n was recorded. The frequency of the v i b r a t i o n seemed to be that of the fundamental l ong i t ud i na l resonant v i b r a t i o n . In the s t r a i n records there i s evidence that there are more f l e x u r a l resonant v ib ra t i ons than ind i ca ted by the l i n e a r theory. These occur at one-half the frequency f o r f l e x u r a l resonant s t r a i n s . A waveform ana lys i s was not c a r r i ed out s ince the s i gna l invo lved i s very sma l l . Some general conclus ions can be drawn i n regards to the r e l a -t i v e value of 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 ong i t ud ina l v i b r a t i o n . At bes t , the l ong i tud ina l s t r a i n at resonance equals the bending plus l ong i t ud ina l s t r a i n f o r transverse resonance; i n genera l , i t i s sma l le r . Fur ther , i t seems that the magnitude of the s t r a ins at resonance decreases as the order of the l ong i t ud ina l frequency inc reases . An experimental i n v e s t i g a t i o n , analogous to that of the bar , was ca r r i ed out f o r a c i r c u l a r p la te suspended i n the center and exc i ted 50 t ransverse l y . Two coupled resonant v ib ra t ions i n the r a d i a l d i r e c t i o n were recorded. The i r frequency r a t i o was 1:2; they occurred respec t i ve l y at 1 and 1/2 times the frequency of the transverse exc i t a t i on f o r the fundamental f r e -quency fo r r a d i a l resonant v i b r a t i o n . No nonl inear coupled f l e x u r a l v i b r a t i on at one-half the frequency of resonant f l e x u r a l ' l i n ea r * v i b r a t i o n were i d e n t i -f i e d . Conclus ion From the i n v e s t i g a t i o n , the fo l low ing conclus ions were drawn: (1) The an t i c ipa ted l ong i t ud ina l v ib ra t i ons e x i s t , when a bar i s exc i ted t ransve rse l y . (2) There i s some evidence that the an t i c ipa ted f l e x u r a l v ib ra t i ons of the beam at twice the frequency of the exc i t a t i on e x i s t . The i r i n f l uence , however, on the transverse v i b r a t i o n , even at t h e i r resonant cond i t i on , was extremely sma l l . (3) The der ived d i f f e r e n t i a l equations of motion fo r the beam are cor rec t i n so f a r as the p red i c t i ons of f requencies of coupled v i b r a t i o n are 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 coupl ing terms i s co r r e c t . (5) Exper imenta l ly , i t was determined that h igher l ong i t ud ina l v i b ra t i ons than the second do occur . A t h i r d resonant l ong i t ud ina l v i b r a t i on was recorded. (6) Resonant v i b ra t i ons occur at the inverse r a t i o of t h e i r f requenc ies . That i s , f o r the beam, the l ong i t ud ina l v i b r a t i o n frequency r a t i o i s 1 :2, the resonant l ong i t ud ina l v ib ra t i ons occur r e spec t i v e l y at 1 and 1/2 times the transverse exc i t a t i on at the fundamental l ong i t ud ina l resonant v i b r a t i o n . (7) I t seems tha t , at bes t , the amplitude fo r resonant l ong i tud ina l v i b r a t i o n 51 s t r a i n equals the amplitude fo r resonant a x i a l p lus bending s t r a i n . In genera l , i t i s smal le r . Nevertheless , the amplitude fo r resonant l o n g i -t ud ina l v i b r a t i o n s t r a i n i s l a rge r than the amplitude fo r l ong i t ud ina l p lus bending s t r a i n at near-by non-resonant cond i t i on . 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 f l e x u r a l resonant s t r a i n s . For the p l a t e , two r a d i a l coupled v ib ra t i ons were recorded. The i r f r e -quency r a t i o was 1:2; they occurred at 1 and 1/2 times the transverse frequency f o r the breath ing mode. 52 BIBLIOGRAPHY H a r r i s , C.M. and Crede, C . E . , "Shock and V ib ra t i on Handbook", v o l . 