Lesson 4: Mechanical vs Electrical Analogies
Estimated time: 30-40 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- State the complete force-voltage analogy between mechanical and electrical systems
- Translate a mechanical problem into its electrical equivalent and vice versa
- Identify which physical quantities map to each other
- Use the analogy to transfer solution techniques between domains
- Solve a problem in one domain using insight from the other
The Core Equations Side by Side
The mathematical structure is identical:
Mechanical: mx'' + cx' + kx = F(t)
Electrical: Lq'' + Rq' + (1/C)q = E(t)
Every concept, solution method, and qualitative behavior in one system has a direct counterpart in the other.
The Force-Voltage Analogy Table
Force-Voltage Analogy: Each mechanical quantity maps to an electrical quantity so that the governing equations are structurally identical.
| Mechanical | Symbol | Electrical | Symbol |
|---|---|---|---|
| Displacement | x | Charge | q |
| Velocity | x' = v | Current | q' = i |
| Mass | m | Inductance | L |
| Damping coefficient | c | Resistance | R |
| Spring constant | k | 1/Capacitance | 1/C |
| Applied force | F(t) | Applied voltage (EMF) | E(t) |
| Kinetic energy ½mv² | Magnetic energy ½Li² | ||
| Potential energy ½kx² | Electric energy ½q²/C |
Damping Classification in Both Domains
| Type | Mechanical Condition | Electrical Condition |
|---|---|---|
| Overdamped | c² > 4mk | R² > 4L/C |
| Critically damped | c² = 4mk | R² = 4L/C |
| Underdamped | c² < 4mk | R² < 4L/C |
Natural Frequency: Mechanical: ω0 = √(k/m). Electrical: ω0 = 1/√(LC). Both describe oscillation in the absence of damping.
Worked Examples: Same Equation, Two Interpretations
Example 1: One Equation, Two Systems
Consider u'' + 6u' + 8u = 0, u(0) = 3, u'(0) = 0.
Mechanical reading: m=1 kg, c=6 Ns/m, k=8 N/m. Mass displaced 3 m from equilibrium, released from rest.
Electrical reading: L=1 H, R=6 Ω, 1/C=8 (so C=0.125 F). Initial charge q(0)=3 C, initial current i(0)=0.
Solution: r²+6r+8 = (r+2)(r+4)=0. D = 36-32 = 4 > 0: overdamped.
u = c1e-2t + c2e-4t. u(0) = c1+c2=3. u'(0)=-2c1-4c2=0, so c1=2c2. Then 3c2=3, c2=1, c1=2. u(t)=2e-2t+e-4t.
Physical interpretation: In both systems, no oscillation; the response decays monotonically to zero.
Example 2: Translating a Problem
A spring-mass system has m = 0.5 kg, c = 3 Ns/m, k = 4 N/m. Find the analogous RLC circuit.
Step 1: L = m = 0.5 H.
Step 2: R = c = 3 Ω.
Step 3: 1/C = k = 4, so C = 0.25 F.
Answer: L = 0.5 H, R = 3 Ω, C = 0.25 F. Both systems satisfy 0.5u'' + 3u' + 4u = 0.
Example 3: Resonance in Both Domains
A mechanical system with m=1, k=25 is driven at ω=5 with no damping. What is the analogous circuit?
Mechanical: x'' + 25x = F0cos(5t). ω0 = 5 = ω. Pure resonance.
Electrical analog: L=1 H, 1/C=25 so C=0.04 F, R=0. ω0=1/√(1·0.04) = 1/0.2 = 5. Driving E(t)=E0cos(5t) at the natural frequency. This LC circuit also exhibits resonance: charge grows without bound as qp = (E0/10) t sin(5t).
Why the Analogy Matters
The analogy is more than a mathematical curiosity. It allows engineers to:
- Prototype cheaply: Test a mechanical design using an electrical circuit (much easier to build and modify)
- Transfer intuition: Understanding damping in circuits helps predict mechanical behavior
- Unify theory: One set of solution techniques covers both domains
- Analog computers: Historically, differential equations were solved using RLC circuits
Extending the Analogy: Energy
Energy Correspondence: Kinetic energy ½mv² corresponds to magnetic energy ½Li². Potential energy ½kx² corresponds to electric energy q²/(2C). Energy dissipated by damping (cv²) corresponds to energy dissipated by resistance (Ri²).
Check Your Understanding
1. In the force-voltage analogy, what electrical component corresponds to a heavier mass?
2. A circuit has L=2, R=0, C=1/18. What is the natural frequency? What spring-mass system is analogous?
3. If adding oil to a mechanical system increases damping, what is the electrical equivalent?
Key Takeaways
- The force-voltage analogy maps m↔L, c↔R, k↔1/C, x↔q, v↔i, F↔E.
- Both systems have the same characteristic equation, same damping classification, and same resonance behavior.
- Natural frequency: mechanical ω0=√(k/m), electrical ω0=1/√(LC).
- The analogy allows cross-domain prototyping and transfers intuition between physics and circuit theory.
- Energy is also analogous: kinetic↔magnetic, potential↔electric, dissipated by c↔dissipated by R.