Damped Harmonic Oscillator

Model Function

Following is the form of the model function for 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 is obtained using the method known as 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, rho = 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>