Example: m = 1, k = 100, b = 1
Friction will damp out the oscillations of a macroscopic system, unless the oscillator is driven. If the speed of a mass on a spring is low, then the drag force R due to air resistance is approximately proportional to the speed, R = -bv. The total force on the object then is
F = -kx - bv.
The equation of motion, F = ma, becomes md2x/dt2 = -kx - bdx/dt.
The solution to this differential equation is
x(t) = Aexp(-bt/2m)cos(ωdampt + φ),
with ωdamp2 = k/m - (b/2m)2,
as long as b2 < 4mk, i.e. as long as the drag force is not too large. The oscillatory character of the motion is preserved, but the amplitude decreases with time.
Link: Damped simple harmonic motion (interactive)
The amplitude of a lightly damped oscillator decreases by 5% during each
cycle. What percentage of the mechanical energy of the oscillator is lost
in each cycle?
The mechanical energy of any oscillator is proportional to the square of the amplitude. It is the sum of the kinetic energy ½mv2 and the elastic potential energy Us = ½kx2. When the oscillator reaches its maximum displacement, then its mechanical energy is all potential energy. We therefore have E = ½kA2.
When A decreases by 5%, then A2/A1 = 0.95 and E2/E1 = (A2/A1)2 = 0.9025. 9.75% of the mechanical energy is lost each cycle.
Amplitude versus driving frequency
A driving force with the natural resonance frequency of the oscillator can
efficiently pump energy into the system.
Assume a driving force F = F0cosωextt. The total force on the object then is F = F0cos(ωextt) - kx - bv.
The equation of motion, F = ma, becomes md2x/dt2 = F0cos(ωextt) - kx - bdx/dt.
After a steady state has been reached, the position varies as a function of time as
x(t) = Acos(ωt + φ).
Here ω = ωext is the angular frequency of the driving force. The amplitude of the motion is given by
A = (F0/m)/[(ω02 - ω2)2 + (bω/m)2]½,
where ω02 = k/m. ω0 is he natural frequency of the undamped oscillator. When the frequency of the driving force is very close to the natural frequency and the drag force is small, then the denominator in the above expression becomes very small, and the amplitude becomes very large. This dramatic increase in amplitude is called resonance, ω = ω0 is called the resonance frequency.
Calculate the resonance frequency of
(a) a 3 kg mass attached to a spring of force constant 240 N/m and
(b) a simple pendulum 1.5 m in length.
(a) ω02 = k/m = 240/(3 s2), ω0 = 8.94/s is the resonance frequency.
(b) ω02 = g/L = 9.8/(1.5 s2), ω0 = 2.55/s is the resonance frequency.
If we push a child on a swing once during each period, and we always push in
the same direction, we will increase the amplitude of the swing, until the
positive work we do on the swing equals the negative work done on the swing by
the frictional forces.
If we drive a harmonic oscillator with a driving force with the natural resonance frequency of the oscillator, then the amplitude can increase enormously, even if the work done during each cycle is very small. The amplitude of an ideal harmonic oscillator increases forever. Friction limits the maximum amplitude of a real oscillator. The amplitude may become large enough for the system to become an anharmonic oscillator. Then the driving force is no longer in phase with the oscillations, and it sometimes does negative work and reduces the amplitude. Often, however the amount of energy put into the system is large enough to damage or break the system.