Laplace, Fourier and Frequency-Domain Methods

Introduction

Many machine-technology systems are governed by differential equations. A rotating shaft accelerates after a motor is switched on, a cutting tool vibrates under periodic excitation, and a control system responds to changes in position or speed. Solving these problems entirely in the time domain can be difficult, especially when inputs are discontinuous or periodic. Transform methods provide an alternative: they convert differentiation and convolution into algebraic multiplication, allowing system behaviour to be analysed through poles, zeros, spectra and frequency response.

This chapter develops three closely related tools:

• the Laplace transform for initial-value problems and transient response;
• Fourier series for periodic signals and vibration harmonics; and
• the Fourier transform for non-periodic signals and continuous frequency spectra.

These methods are central to signal processing, vibration diagnosis, control engineering, system identification and stability analysis. Their usefulness depends on understanding not only how to calculate transforms, but also when the underlying assumptions are valid.

Learning objectives

By the end of this chapter, you should be able to:

1. Define and calculate basic Laplace transforms and inverse transforms.
2. Use operational properties to solve linear differential equations with initial conditions.
3. Interpret poles, zeros and transfer functions in engineering systems.
4. Construct Fourier series for periodic waveforms.
5. Explain the meaning of amplitude, phase and power spectra.
6. Apply Fourier transforms to non-periodic signals and frequency response.
7. Analyse vibration and control applications, including resonance and stability.
8. Recognise sampling, noise, modelling and interpretation limitations.

1. Transform methods and linear systems

A transform maps a function from one mathematical domain to another. For a time signal x(t), the time domain describes how the signal changes with time. The frequency domain describes the sinusoidal components required to construct that signal.

For a linear time-invariant (LTI) system, the relationship between input x(t) and output y(t) can often be represented by a differential equation. Linearity means that the response to a sum of inputs is the sum of the individual responses. Time invariance means that delaying the input delays the output by the same amount. These properties make transform methods particularly powerful.

The response of an LTI system is commonly separated into:

• zero-input response, caused by stored initial energy;
• zero-state response, caused by the applied input; and
• total response, which is the sum of the two.

In mechanical systems, stored energy may occur in a moving mass or a deflected spring. In electrical and control systems, it may occur in inductors, capacitors or integrators.

2. The Laplace transform

The one-sided, or unilateral, Laplace transform of f(t) is

where t is time, s = σ + jω is the complex transform variable, σ is the real part, ω is angular frequency in rad/s, and j = √(−1). The lower limit 0⁻ is useful when signals include impulses at t = 0. For ordinary initial-value problems, 0 is usually sufficient.

The transform exists when the integral converges. A sufficient engineering condition is that f(t) be piecewise continuous and of exponential order. The region of convergence (ROC) specifies the values of s for which the integral is finite. In causal engineering systems, the ROC is commonly a right half-plane to the right of the rightmost pole.

Basic transform pairs

Here δ(t) is the Dirac impulse and ω₀ is an angular frequency. The dimensions of s are inverse time, so transform expressions must be interpreted with the units of the original function included.

Important properties include linearity,

where a and b are constants, and differentiation,

For the second derivative,

The derivative formulas are especially important because they incorporate initial conditions directly. The assumptions are that the indicated derivatives exist piecewise and that the transforms converge.

Time shifting and convolution

If u(t−a) is the unit-step function delayed by a seconds, then

A delay in time therefore becomes multiplication by e^(−as) in the s-domain.

The convolution of f and g is

and the convolution theorem states

For an LTI system with impulse response h(t), the output is y(t) = h(t) ∗ x(t), so Y(s) = H(s)X(s). This is the basis of transfer-function analysis.

3. Inverse Laplace transforms and partial fractions

The inverse transform recovers f(t) from F(s). In elementary engineering problems, partial fractions are often the most efficient method. For example, consider

Write

Multiplication by s(s + 2) gives 5 = A(s + 2) + Bs. Setting s = 0 gives A = 5/2. Setting s = −2 gives B = −5/2. Therefore,

and

The pole at s = 0 produces a constant final component, while the pole at s = −2 produces a decaying transient.

