Order Reduction of Parametrically Excited Linear and Nonlinear Structural Systems

2005 ◽  
Vol 128 (4) ◽  
pp. 458-468 ◽  
Author(s):  
Venkatesh Deshmukh ◽  
Eric A. Butcher ◽  
S. C. Sinha
Keyword(s):  
Nonlinear Systems ◽  
Order Reduction ◽  
Second Order ◽  
Degree Of Freedom ◽  
Structural Systems ◽  
Parametrically Excited ◽  
Reduced Models ◽  
Linear And Nonlinear ◽  
Time Invariant ◽  
Time Periodic

Order reduction of parametrically excited linear and nonlinear structural systems represented by a set of second order equations is considered. First, the system is converted into a second order system with time invariant linear system matrices and (for nonlinear systems) periodically modulated nonlinearities via the Lyapunov-Floquet transformation. Then a master-slave separation of degrees of freedom is used and a relation between the slave coordinates and the master coordinates is constructed. Two possible order reduction techniques are suggested. In the first approach a constant Guyan-like linear kernel which accounts for both stiffness and inertia is employed with a possible periodically modulated nonlinear part for nonlinear systems. The second method for nonlinear systems reduces to finding a time-periodic nonlinear invariant manifold relation in the modal coordinates. In the process, closed form expressions for “true internal” and “true combination” resonances are obtained for various nonlinearities which are generalizations of those previously reported for time-invariant systems. No limits are placed on the size of the time-periodic terms thus making this method extremely general even for strongly excited systems. A four degree-of-freedom mass- spring-damper system with periodic stiffness and damping as well as two and five degree-of-freedom inverted pendula with periodic follower forces are used as illustrative examples. The nonlinear-based reduced models are compared with linear-based reduced models in the presence and absence of nonlinear resonances.

scholarly journals Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications

2005 ◽  
Vol 41 (1-3) ◽  
pp. 237-273 ◽  
Author(s):  
S. C. SINHA ◽  
SANGRAM REDKAR ◽  
VENKATESH DESHMUKH ◽  
ERIC A. BUTCHER
Keyword(s):  
Nonlinear Systems ◽  
Order Reduction ◽  
Parametrically Excited

Order Reduction of Nonlinear Time Periodic Systems Using Invariant Manifolds

2003 ◽  
Author(s):  
S. C. Sinha ◽  
Sangram Redkar ◽  
Eric A. Butcher ◽  
Venkatesh Deshmukh
Keyword(s):  
Center Manifold ◽  
Invariant Manifolds ◽  
Basic Problem ◽  
Equations Of Motion ◽  
Order Reduction ◽  
Center Manifold Theory ◽  
Manifold Theory ◽  
Time Invariant ◽  
Time Periodic ◽  
Linear And Nonlinear Systems

The basic problem of order reduction of linear and 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 applications to real problems. Order reduction possibilities and results for various cases including ‘parametric’, ‘internal’, ‘true internal’ and ‘combination’ resonances are discussed.

Symbolic Bifurcation Boundaries and Time-Dependent Resonance Sets of Time-Periodic Nonlinear Systems

1999 ◽  
Author(s):  
Eric A. Butcher ◽  
S. C. Sinha
Keyword(s):  
Nonlinear Systems ◽  
Parameter Space ◽  
Local Stability ◽  
Degree Of Freedom ◽  
Time Dependent ◽  
Periodic Systems ◽  
Stability Boundaries ◽  
System Parameters ◽  
Parameter Dependent ◽  
Time Periodic

Abstract A recent computational technique is utilized for symbolic computation of local stability boundaries and bifurcation surfaces for nonlinear multidimensional time-periodic dynamical systems as an explicit function of the system parameters. This is made possible by the recent development of a symbolic computational algorithm for approximating the parameter-dependent fundamental solution matrix of linear time-periodic systems. By evaluating this matrix at the end of the principal period, the parameter-dependent Floquet Transition Matrix (FTM), or the linear part of the Poincaré map, is obtained. The subsequent use of well-known criteria for the local stability and bifurcation conditions of equilibria and periodic solutions enables one to obtain the equations for the bifurcation surfaces in the parameter space as polynomials of the system parameters. Because this method is not based on expansion in terms of a small parameter, it can successfully be applied to periodic systems whose internal excitation is strong. In addition, the time-dependent normal forms and resonance sets for one and two degree-of-freedom time-periodic nonlinear systems are analyzed. For this purpose, the Liapunov-Floquet (L-F) transformation is employed which transforms the periodic variational equations into an equivalent form in which the linear system matrix is constant. Both quadratic and cubic nonlinearities are investigated, and all possible cases for the single degree-of-freedom case are studied. The above algorithm for computing stability boundaries may also be employed to compute the time-dependent resonance sets of zero measure in the parameter space. Two illustrative example problems, viz., a parametrically excited simple pendulum and a double inverted pendulum subjected to a periodic follower force, are included.

