FREE VIBRATION OF NONLINEAR SINGLE DEGREE OF FREEDOM NONCONSERVATIVE SYSTEMS

In this lecture discussion on the vibration of a linear single degree of freedom system with viscous, Coulomb damping, quadratic dumping and will be carried out using method of averaging.

System with viscous damping

Let us consider a single degree of freedom system with viscous damping. The equation of motion of this system with mass m, stiffness k and damping c can be written as

........................................................................................................... (6.3.1)

The same system can be written using the term natural frequency , damping ratio ζ as

......................................................................................................(6.3.2)
.....................................................................(6.3.3)

By using Krylov-Bogoliubov method of averaging for an under damped system ( ζ <1 ) the solution can be written as

..........................................................................................................(6.3.4)

where

...............................................................(6.3.5) ...........................................................(6.3.6)

Substituting expression for from Eq. (6.3.3) in Eq. (6.3.5) and Eq. (6.3.6), one obtains

.........................................................................................(6.3.7)
and
. ........................................................................................( 6.3.8)

Solving Eq. (6.3.7) and Eq. (6.3.8) yields

................................................................(6.3.9)

Here and are the initial displacement and phase of the response. Substituting Eq. (6.3.9) in Eq.
( 6.3.4) one obtains the following equation.

...................................................................... (6.3.10)