2. Multidomain System Modelling and Simulation

Introduction and context

Mechatronic systems combine mechanical structures, electrical circuits, sensors, actuators, controllers, fluid power, heat transfer and computation. Their behaviour is therefore not adequately represented by a single-domain model. A servo axis, for example, may contain a motor and power converter, a flexible transmission, a hydraulic brake, temperature-dependent resistance, sensors and a digital controller. Design decisions in one domain can strongly influence performance in another. Increasing motor current may improve acceleration but increase heat generation; increasing hydraulic pressure may reduce actuator size but raise leakage and safety requirements.

Multidomain modelling provides a disciplined way to represent these interactions mathematically. Simulation then uses the model to predict responses before hardware is built or modified. The purpose is not to reproduce every physical detail. Rather, a model should retain the phenomena that matter for the design question while remaining sufficiently simple to analyse, identify and compute.

This chapter develops a unified approach to modelling mechanical, electrical, hydraulic, pneumatic and thermal subsystems. It compares differential-equation, transfer-function, state-space, block-diagram and bond-graph representations, and explains how numerical simulation supports analysis and design. Particular attention is given to assumptions, parameter uncertainty, model validation and the responsible use of digital tools.

Learning objectives

After studying this chapter, the reader should be able to:

1. Formulate governing equations for representative mechanical, electrical, hydraulic, pneumatic and thermal systems.

2. Select appropriate system variables, coordinates, boundary conditions and simplifying assumptions.

3. Convert physical equations into transfer-function, state-space, block-diagram and bond-graph models.

4. Explain causality, energy exchange and coupling in multidomain systems.

5. Choose numerical integration methods and interpret simulation results critically.

6. Distinguish verification, validation, calibration and uncertainty analysis.

7. Apply modelling methods to a practical mechatronic design problem.

8. Recognise modelling errors, numerical instability, ethical responsibilities and emerging digital approaches.

Modelling as an engineering argument

A model is a purposeful representation of a real system. Its validity is relative to a question, operating range and required accuracy. A rigid-body model may be appropriate for estimating positioning time but inappropriate for predicting vibration caused by structural flexibility. Similarly, a linear electrical model may be sufficient around a nominal operating point but inadequate when magnetic saturation or switching nonlinearities dominate.

A sound modelling process begins by defining the system boundary and the intended outputs. The modeller identifies energy domains, inputs, outputs, stored-energy variables, dissipative elements and constraints. Conservation laws are then applied, parameters are assigned, and assumptions are recorded. The result should be traceable: another engineer should be able to determine how each equation follows from the physical system.

Important distinctions are useful. Verification asks whether the equations and their implementation have been solved correctly. Validation asks whether the model represents the real system adequately for its intended use. Calibration estimates uncertain parameters from experimental data; it does not by itself prove that the model is valid. Model credibility depends on all four activities, together with documented limitations.

Governing descriptions across physical domains

Mechanical systems

For translational motion, Newton's second law gives

\[

\sum F = m\ddot{x}.

\]

A mass–spring–damper system is represented by

\[

m\ddot{x}+c\dot{x}+kx=F(t),

\]

where \(m\) is mass, \(c\) damping, \(k\) stiffness and \(F\) applied force. For rotational motion,

\[

J\ddot{\theta}+b\dot{\theta}+k_t\theta=T(t),

\]

where \(J\) is moment of inertia, \(b\) rotational damping and \(k_t\) torsional stiffness. A gear train couples torque and speed through its ratio while transforming reflected inertia and stiffness. Backlash, Coulomb friction, saturation and flexible shafts introduce nonlinear or piecewise behaviour.

Lagrange's method is often more convenient for systems with several coordinates. With generalized coordinates \(q\),

\[

\frac{d}{dt}\left(\frac{\partial L}{\partial \dot q_i}\right)-\frac{\partial L}{\partial q_i}=Q_i,

\]

where \(L=T-V\) is kinetic energy minus potential energy and \(Q_i\) represents non-conservative generalized forces. This approach is especially useful for robot manipulators and coupled mechanisms.

