Solution methods for a system of nonlinear equations containing a convolution term are investigated in both frequency and time domains. Frequency domain analysis formulas are developed by using both the harmonic balance method (HBM) and trigonometric collocation method (TCM). A new technique, called alternation frequency/coefficient (AFC) technique, is proposed for both HBM and TCM. Its unique feature is that it can be used to evaluate any complicated multi-valued nonlinear frequency-response curves and to search for specified types of solutions with relatively few difficulties. Errors of HBM and TCM solutions are analyzed. Based on the Floquet method, an approximate algorithm for the stability analysis of periodic solutions is suggested. Time domain analysis formulas are developed using the Newmark-ß (NMK) integration method and Simpson's rule. The formulas provide a technique for performing time domain response analysis as well as stability analysis. Newton-Raphson's iteration scheme is used in both frequency and time domain analysis algorithms.
Numerical results of five examples show that HBM gives higher numerical accuracy, while TCM results in higher computing speed. The AFC technique proves to be very effective in evaluating complicated nonlinear responses. The effect of the convolution term is discussed. Nonlinear dynamic behaviors of a three-degree-of-freedom tension leg platform (TLP) model are studied for the total wave frequency, different frequency and sum frequency regions. Effects of different parameters are discussed. The nonlinearity in the model is primarily of geometrical nature due to tethers. Subharmonic responses of order 1/2 are found near double the surge natural frequency. The magnitudes of the subharmonics are much larger than those of linear and superharmonic responses. It is noted that large heave motion in the low frequency region is mainly due to the TLP's setdown, while tether elongation plays a major role in the heave motion within the high frequency region. Within either the low or high frequency regions where radiation damping is small, viscous damping has a very important effect on TLP motions.