Introduction to Linear, Time-Invariant, Dynamic Systems for Students of Engineering

About this book

William L. Hallauer, Jr. develops the modeling and analysis of linear, time-invariant dynamic systems for engineering students. The seventeen chapters connect ordinary differential equations, Laplace transforms, time response, frequency response, mechanical and electrical models, feedback, and stability. The source is the 2016 text and uses MATLAB to calculate and display results. Treat software as a means of testing a reasoned model: identify assumptions, units, initial conditions, and the physical meaning of an output before trusting a graph. The same mathematical form can describe very different devices, but only within the assumptions used to derive it.

Chapter-by-chapter summaries

1. Introduction and first- and second-order examples

Simple dynamic models connect physical laws with differential equations and response plots. The order of an equation reflects the states needed to describe its evolution, not simply its apparent complexity. Separate initial-condition response from forcing response. Trace a mass-spring example from a force balance into a computed history, checking units and signs. MATLAB calculations are most useful when you can predict the broad behaviour before executing them and explain why parameter changes alter the resulting curve.

2. Complex numbers, Laplace transforms, and partial fractions

Complex arithmetic supports oscillation and frequency analysis, while Laplace transforms convert suitable differential equations into algebraic relationships. Initial conditions enter the transformed problem and should not be silently discarded. Partial fractions make inverse transforms interpretable through simpler components. Check a transformed solution by recovering its time behaviour or substituting it into the original equation. A transfer function is a particular input-output description under its initial-condition convention, not an unrestricted replacement for every solution of the differential equation.

3. Mechanical units and first-order transients

Low-order mechanical systems provide concrete models for inertia, damping, and applied forces or torques. Dimensional consistency links parameters with time constants and response magnitudes. Establish coordinates and positive directions before balancing forces. Interpret a first-order time constant as a scale for approach, not a finite instant at which the transient disappears. Compare mathematically similar translational and rotational models while preserving their different physical units and meanings.

4. First-order frequency response

Sinusoidal forcing leads to steady-state amplitude and phase relationships described through complex transfer functions. Frequency response differs from a complete startup history because it omits decayed transients. Compare low- and high-frequency limits and check whether the predicted phase agrees with the physical model. Plot magnitude and phase against frequency, keeping radians per second distinct from cycles per second. The chapter connects a compact complex calculation with observable changes in an output sinusoid.

5. Electrical components and circuits

Resistors, capacitors, inductors, and operational-amplifier models provide electrical realizations of dynamic equations. Circuit laws and constitutive relationships must be combined before a transfer function is interpreted. State voltage polarities and current reference directions consistently. Compare an electrical model with a mechanical analogue while identifying which quantities correspond. Ideal component assumptions simplify analysis but do not establish unlimited real-device performance. Verify limiting behaviour, including long-time DC conditions, as a check on a derived circuit model.

6. Convolution and general first-order response

Convolution assembles a forced response from the system’s impulse-response behaviour over time. Each contribution depends on when an input acts and how long its effect has evolved. Distinguish the integration variable from the observation time and preserve the initial-condition term where needed. Work a simple input analytically, then compare numerical integration. A numerical result depends on time resolution and implementation; agreement with a plot alone is weaker than checking a case with a known solution.

7. Undamped second-order systems

Inertia and stiffness create natural oscillation, with a natural frequency determined by the model parameters. Free and forced responses can coexist. Track amplitude and phase from initial displacement and velocity rather than assuming a universal sinusoid. The ideal undamped model does not lose energy, so persistent motion is an expected property, not a simulation error. Compare this idealization with a measured response and identify which missing physical process would explain decay.

8. Pulses, impulses, and convolution sums

Pulse inputs and the Dirac delta idealization clarify short-duration excitation and impulse response. An impulse is characterized by integrated input, not an ordinary finite peak value at one instant. The initial-value theorem and convolution sums connect transform methods with time-domain computation. Compare narrower pulses of equal area and assess when the impulse model becomes useful. Keep numerical step size separate from the physical pulse duration so the approximation does not change inadvertently during discretization.

9. Damped second-order transients

Damping changes the characteristic roots and the form of free and forced responses. Underdamped, critically damped, and overdamped cases require different interpretations. Damping ratio and natural frequency describe related but distinct features. Estimate parameters from a decaying record while stating model and measurement assumptions. A smooth-looking response is not proof of critical damping, and decay rate alone does not fully specify oscillation. Connect each mathematical parameter with the behaviour it actually controls.