Electrical systems

Kirchhoff's current and voltage laws provide the basic equations. For a series RLC circuit driven by voltage \(v(t)\),

\[

L\frac{di}{dt}+Ri+\frac{1}{C}\int i\,dt=v(t).

\]

The capacitor stores electrical energy, the inductor stores magnetic energy and the resistor dissipates energy. Electromechanical conversion is represented in a DC motor by

\[

V=L_a\frac{di}{dt}+R_ai+k_e\omega,

\]

\[

J\frac{d\omega}{dt}+b\omega=k_ti-T_L.

\]

Here, back electromotive force couples mechanical speed to the electrical equation, while electromagnetic torque couples current to the mechanical equation. In SI units, the torque and back-emf constants are numerically equal under conventional assumptions, although their physical units differ.

Power electronic switches produce discontinuous voltages and currents. An averaged converter model may be useful for control design, whereas a switching model is required for semiconductor stress, ripple or electromagnetic-interference analysis. The modelling choice must match the design purpose.

Hydraulic systems

For a hydraulic cylinder, force is approximately

\[

F=A p,

\]

where \(A\) is piston area and \(p\) pressure difference. Compressibility gives a continuity relation such as

\[

\dot p=\frac{\beta_e}{V}(Q-A\dot x-C_l p),

\]

where \(\beta_e\) is effective bulk modulus, \(V\) fluid volume, \(Q\) inflow and \(C_l p\) leakage flow. The ideal valve relation is commonly nonlinear, for example

\[

Q=C_d A_v\sqrt{\frac{2\Delta p}{\rho}},

\]

with discharge coefficient \(C_d\), valve opening area \(A_v\), pressure drop \(\Delta p\) and density \(\rho\). Hydraulic lines may introduce resistance, inertance and compliance. Air entrainment, hose elasticity and valve dynamics can significantly alter the response.

Pneumatic systems

Pneumatic models must account for gas compressibility and often temperature variation. Under an isothermal approximation, pressure and volume satisfy \(pV=mRT\). A chamber balance can be expressed as a mass-flow equation coupled with piston motion. Orifice flow is commonly piecewise: it depends on upstream and downstream pressure and changes form when flow becomes choked. Because air is highly compressible, pneumatic actuators tend to have lower stiffness and more pronounced delays than comparable hydraulic actuators. Friction, dead zones and pressure-dependent leakage are often important.

Thermal systems

A lumped thermal capacitance model follows the energy balance

\[

C_{th}\frac{dT}{dt}=\dot Q_{in}-\frac{T-T_a}{R_{th}},

\]

where \(C_{th}=mc_p\) is thermal capacitance, \(R_{th}\) thermal resistance, \(T_a\) ambient temperature and \(\dot Q_{in}\) heat input. Heat may arise from Joule losses \(i^2R\), friction, fluid throttling or semiconductor switching. Thermal parameters often vary with temperature, and cooling may involve conduction, convection and radiation. A thermal model coupled to an electrical motor model can predict temperature rise and impose a current derating constraint.

Representations used in analysis and design

Transfer functions

For a linear time-invariant system with zero initial conditions, the transfer function is

\[

G(s)=\frac{Y(s)}{U(s)}.

\]

For the mass–spring–damper system,

\[

\frac{X(s)}{F(s)}=\frac{1}{ms^2+cs+k}.

\]

Transfer functions reveal poles, zeros, natural frequency, damping ratio, bandwidth and steady-state gains. They are valuable for classical control design and frequency-domain reasoning. They are less suitable for nonlinear systems, systems with significant initial conditions, multiple energy storage variables or strongly coupled multidomain behaviour.

Linearisation around an equilibrium is often used to obtain a local transfer function. If \(\dot z=f(z,u)\) and \(y=g(z,u)\), small perturbations give

\[

\delta\dot z=A\delta z+B\delta u,\qquad \delta y=C\delta z+D\delta u,

\]

where the matrices are Jacobians evaluated at the operating point. A model valid near one speed, pressure or temperature may be misleading elsewhere.

