1. Chapter 11
Vibrations and SHM

2. 11-1 Simple Harmonic Motion
SHM is any periodic (= time per cycle) vibrating system where the restoring force is directly proportional to the negative of the displacement.
This means, unlike a in Mech, a in SHM is constantly changing!
Ex: mass on a spring
simple* pendulum
UCM, if viewed as a shadow
Objects in SHM are called Simple Harmonic Oscillators (SHO)

3. 11-1 Simple Harmonic Motion
If the spring is hung vertically, the only change is in the equilibrium position, which is at the point where the spring force equals the gravitational force.

4. 11-1 Simple Harmonic Motion
Recall:
Displacement (x) is measured from the equilibrium point
Amplitude (A) is the maximum displacement
A cycle is a full to-and-fro motion; (fig shows 1 full cycle)
Period (T) is the time required to complete one cycle
Frequency (f) is the number of cycles completed per second

5. 11-2 Energy in the SHO
If the mass is at the limits of its motion, x = A;
the energy is all potential.
at a) & c)
If the mass is at the equilibrium point, x = 0;
the energy is all kinetic.
at b)
So a general expression to find total NRG in an ideal system is:
at d)… or anywhere… =

6. The Period & Frequency of SHOs
The period and frequency of a bobbing mass:
larger m  larger T
larger k  smaller T
Note neither T nor f dependent upon amplitude!
The period and frequency of a swinging pendulum:
larger L  larger T
larger g  smaller T
Note, if amp is small, not T nor f dependent upon mass!

7. More Useful Equivalencies…

8. 11-4 The Simple Pendulum
The swing motion of this lamp, hanging by a very long cord from the ceiling of the cathedral of Pisa, is said to have been observed by Galileo and to have inspired him to the conclusion that the period of a pendulum does not depend on amplitude.

9. 11-3 Sinusoidal Nature of SHM
It is easy to see how a bobbing mass would create a cosine or sine curve – it’s actually the mass’s x vs t graph
The bottom curve is the same, but shifted ¼ period so that it is a sine function rather than cosine.

10. 11-3 Sinusoidal Nature of SHM
If we look from the edge of the table at an object in UCM, we can see how the cosine function is associated with the object’s position as a function of time:
cos  = x/A
or more commonly written
x = A cos  where  = t
so x = A cos (t)
where  = 2f
so x = A cos (2ft)
or x = A cos (2t/T)

11. 11-3 Sinusoidal Nature of SHM
The velocity and acceleration can be calculated as functions of time and are plotted at left.
How do the curves make sense relative to each other?

12. 11-5 Damped Harmonic Motion
Damped harmonic motion is harmonic motion in real life… with internal friction or drag force. Note, its amplitude decreases while its T & f stay constant.
In systems where continued oscillations are a problem, just the proper amount of damping, called critical damping, is desired. Like in…

13. 11-5 Damped Harmonic Motion
shock absorbers in cars
door closing mechanisms
large buildings in earthquake prone areas

14. 11-6 Forced Vibrations; Resonance
Forced vibrations occur when there is a periodic driving force. This force may or may not have the same period as the natural frequency (f0) of the system.
f0 is based on an object’s elasticity (see Ch 9) – every object has frequencies at which it will naturally oscillate if an external force (force vibration) is applied.
Resonance: If external frequency = natural frequency, the amplitude becomes quite large.
The sharpness of the resonant peak depends on the damping.
If the damping is small (A), it can be quite sharp; if the damping is larger (B), it is less sharp.

15. 11-6 Forced Vibrations; Resonance
Like damping, resonance can be wanted or unwanted.
Examples that depend upon resonance:
Musical instruments
TV/radio receivers
Child on a swing
Shattering a crystal goblet
Examples of unwanted resonance:
Soldiers marching across a bridge
Spectators stomping/clapping on stadium risers
Tall buildings & long bridges in an earthquake
Wind on the Tacoma Narrows Bridge
Container ships rolling on high seas
(see pictures)