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System Dynamics · Module 5 of 10

Energy Methods, Generalized Variables, and Analogies

Every domain has an effort that drives and a flow that responds, and their product is power. Naming them makes the analogies exact, so a mechanical system and its electrical twin share one set of equations.

Readiness check

This module unifies the four domains. Tick only what you can do closed-notes.

Recall that power is energy per unit time.
Write the energy stored in a spring, ½kx², and a mass, ½mv².
Recall the energy in a capacitor and an inductor.
Map mass, damper, spring to inductance, resistance, capacitance.
Recall ω_n = √(k/m) and 1/√(LC).

0 or 1 weak itemsContinue with this module.

2 weak itemsRevisit the force-voltage analogy in Module 3.

3 or more weak itemsReview energy storage in Module 2.

The core idea

Each domain has an effort variable and a flow variable whose product is power. Inertia stores energy in flow, compliance stores it in effort, and resistance dissipates. Because the forms match, one model serves every domain.

power = effort × flowinertia: ½ I f²; compliance: ½ C e²force↔voltage, velocity↔current

Energy methods give system dynamics a single language. In every domain one variable is an effort that drives, force, voltage, pressure, temperature, and another is a flow that responds, velocity, current, volume rate, entropy rate. Their product is power, which is what crosses between elements. Each element plays one of three roles: an inertia stores energy in its flow (½ I f², like a mass or an inductor), a compliance stores energy in its effort (½ C e², like a spring or a capacitor), and a resistance dissipates power. Because these roles and their energy expressions are identical across domains, a mechanical system has an exact electrical analog with the same equations and the same ω_n and ζ. This is the basis of the bond-graph view that unifies mechatronic modeling.

The skill works when: you identify the effort and flow, classify each element, and map it to the analogous domain.

The skill breaks down when: effort and flow are swapped, or a mass is matched to a capacitor instead of an inductor.

The concept. Effort and flow carry power into every element. Whether the domain is mechanical, electrical, fluid, or thermal, the element either stores energy in flow, stores it in effort, or dissipates it.

The skills, taught in order

Five skills build the generalized variables, the energy in each element, and the analogies they create.

5.1 Effort and flow variables

In each domain, one variable is the effort and one the flow, and their product is power. Mechanical: force and velocity. Electrical: voltage and current. Fluid: pressure and volume rate. Thermal: temperature and entropy rate. Choosing them correctly is the first step of energy-based modeling.

5.2 Generalized inertia and compliance

An inertia element relates effort to the rate of flow (e = I ḟ) and stores ½ I f²: a mass or an inductor. A compliance relates flow to the rate of effort (f = C ė) and stores ½ C e²: a spring or a capacitor. These two store energy; a resistance only dissipates.

Mechanical	Electrical	Role	Stored energy
mass m	inductance L	inertia	½ m v² = ½ L i²
compliance 1/k	capacitance C	compliance	½ k x² = ½ C v²
damper c	resistance R	dissipation	none

The force-voltage analogy element by element. A mass pairs with an inductance, a spring with a capacitance, a damper with a resistance.

5.3 Stored energy and dissipation

The energy in an inertia depends on its flow squared; in a compliance, on its effort squared. A resistance converts power irreversibly to heat. Tracking where energy is stored and where it leaves is a powerful check on any model.

5.4 The cross-domain analogies

The force-voltage analogy maps mass to inductance, compliance to capacitance, and damper to resistance, so a mechanical and an electrical system with matched parameters have identical dynamics, the same ω_n and ζ. The same mapping reaches the fluid and thermal domains.

5.5 Energy-based modeling

Working in effort and flow lets one method, the bond-graph approach of the source text, model a mechatronic system spanning several domains in one diagram. It guarantees that power is conserved at every junction and makes the state equations fall out systematically.

Engineering connection: a loudspeaker, a motor, and a hydraulic actuator are all multi-domain devices whose models are cleanest in the effort-flow language.

Worked example 1: stored energy and the inertia analogy

A spring of stiffness 400 N/m is stretched 0.05 m; find its stored energy. Separately, a 4 kg mass moves at 3 m/s; find its kinetic energy and confirm the energy of an analogous inductor with L = 4 H carrying 3 A.

Figure 1. The spring stores energy in its effort (½kx²); the mass stores energy in its flow (½mv²), exactly matching an inductor of the same numerical inertia and flow.

ProblemFind the spring energy, the mass kinetic energy, and the analogous inductor energy in Figure 1.
Given / findk = 400 N/m, x = 0.05 m; m = 4 kg, v = 3 m/s; L = 4 H, i = 3 A. Find each energy.
AssumptionsLinear elements; the force-voltage analogy maps mass to inductance.
ModelSpring stores ½kx² (effort-type); mass stores ½mv²; inductor stores ½Li² (both flow-type).
EquationsE_spring = ½ k x²E_mass = ½ m v²E_L = ½ L i²
SolveE_spring = ½ × 400 × 0.05² = ½ × 400 × 0.0025 = 0.5 J. E_mass = ½ × 4 × 3² = ½ × 4 × 9 = 18 J. E_L = ½ × 4 × 3² = 18 J.
CheckThe mass and the inductor store identical energy because both use ½(inertia)(flow)² with the same numbers, confirming the mass-inductance analogy.
ConclusionA spring stores energy through its effort and a mass through its flow; the mass and inductor match term for term, which is why the analogy preserves dynamics.

