2. Mathematical Modeling of Mechanical Systems

Introduction and Context

Mathematical modeling is the process of representing the physical behaviour of a system through variables, parameters, equations, and constitutive relations. In machine dynamics and control, a model converts a real mechanical, electromechanical, hydraulic, or pneumatic device into a form that can be analysed, simulated, designed, and controlled.

A model is neither the physical system itself nor an exact copy of it. It is a purposeful approximation. The appropriate model depends on the question being asked. A rigid-body model may be adequate for predicting the motion of a robot arm, whereas a flexible-body model is necessary for studying vibration or control spillover. Similarly, a hydraulic actuator may be represented by a simple first-order relation for preliminary sizing, but a high-fidelity model must include compressibility, leakage, valve dynamics, and nonlinear friction.

The central task is therefore not merely to write equations. It is to decide which physical effects are important, select suitable generalized coordinates, state assumptions explicitly, and assess the uncertainty introduced by simplification. This chapter develops lumped-parameter modeling methods using Newton–Euler, work–energy, Lagrange, and Hamiltonian formulations. It then extends these ideas to translational, rotational, electromechanical, hydraulic, and pneumatic systems.

Learning Objectives

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

1. Explain the purpose, scope, and limitations of lumped-parameter models.

2. Select system variables, generalized coordinates, parameters, and boundary conditions.

3. Derive equations of motion using Newton–Euler, work–energy, Lagrange, and Hamiltonian methods.

4. Model translational and rotational mechanical systems, including damping, stiffness, gears, and constraints.

5. Formulate basic electromechanical, hydraulic, and pneumatic models.

6. Linearize nonlinear models and express them in state-space form.

7. Identify sources of model uncertainty and validate a model against physical evidence.

8. Apply modeling principles to the analysis and control of an industrial actuator.

2.1 What a Mathematical Model Represents

A dynamic system model describes how inputs, internal variables, and disturbances produce outputs as time evolves. A general representation is

\[

\dot{x}=f(x,u,d,p,t), \qquad y=g(x,u,d,p,t),

\]

where \(x\) is the state vector, \(u\) is the controlled input, \(d\) represents disturbances, \(p\) contains physical parameters, and \(y\) is the measured output. For a linear time-invariant system,

\[

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

\]

The state is the minimum set of variables required, together with the input, to determine future behaviour. In a mass–spring–damper system, displacement and velocity are commonly selected as states. In a motor, angular position and speed may be mechanical states, while armature current is an electrical state.

A lumped-parameter model assumes that physical quantities such as mass, compliance, damping, resistance, capacitance, or inductance can be concentrated at discrete locations. This contrasts with a distributed-parameter model, in which displacement, temperature, pressure, or electromagnetic fields vary continuously over space. Lumped models are especially effective when spatial dimensions are small relative to the wavelengths or characteristic propagation distances of interest, and when the dominant dynamics can be represented by a finite number of energy-storage and energy-dissipation elements.

Important modeling distinctions include:

• Input: an externally imposed force, torque, voltage, pressure, flow, or command.
• Output: a quantity of interest, such as position, speed, force, pressure, or current.
• Parameter: a property such as mass, inertia, stiffness, resistance, or viscosity.
• Constraint: a relation restricting possible motion, such as a gear ratio or rigid linkage.
• Disturbance: an uncontrolled influence, such as load torque, friction variation, or supply fluctuation.
• Constitutive law: a relation describing an element, for example \(F=kx\), \(F=c\dot{x}\), or \(V=L\dot{i}\).

A sound model begins with a clearly stated purpose. Modeling for controller design generally requires a model accurate in the frequency range of closed-loop operation. Modeling for structural failure may require stress distributions and local geometry that a low-order control model deliberately ignores.

