- Email: [email protected]
Nonlinear free bending vibrations of plates
Compufers & Strucrures Vol. 21, No. 5, pp. 887-891. Printed in Great Britain.
NONLINEAR
1985 0
FREE BENDING PLATES
VIBRATIONS
A. OVUNC University of Southwestern Louisiana, U.S.A.
0045-7949/85 $3.00 + .oo 1985 Pergamon Press Ltd.
OF
BULENT
Department
of Civil Engineering,
(Received
16 January
Louisiana, Lafayette,
1984)
Abstract-A general solution for the Helmholtz differential equations is obtained in the complex domain and applied to the nonlinear, free, bending vibrations of plates. The analysis is based on the decoupled nonlinear von Karman field equations by Berger assumption for the large deformations of plates. The decoupled differential equation in terms of the deflection function is a fourth order Helmholtz differential equation. Its solution, called the dynamic deflection function, is obtained in the complex domain by means of newly defined first and second kind and modified Bessel functions. The dynamic deflection function can be applied to any plates having any shape and any boundary condition under any arbitrary dynamic loads. For plates with smooth boundary, the parameters of the dynamic deflection function are determined from the boundary conditions of the plates and the initial conditions of the vibrations. The analyses of plates with piece-wise smooth boundaries are obtained on the mapped planes. The nonlinear, free vibration of circular plates are investigated by the dynamic deflection function. The effect of stretching on the natural circular frequencies are illustrated. INTRODUCTION
A general dynamic deflection function is obtained for the decoupled von Karman field equations for large deformations. The decoupling of the differential equations is based on the Berger assumption. The refinements to include the effect of shear and rotatory inertia have been introduced in the bending vibration of plates [8, 11, 241. A theory for the flexural motions of elastic plates has been developed by taking into account the effect of transverse normal strains and transverse normal stresses, together with rotatory inertia and shear [26], by assuming the form used in the static theory of bending [7,22]. The nonlinear vibration of thin, elastic, isotropic plates has been investigated by many authors using von Karman field equations [5, 281. A first attempt has been made by Berger to decouple the von Karman equations [41. The coupled von Karman equations have been substituted for by a simple set of coupled nonlinear equations [ 13, 271. The Berger approximation has been used to derive the equations of motion for rectangular plates with a circular cutout 1211. Based on an averaging technique to satisfy in-plane boundary conditions, as in [19], and also the von Karman field equations, the effect of shear deformation and rotatory inertia on the large amplitude vibration of simple supported rectangular plates has been studied by using Runga Kutta technique to integrate the differential equations [17]. Several authors have questioned the validity of the Berger assumption from the point of view of accuracy [14, 19, 251. Using certain well-known results from the two-dimensional theory of elasticity, a plausible explanation for the origin of the Berger approximation has been suggested, and arguments have been raised to show further that other Bergerlike approximations can be developed [201. An anal-
ysis of the large amplitude vibrations of non-uniform rectangular plates has been presented by neglecting the second invariant of the middle surface strain in the expression of the total potential energy of the plates [ 1,2, 181. The complex variable theory has been applied to the linear vibration of thin, elastic plates [lo, 12, 15, 231. Following the Berger approximation, large amplitude vibrations of irregular plates have been investigated by means of a conformal mapping technique [3, 6, lo]. Herein, the general solution, as the dynamic stress and dynamic displacement functions, derived for the in-plane and bending vibrations of linear, elastic, thin plates [15, 161, is extended to the nonlinear vibrations of plates. The differential equation, in terms of deflection obtained by decoupling the von Karman field equations by means of Berger assumption, is a fourth order Helmholtz differential equation. The general solution of the Helmholtz differential equation for deflection constitutes the dynamic deflection function. The dynamic deflection function is expressed in the complex domain, in series form, in terms of newly defined first and second kind Bessel functions. The unknown coefficients and the parameters in the series are determined from the boundary conditions expanded into Fourier or Fourier-Bessel series and from the initial conditions of the vibrations.
