Download 4.3 Forced Oscillations and Resonance

4.3 Forced Oscillations and Resonance
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If a mass on the end of a spring is pulled
down and released it will continue to
oscillate until something retards it’s
motion.
Damped harmonic motion is harmonic
motion with a frictional or drag force. If
the damping is small, we can treat it as an
“envelope” that modifies the undamped
oscillation.
However, if the damping is large, it no
longer resembles SHM at all.
A: underdamping: there are a few small oscillations before the
oscillator comes to rest.
B: critical damping: this is the fastest way to get to equilibrium.
C: overdamping: system comes to rest without oscillating, but does
not achieve in the shortest possible time.
http://www.patana.ac.th/secondary/science/anrophysics/ntopic4/com
mentary.htm
There are systems where damping is unwanted, such as clocks and
watches.
Then there are systems in which it is wanted, and often needs to be as close to critical damping as
possible, such as automobile shock absorbers and earthquake protection for buildings.
Examples
Under-damping – mass on a stiff spring
Critical damping – door closing damper
Damping is caused my dissipative forces, such as air viscosity, and work is taken from the energy of
oscillation.
Damping is the process whereby energy is taken from the oscillating system
For example a playground swing
– If you push it will oscillate
– It will eventually slow down as energy is lost to friction
– Energy needs to be supplied to keep it oscillating, that comes from you!
Forced vibrations occur when there is a periodic driving force. This force may or may not have the same
period as the natural frequency of the system.
If the frequency is the same as the natural frequency, the amplitude becomes quite large. This is called
resonance.
The oscillations so far have been free oscillations, or natural oscillations, which the system has been
given some energy and left alone.
The frequency of oscillation depends on the inertia and elasticity factors of the system
For example
– Guitar string, it will always play the same notes regardless of how hard you pluck it
– A child’s swing, it will always swing at the same rate regardless of how hard you push it
This is called the natural frequency, f0
Previously the oscillations have been given a single push to start them moving
Often oscillations are subjected to a constant force, called the driving force, f
The effect that the driving for has depends on its frequency
The damping of the system has these effects:
Amplitude
– The amplitude is decreases with damping (cuts down the sharp peak)
– The maximum amplitude is at a frequency less than the natural frequency
•
Energy
– The power of the driver is controlled by
damping
• Wine glass demo
• The glass can be forced to vibrate at it’s natural
resonant frequency.
• Resonance occurs when these forced vibrations
reach maximum amplitude.
• Microwaves – natural resonant frequency of 2450
MHz.
• Gallstones, Kidney stones
• Tacoma Narrows Bridge, Troops Marching
• Quartz Oscillators
-A quartz feels a force if placed in an electric field
and will oscillate when removed.
-Appropriate electronics are added to generate an oscillating voltage from the mechanical movements of
the crystal and this is used to drive the crystal at its own natural frequency.
-These devices provide accurate clocks for microprocessor systems.
• Musical Instruments
-Produce their sounds by arranging for column of air or a string to be driven at its natural frequency,
which causes the amplitude of the oscillations to increase.
• Resonance occurs when the an oscillator is acted upon by a driving force that has the same frequency as
the natural frequency
• The driving force easily transfers its energy to the oscillator
• From the picture the amplitude of oscillation will become very high
• This can be a useful and sometimes very bad
• Electricity, tuning a radio
– The natural frequency of the radio circuit is made equal to the incoming electromagnetic wave
by changing its capacitance
– The electrons in the circuit will oscillate with the incoming electromagnetic wave.
– The electric current will oscillate and this can be turned into sound, through a speaker
• Microwave ovens
– Microwaves are produced at the same frequency as the natural frequency of water molecules
– Water molecules absorb the energy from the microwaves and transfer their energy to the food in
the form of thermal energy
• A Driving force at resonance increases the oscillations, sometimes this is unwanted
• Structures
– Tacoma Narrows bridge, this bridge was destroyed as the wind (driving force) was at the same as
the natural frequency. The bridge vibrated and shook itself apart
An additional unwanted resonance would be
– Tower blocks, the same effect as the bridge the wind, or earthquakes, can cause vibrations to
destroy the buildings
– Vibrations in machinery, if the driving force equals the natural frequency the amplitude may get
dangerously high. Ex. At a particular speed in a truck’s rear view mirror can be seen to vibrate
This can be stopped by designing the building
with heavy damping
– High stiffness, Large mass,
Shape, Good at absorbing energy
The simple harmonic motion of a mass
on a spring is an example of an energy
transformation between potential energy and
kinetic energy.