FREE VIBRATION OF A RECTANGULAR PLATE WITH DAMPING CONSIDERED*

Transcription

1 361 FREE VIBRATION OF A RECTANGULAR PLATE WITH DAMPING CONSIDERED* BY MILOMIR M. STANISIC Armour Research Foundation of Illinois Institute of Technology Introduction. This paper presents a method for calculating the natural frequencies of the normal modes of free vibration of a rectangular plate, fixed along each edge, with arbitrary shape ratio. Viscous damping of the plate material is considered. Generally speaking, the objects of this paper are (a) to present a simple and practical solution for the stated problem, (b) to develop this solution in closed form so that a designer can evaluate the natural frequencies of a clamped plate of any shape ratio in a relatively short time, and (c) to make possible the determination of the damping coefficient by comparing experimental and theoretical values. The theory developed is subjected to the following restrictions: (1) the plate is composed of a material which follows Hooke's law; (2) the deflection of the plate is small compared to its thickness; (3) the thickness of the plate is small compared to its lateral dimensions. The method which is used to solve this problem is that of Galerkin1 [1] and belongs to the same general class as those of Rayleigh and Ritz. This method can be used (a) to determine an approximate solution of a differential equation with given boundary conditions admitting only the functions which satisfy the prescribed boundary condition exactly, and (b) to treat the problems which belong to non-conservative systems. This method was described by E. P. Grossman [2] and also by W. J. Duncan [3], [4]. The degree of accuracy expected can be increased by increasing the number of independent functions which are used in the solution. W. J. Duncan has shown in his papers that when the functions are well chosen an excellent approximation can be obtained by use of a very small number of admissible functions. Since our system is non-conservative this method is most convenient for the solution of the problem. Solution of the problem. If damping forces are proportional to velocity, the motion of the plate is governed by the following partial differential equation: V4w(x, y, t) + wtl(x, y, t) = w,(x, y, t), (1) where w(x, y, t) is the transverse deflection of the plate, k is the damping coefficient, h is thickness of the plate, p is the mass density of the plate material and D = Eh3/12 (1 v2) is the flexural rigidity of the plate. For a uniform plate vibrating harmonically with an amplitude <f>(x, y) we can write w(x, y, t) = <j>(x, y) exp ( at) cos wt, (2) *Received Dec. 16, Presented at Eighth International Congress on Theoretical and Applied Mechanics, August 2th to 28th, 1952, Istanbul, Turkey. 'Numbers in square brackets refer to the Bibliography at the end of the paper.

2 362 MILOMIR M. STANISIC [Vol. XII, No. 4 where co is the angular frequency. Equations (1) and (2) lead to VV(i, y) cos ut + <t>(x, y)< 77 ph v.«, 2 co ; «7^j k i cos cot D K ' D + I ^2 2 7T jy a aco co sin co^ =. (3) Since Eq. (3) must be satisfied for all values of t we obtain (w/d) (2pha k) =. Therefore Now, Eq. (3) becomes * " 27h' (4) VV(x, 2/) ~ ^ (a2 CO2) J<Kz, 2/) = or where VV(z, 2/) - X</)(x, y) =, (5) (6) Fig. 1. Coordinate System for the Plate Assume the solution of Eq. (5) to be given in the form <*>, j)=ei ar.xry. (r, s = 1, 2, 3, ), (7) r «* 1 a " 1 in which the X, and Y, are admissible functions that satisfy only geometrical boundary conditions, but need not satisfy any "natural boundary conditions", and ar, are the amplitude coefficients. Since the function 4>(x, y) is an approximate solution of the problem, then the error caused in the solution is of the magnitude m n «= E ar.[v4(xrf,) - \XrY.]- (8) r = 1 «= 1 The criterion that the approximation is best if e tends to zero requires, as stated by Galerkin, that the integrals J = I f f" exvyq dx dy I J (P, Q = 1, 2, 3, ) (9)

3 1955] FREE VIBRATION OF A RECTANGULAR PLATE 363 be a minimum. The lateral dimensions of the plate are a and b. Equations (8) and (9) lead to E E or. [' f [v4(xrys) - xxry.]a\y, dx ay = o (io) r = ] s = l J J which can be written in the form m n /% a E Ear. r-1 s = l J ' nb [Xr.»«F, + 2 Xr. Y,. + XrY.,vyvv - \XrY.]X,Y. dx dy =. (11) Equation (11) represents a system of linear homogeneous equations in the unknown coefficients a,,. The natural frequencies X,, X2, are determined from the condition that the determinant of the system must vanish. Let the appropriate characteristic functions Xr, Y be given, following Young [5], in the form = (cosh M.j cos ii, ^ p,(s'mh sin n, jj, (12) such that if v = r, = Xr, = x, I = a; if v = s, ^. = y., = y, I = b. The function > ( ) has to satisfy the following boundary conditions:.() = -9.(1) = *,{() = *, ( =. (13) Then cosh n, cos 1 =, (14) cosh p. - cos ix. srnh ii, sm n. Pi l The values of /3» and n, are presented in Table I. Since the characteristic functions are TABLE I Values of and n , , , , v > 6 1. (2v + 1) r/2 (2v + l)1 i4/16

