A motor-driven crank imparts sinusoidal oscillatory motion to a mass-spring via a string that passes over a pulley, from which the mass-spring is suspended. (The mass is a 500-gram mass.) You can vary the speed at which the crank rotates, and thus the frequency at which you drive the mass-spring. When this frequency is far from the natural frequency at which the mass-spring oscillates, the motion of the mass – the amplitude of the oscillation of the mass-spring – remains small. As the frequency at which you drive the mass-spring approaches the natural frequency, the motion of the mass becomes greater. When the driving frequency matches, or is resonant with, the natural frequency, the amplitude of oscillation of the mass-spring grows dramatically.
To set the mass-spring oscillating, turn the power/range switch on the control box one click, to “LO.” If you turn it two clicks, to “HI,” the driving frequency will always be higher than the natural frequency of oscillation of the mass-spring, and you will never achieve resonance. Use the large dial to vary the speed of the motor, and thus the frequency at which you drive the oscillation of the mass-spring. An arrow next to the scale around the dial, indicates the location of the setting at which resonance occurs. The length of the crank determines the amplitude of the driving oscillation. This and the height of the pulley are set so that at resonance, the mass strikes the floor on most cycles. This takes just enough energy out of the system to prevent (usually) the mass-spring from jerking the string off the pulley, so that at resonance, you can watch for a long time as the mass oscillates over a very great distance. If the string comes off the pulley and drops onto the pivot, stop the apparatus immediately, as the string can be torn if it keeps going. The resonance peak is fairly narrow, and because the resonant frequency is moderate, it takes some time for the amplitude of the oscillation to build up when you get the system close to resonance. Once you see the motion becoming great, it may take a bit of patience to find the maximum.
The page for demonstration 40.12 -- Mass-springs with different spring constants and masses, describes the motion of a freely oscillating mass-spring. The page for demonstration 40.33 -- Mass-springs damped in oil, water, describes the motion of damped oscillating mass-springs. This demonstration shows the behavior of a driven mass-spring. If we include damping, then the equation that describes this motion is
m(d2x)/(dt2) + c(dx/dt) + kx = F0 cos ωt,
where F0 cos ωt = FD(t), the periodic driving force. If we divide through by m, define the damping factor γ = c/2m, and note that the natural frequency of the pendulum is ω0 = √(k/m), this yields the solution
x = Ae-γt sin (ω′t + φ) + ((F0/m)/[(ω02 - ω2) + 4γ2ω2]1/2) cos (ωt - δ).
The first term on the right, which describes the motion of a damped, free oscillator, decays over time, and is thus called the transient part of the solution. The amplitude factor A equals (x0 - C cos δ)/sin φ, and ω′ = √(ω02 - γ2). The second term on the right does not decay with time, and is called the steady-state part of the solution. We can see that if the damping is large, the amplitude of oscillation is small. Conversely, if the damping is small, the amplitude can be large, depending on the other factors in the equation. We also see that if the frequency of the driving force is much different from the natural oscillating frequency of the mass-spring, the amplitude of the motion is small, and that as the driving frequency approaches the natural oscillating frequency, the amplitude increases. The phase factor δ gives the relationship – the lag – between the driving force and the motion of the mass-spring, and it equals tan-1 [2γω/(ω02 - ω2)].
You can see this in the motion of the mass relative to that of the suspension point of the spring. If the driving frequency is much lower than the natural frequency of oscillation of the mass-spring, the motion of the mass is almost in phase with that of the suspension point (δ ≈ 0). As ω increases, the phase angle, and thus the lag between the motion of the mass and that of the suspension point, slowly increases. (Since δ is subtracted from ωt, it is a negative phase angle.) As ω approaches ω0, the phase angle increases more and more quickly. When ω = ω0, δ = 90°; the motion of the mass lags that of the suspension point by one quarter cycle. As ω increases further, δ continues to increase. When ω is no longer close to ω0, δ inreases more and more slowly. When ω greatly exceeds ω0, δ approaches 180°.
If we plot the amplitude of the steady-state term as a function of the driving frequency, the curve we obtain is low at points far from ω0 on either side, and rises from either side as ω approaches ω0, to reach a maximum at ω0. The breadth of the peak – its full width at half maximum – at ω0 depends on the damping, γ. For small γ, this width equals γ. Its full width at half maximum is Q = ω0/γ. (This quantity is often called the quality.) Thus, as you move the dial toward the setting that corresponds to the natural frequency of the mass-spring, the motion becomes greater and greater.
For this particular system, damping is very small, and the (ω02 - ω2) in the denominator of the amplitude factor of the steady-state term dominates. Thus, the amplitude can grow very large when resonance occurs.
References:
1) Braun, Martin. Differential Equations and Their Applications (New York: Springer-Verlag, 1978) pp. 161-164.
2) Feynman, Richard P., Leighton, Robert B. and Sands, Matthew. The Feynman Lectures on Physics, Volume I (Reading, Massachusetts: Addison-Wesley Publishing Company, 1963) pp. 23-3 to 23-5.
3) Resnick, Robert and Halliday, David. Physics, Part One, Third Edition (New York: John Wiley and Sons, 1977), pp. 323-325.
4) http://hyperphysics.phy-astr.gsu.edu/hbase/oscdr.html#c1 and sections and links below.
5) http://hyperphysics.phy-astr.gsu.edu/hbase/oscdr2.html#c1 and links therein/sections below.