Chapter 11 Vibrations and SHM
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)
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.
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
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… =
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!
More Useful Equivalencies…
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.
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.
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 = 2f so x = A cos (2ft) or x = A cos (2t/T)
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?
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…
11-5 Damped Harmonic Motion shock absorbers in cars door closing mechanisms large buildings in earthquake prone areas
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.
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)