Static analysis of spatial parallel manipulators by means of the principle of virtual work

Abstract The dynamic analysis of spatial four-degree-of-freedom parallel manipulators is presented in this article. First, expressions for the position, velocity and acceleration of each link constituting the manipulators are obtained. Then, the principle of virtual work is used to derive the generalized input forces of the manipulators. The corresponding algorithm is implemented and numerical examples are given in order to illustrate the results. The results obtained are verified using the classical Newton-Euler approach.

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Determination of Bearing Reactions of Spatial Linkages and Manipulators

Journal of Mechanical Design ◽

10.1115/1.2912589 ◽

1990 ◽

Vol 112 (2) ◽

pp. 168-174 ◽

Cited By ~ 2

Author(s):

F. L. Litvin ◽

J. Tan

Keyword(s):

Virtual Work ◽

Parallel Manipulators ◽

System Of Equations ◽

Principle Of Virtual Work ◽

Simultaneous Solution ◽

Combined Application ◽

Spatial Linkages ◽

Virtual Velocity ◽

D'alembert's Principle

Application of D’Alembert’s principle for determination of dynamic bearing reactions in joints of spatial linkages and parallel manipulators needs the simultaneous solution of a large system of equations. The authors of this paper propose an approach that is a combined application of principle of virtual work and D’Alembert’s principle. The main advantages of the proposed approach are: (1) reduction of the number of equations that have to be solved simultaneously, and (2) simplification of the expressions for the relative virtual velocity. The proposed approach is illustrated with the example of a 7-bar linkage and its application is explained with the crank-slider linkage.

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Solving active wrench of limited-degree of freedom parallel manipulators based on translational/rotational Jacobian matrices

Proceedings of the Institution of Mechanical Engineers Part K Journal of Multi-body Dynamics ◽

10.1243/14644193jmbd181 ◽

2009 ◽

Vol 223 (3) ◽

pp. 221-229 ◽

Cited By ~ 2

Author(s):

Y Lu ◽

B Hu ◽

J Yu

Keyword(s):

Virtual Work ◽

Parallel Manipulators ◽

Degree Of Freedom ◽

Geometric Constraints ◽

Principle Of Virtual Work ◽

Jacobian Matrices ◽

Limited Degree

A methodology is proposed for unified solving active wrench of the limited-degree of freedom (DOF) parallel manipulators (PMs). First, the geometric constraints and the inverse displacement kinematics are analysed. Second, the formulae for unified solving the inverse/forward velocity and the translational/rotational Jacobian matrices and inverse/forward Jacobian matrices are derived. Third, the analytic formulae for unified solving the active wrench of limited-DOF PMs are derived based on the principle of virtual work. Finally, a 3-DOF PM with linear/rotational active legs is presented to illustrate the use of the methodology.

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Solving the Inverse Dynamics of Parallel Manipulators by the Principle of Virtual Work

Volume 1B: 25th Biennial Mechanisms Conference ◽

10.1115/detc98/mech-5865 ◽

1998 ◽

Author(s):

Lung-Wen Tsai

Keyword(s):

Parallel Manipulator ◽

Inverse Dynamics ◽

Virtual Work ◽

Equations Of Motion ◽

Linear Equations ◽

Parallel Manipulators ◽

Principle Of Virtual Work ◽

Dynamical Equations ◽

Systematic Methodology ◽

Stewart Gough Platform

Abstract This paper presents a systematic methodology for solving the inverse dynamics of parallel manipulators. Based on the principle of virtual work and the concept of link Jacobian matrices, a methodology for deriving the dynamical equations of motion is developed. It is shown that the dynamics of a parallel manipulator can be reduced to solving a system of six linear equations. To demonstrate the methodology, the dynamical equations of a Stewart-Gough platform are derived. A computer algorithm is developed and several different trajectories of the moving platform are simulated.

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Analyses of the stiffness and elastic deformation of a 2(3-SPR) serial—parallel manipulator

Proceedings of the Institution of Mechanical Engineers Part K Journal of Multi-body Dynamics ◽

10.1243/14644193jmbd199 ◽

2009 ◽

Vol 223 (3) ◽

pp. 189-198 ◽

Cited By ~ 1

Author(s):

Y Lu ◽

B Hu ◽

J Yu

Keyword(s):

Elastic Deformation ◽

Machine Tools ◽

Parallel Manipulator ◽

Virtual Work ◽

Parallel Manipulators ◽

Robot Arm ◽

Principle Of Virtual Work ◽

Compliance Matrix ◽

End Effector ◽

Machining Speed

Stiffness is one of the important indices for evaluating the performances of serial—parallel manipulators (S—PMs), particularly when the S—PMs are used as machine tools and the robot arm/leg, and higher stiffness allows higher machining speed with higher accuracy of the end-effector. In this article, the stiffness and the elastic deformation of a 2(3-SPR) S—PM are studied systematically. First, a 2(3-SPR) S—PM, including an upper 3-SPR parallel manipulator (PM) and a lower 3-SPR PM, is constructed, and its characteristics are analysed. Second, some formulae for solving the elastic deformation and the compliance matrix of the active legs are derived from the available kinematics/statics of this S—PM. Third, based on the principle of virtual work and the compliance matrix of the active legs, the elastic deformation and the total stiffness matrix of this S—PM are solved and analysed.

