Mechanical Vibrations

Multiple Degree-of-Freedom Systems

GOVERNING EQUATIONS

FREE RESPONSE CHARACTERISTICS

ROTATING SHAFTS ON FLEXIBLE SUPPORTS

Force-Balance Method

Moment-Balance Method

This system consists of linear springs, linear dampers, and translating inertia elements.

In the case of single degree-of-freedom systems, when the forcing is absent, the responses exhibited by a multi-degree-of-freedom system are called free responses.

Undamped Systems

Many types of rotating machinery are mounted on a shaft, which in turn is

mounted to a supporting structure with flexibility. This flexibility is modeled

as springs.

Kinematics and Kinetic Energy

To obtain the equations of motion, we use Lagrange’s equations. We first

construct the kinetic energy and potential energy of the system. In order to go from the Oxyz frame to the Cxyz frame, an Euler-angle sequence consisting of a w rotation about the O-x axis, followed by a u rotation about the intermediate y axis, and a subsequent rotation of q about the O-z or C-z axis is used.