State-space models

A state is a minimum set of variables that, together with future inputs, determines future behaviour. A general nonlinear model is

\[

\dot x=f(x,u,t),\qquad y=g(x,u,t).

\]

The linear form is

\[

\dot x=Ax+Bu,\qquad y=Cx+Du.

\]

For a DC motor, a useful state vector is \(x=[i\ \omega]^T\), giving

\[

\dot x=

\begin{bmatrix}

-R_a/L_a & -k_e/L_a\\

k_t/J & -b/J

\end{bmatrix}x+

\begin{bmatrix}1/L_a\\0\end{bmatrix}V+

\begin{bmatrix}0\\-1/J\end{bmatrix}T_L.

\]

State-space form naturally accommodates multiple inputs and outputs, initial conditions, disturbance inputs, nonlinearities and model reduction. The selected states should be physically interpretable where possible. Redundant states or algebraic constraints can create numerical difficulties.

Block diagrams

A block diagram displays signal flow and functional relationships. Integrators represent energy storage, gains represent physical parameters or transformations, and summing junctions express force, voltage, flow or energy balances. Block diagrams are intuitive for control architecture, but they can obscure physical conservation if signals are rearranged without careful attention to units and signs. Every signal should have a defined meaning and unit.

Bond graphs

Bond graphs represent power exchange rather than merely signal flow. A bond carries effort and flow variables whose product is power. Examples include force and velocity in mechanics, voltage and current in electrical systems, pressure and volumetric flow in hydraulics, and temperature difference and entropy-related flow in thermal analogies.

The principal elements are storage elements \(I\) and \(C\), dissipative elements \(R\), sources \(Se\) and \(Sf\), transformers \(TF\), gyrators \(GY\), and junctions 0 and 1. A 0-junction imposes common effort and a 1-junction imposes common flow. Bond graphs expose common physical structure across domains and help identify state variables, power paths and improper causal assignments. They are particularly useful when a mechatronic system contains several energy domains.

Numerical simulation and computational practice

Once equations are established, simulation requires initial conditions, input histories, parameter values and a numerical solver. Explicit Euler integration is simple:

\[

x_{k+1}=x_k+h f(x_k,u_k,t_k),

\]

but it may become inaccurate or unstable for large steps or stiff systems. Runge–Kutta methods generally provide better accuracy for smooth ordinary differential equations. Variable-step solvers adjust the step size according to estimated local error. Implicit methods are often preferred for stiff systems, such as systems combining fast electrical dynamics with slow thermal dynamics or highly compliant fluid networks.

A credible simulation includes convergence checks. Reducing the time step should not materially change the engineering conclusion. Units, signs, initial conditions and conservation balances should be checked independently. Event handling is needed for impacts, valve switching, saturation and contact. Algebraic loops and discontinuities should be treated explicitly rather than hidden inside numerical blocks.

Simulation studies commonly include step, ramp, sinusoidal, disturbance and operating-cycle inputs. Outputs may include position error, current, force, pressure, temperature, energy consumption and constraint violations. Results should be interpreted against physical limits, not merely plotted for visual appeal.

A disciplined multidomain workflow

A practical workflow is as follows:

1. Define purpose, boundaries, inputs, outputs and required accuracy.

2. Draw the physical architecture and identify interfaces between domains.

3. Select coordinates and energy-storage variables.

4. Write conservation equations and constitutive relations.

5. State assumptions, parameter sources and operating range.

6. Choose a representation: transfer function, state space, block diagram, bond graph or a combination.

7. Implement the model with dimensional checks and clear parameter management.

8. Verify equations, solver settings and limiting cases.

9. Calibrate uncertain parameters using independent experimental data where possible.

10. Validate predictions across relevant operating conditions and document residual errors.

11. Perform sensitivity and uncertainty analysis.

12. Use the model for design decisions, while retaining a record of its limitations.

Sensitivity analysis can be local, using derivatives such as \(\partial y/\partial p\), or global, using parameter sampling. Monte Carlo simulation propagates distributions of uncertain parameters through the model. Interval or worst-case analysis is useful where safety limits matter more than expected performance. Correlated uncertainties and measurement error must not be ignored.

