0
RESEARCH PAPERS

Model Reduction of Systems With Localized Nonlinearities

[+] Author and Article Information
Daniel J. Segalman

 Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM 87185-0847djsegal@sandia.gov

J. Comput. Nonlinear Dynam 2(3), 249-266 (Jan 23, 2007) (18 pages) doi:10.1115/1.2727495 History: Received September 05, 2006; Revised January 23, 2007

An approach to development of reduced order models for systems with local nonlinearities is presented. The key of this approach is the augmentation of conventional basis functions with others having appropriate discontinuities at the locations of nonlinearity. A Galerkin solution using the above combination of basis functions appears to capture the dynamics of the system very efficiently—employing small basis sets. This method is particularly useful for problems of structural dynamics, but may have application in other fields as well. For problems involving small amplitude dynamics, when one employs as a basis the eigenmodes of a reference linear system plus the discontinuous (joint) modes, the resulting predictions, though still nonlinear, are approximated well as linear combinations of the eigenmodes. This is in good agreement with the experimental observation that jointed structures, though demonstrably nonlinear, manifest kinematics that are well described using eigenmodes of a corresponding system where the joints are replaced by linear springs.

Copyright © 2007 by American Society of Mechanical Engineers
Your Session has timed out. Please sign back in to continue.

References

Figures

Grahic Jump Location
Figure 1

A simple serial system of eleven unit masses connected to each other and ground by unit springs. A cubic spring is placed between the fifth and sixth masses.

Grahic Jump Location
Figure 2

The response of a joint with a small cubic nonlinearity appears almost linear so long as the loads are also small

Grahic Jump Location
Figure 3

The kinetic energy of the system with a small cubic nonlinearity resulting from a triangularly shaped impulse. The Galerkin solution employing various numbers of eigenmodes of the reference linear system provides a reasonably good approximation to this slightly nonlinear system.

Grahic Jump Location
Figure 4

The response of a joint with a cubic nonlinearity appears extremely nonlinear when the loads are large. In this case the peak dimensionless load is 0.5.

Grahic Jump Location
Figure 5

The kinetic energy of the system with a large cubic nonlinearity resulting from a triangularly shaped impulse. The Galerkin solution employing various numbers of eigenmodes of the reference linear system does not provide a good approximation to this nonlinear system unless the number of modes equals the total number of degrees of freedom of the physical system.

Grahic Jump Location
Figure 19

At a higher level of base excitation (A0=0.02), the use of a joint mode makes a more noticeable improvement in convergence

Grahic Jump Location
Figure 20

As was the case in the resonance calculation of Fig. 1, though the presence of the joint mode among the basis vectors of the Galerkin calculation greatly accelerates convergence, the amplitude of the generalized acceleration associated with that vector is actually fairly small in this base excitation problem (A0=0.02)

Grahic Jump Location
Figure 21

When the system is subject to a high amplitude (A0=0.05) base excitation, the joint is brought into macroslip and force levels in the joint are saturated at FS

Grahic Jump Location
Figure 22

The magnitude of displacement across the joint resulting from a high amplitude (A0=0.05) base excitation corresponds to the above portion of the motonic force–displacement curve for the joint. Also shown is the tangent stiffness at zero load. The nonlinearity manifest at these excitation levels is large.

Grahic Jump Location
Figure 23

The first POD mode of the full nonlinear spatial solution and the first eigenmode of the reference linear system are quite different in the vicinity of the joint when the system is subject to a high amplitude (A0=0.05) base excitation. The softening nature of the joint is responsible for the large displacement in the POD mode at the location of the joint.

Grahic Jump Location
Figure 24

Macroslip causes frequency responses of the structure well above that of the base excitation—which was tuned to the first resonance of the reference linear system

Grahic Jump Location
Figure 25

The Galerkin solution employing seven eigenmodes of RLS augmented by one joint predicts well the kinetic energy of the jointed system subject to a large amplitude impulse. A large number of elastic modes in addition to the joint mode are necessary to capture the high-frequency response of the system. The necessity of including the joint mode is illustrated by comparison to the predictions made without one. Predictions of a “ruthlessly reduced” model based on three elastic eigenmodes and one joint mode is also shown.

Grahic Jump Location
Figure 26

In this problem of macroslip the generalized acceleration of the joint coordinate is no longer small

Grahic Jump Location
Figure 27

A “ruthlessly reduced” analysis using only three elastic eigenmodes and one joint mode results in accelerations of the rightmost mass that have the appearance of a low-pass filter of the full spatial solution

Grahic Jump Location
Figure 28

When the “ruthlessly reduced” analysis using only three elastic eigenmodes and one joint mode and the full spatial solution are seen through a low pass filter, they appear very similar

Grahic Jump Location
Figure 6

The first proper orthogonal decomposition (POD) mode of the full history for the case of large cubic nonlinearity and the first eigenmode of the reference linear system (RLS). These modes show a marked difference at the location of the nonlinear spring. Because there is a stiffening spring between the fifth and sixth masses, the POD mode shows less deformation at that location than is the case of the eigenmode of the RLS.

Grahic Jump Location
Figure 7

Two manner of joint modes: the sensitivity of the first eigenmode of the reference linear system with respect to stiffness at the location of the nonlinear spring manifests a discontinuity there and a Milman–Chu joint mode

Grahic Jump Location
Figure 8

Convergence of the Galerkin procedure is greatly enhanced when the basis includes an eigenmode sensitivity vector. In this case there are four eigenmodes of the reference linear system and one eigenmode sensitivity vector.

Grahic Jump Location
Figure 9

Convergence of the Galerkin procedure is greatly enhanced when the basis includes an joint vector. In this case there are one eigenmode of the reference linear system and one Milman–Chu joint vector.

Grahic Jump Location
Figure 10

Force/displacement curves of mechanical joints manifest small regions of microslip where force–displacement appears linear, though some amount of dissipation accompanies any load. As the load increases, the tangent stiffness decreases until macroslip initiates.

Grahic Jump Location
Figure 11

The mathematical complexity of the joint is simplified by approximating the whole interface by a single scalar constitutive equation for each of the six relative degrees of freedom. In the illustration shown here all of the nodes on each side of the interface are held rigid and connected to a single joint node.

Grahic Jump Location
Figure 18

Even when the system is subject to a very low amplitude (A0=0.005) base excitation, the use of a joint mode makes a noticeable improvement in convergence

Grahic Jump Location
Figure 16

The history of force across the joint resulting from a very low amplitude (A0=0.005) base excitation

Grahic Jump Location
Figure 17

The magnitude of displacement across the joint in the numerical experiment of Fig. 1 corresponds to this monotonic-pull force–displacement curve. Also shown is the tangent stiffness at zero load.

Grahic Jump Location
Figure 12

Convergence of the Galerkin procedure is greatly enhanced by the presence of a joint vector in this problem involving the structure shown in Fig. 1, F0=0.5, and a nonlinear Iwan joint model. In this case there are three eigenmodes of the reference linear system and one joint vector.

Grahic Jump Location
Figure 13

Though the presence of the joint mode among the basis vectors of the Galerkin calculation greatly accelerates convergence, the amplitude of the generalized acceleration associated with that vector is actually fairly small in this problem

Grahic Jump Location
Figure 14

A nonlinear eleven-mass system excited at its base

Grahic Jump Location
Figure 15

The Morlet wavelet with ω=4

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In