2.2 Conservation Laws and Element Relations

Most mechanical models are built from conservation of mass, momentum, and energy. Newton's second law gives the translational momentum balance,

\[

\sum F=m\ddot{x},

\]

and the rotational form is

\[

\sum T=J\ddot{\theta},

\]

where \(m\) is mass, \(J\) is mass moment of inertia, \(x\) is displacement, and \(\theta\) is angular displacement.

The work–energy relation follows from the power balance. Kinetic energy is

\[

T=\frac{1}{2}m\dot{x}^{2}+\frac{1}{2}J\dot{\theta}^{2},

\]

while potential energy for a linear spring is

\[

V=\frac{1}{2}kx^{2}.

\]

A viscous damper dissipates power

\[

P_d=c\dot{x}^{2},

\]

and does not store recoverable energy. In rotational motion, the corresponding relations are \(T=\frac12J\omega^2\) and \(P_d=b\omega^2\).

The principle of virtual work is useful for systems with constraints. If a system is in static equilibrium, the total virtual work of applied forces for every allowable virtual displacement is zero. In dynamics, inertial effects can be included through d'Alembert's principle, converting a dynamic problem into an equivalent virtual-work statement.

2.3 Newton–Euler Modeling

The Newton–Euler method isolates bodies and writes force and moment balances. It is direct, physically transparent, and particularly effective for systems with few bodies or when reaction forces are required.

Consider a horizontal mass connected to a spring and damper and acted upon by an external force \(F(t)\). Taking positive displacement to the right,

\[

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

\]

or

\[

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

\]

For a rotational shaft with inertia \(J\), viscous friction \(b\), torsional stiffness \(k_t\), applied torque \(T_a\), and load torque \(T_L\),

\[

J\ddot{\theta}+b\dot{\theta}+k_t\theta=T_a-T_L.

\]

If the spring is connected between two rotating inertias, the elastic torque depends on relative angle:

\[

T_s=k_t(\theta_1-\theta_2).

\]

The free-body diagram must use consistent sign conventions. A frequent modeling error is to treat an internal force as having the same sign on both bodies; Newton's third law requires equal and opposite action–reaction forces.

For rigid gears with pitch radii \(r_1\) and \(r_2\), no-slip kinematics gives

\[

r_1\theta_1+r_2\theta_2=0,

\]

with the sign determined by gear orientation. If an ideal gear has ratio \(N=\omega_1/\omega_2\), torque conservation gives approximately \(T_2=N T_1\), subject to the chosen convention. Real gears introduce efficiency losses, backlash, compliance, and friction.

2.4 Work–Energy and Lagrange Formulations

The work–energy approach is often more convenient than repeated force balances when many forces are internal or when the system contains constrained motion. For a conservative system,

\[

T+V=\text{constant}.

\]

With nonconservative generalized work, the rate form is

\[

\frac{d}{dt}(T+V)=P_{\text{input}}-P_{\text{dissipation}}.

\]

Lagrange's method provides a systematic formulation in generalized coordinates \(q_i\). Define the Lagrangian

\[

L(q,\dot q,t)=T(q,\dot q,t)-V(q,t).

\]

For each coordinate,

\[

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

\]

where \(Q_i\) is the generalized nonconservative force. For viscous damping, it is convenient to define the Rayleigh dissipation function

\[

\mathcal{R}=\frac12\sum c_{ij}\dot q_i\dot q_j,

\]

and write

\[

\frac{d}{dt}\left(\frac{\partial L}{\partial \dot q_i}\right)-\frac{\partial L}{\partial q_i}+\frac{\partial\mathcal{R}}{\partial\dot q_i}=Q_i^{\text{applied}}.

\]

For a two-degree-of-freedom translational system with masses \(m_1,m_2\), coordinates \(x_1,x_2\), springs attached to ground with stiffnesses \(k_1,k_2\), and coupling stiffness \(k_c\),

\[

T=\frac12m_1\dot{x}_1^2+\frac12m_2\dot{x}_2^2,

\]

\[

V=\frac12k_1x_1^2+\frac12k_2x_2^2+\frac12k_c(x_1-x_2)^2.