The final-value theorem is

provided all poles of sF(s) lie in the open left half-plane, except possibly a simple pole at the origin. The theorem must not be applied to unstable or sustained oscillatory systems without checking this condition.

The initial-value theorem is

when the required limit exists and impulses or discontinuities are treated correctly.

4. Differential equations and transfer functions

Consider a mass–spring–damper system with mass m, damping coefficient c and stiffness k. Newton’s law gives

where x(t) is displacement in metres and F(t) is force in newtons. The units are consistent: mẍ, cẋ and kx all have units of newtons.

Taking the Laplace transform with zero initial conditions gives

Thus the displacement transfer function is

The poles are the roots of ms² + cs + k = 0. For m > 0, c > 0 and k > 0, both poles have negative real parts, so the free response decays and the system is asymptotically stable.

It is often useful to define the natural frequency and damping ratio:

where ω_n is in rad/s and ζ is dimensionless. The denominator can then be written as

The cases are:

• 0 < ζ < 1: underdamped, with decaying oscillations;
• ζ = 1: critically damped, with the fastest non-oscillatory response;
• ζ > 1: overdamped, with two decaying real modes; and
• ζ = 0: undamped, with sustained oscillation in the ideal model.

Worked example: step response

A unit-step force has transform F(s) = 1/s. Let m = 1 kg, c = 4 N·s/m and k = 3 N/m. Then

Partial fractions give

Therefore,

At t = 0, x(0) = 1/3 − 1/2 + 1/6 = 0, as expected for zero initial displacement. The final displacement is 1/3 m, which agrees with the static value F₀/k = 1/3 m for a unit force.

5. Poles, zeros and stability

A rational transfer function is commonly written as

where K is a gain, zᵢ are zeros and pᵢ are poles. Poles determine natural modes and strongly influence stability. For a continuous-time LTI system, asymptotic stability requires every pole to have a strictly negative real part. A pole on the imaginary axis may represent sustained oscillation and is not asymptotically stable; a pole in the right half-plane indicates instability.

For higher-order characteristic polynomials, the Routh–Hurwitz criterion can determine stability without explicitly calculating all roots. In feedback control, the closed-loop transfer function for forward path G(s) and feedback H(s) is

for negative feedback. The characteristic equation is 1 + G(s)H(s) = 0. Sign conventions must be stated because positive feedback changes the denominator.

6. Fourier series for periodic signals

A periodic function f(t) with period T satisfies f(t + T) = f(t). Its fundamental angular frequency is

The trigonometric Fourier series is

with coefficients

and

The integration may be performed over any complete period, assuming the signal is integrable. For an even function, all b_n are zero. For an odd function, a₀ and all a_n are zero.

The complex form is

where

The complex form is compact and useful in systems and spectral analysis.

Example: square wave

Consider an odd square wave of amplitude A, with f(t) = A for 0 < t < T/2 and f(t) = −A for −T/2 < t < 0. Its average is zero and, because it is odd, only sine terms occur. The result is

Only odd harmonics are present, and their amplitudes decrease as 1/n. At a jump discontinuity, the Fourier series converges to the midpoint of the jump. Near a discontinuity, finite sums exhibit the Gibbs phenomenon: oscillations remain even as more terms are added, although their width decreases.

Parseval’s relation connects time-domain mean-square value and Fourier coefficients:

This relation is important when comparing vibration or electrical signal power across harmonics.

7. Fourier transforms and spectra

For a suitably integrable, non-periodic signal x(t), the Fourier transform is

and the inverse transform is

Here ω is angular frequency in rad/s. If frequency f in hertz is used, then ω = 2πf. The transform convention must be kept consistent because alternative conventions distribute the factor 2π differently.

The magnitude |X(ω)| indicates the strength of frequency components, while arg X(ω) indicates phase. For real signals, X(−ω) = X*(ω), so the magnitude spectrum is even and the phase has corresponding symmetry.

