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System Dynamics · Module 1 of 10

Introduction to System Dynamics and Modeling

A model is a deliberate simplification. Before any equation, you choose a boundary, lump the real hardware into a few ideal elements, and decide what behaviour you are trying to predict.

Readiness check

This module opens the course. Tick only what you can do closed-notes.

Write Newton's second law for a single mass.
Recall Hooke's law F = kx for a spring.
State that a derivative is a rate of change.
Recall the natural frequency ω = √(k/m) of a spring-mass system.
Set up and solve a simple second-order differential equation.

0 or 1 weak itemsContinue with this module.

2 weak itemsRevisit vibration basics in Dynamics and Vibrations, Module 10.

3 or more weak itemsReview differential equations in Mathematics, Module 10.

The core idea

A dynamic model lumps a real system into ideal elements that store or dissipate energy, then writes how its variables change in time. The spring-mass oscillator is the prototype: its natural frequency follows from one balance of stiffness against inertia.

m ẍ + k x = 0ω_n = √(k/m)f_n = ω_n/(2π), T = 1/f_n

System dynamics studies how a system's variables evolve over time. The first move is always modeling: draw a boundary around what you care about, then replace the real hardware with a small set of ideal lumped elements, ones that store energy (a mass, a spring) or dissipate it (a damper). Each element obeys a simple constitutive law, and a physical balance, Newton's law, a circuit law, a mass balance, links them into a differential equation. The number of independent energy stores sets the order of that equation and the number of degrees of freedom. The undamped spring-mass system is the simplest example with dynamics: inertia and stiffness trade energy back and forth, and the rate of that exchange is the natural frequency ω_n = √(k/m), a number that reappears in every later module.

The skill works when: you name the energy-storing elements, write one balance law, and reduce it to a differential equation in your chosen variable.

The skill breaks down when: the model keeps detail that does not matter, or omits an energy store that does.

The concept. The mass stores kinetic energy, the spring stores potential energy, and they exchange it cyclically. The natural frequency is the rate of that exchange, set only by the ratio of stiffness to inertia.

The skills, taught in order

Five skills frame the modeling process and lead to the natural frequency of the prototype system.

1.1 The system boundary

Modeling begins by drawing a boundary: deciding what is inside the system, what acts on it from outside as an input, and what you will read as an output. Everything outside the boundary enters only through those inputs. A good boundary keeps what matters and excludes what does not.

1.2 Lumped elements

Inside the boundary, real hardware is replaced by ideal lumped elements: a mass for inertia, a spring for stiffness, a damper for dissipation, and their electrical, fluid, and thermal counterparts. Lumping assumes each effect acts at a point, valid when the system is small compared with the speed of the signals in it.

1.3 The modeling process

With elements chosen, a physical law connects them: Newton's second law for mechanics, Kirchhoff's laws for circuits, a mass or energy balance for fluids and thermals. The result is a differential equation relating the output to the input, the dynamic model.

Element	Role	Constitutive law	Stores
Mass	inertia	F = m a	kinetic energy
Spring	stiffness	F = k x	potential energy
Damper	dissipation	F = c v	nothing (dissipates)

The three mechanical elements. The two energy stores set the order of the model; the damper only removes energy.

1.4 Degrees of freedom and order

The number of independent energy-storing elements sets the order of the model and the count of state variables. A single spring-mass has two stores, position and velocity, so it is second order with one degree of freedom. Counting stores predicts the model's complexity before any algebra.

1.5 The natural frequency

For the undamped spring-mass, Newton's law gives m ẍ + k x = 0, whose solution oscillates at ω_n = √(k/m). The frequency in hertz is f_n = ω_n/(2π) and the period T = 1/f_n. This single number characterises the system's free response.

Engineering connection: the same ω_n sets the resonance of a structure, the speed of a sensor, and the bandwidth of a controlled machine, which is why it opens the course.

Worked example 1: natural frequency of a spring-mass

A 2 kg mass hangs on a spring of stiffness 200 N/m. Treating it as an undamped oscillator, find the natural frequency in rad/s and hertz, and the period.

Figure 1. The undamped oscillator's frequency depends only on stiffness and mass. A stiffer spring or a lighter mass raises ω_n.

ProblemFind ω_n, f_n, and T for the spring-mass in Figure 1.
Given / findm = 2 kg, k = 200 N/m. Find ω_n, f_n, T.
AssumptionsIdeal massless spring; no damping; small motion, so the spring is linear.
ModelNewton's law gives m ẍ + k x = 0, so ω_n = √(k/m), then f_n and T follow.
Equationsω_n = √(k/m)f_n = ω_n/(2π)T = 1/f_n
Solveω_n = √(200/2) = √100 = 10 rad/s. f_n = 10/(2π) = 1.59 Hz. T = 1/1.59 = 0.628 s.
CheckUnits: √(N/m ÷ kg) = √(s⁻²) = s⁻¹. The period 2π√(m/k) = 2π√(0.01) = 2π(0.1) = 0.628 s agrees.
ConclusionThe system oscillates freely at 10 rad/s, about 1.6 cycles per second. Inertia and stiffness alone set it; the mass value is what the modeling captured.