PROCEDUREOF ANALYSIS
Based on the Berger assumption, the decoupled von Karman field equations for large deformations of plates can be written as,
887
PAW
0
-!!AW +pha2w,=O D
’
D
at2
(1)
888
BULENT
A.OVIJNC
with Nh* -=-
120
au +;+$g’+ ax
(?)‘I (2)
where p, h, and D are the mass per unit volume, the thickness, and the flexural rigidity of the plate, and u, v, and W0 are the two in-plane displacements orthogonal to each other and the deflection, N = Nx + No 1+u’
+ $$
= 0
(3)
$ (N, + NY) = 0
(da)
;
(Jb)
(N, + NY) = 0
AAW-2a:AW-a;W=O
(9)
where, for eqn (7) N
2al*=_ D’ for eqn (8)
N, and NY are the normal section forces and u is the Poisson’s ratio. An alternate approach has been proposed and called the alpha method [25]. In the alpha method, the extensional strain energy due to the second stress invariant has been neglected and the uncoupled equations have been obtained through a variation, as follows, AAW,, - E;AWO
where o is the natural circular frequency of the plate. Both eqns (7) and (8) are fourth order Helmholtz differential equations, and they can be represented by the following single equation,
1 +vN = 1 -vD’
24
and for both equations h a24-p - -co*. D
Equation (9) can be expressed main as,
in the complex do-
a4w __$!GG_~w=o. az2a2*
(10)
The general solution of the above Helmholtz tion is,
where,
equa-
(11)
W, = % V’, (z, t)jT {A,] Nh* -= 120
s(N,
+ NY).
where
The difference in the differential equations of motion [eqns (1) and (3)], which manifests itself in the coefficient of the A W, term, indicates that the two methods represent different approximations in the strain energy of extension. Equations (1) and (3) can be resolved into spatial and temporal components by assuming that the displacement function be written in separable variable form, Wo (x, Y, t) = w (& Y) f(r).
(5)
{Pn(Z, Z)]’ = (z”S,(z, 2) z”YJz, Z) znL(z, 2) z”K,(z, 5) and {A,JT = (a, b, c, 4,) is the vector of integration
constants. The functions j,(z, Z), %,(z, Z), Tn(z, Z), K,(z, i) are newly defined first and second kind and modified Bessel functions. They are defined as follows,
The temporal part is same for eqns (1) and (3), j(t)
+ Cl? f(t)
= 0.
*k&k
PI J”(Z,
a
=
Ix
(6)
22kk!lY(n + k + 1)
(_ I)kp$k&k T”(Z, Z) = x
The spatial parts become, for eqn (l), AAW-D !AW_&,*W=O D
Tn,(z, 2) = 5,(z,-3 Lnz (7) -
and for eqn (3), AAW-l+uNI-11EAW-$,02W=0 _ _-
22kk!T(n + K + 1)
Z(z,
(8)
3
#(n z 22kk!T(nPfk=hk + k + 1)
(‘2) + k)
= ?,(z, t) Lnz (_ l)kp$k:w +(n + k) -I: 22kk!r(n + k + 1)
889
Nonlinear free bending vibrations of plates
The two geometrical
and
conditions
are:
p: = (CXy: + c&“2 + LX:
on the deflection:
W, Ip=, = 0
p: = (a: + a;)‘”
on the slope:
-
+(n + k) =
-!_ + n+l
- (Y:
an
1 + .. . + n + k’
1 n+2
awn
and the two stress conditions
The general solution for the deflection W, [eqn (1 111 on the moment: is called the dynamic deflection function.
=o
(17)
v, lP=t = Q, + j&e
= 0.
APPLICATIONS
In order to avoid the conformal mapping and have simple derivations, plates with smooth boundary, namely circular plates, are considered in the applications. In order that the deflection and the stresses be finite at the center of circular plates, the coefficients B, and D, of the functions y,,(z, 2) and K,(z, Z) must be zero. Thus, the dynamic displacement function [eqn (1 l)] reduces to, + &17) S,(z, 2) + (C,Zn + C,??) ZJZ, 2).
+ +
=
CJn(Ynir
~Jn(~ni,
R[an ~1)
COS +
b,
~e(Jn@niv sin
The integration constants are determined from the given of any two of the above boundary conditions and the initial conditions. Along the boundary, the initial conditions may be related to the argument 8. If so, the initial conditions can be expanded into Fourier-Bessel series. The coefficient of the temporal part of the deflection function [eqn (14)] can be expressed in terms of the integration constants a, and b,. The two boundary and two initial conditions can be written in matrix form.
(13)
p)l (e, CO.7Wnit
+
g,
p) sin
Git)
(19)
In order to have a non-trivial solution, the determinant of the matrix of the coefficients of the integration constants must be zero,
PI
MJn(hni,
h=l
(18)
For circular plates, the total dynamic deflection function, expressed in polar coordinates and in nondimensional form, is WOn
(16)
are:
A!&Ip=l = 0
on the shear:
w, = (a,.?
p=l
(13
(14) Det 1A 1 = 0
(20)
where
hni = PIR,
Yni =
P2R
and R is the radius of the circular plate. Herein, the product of the newly defined function by p”, on 5,
(Lit
P” 1r1 (Yni,
which provides the frequency equation. If the initial conditions and the two boundary conditions are independent of the argument 8, the deflection function [eqn (14)] can be written in separable variable form:
PI
=
Jn
(hi,
P)
=
IFI (Yni,
Bessel
Won
=
PI
I &h
PI
become the regular first kind and modified first kind Bessel functions. For the single valuedness of the deflection function W,, , the parameter n must be an
6.
integer.
