Sources and References

These notes synthesize material from standard references on lumped-parameter system dynamics. MTE 351 is cross-listed with SYDE 351 (Systems Models 1); the two offerings share content, problem style, and assessment scope. The exposition draws on:

• Rowell, D. and Wormley, D. N., System Dynamics: An Introduction. Prentice Hall.
• Karnopp, D., Margolis, D., and Rosenberg, R., System Dynamics: Modeling, Simulation, and Control of Mechatronic Systems. Wiley.
• Shearer, J. L., Murphy, A. T., and Richardson, H. H., Introduction to System Dynamics. Addison-Wesley.
• Ogata, K., System Dynamics. Pearson.
• Palm, W. J., System Dynamics. McGraw-Hill.
• MIT OpenCourseWare 2.004 Dynamics and Control II and 2.151 Advanced System Dynamics and Control.
• Stanford ENGR 105 lecture materials on lumped modelling.
• Cannon, R. H., Dynamics of Physical Systems. Dover reissue.

All discussion is paraphrased; no offering-specific instructor materials are cited.

Chapter 1: The Philosophy of Lumped-Parameter Modelling

1.1 Why We Model

Engineering systems — a car suspension, an op-amp amplifier, a hydraulic press, a thermal reactor — are physical objects in which energy flows, converts, and dissipates. A model is a mathematical surrogate that predicts the important behaviour of such a system without reproducing every microscopic detail. In MTE 351 the object of study is the lumped-parameter dynamic model: a finite set of ordinary differential equations (ODEs) whose solution captures how a handful of chosen variables evolve in time.

The philosophy rests on three commitments. First, the engineer chooses a system boundary and declares everything inside it to be describable by a finite number of energy-storing and energy-dissipating elements. Second, the engineer assumes that signals propagate instantaneously within each element — spatial distribution is absorbed into lumped coefficients. Third, the behaviour at the boundary is captured by a small number of input and output variables.

1.2 Through and Across Variables

Every energy domain covered in this course admits a pair of generalized variables whose product is power. One variable is measured through an element (it has the same value at both terminals — think of current in a wire, force in a rod, volumetric flow in a pipe, heat flow across a wall). The other is measured across an element (it is a difference between two terminals — think of voltage, velocity, pressure, temperature).

Domain	Through variable	Across variable	Power product
Electrical	current \( i \) (A)	voltage \( v \) (V)	\( v\,i \) (W)
Mechanical translation	force \( F \) (N)	velocity \( v \) (m/s)	\( F\,v \) (W)
Mechanical rotation	torque \( T \) (N·m)	angular velocity \( \omega \) (rad/s)	\( T\,\omega \) (W)
Fluid	volume flow rate \( Q \) (m\(^3\)/s)	pressure \( P \) (Pa)	\( P\,Q \) (W)
Thermal (pseudo)	heat flow \( q \) (W)	temperature \( \theta \) (K)	—

The thermal domain is a pseudo-bond: heat flow times temperature is not power, because temperature is an intensive quantity and entropy is the true through-companion of temperature. Nevertheless the analogy is so useful for network construction that every textbook keeps it, with the caveat that thermal energy storage is non-conservative in the bond-graph sense.

1.3 The Three Constitutive Laws

For every domain three elemental laws recur:

1. Storage of across-type energy (capacitor, mass in translation with velocity as across, rotational inertia, fluid capacitance, thermal capacitance). An element of this kind integrates the through variable.
2. Storage of through-type energy (inductor, translational spring, torsional spring, fluid inertance). An element of this kind integrates the across variable.
3. Dissipation (resistor, damper, fluid resistance, thermal resistance). The through and across variables are algebraically related.

1.4 Continuity and Compatibility

Two topological laws govern how elements combine.

Continuity is the generalization of Kirchhoff’s current law: the sum of through-variables entering any node equals zero. In mechanics this is d’Alembert’s principle at a massless junction; in fluids it is conservation of volume at a tee; in thermal networks it is conservation of heat flow at a surface.

Compatibility is the generalization of Kirchhoff’s voltage law: the sum of across-variable differences around any closed loop equals zero. In mechanics it states that the relative velocities around a kinematic loop sum to zero; in fluids it is the single-valuedness of pressure.

Together, continuity and compatibility are the graph-theoretic skeleton of every lumped model. Once an engineer has drawn the linear graph (nodes for distinct values of the across variable, branches for elements carrying the through variable), the equations of motion follow by rote.

