What is Resonance?

All objects vibrate at a particular frequency called its natural frequency. Resonance occurrs when an outside force such as an electric field or a sound wave oscillates at the same frequency as the object. When this matching of frequencies happens the amplitude of the object's oscillation increases substantially. In fact, if it was not for damping, the amplitude would increase infinitally! A damping force is anything that removes energy from a system and changes it into another form of energy such as heat or sound. Damping shifts the resonant frequency away from the natural frequency, thereby eliminating the threat of an infinate amplitude of oscillation.

In this experiment the length of the rod, l, was fixed. The mass rotated about a fixed center. It's position at any time can be represented by l*θ, where θ is the angle between the rod's current position and its rest position. The acceleration of the mass, therefore, is l * θ″. There is no outside force (other than gravity) driving the system, so forcing does not need to be accounted for. It is unfortunate that we could not test the effects of forcing upon the system since resonance really implies that a forcing frequency exists to interact with the system.

Derivation of the formula of the pendulum

Assumptions:
Gravity is constant, no damping, no forcing.

Using Newton's law we obtain:
Fg = ma = -mg * sin (θ)
ml * θ″ = -mg sin (θ)
θ″ + (mg/ml) sin (θ) = 0
θ″ + (g/l) sin (θ) = 0

Small angle case:
sin (θ) ~= θ
θ″ + (g/l) * θ = 0
θ(t) = C₁ * sin (√(g/l)) + C₂ * cos (√(g/l))
θ(t) = Rcos(&radic(g/l) + φ)
θ(t) = Rcos(ω_o + φ)
where ω_o = √(g/l)

Data and Analysis

I: Is ω_o experimentally correct for small angles?

Experimentally, ω_o for small angles was approximately the same. The equation ω_o = (2π)/T was used to find ω_o. T was found using the graphs. The calculated ω_o was found using this equation:

ω_o = 2π/T, where T = 4√(l/g) * ∫ dφ/(√(1-k²sin²φ)

The experimental ω_o was about the same as the calculated one. Click here for the a plot of the amplitude of the mass for the 3° trial.

II: As you make θ bigger, what happens to ω?

As the initial angle becomes larger, ω_o becomes smaller. Click here for a plot of the amplitude of the mass vs. time for the 75° trial.

III: True that it is independent of mass?

 *From Table1, † From Table2

With the mass removed the period of oscillation were approximately equal.

IV: Does the damping term cause the oscillations to decay in an exponential envelope?

Through this experiment, we found that it is truly an exponential envelope. Click here to see a plot of amplitude vs. time that shows the effect of damping in the experimental system.

V: How good are your assumptions?

Due to the assumptions, there is a slight margin of error. The Experimental ω_o and the Calculated ω_o differ slightly. The assumption that made the greatest imapact in the observed error was that we did not account for the damping factor. The equation that was used to determine the calculated period:

ω_o = 2π/T, where T = 4√(l/g) * ∫ dφ/(√(1-k²sin²φ)

does not account for the presence of damping. As seen here (same graph as Part IV) the experiment was obviously not set in an environment free from damping.

The results also indicate that damping was a larger factor for smaller initial angles than larger ones. This conclusion can be reached by noting the difference between the calculated ω_o and the ω_o found experimentally as θ increases. The difference in the 1° case is .63 while the difference in the 103° run was .09. This indicates that the damping factor for the experiment was a function of velocity and/or position i.e. the friction at the pivot point decreased with increased angular velocity or a change in the position of the pivot rod.

Although the frequency shift due to damping would cause a deviation between a calculated and experimental value this shift is in the opposite direction from what the data shows. This indicates that the experiment was flawed in at least one more way. The software that was used to record the data may have been slightly off in its calculations, which would account for the slight frequency offset.

The other assumptions (no forcing and constant gravity) played a much smaller role in the outcome of the experiment. The difference in the force of gravity over the 1.2167 ft radius of the rod was negligable. Also negligable were any unwanted outside forces acting on the system such as air currents.

As for linearizing the equation of motion for small angles, the results were accurate. The deviation from θ to sin(θ) is very small and therefore negligable.

Links:

A few examples of resonance in action!

A demonstrative web applet

Jeffrey Price and Laura Akl are second year students attending the University of Delaware. This presentation was prepared for MATH341 (Differential Equations and Linear Algebra I)

Last update: 12.19.2002