Mathematical Modeling of Physical Systems
Control theory works on equations, not hardware. Before we can analyse a motor, a suspension, or a circuit, we must translate it into a differential equation that captures how it stores and dissipates energy. This chapter builds those models for mechanical and electrical systems, links them through the electrical analogies, derives the all-important DC-motor model, and shows how to tame nonlinear elements by linearizing about an operating point.
- How to write the differential equation of a physical system from its energy-storing and dissipating elements.
- Modeling translational (mass–spring–damper) and rotational (inertia–torsion–friction) mechanical systems.
- Modeling electrical RLC networks with Kirchhoff's laws.
- The force–voltage and force–current analogies that make a circuit and a mechanism the same equation.
- The armature-controlled DC motor — the workhorse model behind half the examples in this book.
- How to linearize a nonlinear element about an operating point so the linear tools of the course apply.
Why We Model — and What a Good Model Is
A mathematical model is a set of equations relating a system's inputs to its outputs. For the linear, lumped-parameter systems of this course, that set is an ordinary differential equation (or, after the Laplace transform of Chapter 3, a transfer function). Every model rests on three kinds of element: those that store energy, those that dissipate it, and the sources that supply it.
The art of modeling is choosing a model that is simple enough to analyse but rich enough to be useful. We assume parameters are lumped (a spring has no mass, a mass no springiness), linear (force proportional to displacement), and time-invariant (constants do not drift). These assumptions are what let the rest of the course exist; Chapter 29 returns to relax them.
Write a balance law — Newton's second law for motion, Kirchhoff's laws for circuits — element by element, and the differential equation assembles itself.
Translational Mechanical Systems
Three elements describe straight-line motion: a mass \(M\) stores kinetic energy and opposes acceleration; a spring of stiffness \(K\) stores potential energy and opposes displacement; a damper (dashpot) of coefficient \(B\) dissipates energy and opposes velocity. Their force laws are:
Newton's second law says the applied force equals the sum of the element forces. Summing them for the mass gives the system's governing equation — a second-order linear ODE that reappears, in disguise, throughout the course:
Rotational Mechanical Systems
Rotation mirrors translation exactly, with angle \(\theta\) replacing displacement and torque \(T\) replacing force. The moment of inertia \(J\) opposes angular acceleration, the torsional spring \(K\) opposes twist, and rotational friction \(B\) opposes angular velocity:
Electrical Systems
Electrical networks are modeled with three elements and Kirchhoff's two laws. A resistor dissipates, an inductor and a capacitor store energy in magnetic and electric fields respectively:
Applying Kirchhoff's voltage law around the loop and writing the capacitor voltage \(v_C\) as the output gives a second-order ODE with exactly the shape of the mechanical system above:
The Electrical Analogies
Because the mechanical and electrical equations share a shape, we can map one onto the other and solve a mechanism by solving its equivalent circuit. Two consistent mappings exist. In the force–voltage (F–V) analogy, force corresponds to voltage; in the force–current (F–I) analogy, force corresponds to current.
| Mechanical (translational) | Force–Voltage analogy | Force–Current analogy |
|---|---|---|
| Force \(f\) | Voltage \(v\) | Current \(i\) |
| Mass \(M\) | Inductance \(L\) | Capacitance \(C\) |
| Damper \(B\) | Resistance \(R\) | Conductance \(1/R\) |
| Spring \(K\) | Reciprocal capacitance \(1/C\) | Reciprocal inductance \(1/L\) |
| Displacement \(x\) | Charge \(q\) | Flux linkage \(\psi\) |
| Velocity \(\dot{x}\) | Current \(i\) | Voltage \(v\) |
Electromechanical Systems: the DC Motor
The armature-controlled DC motor couples an electrical loop to a mechanical load and appears in countless control problems — servos, robots, disk drives. Two physical effects bridge the domains: the torque is proportional to armature current, and the spinning armature generates a back-EMF proportional to speed.
Kirchhoff's voltage law on the armature loop and Newton's law on the shaft give two coupled equations:
where \(\Omega = s\Theta\) is the angular speed. Neglecting the small armature inductance \(L_a\) collapses this to a clean first-order lag — the model used in most speed-control examples to come.
Linearization About an Operating Point
Real elements are often nonlinear — a valve's flow varies as the square root of pressure, a pendulum's restoring torque follows \(\sin\theta\). The linear tools of this course still apply if we work with small deviations about a steady operating point \((\bar{x},\bar{u})\). Expand the nonlinear law \(f(x)\) in a Taylor series and keep only the first-order term:
The slope at the operating point becomes an effective linear gain. For a pendulum, \(\sin\theta \approx \theta\) for small angles; for the valve \(q = c\sqrt{p}\), the linear gain is \(\tfrac{dq}{dp}=\tfrac{c}{2\sqrt{\bar{p}}}\). Provided deviations stay small, the linearized model predicts behaviour faithfully — and lets every method in Parts 2–6 proceed.