Chapter 2: Mechanical Translational Systems

2.1 The Three Translational Elements

A translational system is built from three idealizations:

• Mass \( m \). Newton’s second law \( F = m\,\dot v \) relates the through variable (force) to the derivative of the across variable (velocity) measured between the mass and an inertial reference. Kinetic energy stored is \( \tfrac{1}{2} m v^2 \).
• Spring \( k \). The through variable (force) is proportional to the integral of the relative velocity across the spring: \( F = k\,(x_1 - x_2) \), or equivalently \( \dot F = k\,(v_1 - v_2) \). Potential energy stored is \( F^2/(2k) \).
• Damper \( b \). The through variable is proportional to the relative across variable: \( F = b\,(v_1 - v_2) \). Energy dissipates at rate \( F\,(v_1-v_2) \).

Each element has two terminals; only for the mass is one terminal tied to the inertial reference.

2.2 Free-Body Diagrams vs Linear Graphs

The traditional free-body method isolates each mass, sums forces, and writes \( m\,\dot v = \sum F \). The linear-graph method treats each distinct velocity as a node and each element as a branch; applying continuity at each non-reference node gives the equations directly.

Consider a mass \( m \) on a frictionless surface, connected to a wall by a spring \( k \) and a damper \( b \), with an external force \( F_a(t) \) applied to the mass.

\[ m\,\dot v + b\,v + k\,x = F_a(t), \qquad \dot x = v. \]

This is the canonical second-order oscillator. It will reappear in every domain.

2.3 Undamped Natural Frequency and Damping Ratio

Divide by \( m \):

\[ \ddot x + 2\zeta\omega_n\,\dot x + \omega_n^2\,x = \frac{F_a(t)}{m}, \]

with

\[ \omega_n = \sqrt{k/m}, \qquad \zeta = \frac{b}{2\sqrt{k\,m}}. \]

The undamped natural frequency \( \omega_n \) is the frequency at which the mass-spring pair would oscillate if the damper were removed. The dimensionless damping ratio \( \zeta \) compares actual damping to critical damping \( b_c = 2\sqrt{k m} \). For \( 0 < \zeta < 1 \) the system is underdamped (decaying oscillation); at \( \zeta = 1 \) critically damped (fastest non-oscillatory return); for \( \zeta > 1 \) overdamped (two real decay modes).

2.4 Multi-Mass Systems

For \( N \) masses the state is the \( 2N \)-tuple of positions and velocities. Consider two masses \( m_1, m_2 \) coupled by a spring \( k \) and a damper \( b \), with \( m_1 \) driven by \( F(t) \) and \( m_2 \) anchored to the wall by a spring \( k_2 \):

\[ m_1\,\ddot x_1 + b(\dot x_1 - \dot x_2) + k(x_1 - x_2) = F(t), \]\[ m_2\,\ddot x_2 + b(\dot x_2 - \dot x_1) + k(x_2 - x_1) + k_2\,x_2 = 0. \]

This is the quarter-car suspension stripped to essentials: \( m_1 \) is the sprung mass (chassis), \( m_2 \) the unsprung (wheel), \( k \) and \( b \) are the suspension, \( k_2 \) is the tire. The road input enters through the base of \( k_2 \).

Chapter 3: Mechanical Rotational Systems

3.1 The Three Rotational Elements

Rotation swaps translational quantities for their angular counterparts:

• Rotational inertia \( J \): \( T = J\,\dot\omega \).
• Torsional spring \( k_\tau \): \( T = k_\tau\,(\theta_1-\theta_2) \).
• Rotational damper \( B \): \( T = B\,(\omega_1-\omega_2) \).

3.2 Gear Trains and Lever Arms

A rigid gear pair with teeth ratio \( N = N_2/N_1 \) imposes kinematic compatibility \( \omega_2 = \omega_1/N \) and, for an ideal gear, the through-variable transformation \( T_2 = N\,T_1 \). Ideal gears preserve power: \( T_1\omega_1 = T_2\omega_2 \). Reflected inertia transforms as \( N^2 \); reflected damping and stiffness likewise.

3.3 Shaft Dynamics

A long shaft is not rigid. Two disks \( J_1, J_2 \) joined by a shaft of torsional stiffness \( k_\tau \) and internal damping \( B_s \) produce

\[ J_1\,\ddot\theta_1 + B_s(\dot\theta_1-\dot\theta_2) + k_\tau(\theta_1-\theta_2) = T(t), \]\[ J_2\,\ddot\theta_2 + B_s(\dot\theta_2-\dot\theta_1) + k_\tau(\theta_2-\theta_1) + B_L\,\dot\theta_2 = 0. \]

The system has two natural frequencies: a rigid-body mode at \( 0 \) and a torsional mode at \( \sqrt{k_\tau(J_1+J_2)/(J_1 J_2)} \).

Chapter 4: Electrical Networks

4.1 Elements

• Resistor: \( v = R\,i \), energy dissipated at rate \( v\,i = R i^2 \).
• Capacitor: \( i = C\,\dot v \), energy \( \tfrac{1}{2} C v^2 \).
• Inductor: \( v = L\,\dot i \), energy \( \tfrac{1}{2} L i^2 \).

