Welcome to AE Resources

Converted document

3.2 For each discrete body, draw a free-body diagram, carefully accounting for all forces and moments.

Be careful to draw all relevant forces and moments and make note of direction. If the force or moment is one that acts between two bodies in the system, use Newton's third law to ensure that the force or moment on the first body acts equal and opposite to that on the other body.

3.3 For each discrete body, formulate equations of motions for forces in each direction using Newton's second law.

If the free-body diagram was drawn correctly, this part should be simple. In general, the equation must be written for all three spatial dimensions. However, if in the problem the translation is constrained to only one or two dimensions, the equations do not need to be written in the unused direction(s). Newton’s second law for translation is

(3) ⎲⎳F = (dP)/(dt) = (d(mv))/(dt)

where P is the translational momentum vector and v is the translational velocity vector. For constant mass systems, Eq. 3↑ reduces to the familiar form

(4) ⎲⎳F = ma

where a is the translational acceleration.

3.4 For each discrete body, formulate equations of motion for moments about each axis using Newton’s second law.

In general, the equation must be written for all three spatial dimensions. However, if in the problem the angular motion is constrained to only one or two dimensions, the equations do not need to be written in the unused direction.

(5) ⎲⎳M = (dL)/(dt) = (d(Iω))/(dt)

where M is an applied moment, resolved at the center of mass of the body, L is the angular momentum vector, ω is the angular velocity vector, and I is the moment of inertia tensor. For constant inertia systems, Eq. 5↑ reduces to the familiar form

(6) ⎲⎳M = Iα

where α is the angular acceleration. Note that the moment of inertia is a tensor quantity associated with direction. As a consequence, Eq. 6↑ can only be used when the axes relative to which I is defined do not change with time (otherwise I would not be constant). For this reason, it is common to use a body-fixed axis system in problems including rotation.

3.5 Use the constraint relations to reduce the number of equations.

The constraint relations should have been determined explicitly before writing the equations of motion. These are algebraic relations which are used to eliminate variables and the number of equations. After this process, there should be as many dependent variables and equations as the number of unconstrained degrees of freedom, N.

← Previous Page