Order reduction of structural dynamic systems with static piecewise linear nonlinearities

Abstract

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.

Order reduction of forced nonlinear systems using updated LELSM modes with new Ritz vectors

“…The local equivalent linear stiffness method (LELSM) has been previously utilized in [6] and [32,33]. This method is used here for finding the local equivalent linear stiffnesses of the nonlinear springs.…”

“…Extensions of Guyan reduction have been proposed that include inertial as well as stiffness effects in the order reduction transformation [2]. These linear-based Guyanlike order reduction techniques have also been applied to nonlinear dynamic systems with weak static and damping nonlinearities [3][4][5][6].…”

“…Most recently, a new technique for order reduction of dynamic systems with static piecewise linear nonlinearities was proposed based on previously developed methods for approximation of the NNM frequencies and mode shapes via either an amplitudeindependent piecewise modal method (PMM) or an amplitude-dependent local equivalent linear stiffness method (LELSM) [6,32,33]. The technique was further improved over the previously developed methods for order reduction of large degree-of-freedom unforced nonlinear systems with grounded nonlinearities via iteration of the LELSM modes.…”

Experimental evidence of bifurcating nonlinear normal modes in piecewise linear systems

“…(3), can be found in [18]. Some fundamental properties are recalled in the following [18,19,22,23]: i) since the nonlinear responses are independent of the energy level, as said before, the trajectories of NNM with the same frequency and different energy levels form a set of homothetic curves: thus, by fixing a certain level of the total energy, the initial conditions can be expressed by only one independent parameter; ii) the trajectories of NNMs in the configuration space are open curves and generally do not pass through the origin; iii) for most nonlinear systems the number of NNMs is at least equal to the number of normal modes of the underlying linear system: we denote these as fundamental NNMs with fundamental frequencies ω 1 and ω 2 ; iv) the nonlinear frequency ratio r= ω 2 /ω 1 only depends on the mass and stiffness ratios as well as on damage parameters ε i , being r 0 the frequency ratio of the undamaged system; v) as the level of nonlinearity increases, additional NNMs (superabundant) generated by bifurcation mechanisms can be observed.…”

Closed-form periodic solutions for piecewise-linear vibrating systems

“…In many cases, the corresponding periodic solutions can be combined of different pieces of linear solutions valid for two different subspaces of the configuration space characterized by constant but different stiffness matrixes [2,[4][5][6][7]. It is always preferable though to deal with closed-form solutions, especially when the solutions are involved in further transformations, for instance-perturbation procedures [8].…”