DYNAMIC BEHAVIOUR OF HIGHLY PRESTRESSED ORTHOTROPIC RECTANGULAR PLATE UNDER A CONCENTRATED MOVING MASS

Abstract

In this paper the dynamic response to concentrated moving masses of highly prestressed orthotropic 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 masses. Such vibrational problems defile conventional methods of solution. 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 being 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 and at the boundary called the inner solution. Thereof, the Van Dyke asymptotic matching principle which produces the unknown integration constants in the outer and inner expansions is applied. Thereafter, the inverse Laplace transformation 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 uniformly valid solution in the entire domain of definition of the plate problem. Similarly, the resonant states and the corresponding critical speeds are obtained. The analysis of this result is then shown in plotted curves. Graphical interpretation of the results show that the critical speeds at the respective resonant 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, a 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.