Important properties include:

• time differentiation: 𝓕{dx/dt} = jωX(ω);
• time shift: 𝓕{x(t−t₀)} = e^(−jωt₀)X(ω);
• convolution: 𝓕{x∗h} = X(ω)H(ω); and
• Parseval’s relation: ∫|x(t)|²dt = (1/2π)∫|X(ω)|²dω.

The Fourier transform of an LTI system’s impulse response h(t) is its frequency response H(jω). For sinusoidal input at frequency ω, the steady-state output has amplitude multiplied by |H(jω)| and phase shifted by arg H(jω). This describes filtering and resonance directly.

8. Frequency response and vibration analysis

For the mass–spring–damper system,

The displacement magnitude per unit force is

and the phase, using the correct quadrant, is

At low frequency, stiffness dominates and |H(jω)| is approximately 1/k. Near the natural frequency, the response may become large if damping is small. At high frequency, inertia dominates and the magnitude decreases approximately as 1/(mω²).

A vibration spectrum can reveal:

• shaft rotational frequency and its harmonics;
• bearing defect frequencies;
• gear-mesh frequency;
• structural natural frequencies; and
• broadband energy associated with friction, turbulence or damage.

The spectrum alone does not prove causation. Phase, operating speed, sensor location, load condition and time-domain evidence should also be considered.

9. Illustrative case: diagnosing a machine vibration

Background

This illustrative case concerns a belt-driven machine tool whose operator reports increasing vibration during operation. An accelerometer is mounted near the drive-end bearing, and measurements are collected at a sampling rate of 10 kHz.

Problem

The shaft rotates at 1,800 revolutions per minute. The maintenance team must determine whether the dominant vibration is caused by imbalance, a structural resonance or a bearing defect.

Analysis

The rotational frequency is

A discrete Fourier transform of the acceleration record shows a strong peak at 30 Hz and smaller peaks at 60 Hz and 90 Hz. A run-up test, in which speed is gradually increased, shows that the 30 Hz component grows sharply when the rotational frequency approaches 31 Hz. The phase changes rapidly through this region, indicating resonance. Measurements at different locations show a large response near the bearing housing but not at the motor casing.

The 30 Hz peak is consistent with once-per-revolution excitation, commonly associated with imbalance or eccentricity. The rapid amplitude increase near 31 Hz indicates that the machine structure has a natural frequency close to this operating condition. The harmonics may arise from nonlinearities, misalignment or waveform distortion.

Approach and solution

The team balances the rotating assembly, checks shaft alignment and verifies bearing seating. A modal test estimates the structural natural frequency. A small increase in support stiffness shifts the resonance above the normal operating range. After the repair, the 30 Hz peak falls substantially and the run-up test shows no excessive amplification at normal speed.

Outcome

The evidence supports rotational excitation amplified by structural resonance rather than a bearing defect as the primary cause. Continued monitoring is recommended because a future increase in harmonics or broadband energy may indicate a different fault mechanism.

Learning points

1. Convert rotational speed to frequency before interpreting a spectrum.
2. A spectral peak identifies a frequency component, not automatically its cause.
3. Resonance is identified by the relationship between excitation frequency, amplitude and phase.
4. Sampling rate, anti-alias filtering, windowing and measurement location affect results.
5. Maintenance decisions should combine spectral data with physical inspection and operating context.

10. Sampling and digital implementation

Digital instruments do not measure a continuous spectrum directly. If a signal is sampled at frequency f_s samples per second, the Nyquist frequency is f_s/2. Components above this frequency can be misrepresented as lower frequencies through aliasing. An anti-alias low-pass filter should therefore attenuate frequencies above the usable band before sampling.

For N equally spaced samples collected at sampling frequency f_s, the record duration is approximately N/f_s, and the frequency-bin spacing is

Longer records improve frequency resolution but may reduce the assumption that the signal is stationary. Window functions reduce leakage when the record does not contain an integer number of cycles, but they broaden peaks and alter amplitude estimates. Amplitude correction and consistent measurement procedures are therefore necessary.

The discrete Fourier transform (DFT) is

where x[n] is the sampled sequence and k = 0, 1, …, N − 1. The fast Fourier transform (FFT) is an efficient algorithm for computing the DFT; it is not a different mathematical transform.

