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# A Simple Treatment of Constraint Forces and Constraint Moments in the Dynamics of Rigid Bodies

[+] Author and Article Information
Oliver M. O'Reilly

Department of Mechanical Engineering,
University of California at Berkeley,
Berkeley, CA 94720-1740
e-mail: oreilly@berkeley.edu

Arun R. Srinivasa

Department of Mechanical Engineering,
Texas A&M University,
College Station, TX 77843-3123
e-mail: asrinivasa@tamu.edu

For example, the Boltzmann-Hamel equations, the Gibbs-Appell equations or Kane's equations [2,5-7].

Notable discussions on constraint forces in systems of particles in analytical mechanics include Gantmacher [8] and Planck [9], however, apart from Refs. [10-12], discussions of constraint (or reaction) moments are notably absent from textbooks.

An example of such a calculation can be found in Ref. [20].

This identification ensures that Lagrange's equations of motion (34) are equivalent to the Newton-Euler balance laws $F=mv¯·$ and $M=H·$ where H is the angular momentum of the rigid body of mass m relative to $X¯$ (see, e.g., Refs. [2], [11], [24], and [25] for further details on the equivalence).

With relation to other treatments of constraint forces, it is easy to observe from Eqs. (27) and (39) that the combined virtual work of Fc and Mc will be zero as expected.

Manuscript received March 1, 2014; final manuscript received July 17, 2014; published online August 25, 2014. Editor: Harry Dankowicz.

Appl. Mech. Rev 67(1), 014801 (Aug 25, 2014) (8 pages) Paper No: AMR-14-1028; doi: 10.1115/1.4028099 History: Received March 01, 2014; Accepted July 17, 2014; Revised July 17, 2014

## Abstract

In this expository article, a simple concise treatment of Lagrange's prescription for constraint forces and constraint moments in the dynamics of rigid bodies is presented. The treatment is suited to both Newton–Euler and Lagrangian treatments of rigid body dynamics and is illuminated with a range of examples from classical mechanics and orthopedic biomechanics.

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## Figures

Fig. 1

A circular disk of radius R moving on a horizontal plane. The position vector of the instantaneous point of contact XP of the disk and the plane relative to the center of mass X¯ of the disk is always along e2". A set of 3-1-3 Euler angles are used to parameterize the rotation of the disk.

Fig. 2

A rigid body rotating in a plane about a fixed point XA. The constraint force Fc and constraint moment Mc, which ensure that the point XA remains fixed and the axis of rotation is constrained to be E3 by a revolute joint at XA, respectively, are also shown.

Fig. 3

A cylinder sliding on a smooth horizontal surface. The constraint force Fc and constraint moment Mc, which ensure that the center of mass X¯ moves in the horizontal plane and the cylinder does not rotate into the plane, respectively, are also shown.

Fig. 4

A rigid body rolling without slipping on a rough horizontal plane. The constraint force Fc acting at the instantaneous point of contact XP (where vP = 0) can be decomposed into a normal force N and friction force Ff: Fc = N + Ff.

Fig. 5

A whirling rigid body. The constraint force Fc and constraint moment Mc are also shown.

Fig. 6

Equipollence of (a) a constraint force Fc acting at XC and a constraint moment Mc = λhC and (b) the same force acting at XA and a different constraint moment M = λhC + (xC-xA) × Fc

Fig. 7

Schematic of the right knee joint showing the proximal {p1,p2,p3} and distal {d1,d2,d3} bases which corotate with the femur and tibia, respectively. The Euler and dual Euler basis vectors associated with the rotation of this joint and the condyles C and D are also shown. This figure is adapted from Ref. [18].

Fig. 8

Schematic of the normal forces at the condyles. For the illustrated case θ < 0. This figure is adapted from Ref. [18].

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