Illustrative case study: electro-hydraulic positioning axis

Background

Consider an illustrative industrial positioning axis in which a servo motor drives a hydraulic pump, and the pump supplies a cylinder that moves a carriage. The design objective is accurate positioning with acceptable settling time, power consumption and temperature rise.

Situation and modelling problem

A purely mechanical model predicts carriage motion but cannot explain pressure ripple. A purely hydraulic model predicts force but not motor current or thermal loading. The system must therefore combine motor electrodynamics, pump flow, hydraulic compressibility, cylinder mechanics, friction and heat generation.

Analysis and approach

Let motor current \(i\), angular speed \(\omega\), chamber pressure \(p\), carriage position \(x\), and motor temperature \(T\) be states. A simplified model is

\[

L_a\dot i=V-R_ai-k_e\omega,

\]

\[

J\dot\omega=k_ti-b\omega-T_p,

\]

\[

Q_p=D\omega-C_p p,

\]

\[

\dot p=\frac{\beta_e}{V_h}(Q_p-A\dot x-C_l p),

\]

\[

m\ddot x=Ap-c\dot x-F_f,

\]

\[

C_{th}\dot T=i^2R_a-\frac{T-T_a}{R_{th}}.

\]

Here, pump displacement is \(D\), pump leakage coefficient is \(C_p\), hydraulic volume is \(V_h\), and friction is represented initially by an equivalent viscous term. The model is implemented in state-space form. A more detailed model can later introduce valve nonlinearities, Coulomb friction, saturation and temperature-dependent resistance.

The model is first verified by checking that zero input and zero load produce physically consistent equilibrium behaviour. A constant-speed test checks whether predicted flow and pressure satisfy pump and continuity relations. Experimental current, position and pressure data are then used for calibration, but validation is performed using a different motion profile. Sensitivity analysis shows that bulk modulus and leakage strongly influence settling time, while thermal resistance dominates long-duration temperature prediction.

Outcome and engineering decision

Simulation indicates that aggressive position gains reduce tracking error but excite hydraulic compliance and pressure oscillation. A moderate gain with feedforward flow, pressure feedback and a current limit provides a better compromise. The thermal model shows that short transients are acceptable but repeated high-duty cycles require a lower continuous current or improved cooling. The final design therefore includes a pressure relief function, software current limiting and a temperature-based derating strategy.

Learning points

The case demonstrates that multidomain coupling is not an optional refinement when one domain determines the operating limits of another. It also shows why a model should evolve in stages: begin with dominant physics, validate it, and add detail only when discrepancies or design questions justify the added complexity.

Challenges, limitations and professional responsibilities

Common modelling errors include confusing force with power flow, using inconsistent unit systems, neglecting initial conditions, linearising far from an operating point, treating fitted parameters as universal constants, and validating only against the data used for calibration. Overparameterised models may fit measurements while predicting poorly outside the test set. Underparameterised models may be robust but unable to answer the design question.

Numerical results can also be deceptive. Solver tolerances do not correct incorrect equations. A visually smooth response may conceal violation of energy conservation. A small time step cannot compensate for an incorrect boundary condition or unmodelled delay. Digital simulation should therefore be accompanied by dimensional analysis, limiting-case tests, conservation checks and comparison with measured data.

In safety-critical applications, model assumptions must be reviewed independently and safety factors must not be replaced by optimistic simulation. Configuration control is needed so that the model, parameters, software version and test data remain traceable. Proprietary data and cyber-physical models require access control. Engineers should disclose uncertainty, avoid presenting simulated performance as demonstrated performance, and retain responsibility for decisions assisted by automated or AI-based modelling tools.

