Engineering Acoustics/Solution Methods for Electro-Mechanical Analogies

After drawing the electro-mechanical analogy of a mechanical system, it is always safe to check the circuit. There are two methods to accomplish this:

Review of Circuit Solving Methods

Kirchkoff's Voltage law

"The sum of the potential drops around a loop must equal zero."

$v_{1}+v_{2}+v_{3}+v_{4}=0\displaystyle$

Kirchkoff's Current Law

"The Sum of the currents at a node (junction of more than two elements) must be zero"

$-i_{1}+i_{2}+i_{3}-i_{4}=0\displaystyle$

Hints for solving circuits:

Remember that certain elements can be combined to simplify the circuit (the combination of like elements in series and parallel)

If solving a circuit that involves steady-state sources, use impedances. Any circuit can eventually be combined into a single impedance using the following identities:

Impedances in series: $Z_{\mathrm {eq} }=Z_{1}+Z_{2}+\,\cdots \,+Z_{n}.$

Impedances in parallel: ${\frac {1}{Z_{\mathrm {eq} }}}={\frac {1}{Z_{1}}}+{\frac {1}{Z_{2}}}+\,\cdots \,+{\frac {1}{Z_{n}}}.$

Dot Method: (Valid only for planar network)

This method helps obtain the dual analog (one analog is the dual of the other). The steps for the dot product are as follows:

 1) Place one dot within each loop and one outside all the loops. 

 2) Connect the dots. Make sure that there is only one line through each element and that no lines cross more than one element.

 3) Draw in each line that crosses an element its dual element, including the source. 

 4) The circuit obtained should have an equivalent behavior as the dual analog of the original electro-mechanical circuit.

Example:

The parallel RLC Circuit above is equivalent to a series RLC driven by an ideal current source

Low-Frequency Limits

This method looks at the behavior of the system for very large or very small values of the parameters and compares them with the expected behavior of the mechanical system. For instance, you can compare the mobility circuit behavior of a near-infinite inductance with the mechanical system behavior of a near-infinite stiffness spring.

	Very High Value	Very Low Value
Capacitor	Short Circuit	Open Circuit
Inductor	Open Circuit	Closed Circuit
Resistor	Open Circuit	Short Circuit

Additional Resources for solving linear circuits

Thomas & Rosa, "The Analysis and Design of Linear Circuits", Wiley, 2001

Hayt, Kemmerly & Durbin, "Engineering Circuit Analysis", 6th ed., McGraw Hill, 2002

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