The integration constants a,, b,, c,, d, and the two parameters h,i and Yni, which are related to the natural circular frequencies, Wni, are to be determined from the boundary conditions of the plate and the initial conditions of the vibration. The boundary conditions may be any combination of the two geometrical and two stress conditions. Along the boundary, p = 1.
tt t L
R[cJn(Ani, PI + dnZn(ynivP)I(an COSne
+ b, sin ~6) (e, cos Onit + g, sin o,ir).
'
+
-f +
t
if-t
2.
t--t
+
t
0
.:
-+-I
r
.4
.6
.
I.
Fig. 1. Variation of rigidities.
(21)
890
1
0.
BULENTA.
OVUNC
I
I
1-0
I
!
I.
2.
3.
I.
2.
P,R 3.
n=l
n=O
Fig. 2. Variation of azR versus a&
Herein, the parameters n and i show the number of the nodal diameters and nodal circles, respectively. The given two boundary conditions can be expressed in matrix form as follows, (A,, cos
[eqns (9) (1% (14)1, one has, N
r, %i
ne
Ph 0D
112
(1 - r2)‘”
1 m -- j
[
(1 + r2)lJ2 -
(1 -
r2)“2
I
(2.5) + b, sin ne) (e, cos W,it + g, sin WJ)
= (0).
In order to have a non-trivial solution, the determinant of the matrix of the coefficients of the integration constants must be zero, fi” (J,, L) G”’ (J,, I,) - f12’ (J,, I,) G”’ (In, I,) = 0
(22)
which provides the frequency equation. For clamped edge circular plates, the frequency equation (22) becomes,
where Ani and Yni are functions of the natural circular frequencies, U,i [eqns (9), (12), (14)]. The values of wni which satisfy the above eqn (23), are the natural circular frequencies of the vibration The effect of the stretching rigidity versus the bending rigidity can be investigated by considering the ratio of the axial force and the bending terms of the differential equation of motion [eqn (7)). The ratio of the arguments of the modified first kind and the f&t kind Bessel functions is defmed as parameter r, Yni -=r A ni
By substituting
(24)
ynj and hEi by their expressions
The numerator and the denominator are considered as the stretching and the bending rigidities, respectively. Figure 1 shows the variation of the rigidities [eqn (25)] versus the parameter r, [eqn (24)l. The effect of stretching is smaller when the axial force, N, is smaller or the parameter r is closer to one. The effect of stretching on the natural circular frequencies is also investigated by means of the variation of the argument CYZR [eqns (9), (12), (14)l versus argument u,R, [eqn (24)]. Although in Fig. 2 the variation of azR are given for n and i up to three, the values of the argument azR or the related natural circular frequencies and the corresponding modal shapes can be obtained up to any value of a and i. The natural circular frequencies for linear analysis is determined by neglecting the effect of stretching or by taking r equal to one. The natural circular frequencies for the linear analysis are the same exact values given in 1291 and very close to those given in [9].
CONCLUSION
A general expression for the solution of the Helmholtz differential equation is obtained in the complex domain. For the bending vibration of plates of any shape under any boundary and initial conditions, the general solution of the equation of motion in terms of deflection, which is called the dynamic deflection function, can be easily determined. The natural circular frequencies and the corresponding modal shapes can be evaluated up to
Nonlinear free bending vibrations of plates
any desired number
by means of the dynamic
de-
flection function. Forced vibration of plates can be obtained either by modal analysis or by numerical integration. As an application, the nonlinear free bending vibration of circular plates is investigated. The effect of stretching on the bending vibration is illustrated for clamped edge circular plates. Acknowledgments-Gratitude is expressed to the Computing Center of the University of Southwestern Louisiana for making their computer Honeywell 68/80 (Multics) available, and to Miss Debra Boudreaux for her conscientious typing of the manuscript. REFERENCES 1. F. C. Apple and N. R. Byers, Fundamental
frequency of simply supported rectangular plates of linearly varying thickness. J. Appl. Mech. 32, 163-167 (1965). 2. M. M. Banarjee, On the analysis of large amplitude vibrations of non-uniform rectangular plates. J.-Sound and Vibr. 58(4).,_ 545-553
(1978).
