Question

Briefly explain the steps, tools, and techniques that you would take to analytically investigate a vibrations problem.

Answer 1

Analytical solutions in vibrations are the most advantageous, but not always comfortably obtainable, as in the instance of the plates with different boundary conditions. There are two ways of solving a vibration problem analytically. The first is called as the superposition method and the second is the integral transform method.

Superposition Method:

This dynamic approach provides the principal values for the natural frequencies of plates with different view ratios in contrast with other computational methods. Solutions fulfil the governing differential equation and all boundary conditions. If the solutions are potential to sum infinity terms then the solutions will be acurate. Hence, the superposition method is called an analytical method of solution as it is closed form of an infinite series. The primary objective is to use free vibrations mode shapes for the reason of uncoupling equations of motion. The uncoupled equations are in expression of another variable called as the modal coordinates. Solution for the modal coordinates can be acquired by solving every equation separately. A superposition of the modal coordinates then provides solution of the actual equations. It is not needed to utilise all mode shapes for most realistic problems. Viable estimated solutions can be acquired with superposition method with just first some of the mode shapes. The method is not realistic for huge apparatus as two unknown coefficients have to be present for every mode shape.

Integral Transform Method:

Different sorts of integral transform method have been utilised prosperously to solve variety of mixed boundary value problems in the real world. With the integral transform method, many problems are generally minimized to the terminal equation type of integral equations based on the integral transform methods and handful of their solutions are previously been supplied. If the closed-form solutions are not realistic to be solved owing to the complication of the kernel then the numerical investigations will be formulated there. Integral Transform Method report the performance of the medium accurate, but they are solely relevant for preferably easy geometries and material representation. The Integral Transform Method directing to analytical solutions of the Lam´e differential equation is in a position to report the performance of the infinite medium unconditionally but is obtained under specific generalization. The material has to be uniform and consistent with linear elastic material performance, the induction of surfaces is achievable for elementary geometries and thickness in the material should be only be considered into account if they are parallel to the corresponding surfaces. The Lam´e differential equation is decoupled utilising the Helmholtz method and changed from partial into ordinary differential equations by a Fourier transformation. The ordinary differential equations can be explained with the boundary conditions of the apparatus.