4.2 Kirchhoff’s Laws

KCL (continuity): at every node, \( \sum i = 0 \). KVL (compatibility): around every loop, \( \sum v = 0 \). These are the graph-theoretic laws of Section 1.4 specialized to electrical circuits.

4.3 Mesh and Node Methods

Two systematic approaches exist.

Mesh analysis chooses a set of independent loop currents and writes KVL around each loop; the number of equations equals the number of independent loops (edges minus nodes plus one, in graph-theoretic terms).

Nodal analysis chooses a ground node, assigns node voltages to the remaining nodes, and writes KCL at each non-ground node; the number of equations equals nodes minus one.

For MTE 351 the preferred technique — because it generalizes cleanly to the state-space formulation later — is nodal analysis with the normal tree described in Chapter 9.

4.4 Canonical Second-Order RLC Example

A series RLC loop driven by source \( v_s(t) \):

\[ L\,\ddot q + R\,\dot q + q/C = v_s(t), \]

where \( q \) is the capacitor charge. Comparing with the mechanical canonical form, \( L \leftrightarrow m \), \( R \leftrightarrow b \), \( 1/C \leftrightarrow k \). This is the force-voltage analogy. Its dual, the parallel RLC with a current source, gives the force-current analogy: \( C \leftrightarrow m \), \( 1/R \leftrightarrow b \), \( 1/L \leftrightarrow k \).

Chapter 5: Fluid Systems

5.1 Hydraulic Elements

• Fluid resistance \( R_f \): a pressure drop proportional to volume flow through a restriction, \( \Delta P = R_f\,Q \), valid for laminar flow. Turbulent restrictions give \( \Delta P = R_t\,Q^2 \) and must be linearized about an operating point.
• Fluid capacitance \( C_f \): an open tank of cross-section \( A \) stores volume at a rate \( Q = A\,\dot h \); the pressure at its base is \( P = \rho g h \), so \( C_f = A/(\rho g) \) and \( Q = C_f\,\dot P \).
• Fluid inertance \( I_f \): a slug of fluid in a pipe of length \( \ell \) and cross-section \( A \) has inertia \( I_f = \rho\ell/A \), with \( \Delta P = I_f\,\dot Q \).

5.2 A Two-Tank Example

Consider tanks of areas \( A_1, A_2 \) connected by a laminar valve \( R_1 \); tank 2 drains through \( R_2 \) to atmosphere. Volumetric inflow to tank 1 is \( Q_\text{in}(t) \). Writing continuity at each tank and the linear valve law:

\[ A_1\,\dot h_1 = Q_\text{in} - \frac{\rho g}{R_1}(h_1 - h_2), \]\[ A_2\,\dot h_2 = \frac{\rho g}{R_1}(h_1 - h_2) - \frac{\rho g}{R_2}\,h_2. \]

With \( (h_1,h_2) \) as states, this is a first-order matrix system. In compact form

\[ \begin{bmatrix} \dot h_1 \\ \dot h_2 \end{bmatrix} = A\,\begin{bmatrix} h_1 \\ h_2 \end{bmatrix} + B\,Q_\text{in}, \]

where \( A \) is a 2×2 matrix assembled from \( \rho g/(A_i R_j) \) terms.

5.3 Pneumatic Systems

For gases under moderate pressure variation, the capacitance of a volume \( V \) is \( C_g = V/(RT) \) (mass form) or \( V/(nP_0) \) for polytropic small-signal behaviour. Inertance and resistance are defined analogously. Compressibility makes the pneumatic analogue to the mechanical spring natural: a gas volume stores work as pressure.

Chapter 6: Thermal Systems

6.1 Thermal Resistance and Capacitance

• Thermal resistance \( R_\theta \): \( q = (\theta_1-\theta_2)/R_\theta \). Conductive resistance is \( L/(kA) \); convective is \( 1/(hA) \).
• Thermal capacitance \( C_\theta \): \( q_\text{in} = C_\theta\,\dot\theta \), with \( C_\theta = m\,c_p \).

There is no thermal inductor. Heat does not accumulate “flow-wise” the way electrical current or fluid momentum does: this is the mark of the pseudo-bond.

6.2 A Lumped Thermal Plant

A room of capacitance \( C_r \) loses heat to ambient through \( R_\text{wall} \) and receives power \( q_\text{in}(t) \) from a heater:

\[ C_r\,\dot\theta = q_\text{in}(t) - \frac{\theta - \theta_\infty}{R_\text{wall}}. \]

The time constant \( \tau = R_\text{wall}\,C_r \) sets how fast the room equilibrates. Two-node thermal systems (room plus building envelope) give two time constants and the classic slow-fast decomposition.

