The Principle of Virtual Work

2017 ◽  
Author(s):  
Jennifer Coopersmith
Keyword(s):  
Virtual Work ◽  
Worked Examples ◽  
Black Box ◽  
Soap Bubble ◽  
Newtonian Mechanics ◽  
Principle Of Virtual Work ◽  
Constraint Forces

The meaning behind the mysterious Principle of Virtual Work is explained. Some worked examples in statics (equilibrium) are given, and the method of Virtual Work is compared and contrasted with the method of Newtonian Mechanics. The meaning of virtual displacements is explained very carefully. They must be ‘small’, happen simultaneously, and do not cause a force, result froma force, or take any time to occur. Counter to intuition, not all the actual displacements can be allowed as virtual displacements. Some examples worked through are: Feynman’s pivoting (cantilever) bar, a “black box,” a weighted spring, a ladder, a capacitor, a soap bubble, and Atwood’s machine. The links between mechanics and geometry are demonstrated, and it is shown how the reaction or constraint forces are always perpendicular to the virtual displacements. Lanczos’s Postulate A and its astounding universality are explained.

scholarly journals Generalized Lagrange-D’Alembert principle

2012 ◽  
Vol 91 (105) ◽  
pp. 49-58
Author(s):  
Djordje Djukic
Keyword(s):  
Dynamic Equilibrium ◽  
Virtual Work ◽  
Analytical Mechanics ◽  
Principle Of Virtual Work ◽  
Constraint Forces ◽  
Lagrange Equations ◽  
Virtual Displacement ◽  
Generalized Coordinates ◽  
D’Alembert Principle ◽  
Applied Forces

The major issues in the analysis of the motion of a constrained dynamic system are to determine this motion and calculate constraint forces. In the analytical mechanics, only the first of the two problems is analyzed. Here, the problem is solved simultaneously using: 1) Principle of liberation of constraints; 2) Principle of generalized virtual displacement; 3) Idea of ideal constraints; 4) Concept of generalized and ?supplementary" generalized coordinates. The Lagrange-D?Alembert principle of virtual work is generalized introducing virtual displacement as vectorial sum of the classical virtual displacement and virtual displacement in the ?supplementary" directions. From such principle of virtual work we derived Lagrange equations of the second kind and equations of dynamical equilibrium in the ?supplementary" directions. Constrained forces are calculated from the equations of dynamic equilibrium. At the same time, this principle can be used for consideration of equilibrium of system of material particles. This principle simultaneously gives the connection between applied forces at equilibrium state and the constrained forces. Finally, the principle is applied to a few particular problems.

Inverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual work

Robotica ◽  
2009 ◽  
Vol 27 (2) ◽  
pp. 259-268 ◽  
Author(s):  
Yongjie Zhao ◽  
Feng Gao
Keyword(s):  
Parallel Manipulator ◽  
Inverse Dynamics ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Principle Of Virtual Work ◽  
Constraint Forces ◽  
Dynamical Equations ◽  
Robot System ◽  
Lead Screw ◽  
Moving Platform

SUMMARYIn this paper, the inverse dynamics of the 6-dof out-parallel manipulator is formulated by means of the principle of virtual work and the concept of link Jacobian matrices. The dynamical equations of motion include the rotation inertia of motor–coupler–screw and the term caused by the external force and moment exerted at the moving platform. The approach described here leads to efficient algorithms since the constraint forces and moments of the robot system have been eliminated from the equations of motion and there is no differential equation for the whole procedure. Numerical simulation for the inverse dynamics of a 6-dof out-parallel manipulator is illustrated. The whole actuating torques and the torques caused by gravity, velocity, acceleration, moving platform, strut, carriage, and the rotation inertia of the lead screw, motor rotor and coupler have been computed.

