The Spectrum of a Simple Nonlinear System

The recent advance in space and communicational technologies has led engineers to numerous difficult but fascinating problems in regard to the structural dynamics in random environments. For example, Hempstead and Lax have investigated noise in self-sustained oscillation; 1 , 2 Ariaratnam and Sanker have studied the dynamic snapthrough of shallow, arch-type aircraft components under stochastic pressure. 3 In this paper the random vibration of a simple mass with nonlinear clamping is studied. The nonlinearity of the system is introduced to linear viscous clamping by adding to it an extra term which is inversely proportional to the first power of the current velocity. Emphasis of the analysis is placed on finding the power spectral density of the random motion. Two different approaches are used to obtain the desired solution. First, the exact spectrum is found by solving the associated nonstationary Fokker-Planck equation in terms of the eigenfunction expansion of the degenerate ordinary differential equation. Second, approximate solutions are obtained by using the equivalent linearization technique by which the original nonlinear system is converted to an equivalent linear one. The equivalent linear system, constructed by the least mean square error criterion and based on the small nonlinearity assumption, is then solved by standard linear theory. 2259