# Engineering Acoustics/Characterizing Damped Mechanical Systems

## Characterizing Damped Mechanical Systems

Characterizing the response of Damped Mechanical Oscillating system can
be easily quantified using two parameters. The system parameters are the
resonance frequency ($'''w  resonance'''$ and the damping of the system
$'''Q  (quality factor)or B  (Temporal Absorption''')$. In practice,
finding these parameters would allow for quantification of unknown
systems and allow you to derive other parameters within the system.

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Using the mechanical impedance in the following equation, notice that
the imaginary part will equal zero at resonance.

($Z_m = F/u = R_m + j(w*M_m - s/w)$)

Resonance case:($w*M_m = s/w$)

## Calculating the Mechanical Resistance

The decay time of the system is related to 1 / B where B is the Temporal
Absorption. B is related to the mechancial resistance and to the mass of
the system by the following equation.

$B = Rm / 2*Mm$

The mechanical resistance can be derived from the equation by knowing
the mass and the temporal absorption.

## Critical Damping

The system is said to be critically damped when:

$Rc = 2*M*sqrt(s/Mm) = 2*sqrt(s*Mm) = 2*Mm*wn$

A critically damped system is one in which an entire cycle is never
completed. The absorption coefficient in this type of system equals the
natural frequency. The system will begin to oscillate, however the
amplitude will decay exponentially to zero within the first oscillation.

## Damping Ratio

$Damping Ratio = Rm/Rc$

The damping ratio is a comparison of the mechanical resistance of a
system to the resistance value required for critical damping. Rc is the
value of Rm for which the absorption coefficient equals the natural
frequency (critical damping). A damping ratio equal to 1 therefore is
critically damped, because the mechanical resistance value Rm is equal
to the value required for critical damping Rc. A damping ratio greater
than 1 will be overdamped, and a ratio less than 1 will be underdamped.

## Quality Factor

The Quality Factor (Q) is way to quickly characterize the shape of the
peak in the response. It gives a quantitative representation of power
dissipation in an oscillation.

$Q = wresonance / (wu - wl)$

Wu and Wl are called the half power points. When looking at the response
of a system, the two places on either side of the peak where the point
equals half the power of the peak power defines Wu and Wl. The distance
in between the two is called the half-power bandwidth. So, the resonant
frequency divided by the half-power bandwidth gives you the quality
factor. Mathematically, it takes Q/pi oscillations for the vibration to
decay to a factor of 1/e of its original amplitude.

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