The analysis of continuous systems with nonlinearities in their domain have previously been limited to either numerical approaches, or analytical methods that are constrained in the parameter space, boundary conditions, or order of the system. The present analysis develops a robust method for studying continuous systems with arbitrary boundary conditions and nonlinearities using the assumption that the nonlinear constraint can be modeled with a piecewise-linear force-deflection constitutive relationship. Under this assumption, a superposition method is used to generate homogeneous boundary conditions, and modal analysis is used to find the displacement of the system in each state of the piecewise-linear nonlinearity. In order to map across each nonlinearity in the piecewise-linear force-deflection profile, a variational calculus approach is taken that minimizes the L2 energy norm between the previous and current states. To illustrate this method, a leaf spring coupled with a connector pin immersed in a viscous fluid is modeled as a beam with a piecewise-linear constraint. From the results of the convergence and parameter studies, a high correlation between the finite-time Lyapunov exponents and the contact time per period of the excitation is observed. The parameter studies also indicate that when the system's parameters are changed in order to reduce the magnitude of the velocity impact between the leaf spring and connector pin, the extent of the regions over which a chaotic response is observed increases.
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