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
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
It is clear that this has achieved the desired effect: the oscillation at frequency
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
and the kinetic energy is
The only elastic modulus entering here is Young’s modulus
because this expression apart from the factor
So the time dependence of a free vibration in this mode is
From the definition of Q-factor, the value for this mode is
All modes of this beam will have the same Q-factor, except that the material property
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:
and
where
Once we introduce damping, all four of the stiffness constants become complex:
Now following through the same argument based on Rayleigh’s principle, we deduce that for mode
where the dimensionless constants
It follows immediately from the Rayleigh quotient for the original, undamped plate that
These constants capture the partitioning of potential energy, and hence energy dissipation rate, between the four terms associated with
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
It is useful to see some measured values of
Because
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).