1, McGraw-Hill Book C o . , N.Y. 1961. Jacobsen, L.S. and Ayre, R.S., "Engineer ing V i b r a t i o n s " , McGraw-Hill Book C o . , N.Y. 19587^ Love, A . E . H . , "A T rea t i se on the Mathematical Theory of E l a s - t i c i t y " , 3 r d e d . , Cambridge, Un i ve r s i t y P ress , 1920. McLeod, A . J . and B ishop, R .E .D. , "The Forced V ib r a t i on of C i r c u l a r F l a t P l a t e s " , Mechanical Engineer ing Sc ience , monograph no. 1, The I n s t i t u t i o n of Mechanical Engineers, London, March, 19^5. Met t l e r , E. , Dynamic Buck l ing , "Handbook of Engineer ing  Mechanics" , 1s t e d . , F luegge, W., e d i t o r , McGraw-Hill Book Company, Inc . 1962. Pearson, K., "Memoir on the F lexure of Heavy Beams Subjected to Continuous Systems of Load" , Quar ter ly Journal of Pure  and Appl ied Mathematics, v o l . 24, Longmans, Green, and C o . , London, 1890. Pearson, K., and F i l o n , L .H .G . , "On the F lexure of Heavy Beams Subjected to Continuous Systems of Load" , Quar ter ly Journal  of Pure and Appl ied Mathematics, v o l . 3 1 , Longmans, Green and C o . , London, 1900. S t r u t t , J .W. , Lord Ray le igh , "Theory of Sound", v o l . 1 and 2 , Dover Pub l i c a t i ons , N.Y., 19k~5~. Thomson, W.T., " V i b r a t i on Theory and A p p l i c a t i o n s " , P rent i ce-H a l l , I n c . , N. J . , 1 9 6 5 . Timoshenko, S. and Woinowsky-Krieger, S . , "Theory of P la tes  and S h e l l s " . McGraw-Hill Book Company, Inc . , 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 expressions are der ived and graph ica l i n t e rp re t a t i ons shown fo r the add i t i on of two s inuso ids of d i f f e r en t ampl i tude, d i f f e r en t f requenc ies , and i n phase such that both funct ions cross the time ax is at the same ins tant of t ime, and that an ins tant l a t e r both funct ions are e i the r p o s i t i v e , or negat ive . Part 1 D i f f e r en t amplitude One frequency being twice the other In phase We consider the func t ion V = A s i n 9 + B s i n 26 = (A * 2B cos G) s i ne (l6) = (Modulated Ampl i tude) (C i rcu lar Function) Equation (l6) represents a modulated amplitude r o t a t i ng at the frequency of the fundamental f unc t i on , that i s , regard less how large the i n d i v i d u a l amp l i -tudes a re , the r e s u l t i n g waveform has one per iod equal to that of the funda-mental f unc t i on . The maxima and the minima of the r e s u l t i n g funct ion are found from - - k <"> 5^ ^ £ 2 = _ s i n 29 - A s i n 6 (l8) As good as the above approach i s , i t does not a f fo rd an easy-eva luat ion of the func t ion (17) and (18) above. For t h i s reason, the g r a p h i -c a l eva luat ion of the o r i g i n a l funct ion i s shown i n F i g . 12. Everything which has been der ived above, a n a l y t i c a l l y , can be v e r i f i e d i n these f i g u r e s . AMPLITUDE RATIO |:| F i g . 12. Graphica l Add i t ion bf S inusoids - Frequency Rat io 2 The frequency r a t i o i s def ined as fo l lows Frequency of 'Harmonic' Frequency of Fundamental and the d e f i n i t i o n of the amplitude r a t i o i s Amplitude of 'Harmonic' Amplitude of Fundamental The important items which deserve a t t en t ion i n F i g . 12 a r e , f i r s t , the waveform invo l ved , and, second, the r e l a t i v e magnitude of the two peaks invo lved . As the amplitude of the harmonic, twice the frequency of the 55 fundamental, i s i nc reased , l eav ing the other unchanged, the r e l a t i v e amp l i -tude of the second peak inc reases . As long as the s i ng l e frequency i s f i n i t e , a r e l a t i v e increase of the amplitude of the second harmonic funct ion w i l l cause the amplitude of the second peak to approach the amplitude of the f i r s t peak. However, i t w i l l never equal i t . Th i s i s an important