A. Free vibration  = 0

1. Chapter 8 Vibration A. Free vibration  = 0 k Undamped free vibration m x 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

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

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

5. Case 2: Critical damping when  = 1 += - = -n • General solution : x = (A1+ A2t )exp(-nt) • x approaches zero quickly without oscillation.

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

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

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

10. 

12. Maximum M occurs at: The resonance frequency is

13. tan  = 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(+)

14. 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.

15. 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.

16. The amplitude of the steady-state vibration is:

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