1. Damped and driven oscillations
Physics 223, Spring 2018

2. Damped oscillations
Real-world systems have some dissipative forces that decrease the amplitude.
The decrease in amplitude is called damping and the motion is called damped oscillation.

3. Damped Oscillations
For damped oscillations, simplest case is when the damping force is proportional to the velocity of the oscillating object:
In this case, amplitude decays exponentially:
Drag force:
Equation of motion:

4. Damped oscillations and the complex plane
The standard solution for the equation of motion of the damped oscillator:
becomes, in the complex plane:
The solution does not look complex, but this depends on the relative sizes of (b/2m)2 and w02

5. Damped Oscillations
Ampl.
Damping factor
Sinusoidal
variation
Note that damping reduces the oscillation frequency.
Three important cases:
a = underdamped
b = critically damped
c = overdamped

6. Damped oscillations and the complex plane
If b = 0 (no damping), then
If (b/2m)2 < w02 (underdamping), then the argument of the exponential is imaginary, and:
e–(b/2m)^2

7. Damped Oscillations The damped frequency is given by: Light Damping
(small b/m)
Heavy Damping
(large b/m)
Critical Damping

8. Damped Oscillations
If (b/2m)2 > w02 (overdamping), then the argument of the exponential is real (like the critically damped case), and a frequency can be calculated, though it doesn’t really mean anything (there are no oscillations). Any case where the argument of the exponential is real, there are no oscillations. t
x(t)
long-term behavior

9. Would you want the suspension system in a car to be underdamped, critically damped, or overdamped?
Critically or slightly underdamped. Why? Vibration minimized in amplitude and duration.

10. Application Shock absorbers: want critically damped (no oscillations)
not overdamped (would have a slow response time)

11. Energy in Underdamped Systems
The oscillator’s mechanical energy decays exponentially with time constant t.

12. Example: A Damped Pendulum
A 500 g mass on a 50 cm long string oscillates as a pendulum. The amplitude of the pendulum is observed to decay to ½ of its initial value after 35 s.
What is the time constant t of the damped oscillator?
At what time t1/2 will the energy of the system have decayed to ½ of its initial value?

13. Forced Oscillations
If we apply a periodically varying driving force to an oscillator (rather than just leaving it to vibrate on its own) we have a forced (driven) oscillation.
Free Oscillation with damping:
Forced Oscillation with damping:
where wd = frequency of the external driver

14. Forced Oscillations & Resonance
For the forced oscillation case, when the denominator decreases, the amplitude of vibration increases. In the case when k = md2 (i.e., when the driving frequency equals the natural frequency of the undamped oscillator ), then this is the resonance condition.
In this case, the amplitude goes to infinity. The way to think about this is that the driving force pumps energy into the oscillator (like pushing a child on a swing), and if there is no dissipation, there is no loss of energy and the energy grows to become infinite!

15. Driven (Forced) Oscillations
Now, suppose we drive a damped mechanical oscillator driven by an external force F(t) = Fd cos(w t). Then the linear inhomogeneous 2nd order differential equation describing the system’s motion is:
This system will show the property of resonance. The oscillation amplitude will depend on the driving frequency w, and will have its maximum value when: i.e., when the system is driven at its natural resonant frequency w0.

16. Sinusoidal Driving Force
With the equation in this form, we seek a solution of the form
which after substitution into
yields
We can now solve for the constant C, and rewrite the constant as a complex number, and then solve for the amplitude, A, by multiplying by complex conjugates.

17. Resonance Equation of Motion:
The solution of this equation of motion for driven oscillations is:
“resonance denominator”
“resonance phase”
When w2=k/m, the first term in G vanishes and the amplitude of the oscillation is a maximum. This is the resonance condition.
The width of the resonance curves depends on b, i.e., on the amount of damping. Wider curves with smaller resonance curves correspond to more damping and larger values of b.

18. Forced Oscillations & Resonance
When driving frequency = natural frequency of oscillator, amplitude is maximum.
We say the system is in RESONANCE
“Sharpness” of resonance peak described by quality factor (Q)
High Q = sharp resonance
Damping reduces Q

19. The just-finished Tacoma Narrows bridge collapsed in November 1940
The just-finished Tacoma Narrows bridge collapsed in November Those are 8-foot high steel I-beams that are twisting in the photo on the left—the twisting phenomenon became apparent soon after construction. Winds were a steady 40 mph that morning it collapsed.
The just-finished Tacoma Narrows bridge collapsed in November Those are 8-foot high steel I-beams that are twisting in the photo on the left—the twisting phenomenon became apparent soon after construction. Winds were a steady 40 mph that morning it collapsed. Analysis by scientists at Cal Tech and U of Washington showed that as wind passed over the cables, vortices formed behind the cables which gave the cables a tiny push at one of the resonant frequencies of the bridge. The rebuilt bridge still uses the same vertical supports you see here.

20. Vortex shedding as smoke streams flow past a vertical rod (on the right). Such little vortexes hit the Tacoma Narrows bridge cables and excited one of the bridge’s resonant frequencies, eventually causing its collapse.
Vortex shedding as smoke streams flow past a vertical rod (on the right). Such little vortexes hit the Tacoma Narrows bridge cables and excited one of the bridge’s resonant frequencies, eventually causing its collapse.

21. Forced oscillations and resonance
This lady beetle flies by means of a forced oscillation.
Unlike the wings of birds, this insect’s wings are extensions of its exoskeleton.
Muscles attached to the inside of the exoskeleton apply a periodic driving force that deforms the exoskeleton rhythmically, causing the attached wings to beat up and down.
The oscillation frequency of the wings and exoskeleton is the same as the frequency of the driving force.

22. In the LRC circuit, Q(t) acts just like x(t)
In the LRC circuit, Q(t) acts just like x(t)! underdamped, critically damped, overdamped

23. Driven (and resonance): Vdriving = Vmax coswdt