Electric Circuit Analogs

The vibrating system and its characteristics can be represented by an electric circuit. Now let’s consider the electrical circuit which is shown in the next figure.

 

Figure 1 - Electric Circuit

The system consists of inductor L, a resistor R and a capacitor C. When the voltage E(t) is applied to the system, it caused a current of magnitude i to flow through the circuit. As the current flows past the inductor the voltage drop is L(di/dt). When the current flows through the resistor the voltage drop is Ri, and when it arrives at the capacitor the drop is 

$$\frac{1}{C}\int{idt}$$

Since the current cannot flow past a capacitor, it’s only possible to measure the charge q acting on the capacitor. The charge can be related to the current by the equation

$$i=\frac{dq}{dt}$$

Thus the voltage drops, which occur across the inductor, resistor, and capacitor becomes

$$L\frac{{{d}^{2}}q}{d{{t}^{2}}},R\frac{dq}{dt},\frac{q}{C}$$

By applying the Kirchhoff’s law which states that the applied voltage balances the sum of the voltage drops around the circuit. Therefore:

$$L\frac{{{d}^{2}}q}{d{{t}^{2}}}+R\frac{dq}{dt}+\frac{1}{C}q=E(t)$$

As you can see the previous differential equation which represents the change of voltage in electrical system is similar to the differential equation which describes the motion of Viscous Dampers Forced Vibration system.

By comparing these to differential equation we can see that these equation have the same form, and hence mathematically the procedure of analyzing an electric circuit is the same as that of analyzing a vibration mechanical system

$$m\frac{{{d}^{2}}x}{d{{t}^{2}}}+c\frac{dx}{dt}+kx=F(t)$$

From this we can derive the analogs between two equations and we will show them in the following table.

Electrical		Mechanical
Electric Charge	q	Displacement	x
Electric Current	i	Velocity	dx/dt
Voltage	E(t)	Applied Force	F(t)
Inductance	L	Mass	m
Resistance	R	Viscous damping coefficient	c
Reciprocal of capacitance	1/C	Spring stiffness	k