ME 482: Mechanical Vibrations (062) Dr. M. Sunar

Vibrations FreeForced UndampedDampedUndampedDamped

Undamped Free Vibrations of Single Degree of Freedom Systems Single Degree of Freedom System: System with one mass or inertia

The system is said to have a single degree-of-freedom, because it has only one mass and the variable x(t) is enough to describe the motion. Equation of Motion: Or, in another form: Example 1: Spring-Mass System

The system is again said to have a single degree-of-freedom, because it has only one inertia (and mass) and the variable  (t) is enough to describe the motion. Equation of Motion: Or, in another form: Example 2: Compound Pendulum  L = Length, m = mass

The system is said to have a single degree-of-freedom, because it has only one moment of inertia and the variable  (t) is enough to describe the motion. Equation of Motion: Or, in another form: Example 3: Shaft and Disk

Equation of Motion: Or, in another form: Damped Free Vibrations of Single Degree of Freedom Systems A Single Degree of Freedom System with Viscous Damping:

We define the following: Natural Frequency = Damping Ratio = Damped Natural Frequency = Damped Period = Solution: Depends on the values of  0<  < 1 : Underdamped  = 1 : Critically Damped  > 1 : Overdamped

We get: Assume the solution as: IC are: 0<  < 1 : Underdamped

We also define, Logarithmic Decrement =  = Natural Log of Ratio of Amplitudes of Successive Cycles

 = For small values of ,  = 2  In general, we have  = Furthermore,  = Or,  =

We get: Assume the solution as: IC are:  = 1 : Critically Damped

We get: Assume the solution as: IC are:  > 1 : Overdamped