3. M. M. Banarjee and S. Datta, Large amplitude vibrations of thin elastic plates by the method of conformal transformation. Ins. J. Mech. Sci. 21.689-696 (1979). 4. H. M. Berger, A new approach to theanalysis of large deflections of plates. J. Appl. Mech. (22) 465-472 (1955). 5. N. H. Chu and G. Herrmann, Influence of large amplitude on free flexural vibrations of rectangular elastic plates. .Z. Appl. Mech. 23, 532-540 (1956). 6. S. Datta, Large amplitude vibrations of irregular plates placed on an elastic foundation. Znt. J. NonLin. Mech. 11, 337-345 (1976). 7. C. L. Dym and I. H. Shames, Solid Mechanics, A Variable Approach, McGraw-Hill, Inc., New York
(1973). 8. H. Hencky, Uber die Berucksightigung der Schubverserung in ebenen Platten. Zng. Arch. 16, 72-76 (1947).
9. A. Kalnins and C. L. Dym, Vibration, Beams, Plates, Shells, pp. 60-67. Dowden, Hutchinson and Ross, Inc. (1976). 10. P. A. Laura and A. J. Faulstich, An application of conformal mapping to the determination of the natural frequency of membranes of regular polygonal shape. Proc. 9th Midwestern Mech. Conf., Univ. of Wisconsin, . . . _ _16-18, 1965). - - 55-163 _ _ .. (Aug. 11. K. U. Mindhn, lntluence ot rotatory mertla and shear on flexural motion of isotropic elastic plates. J. Appl. Mech.
Trans. of ASME
73, 31-38 (1951).
12. K. Munakata, On the vibration and elastic stability of a rectangular plate clamped at its four edges. J. Math. Phys 31(l),
69-74 (1952).
891
13. W. A. Nash and J. R. Modeer, Certain approximate analysis of the nonlinear behaviour of plates and shells. Proc. Symp. Theory Thin EZastic Shells. Delft, Interscience, New York, 331-354 (1960). 14. J. L. Nowinski and H. Ohnabe, On certain inconsistencies in Berger equations for large deflections of elastic plates. Znt. J. k4ech. Sci. 14,165-176 (1972). 15. B. A. Ovunc and G. Z. Vovadiis. Anolication of Complex Variable Theory to the Free Vibration of Plates, Boundary Elements. Proc. of5th Znt. Conf. on Boundary Element Methods, Hiroshima, Japan, Nov. 8-l 1, 1983, pp. 709-718. Springer Verlag, (1983). l6 . B. A. Ovunc, Forced Bending Vibration of Plates by Numerical Integration, Proc., 2lst Annual Meeting, SES, Virginia Polytechnic Inst. and State Univ. Blacksburg, Virginia, Oct. 15-17, 1984, pp. 109-l 10. 17. G. Prathap and K. A. V. Pandalai, Non-linear vibrations of transversely isotropic rectangular plates. Znt. J. Non-Lin. Mech. 13, 285-294 (1979). lg. C. Prasad, R. K. Jam, and S. R. Soni, Effect of trans-
verse shear and rotatory inertia on vibrations of infinite strip of variable thickness. J. Phys. Sot. Japan 33, 1156-1159 (1972). 19. G. Pratap and T. K. Varadan, On the non-linear vibration of rectangular plates. .Z. Sound and Vibr. 56, 521-530
(1978).
20. G. Pratap, On the Berger approximation: A critical reexamination. .Z. Sound and Vibr. 60, 149-154 (1979). 21. J. Ramachandran and D. V. Reddy, Nonlinear vibrations of rectangular plates with cutouts. AZAA 10, 1709-1710 (1972). 22. E. Reissner, The effect of transverse shear deformation on the bending of elastic plates. J. AppI. . _ Mech.
Trans. ASME 67, A-69 (1945). 23. P. A. Shahadv. R. Passarelli and P. A. Laura. Com-
plex variable theory and its application to the’ determination of the fundamental frequency of vibrating plates. J. Acous. Sot. Am. 42, 806-809 (1967). 24. Y. S. Uflyand, The propagation of waves in the transverse vibrations bars and plates. Akad. Nauk SSR, Prikl. Mat. Mekh. 12, 287-300 (1948). 25. C. P. Vendhan, A study of the Berger equations
applied to the non-linear vibration of elastic plates. ht.
J. Mech. Sci. 17, 461-468 (1975). 26. G. Z. Voyadjis and M. H. Baluch, Refined theory for flexural motions of isotropic elastic plates. J. Sound and Vibr. 76, 57-64 (1981). 27. T. Wah, Large amplitude flexural vibrations of rectangular plates. Znt. J. Mech. Sci. 5, 425-434 (1963). 28. N. Yamaki, Influence of large amplitudes on flexural vibrations ofelastic plates. ZAMM41,501-510 (1961). 29. A. Zeytinci, A genera1 approach for the vibrations of thin elastic circular plates, considering initial conditions. Doctoral thesis, Istanbul Tekn. Univ. Ins. Fak. (1980).
