Driven Damped Harmonic Oscillation

(99) |

The equation of motion of the system then becomes [cf., Equation (63)]

(100) |

where is the damping constant, and the undamped oscillation frequency. Suppose, finally, that the piston executes simple harmonic oscillation of angular frequency and amplitude , so that the time evolution equation of the system takes the form

We shall refer to the preceding equation as the

We would generally expect the periodically driven oscillator shown in Figure 9 to eventually settle down to a steady (i.e., constant amplitude) pattern of oscillation, with the same frequency as the piston, in which the frictional energy loss per cycle is exactly matched by the work done by the piston per cycle. (See Exercise 9.) This suggests that we should search for a solution to Equation (101) of the form

(102) |

Here, is the amplitude of the driven oscillation, whereas is the

(103) | ||

(104) |

Equation (101) becomes

(105) |

However, and (see Appendix B), so we obtain

(106) |

The only way in which the preceding equation can be satisfied at all times is if the (constant) coefficients of and separately equate to zero. In other words, if

These two expressions can be combined to give

This follows because Equation (108) gives

(111) |

and so

(112) | ||

(113) |

(See Appendix B.) Hence, substitution into Equation (107) gives Equation (109).

Let us investigate the dependence of the amplitude,
, and phase lag,
, of the driven oscillation on the driving frequency,
. This is most easily done graphically. Figure 10 shows
and
plotted as functions of
for
various different values of
. In fact,
,
,
,
, and
correspond to the solid, dotted, short-dashed, long-dashed,
and dot-dashed curves, respectively. It can be seen that as the amount of
damping in the system is decreased the amplitude of the response becomes
progressively more peaked at the system's natural frequency of oscillation,
. This effect is known as *resonance*, and
is termed the *resonant frequency*. Thus,
a weakly damped oscillator (i.e.,
) can be driven to large amplitude by the application of a relatively
small amplitude external driving force that oscillates at a frequency close to the resonant frequency. The response of the oscillator is in phase (i.e.,
)
with the external drive for driving frequencies well below the resonant
frequency, is in phase quadrature
(i.e.,
)
at the resonant frequency, and is in anti-phase (i.e.,
)
for frequencies well above the resonant frequency.

According to Equations (89) and (109),

In other words, if the driving frequency matches the resonant frequency then the ratio of the amplitude of the driven oscillation to that of the piston oscillation is equal to the quality factor, . Hence, can be interpreted as the resonant amplification factor of the oscillator. Equations (109) and (113) imply that, for a weakly damped oscillator (i.e., ) which is close to resonance [i.e., ],

This follows because . Hence, the width of the resonance peak (in angular frequency) is , where the edges of the peak are defined as the points where the driven amplitude is reduced to of its maximum value: that is, . The phase lag at the low and high frequency edges of the peak are and , respectively. Furthermore, the fractional width of the peak is

(116) |

We conclude that the height and width of the resonance peak of a weakly damped ( ) harmonic oscillator scale as and , respectively. Thus, the area under the resonance peak stays approximately constant as varies.

The force exerted on the system by the piston is

(117) |

Hence, the instantaneous power absorption from the piston becomes

(118) |

The average power absorption during an oscillation cycle is

(119) |

because and . Given that the amplitude of the driven oscillation neither grows nor decays, the average power absorption from the piston during an oscillation cycle must equal the average power dissipation due to friction. (See Exercise 9.) Making use of Equations (114) and (115), the mean power absorption when the driving frequency is close to the resonant frequency is

Thus, the maximum power absorption occurs at the resonance (i.e., ), and the absorption is reduced to half of this maximum value at the edges of the resonance (i.e., ). Furthermore, the peak power absorption is proportional to the quality factor, , which means that it is inversely proportional to the damping constant, .