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On the Mechanical Behavior of the Orthotropic Cord-Rubber Composite

Tire Science and Technology ◽

10.2346/1.2167223 ◽

1976 ◽

Vol 4 (4) ◽

pp. 219-232 ◽

Cited By ~ 2

Author(s):

Ö. Pósfalvi

Keyword(s):

Mechanical Behavior ◽

Uniaxial Tension ◽

Elastic Properties ◽

Large Deformations ◽

Virtual Work ◽

Poisson's Ratio ◽

Principle Of Virtual Work ◽

Effective Elastic Properties ◽

Rubber Composite ◽

Mooney Equation

Abstract The effective elastic properties of the cord-rubber composite are deduced from the principle of virtual work. Such a composite must be compliant in the noncord directions and therefore undergo large deformations. The Rivlin-Mooney equation is used to derive the effective Poisson's ratio and Young's modulus of the composite and as a basis for their measurement in uniaxial tension.

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Discrete element model for general polyhedra

Computational Particle Mechanics ◽

10.1007/s40571-021-00415-z ◽

2021 ◽

Author(s):

Alfredo Gay Neto ◽

Peter Wriggers

Keyword(s):

Frictional Contact ◽

Virtual Work ◽

Particle Surface ◽

Element Model ◽

Discrete Element ◽

Discrete Element Model ◽

Principle Of Virtual Work ◽

Dynamics Model ◽

Railway Ballast ◽

Multibody Dynamics Model

AbstractWe present a version of the Discrete Element Method considering the particles as rigid polyhedra. The Principle of Virtual Work is employed as basis for a multibody dynamics model. Each particle surface is split into sub-regions, which are tracked for contact with other sub-regions of neighboring particles. Contact interactions are modeled pointwise, considering vertex-face, edge-edge, vertex-edge and vertex-vertex interactions. General polyhedra with triangular faces are considered as particles, permitting multiple pointwise interactions which are automatically detected along the model evolution. We propose a combined interface law composed of a penalty and a barrier approach, to fulfill the contact constraints. Numerical examples demonstrate that the model can handle normal and frictional contact effects in a robust manner. These include simulations of convex and non-convex particles, showing the potential of applicability to materials with complex shaped particles such as sand and railway ballast.

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Analytical Modelling of the Dynamics of the Four-Bar Mechanism: A Comparison of Some Methods

Volume 5A: 42nd Mechanisms and Robotics Conference ◽

10.1115/detc2018-86204 ◽

2018 ◽

Author(s):

J. P. Meijaard ◽

V. van der Wijk

Keyword(s):

Virtual Work ◽

Equations Of Motion ◽

Differential Algebraic Equations ◽

Force Balance ◽

Dynamic Force ◽

Algebraic Equations ◽

Principle Of Virtual Work ◽

Principal Vector ◽

Reaction Forces ◽

Principal Points

Some thoughts about different ways of formulating the equations of motion of a four-bar mechanism are communicated. Four analytic methods to derive the equations of motion are compared. In the first method, Lagrange’s equations in the traditional form are used, and in a second method, the principle of virtual work is used, which leads to equivalent equations. In the third method, the loop is opened, principal points and a principal vector linkage are introduced, and the equations are formulated in terms of these principal vectors, which leads, with the introduced reaction forces, to a system of differential-algebraic equations. In the fourth method, equivalent masses are introduced, which leads to a simpler system of principal points and principal vectors. By considering the links as pseudorigid bodies that can have a uniform planar dilatation, a compact form of the equations of motion is obtained. The conditions for dynamic force balance become almost trivial. Also the equations for the resulting reaction moment are considered for all four methods.

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Optimal Design of Bi- and Multi-Stable Compliant Cellular Structures

Volume 5A: 42nd Mechanisms and Robotics Conference ◽

10.1115/detc2018-85707 ◽

2018 ◽

Author(s):

Quantian Luo ◽

Liyong Tong

Keyword(s):

Optimal Design ◽

Virtual Work ◽

Reaction Force ◽

Nonlinear Finite Element ◽

Cellular Structures ◽

Stress And Strain ◽

Principle Of Virtual Work ◽

Stable States ◽

Threshold Method ◽

Element Analysis

This paper presents optimal design for nonlinear compliant cellular structures with bi- and multi-stable states via topology optimization. Based on the principle of virtual work, formulations for displacements and forces are derived and expressed in terms of stress and strain in all load steps in nonlinear finite element analysis. Optimization for compliant structures with bi-stable states is then formulated as: 1) to maximize the displacement under specified force larger than its critical one; and 2) to minimize the reaction force for the prescribed displacement larger than its critical one. Algorithms are developed using the present formulations and the moving iso-surface threshold method. Optimal design for a unit cell with bi-stable states is studied first, and then designs of multi-stable compliant cellular structures are discussed.

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