Nonlinear analysis of chatter vibration in a cylindrical transverse grinding process with two time delays using a nonlinear time transformation method

Abstract

A nonlinear dynamic system of cylindrical transverse grinding process is studied in this paper. The system consists of a grinding wheel and a workpiece, which are connected to the base by spring-damper elements, interacting with nonlinear normal forces. This two DOF model includes two time delays originated from the regenerative effects of the workpiece and the grinding wheel. Bifurcation points are located using a numerical algorithm by which we can find all the eigenvalues in a given rectangular region on the complex plane for the delayed differential equations. Supercritical bifurcation has been found for some sets of system parameter values. The amplitudes of the limit cycles are predicted using a nonlinear time transformation method, which is similar to the harmonic balance approach in that a periodic solution is approximated by a Fourier series. However, the main difference is that a nonlinear time ϕ is introduced in the Fourier series rather than the physical time t. The analytical solutions of stable limit cycles up to the third harmonics are compared with numerical simulations for the retarded system. It is shown that the proposed method gives accurate approximate solutions.