NONLINEAR
1985 0
FREE BENDING PLATES
VIBRATIONS
A. OVUNC University of Southwestern Louisiana, U.S.A.
0045-7949/85 $3.00 + .oo 1985 Pergamon Press Ltd.
OF
BULENT
Department
of Civil Engineering,
(Received
16 January
Louisiana, Lafayette,
1984)
Abstract-A general solution for the Helmholtz differential equations is obtained in the complex domain and applied to the nonlinear, free, bending vibrations of plates. The analysis is based on the decoupled nonlinear von Karman field equations by Berger assumption for the large deformations of plates. The decoupled differential equation in terms of the deflection function is a fourth order Helmholtz differential equation. Its solution, called the dynamic deflection function, is obtained in the complex domain by means of newly defined first and second kind and modified Bessel functions. The dynamic deflection function can be applied to any plates having any shape and any boundary condition under any arbitrary dynamic loads. For plates with smooth boundary, the parameters of the dynamic deflection function are determined from the boundary conditions of the plates and the initial conditions of the vibrations. The analyses of plates with piece-wise smooth boundaries are obtained on the mapped planes. The nonlinear, free vibration of circular plates are investigated by the dynamic deflection function. The effect of stretching on the natural circular frequencies are illustrated. INTRODUCTION
A general dynamic deflection function is obtained for the decoupled von Karman field equations for large deformations. The decoupling of the differential equations is based on the Berger assumption. The refinements to include the effect of shear and rotatory inertia have been introduced in the bending vibration of plates [8, 11, 241. A theory for the flexural motions of elastic plates has been developed by taking into account the effect of transverse normal strains and transverse normal stresses, together with rotatory inertia and shear [26], by assuming the form used in the static theory of bending [7,22]. The nonlinear vibration of thin, elastic, isotropic plates has been investigated by many authors using von Karman field equations [5, 281. A first attempt has been made by Berger to decouple the von Karman equations [41. The coupled von Karman equations have been substituted for by a simple set of coupled nonlinear equations [ 13, 271. The Berger approximation has been used to derive the equations of motion for rectangular plates with a circular cutout 1211. Based on an averaging technique to satisfy in-plane boundary conditions, as in [19], and also the von Karman field equations, the effect of shear deformation and rotatory inertia on the large amplitude vibration of simple supported rectangular plates has been studied by using Runga Kutta technique to integrate the differential equations [17]. Several authors have questioned the validity of the Berger assumption from the point of view of accuracy [14, 19, 251. Using certain well-known results from the two-dimensional theory of elasticity, a plausible explanation for the origin of the Berger approximation has been suggested, and arguments have been raised to show further that other Bergerlike approximations can be developed [201. An anal-
ysis of the large amplitude vibrations of non-uniform rectangular plates has been presented by neglecting the second invariant of the middle surface strain in the expression of the total potential energy of the plates [ 1,2, 181. The complex variable theory has been applied to the linear vibration of thin, elastic plates [lo, 12, 15, 231. Following the Berger approximation, large amplitude vibrations of irregular plates have been investigated by means of a conformal mapping technique [3, 6, lo]. Herein, the general solution, as the dynamic stress and dynamic displacement functions, derived for the in-plane and bending vibrations of linear, elastic, thin plates [15, 161, is extended to the nonlinear vibrations of plates. The differential equation, in terms of deflection obtained by decoupling the von Karman field equations by means of Berger assumption, is a fourth order Helmholtz differential equation. The general solution of the Helmholtz differential equation for deflection constitutes the dynamic deflection function. The dynamic deflection function is expressed in the complex domain, in series form, in terms of newly defined first and second kind Bessel functions. The unknown coefficients and the parameters in the series are determined from the boundary conditions expanded into Fourier or Fourier-Bessel series and from the initial conditions of the vibrations.
PROCEDUREOF ANALYSIS
Based on the Berger assumption, the decoupled von Karman field equations for large deformations of plates can be written as,
887
PAW
0
-!!AW +pha2w,=O D
’
D
at2
(1)
888
BULENT
A.OVIJNC
with Nh* -=-
120
au +;+$g’+ ax
(?)‘I (2)
where p, h, and D are the mass per unit volume, the thickness, and the flexural rigidity of the plate, and u, v, and W0 are the two in-plane displacements orthogonal to each other and the deflection, N = Nx + No 1+u’
+ $$
= 0
(3)
$ (N, + NY) = 0
(da)
;
(Jb)
(N, + NY) = 0
AAW-2a:AW-a;W=O
(9)
where, for eqn (7) N
2al*=_ D’ for eqn (8)
N, and NY are the normal section forces and u is the Poisson’s ratio. An alternate approach has been proposed and called the alpha method [25]. In the alpha method, the extensional strain energy due to the second stress invariant has been neglected and the uncoupled equations have been obtained through a variation, as follows, AAW,, - E;AWO
where o is the natural circular frequency of the plate. Both eqns (7) and (8) are fourth order Helmholtz differential equations, and they can be represented by the following single equation,
1 +vN = 1 -vD’
24
and for both equations h a24-p - -co*. D
Equation (9) can be expressed main as,
in the complex do-
a4w __$!GG_~w=o. az2a2*
(10)
The general solution of the above Helmholtz tion is,
where,
equa-
(11)
W, = % V’, (z, t)jT {A,] Nh* -= 120
s(N,
+ NY).