Chapter 7: Electromechanical Coupling

7.1 The Ideal Transformer and Gyrator

Two two-port elements bridge domains. A transformer relates across-to-across and through-to-through by a single ratio \( n \):

\[ v_2 = n\,v_1, \qquad i_2 = i_1/n. \]

A gyrator cross-couples across and through:

\[ v_2 = r\,i_1, \qquad v_1 = r\,i_2. \]

Transformers preserve “kind” (across maps to across); gyrators swap it. This is the algebraic heart of the bond-graph representation in Chapter 9.

7.2 The DC Motor

A permanent-magnet DC motor has armature resistance \( R_a \), armature inductance \( L_a \), back-emf constant \( K_b \), torque constant \( K_t \), rotor inertia \( J \), and bearing damping \( B \). In SI units with a single magnetic circuit \( K_b = K_t = K \). The electromechanical port is a gyrator of modulus \( K \):

\[ v_\text{emf} = K\,\omega, \qquad T_\text{em} = K\,i_a. \]

The circuit-side loop (KVL) and the mechanical-side free body (continuity of torque) yield

\[ L_a\,\dot i_a + R_a\,i_a + K\,\omega = v_\text{in}(t), \]\[ J\,\dot\omega + B\,\omega = K\,i_a - T_L(t). \]

With input \( v_\text{in} \) and load torque \( T_L \), this is a two-state linear system in \( (i_a,\omega) \). For many applications \( L_a \) is negligible, reducing to a first-order relation between voltage and speed with time constant \( J R_a/(R_a B + K^2) \).

7.3 Solenoids and Voice Coils

A solenoid is similarly a gyrator, but between current and translational velocity. A voice-coil actuator in a loudspeaker obeys \( F = B\ell\,i \) and \( v_\text{emf} = B\ell\,\dot x \), where \( B \) is the magnetic flux density in the gap and \( \ell \) the coil length. The coupling modulus is \( B\ell \).

Chapter 8: Analogies Between Domains

8.1 The Systematic Table

Analogies are not poetic — they are structural. Once through and across are fixed, every element in every domain has an analogue, and every equation derived in one domain carries over to every other.

Quantity	Mechanical (force-current)	Electrical	Fluid	Thermal
Across	velocity \( v \)	voltage \( v \)	pressure \( P \)	temperature \( \theta \)
Through	force \( F \)	current \( i \)	flow \( Q \)	heat flow \( q \)
Across-storage	mass \( m \)	capacitor \( C \)	tank \( C_f \)	thermal cap. \( C_\theta \)
Through-storage	spring compliance \( 1/k \)	inductor \( L \)	inertance \( I_f \)	(none)
Dissipation	damper \( b \)	resistor \( R \)	fluid R \( R_f \)	thermal R \( R_\theta \)

8.2 Why the Analogy Works

Because continuity and compatibility are graph-theoretic, and because each element is a two-terminal device with a constitutive law of one of three forms, the structure of the equations is domain-independent. A physics-agnostic computer algebra package could accept a linear-graph description with element laws tagged by domain and produce correct equations without ever “knowing” what the domain means.

Chapter 9: Graph-Theoretic Formulation and the Normal Tree

9.1 Linear Graph

A linear graph represents a lumped system as a set of nodes (one per distinct value of the across variable, plus a reference) and branches (one per two-terminal element). The graph is oriented: each branch has a designated positive direction for the through variable.

9.2 Tree and Cotree

A tree is a maximal set of branches that connects every node without forming a loop. In a graph with \( N \) nodes and \( B \) branches, a tree has exactly \( N-1 \) branches; the remaining \( B-(N-1) \) are links or cotree branches.

A normal tree obeys a priority for which elements to place in the tree:

1. Across-variable sources (voltage, velocity sources) — must be in the tree.
2. Across-storage elements (capacitors, masses, tanks, thermal capacitances) — placed in the tree whenever possible.
3. Resistive elements — placed next to fill remaining slots.
4. Through-storage elements (inductors, springs, fluid inertances) — placed as links whenever possible.
5. Through-variable sources (current, force sources) — must be links.

9.3 State Variables from the Tree

The state variables of a lumped model are:

• the across variable of each tree-branch across-storage element (capacitor voltages, mass velocities, tank heads, thermal temperatures);
• the through variable of each link through-storage element (inductor currents, spring forces, fluid inertance flows).

The number of state variables equals the order of the system. This procedure automatically discards redundant energy storages (two capacitors in parallel count once) and captures the minimum set needed.

9.4 Fundamental Cut-Sets and Loops

Every tree branch, together with some links, defines a fundamental cut-set: continuity applied to this cut yields one equation relating the tree branch’s through variable to the links’ through variables.

Every link, together with some tree branches, defines a fundamental loop: compatibility applied to this loop yields one equation relating the link’s across variable to the tree branches’ across variables.

