The mathematics behind experimental modal analysis starts from an expression for frequency response functions that we discussed back in Chapter 2. A rather general expression for any mechanical frequency response function of a vibrating structure can be written in the form

$$G(x,y,\omega) = (i \omega)^\alpha \sum_n \dfrac{u_n(x) u_n(y)}{\omega_n^2+2i\omega \omega_n \zeta_n-\omega^2} \tag{1}$$

where the sum is over modes labelled by $n$.

In this formula, the $n$th mode is assumed to have mode shape $u_n$, resonant frequency $\omega_n$ and damping ratio $\zeta_n$. The damping ratio is related to the modal loss factor by $\eta_n = 2 \zeta_n$, and to the modal Q-factor by $Q_n=1/\eta_n$. The notation $u_n(x)$ is a shorthand to denote “the amplitude of the mode shape at the excitation position $x$, in the relevant direction of the applied driving force”. In a similar way, $u_n(y)$ denotes the corresponding modal amplitude at the position and in the direction of the measuring sensor. The mode shapes must be normalised in a particular way in order for this formula to work: see section 2.2.5 for the details. The initial factor $(i \omega)^\alpha$ determines which particular frequency response function we are dealing with: $\alpha=0$ corresponds to a measurement of displacement (known as *receptance*), $\alpha=1$ to a measurement of velocity (i.e. *admittance* or *mobility*), and $\alpha=2$ to a measurement of acceleration (i.e. *accelerance*).

We will look first at the behaviour of a single term from this modal summation: define

$$G_n = \dfrac{(i \omega)^\alpha K_n}{\omega_n^2+2i\omega \omega_n \zeta_n-\omega^2} \tag{2}$$

where

$$K_n= u_n(x) u_n(y) . \tag{3}$$

We can simplify this term by factorising the quadratic in the denominator, then expanding $G_n$ in partial fractions. We can write

$$\omega_n^2+2i\omega \omega_n \zeta_n-\omega^2 = – (\omega \mathrm{~} – \mathrm{~} \bar{\omega}_n)(\omega + \bar{\omega}^*_n) \tag{4}$$

where $.^*$ denotes the complex conjugate, and the complex root $\bar{\omega}_n$ is given by

$$\bar{\omega}_n=\omega_n (\cos \phi_n + i \zeta_n) \approx \omega_n (1 + i \zeta_n) \tag{5}$$

where $\phi_n$ is defined by

$$\sin \phi_n = \zeta_n \tag{6}$$

and the final approximate expression is for the case of small damping $\zeta_n \ll 1$, so that $\phi_n \approx \zeta_n$ and $\cos \phi_n \approx 1$.

Now we look for an expansion in partial fractions. The details depend on the chosen value of $\alpha$, although we will see that the most important aspect of the subsequent behaviour is the same for all values of $\alpha$. For the purpose of demonstrating the method, we will analyse the case $\alpha = 1$. We then have

$$G_n = -\dfrac{i \omega K_n}{(\omega \mathrm{~} – \mathrm{~} \bar{\omega}_n)(\omega + \bar{\omega}^*_n)} \tag{7}$$

and we look for an expression of the form

$$G_n = \dfrac{A}{\omega \mathrm{~} – \mathrm{~} \bar{\omega}_n} + \dfrac{B}{\omega + \bar{\omega}^*_n}$$

$$=\dfrac{A(\omega + \bar{\omega}^*_n) + B(\omega \mathrm{~} – \mathrm{~} \bar{\omega}_n)}{(\omega \mathrm{~} – \mathrm{~} \bar{\omega}_n)(\omega + \bar{\omega}^*_n)} \tag{8}$$

where $A$ and $B$ are two constants. To make this expression agree with with equation (7), we need to satisfy the two conditions

$$-iK_n = A + B \tag{9}$$

and

$$A \bar{\omega}^*_n – B\bar{\omega}_n = 0 . \tag{10}$$

Solving these simultaneous equations gives

$$A = -\dfrac{iK_n}{2}(1+ i\tan \phi_n) \tag{11}$$

and

$$B = -\dfrac{iK_n}{2}(1- i\tan \phi_n) , \tag{12}$$

so that

$$G_n = -\dfrac{iK_n}{2} \left[\dfrac{1+ i\tan \phi_n}{\omega \mathrm{~} – \mathrm{~} \bar{\omega}_n} + \dfrac{1- i\tan \phi_n}{\omega + \bar{\omega}^*_n} \right] \tag{13}$$

$$\approx -\dfrac{iK_n}{2} \left[\dfrac{1+ i\zeta_n}{\omega \mathrm{~} – \mathrm{~} \bar{\omega}_n} + \dfrac{1- i\zeta_n}{\omega + \bar{\omega}^*_n} \right] \tag{14}$$

for small damping.

