Differential Quadrature Method for Stability Analysis of Dynamic Systems With Multiple Delays: Application to Simultaneous Machining Operations

Abstract

This paper proposes a semi-analytical solution via the differential quadrature method (DQM) for stability analysis of linear multi-degree-of-freedom second-order systems with multiple delays. The vibration systems are reformulated as a delayed differential equation (DDE) in state–space form. With the derivative of the state vector with respect to time at an arbitrary discrete-time point being expressed as a linear weighted sum of the values of the state vector, the original DDE is approximated by a set of algebraic equations, leading to the Floquet transition matrix. Based on Floquet theory, the stability of the systems is then determined by checking the eigenvalues of the transition matrix. The computational accuracy and efficiency are demonstrated through comparison with existing methods via numerical examples. As an application, the proposed method is employed to predict chatter stability in simultaneous machining operations, providing a reference for the choice of machining parameters.