Analytical Jacobian and sensitivity analysis of path-following for harmonically forced mechanical systems with non-smooth differential hysteresis

We investigate the performance of an efficient analytical framework for computing the Jacobian of the Poincaré map in harmonically forced mechanical systems whose rate-independent hysteresis is described by first-order differential, piecewise-smooth constitutive laws. The general formulation is further extended to the autonomous problem by exploiting a fourth-order Runge–Kutta integration scheme. Closed-form expressions for the sensitivity matrices are derived, leading to simplified evaluations with respect to both periodic solutions and control parameters. The proposed extended approach is then applied to a single-degree-of-freedom system featuring symmetric hysteresis loops, simulated using both the Bouc–Wen and the differential Vaiana–Rosati models, to confirm that the frequency response curves obtained via path-following continuation coincide. Furthermore, numerical investigations are used to assess the effectiveness of the proposed strategy for systems having one or two degrees of freedom. In the former case the number of integration points, fixed step sizes, and adaptive correction strategies are investigated towards accuracy and computational efficiency. Finally, as a representative example of multi-degree-of-freedom systems, the efficiency of the analytical Jacobian formulation is evaluated against its numerical counterpart for two-degrees-of-freedom systems, highlighting the associated trade-offs and convergence characteristics.