\]

The resulting matrix equation is

\[

M\ddot{x}+Kx=f,

\]

where

\[

M=\begin{bmatrix}m_1&0\\0&m_2\end{bmatrix},\qquad

K=\begin{bmatrix}k_1+k_c&-k_c\\-k_c&k_2+k_c\end{bmatrix}.

\]

Lagrange's method is powerful because ideal constraint forces do no virtual work and therefore need not be calculated explicitly. This makes it well suited to robotic manipulators, vehicle models, mechanisms, and multibody systems. Its disadvantage is that an inappropriate coordinate choice can produce complicated expressions or hidden singularities.

2.5 Hamiltonian Viewpoint

Hamiltonian mechanics uses generalized coordinates and momenta rather than velocities. The generalized momentum is

\[

p_i=\frac{\partial L}{\partial\dot q_i},

\]

and the Hamiltonian is

\[

H(q,p,t)=\sum_i p_i\dot q_i-L.

\]

For a conservative mechanical system, \(H=T+V r\) and commonly represents total energy. Hamilton's equations are

\[

\dot q_i=\frac{\partial H}{\partial p_i},\qquad

\dot p_i=-\frac{\partial H}{\partial q_i}.

\]

For a particle, \(p=m\dot{x}\), and

\[

H=\frac{p^2}{2m}+V(x).

\]

Hamiltonian models are valuable in advanced control, nonlinear dynamics, passivity analysis, port-Hamiltonian systems, and energy-based control. They make energy exchange explicit and can support structure-preserving numerical simulation. Damping and external inputs must be added through generalized forces or dissipative ports; consequently, the Hamiltonian is not necessarily conserved in a real system.

2.6 Translational and Rotational Lumped Models

The standard translational elements are inertia, compliance, and dissipation. Their force relations are

\[

F_m=m\ddot{x},\qquad F_k=kx,\qquad F_c=c\dot{x}.

\]

The rotational analogues are

\[

T_J=J\ddot{\theta},\qquad T_k=k_t\theta,\qquad T_b=b\dot{\theta}.

\]

Coulomb friction is modeled approximately by

\[

F_f=F_c\operatorname{sgn}(\dot{x}),

\]

although this expression is discontinuous at zero speed. Stribeck, Dahl, LuGre, and other friction models can describe pre-sliding, breakaway, and velocity-dependent effects more realistically, but they introduce additional parameters and possible identifiability problems.

For a single-degree-of-freedom oscillator,

\[

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

\]

The natural frequency and damping ratio are

\[

\omega_n=\sqrt{\frac{k}{m}},\qquad

\zeta=\frac{c}{2\sqrt{km}}.

\]

These quantities predict whether the response is oscillatory, critically damped, or overdamped. In multi-degree-of-freedom systems, the undamped natural frequencies follow from

\[

\det(K-\omega^2M)=0.

\]

This eigenvalue problem is central to vibration analysis, modal reduction, and avoiding resonance in machine design.

2.7 Electromechanical Modeling

An electromechanical actuator couples an electrical subsystem to a mechanical subsystem. For a separately excited DC motor, the armature equation is

\[

V=Ri+L\frac{di}{dt}+K_e\omega,

\]

where \(V\) is armature voltage, \(R\) resistance, \(L\) inductance, \(i\) current, \(K_e\omega\) back electromotive force, and \(\omega=\dot\theta\). The electromagnetic torque is

\[

T_m=K_t i.

\]

The mechanical equation is

\[

J\dot\omega+b\omega=T_m-T_L.

\]

In SI units for an ideal motor, the numerical values of \(K_e\) and \(K_t\) are equal when compatible units are used. The state vector may be selected as \(x=[i,\omega,\theta]^T\). Combining the equations gives a state-space model suitable for simulation and controller design.

