Nonlinear Dynamics

, Volume 64, Issue 1–2, pp 97–115 | Cite as

Computational dynamics of a 3D elastic string pendulum attached to a rigid body and an inertially fixed reel mechanism

  • Taeyoung Lee
  • Melvin Leok
  • N. Harris McClamroch
Open Access
Original Paper


A high fidelity model is developed for an elastic string pendulum, one end of which is attached to a rigid body while the other end is attached to an inertially fixed reel mechanism which allows the unstretched length of the string to be dynamically varied. The string is assumed to have distributed mass and elasticity that permits axial deformations. The rigid body is attached to the string at an arbitrary point, and the resulting string pendulum system exhibits nontrivial coupling between the elastic wave propagation in the string and the rigid body dynamics. Variational methods are used to develop coupled ordinary and partial differential equations of motion. Computational methods, referred to as Lie group variational integrators, are then developed, based on a finite element approximation and the use of variational methods in a discrete-time setting to obtain discrete-time equations of motion. This approach preserves the geometry of the configurations, and leads to accurate and efficient algorithms that have guaranteed accuracy properties that make them suitable for many dynamic simulations, especially over long simulation times. Numerical results are presented for typical examples involving a constant length string, string deployment, and string retrieval. These demonstrate the complicated dynamics that arise in a string pendulum from the interaction of the rigid body motion, elastic wave dynamics in the string, and the disturbances introduced by the reeling mechanism. Such interactions are dynamically important in many engineering problems, but tend be obscured in lower fidelity models.


Lagrangian mechanics Geometric integrator Variational integrator String pendulum Reel mechanism Rigid body 


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Copyright information

© The Author(s) 2010

Authors and Affiliations

  • Taeyoung Lee
    • 1
  • Melvin Leok
    • 2
  • N. Harris McClamroch
    • 3
  1. 1.Department of Mechanical and Aerospace EngineeringFlorida Institute of TechnologyMelbourneUSA
  2. 2.Department of MathematicsUniversity of CaliforniaSan DiegoUSA
  3. 3.Department of Aerospace EngineeringUniversity of MichiganAnn ArborUSA

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