EOTATORY INERTEA EFFECTS OF ATTACHED MASSES ON THE ViBRATION FREQUENCIES OF BEAMS AND PLATES TENS“ gear the Degree of pk. D. MICHIGAN STATE UNIVERSITY Salil Kumar Das 1962 VVr“- ‘ TUI}-!C.. This is to certify that the thesis entitled Rotatory Inertia Effects of Attached Masses on the Vibratioanequencies or Beams and Plates presented by Salii Kumar Das has' been accepted towards fulfillment of the requirements for Ph.D. degree inA g glied Mechanics Xbflf/V . ( Dr G e or g eMQPngxagessor mew 0-169 LIBRARY Michigan State University ABSTRACT ROTATORY INERTIA EFFECTS OF ATTACHED MASSES ON THE VIBRATION FREQUENCIES OF BEAMS AND PLATES by Salil Kumar Das This investigation concerns itself with the effect of rotatory inertia . of attached masses on the vibration frequencies of beams and plates. The equations which are derived, are quite general and can be used for any number of attached masses. The usual assumptions of Hooke’s Law, isotropy of material and small deflection theory are assumed in deriving the general equations. In the case of plates, rotatory inertia and shear deformation of the plate are neglected, whereas, for the beam, only part of the shear is neglected. In the latter case, the resulting equation is compared with Timoshenko's reduced equation and is shown to give the same result. Solutions were obtained with the help of digital computer and an accuracy of about five places was realized. The tables of values, given in this work, contain only the frequencies, the mode shapes being omitted because of prohibitive amount of space required to tabulate them. As may be expected, the solutions obtained are approximate, rather than exact. In order to verify these results, a series of experi- ments were performed and the calculated frequencies compares with the measured ones. The agreement is very encouraging and seems to be quite adequate for most practical purposes. In Chapter V, a method is developed, that can be used for many problems which involve concentrated masses. A few examples are worked out and results are compared with values from other chapters. The agreement seems to be quite good. ROTATORY INERTIA EFFECTS OF ATTACHED MASSES ON THE VIBRATION FREQUENCIES OF BEAMS AND PLAT ES BY Salil Kumar Das A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCT OR OF PHILOSOPHY Department of Applied Mechanics 1962 ACKNOWLEDGMENTS The author wishes to express his sincere gratitude to Dr. George E. Mase for his guidance and inspiration throughout this project. Sincere appreciations are expressed to Dr. C. A. Tatro for his valuable help in instrumentation and to Dr. William A. Bradley for his help in several aspects of this project. Sincere thanks are due to the members of the guidance committee, Dr. Charles O. Harris, Dr. Charles P. Wells, and Dr. Francis S. Tse. ‘ Thanks are also due to Mr. Donald Childs and his staff for their cooperation, patience and valuable suggestions in the experi- mental part of this project. Finally, the author wishes to thank his colleagues in the department for their encouragement and many valuable suggestions. 3:: >3 3}: :k 3:: >§< 3:: :1: :{r g: a}: ::< >1: >1: i: ii TABLE OF CONTENTS CHAPTER Page I INTRODUCTION ...................... 1 Historical background ................ 3 11 GENERAL THEORY .................... 5 (a) Plate ....................... 5 (b) Beam ....................... 10 (c) Uniform beam ................ . . 13 III NUMERICAL EXAMPLES ................. 15 (a) Plate ....................... 15 (1) Without attached mass ........... 16 (2) With attached mass ............ 18 (b) Cantilever beam ................. 21 (1) Natural frequency without rotatory inertia and shear deformation ....... 21 (2) Natural frequency with rotatory inertia and shear deformation ........... 25 (3) Frequency with mass, rotatory inertia and shear deformation ........... 25 (c) Uniform beam . . ................ 28 (1) A mass at mid-point of the beam ..... 28 (2) Two equal masses at quarter points from the ends .................. 4O (3) One mass at quarter point from one end . 43 IV EXPERIMENTAL RESULTS ......... . ...... 46 (a) Plate ....................... 49 (b) Cantilever beam ................. 54 (c) Uniform beam .................. 58 (d) Discussion of the results ............ 67 V REDUCED MASS SYSTEM ................. 69 VI SUMMARY AND CONCLUSIONS .............. 78 BIBLIOGRAPHY ...................... 83 iii TABLE 10. ll. 12. l3. 14. 15. LIST OF TABLES . Natural frequencies of plate. . Matrices for natural frequencies of plate. . Frequencies of plate with attached mass . . . . . . . . . Frequencies of plate with attached mass . . Natural frequencies of cantilever beam. . . Frequencies of cantilever beam with attached mass . . Frequencies of cantilever beam with attached mass . . Frequencies of cantilever beam with attached mass . . Frequencies of simply supported uniform beam with massatcenter............. ..... Frequencies of simply supported uniform beam with two equal masses at quarter points . . Frequencies of simply supported uniform beam with mass at quarter point . . . . . . ........ Experimental frequencies of plate without attached mass 0 O .......... O O I O O O O O O Q C 0 Experimental frequencies of plate with attached mass . Experimental frequencies of cantilever beam with and without attached mass . . . . ...... . . . . Experimental frequencies of uniform simply supported beam with mass at the center . . iv Page 18 19 22 23 26 29 30 31 38 41 44 52 53 59 63 LIST OF TABLES - Continued TABLE 16. 17. 18. 19. 20. Experimental frequencies of uniform simply supported beam with mass at quarter point. First mode frequencies for a simply supported beam with a concentrated mass at the center, obtained by reduced mass method. . . . . . . . ..... Third mode frequencies for a simply supported beam with a concentrated mass at the center, obtained by reduced mass method. . . . .......... Third mode frequencies for a simply supported uni- form beam with a concentrated mass at quarter point, obtained by reduced mass method ............ First mode frequencies for a simply supported beam with a concentrated mass at quarter point, obtained by reduced mass method. . . Page 65 72 74 75 77 LIST OF FIGURES FIGURE 1. 10. 11. 12. 13. 14. 15. Variable thickness plate . . . ............ . Variable thickness cantilever beam .......... . Frequency distribution for cantilever beam ..... . Mass arrangements for simply supported beam. . . . . Frequency distribution for simply supported beam; mass at center ..................... . Frequency distribution for simply supported beam; two masses at quarter points ............ . Frequency distribution for simply supported beam; mass at quarter point . . . . ............. . Schematic diagrams of supports ............ . Plate with rod attached . . . . . . . . . . . ..... Masses on plate .................... Clamp for cantilever beam . ...... . . . . . . . Clamp for simply supported beam .......... Masses for cantilever beam . ............. Masses for simply supported beam .......... Mass arrangements for reduced mass system . . . . vi 32 33 39 42 45 47 50 51 55 55 57 62 71 LIST OF APPENDICES APPENDIX Page A Work done by rotatory inertia of attached masses onplate.................. ..... . 87 B Verification of plate equation by Ritz method . . . . 90 C Work done by rotatory inertia of attached masses on beam . . ..................... . 92 D Work done by rotatory inertia and shear deforma- tion in beam. . . . . . . . . . . . . . . . ..... . 93 E Vibration of a wedge . . . . . . . . ........ . 99 vii Plate: D U (X. y) E h(x. y) V NOMENCLATURE Eh’ Plate rigidity defined by 3% . Work done. Subscripts with U refer to work done by particular type of forces. Coordinates on plate surface. Modulus of Elasticity. Thickness of plate at any point (x, y). Poisson's ratio. w (x, y, t) Deflection of plate in z direction. fl Time. Mass density of plate material, assumed to be constant. Distributed load per unit area. Potential energy. Kinetic energy. Plate deflection defined by Equation (5). Circular frequency, radians per sec. Shear Modulus. Function of time defined by Equation (5). Mass of load placed at xk, yk. Moment of inertia of a mass about n axis, placed at Xk. Yk; n = x or y. Length of the sides ‘of a square plate. Ratio of weight of attached mass to that of the plate. A non-dimensional inertia parameter. viii NOMENC LATUR E - C ontinu ed Beam: E U 0k 5k X (knL)iz kL Modulus of Elasticity. Work done. Subscripts with U refer to work done by particular forces. Deflection of the center line of beam. Length of beam between supports. Distance along center line of beam from left hand support. Mass density of beam material, assumed to be constant. Moment of inertia of beam sections about plane of bending. Normal function of the corresponding uniform beam for the mt— mode. Coefficients for series expansion of y. Circular frequency, radians per sec. Potential energy of the vibrating beam. Beam cross section. Mass moment of inertia of kt_hmass about plane of bending. Shape factor for cross-section of beam. Mass of ktll- mass. Mass of beam. 14.1; mb Rk Inertia parameter. This equals to — for an attached mass in the shape of a disc whose radius is Rk. For other shapes 2 4 of load, Bk ; , mas 3 ratio . fly- ' a fre uenc arameter zpzmbL3 ' q y p ' kL value for the nt—h mode, obtained from a 12 terms expansion of y. I 2 3 4 r-n-b—p-L- ; Ibo is moment of inertia of beam at x = 0. EIbo ix CHAPTER I INTRODUCTION Inertia is an inherent property of matter in motion. When a body oscillates about a point or line, the inertia in question is called rotatory inertia. If a mass is attached to a beam or plate, it is well-known that the natural frequency of the system is reduced due to translatory inertia of the mass. If the rotatory inertia of the mass is also taken into account, keeping the mass constant, the frequency is further reduced. In this investigation, a study has been made to determine the effect of rotatory inertia of attached masses on the frequencies of vibration of beams and plates. In the general theory, rotatory inertia and shear deformation of the plate are neglected, but for the beam, only part of the shear deformation is neglected. The assumptions of Hooke's Law, isotropy of material and small deflection theory, are made in the derivation of the fundamental equations. These equations can be applied to any type of variable thickness beams or plates with an arbitrary number of masses attached to the systems at different points. In order to investigate this effect of rotatory inertia, several approaches may be taken. In the case of a beam, having one or two masses, the classical approach may be used. For this case, each mass is replaced by the equivalent shear force and bending moment and these, in