Current Concepts in Engineering Research and Technology Vol. 3

The stochastic response of nonlinear dynamic systems is a significant concern in engineering vibration analysis, particularly when analytical solutions are difficult to obtain. This chapter examines the response of the Duffing oscillator subjected to white-noise random forcing by using an equivalent-linearisation approach supported by transformation methods and numerical evaluation. The formulation first considers a general n-degree-of-freedom system acted on by multiple random excitations, where the response is described through mechanical transfer functions, auto-spectral density functions and cross-spectral density functions. The analysis then reduces the framework to a single-degree-of-freedom Duffing oscillator with nonlinear restoring force, allowing the equivalent linear parameters, displacement variance, velocity variance and power spectral density of the response to be evaluated. Numerical examples are presented for two sets of system parameters, showing how changes in mass, stiffness, damping and nonlinear coefficient influence the equivalent frequency parameter and the form of the response spectrum. The results indicate that equivalent linearisation provides a tractable approximation for estimating the principal resonant behaviour of a nonlinear oscillator under the stated assumptions. However, the interpretation remains restricted to the considered parameter ranges, the use of Gaussian white-noise excitation and the simplified representation of nonlinear stiffness. The chapter therefore presents the method as an illustrative analytical and computational framework for stochastic vibration problems rather than as a universal solution for all nonlinear random systems.