Random response evaluation of mono-stable and bi-stable Duffing systems with Dahl friction

The random responses of Duffing systems with dynamic friction are investigated, and a uniform procedure is established to derive the explicit expressions of stationary probability densities for mono-stable potential and bi-stable potential cases. The Dahl friction model is adopted to describe the dynamic friction behavior, and the friction force is first expressed as a time-domain integration of the velocity function by directly integrating the auxiliary differential equation with non-smooth term. By introducing the associated generalized harmonic transformations for mono-stable potential and bi-stable potential cases, the friction force is further simplified and only approximately depends on the present velocity, amplitude and phase. The friction force is then mandatorily separated as conservative and dissipative components through the quasi-linearization technique. At last, the mono-stable and bi-stable Duffing systems with dynamic friction are equivalent to approximate nonlinear systems with modified damping and modified stiffness. The stationary probability densities of the original friction systems are evaluated by those of the approximate nonlinear systems, which can be analytically expressed. The effectiveness and accuracy of the uniform procedure are verified by comparison with the Monte Carlo simulation for mono-stable potential (including hardening stiffness and softening stiffness) and bi-stable potential cases.