WEEK-3: WEEK-3: Free Vibration of SDOF Free Vibration of SDOF

systemssystemsAssoc. Prof. Dr. Abdullah Assoc. Prof. Dr. Abdullah SeçginSeçgin

Free vibration of SDOF systems

•Natural frequency

•Damping Ratio

•Equation of motion

•Standart form of equation of motion

•Assume following initial conditions

•Assume a harmonic solution

Free vibration of SDOF systems•Equation of motion •Harmonic solution

• Types of Free Vibrations

Free vibration of SDOF systems•General harmonic solution

•Applying initial conditions

Example:

Example:

1.Underdamped vibration

•By applying initial conditions and expand with cos and sine terms gives

1.Underdamped vibration

1.Underdamped vibration

Example: Governing equation of motion of an underdamped single degree of freedom system is given as

a) Find the natural frequencyb) Find the damping ratioc) Find the damped natural frequency of the systemd) Find the free vibration response amplitude and its phase for the following initial conditions:

e) Draw roughly the amplitude-time graphics of the response

0)0(,15.0)0( xmx

Example: Governing equation of motion of an underdamped single degree of freedom system shown below is given as

a) Find all vibration parameters

Solution: