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1

Approximate formulation of the rigid body motions of an elastic rectangle under sliding boundary conditions

Şahin Önur, Erbaş Barış, Wilson Brent

Acta Mechanica et Automatica

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2021

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Vol. 15, no. 2

82--90

EN

Low-frequency analysis of in-plane motion of an elastic rectangle subject to end loadings together with sliding boundary conditions is considered. A perturbation scheme is employed to analyze the dynamic response of the elastic rectangle revealing nonhomogeneous boundary-value problems for harmonic and biharmonic equations corresponding to leading and next order expansions, respectively. The solution of the biharmonic equation obtained by the separation of variables, a consequence of sliding boundary conditions, gives an asymptotic correction to the rigid body motion of the rectangle. The derived explicit approximate formulae are tested for different kinds of end loadings together with numerical examples demonstrating the comparison against the exact solutions.

2

Influence of Vibration of Links on Mechanism Motion

Hać M.

Machine Dynamics Research

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2018

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Vol. 42, No. 4

93--100

EN

In this paper a finite element method with 2D beam elements is derived for flexible planar linkages for the case of mutual dependence between rigid and elastic motions. For this purpose both rigid body and elastic degrees of freedom are considered as generalized coordinates during the derivation procedure and are substituted to the Gibbs-Appel equation. This procedure requires formulation of the shape functions for rigid body motion of finite elements. The numerical calculations are conducted for the case of planar slidercrank mechanism with the goal of determining the influence of the vibration of links on the mechanism motion. The results of numerical simulation show that for transient analysis the influence for large displacement mechanism motion has limited impact.

3

Rigid-Body Motion of Planar Mechanisms by Finite Element Method With Comparison to the Multibody Approach

Hać M., Łazęcki J.

Machine Dynamics Research

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2016

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Vol. 40, No. 3

15--24

EN

In this paper the rigid-body motion of planar mechanisms (e.g. rigid rod, robotic arm, slider-crank mechanism) is considered by using two different approaches: FEM with modified truss elements named here as a stick model and the multibody method used in the engineering software (e.g. Adams, Modelica, etc.). The analysis shows that the stick model of mechanisms consisting of modified truss-type elements gives reliable results when compared to the solid model of multibody approach by using the Adams software. The presented method of description of motion of rigid mechanism can be successfully used as a preliminary calculation of mechanism motion. The method is illustrated with planar examples of mechanisms, however with the use of proper elements the method can be extended to the 3D case.

4

Euler-Poincare reduction of externally forced rigid body motion

Wiśniewski R., Kulczycki P.

Control and Cybernetics

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2004

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Vol. 33, no 2

297-310

EN

If a mechanical system experiences symmetry, the Lagrangian becomes invariant under a certain group action. This property leads to substantial simplification of the description of movement. The standpoint in this article is a mechanical system affected by an external force of a control action. Assuming that the system possesses symmetry and the configuration manifold corresponds to a Lie group, the Euler-Poincare reduction breaks up the motion into separate equations of dynamics and kinematics. This becomes of particular interest for modeling, estimation and control of mechanical systems. A control system generates an external force, which may break the symmetry in the dynamics. This paper shows how to model and to control a mechanical system on the reduced phase space, such that complete state space asymptotic stabilization can be achieved. The paper comprises a specialization of the well-known Euler-Poincare reduction to a rigid body motion with forcing. An example of satellite attitude control illustrates usefulness of the Euler-Poincare reduction in control engineering. This work demonstrates how the energy shaping method applies for Euler-Poincare equations.