The coupling is governed by power conversion. Electrical input power is \(Vi\), electromechanical conversion power is approximately \(T_m\omega\), and losses include copper loss \(Ri^2\), viscous friction, brush loss, and magnetic losses. Ignoring one of these terms may be acceptable for a narrow operating range but can produce incorrect predictions of heating, efficiency, or transient torque.

Other actuator models include permanent-magnet synchronous motors, induction motors, solenoids, piezoelectric actuators, and voice-coil drives. Their detailed equations differ, but the modeling principle is the same: identify energy-storage variables, conversion laws, dissipation, inputs, and outputs.

2.8 Hydraulic and Pneumatic Models

Hydraulic systems are commonly described using pressure, flow, displacement, and force. For an incompressible ideal fluid, continuity requires that inflow and outflow balance. For a chamber with effective bulk modulus \(\beta_e\), volume \(V\), flow imbalance \(Q\), and leakage \(C_t p\), a simplified pressure equation is

\[

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

\]

where \(A\dot{x}\) is the actuator volume rate. The piston force is

\[

F=A p-b\dot{x}-F_L,

\]

with load force \(F_L\). Valve flow is nonlinear and may be represented near an operating point by

\[

Q=K_q u-K_c p,

\]

where \(u\) is a valve command, \(K_q\) is flow gain, and \(K_c\) is pressure-flow coefficient. At larger signal amplitudes, square-root pressure-flow relations and saturation must be retained.

Pneumatic systems differ because air is highly compressible and its temperature may change. A chamber model based on mass conservation is

\[

\dot m=\dot m_{in}-\dot m_{out},

\]

with pressure related to mass and temperature through the ideal-gas law,

\[

pV=mRT.

\]

Under an isothermal assumption, \(T\) is constant; under an adiabatic assumption, compression and expansion occur without heat transfer. Real pneumatic systems may lie between these limits. Valve flow is usually choked or unchoked depending on pressure ratio, making the flow relation strongly nonlinear.

Hydraulic models are often dominated by high stiffness and fast pressure dynamics, whereas pneumatic models commonly exhibit compliance, delay, friction, and load-dependent behaviour. These differences influence actuator sizing, stability margins, and controller architecture.

2.9 Linearization and State-Space Representation

Many physical systems are nonlinear. Let

\[

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

\]

Around an equilibrium \((x_0,u_0)\), define perturbations \(\delta x=x-x_0\) and \(\delta u=u-u_0\). First-order Taylor expansion gives

\[

\delta\dot{x}=A\delta x+B\delta u,

\qquad

\delta y=C\delta x+D\delta u,

\]

where

\[

A=\left.\frac{\partial f}{\partial x}\right|_{x_0,u_0},\quad

B=\left.\frac{\partial f}{\partial u}\right|_{x_0,u_0}.

\]

Linearization is local. It does not represent saturation, backlash, dead zones, large-angle motion, valve square-root behaviour, or friction reversal unless these effects are treated separately. A transfer function may be derived from a linear state-space model, but state-space form is generally more informative for multi-input, multi-output, nonlinear, and constrained systems.

2.10 Model Assumptions, Uncertainty, and Validation

Every model contains assumptions. Common assumptions include rigid links, small displacement, constant parameters, negligible leakage, linear damping, no backlash, uniform temperature, and ideal sensors. Assumptions should be recorded because they define the validity range of the model.

Uncertainty may be classified as:

• Parametric uncertainty: imprecise mass, stiffness, resistance, friction, or damping values.
• Structural uncertainty: omitted physical effects such as flexibility, hysteresis, or delay.
• Measurement uncertainty: sensor noise, bias, quantization, and calibration error.
• Operating uncertainty: changing load, temperature, supply pressure, wear, or lubrication.

Model validation compares predicted and measured behaviour under conditions not used exclusively for parameter fitting. Useful tests include step response, frequency response, free-decay testing, static load testing, and energy or power balance. Parameter identification should be designed so that parameters are observable from the selected inputs and outputs. For example, stiffness and mass cannot always be distinguished from a single low-frequency displacement measurement.

