Oscillations Summary Free Oscillations: Dierential Equation: Solution:
d2 x dt2
+ 2x = 0
x(t) = A cos(t + )
Common Oscillators: Mass on Spring: Simple Pendulum: Physical Pendulum: Uncommon Oscillators: Damped Oscillations: Solution: x(t) = A0 e 2m cos(t + ) A(t) = A0 e 2m =
2 0 ( 2b )2 m
bt bt
0 = 0 =
k m g L mgd I
0 =
Use Forces or Work/Energy to obtain 0
The amplitude is time-dependent !
Frequency Shift ! System oscillates at
Classication: if is real, the system is underdamped. If it is imaginary, the system is overdamped. If it is zero, the system is critically damped. Energy (Lightly Damped): E = 1 kA(t)2 2 Driven Oscillations: Solution: x(t) =
F0 2 m2 (0 2 )2 +(b )2
(or similar)
cos( t + )
is the angular frequency of the driver. Phase dierence between driver and oscillation: resonant = 0
b 1 2( 2m0 )2
tan( ) =
b 2 m(0 2 )
...
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