Systematic bookkeeping of cut-set and loop equations, combined with the element constitutive laws, produces the state equations with no manipulation required.

Chapter 10: State-Space Representation

10.1 The Canonical Form

A linear time-invariant lumped system in state-space form is

\[ \dot{\mathbf x}(t) = A\,\mathbf x(t) + B\,\mathbf u(t), \]\[ \mathbf y(t) = C\,\mathbf x(t) + D\,\mathbf u(t). \]

Here \( \mathbf x \in \mathbb R^n \) is the state vector (the variables identified by the normal-tree procedure), \( \mathbf u \in \mathbb R^p \) is the input vector, \( \mathbf y \in \mathbb R^q \) is the output vector. The matrices \( A, B, C, D \) have dimensions \( n\times n, n\times p, q\times n, q\times p \).

10.2 Worked Example: Mass-Spring-Damper

Choose \( x_1 = x \) (position), \( x_2 = v \) (velocity), input \( u = F_a \), output \( y = x \). Then

\[ \dot x_1 = x_2, \]\[ \dot x_2 = -\frac{k}{m}\,x_1 - \frac{b}{m}\,x_2 + \frac{1}{m}\,u, \]

giving

\[ A = \left[\begin{array}{cc} 0 & 1 \\ -k/m & -b/m \end{array}\right], \quad B = \left[\begin{array}{c} 0 \\ 1/m \end{array}\right], \quad C = \left[\begin{array}{cc} 1 & 0 \end{array}\right], \quad D = 0. \]

10.3 Worked Example: DC Motor

With states \( x_1 = i_a \), \( x_2 = \omega \), input \( u = v_\text{in} \) (ignoring load torque for brevity),

\[ A = \left[\begin{array}{cc} -R_a/L_a & -K/L_a \\ K/J & -B/J \end{array}\right], \quad B = \left[\begin{array}{c} 1/L_a \\ 0 \end{array}\right]. \]

10.4 Why State Space?

State-space form generalizes to multi-input multi-output (MIMO), nonlinear, and time-varying systems; it is the natural home for numerical integration, modern control, estimation, and optimization. Transfer-function thinking and state-space thinking coexist throughout this course; for analysis problems set in the Laplace domain the transfer function is often quicker, but as soon as the system has more than one input or more than one output the state-space view is indispensable.

Chapter 11: Linearization About an Operating Point

11.1 Motivation

Many elements are linear only over a limited range: the fluid valve \( Q = C_d A_v \sqrt{\Delta P} \) is square-root, the radiator law is quartic, a pendulum obeys \( \sin\theta \). For small perturbations about a chosen operating point (equilibrium or trim condition), a Taylor expansion yields a linear model whose behaviour approximates the full nonlinear one.

11.2 Mechanics of Linearization

Given nonlinear dynamics \( \dot{\mathbf x} = \mathbf f(\mathbf x,\mathbf u) \) and a trim \( (\mathbf x_0,\mathbf u_0) \) at which \( \mathbf f(\mathbf x_0,\mathbf u_0)=0 \), write \( \mathbf x = \mathbf x_0 + \delta\mathbf x \), \( \mathbf u = \mathbf u_0 + \delta\mathbf u \). Expanding,

\[ \dot{\delta\mathbf x} = \left.\frac{\partial \mathbf f}{\partial \mathbf x}\right|_0 \delta\mathbf x + \left.\frac{\partial \mathbf f}{\partial \mathbf u}\right|_0 \delta\mathbf u + \mathcal O(\delta^2). \]

The Jacobians at the operating point become the \( A \) and \( B \) matrices of the linearized model.

11.3 Worked Example: Orifice

A tank drains through an orifice of area \( A_v \), with \( Q = C_d A_v\sqrt{2 g h} \). Trim: inflow \( Q_0 \) balances outflow, giving \( h_0 = Q_0^2/(2 g (C_d A_v)^2) \). Linearized resistance about \( h_0 \):

\[ R_\text{lin} = \left.\frac{dh}{dQ}\right|_{h_0} = \frac{2\,h_0}{Q_0}. \]

Thus a small-signal tank model replaces \( Q = \sqrt{2gh}/R^\ast \) by \( \delta Q = \delta h / R_\text{lin} \) and proceeds as before. All the machinery of state-space and transfer-function analysis now applies — but only locally.

Chapter 12: Solution of the State Equations

12.1 The Matrix Exponential

For \( \dot{\mathbf x} = A\mathbf x \) with \( \mathbf x(0) = \mathbf x_0 \), the solution is

\[ \mathbf x(t) = e^{A t}\,\mathbf x_0, \]

where \( e^{A t} = \sum_{k=0}^\infty (A t)^k/k! \). The matrix exponential generalizes \( e^{a t} \) to the multivariable case and is the single most important object in linear system dynamics.