A model can be verified mathematically—checking equations, units, limiting cases, and numerical implementation—but validation requires comparison with the real system. Dimensional consistency is a powerful first check. If a term added to force has units of torque, the formulation is incorrect. Limiting-case checks are also valuable: setting damping to zero should recover an undamped model, and setting load torque to zero should produce the expected free response.

2.11 An Illustrative Case: Electro-Hydraulic Positioning Axis

Background

Consider an industrial positioning axis consisting of a servo valve, hydraulic cylinder, moving load, and position sensor. The purpose is to design a preliminary position controller and predict the effect of load changes. This is an illustrative case rather than a report of a particular commercial machine.

Situation and Modeling Challenge

The cylinder must move a load of effective mass \(M\). The valve command \(u\) controls flow, but pressure dynamics, leakage, seal friction, and oil compressibility affect motion. A purely kinematic model would predict position from flow but would fail to predict compliance and force disturbances.

Modeling Approach

Let \(x\) be piston displacement, \(v=\dot{x}\) velocity, and \(p_L\) load pressure. A linearized valve relation is assumed:

\[

Q_L=K_q u-K_c p_L.

\]

The continuity equation for the effective chamber pair is approximated by

\[

\dot p_L=\frac{4\beta_e}{V_t}(Q_L-A v-C_t p_L),

\]

where the factor 4 represents an equivalent symmetric-cylinder approximation, \(V_t\) is total effective fluid volume, and \(C_t\) is leakage coefficient. The mechanical balance is

\[

M\dot v=A p_L-bv-F_f(v)-F_L.

\]

Finally,

\[

\dot x=v.

\]

For small velocity excursions, friction may be linearized as \(F_f(v)\approx b_fv\), producing the state vector \(z=[x,v,p_L]^T\). The model can then be written in state-space form and used to estimate response speed, pressure excursions, and load sensitivity.

Analysis and Solution

The model reveals three interacting energy domains: valve flow supplies fluid compressive energy, pressure produces mechanical force, and the load stores kinetic energy. Increasing \(K_q\) improves command authority but may increase pressure transients. Increasing effective volume reduces pressure rate and therefore slows the axis. Leakage reduces static force capability and changes the low-frequency gain. Friction creates a dead zone and may cause limit cycles under high-gain position control.

A practical design would first identify the operating point, estimate \(\beta_e,V_t,K_q,K_c,C_t,M\), and measure friction. A low-order linear model may be used for initial controller design, followed by nonlinear simulation containing valve saturation, pressure limits, friction, and load variation. Experimental step tests at several loads would then be used for validation.

Outcome and Learning Points

The case demonstrates why a useful actuator model must include both mechanical and fluid states. A model based only on piston geometry cannot predict pressure instability, compliance, or force limitation. It also shows that model order should be selected according to the design decision: a controller may use a reduced model, while safety analysis requires pressure saturation and relief-valve behaviour.

2.12 Professional Practice, Digital Tools, and Emerging Directions

Modeling is increasingly integrated with computer-aided engineering, multibody dynamics, finite-element reduction, system identification, and digital twins. A high-fidelity finite-element model may be reduced to a few dominant modes for control design. Sensor data can update uncertain parameters during operation, provided the identification procedure is safe and the data are sufficiently informative.

Automated symbolic tools can derive Lagrange equations, generate Jacobians, and produce state-space implementations. Numerical solvers support nonlinear simulation, but solver output should not be accepted without checking stiffness, tolerances, discontinuities, and conservation errors. Data-driven and machine-learning models can represent friction, hysteresis, or unmodeled dynamics, but they should complement—not obscure—physical constraints. A learned model that violates conservation laws can produce unsafe control decisions.

Professional practice requires traceability. Assumptions, parameter sources, calibration procedures, software versions, uncertainty bounds, and validation tests should be documented. Safety-critical models should be independently reviewed, subjected to worst-case analysis, and kept separate from unverified experimental code. Ethical engineering requires that model limitations not be concealed when decisions affect personnel safety, equipment integrity, or environmental performance.