Stability and Control of a Parametrically Excited Rotating Beam

1998 ◽  
Vol 120 (4) ◽  
pp. 462-470 ◽  
Author(s):  
S. C. Sinha ◽  
Dan B. Marghitu ◽  
Dan Boghiu
Keyword(s):  
Equations Of Motion ◽  
System Stability ◽  
Flexible Beam ◽  
Parametrically Excited ◽  
Stability And Control ◽  
Rotation Parameters ◽  
The Stability ◽  
And Control ◽  
Time Invariant ◽  
Time Periodic

In this paper the stability and control of a parametrically excited, rotating flexible beam is considered. The equations of motion for such a system contain time periodic coefficients. Floquet theory and a numerical integration are used to evaluate the stability of the linearized system. Stability charts for various sets of damping, parametric excitation, and rotation parameters are obtained. Several resonance conditions are found and it is shown that the system stability can be significantly changed due to the rotation. Such systems can be used as preliminary models for studying the flap dynamics and control of helicopter rotor blades and flexible mechanisms among other systems. To control the motion of the system, an observer based controller is designed via Lyapunov-Floquet transformation. In this approach the time periodic equations are transformed into a time invariant form, which are suitable for the application of standard time invariant controller design techniques. Simulations for several combinations of excitation and rotation parameters are shown.

Order Reduction and Control of Large-Scale Linear Time Periodic Dynamical Systems

1999 ◽  
Author(s):  
Venkatesh Deshmukh ◽  
S. C. Sinha
Keyword(s):  
Large Scale ◽  
Linear Time ◽  
Parametric Instability ◽  
Order Reduction ◽  
System Stability ◽  
Order System ◽  
Control Laws ◽  
Reduced Order ◽  
Time Invariant ◽  
Time Periodic

Abstract This paper provides methodology for designing reduced order controllers for large-scale, linear systems represented by differential equations having time periodic coefficients. The linear time periodic system is first converted into a form in which the system stability matrix is time invariant. This is achieved by the application of Liapunov-Floquet transformation. Then a system called an auxiliary system is constructed which is a completely time invariant. Order reduction algorithms are applied to this system to obtain a reduced order system. The control laws are calculated for the reduced order system by minimizing the least square error between the auxiliary and the transformed system. These control laws when transformed back to time varying domain provide the desired control action. The schemes formulated are illustrated by designing full state feedback and output feedback controllers for a five mass inverted pendulum exhibiting parametric instability.

Model Order Reduction for Pre-Stressed Harmonic Analysis of Micromechanical Beam Resonators

2010 ◽  
Vol 40-41 ◽  
pp. 739-743
Author(s):  
Jian Shi ◽  
Ben Lian Xu
Keyword(s):  
Harmonic Analysis ◽  
Model Order Reduction ◽  
Substantial Reduction ◽  
Order Reduction ◽  
Second Order ◽  
Order System ◽  
Element Analysis ◽  
Model Order ◽  
Reduced Models ◽  
Dimensional Approximation

A second-order system model order reduction method for pre-stressed harmonic analysis of electrostatically actuated microbeams is demonstrated, which produces a low dimensional approximation of the original system and enables a substantial reduction of simulation time. The moment matching property for second-order dynamic systems is studied and the block Arnoldi algorithm is adopted for the generation of the Krylov subspace, which extracts the low order model from the discretized system assembled through finite element analysis. The difference between two successive reduced models suggests the choice of the order for the reduce model. A detailed comparison research among the full model and the reduced models is performed. The research results confirm the effectiveness of the presented method.

Some Techniques for Order Reduction of Nonlinear Time Periodic Systems

2003 ◽  
Author(s):  
Sangram Redkar ◽  
S. C. Sinha ◽  
Eric A. Butcher
Keyword(s):  
Nonlinear Systems ◽  
Invariant Manifold ◽  
Equations Of Motion ◽  
Linear Method ◽  
Order Reduction ◽  
Reduced Order ◽  
Reduction Techniques ◽  
Nonlinear Projection ◽  
Short Period ◽  
Time Periodic

In this paper, some techniques for order reduction of nonlinear systems with time periodic coefficients are introduced. The equations of motion are first trasformed using the Lyapunov-Floquet transformation such that the linear parts of the new set of equations are time-invariant. To reduce the order of this transformed system three model reduction techniques are suggested. The first approach is simply an application of the well-known linear method to nonlinear systems. In the second technique, the idea of singular perturbation and noninear projection are employed, whereas the concept of invariant manifold for time-periodic system forms the basis for the third method. A discussion of nonlinear projection method and time periodic invariant manifold technique is included. The invariant manifold based technique yields a ‘reducibility condition’. This is an important result due to the fact that various types of resonance are present in such systems. If the ‘reducibility condition’ is satisfied only then a nonlinear order reduction is possible. In order to compare the results obtained from various reduced order modeling techniques, an example consisting of two parametrically excited coupled pendulums is included. Reduced order results and full-scale dynamics are used to construct approximate and exact Poincare´ maps, respectively, because it portrays the long-term behavior of system dynamics. This measure is more convincing than just comparing the time traces over a short period of time. It is found that the invariant manifold yields the most accurate results followed by the nonlinear projection and the linear techniques.

Sign in / Sign up

Export Citation Format

Share Document