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.

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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