Digital applications and emerging directions

Modern engineering environments integrate equation-based modelling, finite-element analysis, computational fluid dynamics, control simulation and hardware-in-the-loop testing. Reduced-order models permit rapid optimisation while high-fidelity models are reserved for final verification. Co-simulation allows specialised solvers to exchange variables across domain boundaries, although communication delays and inconsistent time stepping must be managed.

Digital twins extend simulation by connecting a model to operational data. Online parameter estimation can track wear, leakage or thermal resistance, while anomaly detection can compare measured residuals with model predictions. Machine-learning surrogates may accelerate expensive simulations, but they require representative training data and physical plausibility checks. Hybrid models that combine conservation laws with data-driven corrections are often more interpretable than unconstrained black-box models.

Best practices

A professional multidomain model should have a clearly stated purpose, explicit assumptions, consistent units, modular subsystems and documented interfaces. Parameters should have sources, uncertainty ranges and identification procedures. Models should be tested at limiting cases and validated over the intended operating envelope. The simplest model that answers the question is generally preferable, but simplification must never remove a phenomenon that governs safety, stability or the decision being made.

Chapter Summary

Multidomain system modelling represents the interaction of mechanical, electrical, hydraulic, pneumatic and thermal phenomena through conservation laws and constitutive relations. Mechanical systems are commonly described through Newtonian or Lagrangian equations; electrical systems through Kirchhoff laws; fluid systems through continuity, momentum and flow relations; and thermal systems through energy balances. Transfer functions support linear frequency-domain analysis, state-space models support coupled and nonlinear simulation, block diagrams clarify functional signal flow, and bond graphs expose power exchange and cross-domain structure.

Numerical simulation requires appropriate solvers, initial conditions, inputs, event treatment and convergence checks. A reliable workflow defines purpose, formulates equations, states assumptions, verifies implementation, calibrates parameters, validates against independent evidence and analyses uncertainty. Multidomain models support design, control, diagnostics, thermal management and digital twins, but they remain conditional representations rather than substitutes for engineering judgement.

Key Terms

Model: A purposeful mathematical or computational representation of a physical system.

Multidomain system: A system in which energy or information is exchanged among two or more physical domains.

State variable: A variable that, with inputs, defines the future evolution of a system.

Transfer function: The Laplace-domain ratio of output to input for a linear system with zero initial conditions.

State-space model: A first-order differential-equation representation using state, input and output variables.

Linearisation: Approximation of a nonlinear model by its first-order behaviour near an operating point.

Causality: The assignment of which variable in a physical relation is determined from another.

Bond graph: A power-based graphical representation using effort, flow and energy-storage elements.

Effort variable: A domain-specific intensive variable, such as force, voltage or pressure.

Flow variable: A domain-specific rate variable, such as velocity, current or volumetric flow.

Constitutive relation: An equation describing the behaviour of a physical element, such as resistance or stiffness.

Verification: Determination of whether a model and its implementation have been solved correctly.

Validation: Determination of whether a model adequately represents the real system for its intended use.

Calibration: Estimation of model parameters using observations or experimental data.

Sensitivity analysis: Study of how changes in parameters affect model outputs.

Uncertainty propagation: Assessment of how uncertain inputs or parameters influence predicted results.

Stiff system: A system containing dynamics with widely separated time scales that restrict explicit numerical steps.

Reduced-order model: A simplified model retaining dominant behaviour while reducing computational effort.

Digital twin: A model connected to data from a physical asset for monitoring, prediction or optimisation.