where
The difference in the differential equations of motion [eqns (1) and (3)], which manifests itself in the coefficient of the A W, term, indicates that the two methods represent different approximations in the strain energy of extension. Equations (1) and (3) can be resolved into spatial and temporal components by assuming that the displacement function be written in separable variable form, Wo (x, Y, t) = w (& Y) f(r).
(5)
{Pn(Z, Z)]’ = (z”S,(z, 2) z”YJz, Z) znL(z, 2) z”K,(z, 5) and {A,JT = (a, b, c, 4,) is the vector of integration
constants. The functions j,(z, Z), %,(z, Z), Tn(z, Z), K,(z, i) are newly defined first and second kind and modified Bessel functions. They are defined as follows,
The temporal part is same for eqns (1) and (3), j(t)
+ Cl? f(t)
= 0.
*k&k
PI J”(Z,
a
=
Ix
(6)
22kk!lY(n + k + 1)
(_ I)kp$k&k T”(Z, Z) = x
The spatial parts become, for eqn (l), AAW-D !AW_&,*W=O D
Tn,(z, 2) = 5,(z,-3 Lnz (7) -
and for eqn (3), AAW-l+uNI-11EAW-$,02W=0 _ _-
22kk!T(n + K + 1)
Z(z,
(8)
3
#(n z 22kk!T(nPfk=hk + k + 1)
(‘2) + k)
= ?,(z, t) Lnz (_ l)kp$k:w +(n + k) -I: 22kk!r(n + k + 1)
889
Nonlinear free bending vibrations of plates
The two geometrical
and
conditions
are:
p: = (CXy: + c&“2 + LX:
on the deflection:
W, Ip=, = 0
p: = (a: + a;)‘”
on the slope:
-
+(n + k) =
-!_ + n+l
- (Y:
an
1 + .. . + n + k’
1 n+2
awn
and the two stress conditions
The general solution for the deflection W, [eqn (1 111 on the moment: is called the dynamic deflection function.
=o
(17)
v, lP=t = Q, + j&e
= 0.
APPLICATIONS
In order to avoid the conformal mapping and have simple derivations, plates with smooth boundary, namely circular plates, are considered in the applications. In order that the deflection and the stresses be finite at the center of circular plates, the coefficients B, and D, of the functions y,,(z, 2) and K,(z, Z) must be zero. Thus, the dynamic displacement function [eqn (1 l)] reduces to, + &17) S,(z, 2) + (C,Zn + C,??) ZJZ, 2).
+ +
=
CJn(Ynir
~Jn(~ni,
R[an ~1)
COS +
b,
~e(Jn@niv sin
The integration constants are determined from the given of any two of the above boundary conditions and the initial conditions. Along the boundary, the initial conditions may be related to the argument 8. If so, the initial conditions can be expanded into Fourier-Bessel series. The coefficient of the temporal part of the deflection function [eqn (14)] can be expressed in terms of the integration constants a, and b,. The two boundary and two initial conditions can be written in matrix form.
(13)
p)l (e, CO.7Wnit
+
g,
p) sin
Git)
(19)
In order to have a non-trivial solution, the determinant of the matrix of the coefficients of the integration constants must be zero,
PI
MJn(hni,
h=l
(18)
For circular plates, the total dynamic deflection function, expressed in polar coordinates and in nondimensional form, is WOn
(16)
are:
A!&Ip=l = 0
on the shear:
w, = (a,.?
p=l
(13
(14) Det 1A 1 = 0
(20)
where
hni = PIR,
Yni =
P2R
and R is the radius of the circular plate. Herein, the product of the newly defined function by p”, on 5,
(Lit
P” 1r1 (Yni,
which provides the frequency equation. If the initial conditions and the two boundary conditions are independent of the argument 8, the deflection function [eqn (14)] can be written in separable variable form:
PI
=
Jn
(hi,
P)
=
IFI (Yni,
Bessel
Won
=
PI
I &h
PI
become the regular first kind and modified first kind Bessel functions. For the single valuedness of the deflection function W,, , the parameter n must be an
6.
integer.