10. Second-order frequency response and beating

Resonance, phase, damping, and suddenly applied sinusoidal excitation explain why steady-state and transient interpretations can differ. Beating reflects interference between oscillatory contributions rather than a new forcing frequency. Compare excitation near and far from natural frequency and distinguish finite damped peaks from ideal undamped resonance. Check whether the frequency-response curve describes amplitude or another normalized quantity before reading a value from it. Interpret startup histories alongside, rather than in place of, steady-state response.

11. Rigid-body translation and rotation

Mechanical systems involving plane translation and rotation require consistent kinematics and multiple balances. Coupling can appear because a force affects both translational and rotational motion. Draw coordinates, constraints, force locations, and moment reference points before deriving equations. An apparently extra degree of freedom may be removed by a genuine constraint, while an unjustified constraint can delete a real mode. Verify dimensions across matrix equations and interpret the physical source of each coupling term.

12. Two-degree-of-freedom vibration modes

Coupled undamped systems can be described through natural frequencies and mode shapes. A mode represents a coordinated pattern of motion, not an independently moving component. Initial conditions determine how modes combine in an observed response. Calculate a simple case and reconstruct physical coordinates from modal contributions. Distinguish normalization of a mode shape from its physical excitation amplitude. The chapter shows why a measurement at one location may fail to reveal every mode equally clearly.

13. Laplace block diagrams

Block diagrams represent input-output relationships and prepare for feedback analysis. Algebraic combinations must preserve signal directions and summing signs. Reduce a diagram step by step and compare its transfer function with equations written directly from the signals. Moving a summing point or branch requires compensating changes; the drawing is an algebraic representation, not merely a flowchart. Keep internal signals visible where possible so a simplified external relationship does not conceal an important model assumption.

14. Introduction to feedback control

Feedback compares or uses measured output to influence input, illustrated through rotational-position control. Closed-loop behaviour depends on the plant, controller, sensor, and sign of the return path. Derive the loop equations before interpreting a denominator. Feedback can improve performance while introducing stability and sensitivity questions. Compare commanded output, actual output, and error as separate signals. A smaller error for one input does not establish satisfactory behaviour for every disturbance or parameter variation.

15. Proportional, integral, and derivative control

Controller terms act on different aspects of error history. Proportional action responds to current error, integration accumulates it, and derivative action relates to its change. Compare how each affects transient response and long-term error in the source’s models. Adding terms does not automatically improve every criterion. State the performance objective and examine poles, response, and implementation assumptions together. Ideal derivative and integral expressions require careful interpretation when measurements or practical limits are introduced.

16. Time-response stability criteria

Characteristic roots determine whether natural responses decay, persist, or grow under the relevant linear model. Stability is distinct from an attractive transient or small tracking error. Analyze the characteristic polynomial and parameter ranges before simulating one case. Root-locus reasoning shows how changes can move a system between qualitatively different regimes. A finite simulation cannot prove indefinite stability, especially when slow growth is present. Connect each conclusion with the definition of stability being applied.

17. Frequency-response stability criteria

Frequency-domain analysis relates open-loop response to closed-loop stability and robustness. Magnitude and phase information must be interpreted together under the method’s assumptions. Margins describe distance from particular critical conditions, not a guarantee against every unmodeled effect. Compare a frequency-domain conclusion with characteristic roots for a tractable example. The final chapter connects earlier response tools with feedback assessment, emphasizing a justified model and careful reading of what a plotted criterion actually establishes.

Study instructions

1. Derive models from balances with a drawing, units, coordinate choices, and stated idealizations.
2. For every response, separate forcing and initial conditions and predict limiting behaviour before calculation.
3. Verify one numerical example per chapter analytically or through an independent formulation.
4. Compare time-domain, transform, and frequency-domain conclusions on the same system, recording the assumptions each uses.

Questions for reflection

• When does a transfer function omit information needed for a complete response?
• Why can a stable system still have undesirable performance?
• How could an observation point hide a vibration mode?

Sources and coverage

This original reading guide follows the linked source edition’s chapter, module, unit, or lesson structure. Consult the source for full arguments, cases, worked examples, exercises, illustrations, and references. Source: 2016 text by William L. Hallauer, Jr. The guide covers all seventeen chapters; appendices and worked problems remain in the source. Read the source edition.