Applicability of the equivalent linearization for thermally buckled plate random vibration

Abstract

A single-mode model is used for the random vibration of the simply-supported and clamped plates undergoing thermal buckling due to the immovable edge constraint imposed on all plate edges. It is shown that the equivalent linearization technique gives the mean square transverse displacement and RMS strain and stress dmibutions on the simply-supported and clamped plates, which are much larger (by more than 50% in some numerical examples) than the FokkerPlanck values. Yet, it is a fortuitous accident that the equivalent linearization can be used at all, which essentially yields a wrong variance. Perhaps, the saving grace is that total moments are dominated by the snapthrough displacement and not by fluctuations. 1. Problem formulation We have begun in~esti~ationl-~ of the simplysupported and clamped isotropic plates undergoing thermal buckling due to the immovable edge constraint imposed on all plate edges. Much can be learned from a prototype dimensionless single-mode equation for displacement q which is the starting point of Ref [2]. First, q is the inertial term and pq represents the viscous damping, where the overhead dot denotes d/dt. Note that ko(l s)q is the combined stiffness consisting of structural koq and thermal -skoq contributions, where the parameter s subsumes the uniform plate temperature rise above room temperature and the temperature variation over the mid-plate plane. We have s=O in the This paper is declared a wok of the US Government and is not subject to copyright protection in the United States. * Research Engineer nonthermal case, and s=l corresponds to the critical buckling temperature. Next, aq3 is the cubic stiffness representing geometric nonlinearity of membrane stretching. Finally, fo is the thermal moment induced by the temperature gradient through the plate thickness. This together with the external forcing f ( t ) constitutes the combined forcing. In further detail, we have for a simply-supported 2 2 plate' k0=(y +I) , ~T~[~-~(~-P)G,BI. 2 2 2 4 f0+y +I) 6,T0 0 4 , eY(1-p )(Y +W%Y 4+2~~2+1)1 and ko-=+(y4+2 y2I3+l). sTo[l +(1-~~v(ly2(y2+1)-2) 161, fo+y4+2y213t1~,~o 16, e ( Y ~ + Y ~ + ~ P N -$ (1-p21y (y2+y-2~(yy-1)-2+(Y+4Y-1)-2+(4y+y-1)-21 I for a clamped plate. Here, y =bla is the aspect ratio of plate sides a and b, andp is Poisson's ratio. The uniform plate temperature To is measured in units of the critical buckling temperature. The maximum temperature variation on the mid-plate plane is denoted by To& and, similarly, T06, is the maximum temperature gradient across the plate thickness, where 6, and Sg are the scale factors. For instance, 6,=0 refers to no temperature variation over the mid-plate plane and 6, =O to zero temperature gradient across the plate thickness. For the typical y =1 and p 4 to be considered in this paper, we see that the ratio of To to T06, is 1:0.085 for a simply-supported and 1 :O. 14 for a clamped plate. This therefore means that the uniform plate temperature rise contributes to thermal buckling much more significantly than the superposed temperature variation on the mid-plate plane. Besides, we also see that fo= 0.17TOSg and fo= 0.44TOag for the simply-supported and clamped plates, respectively. Of course , these numerical comparisons are dnectly dependent upon the particular profiles for the temperature variation and gradient chosen in Ref [I]. Finally, the viscous damping coefficient /3=&/&will be adopted with 6 =0.04 for all computation to be presented here. Next, it is necessary to transform away the nonzero fo in the right-hand side of Eq (1). To do this, we define a static (snap-through) displacement by