Electrical Analogy of Mechanical Oscillators

Athul Kurian
Electrical Analogy of Mechanical Oscillators

Basic Components:

  1. Mass (m):
    • In the electrical analogy, mass is represented by an inductor (\(L\)).
    • The inductor stores energy in its magnetic field, similar to how mass stores energy in motion.
  2. Spring (k):
    • The spring is represented by a capacitor (\(C\)).
    • The capacitor stores energy in an electric field, analogous to how a spring stores energy in its deformation.
  3. Damper (b):
    • The damper is represented by a resistor (\(R\)).
    • The resistor dissipates energy, which is similar to how a damper dissipates mechanical energy in the form of heat.

Equations:

  1. Equation of Motion for Mechanical System: \[ m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = F(t) \]
  2. Equation for Electrical Analogy: \[ L \frac{di}{dt} + R i + \frac{1}{C} \int i \, dt = V(t) \]
    • \( i \): Current (analogous to velocity in mechanics)
    • \( V(t) \): Voltage (analogous to force in mechanics)

Analysis:

  1. Impedance (\(Z\)):
    • In the electrical domain, impedance plays a role similar to resistance in the mechanical domain.
    • Impedance (\(Z\)) is given by \[ Z = R + j(\omega L - \frac{1}{\omega C}) \], where \(\omega\) is the angular frequency.
  2. Resonance:
    • Resonance in the mechanical system corresponds to a condition where the impedance is purely resistive in the electrical system.
    • This occurs when \(\omega L = \frac{1}{\omega C}\).
  3. Transfer Function:
    • The transfer function (\(H(s)\)) relates the output to the input in the Laplace domain.
    • For the electrical analogy, \[ H(s) = \frac{V(s)}{I(s)} = \frac{1}{ms^2 + bs + k} \], where \(s\) is the complex frequency variable.

Example:

Consider a simple mass-spring-damper system with mass \(m = 1 \, \text{kg}\), damping coefficient \(b = 0.5 \, \text{N.s/m}\), and spring constant \(k = 10 \, \text{N/m}\).

The electrical analogy would have \(L = 1 \, \text{H}\), \(R = 0.5 \, \Omega\), and \(C = 0.1 \, \text{F}\).

Summary:

The electrical analogy simplifies the analysis of mechanical systems by using familiar electrical components. This method is particularly useful in control systems and electrical engineering for understanding and designing systems with mechanical components. It provides a bridge between mechanical and electrical engineering, allowing engineers to leverage their knowledge in both domains.

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