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 md^{2}x/dt^{2} = -kx -
bdx/dt.

The solution to this differential equation is

x(t) = Aexp(-bt/2m)cos(ω_{damp}t + φ),

with ω_{damp}^{2 }= k/m - (b/2m)^{2},

as long as b^{2 }< 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?

Solution:

The mechanical energy of any oscillator is proportional to the square of the
amplitude. It is the sum of the kinetic energy ½mv^{2} and the
elastic potential energy U_{s} = ½kx^{2}. When the
oscillator reaches its maximum displacement, then its mechanical energy is all
potential energy. We therefore have E = ½kA^{2}.

When A decreases by 5%, then A_{2}/A_{1} = 0.95 and E_{2}/E_{1}
= (A_{2}/A_{1})^{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 = F_{0}cosω_{ext}t.
The total force on the object then is F = F_{0}cos(ω_{ext}t) - kx - bv.

The equation of motion, F = ma, becomes md^{2}x/dt^{2}
= F_{0}cos(ω_{ext}t)
- 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 = ω_{ext} is the angular frequency of the driving force. The amplitude of the
motion is given by

A = (F_{0}/m)/[(ω_{0}^{2 }- ω^{2})^{2
}+ (bω/m)^{2}]^{½},

where ω_{0}^{2 }= k/m. ω_{0} 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 = ω_{0} 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.

Solution:

(a) ω_{0}^{2 }= k/m = 240/(3 s^{2}), ω_{0} = 8.94/s is the
resonance frequency.

(b) ω_{0}^{2 }= g/L = 9.8/(1.5 s^{2}), ω_{0} = 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.

Links:

- Driven harmonic oscillator (interactive)
- Driven oscillator (Youtube)
- Forced Oscillations