1. Damped Oscillations
(Serway )
Physics 1D03 - Lecture 35

2. Simple Pendulum L θ T mg Recall, for a simple pendulum we have the
following equation of motion: L θ T
Which give us: mg
Hence:
or:
Application - measuring height finding variations in g → underground resources
Physics 1D03 - Lecture 35

3. SHM and Damping t x
SHM: x(t) = A cos ωt Motion continues indefinitely. Only conservative forces act, so the mechanical energy is constant. x t
Damped oscillator: dissipative forces (friction, air resistance, etc.) remove energy from the oscillator, and the amplitude decreases with time.
Physics 1D03 - Lecture 35

4. f = -bv where b is a constant damping coefficient
A damped oscillator has external nonconservative
force(s) acting on the system. A common example
is a force that is proportional to the velocity.
f = -bv where b is a constant damping coefficient
F=ma give:
eg: green water (weak damping)
For weak damping (small b), the solution is: x t
A e-(b/2m)t
Physics 1D03 - Lecture 35

5. Without damping: the angular frequency is
With damping:
Effectively, the frequency ω is slower with damping, and
the amplitude gets smaller (decays exponentially) as time goes on:
Physics 1D03 - Lecture 35

6. Question: What is the value of b/(2m)?
Example:
A mass on a spring oscillates with initial amplitude 10 cm. After 10 seconds, the amplitude is 5 cm.
Question: What is the value of b/(2m)?
Question: What is the amplitude after 30 seconds?
Physics 1D03 - Lecture 35

7. Example: A pendulum of length 1. 0 m has an initial
Example: A pendulum of length 1.0 m has an initial amplitude 10°, but after a time of1000 s it is reduced to 5°. What is b/(2m)?
Physics 1D03 - Lecture 35

8. Types of Damping Since: water
We can have different cases for the value under
the root: >0, 0 or <0. This leads to three types of
damping !
Eg: Strong damping (b large): there is no oscillation when:
axle grease?
Physics 1D03 - Lecture 35

9. x(t) t : “Underdamped”, oscillations with decreasing amplitude
: “Critically damped”
: “Overdamped”, no oscillation
x(t)
overdamped
critical damping
Critical damping provides the fastest dissipation of energy. t
underdamped
Physics 1D03 - Lecture 35

10. Quiz: If you were designing a automobile suspensions:
Should they be:
Underdamped
Critically damped
Overdamped
I don’t need suspension !
Physics 1D03 - Lecture 35

11. and hence is always negative.
Example:
Show that the time rate of change of mechanical energy for a damped oscillator is given by:
dE/dt=-bv2
and hence is always negative.
Physics 1D03 - Lecture 35

12. Damped Oscillations and Resonance
(Serway )
Forced Oscillations
Resonance
Physics 1D03 - Lecture 35

13. Forced Oscillations A periodic, external force pushes on the mass
(in addition to the spring and damping):
This transfers energy into the system
The frequency w is set by the machine applying the force. The system responds by oscillating at the same frequency w. The amplitude can be very large if the external driving frequency is close to the “natural” frequency of the oscillator.
Physics 1D03 - Lecture 35

14. Assume that : x = A cos (wt + f) ; then the
is called the natural frequency or resonant frequency of the oscillation.
Newton’s 2nd Law:
Assume that : x = A cos (wt + f) ; then the
amplitude of a drive oscillator is given by:
Physics 1D03 - Lecture 35

15. So, as the amplitude, A, increases !
If the external “push” has the same frequency
as the resonant frequency ω0. The driving
force is said to be in resonance with the system. A
Physics 1D03 - Lecture 35

16. Resonance occurs because the driving force changes direction at just the same rate as the “natural” oscillation would reverse direction, so the driving force reinforces the natural oscillation on every cycle.
Quiz: Where in the cycle should the driving force be at its maximum value for maximum average power?
When the mass is at maximum x (displacement)
When the mass is at the midpoint (x = 0)
It matters not
Physics 1D03 - Lecture 35

17. A 2.0 kg object attached to a spring moves without
Example
A 2.0 kg object attached to a spring moves without
friction and is driven by an external force given by:
F=(3.0N)sin(8πt)
If the force constant of the spring is 20.0N/m, determine
the period of the motion
the amplitude of the motion
Physics 1D03 - Lecture 35

18. Physics 1D03 - Lecture 35

19. Summary
An oscillator driven by an external periodic force will oscillate with an amplitude that depends on the driving frequency. The amplitude is large when the driving frequency is close to the “natural” frequency of the oscillator.
For weak damping, the system oscillates, and the amplitude decreases exponentially with time.
With sufficiently strong damping, the system returns smoothly to equilibrium without oscillation.
Problems, Chapter 15 42, 43, 70 (6th ed – Chapter 13) 42, 70
Physics 1D03 - Lecture 35