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Research Papers

# Dynamics of Mechanical Systems and the Generalized Free-Body Diagram—Part I: General Formulation

[+] Author and Article Information
József Kövecses

Department of Mechanical Engineering and Centre for Intelligent Machines, McGill University, 817 Sherbrooke St. West, Montréal, Québec, H3A 2K6, Canadajozsef.kovecses@mcgill.ca

The transformation here is illustrated for velocity components. However, it can also be defined for other kinematic quantities (e.g., acceleration components) in the same form as long as the elements of the arrays can be interpreted as components of a generalized geometric vector in the tangent space.

Inertial forces can of course also be established in many other ways (e.g., using the kinetic energy function). However, the most compact and general form may be given with the Gibbs–Appell function.

For the Cholesky factorization, in particular, it can be shown that the algorithm contains operations that are consistent with the possibly different physical units of the elements of $M$.

In this case, $sc$ and $uc$ are $m×1$ arrays, $sa$ and $ua$ are $(n−r)×1$ arrays, $s$ is an $(n−r+m)×1$ array, and $R$ is an $(n−r+m)×n$ matrix.

For local parametrizations, in the following, we will assume that such a selection of $A$ is already done. For singular transformations this can be achieved as $A=ΓG$.

This experimental test-bed was built by QUANSER.

J. Appl. Mech 75(6), 061012 (Aug 20, 2008) (12 pages) doi:10.1115/1.2965372 History: Received November 25, 2007; Revised May 21, 2008; Published August 20, 2008

## Abstract

In this paper, we generalize the idea of the free-body diagram for analytical mechanics for representations of mechanical systems in configuration space. The configuration space is characterized locally by an Euclidean tangent space. A key element in this work relies on the relaxation of constraint conditions. A new set of steps is proposed to treat constrained systems. According to this, the analysis should be broken down to two levels: (1) the specification of a transformation via the relaxation of the constraints; this defines a subspace, the space of constrained motion; and (2) specification of conditions on the motion in the space of constrained motion. The formulation and analysis associated with the first step can be seen as the generalization of the idea of the free-body diagram. This formulation is worked out in detail in this paper. The complement of the space of constrained motion is the space of admissible motion. The parametrization of this second subspace is generally the task of the analyst. If the two subspaces are orthogonal then useful decoupling can be achieved in the dynamics formulation. Conditions are developed for this orthogonality. Based on this, the dynamic equations are developed for constrained and admissible motions. These are the dynamic equilibrium equations associated with the generalized free-body diagram. They are valid for a broad range of constrained systems, which can include, for example, bilaterally constrained systems, redundantly constrained systems, unilaterally constrained systems, and nonideal constraint realization.

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## Figures

Figure 5

Trajectory of the center point of the wand (units are in mm)

Figure 6

Kinetic energy decomposition for one-dimensional SCM based on experimental data (units: time in s and energy in J)

Figure 7

Kinetic energy decomposition for two-dimensional SCM based on experimental data (units: time in s and energy in J)

Figure 1

Generalized force balance

Figure 2

Subspaces, global and local parametrizations

Figure 3

A generalized particle on a circular block

Figure 4

Dual-pantograph device

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