With forcing,

\[ \mathbf x(t) = e^{A t}\,\mathbf x_0 + \int_0^t e^{A(t-\tau)}\,B\,\mathbf u(\tau)\,d\tau. \]

The integral is the convolution of input with the state transition kernel.

12.2 Computing \( e^{A t} \)

Three standard techniques:

1. Diagonalization. If \( A = V\Lambda V^{-1} \) with distinct eigenvalues, then \( e^{A t} = V\,e^{\Lambda t}\,V^{-1} \), and \( e^{\Lambda t} \) is diagonal with entries \( e^{\lambda_i t} \).
2. Laplace transform. \( e^{A t} = \mathcal L^{-1}\{(sI-A)^{-1}\} \). This works for repeated eigenvalues too.
3. Cayley-Hamilton. Since \( A \) satisfies its own characteristic polynomial of degree \( n \), \( e^{A t} \) can be written as a finite polynomial in \( A \) with scalar time-dependent coefficients.

12.3 Laplace Transform of the State Equation

Taking the Laplace transform of \( \dot{\mathbf x}=A\mathbf x+B\mathbf u \), assuming \( \mathbf x(0)=\mathbf 0 \),

\[ s\,\mathbf X(s) = A\,\mathbf X(s) + B\,\mathbf U(s), \]\[ \mathbf X(s) = (sI-A)^{-1} B\,\mathbf U(s). \]

The output is \( \mathbf Y(s) = G(s)\,\mathbf U(s) \) with transfer function matrix

\[ G(s) = C(sI-A)^{-1} B + D. \]

Each entry \( G_{ij}(s) \) is a proper rational function of \( s \) whose poles are eigenvalues of \( A \).

Chapter 13: Time-Domain Response

13.1 Impulse and Step Response

For a single-input single-output system with transfer function \( G(s) \), the impulse response is \( g(t) = \mathcal L^{-1}\{G(s)\} \), and any response is the convolution \( y(t)=\int_0^t g(t-\tau) u(\tau)\,d\tau \). The step response is \( y_\text{step}(t) = \mathcal L^{-1}\{G(s)/s\} \).

13.2 First-Order Response

For \( G(s) = K/(\tau s + 1) \) and a unit step,

\[ y(t) = K\,\left(1 - e^{-t/\tau}\right). \]

After \( t = \tau \) the response has reached 63.2% of steady state; after \( 4\tau \), 98.2%; after \( 5\tau \), essentially settled. This is the universal first-order signature of RC circuits, single-tank fluid systems, and lumped thermal plants.

13.3 Second-Order Response

For \( G(s) = \omega_n^2/(s^2 + 2\zeta\omega_n s + \omega_n^2) \) and a unit step, the response depends on \( \zeta \):

• Underdamped (\( 0<\zeta<1 \)): damped oscillation with envelope \( e^{-\zeta\omega_n t} \) and ringing frequency \( \omega_d = \omega_n\sqrt{1-\zeta^2} \). Peak overshoot \( M_p = e^{-\pi\zeta/\sqrt{1-\zeta^2}} \). Peak time \( t_p = \pi/\omega_d \). 2% settling time \( \approx 4/(\zeta\omega_n) \).
• Critically damped (\( \zeta=1 \)): fastest return without overshoot, \( y(t) = 1-(1+\omega_n t) e^{-\omega_n t} \).
• Overdamped (\( \zeta>1 \)): sum of two exponentials with time constants \( 1/(\zeta\omega_n \pm \omega_n\sqrt{\zeta^2-1}) \).

Chapter 14: Frequency Response

14.1 Steady-State Sinusoidal Response

For a stable LTI system driven by \( u(t) = U\cos(\omega t) \), the steady-state output is

\[ y_\text{ss}(t) = |G(j\omega)|\,U\,\cos\big(\omega t + \angle G(j\omega)\big). \]

The complex-valued function \( G(j\omega) \) is the frequency response: its magnitude gives the amplitude ratio and its angle the phase shift.

14.2 Bode Plots

The Bode magnitude plot graphs \( 20\log_{10}|G(j\omega)| \) in decibels versus \( \log\omega \); the Bode phase plot graphs \( \angle G(j\omega) \) in degrees versus \( \log\omega \). Straight-line approximations are built from a few asymptotic rules:

• A factor \( K \) contributes a constant \( 20\log_{10}|K| \) dB and 0° phase.
• A pole at the origin \( 1/s \) contributes \( -20 \) dB/decade slope and \( -90° \) phase.
• A simple pole \( 1/(\tau s+1) \) contributes 0 dB below the break frequency \( 1/\tau \) and \( -20 \) dB/decade above, with phase going from 0° to \( -90° \) centered at \( 1/\tau \).
• A complex-conjugate pole pair with damping \( \zeta \) gives a \( -40 \) dB/decade slope above \( \omega_n \); the magnitude peaks at \( \omega = \omega_n\sqrt{1-2\zeta^2} \) with height \( 1/(2\zeta\sqrt{1-\zeta^2}) \) for small \( \zeta \).

