Time-domain integration methods of exponentially damped linear systems

In this paper, three new kinds of time-domain numerical methods of exponentially damped systems are presented, namely, the simplified Newmark integration method, the precise integration method, and the simplified complex mode superposition method. Based on the traditional Newmark integration method and transforming the equation of motion with exponentially damping kernel functions into an equivalent second-order equation of motion by using the internal variables technique, the simplified Newmark integration method is developed by using a decoupling technique to reduce the computer run time and storage. By transforming the equation of motion with exponentially damping kernel functions into a first-order state-space equation, the precise integration technique is used to numerically solve the state-space equation. Based on a symmetric state-space equation and the complex mode superposition method, a delicate and simplified general solution of exponentially damped linear systems, completely in real-value form, is developed. The accuracy and efficiency of the developed numerical methods are compared and discussed by two benchmark examples.