deduct ion. I t al lows one to estimate the r a t i o of magnitudes i nvo l ved , without ca r r y ing out a Four i e r ana l y s i s . As a f i n a l a n a l y t i c a l de r i va t i on i n t h i s s e c t i o n , the RMS value of the r e s u l t i n g funct ion w i l l be determined. Th i s i s of value i n the i n t e r p r e t a t i o n of s t r a i n frequency-spectra s i n ce , f o r the sake of minimizat ion of e r r o r s , RMS values were p lo t t ed thereon. V R M S - j l j v 2 ( e ) d e To make the so l u t i on as general as p o s s i b l e , the fo l low ing funct ion i s con -s idered V = A s i n G + B s i n nG where n i s an i n t ege r , n > 1 Subs t i t u t i ng VRMS = sJl (A" + B 2 ) Consequently, the RMS value of a funct ion cons i s t i ng of the superpos i t ion of two s ine func t ions , the frequency of one being an i n t e g r a l mul t ip le of the o ther , i n phase, i s always given by the above express ion. 56 Part 2 Di f f e ren t amplitude One frequency being t h r i c e the other In phase I t can be shown that by and V = A s i n Q + B s i n 3 £ = £ A + B (3 c o s 2 Q - s i n 2 Q ) ] s i n Q = £ Modulated amplitudeJ ( c i r c u l a r funct ion) (18) As be fore , the cond i t ion f o r a maxima, or a minima i s given cos Q | A + B (3 c o s 2 Q - 9 s i n 2 0. ) I = 0 J d f v dQ 2 2 2 - A s i n Q + B s i n Q ( 9 s i n Q - 29 cos Q ) Everyth ing sa id fo r Part 1 has a counterpart i n Part 2. For s i m i l a r reasons, the graphic so l u t i on of the problem, Part 2, i s shown below i n F i g , 13» The d e f i n i t i o n of amplitude r a t i o and frequency r a t i o i s given i n Part 1. AMPLITUDE RATIO 1 = 3 F i g , 13, Graphica l Add i t ion of S inusoids - Frequency Rat io 3 57 APPENDIX B LINEAR EQUATIONS FOR PLATE AND BEAM C i r c u l a r P la te The ' l i n e a r ' equation of motion f o r transverse v i b r a t i o n symmetrical about the center according to (4) i s wr i t ten 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 (Kr) + B Y (Kr) + C I (Kr) + D K (Kr) cos cot ' I o o o o J w h e r e *4 gut and A, B, C, and D are determined by the boundary cond i t i ons . For a c i r c u l a r p la te clamped i n the center , with force exc i -t a t i o n i n the center and outer edge f r e e , the na tura l transverse v i b r a t i on i s given by ( l ) as a) = B I , ^ . ^ / S E C n \l ^  (1 -^ 2) and B1 = 4.35 B 2 = 24.26 B^  = 70.39 B^  = 138.85 The 'b rea th ing mode', or the f i r s t r a d i a l v i b r a t i on can be deduced from the equations l i s t e d i n (3) 5 8 dJ (Kr) - h s y = J o ( K r ) - J i ( K r ) By t r i a l and error Ka = 4 . 1 but K = cp ( 1 - co therefore E f - 1 3 . 9 4 ( 1 0 ) * * H s a Cantilever Bar The l i n e a r equations of motion for the transverse vibration of a beam, vibrating under i t s own weight, assuming a harmonic excitation, i s given by ( 9 ) as ^ - nS = 0 dx 2 where n 4 _ cpco EI \4iose general solution i s y = A cosh nx + B sinh 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 ) i s (r I) n / ET R A D , - ™ co = — r - r - ./ T— /SEC n g 2 U Acp ' and (^E) 2 = 3 . 5 2 (r 2 T ) 2 = 2 2 . 0 (r^L)2 = 6l0?5 ( r k e ) 2 = 1 2 0 . 8 59 The f i r s t l ong i t ud i na l v i b r a t i o n according to (2) i s c a l -cu lated by us ing (2B - 1) „. / E W n = 2l  n X* 60 APPENDIX C APPROXIMATE CONSTANT POWER FREQUENCY SPECTRA F i g . C-14. Tangent ia l S t r a i n Frequency Spectra fo r C i r c u l a r P la te F i g . 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 Fig. C-l6. R a d i a l plus Bending Strain Frequency-Spectra f o r C i r c u l a r P l a t e 63 F i g . C-17. Bending plus A x i a l Frequency Spectra f o r C a n t i l e v e r Bar F i g . C - l 8 . A x i a l S t r a i n Frequency Spectra f o r Cantilever Bar 

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