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TECHNICAL PAPERS

Modal Analysis to Accommodate Slap in Linear Structures

[+] Author and Article Information
Daniel J. Segalman

 Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM 87185-0557djsegal@sandia.gov

Anthony M. Roy1

 Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM 87185-0557

Michael J. Starr

 Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM 87185-0557

1

Current graduate student, California Institute of Technology.

J. Vib. Acoust 128(3), 303-317 (Dec 09, 2005) (15 pages) doi:10.1115/1.2172257 History: Received October 28, 2004; Revised December 09, 2005

The generalized momentum balance (GMB) methods, explored chiefly by Shabana and his co-workers, treat slap or collision in linear structures as sequences of impulses, thereby maintaining the linearity of the structures throughout. Further, such linear analysis is facilitated by modal representation of the structures. These methods are discussed here and extended. Simulations on a simple two-rod problem demonstrate how this modal impulse approximation affects the system both directly after each impulse as well as over the entire collision. Furthermore, these simulations illustrate how the GMB results differ from the exact solution and how mitigation of these artifacts is achieved. Another modal method discussed in this paper is the idea of imposing piecewise constant forces over short, yet finite, time intervals during contact. The derivation of this method is substantially different than that of the GMB method, yet the numerical results show similar behavior, adding credence to both models. Finally, a novel method combining these two approaches is introduced. The new method produces physically reasonable results that are numerically very close to the exact solution of the collision of two rods. This approach avoids most of the nonphysical, numerical artifacts of interpenetration or chatter present in the first two methods.

Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 3

The problem of an elastic rod of initial velocity 2V0 striking an identical quiescent rod is considered

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Figure 14

The dimensionless gap between the rods predicted by the PCIF method for three different time steps for the case of ten modes. The larger time steps admit significant interpenetration while the smaller time steps prevent interpenetration, but at the cost of more dramatic “bounce.”

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Figure 15

The locations of the contact points χA on body A and χB on body B for the exact solution and for the MDA time integration method for the case of ten modes. The time steps in each case were selected to be one-third of the period of the highest mode.

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Figure 17

The locations of the contact points χA on body A and χB on body B for the exact solution and for the MDA time integration method for the cases of time steps of 0.01τ10, 0.1τ10, and 1.0τ10, where τ10 is the period of the highest (tenth) mode

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Figure 18

The locations of the contact points χA on body A and χB on body B for the exact solution and for the MDA time integration method for the case of ten modes for the first third of the physical contact period

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Figure 1

Schematic of two linear bodies undergoing intermittent slap. The development presented here assumes small rotations and translations.

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Figure 2

Normal impact occurs at locations χA and χB with a direction of approach collinear with the normal to both bodies at the point of impact. This is not the case for oblique impact.

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Figure 9

Total system energy versus time for the GMB method for the case of ten modes and three values of CR: 0.5, 0.75, and 1.0. The curves associated with dissipative contact are terminated when the intercontact interval becomes so small that further integration is prevented by the limits of machine precision. As expected, there is some energy loss from impacts with CR<1.

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Figure 10

The locations of the contact points χA on body A and χB on body B for the exact solution and for the PCIF method for cases of 5, 10, and 20 modes. The time steps in each case were selected to be one-tenth of the period of the highest mode.

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Figure 11

The time distribution of contact forces for the PCIF method for the case of ten modes and three values of time step

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Figure 12

The locations of the contact points χA on body A and χB on body B for the exact solution and for the PCIF method for the cases of time steps of 0.01τ10, 0.1τ10, and 1.0τ10, where τ10 is the period of the highest (tenth) mode

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Figure 13

The locations of the contact points χA on body A and χB on body B for the exact solution and for the PCIF method for the case of ten modes for the first fourth of the physical contact period

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Figure 16

The time distribution of impulses and contact forces for the MDA time integration method for the case of ten modes and dt=0.1τ10. The impulses are normalized by (MAVA+MBVB)∕20 and the contact forces are normalized by EAV0∕(2c).

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Figure 19

Total system energy versus time for the MDA time integration method for the cases of 5, 10, and 20 modes and CR=0

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Figure 4

The locations of the contact points χA on body A and χB on body B for the exact solution and for the GMB method for cases of 5, 10, and 20 modes. In these calculations, the local coefficient of restitution CR is unity.

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Figure 5

The distribution of dimensionless stress 2cσ∕(V0E) at dimensionless time tc∕L for the exact solution and for the GMB method for cases of 5, 10, and 20 modes. In these calculations, the local coefficient of restitution CR is unity.

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Figure 6

The time distribution of impulses for the GMB method for the case of ten modes and the local coefficient of restitution CR equal to unity

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Figure 7

The length of time intervals between contacts for the cases of ten modes and three values of local coefficient of restitution are shown. For CR=1 the intervals are approximately uniform. For the cases of CR=0.75 and CR=0.5 the time intervals appear to decrease as geometric sequences until further time integration is prevented by the limits of machine precision.

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Figure 8

The locations of the contact points χA on body A and χB on body B for the GMB method for the case of 10 modes and two values of CR: CR=1 and CR=0.75, over the first fourth of contact period

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