Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications (original) (raw)

Order Reduction of Parametrically Excited Nonlinear Systems Subjected to External Periodic Excitations

Volume 4: 7th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B and C, 2009

The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered in state space and in direct second order (structural) form. In state space order reduction methods, the equations of motion are expressed as a set of first order equations and transformed using the Lyapunov-Floquet (L-F) transformation such that the linear parts of new set of equations are time invariant. At this stage, four order reduction methodologies, namely linear, nonlinear projection via singular perturbation, post-processing approach and invariant manifold technique, are suggested. The invariant manifold technique yields a unique 'reducibility condition' that provides the conditions under which an accurate nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An alternate approach of deriving reduced order models in direct second order form is also presented. Here the system is converted into an equivalent second order nonlinear system with time invariant linear system matrices and periodically modulated nonlinearities via the L-F and other canonical transformations. Then a master-slave separation of degrees of freedom is used and a nonlinear relation between the slave coordinates and the master coordinates is constructed. This method yields the same 'reducibility conditions' obtained by invariant manifold approach in state space. Some examples are given to show potential applications to real problems using above mentioned methodologies. Order reduction possibilities and results for various cases including 'parametric', 'internal', 'true internal' and 'true combination resonances' are discussed. A generalization of these ideas to periodic-quasiperiodic systems is included and demonstrated by means of an example.

Order Reduction of Nonlinear Time Periodic Systems Using Invariant Manifolds

Volume 5: 19th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C, 2003

The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered. First, the equations of motion are transformed using the Lyapunov-Floquet transformation such that the linear parts of new set of equations are time invariant. At this stage, the linear order reduction technique can be applied in a straightforward manner. A nonlinear order reduction methodology is also suggested through a generalization of the invariant manifold technique via 'Time Periodic Center Manifold Theory'. A 'reducibility condition' is derived to provide conditions under which a nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An example consisting of two parametrically excited coupled pendulums is given to show potential applications to real problems. Order reduction possibilities and results for various cases including 'parametric', 'internal', 'true internal' and 'combination' resonances are discussed. r

Order reduction of nonlinear systems with time periodic coefficients using invariant manifolds

Journal of Sound and Vibration, 2005

Some Techniques for Order Reduction of Nonlinear Time Periodic Systems

Design Engineering, Volumes 1 and 2, 2003

This work reports new approaches for order reduction of nonlinear systems with time periodic coefficients. First, the equations of motion are transformed using the Lyapunov-Floquet (LF) transformation, which makes the linear part of new set of equations time invariant. At this point, either linear or nonlinear order reduction methodologies can be applied. The linear order reduction technique is based on classical technique of aggregation and nonlinear technique is based on 'Time periodic invariant manifold theory'. These methods do not assume the parametric excitation term to be small. The nonlinear order reduction technique yields superior results. An example of two degrees of freedom system representing a magnetic bearing is included to show the practical implementation of these methods. The conditions when order reduction is not possible are also discussed.

Order reduction of nonlinear quasi-periodic systems subjected to external excitations

International Journal of Non-Linear Mechanics, 2022

In his paper, we present order reduction techniques for nonlinear quasi-periodic systems subjected to external excitations. The order reduction techniques presented here are based on the Lyapunov-Perrone (L-P) Transformation. For a class of non-resonant quasi-periodic systems, the L-P transformation can convert a linear quasi-periodic system into a linear time-invariant one. This Linear Time-Invariant (LTI) system retains the dynamics of the original quasi-periodic system. Once this LTI system is obtained, the tools and techniques available for analysis of LTI systems can be used, and the results could be obtained for the original quasi-periodic system via the L-P transformation. This approach is similar to using the Lyapunov-Floquet (L-F) transformation to convert a linear time-periodic system into an LTI system and perform analysis and control. Order reduction is a systematic way of constructing dynamical system models with relatively smaller states that accurately retain large-scale systems' essential dynamics. In this work, reduced-order modeling techniques for nonlinear quasi-periodic systems subjected to external excitations are presented. The methods proposed here use the L-P transformation that makes the linear part of transformed equations time-invariant. In this work, two order reduction techniques are suggested. The first method is simply an application of the well-known Guyan like reduction method to nonlinear systems. The second technique is based on the concept of an invariant manifold for quasi-periodic systems. The 'quasi-periodic invariant manifold' based technique yields' reducibility conditions.' These conditions help us to understand the various types of resonant interactions in the system. These resonances indicate energy interactions between the system states, nonlinearity, and external excitation. To retain the essential dynamical characteristics, one has to preserve all these 'resonant' states in the reducedorder model. Thus, if the 'reducibility conditions' are satisfied then only, a nonlinear order reduction based on the quasi-periodic invariant manifold approach is possible. It is found that the invariant manifold approach yields good results. These methodologies are general and can be used for parametric study, sensitivity analysis, and controller design. The reducibility of a linear quasi-periodic system has been the subject of research in the scientific community. Many excellent references discuss the reducibility of quasi-periodic systems [20-26]. Recently, Waswa and Redkar presented a technique based on L-F transformation, state augmentation, and normal form to reduce linear quasi-periodic system into an LTI system [21]. Very recently, Subramanian and Redkar presented a method to compute L-P transformation based on intuitive state

Order Reduction of nonlinear time periodic systems subjected to external excitations

2009

A New Technique For Reduced-Order Modelling of Linear Time-Invariant System

IETE Journal of Research, 2017

In this paper, a new technique for order reduction of linear time-invariant systems is presented. This technique is intended for both single-input single-output (SISO) and multi-input multi-output (MIMO) systems. Motivated by other reduction techniques, the new proposed reduction technique is based on modified pole clustering and factor division algorithm with the objective of obtaining a stable reduced-order system preserving all essential properties of the original system. The new technique is illustrated by three numerical examples which are considered from the literature. To evaluate the superiority and robustness of the new technique, the results of the proposed technique are compared with other well-known and recently developed order-reduction techniques like Routh approximation and Big Bang-Big Crunch algorithm. The comparison of performance indices shows the efficiency and powerfulness of the new technique.

Efficient model order reduction for dynamic systems with local nonlinearities

Journal of Sound and Vibration, 2014

ABSTRACT In the nonlinear structural analysis, the nonlinear effects are commonly localized and the rest of the structure behaves in a linear manner. Considering this fact, this research work proposes a harmonic balance solution in order to determine the nonlinear response of the structures. The solution is simplified by using an exact dynamic reduction along with the modal expansion technique. This novel approach, which is applicable to both discrete and continuous systems, converts the system equations of motion in each harmonic to a small set of nonlinear algebraic equations. The full set of system equations is reduced to a discrete system with a few generalized degrees of freedom (DOFs) confined to the localized nonlinear regions. The resultant reduced order model is shown to be accurate enough for determining the periodic response. To demonstrate the capability of the proposed method, numerical case studies for continuous and discrete systems, including systems with internal resonance, have been studied and the outcomes are validated with benchmark studies. In addition, the method is applied in the identification process of an experimental test setup with unknown frictional support parameters, and the results are presented and discussed.

Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications (original) (raw)

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