The integration constants a,, b,, c,, d, and the two parameters h,i and Yni, which are related to the natural circular frequencies, Wni, are to be determined from the boundary conditions of the plate and the initial conditions of the vibration. The boundary conditions may be any combination of the two geometrical and two stress conditions. Along the boundary, p = 1.
tt t L
R[cJn(Ani, PI + dnZn(ynivP)I(an COSne
+ b, sin ~6) (e, cos Onit + g, sin o,ir).
'
+
-f +
t
if-t
2.
t--t
+
t
0
.:
-+-I
r
.4
.6
.
I.
Fig. 1. Variation of rigidities.
(21)
890
1
0.
BULENTA.
OVUNC
I
I
1-0
I
!
I.
2.
3.
I.
2.
P,R 3.
n=l
n=O
Fig. 2. Variation of azR versus a&
Herein, the parameters n and i show the number of the nodal diameters and nodal circles, respectively. The given two boundary conditions can be expressed in matrix form as follows, (A,, cos
[eqns (9) (1% (14)1, one has, N
r, %i
ne
Ph 0D
112
(1 - r2)‘”
1 m -- j
[
(1 + r2)lJ2 -
(1 -
r2)“2
I
(2.5) + b, sin ne) (e, cos W,it + g, sin WJ)
= (0).
In order to have a non-trivial solution, the determinant of the matrix of the coefficients of the integration constants must be zero, fi” (J,, L) G”’ (J,, I,) - f12’ (J,, I,) G”’ (In, I,) = 0
(22)
which provides the frequency equation. For clamped edge circular plates, the frequency equation (22) becomes,
where Ani and Yni are functions of the natural circular frequencies, U,i [eqns (9), (12), (14)]. The values of wni which satisfy the above eqn (23), are the natural circular frequencies of the vibration The effect of the stretching rigidity versus the bending rigidity can be investigated by considering the ratio of the axial force and the bending terms of the differential equation of motion [eqn (7)). The ratio of the arguments of the modified first kind and the f&t kind Bessel functions is defmed as parameter r, Yni -=r A ni
By substituting
(24)
ynj and hEi by their expressions
The numerator and the denominator are considered as the stretching and the bending rigidities, respectively. Figure 1 shows the variation of the rigidities [eqn (25)] versus the parameter r, [eqn (24)l. The effect of stretching is smaller when the axial force, N, is smaller or the parameter r is closer to one. The effect of stretching on the natural circular frequencies is also investigated by means of the variation of the argument CYZR [eqns (9), (12), (14)l versus argument u,R, [eqn (24)]. Although in Fig. 2 the variation of azR are given for n and i up to three, the values of the argument azR or the related natural circular frequencies and the corresponding modal shapes can be obtained up to any value of a and i. The natural circular frequencies for linear analysis is determined by neglecting the effect of stretching or by taking r equal to one. The natural circular frequencies for the linear analysis are the same exact values given in 1291 and very close to those given in [9].
CONCLUSION
A general expression for the solution of the Helmholtz differential equation is obtained in the complex domain. For the bending vibration of plates of any shape under any boundary and initial conditions, the general solution of the equation of motion in terms of deflection, which is called the dynamic deflection function, can be easily determined. The natural circular frequencies and the corresponding modal shapes can be evaluated up to
Nonlinear free bending vibrations of plates
any desired number
by means of the dynamic
de-
flection function. Forced vibration of plates can be obtained either by modal analysis or by numerical integration. As an application, the nonlinear free bending vibration of circular plates is investigated. The effect of stretching on the bending vibration is illustrated for clamped edge circular plates. Acknowledgments-Gratitude is expressed to the Computing Center of the University of Southwestern Louisiana for making their computer Honeywell 68/80 (Multics) available, and to Miss Debra Boudreaux for her conscientious typing of the manuscript. REFERENCES 1. F. C. Apple and N. R. Byers, Fundamental
frequency of simply supported rectangular plates of linearly varying thickness. J. Appl. Mech. 32, 163-167 (1965). 2. M. M. Banarjee, On the analysis of large amplitude vibrations of non-uniform rectangular plates. J.-Sound and Vibr. 58(4).,_ 545-553
(1978).