Review Questions

Short Answer Questions

1. Why is model validity always related to a purpose and operating range?

2. What physical quantities are coupled in a DC motor model?

3. State the difference between verification and validation.

4. What is the role of a thermal capacitance in a lumped thermal model?

5. Why are bond graphs useful for multidomain systems?

Long Answer Questions

1. Derive the transfer function of a mass–spring–damper system and interpret its poles.

2. Explain the formulation of a DC motor in state-space form, including load torque as a disturbance.

3. Describe a complete workflow for developing and validating a multidomain mechatronic model.

4. Compare transfer functions, state-space models, block diagrams and bond graphs in terms of strengths and limitations.

5. Discuss numerical stability, stiffness, step-size selection and event handling in simulation.

Analytical / Application Questions

1. A motor-driven actuator exhibits excessive temperature rise during repeated cycles although its position response is satisfactory. Propose a coupled electrical–mechanical–thermal model and identify the parameters requiring validation.

2. A pneumatic positioning system shows oscillation after a command step. Explain which physical effects should be investigated and how simulation and experiments could distinguish them.

3. Develop a modelling and uncertainty-analysis plan for the electro-hydraulic axis described in the case study. Specify the measurements needed for calibration and independent validation.

22. Multiple Choice Questions with Answers

Question 1

Which representation most directly exposes power exchange between different physical domains?

A. Transfer function

B. Bond graph

C. Root-locus plot

D. Static block diagram

Correct Answer: B. Bond graph

Question 2

For a DC motor model, which variable provides the coupling from mechanics to the electrical subsystem?

A. Armature resistance

B. Load torque

C. Back electromotive force proportional to speed

D. Rotor inertia

Correct Answer: C. Back electromotive force proportional to speed

Question 3

A model is calibrated using one experimental data set and then tested using a different motion profile. The second test primarily supports which activity?

A. Validation

B. Linearisation

C. Causality assignment

D. Parameter scaling

Correct Answer: A. Validation

Question 4

Which equation best represents a lumped thermal resistance-capacitance model?

A. F=ma

B. V=L di/dt+Ri

C. Cth dT/dt=Qin-(T-Ta)/Rth

D. Q=CdAv sqrt(2Δp/ρ)

Correct Answer: C. Cth dT/dt=Qin-(T-Ta)/Rth

Question 5

Why may an implicit numerical solver be preferred for a stiff mechatronic model?

A. It removes the need for initial conditions

B. It can remain stable with dynamics having widely separated time scales

C. It automatically validates physical assumptions

D. It converts nonlinear equations into transfer functions

Correct Answer: B. It can remain stable with dynamics having widely separated time scales

Question 6

A transfer function obtained by linearising a hydraulic actuator around one pressure operating point should be used cautiously because

A. transfer functions cannot contain poles

B. the local approximation may not represent behaviour far from that operating point

C. hydraulic systems have no energy storage

D. linear models always require finite-element software

Correct Answer: B. the local approximation may not represent behaviour far from that operating point

Question 7

In a hydraulic cylinder model, fluid compressibility is primarily represented by

A. A pressure-storage or continuity relation involving bulk modulus and volume

B. A gear-ratio equation

C. A thermal radiation equation

D. A motor back-emf equation

Correct Answer: A. A pressure-storage or continuity relation involving bulk modulus and volume

Question 8

Which practice is most appropriate for checking numerical simulation credibility?

A. Increase plotting resolution only

B. Reduce the time step and check convergence, units and conservation balances

C. Fit more parameters to the same data

D. Ignore initial conditions if the input is known

Correct Answer: B. Reduce the time step and check convergence, units and conservation balances

Question 9

In the electro-hydraulic case, increasing controller gain reduced position error but excited pressure oscillations. This illustrates that

A. domains can be designed independently without interaction

B. hydraulic compliance and control dynamics are coupled

C. thermal models are unnecessary

D. pressure is unrelated to actuator motion

Correct Answer: B. hydraulic compliance and control dynamics are coupled

Question 10

Which statement about calibration is correct?

A. Calibration alone proves validity over all operating conditions

B. Calibration estimates parameters from data but does not replace independent validation

C. Calibration is identical to numerical integration

D. Calibration eliminates structural model error

Correct Answer: B. Calibration estimates parameters from data but does not replace independent validation