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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 xi and the hysteretic one, usually characterized by letter eta.

Hysteretic damping

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

loading... (1)

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

loading... (2)
Introducing the angular frequency of the undamped spring-mass, omega_0 = sqrt(k/m), in the previous expression leads to:
loading... (3)

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

loading... (4)

The resonance angular frequency of the damped system (3), omega_r, is obtained solving:

loading... (5)

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

loading... (6)

The right hand side of eq. (6) vanishes when omega equals omega (for omega not null ).
Thus, the resonance angular frequency of the damped system (3), omega_r, is equal to resonance angular frequency of the undamped system omega as the driving angular frequency, omega, 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:

loading... (7)

In Fourier's domain, this expression becomes:

loading... (8)

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

loading... (9)

Introducing the specific damping, xi, as c/k = 2xi/omega_0 one rewrites eq. (9) as:

loading... (10)

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

loading... (11)

The resonance angular frequency omega_r is calculated using the same reasoning as for the hysteric case above, i.e. finding omega so that:

loading... (12)

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

loading... (13)

Except for frequency 0 Hz, omega_r = omega_0 sqrt(1-2xi^2) because the driving frequency appears in the imaginary part. With this model, damping appears in the resonance angular frequency.
Note that xi is well defined only for the harmonic oscillator because xi depends on omega and for a system with multiple freedom degrees xi must be defined for each frequency mode.

Comparison between specific and hysteretic damping

xi and eta have both non-dimensional quantities but the two FRF have different behaviours regarding first their resonance frequency.

Damping model Magnitude of frequency response
eta loading...
xi loading...

At omega = omega_0 it is possible to write:

Damping model Magnitude of frequency response
at omega = omega_0
eta 1/(sqrt(eta^2))
xi 1/(sqrt(4 xi^2))

If the two frequency reponses must be the same at omega = omega_0, then eta = 2 xi. This is true at only one frequency. In practice, these two models are similar for very small damping.

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