A technique for order reduction of dynamic systems in structural form with static piecewise linear nonlinearities is presented. By utilizing two methods which approximate the nonlinear normal mode (NNM) frequencies and mode shapes, reduced-order models are constructed which more accurately represent the dynamics of the full model than do reduced models obtained via standard linear transformations. One method builds a reduced-order model which is dependent on the amplitude (initial conditions) while the other method results in an amplitude-independent reduced model. The two techniques are first applied to reduce two-degree-of-freedom undamped systems with clearance, deadzone, bang-bang, and saturation stiffness nonlinearities to single-mode reduced models which are compared by direct numerical simulation with the full models. It is then shown via a damped fourdegree-of-freedom system with two deadzone nonlinearities that one of the proposed techniques allows for reduction to multi-mode reduced models and can accommodate multiple nonsmooth static nonlinearities with several surfaces of discontinuity. The advantages of the proposed methods include obtaining a reducedorder model which is signal-independent (doesn't require direct integration of the full model), uses a subset of the original physical coordinates, retains the form of the nonsmooth nonlinearities, and closely tracks the actual NNMs of the full model.

We present a technique for using constant-gain linear position feedback control to implement eigenstructure assignment of n-degrees-of-freedom conservative structural systems with piecewise linear nonlinearities. We employ three distinct control strategies which utilize methods for approximating the nonlinear normal mode (NNM) frequencies and mode shapes. First, the piecewise modal method (PMM) for approximating NNM frequencies is used to determine n constant actuator gains for eigenvalue (pole) placement. Secondly, eigenvalue placement is accomplished by finding an approximate single-degree-of-freedom reduced model with one actuator gain for the mode to be controlled. The third strategy allows the frequencies and mode shapes (eigenstructure) to be placed by using a full n 1 n matrix of actuator gains and employing the local equivalent linear stiffness method (LELSM) for approximating NNM frequencies and mode shapes. The techniques are applied to a two-degrees-of-freedom system with two distinct types of nonlinearities: a bilinear clearance nonlinearity and a symmetric deadzone nonlinearity.

A technique for order reduction of nonsmooth vibrating systems in structural form of arbitrary dimension with multiple surfaces of discontinuity is presented. By utilizing methods based on the bilinear frequency relation which approximates the nonlinear normal mode (NNM) frequencies and mode shapes, reduced order models are constructed which retain the form of the nonsmooth nonlinearity of the full model and more accurately represent the NNM dynamics in the full model than do reduced models obtained via linear transformations. The technique is applied to multi-degree-of-freedom systems with nonsmooth nonlinearities of deadzone and saturation type in which the full and reduced models are compared by direct numerical simulation. The advantages of the present technique include obtaining a reduced order model which uses a subset of the original physical coordinates and can easily accommodate large order systems and multiple nonsmooth nonlinearities with several surfaces of discontinuity. These characteristics make the method practical for use in large-scale structural dynamics applications in which the linear part of the model dominates the dynamics.

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