Best-Practice Checklist

Before using a model for design or control, the engineer should:

1. Define the modeling purpose, inputs, outputs, and operating range.

2. Draw the physical system and identify energy-storage, conversion, and dissipation elements.

3. Select coordinates that are independent and physically interpretable.

4. State all idealizations and sign conventions.

5. Check dimensions, limiting cases, energy balance, and initial conditions.

6. Compare alternative formulations when practical—for example, Newton–Euler and Lagrange derivations.

7. Estimate parameter uncertainty and examine sensitivity.

8. Validate predictions using independent experimental data.

9. Retain nonlinearities that affect safety, stability, saturation, or performance.

10. Document the model sufficiently for review, reuse, and future updating.

Chapter Summary

Mathematical modeling provides the foundation for analysing and controlling mechanical systems. Lumped-parameter models represent distributed physical behaviour through discrete masses, inertias, springs, dampers, electrical elements, fluid capacitances, and flow restrictions. Newton–Euler equations are direct and physically transparent; work–energy methods emphasize power and energy; Lagrange's equations provide an efficient formulation for generalized coordinates and constrained multibody systems; Hamiltonian mechanics expresses dynamics through coordinates and momenta and is especially useful for energy-based analysis.

Translational and rotational systems are linked through analogous force–acceleration and torque–angular-acceleration relations. Electromechanical models couple voltage, current, torque, and speed. Hydraulic and pneumatic models require continuity, compressibility, pressure-flow, and actuator-force relations. Nonlinear models may be linearized locally for state-space and control analysis, but saturation, friction, backlash, leakage, and changing parameters can limit the validity of linear approximations.

A model is credible only when its assumptions, uncertainty, and validation evidence are clear. Good modeling practice combines physical insight, mathematical consistency, computational tools, experimental data, and professional judgment.

Key Terms

Lumped-parameter model: A model that concentrates physical properties into discrete elements.

Distributed-parameter model: A model in which variables vary continuously in space and time.

State: A minimum set of variables that determines future system behaviour with the input known.

Generalized coordinate: An independent coordinate describing a system configuration.

Constitutive law: An equation relating variables of a physical element.

Newton–Euler formulation: Motion equations obtained from force and moment balances.

Lagrangian: The difference between kinetic and potential energy, \(L=T-V\).

Hamiltonian: An energy-based function expressed in generalized coordinates and momenta.

Generalized force: A force or torque associated with a generalized coordinate.

Rayleigh dissipation function: A function used to represent viscous energy dissipation.

Inertia: Resistance to acceleration, represented by mass or rotational moment of inertia.

Compliance: Ability to deform under force; its reciprocal is stiffness.

Damping: Energy dissipation associated with relative motion.

State-space model: First-order differential representation using state, input, and output matrices.

Linearization: Approximation of a nonlinear model near an equilibrium point.

Parametric uncertainty: Uncertainty in numerical values of model parameters.

Structural uncertainty: Error caused by omitted or incorrectly represented physical effects.

Model validation: Comparison of model predictions with independent physical measurements.

Identifiability: Ability to determine model parameters from available inputs and outputs.

Port-Hamiltonian model: An energy-based model representing storage, dissipation, and power exchange through system ports.

Review Questions

Short Answer Questions

1. What is meant by a lumped-parameter model?

2. Define a generalized coordinate and give one mechanical example.

3. State the Lagrange equation for a system with generalized coordinate \(q_i\).

4. Why is a hydraulic pressure state required in many actuator models?

5. Distinguish parametric uncertainty from structural uncertainty.

Long Answer Questions

1. Derive the equation of motion of a mass–spring–damper system using Newton's second law and explain each term.

2. Explain how Lagrange's method handles constrained mechanical systems more efficiently than direct force balancing.

3. Develop a state-space model for a DC motor with armature inductance, viscous friction, and load torque.

4. Compare hydraulic and pneumatic actuator models with respect to compressibility, flow behaviour, and dynamic response.

