Limiting behaviors of randomly excited hyperbolic tangent systems

We investigate the steady-state probability density distribution of a large class of random processes by solving the governing Fokker-Planck equation. The random response statistics of a nonlinear single-degree-of-freedom mechanical model with hyperbolic tangent stiffness are discussed in some detail. The probability density of such systems is of the sech-power type which belongs to a class of distributions whose behaviors are carefully examined at the limits where the system parameter b approaches zero and infinity. Other important response statistics such as the mean square response, zero crossings, and peak distributions are also studied.