3. M. M. Banarjee and S. Datta, Large amplitude vibrations of thin elastic plates by the method of conformal transformation. Ins. J. Mech. Sci. 21.689-696 (1979). 4. H. M. Berger, A new approach to theanalysis of large deflections of plates. J. Appl. Mech. (22) 465-472 (1955). 5. N. H. Chu and G. Herrmann, Influence of large amplitude on free flexural vibrations of rectangular elastic plates. .Z. Appl. Mech. 23, 532-540 (1956). 6. S. Datta, Large amplitude vibrations of irregular plates placed on an elastic foundation. Znt. J. NonLin. Mech. 11, 337-345 (1976). 7. C. L. Dym and I. H. Shames, Solid Mechanics, A Variable Approach, McGraw-Hill, Inc., New York
(1973). 8. H. Hencky, Uber die Berucksightigung der Schubverserung in ebenen Platten. Zng. Arch. 16, 72-76 (1947).
9. A. Kalnins and C. L. Dym, Vibration, Beams, Plates, Shells, pp. 60-67. Dowden, Hutchinson and Ross, Inc. (1976). 10. P. A. Laura and A. J. Faulstich, An application of conformal mapping to the determination of the natural frequency of membranes of regular polygonal shape. Proc. 9th Midwestern Mech. Conf., Univ. of Wisconsin, . . . _ _16-18, 1965). - - 55-163 _ _ .. (Aug. 11. K. U. Mindhn, lntluence ot rotatory mertla and shear on flexural motion of isotropic elastic plates. J. Appl. Mech.
Trans. of ASME
73, 31-38 (1951).
12. K. Munakata, On the vibration and elastic stability of a rectangular plate clamped at its four edges. J. Math. Phys 31(l),
69-74 (1952).
891
13. W. A. Nash and J. R. Modeer, Certain approximate analysis of the nonlinear behaviour of plates and shells. Proc. Symp. Theory Thin EZastic Shells. Delft, Interscience, New York, 331-354 (1960). 14. J. L. Nowinski and H. Ohnabe, On certain inconsistencies in Berger equations for large deflections of elastic plates. Znt. J. k4ech. Sci. 14,165-176 (1972). 15. B. A. Ovunc and G. Z. Vovadiis. Anolication of Complex Variable Theory to the Free Vibration of Plates, Boundary Elements. Proc. of5th Znt. Conf. on Boundary Element Methods, Hiroshima, Japan, Nov. 8-l 1, 1983, pp. 709-718. Springer Verlag, (1983). l6 . B. A. Ovunc, Forced Bending Vibration of Plates by Numerical Integration, Proc., 2lst Annual Meeting, SES, Virginia Polytechnic Inst. and State Univ. Blacksburg, Virginia, Oct. 15-17, 1984, pp. 109-l 10. 17. G. Prathap and K. A. V. Pandalai, Non-linear vibrations of transversely isotropic rectangular plates. Znt. J. Non-Lin. Mech. 13, 285-294 (1979). lg. C. Prasad, R. K. Jam, and S. R. Soni, Effect of trans-
verse shear and rotatory inertia on vibrations of infinite strip of variable thickness. J. Phys. Sot. Japan 33, 1156-1159 (1972). 19. G. Pratap and T. K. Varadan, On the non-linear vibration of rectangular plates. .Z. Sound and Vibr. 56, 521-530
(1978).
20. G. Pratap, On the Berger approximation: A critical reexamination. .Z. Sound and Vibr. 60, 149-154 (1979). 21. J. Ramachandran and D. V. Reddy, Nonlinear vibrations of rectangular plates with cutouts. AZAA 10, 1709-1710 (1972). 22. E. Reissner, The effect of transverse shear deformation on the bending of elastic plates. J. AppI. . _ Mech.
Trans. ASME 67, A-69 (1945). 23. P. A. Shahadv. R. Passarelli and P. A. Laura. Com-
plex variable theory and its application to the’ determination of the fundamental frequency of vibrating plates. J. Acous. Sot. Am. 42, 806-809 (1967). 24. Y. S. Uflyand, The propagation of waves in the transverse vibrations bars and plates. Akad. Nauk SSR, Prikl. Mat. Mekh. 12, 287-300 (1948). 25. C. P. Vendhan, A study of the Berger equations
applied to the non-linear vibration of elastic plates. ht.
J. Mech. Sci. 17, 461-468 (1975). 26. G. Z. Voyadjis and M. H. Baluch, Refined theory for flexural motions of isotropic elastic plates. J. Sound and Vibr. 76, 57-64 (1981). 27. T. Wah, Large amplitude flexural vibrations of rectangular plates. Znt. J. Mech. Sci. 5, 425-434 (1963). 28. N. Yamaki, Influence of large amplitudes on flexural vibrations ofelastic plates. ZAMM41,501-510 (1961). 29. A. Zeytinci, A genera1 approach for the vibrations of thin elastic circular plates, considering initial conditions. Doctoral thesis, Istanbul Tekn. Univ. Ins. Fak. (1980).