5. Explain Hamiltonian mechanics and discuss its relevance to energy-based control.

Analytical / Application Questions

1. A rotational actuator has inertia \(J\), viscous damping \(b\), torsional stiffness \(k_t\), and applied torque \(T(t)\). Formulate its equation of motion, state variables, and undamped natural frequency.

2. A two-mass system is connected by a flexible shaft. Determine the potential energy, write the coupled equations using Lagrange's method, and explain how shaft flexibility affects resonance.

3. An electro-hydraulic axis exhibits poor position accuracy under changing loads. Using the model structure developed in the chapter, identify at least three physical causes and propose experiments to distinguish among them.

22. Multiple Choice Questions with Answers

Question 1

Which statement best describes a lumped-parameter model?

A. It represents every field variable as a continuous function of position.

B. It concentrates relevant physical properties into discrete elements.

C. It excludes all dissipative effects from the system.

D. It can be used only for static systems.

Correct Answer: B. It concentrates relevant physical properties into discrete elements.

Question 2

For a mass–spring–damper system driven by force F(t), which equation is correct?

A. m x + c x_dot + k x = F(t)

B. m x_ddot + c x_dot + k x = F(t)

C. m x_ddot + c x + k x_dot = F(t)

D. m x_dot + c x_ddot + k x = F(t)

Correct Answer: B. m x_ddot + c x_dot + k x = F(t)

Question 3

The Lagrangian of a conservative mechanical system is defined as

A. T + V

B. T - V

C. V - T

D. T/V

Correct Answer: B. T - V

Question 4

Which advantage is most associated with Lagrange's formulation?

A. It directly gives every internal reaction force without further calculation.

B. It eliminates the need to define any coordinates.

C. Ideal constraint forces can often be omitted from the equations.

D. It is valid only for linear systems.

Correct Answer: C. Ideal constraint forces can often be omitted from the equations.

Question 5

In a DC motor model, the term K_e omega in the armature equation represents

A. Copper loss

B. Back electromotive force

C. Mechanical load torque

D. Viscous friction torque

Correct Answer: B. Back electromotive force

Question 6

For a hydraulic chamber, increasing effective bulk modulus while holding other quantities fixed generally

A. Makes pressure respond more rapidly to a flow imbalance.

B. Eliminates leakage completely.

C. Makes the fluid incompressible in every practical situation.

D. Removes the actuator's mechanical inertia.

Correct Answer: A. Makes pressure respond more rapidly to a flow imbalance.

Question 7

Which pair consists of canonical Hamiltonian equations?

A. q_dot = partial H/partial q and p_dot = partial H/partial p

B. q_dot = partial H/partial p and p_dot = -partial H/partial q

C. q_dot = -partial H/partial p and p_dot = partial H/partial q

D. q_dot = Hq and p_dot = Hp

Correct Answer: B. q_dot = partial H/partial p and p_dot = -partial H/partial q

Question 8

Linearization of a nonlinear model is generally valid

A. For all operating points and amplitudes.

B. Only near the selected equilibrium or operating point.

C. Only when damping is zero.

D. Only for distributed-parameter systems.

Correct Answer: B. Only near the selected equilibrium or operating point.

Question 9

Which is an example of structural model uncertainty?

A. The measured mass differs slightly from its catalog value.

B. A sensor contains quantization noise.

C. The model omits backlash that exists in the physical transmission.

D. The resistance changes with temperature within a known model.

Correct Answer: C. The model omits backlash that exists in the physical transmission.

Question 10

A model is best validated by

A. Checking only that its equations compile in simulation software.

B. Comparing predictions with independent measurements from the physical system.

C. Increasing the number of parameters until every test is fitted exactly.

D. Removing all nonlinearities before testing it.

Correct Answer: B. Comparing predictions with independent measurements from the physical system.