Kane's equations, Lagrange's equations, and virtual work

1992 ◽  
Vol 15 (1) ◽  
pp. 277-280 ◽  
Author(s):  
Miles A. Townsend
Keyword(s):  
Virtual Work ◽  
Kane’S Equations ◽  
Kane's Equations ◽  
Lagrange’S Equations ◽  
Lagrange's Equations

Kane’s Equations With Undetermined Multipliers—Application to Constrained Multibody Systems

1987 ◽  
Vol 54 (2) ◽  
pp. 424-429 ◽  
Author(s):  
J. T. Wang ◽  
R. L. Huston
Keyword(s):  
Multibody Systems ◽  
Automated Analysis ◽  
Numerical Analyses ◽  
Constrained Multibody Systems ◽  
Controlled Systems ◽  
Kane’S Equations ◽  
Kane's Equations ◽  
Lagrange’S Equations ◽  
Lagrange's Equations

A procedure for automated analysis of constrained multibody systems is presented. The procedure is based upon Kane’s equations and the concept of undetermined multipliers. It is applicable with both free and controlled systems. As with Lagrange’s equations, the multipliers are identified as scalar components of constraining forces and moments. The advantage of using Kane’s equations is that they are ideally suited for development of algorithms for numerical analyses. Also, generalized speeds and quasi-coordinates are readily accommodated. A simple example illustrating the concepts is presented.

Lagrange’s Equations, Hamilton’s Equations, and Kane’s Equations: Interrelations, Energy Integrals, and a Variational Principle

1995 ◽  
Vol 62 (2) ◽  
pp. 505-510 ◽  
Author(s):  
D. L. Mingori
Keyword(s):  
Analytical Mechanics ◽  
Energy Integral ◽  
Governing Equations ◽  
Kane’S Equations ◽  
Kane's Equations ◽  
Lagrange’S Equations ◽  
Energy Integrals ◽  
Hamilton's Equations ◽  
Lagrange's Equations ◽  
Hamilton’S Equations

A new viewpoint is suggested for expressing the governing equations of analytical mechanics. This viewpoint establishes a convenient framework for examining the relationships among Lagrange’s equations, Hamilton’s equations, and Kane’s equations. The conditions which must be satisfied for the existence of an energy integral in the context of Kane’s equations are clarified, and a generalized form of Hamilton’s Principle is presented. Generalized speeds replace generalized velocities as the velocity variables in the formulation. The development considers holonomic systems in which the generalized forces are derivable from a potential function.

Equivalence of Lagrange’s equations for non-material volume and the principle of virtual work in ALE formulation

2019 ◽  
Vol 231 (3) ◽  
pp. 1141-1157 ◽  
Author(s):  
Kai-Dong Chen ◽  
Jia-Peng Liu ◽  
Jia-Qi Chen ◽  
Xiao-Yu Zhong ◽  
Aki Mikkola ◽  
...  
Keyword(s):  
Virtual Work ◽  
Principle Of Virtual Work ◽  
Lagrange’S Equations ◽  
Ale Formulation ◽  
Material Volume ◽  
Lagrange's Equations

Instantaneous impulses in multibody dynamics

2015 ◽  
Vol 22 (4) ◽  
pp. 581-635
Author(s):  
P Lidström
Keyword(s):  
Multibody Dynamics ◽  
Virtual Work ◽  
Mass Matrix ◽  
Full Rank ◽  
Rigid Bodies ◽  
Unilateral Constraints ◽  
Lagrange’S Equations ◽  
Bilateral Constraints ◽  
The Impact ◽  
Lagrange's Equations

This paper considers instantaneous impulses in multibody dynamics. Instantaneous impulses may act on the multibody from its exterior or they may appear in its interior as a consequence of two of its parts interacting by an impact imposed by a unilateral constraint. The theory is based on the Euler laws of instantaneous impulses, which may be seen as a complement to the Euler laws for regular motions. Based on these laws, and specific continuum properties of the quantities involved, local balance laws for momentum and moment of momentum, involving instantaneous impulses and introducing the Cauchy impulse tensor, are derived. Thermodynamical restrictions on the impulse tensor are formulated based on the dissipation inequality. By stating a principle of virtual work for instantaneous impulses, and demonstrating its equivalence to Euler’s laws, Lagrange’s equations are derived. Lagrange’s equations are convenient to use in the case of multibody dynamics containing rigid as well as flexible parts. A central theme of this paper is the discussion of the interaction between parts of the multibody and their relation to geometrical and kinematical constraints. This interaction is severely affected by the presence of friction, which is notoriously difficult to handle. In a preparation for this discussion we first consider the one-point impact between two rigid bodies. The importance of the so-called impact tensor for this problem is demonstrated. In order to be able to handle the impact laws of Poisson and Stonge, an impact process, governed by a system of ordinary differential equations, is defined. Within this model phenomena, such as slip stop, slip start and slip direction reversal, may be handled. For a multibody with an arbitrary number of parts and multiple impacts, the situation is much more complicated and certain simplifications have to be introduced. Equations of motion for a multibody, consisting of rigid parts and in the presence of ideal bilateral constraints and unilateral constraints involving friction, are formulated. Unique solutions are obtained, granted that the mass matrix of the multibody system is non-singular, the constraint matrices satisfy specific full rank conditions and that the friction is not too high.

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