turn, are employed to give the required boundary conditions for the differential equations. The number of simultaneous differential equations that arise from this procedure is one more than the number of attached masses. As may be expected, this method becomes quite laborious when the number of attached masses is more than two, and also, it is difficult to generalize it to an arbitrary number of masses, because the boundary conditions that will have to be satisfied for each differential equation are influenced by the boundary conditions of the system, as well as the locations of the masses. To avoid this difficulty, d'Alembert's principle together with the principle of virtual work is used to find the governing partial dif- ferential equation. This approach leads to a single differential equation. Since the primary interest here is in free vibrations, a harmonic oscillation is assumed, with the result that the governing partial dif- ferential equation is reduced to an ordinary differential equation. The displacement function is then expanded in terms of the normal functions of the corresponding uniform beam. The number of terms taken in the expansion will depend on the'accuracy required of the lower mode frequencies, as well as the number of modes under investigation. Taking a finite number of terms of this expansion, a system of linear algebraic equations are obtained. Finding a solution other than the trivial one demands the vanishing of the determinant of the coefficients of these equations and this gives the frequency equation. . Actually, the set of algebraic equations generate a pair of symmetric matrices, the order of each of which is the same as the number of terms taken in the expansion of the displacement function. These, in turn, can be solved by the usual matrix methods. In the present investigation, the digital computer was used to solve these matrices. The eigenvalues of the matrices gave the frequency functions and the mode shapes were obtained from the eigenvectors. . For the case of a beam or plate without any attached masses, this method generates two symmetric matrices, one from the elasticity terms and the other from the translatory motion, rotatory inertia and shear deformation of the system. Once these matrices are known, the addition of masses to the system add certain terms on the latter matrix. As these terms are functions of the points of application of the loads, there are no integrals involved and as such the computation of these values is quite simple. In Chapter V, an approximate method is derived for finding the frequencies of a system loaded with concentrated masses, when the unloaded frequencies of the original system are known. This method has been introduced previously by D. Young (11):: and applied for the fundamental mode only. The present method differs from that presented in (11) in the sense that there is no trial and error solution.- necessary and also it is applicable for higher modes. Some results by this method are compared with results from other chapters and agree- ment is found to be very good. Historical backg round: The earliest work done on beams with rotatory inertia of load apart from Rayleigh's work, seems to be that of R. M. Davies (1-4). In his papers, Davies considered a uniform cantilever beam with a load at the free end. The effects of shear deformation and rotatory inertia of the beam are included in (4). R- H. Scanlan (5, 6) introduced the effect of rotatory inertia of loads by the usual method for lumped systems, obtaining a matrix in terms of displacements and angles. The same work was further investigated by H- E. Fettis (7) who obtained a variation of about 46% on the second mode of a wing when rotatory inertia of the engine was included. R. F. S. Hearmon and E. H. Adams (8) employed Rayleigh's approximation to the case of a loaded vertical strip to {include the effect of rotatory inertia of the load. References (9) through (17) deal with concentrated and distributed masses on beams, >:< Numbers in parentheses refer to the Bibliography at the end. neglecting rotatory inertia of the loads. So far, no general solution seems to have been presented for loaded beams and plates considering the rotatory inertia of the loads except in (5,6). It is to be noted that in (5, 6) the matrix is of higher order due to the use of slope functions as separate unknowns. Also, it should be pointed out that the accuracy of the results in lumped mass systems depend on the manner in which the mass distribution is assumed, as shown by J. P. Ellington (29). This is not necessary in the present method. Plate Considerable work has been done on vibration of plates but very few publications were found on rotatory inertia effects of masses on plates. A brief bibliography regarding the effect of engine mass on wing vibration may be found on page 361 of (6). G. B. Warburton (18) gives a detail analysis of vibration of uniform plates together with a long bibliography and B. B. Raju (19) gives a fairly complete bibliography of important publications about variable thickness plates. R. E. Roberson in (20) and (21) analyses the vibration of uniform circular plates with the help of Dtrac 6 function to represent the attached mass at the center, the former one for a free plate and the latter for a clamped plate. W. F. Z. Lee and E. Saibel (12) consider the case of a simply supported circular plate with a mass at center and obtain the solution in terms of the normal functions of the plate which are Bessel functions. J. Hansen, E. Warlow-Davis and J. Taylor (28) illustrates an interesting way of analyzing experimentally the effects of engine mass on the flexural and torsional vibrations of a wing. This includes the effects of weight of the engine and also its rotatory inertia. CHAPTER II GENERAL THEORY (a) Plate From classical theory, for a uniform plate, D v4 + h 52‘” — (x t) (1) w p "B—tz— - Cl . Y . If there is no external load q = O and Equation (1) reduces to 52w D V‘ + h = 0 2 W p ‘5? ‘ ’ The general expression of plate vibration, including rotatory inertia and shear deformation of the plate, is given by R. D. Mindlin (22) as 2 3 Z 2 This equation may, perhaps, be solved by a direct application of the Ritz method or the equivalent energy equation may be derived and solved. However, when the plate carries attached masses, it is convenient to use energy principles; otherwise, generalization to an arbitrary number of masses is not easily affected. Following R. D- Mindlin (22), the secondary effects due to rotatory inertia and shear deformation of the plate will be neglected in the following discussion. 1 , From (23), the increment of potential energy of a plate element during vibration is given by .55st <§—:¥—>2+ <3§¥>2+ asses h h 2 Z+<%—WT may?” 52W (6) + 2 1- 2 ’- dx d ( ”(B—E3) } ¢ _ 3”] Considering harmonic oscillations, assume ¢(t) : Sin pt and W(x, y) = E E AmnXmlx) Yn(y) (7) m=l n=l where X and Y are the normal functions of the corresponding uniform beams in'the respective directions and m and n are corresponding mode numbers. Substitution of Equation (7) into Equation (6) gives dz- sz =;.[ffn{(m:5”:15Arm—(13:59Yn)2+(70:51:51.A.mnxmn)z +2 (‘5 Vm=1n=l Amn—zm—Yn)(m§121AmnXm—;9—) +2(1-v)(m°5 5 A dxm dY (8) __ z =ln=1 mu (1;: dy __n_) } Sinz pt dxdy] If a virtual displacement is taken in the form 6 W(x, y) = 5».inin (9) the virtual work done by the elasticity forces becomes a v 5Ue = -O-A_-j 6A” co co dem dzx1 — - [ND { (1.31.31 Amn —z-d?{ Y) (7Y3) +( '5 °5Am nXd—TZYHHX‘f-l-{Qq m=1n= l m dy 1 dy 2 +v( m%lng£°lAm g—EEIzn—Yn)(X-d —,1—) co an szn + ugh?1 n§l Amn Xm ) (-—21-dx Y) (10) a» co 'dX dY dX- dY- - _m_.n. _1 __.1 ' 7- . .. +2(1 vumzzl n§1 Amn d; dy ) (dx dy ) 15m pt dx dy]¢5A1J The inertia force of an element dx dy of the plate is z - p h-gtjw— dx dy. = [ p p2 h W (X, y)dxdy] Sin pt. Therefore the virtual work done by the inertia force of the entire plate is °° °° . 2 = [p psz h (mg, n§1 AmnXmYn)Xin 6Aijdx_dy,] sm pt (11) When the plate is free from attached masses, the total virtual work done by these forces equals zero, from which the natural frequency equation is found to be* 2:: . This same equation has been derived by Ritz method in Appendix B, for the purpose of verification. a: co dZX dei 0° 0° dZYn H D[ {mél n§1 Amm de YnH—T Yj) + (1.1231 nEIAmnXm ayz) Y! m3 on d-’-XEa sz; Xi ddy H + V1 ‘5 AiJ °° Slim d_X.i . .. - z + (Iky mEI n21 Amn dx Yn) (dx YJ) 6 A13] S1n th=Xk,y=yk (14) Adding all the virtual works from Equations (10), (11), (13) and (14), and equating them to zero, the final form of the frequency equation is found to be E co co de dX1 12(l-vz) ffh3[(m§l r112;A ndi Yn)(d Y3) m dZY. +(m2-2-l nz=lArnnX m gym 1dyh) co dZX dZY- E1 Amn de Yn) (Xi dyg ) dz'X +(212Amaniyan lejH dY Q +Z(l-v)( 2 "2°: A de dYn m=l n=1 mn dx )6?ch filldxdy _ 2 a) co . + kél Mk(m§l nél Amn ann)(Xin)}X=Xk.Y"-Yk 1‘ co dYn + 151i 1kx(mz:.__1 n21 AmnXm— dy )(X —-J) . (15) m on + I A 9511‘ Y n) (d——X1 Yj)}_ _] ky (1,331 31 mn dx x-Xk,Y-Yk 10 Eh3 . . where D has been replaced by 12” v ) . Th1s 15 the general frequency equation for a variable section plate with any number of masses attached to the plate at arbitrary points. (b) Beam In this part, as with the plate, the usual assumptions are made regarding Hooke's Law, isotropy and small deflection theory. The general equation contains the effects due to rotatory inertia of the beam and also some part due to shear deformation. These secondary effects are included as in Timoshenko beam theory excepting a few secondorder terms including the fourth—order time derivative function which are neglected. The validity of the resulting equation, for a uniform beam, is compared in Appendix D with that of (23) and they are found to be exactly the same. The strain energy of bending of the bar at any instant 13 E L a2 I ‘ v = "'2“ IOIbIMIEfi—IZ d; (16) Let Y = (t) Y(x) Considering harmonic oscillations, assume an ¢(t) : Sin pt and Y(x) = :51 An Xn where Xn are the normal functions of the corresponding uniform beam. Thus (D y = (n‘El Aan) S1n pt (17) Substitution of Equation (17) into Equation (16) gives _. E °° dzxnz -2 V- T{I:Ib(n§1An'd—x[—) d3iSin pt 11 Taking a variation 6 Ai in one of the coefficients Ai of y, the virtual work done by elasticity forces during the virtual displacement is given by 6U — 6A — E{fLI (E A dZX‘IdZX‘dxb' Z t a A- (18) e—-5Ai 1" b “3373? inhp 1 0 n=1 The inertia force of an element dx of the beam at any instant is 2 -pA g—fi- dx. So the total work done by the inertia force on a virtual displacement 6y (= Xi 6Ai Sin pt) is L 5’- f a U1: - (pA-a—E’de) (Xi 6A,) Sin pt z p p2{ fol“ (A $21 Aan)Xidx I smz pt 6A, (19) From Equations (18) and (19), the frequency equation for a variable thick- ness beam, neglecting rotatory inertia and shear deformation