Chapter 8 Vibration A. Free vibration  = 0 k m x Undamped free vibration n= (k/m)1/2: natural frequency C : amplitude o : initial phase angle xo : initial displacement, xo= C sin(o ) T : period, such that Tn =2, or T = 2 /n f : frequency = 1/T

Damped free vibration : k m Define viscous damping constant c (N s/m) Use a trial solution x = A et : k m x

> Case 1 : Overdamped when  > 1, such that + and - < 0 x decays to zero without oscillation 1, > z overdamped 1 = Critical damped < underdamping

Case 2: Critical damping when  = 1 General solution : x = (A1 + A2 t )exp(-nt) x approaches zero quickly without oscillation.

Case 3: Underdamped < 1 Period d = 2/d

Experimental guideline: Measure  and n Calculate  Calculate viscous damping constant c according to  = c/(2mn)

B. Forced vibration of particles : The equation of motion :



The resonance frequency is Maximum M occurs at: The resonance frequency is

tan  = (3)   n-, tan +,   /2(-) Consider the following regions: (1)  is small, tan > 0,   0+, xp in phase with the driving force (2)  is large, tan < 0,   0-,  = , xp lags the driving force by 90o (3)   n-, tan +,   /2(-)   n+, tan -,   /2(+)

If the driving force is not applied to the mass, but is applied to the base of the system: If b2 is replaced by Fo/m: This can be used as a device to detect earthquake.

Example m =45 kg, k = 35 kN/m, c = 1250 N.s/m, p = 4000 sin (30 t) Pa, A= 50 x 10-3 m2. Determine : (a) steady-state displacement (b) max. force transmitted to the base.

The amplitude of the steady-state vibration is:

The force transmitted to the base is : For max Ftr :