Harmonic response analysis of a circular plate subjected to moving point loads using Green’s function approach

Dynamics of circular plates under moving loads are an important class of problems in manufacturing industry, for example, cutting thin annular plates using point cutting tools and wood/bar/pipe cutting saws. These tools are subjected to excessive vibration during operation. Its dynamic behavior needs to be analyzed rigorously. In the present work, dynamics of a circular plate subjected to moving harmonic point load is analyzed using Kirchhoff’s plate hypothesis. In order to capture the geometric nonlinearity, the von Kármán strain displacement relation is used. Constitutive relations are developed using a complex modulus. Further, extended Hamilton’s principle is used to obtain the nonlinear coupled governing equations in radial, tangential, and transverse directions. Governing equations are discretized using Galerkin’s method. The novelty in the present work is to develop Green’s function incorporating viscous damping to obtain the plate response subjected to arbitrary moving load for general boundary conditions. The proposed method is capable to obtain the plate response due to single/multi point moving loads. In order to verify the accuracy of the proposed method, Green’s function solution is compared with the numerical solution obtained from the RK4 method and shows good agreement. Further, resonance instabilities are studied, which occur when a single point load rotates with critical angular velocity. Subsequently, the response of the circular plate under three-point loads, each having a magnitude of 1/3rd of the single moving point load moving circularly in a constant phase has been discussed. In the end, a parametric study of hysteresis damping on the dimensionless deflection of the plate is carried out.