of the beam is given by L dZXh dZXi n dxz) dxz a) L on A( g Aan)Xidx = E f0 Ib(n§1 A n 1 dx (20) pzpf 0 Let there be k number of masses attached to the beam at points x1, x2, - - - -, xk, having masses M1, M2 - - - --, Mk and moments of inertia II, 12, - - - Ik’ The virtual work done by the masses during translatory motion is k 52 6 Ut : - k§1[MkB_t¥- (6y)]x=x‘k k - 2 2 M 35 -p k=1[ k(n= 1 Aan)Xi]x:Xk Sinz pt 6A1 (21) The virtual work done by the rotatory inertia forces of the masses is f ‘S ee Appendix C 12 6U — >1; [(1 #53 a 5” r"-kzl kxt) (Tn x=xk 2 k dX dX; . z :[p 1321 { Ik(nE1 An—E—dx ) } ] sm pt 6A, (22) x=xk Rotatory inertia torque of an element dx of the beam is,“ 529 53 'prW—dx"plb 53%”! (23) Therefore, the virtual work done is 3 ”Huh“: Ibs‘xbsiz 5 (1%” L on dX dX' . mam. IbIAAn-a-f-I .1; deszpt 6A. <24) Shear torque in an element dx of the beamg" E a az To 6.7% Ti. ”1" So the virtual work done is is 1 6Usb= -p —§— GIOLBB—(Ibg—t’é) 5 (Tile) dx pZ k'G -E-——[f: T (Ib nE 1Aan)----— d-l—i dx] SinZ pt 6 A1 (25) Adding all the virtual work from Equations (18), (19), (21), (22), (24) and (25) and equating it to zero, the frequency equation is obtained as 2[ {LIME AX)X-dx+ :{M(% AX)X'i ppo n=1nn1 k=1 kn=lnn1 x=xk k m an dXi L w dxn dXi 2_:1{Ik(n§1An dx) dx i + p "olb(n§lAn ndx —’dx dx x=x1< a +P'ETE'36 lid—ain'bnm E 1Aan)d dx] a, 2 z =E fi‘IbInglAn ‘gznjng-dx (26) >’ i5 AX )X‘dx+ >1“: {M (‘5 AX )X} o n=l n n 1 k=l kn=l n n 1 x=xk k + k§1{'1k(§1AnE§_n)dxi} 1 — n— x x=xk L ’- d EIb f0 (2 Andan) dxgdx (28) Equation (28) is used in Chapter III to find the frequencies for three cases of a simply supported beam. The frequency values are tabulated and plotted in terms of a reference parameter. Actually, the expansion of the series in Equation (28) was carried up to 12th modes, even though the values given in the tables are up to 6til modes. This was done with 14 a view to improve the accuracy of the lower modes. The secondary effects of rotatory inertia and shear deformation of the beam are expected to be small because of the length of the beam. Due to orthogonal property of the normal modes, Equations (27) and (28) will simplify considerably, but that aspect is shown in each case separately. Also it may be mentioned that Equation (28) can be derived directly from Equation (15) by assuming Yn = Yj = 1, v = O, Ikx = 0, and Iky = 1k. CHA PT ER III NUMERICAL EXAMPLES (a) Plate For this part, .a square plate was used, as in Figure 1. The thick- ness at the middle was 0.125" which decreased gradually to 0. 0625" at the edge. This was the same plate as model E of B. B. Raju (19). x ‘T 0.125” 1 Figure 1. Variable thickness plate. To facilitate computation, this part of the analysis is divided into two sections. The first section contains the evaluation of the natural frequencies of the plate and section 2 contains that of the plate with mass 15 16 attached. It may be mentioned that in (19), finite difference method was used to find the natural frequencies of the plate. (1) Without attached mass. To find the natural frequencies, Equation (12) was used. The plate was assumed to be simply supported. Taking only one term of the series gives, dZX, de sz, dZY HD[{(A11—;2'1Y)(—5—2-x11Y)+(A11X1—71‘1(X1(11)} de, dZY sz dZX + ”1(A11Tz—Y1)(X1d—y'z—l’) ‘1' (P*11)(ld)l (le YIIi dx, dY, dX, dY, = P2 p If h IAIIXIYII (XIYII dx dy (29) For this case X1 = Sin 1T: , Y1 = 51,,ng Substitution of these values in Equation (29) results in 4 . 1T E 3 , 2 TTX , 2 1Ty , z TTX , 2 fly ((va) [ If h {2 Sin _a S1n —a + 2 v(S1n —a Sln _a ) +2(1—v) C0321): C0523 1 dx dy a a =pzp{fthinz-1-EESinz¥}dxdy (30) Let ho be the thickness at the center of the plate and assume— h — 'y. 0 Then 7 has the following relations: IfX§Y_<_a-x 7(X.Y)=(%-+-5 If y: x: a-y va. y) = (%+ %> 0. 5 5 7 _<_ 1.0 The integrals in Equation (30), as well as others to come, were solved in the digital computer with program. EAI-M with 48 divisions 17 between the limits. With this program, an accuracy up to about 7 places has been realized. This program uses quadrature formula (25) Q66, obtained from a 62!} degree polynomial that fits the f(x) values at the seven points indicated, by integrating over the six panels, between x0 and x6. Actually, 49 points were used between limits, whereby eight cycles were necessary to cover the complete range. The following values are shown, as representative examples, that were used for Equation (30) . 1’°2—"y ’3'2—1” 5X =010 000 10 Sm a 10 v Sm a dIaIdIaI . 3 499 4 l 2 EX 1 3 2111‘. 35 X : O 1 1 Io Cos a f0 ‘Y Cos a d(a)d(a) .05392394 (3 ) 1 -"Z “Y 1 ’ Z LX 1‘. X :2 IO Sm —a f0 7 Sm a d(a)d(a) 0.198397736 In (19), the frequency p is expressed in terms of a reference parameter _ ] Do h 1, p0 — W W e e _ E1193 D0 ‘ 12(1-v I With the help of Equation (31) and taking v = a} the first approximation to the fundamental frequency is found to be pZ/poz = 2. 1164356114 or p/po = 14.3582752 The experimental value for this case is 13. 78, a variation of only 4. 19 p. c. Next, nine terms of the series in Equation (12) were taken for i = 1,, 2, 3 andj = l, 2., 3. To get the non-trivial solution, the determinant of the coefficients must vanish and this gives a determinant of the form I[A] - A [B]| = 0 where x = 1%, [A] is the matrix of the elements on the right hand side of Equation (12) and [B] is the matrix of 18 the remaining part. The two matrices are shown in Table 2. Table 1 contains the frequency values obtained through M-Sprogran of MISTIC. Table 1. Natural frequencies of plate. Mode p/po from p/po from Extrapolated p/po from 6 terms 9 terms p/po (19) 1 13.794 13.738 13.693 13.568 2 35.065 34.731 34.464 33.195 3 35.065 34.731 34.464 33.195 4 55.234 55.234 55.234 52.279 5 73.935 71.665 69.849 66.121 6 70.479 70.479 70.479 68.001 7 88.909 83.339 8 88.909 9 124.850 (2) With attached mass. For this part, a mass was assumed to be located at x =(2/3)a and y =(2/3)a. This point was chosen with a view to get higher rotatory inertia effects for all modes under consideration. Equation (15) is the governing equation. Assuming the load to be fixed perpendicular to the neutral surface of the plate, Ix = IV, the rotatory inertias about x and y axes. zz Letlx=I = a_r_n_2[3_3_ wheres: , [3 is a non-dimen’ional Y 4 * inertia parameter , m is the mass of the plate and M is the total attached 2t a2 mass. In this case mp = g p. ___. as ‘ This notation came from the beam analysis, where B = R/L, R being the radius of a disc fitted on the beam and L is the length of the beam. 19 choooao.hm o o w¢o~oaw.mu wwomoow.mn o o o wmhwmmm. o mnemnow.mH o o o o o wowmwam.u o o o mbomnow.m~ o o o mowm¢aw.u o o w¢o~oow.mu o o ovwmmmo.o mnemonw. o o o m~mo>m¢.n w¢omoaw.mu o o wwomonw. ovmmmwo.o o o o mHmOPMv.u o o o o o Awowoow.m o o o o o oommwfiw.n o o o owoavom.m o o o oowmwfiw.u o o o o o mwoawom-m o vmhvmmm. o o m~mo>m¢.n m~m0>m¢.: o o o oomwadv. HE mmmNONH. o o mmm¢ooo.: omowooo.n o o o mfioohoo. o omwmahfi. o o o o o hmowhoo.n o o o omwmmha. o o o wmawnoo.n o o mmo¢ooo.: o o mfiwhowH. mfioohoo. o o o Nwmwmfio.u owowwoo.u o o m~oo>oo. mwwhowa. o o o Nwmwmdo.u o o o o o womowfia. o o o o o hmowhoo.u o o o mmvmmwfi. o o o hmawhoo.u o o o o o mmfimmmfi. o mfioanoo. o o Nvmwmfio.n Nvmwmao.u o o o whwmwofi. E .vomHQ mo wowocosvouw Hmuamc new mmucumz .N smash. 20 Substitution of the above values in Equation (15) gives 2 Z E3 1 3 m a, d Xm dZXi . , 12 PP (l-vz) “2% 7 { (m§1 n§1 Amn dxz Yn) ( dxz Y3) m m dZYn sz' + ‘mél £31 AmnXmV) ”‘1 .172" w o dZXm ,sz- co co sz . de' . + ”(14.21 .ElAmnW Yn’ “1.1—er “mil .331 AmnXm—d—yfI ”$341) . 0° 0° de. dY dX‘ dY- x y _ , __ J _J __; .. _ +2“ v);(IIT§1.nZ—_31 Amn dx dy ) (dx dy ) } d(a)d(a) =~[ fgfgy (mil n; Amn XmYnHXiYfidq-Ng) a) m +§-{ (1(le rg AmnXmYnHXin) dfizaz w w dYn _ de + 4 (m‘El r531 AmnXm dy ”Xi dy ) 2 z - (32) dB a 0" °° de; ‘ dXi + 4 mél n§1 Am“ dx Yn” dx YJ) } 35335.3“-23 mTTX Assuming Xm = Sin , Yn = Sin nTry 9 expanding the series to any number of terms and equating the determinant of the coefficients to zero, a matrix of the form I [A] - X [B]! = 0 is obtained. This, in turn, was solved by M-S program of MISTIC. For this case, a 9 terms expansion was used. It is to be noted that the 9 term expansion is quite small for a problem of this type. But as each integral of Equation (32) used to take a long time (about 4-5- minutes of machine time) it was decided to be satisfied with 9x9 only. An extrapolation (ha) is used to improve the accuracy of the results. 3 In Equation (32), the first term of the right hand side and the complete left hand side has already been evaluated in section 1. To find the other values, the following values of the parameters a. and (3 were used. 1.0 and 2.5 a l3 0.1, 0.2, 0.3 and 0.4 21 Tables 3 and 4 contain. the p/po values, some of which are compared with the experimental values in Table 16. (b) Cantilever beam In this part of the investigation, a cantilever beam, as in Figure 2, was used. The complete analysis is divided into three sections. Sections (1) and (2) contain the evaluation of the natural frequencies of the beam, without and with corrections for rotatory inertia and shear deformation of the beam respectively. To get reasonable variations due to these correction terms, the length of the beam was purposely made \\\\\\\\\ {in/Attached/m ss \ z; (B) (A) k \\ \\\\\\‘.._—— ,3. L I T... Figure 2. Variable thickness cantilever beam. short (20"). The results are shown in Table 5. -In section (3) a mass is assumed to be attached at a point three-fourths the length of the beam away from the fixed end. Tables 6, 7 and 8 show the values of the fre- quencies for different values of o. and (3. , In. Figure 3 the above values are plotted to show the effect of (3 on the frequencies. Numbers on the right give the corresponding mode of vibration. (1) Natural frequency without rotatory inertia and shear deform- ation. For this case, Equation (26) reduces to 22 Table 3. Frequencies of plate with attached mass. Mode 6 terms 9 terms Extra- expansion expansion polated (1:10, [3:01 1 7.39 7.35 7.32 2 23.08 22.68 22.36 3 32.25 32.25 32.25 4 47.55 46.55 45.75 5 61.98 53.20 46.18 (1:10, [3:0 2 1 7.36 7.33 7.31 2 21.08 19.88 18.92 3 25.00 24.27 23.69 4 38.44 34.99 32.23 5 48.38 42.56 37.90 (1:10, [3:0 3 1 7.32 7.29 7.27 2 17.96 16.03 14.49 3 18.85 17.69 16.76 4 33.25 30.79 28.82 5 45.95 41.30 37.58 (1:1 0, 8:0 4 1 7.25 7.22 7.20 2 14.83 12.97 11.48 3 15.03 13.67 12.58 4 31.26 29.43 27.97 5 45.15 40.76 37.25 23 Table 4. Frequencies of plate with attached mass Mode 6 terms 9 terms Extra n n exp — exp — polated (1:2 5, 8:0 1 1 55.05 5.02 5.00 2 21.28 20.58 20.02 3 28.20 27.83 27.53 4 42.09 38.90 36.35 5 50.74 45.39 41.11 a: 2.5, (3 =0.2 1 5.03 5.00 4.98 2 16.82 15.03 13.60 3 18.09 16.92 15.98 4 32.59 30.06 28.04 5 45.82 41.23 37.56 d=2.5, [3: 0.3 1 5.00 4.98 4.96 2 12.70 10.87 9.41 3 12.72 11.67 10.83 4 30.12 28.43 27.08 5 44.80 40.47 37.01 0.: 2.5, 6:0.4 1 4.95 4.93 4.91 ' 2 9.38 8.44 7.69 ’ 3 10.04 8.84 7.88 4 29.39 27.93 26.76 5 44.56 40.33 36.95 24 p201: A (n21 Aan)Xidx = E 1:11) ($21 An (1:?) 