Equation (13) or (14) tells us that our term $G_n(\omega)$ has two *poles* in the complex $\omega$ plane, at

$$\omega = \bar{\omega}_n \approx \omega_n(1 + i \zeta_n) \tag{15}$$

and

$$\omega = -\bar{\omega}_n^* \approx \omega_n(-1 + i \zeta_n) . \tag{16}$$

A pole is a place where the complex function goes to infinity, and these are both *simple poles* where the infinite behaviour takes the form of the inverse of a linear factor like $(\omega \mathrm{~} – \mathrm{~} z)$, for a pole at $z$.

Exactly the same is true for the other possible values of $\alpha$. For $\alpha = 0$, the result corresponding to equations (13) and (14) is

$$G_n = -\dfrac{K_n}{2 \omega_n \cos \phi_n} \left[\dfrac{1}{\omega – \bar{\omega}_n} – \dfrac{1}{\omega + \bar{\omega}^*_n} \right] \tag{17}$$

$$\approx -\dfrac{K_n}{2 \omega_n } \left[\dfrac{1}{\omega – \bar{\omega}_n} – \dfrac{1}{\omega + \bar{\omega}^*_n} \right] . \tag{18}$$

For $\alpha = 2$ the equivalent result is

$$G_n = K_n+\dfrac{K_n \omega_n}{2} \left[\dfrac{(1-2\zeta_n^2)/\cos \phi_n + 2 i \zeta_n}{\omega – \bar{\omega}_n}\right.$$

$$\left. – \dfrac{(1-2\zeta_n^2)/\cos \phi_n – 2 i \zeta_n}{\omega + \bar{\omega}^*_n} \right] \tag{19}$$

$$\approx K_n+\dfrac{K_n \omega_n}{2} \left[\dfrac{1 + 2 i \zeta_n}{\omega – \bar{\omega}_n} – \dfrac{1 – 2 i \zeta_n}{\omega + \bar{\omega}^*_n} \right] . \tag{20}$$

We can now deduce the pattern of poles in the complex $\omega$ plane for our entire frequency response function $G(x,y,\omega)$: it must look something like Fig. 1. For the particular value $n=3$, the pair of poles are indicated by the red and blue stars, at positions that are symmetrical in the vertical axis. All the other values of $n$ contribute similar pairs, shown as green stars. Note that different scales are used for the horizontal and vertical axes, because for small damping the poles are all very close to the real (horizontal) axis. In this figure, the value $\zeta_n = 0.01$ has been used for all modes.

Now suppose we are interested in the behaviour of the frequency response function close to one of the resonance peaks, for example for frequencies close to $\omega_3$. That means we are close to the red star in Fig. 1. If the damping is low enough that the peaks are well separated, that suggests that we might ignore the effect of the neighbouring poles marked by green stars. But if we are to neglect those, we can surely also neglect the pole marked by the blue star — although it is associated with the same vibration mode as the red star, it is more remote in the complex plane than other poles we have already decided to neglect.

Based on this argument, we can hope to obtain a good first approximation to the behaviour from a single term of the form

$$G(\omega) \approx \dfrac{A}{\omega – \bar{\omega}_3} \tag{21}$$

where $A$ is a (complex) constant. Assuming that the damping is small, we can see from equations (14), (18) and (20) that $A$ is approximately a real number for $\alpha = 0$ or $\alpha = 2$, whereas for $\alpha=1$ it is approximately a pure imaginary number.

For simplicity, we look first at the case where $A$ is a positive real number. We also know that $\bar{\omega}_3 \approx \omega_3(1+i \zeta_3)$ from equation (5). The claim now is that if we plot the trajectory of the complex value $G(\omega)$ in the complex plane, as $\omega$ varies, we will obtain a circle. The simplest way to prove this is first to guess where the circle would have to be, then substitute into the equation for that guessed circle and verify that it is indeed satisfied.

We can guess where the circle must be by looking at the behaviour at some particular frequencies. When $\omega=0$, $G=-A/\omega_3(1+i \zeta_3) \approx -A/\omega_3$. In other words, $G$ is approximately a negative real number. At the opposite extreme, as $\omega \rightarrow\infty$, $G \rightarrow A/\omega$. It is a small real number, tending towards zero from the positive side. Now we look at the peak value, which will occur for $\omega \approx \omega_3$. The value there is $G=iA/\omega_3 \zeta_3$: this is a pure imaginary value, and it is very large because $\zeta_3$ is small.

So when the frequency $\omega$ is well below or well above the resonance frequency, $G$ is near the origin in the complex plane, near the real axis and on one side or the other of the origin. But at the peak, $G$ is high up along the imaginary axis. If there is a circle passing through all these points, it must surely be the one which passes through the origin and the peak value, with its centre on the imaginary axis to preserve symmetry on either side of the resonance frequency. So we might be looking for a circle of radius $R=A/2\omega_3 \zeta_3$, centred on the complex value $iR$, as sketched in Fig. 2. The sketch highlights the various features just mentioned: the peak, the trends as $\omega$ tends to zero and infinity, and the circle centre and radius.

