1. 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

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

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

5. Case 2: Critical damping when  = 1
General solution : x = (A1 + A2 t )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. The resonance frequency is
Maximum M occurs at:
The resonance frequency is

13. 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(+)

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 :