22:23 dx (33) Here, D 44,42 - 313), A =9?” -%)Z,1b,= fig -3534 Now let Xi = (195,: gig—(L kip gill—>519”: 31: = z and )‘v = 64 pity} With these notations, Equation (33) becomes 1:12-21:11}; Ann) 4142 = xv 132-214 { n21 An<1i'dz 1 a: n u = xv1012-21‘1nzgl An(knL)z¢n 1 (14141241 dz (35) From Equation (35) it is clear that the effect of rotatory inertia and shear deformation is dependent on the length of the beam, as is well-known. In case of simple supports, knL = rm and as such, the length term in the denominator of the second term on the left is replaced by the corre8ponding wave length term, as in (23). The second and third terms on the left are the only additional ones and substituting these terms, the following values of p/pr, as given in Table 5, were :obtained. (3) Frequency with mass, rotatory inertia and shear deformation. In this section, the same beam as in Figure 2 is used with a mass at x = }; (arbitrary). For this case Equation (26) becomes 26 $93. $383 3.89am 338$ $08.3 34.3.3 8.883 a woman mHNQNéwu oommoéw omwwnJau onomoéau oowmmdv whom; .3» m wma .N mwomvdm doowmdm @mwnodm ohwwodm omfiwoom thomom w mwoé oonmoéfi mwmooda fenodfi odvowda omZmAH hoomwdfl m whmd mwhmmd vammfi Sommd ewwmmd omwmmfi hoommd N :06 oomamé womgg mom~m4 54mg; 34mg; ovmamé H o .9 GM @3209 mayo» o mayo» 0 pom—20m mauve o mayo» p #53352, nmuuxm— E0: um\m Eoum um\m nouaxm Eonm um\m Scum um\m 0602 “3300.200 5C5 um\m ”5300.300 “593:5 um\m .Emoo. soc/33500 mo mmwososvoum fidudumz . .m oBdH 27 p"'[410"-6 (2-—)2(>3___ 1n4,414.14): + {M(>3 11.13.47 14 + I]? ( {10201 A n(knL)¢n')(kiL)¢i' } 3L 71— An(knL) ¢nv)(kiL)¢i'dx 110 254°” + z{1———:‘}f(2-L)(n§_ 1024L 1 E L a) - 2615152k::<'G Io (Z'E)3(n§1An¢n)(kiL)¢i'dx _ $142 [0 (2' £19,121 "°‘n(1;1 An4n14idz +-:- {a4i + °—‘3— (2 An&,T———-=r:__—_—:————L-zr———..—.—-"r—5 ‘\ " -—— ‘H\*—l ' \- \ §\ . -—2 .3‘— T.‘ . . 4 7— \\:‘ \“ \ , 1 1 l 1 7-3 a) T Loaded kL/n17 tn Figure 7. Frequenéy distribution for simply supported beam; mass at quarter point. CHA PT ER IV EXPERIMENTAL RESULTS As mentioned earlier, three cases were investigated experimentally. They are: (a) Plate: This was a square plate, 9"x9", made of aluminum. The thickness at the middle of the plate was 0. 125" which gradually decreased to 0. 0625" at the edges. The mass was fixedatx=E3-a-, y=%. (b) Cantilever beam: The beam was made of aluminum and 20" long. The diameter at the fixed end was 1" which gradually decreased to i” at the free end. The mass was fixed at x -‘ LL. - 4 , (c) Uniform beam: This was a steel bar, 1" in diameter and 35" long between supports. Two cases of this setup were investi- gated: (1) The mass was attached at the mid-point of the beam. (2) The mass was attached at quarter point. The different types of supports, used in the experiments, are shown schematically in Figure 8. Figure 8a is for the plate which rests on the support along four edges. By spreading some special fine grained sand along the edges and vibrating the plate by pulsed air, it was brought to resonance. The side bolts were then gradually tightened until the sand along the edges just stopped vibrating. This gave the condition for simple supports where the edges should not move but can rotate. Due to finite width of the supports, there was friction and this raised the frequencies of the plate . 46 47 // V V /2 ///> 7% (a) Plate (b) Cantileve r beam L. ” f _ 1F 55 I 1 :1'4. . 2 7 143. J1 (c) Simple support Figure 8. Schematic diagrams of supports. 48 Figure 8b shows the support arrangement for the cantilever beam. The beam was fixed on a bearing and the whole assembly was fixed on a heavy steel table with a C-clamp. It was found from1the natural frequencies of the beam that the support was weak. But as no other sup- port arrangement could be built near the compressor, this was taken to serve the purpose. ‘3 The support for the uniform beam was made from three channel iron sections with stiffening ribs, as shown in Figure 8c. Two steel plates were then welded on the two vertical channels and the bearings were bolted to these two plates. Two 1/2" holes were drilled through the beam at the ends and these were honed to fit two ground 1/2" pins. The two pins were then fitted on the bearings with screws. To remove middle plane forces along the beam, one end of one of the holes on the beam was cut out. The whole assembly was then clamped onto the table with two C-clamps. The systems were vibrated with pulsed air, supplied from a com- pressor at about 90 psi. The setup used was the same as used by B. B. Raju (19). To find resonance of the beams, a vibration pickup, in- stead of SR-4 gages, was used. This had the advantages of higher signal strength and also it could be moved to any point along the beam. A SR-4 gage was used in the case of the plate. To vary the rotatory inertia of the attached masses, keeping the mass constant for each value of a, it was necessary to fix clamps on the bars with extensions attached to the clamps. The masses were slid- - on these extensions. As only some part of the attached masses .could be slided, it was necessary to compute the lengths at which the sliding masses had to be fixed to get the required amount of rotatory inertia. The compu- tations are shown in the following pages. 49 Plate In order to attach masses on the plate, the plate was drilled at x = 333- = 3;. A 1/2" diameter magnesium rod was chosen and it was threaded throughout its whole length. This rod and other masses were fixed at desired locations with the help of six nuts. Following are the data for this setup. 2 2 Weight of aluminum plate = mpg = toa (pg) = 0. 675# 3 Weight of threaded magnesium rod = 0.0625# Length of threaded magnesium rod = 5%“. Therefore effective diameter of magnesium rod = 0.463" (at (pg)=. 065#) Weight of six nuts 2 0. 375# Therefore, weight of each nut = . 0625# Height of each nut = 0.482“. With this setup, only two values of 0. were used. They were 1.0 and 2. 5. Following are the data. Weight of rod and six nuts = 0.4375# 0. =1.0; extra weight = 0. 2375# This weight was made from a steel plate, having a 1/2" diameter hole, with dimensions 2%" x 2" x1277" -' c1.=12”.5; extra weight = 1. 25 #. This was also made out of a steel plate with a %-" hole at the middle, having dimensions 315,-?" x 1%" x%". The weight of this plate was 1#. So, this plate, together with the first plate and a washer (0.0125# - height 0. 1"), the total weight was 1,. 25#. It was found that if a single weight of 1. 25# is made, 8 = 0. 2 is not possible to obtain... Let I1. = required moment of inertia; 2 ampa 4 [32 13 . 65 g II 0.82 50 0. 1 1 From Figure 9., rotatory inertia of two nuts = ——0 3 5 . . 0. 171 rotatory 1nert1a of rod : _ 0. 18415 Total rotatory inertia of Figure 9 = g 1 1111“” 1'11I1’r £5 64 —: r o. 7042”/ Figure 9. Plate with rod attached 11:1.0 (8=0.1 and 0.2 are not possible) 1. . . .20., r=fl=02_765+(fl1f g g g So, L = 1.64" (Figure 10a) Similarly, for 8 = 0.4, L = 2. 3" .1 (1: 2.5 Ir=34 5 Bl 8 Z (3: 0.2 11,:1-366 =_1.1799 +0.0625L So, L = 1.728" (Figure 10b) 2 3.074 _0.375 L2 +0.0161 L + 1.06081 8=0.3 1,. g g g g So, L = 2.29" (Figure 10c) 51 Mas ses on plate . e m P / . e . M . . D lav/$2 , .1 _fi >xxye. . L .ruwwyaéll m ,, 1,41%? fix” m iffél _[ L f 0 ) . a ( 52 5.46 _ 0.5248 + 0.02L + 1.1375 Lz 1' g g g g So, L 2.08" (Figure 10d). Table 12 gives the natural frequencies of this plate and Table 13 gives those of the loaded plate. Since frequencies in cycles per second gives a better feel of the phenomenon, they are also tabulated. Table 12. Experimental frequencies of plate without attached mass. Mode Theoretical Experimental Theoretical Experimental Frequency Frequency p/pr p/pr 1 206.5 209 13.693 13.78 2 518 514 34.464 34.2 3 518 514 34.464 34.2 4 831 824 55. 234 54. 8 5 1051.5 1026 69.849 68.2 Note: Since the plate is square and symmetrical, the second and third modes are usually accepted as the same. But due to unsymmetric plac- ing of the attached mass, these two modes are not same for the loaded plate. As such they are assumed here as different. 53 Table 13. Experimental frequencies of plate with attached mass. a (3 Mode Theoretical Experimental Frequency .Frequency 1 0 0.3 1 109.3 113 2 218 214 3 261.5 256 4 434 427 5 564 560 1 O 0.4 1 108.1 112 2 172.7 159 3 189.3 186 4 421 416 5 560 553 2 5 0.2 1 74.9 76 2 204.5 --- 3 240.1 --- 4 422 414 5 563.5 561 2.5 O 3 1 74.6 75 2 141.9 --- 3 163.7 --- 4 407.5 398 5 557 552 2 5 0.4 1 73.7 74 2 113.9 --- 3 118.5 --- 4 403 387 5 556 547 Note: For a. = 2.5 and B = 0.2, 0.3,, 0.4, it was not possible to get any good response for the second and third modes, even with the full pres- sure on. There were some reaponses near about the theoretical values but the presence of subharmonics of higher modes made it very difficult to locate these values accurately and as such they are not filled in. For the value corresponding to the second mode of 0.: 1 and [3: 0.4, it was noticed that the load vibration was very violent. The high variation be- tween theoretical and experimental value seems to be due to slow con- vergence of the series. Higher number of terms in the series apparently must be taken to improve these values. 54 (b) Cantilever beam At the earlier stages, a steel bar was chosen for this part of the experiment. But while checking the natural frequencies of the unloaded bar, it was found that the support was too weak for this purpose (Figure 8). Consequently, an aluminum bar had to be used. The different weights that were made to suit the different values of 0., were no longer found adequate when the beam material was changed. Instead of making the weights again, it was decided to draw curves of theoretical values of frequencies vs. 0. for each [3 and compare the experimental frequencies for those a values that were available from all the weights. Only three 0. values were available and these are 3.64, 6. 18 and 7. 915. The following calculations are made on this basis. The data for this setup are given below. 237, # (Figure 11) 1% # (Figure 2) Weight of clamp Weight of beam {+41 (Figure 13a) %—# (Figure 13b) 311-5 # (Figure 13c) 4&4} (Figure 13d) Weight of clamping piece Weight of first mass Weight of second mass Weight of third mass Referring to Figure 2, Weight of (A) + (B) = 1%; #. So weight of (A) = 0.945#. ~ L2 94.5 ’- Ir : (LI—:2..— 52 : _g_9£_ (49) Clamp : steel (Figure 11) 1%- 9i- 2 «d Rotatory inertia of clamp = p[ [l3 '3- "rsdr + (121? ’92? 1 g Let Ix = Moment of inertia of any mass about an axis x, through its center of gravity. 