Eigenstructure assignment for vibrating systems with nonsmooth nonlinearities

In this study, a technique for eigenstructure assignment of multi-degree-of-freedom vibrating systems with nonsmooth (piecewise linear) nonlinearities is presented. In such systems, only the exact eigenfrequencies and mode shapes (eigenvectors) of individual linear subregions are known, while the composite nonlinear normal modes (NNMs) occur on curved invariant manifolds in the configuration space. Three distinct control strategies which utilize methods for approximating the NNM frequencies and mode shapes are employed. These methods are based on previous extensions of the well-known bilinear frequency relation (BFR) to account for a nonvanishing clearance. The first strategy, which employs the piecewise modal method (PMM) for approximating NNM frequencies, determines n constant actuator gains for an n degree-of-freedom system and results in eigenvalue (pole) placement only. The second strategy, which also results in eigenvalue placement, involves finding an approximate single-degree-of-freedom reduced model with one actuator gain for the mode to be controlled. The third strategy allows the designer to shape the frequencies and mode shapes (eigenstructure) by using a full n x n matrix of actuator gains and employing the local equivalent linear stiffness method (LELSM) for approximating NNM frequencies and mode shapes. These strategies are implemented via constant-gain feedback control and thus do not require the use of gain switching. The techniques are applied to two degree-of-freedom illustrative examples with two distinct types of nonlinearities: a bilinear clearance nonlinearity and a symmetric deadzone nonlinearity.