Linear Harmonic Motion
Model Function
Following is the form of the function which models the motion of a linear damped harmonic
oscillator:
In this function, A is the amplitude of oscillation, delta
is the damping constant, omega is the freqency of
oscillation, and phi is the phase shift of the graph.
Group Discussion
Give some everyday examples of oscillating objects. Which appear to
be damped and which appear to be undamped?
Amplitude
The amplitude of an oscillating object corresponds to the greatest
displacement of that object while in motion.
Damping
The amplitude of any real oscillator such as a pendulum, a spring,
or an oscillating cantilevered beam tends to
decrease with time until the oscillations cease altogether. This
is due to internal friction of the system and air resistance. An
example of damped oscillations is shown below.
The damping constant can be obtained using the
log decrement method.
In this method, a linear regression analysis is done on a semi
log plot of the local maxima on the graph of the oscillator. The
slope of the best fit line is the damping constant.
Natural Frequency
Oscillating objects experience periodic motion; their motion repeats
itself, back and forth, over the same path. The completion of
one repetition of this path from an initial point back to
the same point is called a cycle. The time required
to undergo a cycle is the period. The frequency corresponds to the
number of complete cycles per second. Thus, the relationship between
the frequency, omega, and the period, T, is reciprocal.
That is, omega = 1 / T , where the frequency is expressed in
Hertz and the period is in seconds.
Phase Angle
The phase angle, phi, of a damped harmonic oscillator indicates
how long before or after t = 0 the peak of maximum amplitude
is reached.
Question 1
Given that there is no damping, what is the value
of the phase angle,
phi, in the harmonic oscillator model function,
if at t = 0, the oscillating particle is at
- F(0) = A
- F(0) = 0
- F(0) = -A
- F(0) = (1/2) A
Question 2
Given that there is no damping for a particle
undergoing harmonic motion
with amplitude A, what is the total
distance it travels in
one period?
More: Comparison of Model with Data
Next: Higher Harmonic Motion
Up: Introduction
Previous: Oscillating Cantilevered Beams
Jennifer Powell<jpowell@geom.umn.edu>
Fati Liamidi<liamidi@geom.umn.edu>