55 .8363 potommdm >383 p0w QEdHU .NH madman“ .Emmn. .8333ch HOW @830 .2 6.59th hlao § Q“ l 1 L. 9“. 94.x: “(me E. Ix. —1— _— //' B \1 1 \fi 5; \x - I \l \\ T 1 \s 5‘ \ “w‘ ~ -- - L..__._-.~._ _-_——.. 56 In the following, Ix of the clamping piece (Figure 13a) will be neglected First mass: Figure 13b and clamping piece 13a Total weight on bar = 3125 # ‘So, 0. = 3.64 34 . From Equation (49), Ir = i 52 Comparing with Equation (50), B = O. 1, 0.2 and O. 3 are not possible. 0. Ix of first mass = 722' (51) g z For0=0.4 1r=.___55~0 240.6 +0.722 +1_._ g g g g 80. L = 3.70" Similarly for (3 = O. 5 L = 6.685". Second mass: Figure 13c and clamping piece 13a Total weight on bar = 5. 8431# So, 0:6.18, 1r: 5—824—3—1— 02 IX of second mass = 0°575 (52) g 0'575 + 4—9 (L+0.23)z Inertia at distance L = —— g 16g 2 50, total inertia = _41'23684- ——l°4gO8L + ————3°4:65L (53) With these, the following values are obtained. For (3:0.4 L: 3.72" For (3: 0.5 L=5.335" Third mass: Figure 13d and clamping piece 13a. Total weight on bar = 7.469# So 0. = 7.915,1r = ”6°91? l. 54 (54) IX of third mass = —— 57 zg'kl’firfi” (“’71 I i '1 H; 1 /-’7/9—— ‘3 - X “Q, l 9 I__,.:,;_4/ __.___ “fig: 1 I 4/ A’:t——e,/—/ -——-—--- / [‘Z-_J’ —--r- //' v a“ u f ’, I! u k’sg *1 \—4.85X/-2-’- x17;- ngaxno' (a) (C) /4éhxzuxzu /.> 7" x / a: -77- d———— (b) (d) Figure 13. Masses for cantilever beam. 58 1. 4 Inertia at distance L = 5 + 12— (L + 0. 5)2 g 165 . 1 . . 1 2 50, total inertia = 43 3 2+ 4—————6875L + 5—03 SL (55) g g g Therefore, for (3 = 0.4 L = 3.45" for (3:0.5 L=4.89" Table 14 contains the experimental values obtained and these are compared with the theoretical values obtained from Figure 3. (c) Uniform beam The same clamp and weights were used for both the settings, the first one being for the masses at the center and the second one for that at quarter point away from one end. Following are the data for this part. Beam length between supports = 35" Diameter of beam (uniform) = 1" -Material - Carbon steel drill rod (SAE1096) Young's Modulus = 30.5 x 106 psi Density of beam material = 0. 283# in3. Beam weight = 7%}? Weight of clamp = 211-2- # Weight of clamping piece = «ii-a)! Weight of first mass (a: 0.5) = 332-11} Weight of second mass (0. = 1.0) = 4&4)! Weight of third mass (0. = 2. 5) = 16%}? Weight of fourth mass (0. = 5.0) = 363-11E The following calculations are made on this basis. The same clamping piece as in Figure 13a was used. 59 Table 14. Experimental frequencies of cantilever beam with and without attached mass. a. 8 Mode Theoretical Experimental Frequency Frequency 0 0 1 92.8 91 2 390.8 385 3 963.8 947 4 1795.2 --- 3.64 0.4 1 24.9 25 2 89.75 89 3 664 657 4 858 857 0.5 1 23.45 22.7 2 77 77 3 664 648 4 850 846 6.18 0.4 1 19.5 20 2 70.6 71 3 663 658 4 855 850 0.5 1 18.3 17 2 59.75 60 3 663 655 4 845 839 7.915 0.4 1. 17.3 17 2 61.5 58 3 660 651 4 850 841 0.5 1 15.63 14.7 2 52.8 52 3 660 650 4 846 838 6O Clamp : steel (Figure 12) 1 1 Rotatory inertia of clamp = p[ fl %- Trr3dr + [I 2 ] T 125 11 .4 = 6 (56) 8 2 0 Ir : 3971b}— pl : 8.2—0 GB?- (57) 1 00 a: 0.5 ; From Equation (57) II. = -2g— P12 From Equation (51) IX = 0°;22 1 1 . 0. 3 So, Ir : 200 F52 = 16 4 + 722 +_1:_ 8 8 8 8 For 8=0.4 L=8.9" For 8:0.5 L= 13.7" (not possible). 0. =1.0 ; From Equation (57) Ir = -2-4g—- (32 From Equation (54) 1x = if?- 2400 2 116.4 1.54 11 2 75 2 : —— : + — . 50,11. g s g g +32gL+16g(L+O 5) From this, for {3 = 0.3 L = 4.05" for (3 = 0.4 L :6.90" for B = 0.5 L = 9.46" c = 2.5 ; From Equation (57) II. = 60:0 [37‘ Referring to Figure 14a, _ 2.795 . _ . . . _ . . . 1X1 - —-—g—— , 1x2 — negligible , 1x3 — negligible 14.92 1.565 0.115 8 8 8 6000 l . . . Ir = 02 = 47 065 + 39 172 L+ 16 94 L2 8 8 8 8 61 So, for [3 :0.3 L = 3.87" for8=0.4 L =5.97" for[3=0.5 L =8.06" c1 = 5.0; From Equation (57), Ir = 12:00 (32 Referring to Figure 14b, I =35.5 ; I = 0.426 ; I = 0.404 X1 g X2 g X3 g 26.7 5.11 4.84 ml: _"" ; m2 : ; m3 2 8 8 8 r = 12000 F52 = 243.93 + 59.43 L + 36.99 L2 8 8 8 8 From this, for [3 = 0. 3 L = 4.08" I for 0 =0.4 L=6.06" for [3 = 0.5 L = 7.975". T able 15 contains the frequency values for the system with the mass at the center and Table 16 contains those with the mass at quarter point. 62 x, 5’6 "x3913 —-- ____._. .. _ — X1 __._ _._1 - 1 (a) 5%";(3’x/ .-—-- -— - - x3 — —-——————--——"' — 2 6,;x3nx’u 5313434" (b) Figure 14. Masses for simply supported beam. 63 Table 15. Experimental frequencies of uniform simply supported beam with mass at the center. o. 8 Mode Theoretical Experimental Frequency Frequency 0 5 0.4 1 45.2 46.6 2 145 145 3 466 473 4 483 484 1.0 0.3 1 36.8 37 2 139 139.5 3 440 437 4 477 478 0.4 1 36.8 37 2 110 111 3 440 437 4 457 456 0.5 1 36.8 37 2 90.2 92 3 440 437 4 447 446 2.5 0.3 1 26 . 27 2 94.5 94 3 418 421 4 448 447 0.4 1 26 . 27 2 72.4 71 3 418 421 4 440 439 0.5 1 26 27 2 58. 6 57 3 418 421 4 436.5 433 5.0 0.3 1 19.2 19 2 68. 6 68 3 411 414 4 439 437 Continued 64 Table 15 - Continued 0 fl Mode Theoretical Experimental Frequency Frequency 5.0 0.4 1 19. 2 19 2 52 52 3 411 414 4 436 434 O. 5 1 19. 2 19 2 41.8 41 3 411 414 4 433.5 430 65 Table 16. Experimental frequencies of uniform simply supported beam with mass at quarter point. a 0 Mode Theoretical Experimental Frequency Frequency 0 5 0.4 1 48.75 49 2 187 188 3 252 249 4 627.5 619 1.0 0.3 1 42.3 43 2 165 167 3 244 251 4 617 619 0.4 1 40.7 42 2 147.2 146 3 216 214 4 612 618 0.5 1 38.9 35 2 131 134 3 206 208 4 609 615 2.5 0 3 1 30.9 30 2 119 117 3 204.6 197 4 602 594 0.4 1 29.4 28 3 98.6 100 3 195.8 198 4 600 593 0.5 1 27.65 27 2 85.5 84 3 192.8 188 4 599 591 5.0 0 3 1 23.15 22 2 87.2 83 3 193 191 4 597 590 Table 16 - Continued 66 o. [3 Mode Theoretical Experimental Frequency Frequency 5.0 0.4 1 21. 9 21 2 70. 75 72 3 189 188 4 596 560 0. 5 l 20. 5 19 2 61 64 3 187. 5 185 4 550 555 67 Discussion pf Experimental Results The following remarks seem appropriate before the final results are discussed. In the case of the simply supported uniform beam with a mass at mid-point, the first mode was quite simple to find. The amplitude became fairly large and the scope trace was almost perfectly sinusoidal. The higher modes required judgment on the part of the investigator. As the frequency was increased, the scope trace did not remain sinusoidal until the next higher mode was reached. It was observed that the amplitude of the next higher mode may become smaller than that shown just before the resonant frequency was reached. The only way to obtain the resonant frequency was by noting the trace which should be sinusoidal. In the case of the same beam with mass at the quarter point, the maximum amplitude was seen to be at the fourth mode. The remarks made for the previous case apply to this case also. Subharmonics of the fourth mode presented considerable difficulty which had to be reduced by the use of a filter. One important aspect noticed in this experiment was that if the nozzle was placed at any arbitrary point of the beam, the resonant frequency was higher than the theoretical values. When the nozzle was placed at the point where the amplitude was maximum, the resonant frequency was minimum and gave best agreement to the theo- retical value. This point of maximum amplitude was very close to the value obtained from the eigenvectors. The results obtained experimentally agreed fairly well with the theoretical values. There are several reasons for discrepancies as listed below. (a) Slow convergence of the series: This seemed to be the main reason for the cases of variable section beam and plate. This is obvious from the results of the higher modes. Also, in some cases, particularly 68 the second mode of the loaded plate, the load was observed to vibrate considerably in the plane of the plate. This effects the rotatory inertia of the attached mass. The maximum variation of experimental values from the theoretical values are most evident in this mode. (b) The reasons given in (19) applies to both plate and beams. These are (i) determination of exact resonance, (ii) errors in the read- ing instruments, (iii) actual model differing from the theoretical model, (iv) inaccuracies in the physical constants, (v) support conditions, (vi) vibration of support, (vii) damping in the material, and (viii) effect of air mass. The effects of (ix) large amplitudes, and (x) shear and rotatory inertia of the system are applicable to plate only, because a vibration pick-up, used in the case of beam, reduced the amplitude and shear and rotatory inertia effects were also taken care of. (c) Rotatory inertia Ix of the clamping piece. In the calculation of length to produce certain amount of rotatory inertia, this IM part was neglected. The effect of this is quite small and should be less than a fraction of a percent. ((1) Slackness in tightening bolts. When the masses were fixed on the clamps, they were fastened with two bolts to the clamping piece. It was seen that if these bolts are loose, the frequency was higher because the load had the freedom to remain at the same place instead of moving with the clamp. The contribution due to this appears negligible because the bolts were checked occasionally. (e). Variation of a values. As may be seen (page 58) for the uniform beam, the weights that were used to produce particular values of 0. were a little off. The maximum variation was less than one percent and as such the variation in frequencies should be less than one percent. CHA PT ER V REDUCED MASS SYSTEM From a paper by D. Young (11) it was seen that the effect of a concentrated mass on a beam can be approximated by reducing the mass of the beam at the position of the concentrated mass in such a way that the natural frequency of the beam without concentrated mass is same as a single mass placed at the position of the concentrated mass and assuming the intertia of the entire beam to be zero. It is stated in the above paper that this method is valid only for the fundamental mode. It is the purpose of this chapter to show that the above method can be used for higher modes also. And the same method may be used, with sufficient accuracy, for plates as well. For this method, the natural frequencies of the unloaded beam or plate are needed, which can be derived by standard formulas and also the stiffness coefficients of the beam or plate at the positions of the concentrated masses. These can be derived or in complicated cases, they may be obtained by measurement. The procedure for beam may be described as follows. The same applies for plate as well. Let there be a concentrated mass at g 4— 5_ g point P, at a distance "a" from the a “2.) left support (figure at left). The beam may have any type of supports at the ends. If the first mode frequency of the loaded beam is required, then the mass of the beam is replaced at P by a reduced mass mr, where m1. is unknown. Let the Spring stiffness (force per unit deflection) of the beam at P be k. Then from the elementary formula for a spring mass system, it is known that natural frequency is given by, 69 70 co" = L (58) m1. where w is in radians per sec. In order to use this method, 0) must be known. From Equation(58), m1. = :32- (59) As k depends on the position P, even for the same beam, mr depends on P also. After mr is obtained, this mass is superposed on the concen- trated mass M at P. Then the frequency of the beam with the concentrated mass is given by k k 2_ _ __ p ’ (M+mr) ’ (M+ £2) (60) Example: Simply supported uniform beam with a mass at the center (Figure 15a). From elementary theory 1 _ PL3 ymax " 48EI Therefore, k = spring stiffness at center _ 48E‘I - T 4 E For this beam, of? = grab—II}— So, from Equation(59)., m1. = 0.492767 mb. Let a concentrated mass M be placed at the center and assume a = n:- ' b From Equation(60), 2 _ 431:1 ' P1 ‘ mbL3(o. + 0.492767mb) (61) Table 17 shows the values of p1 for six different values of o. and these values are compared with the values obtained from Equation(40)o with (3 = 0. 