Result. ω_n = 10 rad/s, f_n = 1.59 Hz, T = 0.628 s.

Worked example 2: equivalent stiffness of combined springs

Two springs, k₁ = 300 N/m and k₂ = 200 N/m, support a 5 kg mass. Find the equivalent stiffness and the natural frequency if the springs are in parallel, and again if they are in series.

Figure 2. Parallel springs share the load and add their stiffnesses; series springs share the deflection and add as reciprocals. The arrangement changes the natural frequency for the same mass.

ProblemFind k_eq and ω_n for the springs of Figure 2 in parallel and in series.
Given / findk₁ = 300 N/m, k₂ = 200 N/m, m = 5 kg. Find k_eq and ω_n for each arrangement.
AssumptionsIdeal linear springs; the mass moves in one direction.
ModelParallel springs add directly; series springs add as reciprocals; then ω_n = √(k_eq/m).
Equationsparallel: k_eq = k₁ + k₂series: k_eq = k₁k₂/(k₁ + k₂)ω_n = √(k_eq/m)
SolveParallel: k_eq = 300 + 200 = 500 N/m, ω_n = √(500/5) = √100 = 10 rad/s. Series: k_eq = (300 × 200)/500 = 60000/500 = 120 N/m, ω_n = √(120/5) = √24 = 4.90 rad/s.
CheckThe series stiffness (120 N/m) is below the softer spring (200 N/m), as series always is; the parallel stiffness (500 N/m) exceeds both. Lower stiffness gives a lower natural frequency.
ConclusionHow elements combine changes the model's parameters and therefore its dynamics. Recognising series and parallel stiffness is the mechanical twin of combining resistors.

Result. Parallel: 500 N/m, 10 rad/s. Series: 120 N/m, 4.90 rad/s.

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
Modeling everything	An intractable model with no insight	"What behaviour am I trying to predict?"	Keep only the elements that affect the output of interest.
Missing an energy store	Model order too low for the real response	"How many independent stores are there?"	Count masses and springs; each is a store.
Series and parallel springs swapped	Stiffness larger when it should be smaller	"Do the springs share load or deflection?"	Parallel adds stiffness; series reduces it.
Frequency confused with stiffness	ω_n reported as k	"Did I divide by the mass and take the root?"	ω_n = √(k/m), not k.

Practice ladder

Level 1 · Direct skill

A 4 kg mass on a 256 N/m spring. Find the natural frequency in rad/s.

Show answer

ω_n = √(256/4) = √64 = 8 rad/s.

Level 2 · Mixed concept

The same 4 kg mass now hangs on two 256 N/m springs in parallel. Find the new natural frequency.

Show answer

k_eq = 512 N/m, ω_n = √(512/4) = √128 = 11.3 rad/s. Parallel springs raise the frequency.

Level 3 · Independent problem

A spring-mass has a measured period of 0.5 s and a mass of 1.5 kg. Find the stiffness.

Show answer

ω_n = 2π/T = 2π/0.5 = 12.57 rad/s. k = m ω_n² = 1.5 × 157.9 = 237 N/m.

Transfer task | Real engineering

A sensor mounted on a bracket vibrates at an unwanted resonance. Argue, from ω_n = √(k/m), two physical changes that would raise the resonant frequency away from the disturbance.

What good work looks like

Stiffen the bracket (raise k) or reduce the mounted mass (lower m); either raises ω_n. The argument ties a hardware change directly to the model parameter that sets the frequency.

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that I counted the independent energy stores correctly."

"Give me three systems; I will choose a boundary and the lumped elements for each."

"Build the model for me." Choosing the boundary and elements is the skill.

"What is the natural frequency?" Deriving ω_n from the balance is the point.

Portfolio task

Take a real device, draw a system boundary, lump it into ideal elements, and estimate its natural frequency, then sanity-check the number against how fast the device actually moves.

Must include: a boundary, the lumped elements, a computed ω_n, and a reality check.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. What is the first step in modeling?

Draw a system boundary: decide what is inside, what is an input, and what is an output.

2. What sets the order of a model?

The number of independent energy-storing elements.

3. Write the natural frequency of a spring-mass.

ω_n = √(k/m).

4. How do parallel and series springs combine?

Parallel stiffnesses add; series stiffnesses add as reciprocals.

5. What does a damper store?

Nothing: it only dissipates energy.

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

+1 dayRe-derive ω_n from m ẍ + k x = 0 on a blank page.

+3 daysModel three new devices into lumped elements.

+7 daysCarry the modeling habit into mechanical systems, Module 2.

+30 daysReuse ω_n when reading a measured resonance.

Textbook mapping

This module follows Karnopp, Margolis, and Rosenberg, System Dynamics: Modeling, Simulation, and Control of Mechatronic Systems, 5th edition. Use these references to read further.

Topic in this module	Where to read more
The modeling process and system boundaries	Karnopp, Margolis & Rosenberg, Chapter 1
Lumped elements and energy stores	Karnopp, Margolis & Rosenberg, Chapter 2
Degrees of freedom and natural frequency	Karnopp, Margolis & Rosenberg, Chapter 2

Chapter numbers refer to the 5th edition. The modeling principles are standard, so any recent edition will align closely.