D’Alembert’s Principle

2017 ◽  
Author(s):  
Jennifer Coopersmith
Keyword(s):  
Quantum Mechanics ◽  
Virtual Work ◽  
Inertial Force ◽  
Black Box ◽  
Principle Of Virtual Work ◽  
Conservation Of Energy ◽  
Fundamental Nature ◽  
Worked Example ◽  
The Law ◽  
D'alembert's Principle

It is explained how the mysterious Principle of Virtual Work in statics is extended to the even more mysterious Principle of d’Alembert’s in dynamics. This is achieved by d’Alembert’s far-sighted stratagem: considering a reversed massy acceleration as an inertial force. A worked example is given (the half-Atwood machine or “black box”). Some counter-intuitive aspects are made intuitive by more examples: the Pluto-Charon system of orbiting planets; Newton’s and then Mach’s explanation of Newton’s bucket. Also, it is demonstrated that the law of the conservation of energy actually follows from d’Alembert’s Principle. The reader is alerted to the astoundingly fundamental nature of d’Alembert’s Principle. It is the cornerstone of classical, relativistic, and quantum mechanics. As Lanczos writes: “All the different principles of mechanics are merely mathematically different formulations of d’Alembert’s Principle”.

Symbolic Closed-Form Modeling and Linearization of Multibody Systems Subject to Control

1991 ◽  
Vol 113 (2) ◽  
pp. 124-132 ◽  
Author(s):  
Junghsen Lieh ◽  
Imtiaz-ul Haque
Keyword(s):  
Closed Form ◽  
Multibody Systems ◽  
Space Form ◽  
Virtual Work ◽  
Dynamic Equations ◽  
Principle Of Virtual Work ◽  
Constraint Forces ◽  
Active Suspensions ◽  
Linear Dynamic ◽  
Curved Track

Symbolic closed-form equation formulation and linearization for constrained multibody systems subject to control are presented. The formulation is based on the principle of virtual work. The algorithm is recursive, automatically eliminates the constraint forces and redundant coordinates, and generates the nonlinear or linear dynamic equations in closed-form. It is derived with respect to principal body coordinates and a moving reference frame that allows one to generate the dynamic equations for multibody systems moving along curved track or road. The output equations may be either in syntactically correct FORTRAN form or in the form as derived by hand. A procedure that simplifies the trigonometric expressions, linearizes the geometric nonlinearities, and converts the linearized equations in state-space form is included. Several examples have been used to validate the procedure. Included is a simulation using a seven-DOF automobile ride model with active suspensions.

On the Mechanical Behavior of the Orthotropic Cord-Rubber Composite

1976 ◽  
Vol 4 (4) ◽  
pp. 219-232 ◽  
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.

scholarly journals Discrete element model for general polyhedra

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.

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

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.

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

2012 ◽  
Vol 28 (3) ◽  
pp. 385-401 ◽  
Author(s):  
J. Jesús Cervantes-Sánchez ◽  
José M. Rico-Martínez ◽  
Salvador Pacheco-Gutiérrez ◽  
Gustavo Cerda-Villafaña
Keyword(s):  
Static Analysis ◽  
Virtual Work ◽  
Parallel Manipulators ◽  
Principle Of Virtual Work

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

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.

On the Sufficiency of the Principle of Virtual Work for Mechanical Equilibrium: A Critical Reexamination

1989 ◽  
Vol 56 (3) ◽  
pp. 704-707 ◽  
Author(s):  
John G. Papastavridis
Keyword(s):  
Virtual Work ◽  
Mechanical Equilibrium ◽  
Principle Of Virtual Work ◽  
Holonomic Constraints ◽  
Kinetic Principle ◽  
Sufficiency Conditions

Starting from the general kinetic principle of d’Alembert/Lagrange, an energetic proof of the sufficiency conditions for equilibrium (known as Principle of Virtual Work) is presented. It is clearly demonstrated why to maintain equilibrium requires that, in addition to the familiar vanishing of the virtual work of the impressed forces on the originally motionless system, its geometrical (holonomic) constraints be explicitly time independent (stationary) and its nonintegrable kinematical (nonholonomic) ones be linear and homogeneous in the generalized velocities (catastatic).

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