71 4.}, '8 _+_- .__.. 15. (a) W41 .1. L79 Kwufi LT». LI. LT». 6+ LT. rmlpl (d) (C) Mass arrangements for reduced mass system. Figure 15. 72 T able 17. First mode frequencies for a simply supported beam with a concentrated mass at the center, obtained by reduced mass method. (1 p1 from Equation(6 1) p, from Equation(40) Variation x EI x E1 in p. c. mL3 mL3 0.5 6.9534 6.9661 -0.182 1.0 5.6705 5.6797 -0.162 2.5 4.0048 4.0089 -0.102 5.0 2.9561 2.9580 -0.064 7.5 2.4506 2.4517 —0.044 10.0 2.1388 2.1396 -0.037 If there are n masses or the ntell mode is required, then the beam mass should be replaced by n masses. If there are concentrated masses, then these reduced masses should be placed on the concentrated masses, in order to keep the order of the matrix to a minimum. But if only the nF-hmode is required with lesser number of concentrated masses, then they may be placed anywhere. However, if placed on equal intervals, the computation becomes a little simpler. Example: As an example, the previous example is taken and the third mode is evaluated. (The second mode is same as an unloaded beam, since the rotatory inertia of the mass is neglected.) For this case, the beam mass is divided into three equal masses m at quarter points and r mid-point, as shown in Figure 15b. To find the frequency, the influence coefficient method is used. Denoting by 6nf the deflection at n produced by an unit force at f, the following relations may be obtained from‘any strength of materials book (see Reference 30). 73 611 = 633:18C 512: 621: 532: 523: 22C 613 = 631: 14C 622 - 32C L3 where c — _1536EI Considering A1, A2 and A3 as the amplitudes of the three locations and assuming harmonic oscillations, the following three equations are obtained. 3’ I - mrpzc[18A1 + 22.42 + 14A3] .3” I - mrp2c[22A1 + 32Az + 22A3] A, = mrp2c[14A, + ZZAZ + 18A3] Solving these equations give, )‘1 = 63.1126984, X2 = 4.0, A3 = 0.8873016 wh r )x‘ 1 e e — m ' Since the highest frequency is needed, 1 (k3L)3EI p2 :— : 3 mrk3c mbL from which, mr = 0. 2193994mb. Adding mass M of the concentrated mass to the middle reduced mass, gives ml=mr 0. 2193994mb m2 = mr + M = mb(a +0. 2193994) m3 = m1. 0. 2193994mb With these three new masses, the frequency equations become, 74 1A, = 3.9491892A1+ (220. + 4.8267868)A,_ + 3.0715916A3 1A, 2 4.8267868Al + (326 + 7.0207808)A2 + 4.8267868A3 (62) XA3 = 3.0715916A, + (226 + 4.8267868)Az + 3.9491892A3 The solution of these equations, with o. = 0. 5, give, 1,: 29.744403; 1, = 0.8775976, x, = 0.2971586 EI from which pg = 5168. 957 m . It is interesting to note that the result obtained from Equation(40) was pi = 5162. 10 fig— , a variation of only 0.102%. Considering pf, it is found that pf = 51.640 gig— , whereas, the value obtained from Equation b (40) was 48. 526mEIIJ . This variation is 6.4%, which is fairly high. b From this it is concluded that this method is accurate only for the particular mode for which the unloaded frequency is matched. Table 18 compares the results obtained from both the methods for different values of a. Table 18. Third mode frequencies for a simply supported beam with a concentrated mass at the center, obtained by reduced mass method. 0. p3 from Equation(62) p3 from E uation(40) Variation x _E_I_3_ xfifr in p.c. mbL mbL 0.5 71.895 I 71.848 0.065 1.0 68.459 67.932 0.776 2.5 65.627 64.599 . 1.591 5.0 64.497 63.244 1.981 7. 5 64.095 62.758 2.130 10.0 63.888 62.509 2. 206 75 Next, the third mode of an unsymmetrical case is investigated, as in Figure 15c. As in the previous case, mr = 0. 2193994 mb Adding the concentrated mass M to the left mr, the frequency equations are found to be 1A, = (180. + 3.9491892)A, + 4.8.267868A2 + 330715916143 XAZ = (22a + 4.8267868)A1 + 7.0207808Az + 4.8267868A3 (63) Substituting different values of 0., the following values of frequencies are obtained, as shown in Table 19, and these are compared with the values obtained from Equation (4 7). Table 19. Third mode frequencies for a simply supported uniform beam with a concentrated mass at quarter point, obtained by reduced mass method. (1 p3 from Equation(63) p3 from E uation (47) Variation mbL mbL 0.5 81.689 81.516 0.212 1.0 80.558 80.139 0.523 2.5 79.699 79.049 0.822 5.0 79.374 78.627 0.950 7.5 79.261 78.480 0.995 10.0 79.203 78.404 1.019 This method is helpful in finding the frequencies not only for masses placed at the positions of the reduced masses as explained above, but 76 also can be used for any other positions. In this case, the order of the matrix to be solved increases. This is illustrated by an example below. Example: Let there be a mass at quarter point of the beam as shown in Figure 15d, and assume that the first mode m1. is placed at the middle of the beam. For this case, mr = 0.492767mb. To find the first mode frequency, assume two lumped masses, rnr at middle of the beam and M at quarter point. Assuming stations (1) and (2), as shown in Figure 15d, AAI 18MA1 + zzmrAz (64) 1A, 22MA1 + 32mrAz As mentioned earlier, the order of the matrix has increased, in this case, to 2. If there were n masses and the first mode frequency is desired, then the order of the matrix will be (n + 1). Solving Equations (64) for different values of 0., two frequencies are obtained for each 0., the lower one being the first mode frequency. This way the values shown in Table 20 were obtained and they are compared with the corresponding values from Equation(47).. ‘ The same procedure may be followed for plates also. The natural frequency and spring stiffness of the plate may be measured or calculated, and from this the reduced mass value is obtained. Adding this reduced mass onto the concentrated mass, the loaded frequency can be easily evaluated. 77 Table 20. First mode frequencies of a simply supported beam with a concentrated mass at quarter point, obtained by reduced mass method. (1 p1 from Equation(64) p, from Equation(47) Variation x EI xf—E—T; in p.c. mbL3 mbL 0.5 8.031 7.991 0.501 1.0 6.889 6.851 0.555 2.5 5.109 5.087 0.432 5.0 3.851 3.840 0.286 7.5 3.216 3.210 0.187 10.0 2.818 2.814 0.142 CHAPTER VI SUMMAR Y AND C ONC LU SIONS Summary The effect of rotatory inertia of attached masses on. the vibration frequencies of beams and plates is analyzed by the method of normal mode superposition; this method is also known as the method of undetermined coefficients. General equations are derived. for variable thickness beams‘ and plates with arbitrary number of attached masses. In the numerical examples, only one or two masses are used and the resulting eigen value problem is solved by the use of digital computer. In the experimental part, only one mass was used. The frequency values were obtained experimentally with a pulsed-air vibrator. The theoretical and experimental values are compared and the variations between the two are discussed. In the latter part of this work, a method is developed by which the frequencies of systems, loaded with concentrated masses, can be pre- dicted by the knowledge of the unloaded frequencies of the system. The results from this method are compared with the results from other sections. Conclusions The normal mode superposition method seems to be very well suited for problems concerning vibrations of beams and plates. For uniform systems (beams of plates), the integrals can be evaluated quite easily but for systems with variable thicknesses, the use of the digital computer is essential. 78 79 The accuracy of the results depends mainly on the number of terms taken in the series expansion of the displacement function. It is apparent that when a load is added to a uniform beam, a certain number of terms of the series are needed for certain accuracy. The number of terms needed for the same accuracy will increase if rotatory inertia of the load is taken into account. If in addition to this, the beam happens to be of variable thickness or the uniform beam is replaced by a uniform plate, a still higher number of terms in the series expansion will be necessary. As may be seen, for a plate of variable thickness with attached mass, the number of terms required will be very high. Since extreme accuracy is not the primary object of this investigation, only nine terms were taken. The extrapolation formula used seems to improve the values reasonably well, at least for the higher modes. However, there is some reservation in the mind of the author at using this formula. A point of uncertainty is the way the shear deformation term is introduced. The transformation of the terms of the differential equation to energy forms was necessitated by the requirement of introduction of the loads and rotatory inertia of the loads. Since the shear deformation term introduces very little correction, especially for the lower modes, the effect of error in the assumption for shear deformation will not effect the final results much. Actually it seems to improve the results as expected. Further study in this area seems to be in order. When this work was started, it was assumed that normal functions are the best functions in terms of which the deflections may be represented. But as seen from the wedge problem in Appendix E, it may be concluded to be erroneous. Also, normal functions are difficult to handle, particularly for clamped and free edge conditions. This suggests the necessity of investigating other functions, mainly polynomials, which can be used for problems of this type. This will reduce the amount of time required for numerical calculations . 