Material damping associated with small-amplitude vibration can be analysed by making use of a result known, rather grandly, as *the correspondence principle of linear viscoelasticity* (see for example the textbook by Bland [1]. This principle can be described as follows. For any harmonic response problem, at a frequency $\omega$, first solve the undamped problem. Now look to see where elastic moduli enter the solution (such as Young’s modulus $E$, or the plate stiffnesses $D_1$–$D_4$ introduced in section 10.3.2). The corresponding damped problem is solved simply by replacing them with suitable complex values. So, for example, you would replace $E$ with $E(1+i \eta_E)$. Naturally enough, such things are called “complex moduli”. We have already met this idea briefly, back in section 5.4.4 when we were talking about damping in strings, but now we will use it more systematically.

The result is that the resonance frequencies predicted by the analysis become complex numbers. So for example free vibration which for the undamped problem varied like $e^{i \omega t}$ might now vary like

$$e^{i(\omega + i \Delta)t} = e^{i \omega t} e^{-\Delta t} \tag{1}$$

It is clear that this has achieved the desired effect: the oscillation at frequency $\omega$ now decays exponentially at a rate determined by the imaginary part of the frequency, $\Delta$.

If material damping is small, as it usually is in vibration problems, we can use Rayleigh’s principle together with the viscoelastic correspondence principle to find out how the damping varies from mode to mode of a system. Suppose that we know the expressions for the kinetic and potential energies of the system (without damping). Elastic moduli will usually come into the potential energy, but not into the kinetic energy. Either analytically or, more likely, numerically we calculate the first few modes of the undamped system. Now:

(i) The correspondence principle says that to solve the damped problem we replace elastic moduli in the expression for potential energy with complex moduli. For small damping these will only have small imaginary parts.

(ii) Rayleigh’s principle says that given an approximation to a mode shape we can get a rather good approximation to its natural frequency by evaluating the Rayleigh quotient. The modes of the damped system will be slightly different from the modes of the undamped system, but the undamped mode shapes will still give a good approximation. So we evaluate the Rayleigh quotient using the true expression for the potential energy, with complex moduli, but with the approximate expression for mode shape from the undamped calculation. This gives a good approximation to the complex natural frequency, and hence to the modal damping factor.

For a first example of applying this idea, think about beam vibration. We already know from section 3.3.1 that for a bending beam with displacement $w(x,t)$, bending stiffness $EI$ and mass per unit length $m$, the potential energy is

$$ V = \dfrac{1}{2} \int{EI \left(\dfrac{\partial^2 w}{\partial x^2} \right)^2 dx} \tag{2}$$

and the kinetic energy is

$$T=\dfrac{1}{2} \int{ m \left(\dfrac{\partial w}{\partial t} \right)^2 dx } . \tag{3}$$

The only elastic modulus entering here is Young’s modulus $E$. With material damping we can replace this with $E(1+i \eta_E)$. The factor $\eta_E$ may vary with frequency, but for the purposes of this approximate calculation we can evaluate it at the undamped natural frequency $\omega_n$ corresponding to mode shape $w_n(x)$. The Rayleigh quotient, from equations (2) and (3), gives

$$\omega^2 \approx \dfrac{E(1+i \eta_E) I \int{EI \left(\dfrac{\partial^2 w_n}{\partial x^2} \right)^2 dx}}{m \int{w_n^2 dx}}=(1+i \eta_E) \omega_n^2 \tag{4}$$

because this expression apart from the factor $(1+i \eta_E)$ is the Rayleigh quotient for the undamped problem, which is equal to $\omega_n^2$.

So the time dependence of a free vibration in this mode is

$$e^{i \omega_n t} e^{-\eta_E \omega_n t/2} . \tag{5}$$

From the definition of Q-factor, the value for this mode is

$$Q_n \approx \dfrac{1}{\eta_E} . \tag{6}$$

All modes of this beam will have the same Q-factor, except that the material property $\eta_E$ may vary with frequency, but such variation is usually only slow. If you measure damping as well as frequency in a beam test like the ones described in section 10.3.1, that will immediately give the value of $\eta_E$ for the direction aligned with your beam.

Now we can apply the approach to the damping of a wooden plate. We start from the expressions for potential and kinetic energy, from section 10.3.2:

$$ V = \dfrac{1}{2} \int{\int{h^3 \left[ D_1 \left(\dfrac{\partial^2 w}{\partial x^2} \right)^2 + D_2 \dfrac{\partial^2 w}{\partial x^2} \dfrac{\partial^2 w}{\partial y^2} \right. }} $$

$$ \left. + D_3 \left(\dfrac{\partial^2 w}{\partial y^2} \right)^2 + D_4 \left(\dfrac{\partial^2 w}{\partial x \partial y} \right)^2 \right] dx dy \tag{7}$$

and

$$T=\dfrac{1}{2} \int{\int{ \rho h w^2 dx dy }} \tag{8}$$

where $w(x,y)$ is the displacement of the plate, $h$ is the thickness, $D_1$–$D_4$ are the four stiffness constants and $\rho$ is the density.

