The collocation method is a semi-numerical scheme that has been used quite extensively for linear and nonlinear boundary value problems in solid mechanics. As a part of my Ph.D. work with Professor Harold Nelson at Arizona State University, I adapted the method to periodic solutions with trigonometric comparison functions. The specific application is nonlinear rotating systems. In particular, we looked at systems with concentrated nonlinearities such as would occur at bearings and couplings. We found that the method is highly efficient (vs. numerical simulation) at predicting the amplitudes and frequencies of non-autonomous and autonomous nonlinear systems. It is not limited to small nonlinearities as is the case with perturbation methods.

The following is a list of some related publications.

• C. Nataraj and H. D. Nelson, 1989, "Periodic Oscillations in Rotor Dynamic Systems With Nonlinear Supports: A General Approach," Trans. ASME, Journal of Vibration, Acoustics, Stress and Reliability in Design, Vol. 111, pp. 187-193.
• C. Nataraj and H. D. Nelson, 1989, "The Application of Trigonometric Collocation Method to the Determination of Periodic Solutions in Nonlinear Systems," Proceedings of the 1989 ASME Conference on Vibration and Noise, Montreal.
• C. Nataraj, H. D. Nelson, and N. Arakere, 1985, "An Analytical Study of a Rigid Rotor System With a Coulomb Spline," Instability in Rotating Machinery, NASA CP-2409, pp. 225-233.
• H. D. Nelson, C. Nataraj and W. J. Chen, 1989, Periodic Motion of Mechanical Systems With Nonlinear Components, NASA Report NAG 3-580. C. Nataraj, 1987, Periodic Oscillations in Nonlinear Mechanical Systems, Ph.D. dissertation, Arizona State University.