80 From the curves of frequency vs (3 with o. as parameter, it looks, as if, with higher (3, the frequency remains constant. When 8 is high, however, it means that physically the mass or masses do not rotate. This may be used as a means of application of bending moment or other constraints for further investigation. The same conclusions can be drawn for higher a. In this case, the translatory motion of the mass is reduced. The results presented here from theory seem . to compare quite well with experimental values. To improve the values, two aspects need considerable attention. The most important factor is to use higher number of terms in the theoretical calculations. Even with the best extrapolation formulae, it is not always possible to get very good results. The second part that needs attention is the support conditions in the experimental set- up. The simple. support for the beam was fairly good with careful lubri— cation of the pins and occasional checking of the support screws. But the cantilever support was definitely weak as shown by the results of natural frequencies. The worst case of support was found in the case of plate. Some other design of support seems essential for better verification of the theory. In the reduced mass method, some of the results seem very good whereas some are not so good. Even though the variations are within 2. 5 p.c. , this may perhaps be improved. The only reason that can be offered for discrepancies is error in numerical calculations. Further investigation in this line seems advisable. The following are a few of the items that can be suggested for future investigations. They are given separately for beams, plates, etc. Beams , (1) Variation of cross-sections other than assumed here. If stepped beam is used, the integrations will contain limit points other than from Oto 1. 81 (2) Include the neglected terms for shear deformation. Also a better relation for shear deformation may, perhaps, be derived. One helpful suggestion is to use the normal functions given by T. C. Huang (31). (3) The nature of constraints that can be incorporated by proper selections of a and B quantities, as mentioned earlier in this chapter. (4) Addition of work done by W. F. 2. Lee and E. Saibel (12) to the present work, which opens up a whole variety of problems that can be solved quite easily. This may include the cases of continuous beams, sprung masses, elastic foundation, etc. (5) The effect of stretching of the center line of the beam due to addition of load. (6) Study of visco-elastic beams. (7) Use of finite difference for this type of problems. (8) Forced vibration. Plates (1) Inclusion of shear and rotatory inertia of plate. Procedure outlined in (32) may be helpful in this respect. (2) Use of other end conditions than simple supports. (3) Continuous plates. (4) Use of finite difference. (5) Effect of stretching of the middle plane due to application of loads. (6) Study of viscoelastic plates. (7) Forced vibration. Reduc ed mas s (1) To include rotatory inertia effects of loads. (2) Approximate non-linear behaviour of beams or plates, withload attached. Since the method reduces the whole system to a spring mass system, this study seems possible. 82 In the experimental part, subharmonics and ultraharmonics were observed in almost all cases. One important difference between the two was noticed. Assuming p to be the main frequency, the subharmonics of p had the frequency of p, as measured from the oscilloscope trace, but ultraharmonics had the frequency of the ultraharmonic itself. And this created a little confusion at the early part of the experiment. Comparing with the natural frequency results of B. B. Raju (19) it is observed that the finite difference results gave a lower bound whereas, this method gave an upper bound to.the actual frequency values. The explanation for the latter case seems to follow from Rayleigh's principle but the reason for the lower bound in the former case is not clear. All the same, further investigation should be carried out to establish the validity of this observation. 10. ll. BIBLIOGRAPHY . Davies, R. M. , "The Frequency of Transverse Vibration of a Loaded Fixed-Free Bar, " Philosophical Magazine XXII, 1936, p. 892. . Davies, R. M. , "The Frequency of Transverse Vibration of a Loaded Fixed-Free Bar - II. The Effect of the Rotatory Inertia of the Load." Philosophical Magazine, XXIII, 1937, p. 464. . Davies, R. M. , "The Frequency of Transverse Vibration of a Loaded Fixed-Free Bar - III. The Effect of the Rotatory Inertia of the Bar." Philosophical Magazine XXIII, 1937, p. 563. . Davies, R. M. , "The Frequency of Transverse Vibration of a Loaded Fixed-Free Bar - IV. The Effect of Shearing of the Bar, " Philosophical Magazine XXIII, 1937, p. 1129. . Scanlan, R. H. . "A Note on Transverse Bending of Beams Having Both Translating and Rotating Mass Elements, " Journal of Aeronautical Sciences, July 1948, p. 425. . Scanlan, R. H. and Rosenbaum, R. , Introduction to the Study of Aircraft Vibration and Flutter. Macmillan Co. , New York. . Fettis, H. E. , "Effect of Rotatory Inertia on Higher Modes of Vibration. " Journal of the Aeronautical Sciences, July 1949, p. 445. . Hearmon, R. F. S. and Adams, E. H., "The Flexural Vibrations of an End Loaded Vertical Strip. '1 British Journal of Applied Physics, 6, 8, pp. 280-284, August 1955. . Horvay, G. and Ormondroyd, J. "Static and Dynamic Spring Constants. ” ASME Trans., 1943, p. 220. Horvay G. and Ormondroyd, J. , "Appropriate Lumped Constants of Vibrating Shaft Systems. " ASME Transactions 1943, p.A-220. Young, D. , "Vibration of a Beam with Concentrated Mass, Spring and Dashpot. " Journal of Applied Mechanics, 1948, p. 65. 83 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 84 Lee, W. F. Z. and Saibel, E. , "Free and Forced Vibrations of Constrained Beams and Plates. " Technical Report, Carnegie Institute of Technology, Department of Mathematics, Pittsburgh 13, Penn. Hoppman, W. H. , "Forced Lateral Vibration of Beam Carrying Concentrated Mass. " Journal of Applied Mechanics, 1952, p. 301. Godzevich, I. N. "The Determination of the Natural Frequency of Oscillation of Straight Linear Bars of Variable Sections with Distributed and Concentrated Mass (in Russian). " Trudi. Ural'sk. politekh. in-ta S654, 126-132, 1955. Bronskii, A. P., "Determination of Natural Frequencies in Transverse Vibration of Beams Carrying Concentrated Loads (in Russian). " Uch. Zap. Mask. Gov. Ped. In-ta 49, 135-138, 1956. Cox, H. L. , "Vibration of Axially Loaded Beams Carrying Distributed Masses. " Journal of the Accoustical Society of America, 30, 6, pp. 568-571, June 1958. Prescott, J. , Applied Elasticity, Dover Publications, Inc. , New York. Warburton, G. B. , "The Vibration of Rectangular Plates. " Proceedings of the Institute of Mechanical Engineers, 168, 12, pp. 371-381, 1954. Raju, B. B. , " Bending and Vibration in Plates of Variable Thickness. " Thesis, Michigan State University, East Lansing, Michigan. Roberson, R. E. , "Transverse Vibration of a Free Circular Plate Carrying Concentrated Mass. " Journal of Applied Mechanics, 1951, p. 280. Roberson, R. E. , "Vibration of a Clamped Circular Plate Carrying a Concentrated Mass. " Journal of Applied Mechanics, 1951, p. 349. Mindlin, R. D. , "Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates. " Journal of Applied Mechanics, 1951, pp. 31-38. Timoshenko, S. Vibration Problems in Engineering. D. Van Nostrand Co.‘Inc., Princeton, N. Y. 24. 25. 26. 27. 28. 29. 30. 31. 32. 85 Kirchoff, G. R. , "Ueber die Transversalschwingungen eines Stabes von veranderlichen Querschnitt. " Berliner Monatsberichte, Jahrgang 1879. PP. 815-828. Kunz, K. S. , Numerical Analysis. McGraw-Hill Book Co. Inc., New York, 1957 . Young, D. and Felgar, R. P. "Tables of Characteristic Functions Representing Normal Modes of Vibration of a Beam. " Published by the University of Texas, Austin. Mindlin, R. D. and Deresiewicz, H., "Tirnoshenko's Shear Coefficient for Flexural Vibrations of Beams. " Proceedings of the Second U. S. National Congress of Applied Mechanics. Published by ASME, 1954, p. 175. Hansen, J. , Warlow-Davies, E. and Taylor, J. , "Model Experiments on the Effect of Wing Engines on the Natural Frequencies and Modes of Vibration of Wings. " British Royal Aircraft Establishment“, AD 3120 MT5633, April 1939. Ellington, J. P. , "The Vibration of Segmented Beams. " British Journal of Applied Physics, 7, 8, August 1956, p. 299. Marin, J. and Sauer, J. A. Strength of Materials. The Macmillan Company, New York. Huang, T. C. , "The Effect of Rotatory Inertia and of Shear Defamation on the Frequency and Normal Mode Equations of Uniform Beams with Simple End Conditions. " Journal of Applied Mechanics, paper no. 61, APM-25. Huang, T. C. , "Application of Variational Methods to the Vibration of Plates including Rotatory Inertia and Shear. " Developments in Mechanics, Proceedings of the Seventh Midwestern Mechanics Conference, Distributed by Plenum Press, New York, pp. 61-72. Edited by J. E. Lay and L. E. Malvern. APPENDICES 86 '0‘;'.. f _.'L. Afiwn . APPENDIX A WORK DONE BY ROTATORY INERTIA OF ATTACHED MASSES ON PLATE To include the effect of rotatory inertia of the masses, it should be borne in mind that the masses will, in general, rotate about both x axis as well as y axis and the net effect will be rotation about some other intermediate axis in the x, y plane. To appreciate this physically, consider a right handed Cartesian coordinate system and let a body of mass m be located at point P at a distance ‘- ’ —' ” L below the x, y plane i. e. in z direction. When the plate vibrates, there will be inertia forces generated due to inertia of the mass and this in turn will produce a torque Tn' Let its components be Tx and Ty in the Figure (a) x and y directions respectively. From elementary theory, 030 From Figure (a), it is clear that due to bending in the x direction, Ty will produce work given by - Ty _TBW and due to bending in the y direction, Tx will produce work equal to Tx TW. In Figure (a), the point P moves 6w W . to Q due to Ty— and then from Q to R due to - Tc- . Since both motions are present at the same time and simple harmonic motion is assumed, the mass will move parallel to PR. It is parallel, because the point P will also be moving up and down. Since the force field, generated by the inertia force, is conservative (no damping assumed), the total work . . aw 5w . done by the torques Ty and Tx in mov1ng thru' - W and 17 is same as the work done by the torque Tn in moving the point P thru' PR. 87 88 (An alternative proof of this is given in the next paragraph. *) This gives . . . . W 6W a Simple relation for the work done by the inertia forces as Tx(37) £1.19? ). In this analysis, it was assumed that TX and TV 0 are constant and 6W 5W as small which 2: 7532' ‘8; a is valid for virtual work principle. Tn For a mathematical proof, the following n may be considered. 