Worked Examples
Problem. A 2 kg mass on rollers is tied to a wall by a spring \(K=8\ \mathrm{N/m}\) and a damper \(B=6\ \mathrm{N\,s/m}\). Write its equation of motion under force \(f(t)\).
Solution. Substitute into the translational form:
Dividing by 2 gives \(\ddot{x}+3\dot{x}+4x=\tfrac12 f\); we will read its natural frequency and damping straight off this form in Chapter 8.
Problem. Using the force–voltage analogy, write the electrical equivalent of the mass–spring–damper in Example 1.
Solution. Map \(M\to L\), \(B\to R\), \(K\to 1/C\), \(x\to q\):
The mechanical problem is now a series-RLC loop with \(L=2\) H, \(R=6\ \Omega\), \(C=0.125\) F.
Problem. A motor inertia \(J_1\) drives a load inertia \(J_2\) through a shaft of torsional stiffness \(K\), with friction \(B\) on the load. Write the torque balance on each inertia.
Solution. Apply the rotational law to each body, the shaft transmitting \(K(\theta_1-\theta_2)\):
Coupled equations like these become a single transfer function after the Laplace transform of Chapter 3.
Problem. Neglecting armature inductance (\(L_a \approx 0\)), find the speed-to-voltage transfer function of the DC motor.
Solution. With \(L_a=0\), \(i_a=(e_a-K_b\dot\theta)/R_a\). Substitute into the shaft equation and use \(\Omega=\dot\theta\):
A first-order lag with motor gain \(K_m\) and time constant \(\tau_m\) — the form we will reuse repeatedly.
Problem. The torque on a simple pendulum is \(T=-mgl\sin\theta\). Linearize about the hanging position \(\bar{\theta}=0\).
Solution. The slope of \(\sin\theta\) at \(0\) is \(1\), so \(\sin\theta\approx\theta\):
A linear oscillator valid for small swings; near \(\theta=\pi\) (inverted) the slope flips sign, predicting the instability that makes the inverted pendulum a classic control challenge.
Problem. A valve passes flow \(q=4\sqrt{p}\). Find its linear resistance about \(\bar{p}=4\).
Solution. The small-signal gain is the derivative at the operating point:
Around this point a unit change in pressure gives a unit change in flow — a linear element we can drop straight into a block diagram.
Chapter Summary
Every linear model combines an energy store, a dissipator, and a restorer, balanced against the input.
Translational \(M\ddot x+B\dot x+Kx=f\) and rotational \(J\ddot\theta+B\dot\theta+K\theta=T\) — the same second-order shape.
Series RLC \(L\ddot q+R\dot q+\tfrac1C q=v\): the electrical twin of the mechanical system.
F–V and F–I mappings let one domain stand in for another; recognise the form once, use it everywhere.
\(T_m=K_t i_a\), \(e_b=K_b\dot\theta\): coupled loops give a low-order model, often a first-order lag.
Take the slope at the operating point: \(\delta f = f'(\bar x)\,\delta x\). Small signals make nonlinear systems linear.
Problems
For each item, first identify the energy-storing and dissipating elements, then write the balance law (Newton or Kirchhoff) element by element. Difficulty rises down the list.
- A 5 kg mass with damping \(B=10\ \mathrm{N\,s/m}\) and spring \(K=20\ \mathrm{N/m}\) is driven by \(f(t)\). Write its equation of motion.
- Write the rotational equation for an inertia \(J=0.5\), friction \(B=2\), torsional spring \(K=4\) driven by torque \(T(t)\).
- Derive the governing ODE of a series RLC circuit with output taken as the inductor voltage instead of the capacitor voltage.
- Using the force–current analogy, write the electrical equivalent of the system in Problem 1.
- Two masses \(M_1, M_2\) are joined by a spring \(K\) and damper \(B\), with \(f(t)\) applied to \(M_1\). Write the two coupled equations of motion.
- For a parallel RLC circuit driven by a current source, write the node equation and identify its order.
- Write the two coupled equations of an armature-controlled DC motor without neglecting \(L_a\).
- Show that the field-controlled DC motor (constant armature current) yields a transfer function \(\Theta(s)/E_f(s)\) and find its order.
- Linearize \(f(x)=x^3\) about \(\bar{x}=2\) and state the small-signal gain.
- The drag on a body is \(F_d = c v^2\). Linearize about cruising speed \(\bar v\) and interpret the result as an effective damping coefficient.
- A liquid-level tank has inflow \(q_i\), outflow \(q_o = h/R\) through a valve of resistance \(R\), and area \(A\). Write the tank's differential equation in head \(h\).
- An inverted pendulum has torque \(T = mgl\sin\theta\) about the upright \(\bar\theta=\pi\). Linearize and explain why the resulting model is unstable.