4 364 MILOMIR M. STAXISlC [Vol. XII, No. 4 of the form of Eq. (12), we obtain 4 4 v Y V til v Hfil r j J n,yyyy })* \xyj/ and a a (l if V = r j X Xr dx = a&l, where 5; = < (17) ' if p ^ r Using the abbreviation knowing that and using boundary rb (l if q = s Y,Y. dy = bs"q, where a; = < (18) to if q ^ s. Hpr = o f XJZT, dx, [ \o ~ f J condition Eq. (13), we have H = 6 f" Y,Y,. dy, [ *,*,-, { d = - f i d. (19) TABLE II Values of I <frit \ Numerical values for the integrals, Eq. (19), are presented in Table II. Hence IIVT = -a P Xv,,Xr., dx, (2) II = -b r Y yy.,vdy. (21)

5 1955] FRER VIBRATION OF A RECTANGULAR PLATE 365 Now Eq. (II) becomes E E *r.\ r r lxjcr. Y,Y. + xjcry,y tm ra -I l a. = l 1 I WO.7 n J r, or + 2XvXr.IXY,Y..vy) dx dy - X f XrXpY,Y, dx dyf = E E a J f ^.Y,X, rfx f Y,Y. dy+f X Xr dx f ft F, F, r-1»-l WO & J ^ + 2 f X,Xr. dx f F,F.. dy - X f XBXr cfe f" YqY. dy\ =. (22) '(» ^ 'o»'o J After substitution of the corresponding values, and multiplication of each term by a we obtain where E E ^ T 1 «= 1 Equation (23) may be rewritten as J + fl)a36c' + 2 ^ //_//,] - Xo'bC'} =, (23) Si"' = f" XvXrdx f Y.Y. dy. (24)»' in which m n E E <*r r 1 8 = 1 ~m! 4* A5;r'!> =, (25) (26) Let Then Eq. (25) becomes " fb m: ic + 2 = h,ji. (27) where and <j (ra) dd(/ = ZiuC-AC']^, (28)» = I.1 = 1 11 for p = r and q = s (. if either p ^ r or q 9* s ic' = ^: + (JUl + 2 j HrrH for p = r and 9 = s, = 2 t HDrHq, if either or q ^ s. (29)

6 366 MILOMIR M. STAXISIC [Vol. XII. Xo. 4 Equation (28) is a system of mn linear homogeneous equations in mn unknowns ar, {r 1,2, m; 8 = 1,2, n). Since ars cannot all be zero, the determinant of the system must he zero. This leads, in general, to an algebraic equation of degree mn in A. From Eq. (26) it follows that A D _ ( k \ phab \2phJ Then for any shape ratio <x = h/a, one obtains If k =, then = _(a) L phaa* \2phJ _ (3) (31) A = (32) which is the eigenvalue parameter for the plate with fixed boundary conditions. From Eq. (3) it can be seen that the frequency decreases with an increase in the value of the damping factor k. The zero value of u is obtained for 2 / A\I/2 =? {»hd V <33> The frequencies are calculated for the first three modes of a square plate fixed along each edge, and the results are presented in Table IV. Coefficients for Vibration TABLE III of Damped Square Plate T S E(r<* ' EE (r«) E(r») j(ra) ^(r«) , , , , TABLE IV The Values of A1/2 of Vibration of Square Plate Fixed Along Each Edge r ad a l plica* \2 (k Yn,/2 \2phJ _ Mode 1st 2nd 3rd A1' (D.Young) (35.99) (73.41) (18.27) Table III represents the coefficients for the vibration of a clamped square plate. Using only the first two terms of the series for E^", we obtain approximations for

7 1955] FREE VIBRATION OF A RECTANGULAR PLATE 367 A1/2 which are compared with the D. Young solution [5] (given in parenthesis in Table IV). It can be seen that a good agreement between the results is obtained. 4-S--S / V77777'/ '>77777' / rh// 1st 2nd 3rd Fig. 2. Nodal Line for Square Plate Conclusion. The problem of free vibrations of a rectangular plate fixed along each edge and having internal viscous damping is presented and solved by means of generalized Galerkin's method. By choosing the function ^( ) to satisfy the boundary conditions, Galerkin's method can be applied to a plate with any type of support, even when damping forces are present. The influence of the damping factor on the natural frequency is given by Eq. (3). It can be seen that the natural frequency decreases with increase in the fc-values. Bibliography 1. V. G. Galerkin, Rods and -plates, Vestnik Ingeneroff, 1915, p E. P. Grossman, Oscillations of the tail units of aeroplanes, Transactions of the Central Aero-Hydrodynamical Institute, No W. J. Duncan, Galerkin's method in mechanics and differential equations, Aeronautical Research Committee, Rep. & Memo. No (1937) 4. W. J. Duncan, The principle of the Galerkin method, Aeronautical Research Committee, Rep. & Memo. No (1938) o "D. Young, Vibration of rectangular plates by the Ritz method, J. Appl. Mech. 17, 448 (195)