1 Lecture D32 : Damped Free Vibration Spring-Dashpot-Mass System Spring Force k > 0 Dashpot c > 0 Newton’s Second Law (Define) Natural Frequency and Period (Define) Damping Factor Equation of motion

2 Solution Try Characteristic Polynomial Roots General Solution (superposition)

3 Types of Solutions ζ > 1 (Overdamped) ζ = 1 (Critically Damped) One zero crossing at most !!

4 Types of Solutions (cont’d) ζ< 1 (Underdamped) Damped Natural Frequency (and Period) Solution or,

5 Damping Factor Needs to be estimated experimentally: Measure the ratio of two (or more) successive amplitudes and, Let Since For lightly damped systems,

6 Energy Decay For Undamped harmonic oscillator: For Underdamped oscillator Thus, we expect... valid for lightly damped systems This can be used to estimate ζ if Energy (decay) can be measured. (constant)