One aspect of vibration behaviour which was made explicit by the transfer function formula (11) of section 2.2.5 is that natural frequencies correspond to resonances, in the sense that a small applied force at or near such a frequency causes a large response. Unfortunately, the formula predicts *infinite* response when the forcing frequency matches a natural frequency exactly. This obviously unphysical prediction is the result of the fact that we have ignored energy dissipation. If some energy dissipation, or damping, is included in the analysis, the prediction of infinite response is reduced to one of a large but finite response level.

This problem of infinite response at resonance was already seen in section 2.2.2, for the simple mass-spring oscillator without damping. That model can easily be extended as sketched in Fig. 1, to include a mechanism for energy dissipation: the usual way to do this is to add a third component to the system, called a *dashpot*. The symbol in the diagram is meant to indicate a loose-fitting piston in a cylinder, like the shock absorbers of a car or the devices sometimes seen on the backs of doors so that they close without slamming. When the piston is moved in the cylinder, fluid flow around it causes energy loss via viscosity. In the idealised version suitable for our linear system, the resisting force associated with this fluid flow is assumed to be proportional to the relative velocity of piston and cylinder, with a constant of proportionality called $c$ here.

The equation of motion of this modified system is

$$m \ddot{x}+c\dot{x} +kx=f(t) \tag{1} $$

when driven by an external force $f(t)$. But we first examine free motion, with $f=0$. To find the solution, we can try substituting $x(t)=e^{i \alpha t}$ in the hope of solving for possible values of $\alpha$. Equation (1) then requires

$$[-\alpha^2 m +i \alpha c + k] e^{i \alpha t} =0. \tag{2} $$

The exponential factor can be cancelled, leaving a quadratic equation for $\alpha$ which can be solved with the usual formula:

$$\alpha = \dfrac{ic \pm \sqrt{-c^2 + 4 m k}}{2m}. \tag{3}$$

Provided the damping is light, in the sense that $c^2 << 4mk$, this reduces to

$$\alpha \approx \pm \sqrt{\frac{k}{m}} + \frac{ic}{2m}. \tag{4} $$

Now note that $\sqrt{\frac{k}{m}}=\Omega$, the natural frequency of the undamped oscillator found in section 2.2.2. It is convenient to define a dimensionless quantity $\zeta$ so that

$$\frac{c}{2m} = \zeta \Omega \tag{5} $$

which requires

$$\zeta=\frac{c}{2\sqrt{km}}. \tag{6} $$

Then

$$e^{i \alpha t} \approx e^{\pm i \Omega t} e^{-\zeta \Omega t}. \tag{7} $$

This gives us our general solution for free motion of the damped oscillator: in a similar form to equation (5) of section 2.2.2 it is

$$x(t) \approx R \cos(\Omega t + \phi) e^{-\zeta \Omega t} \tag{8} $$

so that the motion consists of oscillation at frequency $\Omega$ with initial amplitude $R$ and phase offset $\phi$, but the effect of the damping is to make the oscillations die away exponentially with time.

The rate of decay is governed by the value of $\zeta$, called the *damping ratio*. But this is not the only quantity used to characterise damping: two related quantities are worth mentioning straight away. The *loss factor*, often denoted $\eta$, is for this simple problem related by $\eta=2\zeta$. The *quality factor* or *Q-factor* is another commonly-used measure of damping, defined by $Q=1/\eta=1/2\zeta$. Low damping is associated with high Q-factor: the name comes from the field of electrical engineering, where for example a “high-quality” quartz crystal oscillator is one with very low damping so that it gives an accurate frequency reference for your computer’s internal clock.

To understand the link between damping and accuracy of a frequency reference, we need to look at the response of the damped oscillator to sinusoidal excitation. The calculation is very similar to the one in section 2.2.2. If $f(t)=F e^{i \omega t}$, the response must also be sinusoidal at the same frequency: $x(t) = Xe^{i \omega t}$. Substituting in equation (1) and cancelling the exponential factor gives

$$[-\omega^2 m + i \omega c +k]X=F \tag{9}$$

so that the frequency response function is

$$G(\omega) = \frac{X}{F} = \frac{1}{k+i \omega c – \omega^2 m} \tag{10} $$

which can be re-written in the alternative forms

$$G(\omega) = \frac{1/m}{\Omega^2+2i \omega \zeta \Omega – \omega^2 }= \frac{1/k}{1+2i \zeta [\omega/\Omega] – [\omega/\Omega]^2 }. \tag{11} $$

The response function $G(\omega)$ is now complex, and we recall that the physical answer is given by the real part of this complex expression.

Both amplitude and phase shift now vary with frequency: some examples are plotted in Fig. 2, for different values of $\zeta$. The amplitude is plotted on a dB scale, in normalised form such that the value at zero frequency (DC) is unity, or 0 dB. The frequency scale shows the ratio $\omega/\Omega$. The phase lag $\phi$ is plotted in degrees. The blue curves show the undamped result: the amplitude goes off to infinity on this dB scale, and the phase switches abruptly at the resonant frequency from zero (in phase) to $-180^\circ$ (opposite phase). The other curves all show non-zero values of $\zeta$. The amplitude at resonance becomes finite, and the peak value reduces as $\zeta$ rises. The phase curves have the abrupt jump smoothed out, over an increasing wide frequency range as $\zeta$ increases.

A simple calculation reveals a useful formula for the bandwidth associated with damping. At the frequencies $\Omega(1 \pm \zeta)$, the amplitude has fallen from the peak value by a factor $1/\sqrt{2}$, while the phase is $90^\circ \pm 45^\circ$. These two frequencies are called the *half-power points*: the energy in the oscillator is proportional to the square of the amplitude, so it has reduced to 50% of its peak value at these two frequencies. The difference of these two frequencies is the *half-power bandwidth*. The half-power bandwidth tends to zero as damping tends to zero, or as Q-factor tends to infinity. This is the reason that a high-Q oscillator can define a very precise resonant frequency.

To give an intuitive feel for the meaning of the damping ratio, some familiar systems illustrating different orders of magnitude are as follows:

- Tuning fork: $\zeta_n \approx 10^{-4}$;

- Guitar string: $\zeta_n \approx 10^{-3}$;

- Steel structure like a car body: $\zeta_n \approx 10^{-2}$;

- Finger pulled out of glass bottle to make a “pop”: $\zeta_n \approx 10^{-1}$.

The last one gives an impression of being quite highly damped, but $\zeta_n \approx 0.1$ is still a sufficiently small number for the purposes of the “small damping” assumption that underlies the simple approach to damping used here. “Small damping” is the rule, not the exception.

Turning to the more general problem, a simple strategy to incorporate some damping into the response of systems with many degrees of freedom is to add some damping to each modal oscillator separately, drawing inspiration from the damped harmonic oscillator just discussed. Thus, we modify equation (11) of section 2.2.5 to

$$G(j,k,\omega) \approx \dfrac{q_k}{F}=\sum_n \dfrac{u_j^{(n)}u_k^{(n)}}{\omega_n^2+2i\omega \omega_n \zeta_n-\omega^2}. \tag{12}$$

where the $\zeta_n$ are dimensionless numbers called *modal damping ratios*.