Abstract The transient responses of several second-order single-degree-of-freedom mechanical systems in both free and forced vibration are obtained by six numerical methods: two explicit and four implicit ones. Quite a lot is known about the behavior of these numerical methods when applied to specific linear problems, but in the case of non-linear problems relatively little research has been performed due to complexities inherent in the non-linear systems. In the present paper, linear and non-linear, undamped and damped systems are examined. In some cases, the exact solutions are known; this provides a concrete basis for comparing the accuracy of the results. However, in other cases, a converged solution is used as a basis of comparison. The converged solution is obtained by continuing to solve the problem for smaller and smaller time steps until the response converges to a fixed-time curve. The methods are studied for stability and computational efficiency. Stability is measured by the performance of the integration scheme as the solution time step is gradually increased. Numerical efficiency is gauged by the total time required to calculate the system response. The motivation for this analysis is to select a suitable integration scheme for solving non-linear transient problems containing higher degrees of freedom (beams and elastic mechanisms).
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