17 Referring to Figure (b), let there be two Y torques Tn and Tt acting in two normal (Figure (b) directions at point 0. If two variations are given in the n and t directions i. e. 6 Egg) and 6 (-'%%v)’ then the work done by the torques Tn and Tt is 0w 5W r Tn‘S‘TT) ' TEN—6?) Tx Cos 0 + Ty Sine U But T n Tt= - TxSin0+TyCos0 ow ow 0w . m - —B—X COS 9W 51n9 0W: _ awsme + 0w C086 at X By Substituting these in the above expression for Ur and simplifying, the same result is obtained e. g. , Ursz6(—%—V;V) - gap—gig) X Ty—N . Inertia Torque—«g? Inertia force -T){'_—." 3' Figure (c) :1: The writer is indebted to Dr. W. A. Bradley, Professor of Applied MechanicsDepartment, M.S.U. , for this proof. 89 As regards the signs of Tx and Ty, consider Figure (c). Assume that a mass is attached at a certain distance below point P. When the plate vibrates, for positive w, the inertia force will be directed as shown in Figure (c). This force will create a torque as indicated. The com- ponents of this torque in the x and y directions have opposite signs. From this it is apparent that both the terms in Equation (14) have negative signs. APPENDIX B VERIFICATION OF PLATE EQUATION BY RITZ METHOD The Ritz method is used in this section to evaluate Equation (12). In this case, the expression for potential energy remains the same as in Equation (4). The kinetic energy of the vibrating plate is given by ”_"H h (—5_t )zdx dy — .9.— Z °° °° 2 2 - 2 p H h (mg 112;, AmnXmYn) Cos pt dx dy Let p If h(m 7: n2 AmnXmYnV' Coszpt dx dy = Q (13-1) =1 Then T: p Q-Z- Equating the maximum values of V and T, 2 __ 2 Vmax (B-2) p _. Qmax Applying Ritz method, z obp z 0 Aij 6 > 5 _ . or 2 Qmax aAij Vmax ‘ 2 Vmax aAij anax _ 0 (Since Clniax 1: (B-3) p2 But from Equation B-2, Vmax = 3- Qmax’ Substituting in Equation (B-3), 5 2. 5 _ 2 9’43 Vmax " p 3A“ Qmax " b 2 . or [2 V - p ] = 0 (B-4) aAij max Qmax 90 0) 91 Substitution of V from Equation (8) and anax from Equation (B-l) into Equation B 4 and performing the indicated differentiation results in m (D d_2_TXm dSXi a) (D + (mg-=1 nz=1Amnddx§ ) (deii_ a} an dZX ydzy- V{(mz:'1n2:1 Arnn—d—T men) (XIT7L) +(mzlnglAmndedY—41—HdzxiYj)} + kuz: A de dYn l X. 1n=1 mn dx dy d1 de )‘dx dy) 11‘1“” Z ”h(g g A x Y)(X-Y)dd pp m=1n=1mn n 1.1 xy which is the same as Equation(12). APPENDIX C WORK DONE BY ROTAT ORY INERTIA OF ATTACHED MASSES ON BEAM To find the virtual work done by the rotatory «mm-w- inertia of the masses, an arbitrary mass M is taken as shown in Figure (a). Considering an r element of the mass at a distance r) from the V x axis gives Figure (a) d(Torque) = d (Volume)px(-Acceleration) x r) where p = mass density. If the mass rotates through an angle 0 = -§—: about the z axis, then d(Torque) = -(<11V)p(é 77171 _ 03 z --pxt an 3 virtual work done = - pgbggtz dV 772 6 (121‘) 3 Total virtual work done = - IV p bxytz n2 6 (12!: )dV. Considering only rigid mass and uniform density, it is found that 63 b . . . . . p, 53?th and 6 (132‘) is independent of the integration variable. As such, the total work done is given by ”059??? «Hg-E) Iv n’dv = “15¢th 5 (.631) where I = 0 IV 19" dV is the mass moment of inertia. about x axis. 92 APPENDIX D WORK DONE BY ROTATORY INERTIA AND SHEAR DEFORMATION IN BEAM Rotatory Ine rtia To find the virtual work done by the rotatory inertia of the beam, an elemental volume dV at a distance 17 from the bent neutral line of 3 the beam is taken (Figure (a)) Figure (a) d(mass) = pdV = pbd ndx where b = width of beam at x and in °. d(force) = -bp(8n )dn dx d(torque) = -bp8 n‘zdndx. Torque = -p [Aw n‘zbdn)dx = -péibdx (11-1) the virtual work done by the section dx is h d(work) = - pédx [ I: b1’).zdn5(1§')xc )] -7- .-. - pé dx 1b 6 (%) = -pr3—§:'35th d" 5 811’ L .. total virtual work done = - p f0 Ibde6 (g9 Shear Considering an element as in Figure (b) it n 19‘ 11+ 4* L I is seen that in order to include the effects .4 «L‘A‘ffifi‘vd‘ of rotatory inertia and shear deformation of Figure (b) the beam, the fOIIOWing relations must be satisfied. 93 94 6M 8 6" -—5—X + TX (9A) - 1130-535 (D-Z) 6 6’ ‘81“! A) = M 8% where M = Bending moment across a section = - El 6? . b 5x q = Average shear = k'GB = EGGS-)2: - 1y) . Y’= Slope of center line without shear _ _ o 13 — Shear angle at the center - 33E - 1y Substituting the respective values in Equations (D-Z) gives E83215?!) + “8:3 Ibpfi 03-3) 2 Ak'G(%§3§ - 793,5) + k'GB-%‘—A‘= Ar§~§ For a uniform beam, 1;! can be eliminated between these two equations and a single equation may be obtained for y. But when the beam is of variable cross-section, it is difficult to eliminate ‘1’- In the case of the cantilever beam under investigation in which the shear deformation has been included 5A 1T 1T x x A=—(2-—>z , —3—=-—(2-—) 16 L x 8L L (D-4) _ 17 x 4 OI _ _ 11' -35 3 I ‘ 1024 (2 L) ’ 5x ’ 256L (2 L) From Equation (D-4i) it is seen that when L = 20" 6A ('3? max - -0.0393 03-5) 91 = -0.00491 (Bar’s... 95 Since this investigation concerns itself only with modes up to the fourth, the shear angle 8 at the center may be assumed to be small. As such, the term k'Gfi %-g- in the second equation of (D-3) may be neglected in comparison to the first term. This gives by day p 52’}; : - __ D- x 5x2 k'G t2 ( 6) Differentiating the first of Equation (D-3) with respect to x and substituting g—‘f from Equation (D-6), the final form of the equation reduces to E69408?) + 1.3,, -ps;<1b§£%.z>- Es—izszo 8% D-7 + on. 52733,. o4 = 1 1 p3; Btz k'G t It is shown in (23) that the last term is of second order compared to the third and fourth terms. Also, for the lower modes, it is reasonable to 2 assume the fifth term to be small since gig: is small and 3??? should not be too large. Neglecting these two terms, Equation (D-7) reduces to 2 2 z 3 2 2 Efirubifiwgi- p53“. 3. >-§6§sub%¥>=o (D-8) In Equation(D.-8) the third term is the first spatial derivative of the rotatory inertia torque, as may be seen from Equation (D-l). The first term also can be shown to be the first spatial derivative of the elasticity torque (this term is used here to keep the same notation) as follows . 6 U6 = I:(Torque) { 6 (angle)}dx Efl‘ab;(1b§;§) o (18me E1 Lang-:15 )Eumdx 96 Integrating by parts , 2 5Ue=E[Ib%—§5;(5Y)]o Lo-Ef L1b(%¥)5—T(5Y)dx For any standard end conditions, the first term is zero, because for a fixed end '36; (5 Y) = 2 for a simple support 3575 = 0 6" _ for a free end 3:5 — 0 Therefore (25y 52 6Ue=-EfoIb (F)3;f(6y)dx =-E{f:"1b( ndan)ddX; dx)Sin7‘pt 6A1 (D-9) where the series expansion of y and 6y. are substitutederom Chapter II, Part (b). As may be seen, Equation (D-9) is the same as Equation (18). It is difficult to show that the second term is the spatial derivative of the inertia torque for a variable cross-section beam. This appears to be due to the neglected term in the second equation of Equations (D-3). An alternate way of defining the shear angle 0 might remove this difficulty. In the case of a uniform beam, A is constant. Expanding y in terms of an infinite series of normal functions of the beam, it can be shown easily that this represents the inertia torque. This follows easily from the relations of the type gag-i2- = kLXn. From these analogies, it is reasonable to assume that the fourth term is the spatial derivative of the shear torque. With this assumption, the work done by the shear deformation is given by 2 bush: k,"GfoL59—(1b%-t§)o (gig) dx (11-10) 97 Since no example was found for the shear correction on a variable thick- ness beam, the case of an uniform simply supported beam is taken to check the validity of the above assumptions. This case is treated in (23). For this case, Equation(26) reduces to L a) L OD an dX' 7‘ Z . _ 1 “DA f0 (n21 Anxn)x1dx J", “b Io (IE—1A“ dx) dx ‘1" E L 00 an dXi + plbk'G‘ro (IE-=1 n dx ’ dx ‘1" L dSX dX‘Z =E1b{f (; An x—z") 1dx} (1)-11> o n—l But Xn= Sin “5" , L L , f0 Xindx = -2- for 1 =1 = 0 fori=1=J deX1dX _17-7- for,_ 0 dx dx _ 2L 1“] =0 fori#J and I-‘dei dxz ' i4 “4 . . f0 _xz_d -d—x-}- dx- -—2—3L ‘ fOI‘I-J =0 fori#j Substituting these in Equation (D-ll), results in 1211.2 izflz 1411.4 P zi[PA(A —) + pr(Ai"—' 2L _) + p1b_%(Ai—2L)] = EIb(Ai—3'—) (D- 12) Assuming pAL = mass of the beam = mb x = l.”- 1 a2 ._. 2111.1. mb and Ipr = IbI'Xb = yzmb 98 and substituting in Equation (D-lZ) gives 1 "4&2 Z 22 7 1 — X “True it” Considering the last two terms as small compared to unity, the denominator can be expanded in a binomial series. Neglecting all higher order terms, it is found that Tl’za 1 “272 E p‘ 12 [1'3 1 (H’k'o )] (13-13) I which is the same as Equation (140) of (23). APPENDIX E VIBRATION OF A WEDGE This problem was investigated with a view toward checking the convergence of the series of normal functions for a uniform cantilever beam. The problem was investigated by G. R. Kirchoff (24) who obtained an exact solution, neglecting rotatory inertia and shear deforma- tion of the wedge. For this case, Equation (20) {m L __ ’ rFigure (a) represents the frequency equation. From Figure (a) Zb(L-x) 2b3 3 - —— Ib = -— (l - --) Xn = Cos h knx - Cos knx - o. n(Sin3 h knx - Sin knx) The values of knx, on were obtained from (26). i.t = e de ‘1 dflcnadp ¢n With these notations, Equation (20) reduces to p2 [L‘A.(§5 11 ) -d - Ich I z A.nk " "d ,E 1 p0 mfl n¢n¢,x— o h(n _1 nflqfi¢ x (-) where prime represents differentiation with respect to (knx). Assuming only one term of the series in Equation E- l, Zksz . ,_ L [fa-{Mum = 3%?”- Iou-{dex mm The following are the values of these integrals: 99 100 0.19346191 1 x X [0(1 - ‘1'.) («>02 ‘1‘?) 1 3‘. Z n 2 i IOU - L) (01 )d(L) 0.5791407 Substitution in Equation E-Z gives b - p = 6.0833815 —7 I E L —-—3p which is a variation of about 14.4 p. c. to that of (24). It may be recalled that by assuming another series and applying Ritz method, the value obtained in (23) by one term approximation was 5.48. This is a variation of only 3. l p. c. This shows that the use of functions, other than normal functions, may sometimes be profitable but this needs judgment on the part of the investigator. As the normal functions are standard functions, they can be used more effectively, if suitable tables can be prepared for different types of integrals of these functions. To check the convergence, two terms of the series in Equation E-l were taken. The value obtained for the first mode frequency was 5.434991 b . . 2n L3 13:3 , a variation of only 2. 26 p. c. The problem was further investigated with eight terms of the series and the following results were obtained. The exact frequency value for the fundamental mode is 5°2i15 E;- I E 30 First mode frequencyL—— +2165 'E‘; J— fp Second mode frequency'-—--—--152'-—--3—--17 31):: I é); Third mode frequency 31 _—_1r:-—Z:1I!:3".3—p—E The exact solution results for the remaining two modes were not available but they are listed here as reference. ROGM USE OMLY lllfllflll‘LHlfllHlflHlllfl 30 3 1293