If the real and imaginary parts of $G$ are $G=P+iQ$, the equation of that circle would be

$$P^2 + (Q-R)^2 = R^2. \tag{22}$$

Well, we can easily calculate $P$ and $Q$:

$$G=\dfrac{A}{\omega-\omega_3(1+i\zeta_3)}=\dfrac{A[\omega-\omega_3(1-i\zeta_3)]}{|\omega-\omega_3(1+i\zeta_3)|^2} \tag{23}$$

and so

$$P=\dfrac{A(\omega-\omega_3)}{(\omega-\omega_3)^2 + \omega_3^2 \zeta_3^2} \tag{24}$$

and

$$Q=\dfrac{A\omega_3\zeta_3}{(\omega-\omega_3)^2 + \omega_3^2 \zeta_3^2} . \tag{25}$$

Substituting:

$$P^2 + (Q-R)^2 = \dfrac{A^2(\omega-\omega_3)^2}{[(\omega-\omega_3)^2 + \omega_3^2 \zeta_3^2]^2}$$

$$+ \left\lbrace \dfrac{A\omega_3\zeta_3}{(\omega-\omega_3)^2 + \omega_3^2 \zeta_3^2} – \dfrac{A}{2 \omega_3 \zeta_3} \right\rbrace^2 $$

$$=\dfrac{A^2(\omega-\omega_3)^2 + A^2 \omega_3^2 \zeta_3^2-A^2 [(\omega-\omega_3)^2 + \omega_3^2 \zeta_3^2]}{[(\omega-\omega_3)^2 + \omega_3^2 \zeta_3^2]^2}+R^2$$

$$=R^2. \tag{26}$$

So we see that the single-pole expression for $G(\omega)$ does indeed mark out a circle in the complex plane, exactly as sketched in Fig. 2. That sketch also reveals another interesting feature. We have previously said that the usual way to characterise the bandwidth of a resonance peak in a frequency response function is via the *half-power points* on either side of the peak. We now see that this is a very natural choice for a characteristic bandwidth: those half-power points fall at the ends of the horizontal circle diameter, perpendicular to the diameter through the peak value. The two dashed black lines are angled at $\pm45^\circ$ to the imaginary axis, and the angle between them is a right angle. (This is a well-known theorem of geometry: “the angle in a semicircle is a right angle.”) Pythagoras’ theorem then tells us that the two dashed lines both have length $\sqrt{2}R$. So the squared amplitude is indeed reduced from the peak value by a factor of 2 at these points, and also the phase (given by the angles of the dashed lines) leads and lags the phase at the peak by $45^\circ$.

All this discussion has been based on the case where the constant $A$ was a positive real number. It is easy to see what happens for other values of this constant. If it is a negative real number, the circle would go downwards from the origin rather than upwards. If $A$ is complex, the circle would be rotated. If we write $A=|A| e^{i \theta}$, then everything is multiplied by the factor $e^{i \theta}$, producing a rotation by angle $\theta$. In particular, if $A$ is a pure imaginary number, as we expect to be approximately for the case for any frequency response based on measuring velocity ($\alpha = 1$ in terms of the earlier analysis), then $\theta \approx \pm 90^\circ$ so that the circle will be aligned horizontally rather than vertically in the complex plane.

Our approximation was based on ignoring all the other poles of the frequency response. We are far away from the resonance frequencies of all the other poles, so that their frequency response will have low amplitude, and also it will change very little over the narrow range around the peak we are looking at. The net effect, therefore, is to add a (complex) constant to the approximate expression (21), and so the main effect of these neglected poles is simply to give a slight shift in the position of the centre of the circle.

This gives the basis for a technique for extracting modal parameters from a grid of measured frequency response functions, which was very important in the early years of experimental modal analysis when computer processing power was more limited. For any response peak that is well separated from its neighbours, a suitable narrow frequency range can be chosen so that the plot in the complex plane looks convincingly like part of a circle. The computer can then be made to best-fit a circle, with the radius and the coordinates of the centre as fitted variables. Once the circle has been found, the offset of the centre can be ignored, and the single-pole theoretical expression (21) can be best-fitted to the data points to determine the complex values $A$ and $\bar{\omega}_n$. With a grid of measured responses, $\bar{\omega}_n$ should be the same for all of them, so a single best fit using all available data will give the best estimate. The constant $A$, however, will be different at each grid point. That variation maps out the mode shape.

These days, modal analysis software normally uses a range of more sophisticated signal processing techniques to allow the simultaneous fitting of modal parameters to several modes within a chosen frequency band. To an extent, this allows modal information to be extracted even when adjacent resonance peaks start to overlap. However, it is still a useful check on data quality to look for convincing circle plots near isolated response peaks. As modal overlap increases, the reliability of multi-mode fitting procedures decreases, and it requires experience and sophistication to judge when the result become too unreliable to be trusted. We will not delve any further into these techniques here: for more detail see for example the textbook by Maia et al [1].

[1] Nuno M. M. Maia and Júlio M. M. Silva (editors), “Theoretical and experimental modal analysis”, Research Studies Press/Wiley (1997)