Result. Spring 0.5 J; mass 18 J, equal to the analogous inductor's 18 J.

Worked example 2: a mechanical system and its electrical twin

A mass-spring-damper has m = 2 kg, c = 8 N·s/m, and k = 50 N/m. Build its force-voltage electrical analog and confirm both have the same natural frequency.

Figure 2. Mapping mass to inductance, damper to resistance, and stiffness to inverse capacitance produces an electrical circuit with the identical natural frequency, the analogy made concrete.

ProblemBuild the electrical analog of the mechanical system in Figure 2 and compare natural frequencies.
Given / findm = 2 kg, c = 8 N·s/m, k = 50 N/m. Find L, R, C and both ω_n values.
AssumptionsForce-voltage analogy: mass to L, damper to R, compliance 1/k to C.
ModelMap L = m, R = c, C = 1/k; then ω_n,mech = √(k/m) and ω_n,elec = 1/√(LC).
EquationsL = m, R = c, C = 1/kω_n,mech = √(k/m)ω_n,elec = 1/√(LC)
SolveL = 2 H, R = 8 Ω, C = 1/50 = 0.02 F. ω_n,mech = √(50/2) = √25 = 5 rad/s. ω_n,elec = 1/√(2 × 0.02) = 1/√0.04 = 1/0.2 = 5 rad/s.
CheckBoth natural frequencies are 5 rad/s. The damping ratios also match: ζ = c/(2√(km)) = 8/(2√100) = 0.4, and (R/2)√(C/L) = 4 × √(0.01) = 0.4.
ConclusionThe electrical twin reproduces the mechanical dynamics exactly. One analysis, or one simulation, can stand in for the other.

Result. L = 2 H, R = 8 Ω, C = 0.02 F; both ω_n = 5 rad/s.

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
Effort and flow swapped	Power product has wrong units	"Which variable drives, which responds?"	Effort drives (force, voltage); flow responds (velocity, current).
Mass matched to capacitor	Analogy gives wrong dynamics	"Does this element store in flow or effort?"	Mass stores in flow, so it maps to an inductor.
Stiffness mapped to C, not 1/k	Natural frequencies disagree	"Is compliance 1/k or k?"	Compliance is 1/k, mapping to capacitance.
Counting a resistor as a store	Model order too high	"Does this element store energy?"	Resistance and damping dissipate, never store.

Practice ladder

Level 1 · Direct skill

A 3 kg mass moves at 2 m/s. Find its kinetic energy.

Show answer

E = ½mv² = ½ × 3 × 4 = 6 J.

Level 2 · Mixed concept

Give the force-voltage electrical analog of a system with m = 5 kg, c = 20 N·s/m, k = 80 N/m.

Show answer

L = 5 H, R = 20 Ω, C = 1/80 = 0.0125 F.

Level 3 · Independent problem

For that analog, find ω_n both mechanically and electrically and confirm they agree.

Show answer

ω_n,mech = √(80/5) = √16 = 4 rad/s. ω_n,elec = 1/√(5 × 0.0125) = 1/√0.0625 = 1/0.25 = 4 rad/s. They agree.

Transfer task | Real engineering

Explain why an engineer might simulate a mechanical suspension as an electrical circuit, and what the effort and flow variables would be.

What good work looks like

Because the analog has identical dynamics and circuit tools are fast and well developed, with force as the effort (voltage) and velocity as the flow (current). The simulation predicts the mechanical response exactly.

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that I matched each element to the right analogous one."

"Give me three systems; I will name the effort and flow for each."

"Build the analog for me." Mapping the elements yourself is the skill.

"Why do the frequencies match?" Reasoning from the energy forms is the point.

Portfolio task

Take a real system in one domain, build its analog in another, and confirm both share a natural frequency and damping ratio, identifying the effort and flow throughout.

Must include: the effort and flow variables, the element mapping, and matched ω_n and ζ.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. What is power in generalized variables?

The product of effort and flow.

2. What does an inertia store, and a compliance?

Inertia stores ½ I f² in flow; compliance stores ½ C e² in effort.

3. Map mass, damper, spring to electrical elements.

Mass to inductance, damper to resistance, compliance 1/k to capacitance.

4. Why do analogous systems share ω_n?

Their energy expressions and equations are identical in form.

5. Name the mechanical and electrical effort variables.

Force and voltage.

TodayFinish this quiz and Levels 1 and 2 of the ladder.

+1 dayRe-derive an electrical analog from a mechanical system.

+3 daysIdentify effort and flow for three new systems.

+7 daysMove from models to transfer functions, Module 6.

+30 daysReuse the analogies when simulating a multi-domain device.

Textbook mapping

This module follows Karnopp, Margolis, and Rosenberg, System Dynamics: Modeling, Simulation, and Control of Mechatronic Systems, 5th edition, whose energy-based and bond-graph approach this module reflects.

Topic in this module	Where to read more
Effort and flow variables	Karnopp, Margolis & Rosenberg, Chapter 2
Generalized inertia, compliance, and energy	Karnopp, Margolis & Rosenberg, Chapter 3
Cross-domain analogies and bond graphs	Karnopp, Margolis & Rosenberg, Chapters 3 to 5

Chapter numbers refer to the 5th edition, whose unifying effort-flow framework is the heart of this course.