DYNAMIC RESPONSE TO MOVING CONCENTRATED MASS OF HIGHLY PRESTRESSED ORTHOTROPIC RECTANGULAR PLATE-STRUCTURE

Abstract

In this work, the dynamic response to concentrated moving mass of highly prestressed or- thotropic rectangular plate-structure is examined. When the ratio of the bending rigidity to the in-plane loading is small, a small parameter multiplies the highest derivatives in the equation governing the motion of the plate under the action of moving concentrated mass. When the convectional methods of solution is applied to this dynamical problem, the derived solution is not obtained. An approach suitable for the solution of this type of problem is the singular perturbation. To this end, a choice is made of the method of matched asymptotic expansions (MMAE) among others. The application of the singular perturbation scheme in conjunction with the finite Fourier sine transform produces two different but complementary approximations to the solution for small parameter, one be- ing valid in the region where the other fails. One is valid away from the boundary called the outer solution while the other is valid near the boundary called the inner solution. Applying the Van Dyke asymptotic matching principle produces the unknown integration constants in the outer and inner expansions. Thereafter, the inverse Laplace transforma- tion of the obtained results is carried out using the Cauchy residue theorem. This process produces the leading order solution, and the first order correction, to the uniformly valid solution of the plate dynamical problem. The addition of the two results above produces the sought after uniformly valid solution in the entire domain of definition of the plate problem. The processes above is repeated for the dynamic response of highly prestressed orthotropic rectangular plate-structure resting on Pasternak foundation. Subsequently, the resonance states and the corresponding critical speeds are obtained. The analysis of these results are then shown in plotted curves. Graphical interpretation of the results show that the critical speeds at the respective resonance states increase as the value of prestress increase. Thus, the risk of resonance is remote as prestress is increased for any choice of value of rotatory inertia correction factor. Also, lower values of rotatory inertia show variation in the value of critical speed; hence, the possibility of resonance. Similarly, the critical speed increases with shear modulus for various values of prestress. However, as the value of shear modulus increases, critical speed approaches more or less constant value. Thus, design incorporating high value of shear modulus is more stable and reliable. The critical speed increases with material orthotropy for lower values of rotatory inertia correction factor. A comparison between the responses of the plate-structure to moving force and moving mass is made. Investigations reveal that the critical speed for moving mass is higher than the critical speed for moving force irrespective of the parameters con- sidered - rotatory inertia correction factor, shear modulus or prestress. Thus resonance is reached earlier in the latter than in the former.