The Bode plot is the frequency-domain fingerprint of the system: it tells you bandwidth, resonance, and phase margin at a glance.

14.3 Impedance and Admittance

Generalized impedance relates across to through at a pair of terminals:

\[ Z(s) = \frac{\text{across}(s)}{\text{through}(s)}, \qquad Y(s) = 1/Z(s). \]

Electrical: \( Z_R = R \), \( Z_L = sL \), \( Z_C = 1/(sC) \). Mechanical (force-current): \( Z_m = 1/(sm) \) for a mass, \( Z_k = s/k \) for a spring, \( Z_b = 1/b \) for a damper — note the “current-convention” inversion that makes series-parallel composition behave the same as in electrical networks.

Impedance methods let an engineer compose transfer functions by series-parallel combination of element impedances, bypassing explicit ODE derivation. This is the quickest path for ladder networks, hydraulic manifolds, and compound suspensions.

Chapter 15: Transfer-Function Realization

15.1 From Transfer Function to State Space

Given a scalar transfer function

\[ G(s) = \frac{b_m s^m + \cdots + b_1 s + b_0}{s^n + a_{n-1}s^{n-1} + \cdots + a_1 s + a_0}, \]

(with \( m\le n \)), one classical realization is the controllable canonical form:

\[ A = \left[\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & & & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \\ -a_0 & -a_1 & -a_2 & \cdots & -a_{n-1} \end{array}\right], \qquad B = \left[\begin{array}{c} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \end{array}\right]. \]

With \( C \) read off from the numerator coefficients, the triple \( (A,B,C) \) reproduces \( G(s) \). Observable canonical form, Jordan form, and diagonal (modal) form are alternative realizations, each useful for different purposes (control design, numerical conditioning, modal identification).

15.2 Realization is Not Unique

Any similarity transformation \( \tilde A = T A T^{-1} \), \( \tilde B = T B \), \( \tilde C = C T^{-1} \) yields a different state-space description of the same input-output behaviour. The non-uniqueness is a reminder: the state is chosen; the transfer function is intrinsic.

Chapter 16: Bond Graphs in Brief

16.1 Motivation

Bond graphs unify the linear-graph formulation across all domains into a single symbol set. A bond carries a power pair (effort, flow); effort is the across variable, flow is the through variable. Elements are labelled by letter:

• \( R \): dissipators (resistor, damper, valve, thermal resistance).
• \( C \): across-storage (capacitor, spring compliance, tank, thermal capacitance).
• \( I \): through-storage (inductor, mass, fluid inertance).
• \( Se, Sf \): effort and flow sources.
• \( TF \): transformer (gears, levers, ideal transformer).
• \( GY \): gyrator (DC motor, voice coil).
• \( 0 \)-junction: common-effort node (KCL at a node).
• \( 1 \)-junction: common-flow node (KVL around a loop).

16.2 Causality Assignment

Each bond is given a causal stroke indicating which end imposes effort and which imposes flow. Proper causality assignment proceeds outward from sources, through junctions, and reveals the state variables automatically: \( C \) elements in integral causality give effort-state (across-type); \( I \) elements in integral causality give flow-state (through-type). Bond graphs subsume the normal-tree method and extend cleanly to nonlinear and multi-port elements; MTE 351 treats them briefly but they underpin modern multi-domain simulation environments (Modelica, Simscape, 20-sim).

Chapter 17: Sensors and Actuators

17.1 Common Sensors

Sensor	Measures	Output	Transfer character
Potentiometer	position	voltage	proportional, \( v = k_p x \)
Tachometer	angular velocity	voltage	proportional, \( v = K_t \omega \)
Accelerometer (MEMS)	acceleration	voltage	second-order with high \( \omega_n \)
LVDT	displacement	AC voltage	proportional over linear range
Strain gauge	strain	resistance change	linear over small strain
Thermocouple	temperature	mV	nonlinear, linearized per junction
Pressure transducer	pressure	voltage/current	first-order dynamics

Every sensor has its own dynamics (bandwidth, lag) and adds to the model. In careful design the sensor is modelled as an extra transfer-function block cascaded with the plant.

17.2 Common Actuators

• DC servo motor (Chapter 7). Linear gyrator-coupled electromechanical actuator.
• Hydraulic cylinder and servo valve. Large force at moderate speeds; valve gain linearized about operating point.
• Pneumatic cylinder. Compressibility adds a spring-like mode.
• Piezoelectric stack. High bandwidth, very small displacement; modelled as a stiff spring with a charge input.
• Voice coil. Small stroke, high bandwidth, linear.

