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DERIVING EQUATIONS OF MOTION

1 Overview

The dynamics of discrete or continuous systems can be described mathematically by ordinary or partial differential equations. This document describes the theory in deriving the governing equations of motion for discrete dynamic systems. These are ordinary differential equations. After the equations of motion are derived, they can be solved by a variety of analytical or numerical methods, which are not described here.

There are a number of ways to derive governing equations of motion for a dynamic system. These methods can usually be classified as Newtonian-based or energy-based. Both methods result in the same governing equations when applied correctly but have different advantages and disadvantages. Newtonian methods involve developing a free-body diagram for each discrete body, applying Newton’s laws, and reducing the number of equations by explicitly accounting for constraint relations. In energy-based methods, the equations of motion are derived from the kinetic and potential energy by Lagrange’s method or another formulation. Lagrange’s method is probably the most widely used and is described here.

2 Degrees of Freedom and Constraints

Before deriving the equations of motion, it is important to identify the constraints and the number of degrees of freedom in the system. For clarity, from this point on a distinction will be made between degrees of freedom and unconstrained degrees of freedom. The total number of degrees of freedom in a system, or m, is simply a function of the number of discrete bodies in the system, n:

(1) m = 6n

The factor of six in Eq. 1↑ is because a real body can translate and rotate in three dimensions. The total number of equations of motion and dependent variables in the system of equations must be equal to the number of unconstrained degrees of freedom, N. N is related to m and the number of constraints, l, as follows:

(2) N = m − l

Equation 2↑ highlights the importance of identifying the constraints before attempting to develop the equations of motion, as the final number of equations of motion must be equal to N. In many practical problems, a number of constraints can be identified immediately. For example, if the problem is two-dimensional with one rotational degree of freedom, three constraints for each body can be identified immediately for each body; namely, that one of the linear translations and two of the rotations are zero. An example of such a problem is an airfoil undergoing pitch motion. Other constraints are less obvious, and sometimes geometry and algebra may be required to develop the constraint relations.

3 Newtonian Method

The Newtonian method proceeds as follows:

3.1 Identify each discrete body in the system.

A discrete body is one that has its own mass and/or rotational inertia. Other components like springs, dampers, and cables are usually assumed to have no mass in order to simplify the analysis.