Multiple Degree of Freedom Systems: Chapter 8 Examples
Example 8-1
The double pulley shown below (which is supported by a spring with stiffness ) has a moment of inertia about its center . The pulley supports a mass by a light cable which has an effective stiffness . The coordinates and as shown are used to describe the system.
- Determine the equations of motion for this system using Newton’s Law’s.
- If
the equations of motion become
Determine the resulting natural frequencies for this system and sketch the associated mode shapes.
Example 8-2
Determine expressions for the natural frequencies and mode shapes for the system shown. Use the coordinates and as indicated.
Example 8-3
Two simple pendulums of the same length are connected by a light spring with stiffness as shown.
For this system, determine
- the equations of motion,
- the natural frequencies,
- the mode shapes,
- the total response of the system (for ) if the initial conditions at time are
For parts (b)–(d) assume that .
Example 8-4
A rigid beam (mass , length , centroidal moment of inertia ) is supported by springs at its ends as shown. Determine the equations of motion, natural frequencies and mode shapes for this system.
Solve this problem using two different sets of coordinates:
- and ,
- and ,
as illustrated below.
Be sure to identify any in each mode shape.
Example 8-5
An asymmetric machine mounting is supported by both vertical and horizontal springs as shown below.
- Using the coordinates , , and as illustrated, determine the equations of motion for this system.
- Explain how the equations of motion and mode shapes change in each of the following special cases:
- ,
- ,
- and .
Example 8-6
The two degree of freedom system shown consists of a wheel (with mass and centroidal moment of inertia ) which rolls without slipping and a bar (with mass and centroidal moment of inertia ) connected by three springs each of stiffness .
- Using the coordinates (the horizontal position of the center of the wheel) and (the horizontal position of the lower end of the bar) show that if and , the equations of motion for this system become
- Determine the natural frequencies and mode shapes for this system.
Example 8-7
Determine the steady state response for the system shown.