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A Study of the Longterm Dynamics of a Discretization of a Light Viscoelastic Rod Carrying a Heavy Block

Fang, Fang
http://rave.ohiolink.edu/etdc/view?acc_num=akron1303152538

Abstract Details

2011, Master of Science, University of Akron, Applied Mathematics.
A heavy rigid body attached to a relatively light deformable body is a common structure in many mechanical systems. The longterm dynamics of this structure characterize the true features of the structure response to external excitation. Therefore, the study of the longterm dynamics can provide fundamental guidance for engineering design and production. In this thesis, we study a system that describes the longitudinal motion of a light viscoelastic rod carrying a heavy particle. We discretize this problem by replacing the rod with K light particles connected by massless springs. The discretized problem is governed by a system of coupled nonlinear ordinary differential equations. We let ε denote the ratio of the mass of one of the light particles to the mass of the end particle. This introduces a small parameter into the problem. We use a singular perturbation approach developed by O’Malley and Hoppensteadt to analyze this problem. We focus on the K = 1 case; and the longterm dynamics of this case is reduced to the longterm dynamics of the order 1 outer problem based on the matching condition and the approximate solutions. The longterm dynamics of the order 1 outer problem depends on whether the restoring force f of the springs in the discretized system is monotone or non-monotone. For monotone restoring forces, we show that there exists an invariant manifold 𝓜 that attracts all solutions to a system equivalent to the order 1 outer problem. The dynamics on 𝓜 is governed by a classical second-order ordinary differential equation. We also illustrate that the longterm dynamics on 𝓜 determines the longterm dynamics of the order 1 outer problem. For non-monotone restoring forces, the invariant manifold 𝓜 still exists. We show by considering a specific example that 𝓜 fails to attract all solutions.
J. Patrick Wilber, PhD (Advisor)

Recommended Citations

Citations

  • Fang, F. (2011). A Study of the Longterm Dynamics of a Discretization of a Light Viscoelastic Rod Carrying a Heavy Block [Master's thesis, University of Akron]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=akron1303152538

    APA Style (7th edition)

  • Fang, Fang. A Study of the Longterm Dynamics of a Discretization of a Light Viscoelastic Rod Carrying a Heavy Block. 2011. University of Akron, Master's thesis. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=akron1303152538.

    MLA Style (8th edition)

  • Fang, Fang. "A Study of the Longterm Dynamics of a Discretization of a Light Viscoelastic Rod Carrying a Heavy Block." Master's thesis, University of Akron, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=akron1303152538

    Chicago Manual of Style (17th edition)

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© 2011, all rights reserved.
This open access ETD is published by University of Akron and OhioLINK.