Once we introduce damping, all four of the stiffness constants become complex:

$$D_j \rightarrow D_j(1+i \eta_j) \mathrm{,~~~~} j=1,2,3,4 . \tag{9}$$

Now following through the same argument based on Rayleigh’s principle, we deduce that for mode $n$ the modal damping factor, the inverse of the modal Q-factor, is a simple weighted sum of the four $\eta_j$:

$$\dfrac{1}{Q_n} \approx J_1 \eta_1 + J_2 \eta_2 + J_3 \eta_3 + J_4 \eta_4 \tag{10}$$

where the dimensionless constants $J_1$–$J_4$ are defined as follows:

$$J_1 = \dfrac{D_1 \int{\int{h^3 \left(\dfrac{\partial^2 w_n}{\partial x^2} \right)^2 dx dy }}}{\omega_n^2 \int{ \int{ \rho h w_n^2 dx dy }}} \tag{11}$$

$$J_2 = \dfrac{D_1 \int{\int{h^3 \dfrac{\partial^2 w_n}{\partial x^2} \dfrac{\partial^2 w_n}{\partial y^2} dx dy }}}{\omega_n^2 \int{ \int{ \rho h w_n^2 dx dy }}} \tag{12}$$

$$J_3 = \dfrac{D_1 \int{\int{h^3 \left(\dfrac{\partial^2 w_n}{\partial y^2} \right)^2 dx dy }}}{\omega_n^2 \int{ \int{ \rho h w_n^2 dx dy }}} \tag{13}$$

$$J_4 = \dfrac{D_1 \int{\int{h^3 \left(\dfrac{\partial^2 w_n}{\partial x \partial y} \right)^2 dx dy }}}{\omega_n^2 \int{ \int{ \rho h w_n^2 dx dy }}} . \tag{14}$$

It follows immediately from the Rayleigh quotient for the original, undamped plate that

$$J_1 + J_2 + J_3 + J_4 = 1 . \tag{15}$$

These constants capture the partitioning of potential energy, and hence energy dissipation rate, between the four terms associated with $D_1$, $D_2$, $D_3$ and $D_4$ in equation (7).

In terms of measurement, the position is very similar to the previous discussion of stiffness. With beam samples cut along and across the grain you can determined the loss factors associated with the two Young’s moduli. With plate samples, the same three modes which gave simple estimates of $D_1$, $D_3$ and $D_4$ now give direct estimates of $\eta_1$, $\eta_3$ and $\eta_4$. The reason is that for each of those modes, one of the $J_j$ constants is approximately equal to 1, while the other three are approximately zero. Equation (10) then says that the measured loss factor (or inverse Q-factor) is essentially equal to the corresponding $\eta_j$. We found before that $D_2$ didn’t play a very strong role in determining frequencies: the equivalent result is that $\eta_2$ seems to have very little influence on modal Q-factors, to the extent that it is usually not possible to determine a convincing value from measurements.

It is useful to see some measured values of $\eta_1$, $\eta_3$ and $\eta_4$. We can show results for the same two spruce plates whose stiffness values were quoted in section 10.3.2. For the quarter-cut plate, the values were $\eta_1 = 0.0051$, $\eta_3 = 0.0216$ and $\eta_4 = 0.0164$. The corresponding results for the plate with a ring angle close to $45^\circ$ were $\eta_1 = 0.0074$, $\eta_3 = 0.0212$ and $\eta_4 = 0.0139$. Note that for both plates the long-grain loss factor $\eta_1$ was significantly lower than the cross-grain loss factor $\eta_3$: by about a factor of 4 for the quarter-cut plate. This is in line with a very general trend across a wide range of materials: there is a negative correlation of loss factor with stiffness. Stiff materials like ceramics or hardened steel tend to have low damping, soft materials like lead or rubber tend to have high damping.

Because $\eta_1$, $\eta_3$ and $\eta_4$ are significantly different, we can expect mode-to-mode variations in modal damping factors, governed by the values of $J_1$, $J_3$ and $J_4$. As an example, the values given above for the quarter-cut spruce plate were based on measuring the Q-factors of 8 modes. These varied from 40 to 140, a range big enough to make very significant differences in sound. Only 3 modes were needed to determine the values of $\eta_1$, $\eta_3$ and $\eta_4$, and remaining 5 measured Q-factors could be used to cross-check the theory on which this method is based. The detailed variation from mode to mode was reproduced to very satisfactory accuracy: see [2] for details.

Measuring values like the ones just quoted, for lightweight materials like spruce, is quite challenging. The reasons were discussed in section 10.3: it is difficult to support the test sample with adding significant extra damping, and it is also hard to fix any kind of sensor to measure the response without also adding extra damping. Furthermore, a sensor of some kind is needed here: we cannot use Chladni patterns to measure damping, we need some kind of quantitative response measurement in order to determine decay rates or half-power bandwidths in FFT results. A description of how the reported measurements were done is given in [2]. We will return to these issues in section 10.4, when we talk about measuring frequency response functions.

[1] D. R. Bland, “The theory of linear viscoelasticity”, Pergamon Press (1960).

[2] M. E McIntyre and J. Woodhouse, “On measuring the elastic and damping constants of orthotropic sheet materials”, *Acta Metallurgica* **36**, 1397—1416 (1988).