Mechanical damping

The mechanical damping tends to reduce the amplitude of a vibrating structure by molecular interactions inside the solid phase of a porous material. Mainly two models accounting for this damping are used, the the specific damping, usually characterized with letter $ξ$ and the hysteretic one, usually characterized by letter $η$ .

Hysteretic damping

This model of damping is only defined in Fourier's domain.

(- m ω^{2} + k (1 + i η)) \tilde{u} (ω) = \tilde{f} (ω)

(1)

dividing eq. (1) by the stiffness of the spring, $k$ , leads to the following frequency response (ration of applied force over resulting displacement):

\tilde{H} = \frac{1 / k}{- \frac{m}{k} ω^{2} + (1 + i η)}

(2)

Introducing the angular frequency of the undamped spring-mass, $ω_{0} = \sqrt{k / m}$ , in the previous expression leads to:

\tilde{H} = \frac{1 / k}{1 - {(\frac{ω}{ω_{0}})}^{2} + i η}

(3)

From this last equation, one can deduce the magnitude of the frequency response:

| \tilde{H} | = \frac{1 / k}{\sqrt{{(1 - {(\frac{ω}{ω_{0}})}^{2})}^{2} + η^{2}}}

(4)

The resonance angular frequency of the damped system (3), $ω_{r}$ , is obtained solving:

\frac{\partial | \tilde{H} |^{2}}{\partial ω} = 0

(5)

The left hand side of the previous equation is calculating from eq. (4):

\frac{\partial | \tilde{H} |^{2}}{\partial ω} = \frac{- 1 / k^{2} (2 (- 2 \frac{ω}{ω_{0}^{2}}) (1 - {(\frac{ω}{ω_{0}})}^{2}))}{{[{(1 - {(\frac{ω}{ω_{0}})}^{2})}^{2} + η^{2}]}^{2}}

(6)

The right hand side of eq. (6) vanishes when $ω$ , equals $ω_{0}$ (for $ω \neq 0$ ).
Thus, the resonance angular frequency of the damped system (3), $ω_{r}$ , is equal to resonance angular frequency of the undamped system $ω_{0}$ as the driving angular frequency, $ω$ , does not appear in the imaginary part of the frequency response (3).

Specific damping

The so-called fundamental equation of the damped harmonic oscillator in the time domain can be written as:

m \frac{d^{2} u (t)}{d t^{2}} + c \frac{d u (t)}{d t} + k u (t) = f (t)

(7)

In Fourier's domain, this expression becomes:

(- m ω^{2} + i c ω + k) \tilde{u} (ω) = \tilde{f} (ω)

(8)

From equation (8) the frequency response (ratio of force applied over resulting displacement) is:

\tilde{H} = \frac{1 / k}{1 - {(\frac{ω}{ω_{0}})}^{2} + i \frac{c}{k} ω}

(9)

Introducing the specific damping, $ξ$ , as $c / k = 2 ξ / ω_{0}$ one rewrites eq. (9) as:

\tilde{H} = \frac{1 / k}{1 - {(\frac{ω}{ω_{0}})}^{2} + 2 i ξ \frac{ω}{ω_{0}}}

(10)

From eq. (10), the expression of magnitude of the frequency response is deduced:

| \tilde{H} | = \frac{1 / k}{\sqrt{{(1 - {(\frac{ω}{ω_{0}})}^{2})}^{2} + 4 ξ^{2} {(\frac{ω}{ω_{0}})}^{2}}}

(11)

The resonance angular frequency $ω_{r}$ is calculated using the same reasoning as for the hysteric case above, i.e. finding $ω$ so that:

\frac{\partial | \tilde{H} |^{2}}{\partial ω} = 0

(12)

The left hand side of the previous equation is calculating from eq. (11):

\frac{\partial | \tilde{H} |^{2}}{\partial ω} = \frac{- 1 / k^{2} (2 (- 2 \frac{ω}{ω_{0}^{2}}) (1 - {(\frac{ω}{ω_{0}})}^{2}) + 8 ξ^{2} \frac{ω}{ω_{0}^{2}})}{{[{(1 - {(\frac{ω}{ω_{0}})}^{2})}^{2} + 4 ξ^{2} {(\frac{ω}{ω_{0}})}^{2}]}^{2}}

(13)

Except for frequency 0 Hz, $ω_{r} = ω_{0} (1 - 2 ξ^{2})^{1 / 2}$ because the driving frequency appears in the imaginary part. With this model, damping appears in the resonance angular frequency.
Note that $ξ$ is well defined only for the harmonic oscillator because $ξ$ depends on $ω_{0}$ and for a system with multiple freedom degrees $ξ$ must be defined for each frequency mode.

Comparison between specific and hysteretic damping

$ξ$ and $η$ have both non-dimensional quantities but the two FRF have different behaviours regarding first their resonance frequency.

Damping model	Magnitude of frequency response
$η$	$\| \tilde{H} \| = \frac{1 / k}{\sqrt{{(1 - {(\frac{ω}{ω_{0}})}^{2})}^{2} + η^{2}}}$
$ξ$	$\| \tilde{H} \| = \frac{1 / k}{\sqrt{{(1 - {(\frac{ω}{ω_{0}})}^{2})}^{2} + 4 ξ^{2} {(\frac{ω}{ω_{0}})}^{2}}}$

At $ω = ω_{0}$ it is possible to write:

Damping model	Magnitude of frequency response at $ω = ω_{0}$
$η$	$\frac{1 / k}{\sqrt{η^{2}}}$
$ξ$	$\frac{1 / k}{\sqrt{4 ξ^{2}}}$

If the two frequency reponses must be the same at $ω = ω_{0}$ , then $η = 2 ξ$ . This is true at only one frequency. In practice, these two models are similar for very small damping.