Chapter 18: Case Studies

18.1 Quarter-Car Active Suspension

Two masses (chassis \( m_c \), wheel \( m_w \)), two springs (suspension \( k_s \), tire \( k_t \)), one damper \( b_s \), and an actuator force \( F_a(t) \) between chassis and wheel; road input \( z_r(t) \) enters through the base of the tire.

\[ m_c\,\ddot z_c = -k_s(z_c - z_w) - b_s(\dot z_c - \dot z_w) + F_a, \]\[ m_w\,\ddot z_w = k_s(z_c - z_w) + b_s(\dot z_c - \dot z_w) - k_t(z_w - z_r) - F_a. \]

Four states \( (z_c, \dot z_c, z_w, \dot z_w) \). Transfer functions from \( z_r \) to \( z_c \) (ride comfort) and from \( z_r \) to \( (z_w - z_r) \) (tire grip) are the design metrics. Classical passive design trades ride for handling; active suspension uses \( F_a \) to improve both.

18.2 Op-Amp Integrator

An ideal operational amplifier with input resistor \( R \) and feedback capacitor \( C \) realizes \( V_o = -\tfrac{1}{RC}\int V_i\,dt \). The ideal model treats the amplifier as a zero-impedance voltage-controlled voltage source with infinite gain. First-order corrections (finite gain-bandwidth, finite input impedance) yield a realistic transfer function whose dominant pole is at \( -1/(R C A_\text{ol}) \) and whose bandwidth is bounded by gain-bandwidth product.

18.3 Thermal Plant with Sensor Lag

A room modelled as in 6.2, with a thermistor of capacitance \( C_s \) and contact resistance \( R_{cs} \) measuring \( \theta_s \). The measured temperature lags the room temperature:

\[ C_s\,\dot\theta_s = (\theta_r - \theta_s)/R_{cs}. \]

For slow thermal plants the sensor lag may be comparable to the plant time constant and must be accounted for in control design; otherwise the feedback controller sees a delayed plant and may go unstable.

18.4 DC Servo Position System

DC motor from 7.2 driving a load through a gear ratio \( N \); position feedback through a potentiometer \( k_p \); proportional controller \( K_P \) closing the loop. The open-loop transfer function from voltage to position (neglecting \( L_a \)) is

\[ G_\text{ol}(s) = \frac{K/N}{s\,(R_a(J_\text{tot} s + B_\text{tot}) + K^2/N^2)}, \]

which is Type-1 (integrator in the plant). With proportional feedback the closed loop is second order; damping is set by the loop gain \( K_P \). Added derivative action (rate feedback from a tachometer) increases damping without adding low-frequency phase lag.

Chapter 19: Practical Advice and Common Pitfalls

19.1 Sign Conventions

Every element gets a chosen positive direction; once chosen, it must be used consistently in the constitutive law, the cut-set equations, and the state definition. Most errors in hand-worked problems trace to a sign flip between free-body diagram and graph.

19.2 Units

Write every quantity with SI units at the first appearance. A torque constant in oz-in/A mixed with a damping in N·m·s is the recipe for mysteriously wrong Bode plots. The most common offenders: gear ratios (dimensionless, but often given as tooth counts), thermal resistances (K/W vs °C/W — numerically identical for differences), and pressure (Pa vs psi vs bar).

19.3 Identifying Model Validity

A lumped model is valid when (i) the element sizes are small compared with the wavelength of signals at the frequencies of interest, and (ii) the operating amplitudes stay within the linear range of the constitutive laws. Outside these limits, distributed-parameter or nonlinear models are required; MTE 351 teaches the intuition for when to reach for them.

19.4 Numerical Checks

Always check the steady-state value of a simulation against a hand calculation. Always check that the model reduces to a recognized textbook case in appropriate limits (e.g. the DC motor model should collapse to a first-order voltage-speed transfer function when \( L_a \to 0 \)). Always plot impulse and step responses to confirm qualitative behaviour before trusting any quantitative prediction.

Chapter 20: Summary

MTE 351 is an exercise in recognizing that disparate physical systems — a car, a circuit, a tank, a heater, a motor — share a common mathematical skeleton. The skeleton has three pieces: constitutive laws for storage and dissipation; topological laws (continuity and compatibility) enforced by the linear graph; and the state-space and transfer-function machinery that turns those laws into predictions.

A student who finishes the course should be able to look at a mixed-domain system, sketch its linear graph, pick a normal tree, write the state equations, linearize if necessary, transform to the Laplace domain, and interpret the resulting pole-zero pattern and Bode plot. This toolbox is the foundation for every downstream course in controls, mechatronics, robotics, and signal processing — and for professional engineering practice wherever dynamic behaviour matters.