11. Limitations, professional practice and responsible use

Transform methods are powerful but model-dependent. Important limitations include:

• linearity may fail because of backlash, saturation, friction or contact impacts;
• system parameters may vary with temperature, load and wear;
• finite records produce leakage and limited frequency resolution;
• noise can obscure weak components;
• transfer functions may be valid only around a particular operating point; and
• numerical differentiation amplifies high-frequency noise.

Best practice includes documenting sensor type, calibration, sampling rate, filter settings, window, units, operating speed and data-processing software. Engineers should preserve raw data and distinguish measured results from model predictions. Automated or AI-based fault classifiers should be validated against labelled operating conditions, monitored for distribution changes and used with human engineering review. A false alarm can cause unnecessary downtime, whereas a missed fault can create safety risks. Data governance should also protect production information and ensure traceability of maintenance decisions.

Chapter Summary

Laplace transforms convert time-domain differential equations into algebraic equations and incorporate initial conditions through derivative properties. Transfer functions express the input–output behaviour of LTI systems, while poles determine natural modes and stability. Mass–spring–damper systems illustrate transient response, damping, resonance and the relationship between physical parameters and pole locations.

Fourier series represent periodic signals as sums of harmonically related sinusoids. Symmetry simplifies coefficient calculations, and Parseval’s relation connects harmonic content with mean-square energy. Fourier transforms extend spectral analysis to non-periodic signals. The frequency response H(jω) gives the amplitude and phase changes imposed by an LTI system.

In machine technology, these methods support vibration diagnosis, condition monitoring, filter design and control analysis. Reliable interpretation requires attention to sampling, aliasing, leakage, sensor placement, model assumptions and physical evidence. Transform analysis is therefore both a mathematical technique and an engineering reasoning process.

Key Terms

Laplace transform: Integral transform that maps a time function into a complex s-domain function.

Region of convergence: Set of complex s values for which a Laplace transform exists.

Transfer function: Ratio of output transform to input transform under zero initial conditions.

Pole: Value of the transform variable that makes a transfer-function denominator zero.

Zero: Value of the transform variable that makes a transfer-function numerator zero.

Stability: Property describing whether system responses remain bounded or decay appropriately.

Convolution: Integral operation describing the output of an LTI system from its impulse response and input.

Impulse response: Output of a system produced by a unit impulse input.

Fourier series: Representation of a periodic function as harmonically related sinusoids.

Fundamental frequency: Lowest non-zero frequency of a periodic signal.

Harmonic: Integer multiple of a fundamental frequency.

Fourier transform: Representation of a non-periodic signal in the continuous frequency domain.

Spectrum: Distribution of signal magnitude, phase or power over frequency.

Frequency response: Complex steady-state response of an LTI system as a function of frequency.

Resonance: Large response produced when excitation approaches a system natural frequency.

Damping ratio: Dimensionless measure of damping relative to critical damping.

Aliasing: Misrepresentation of high-frequency content as lower-frequency content after sampling.

Nyquist frequency: One-half of the sampling frequency.

Spectral leakage: Spreading of energy between frequency bins caused by finite or non-periodic records.

Fast Fourier transform: Efficient algorithm for computing the discrete Fourier transform.

Review Questions

Short Answer Questions

1. What is the principal advantage of applying a Laplace transform to a linear differential equation with initial conditions?
2. Define a transfer function and state the initial-condition assumption used in its definition.
3. What is the relationship between a periodic signal’s period T and its fundamental angular frequency ω₀?
4. Distinguish between a Fourier magnitude spectrum and a phase spectrum.
5. Why is an anti-alias filter used before analog-to-digital conversion?

Long Answer Questions

1. Derive the transfer function of a mass–spring–damper system and explain how m, c and k affect its poles.
2. Explain the Fourier-series coefficient formulas and show how even and odd symmetry simplifies them.
3. Compare Laplace transforms, Fourier series and Fourier transforms in terms of signal type, purpose and engineering application.
4. Explain the meaning of poles and zeros in a feedback-control system and discuss their relationship to stability.
5. Describe spectral leakage, windowing, frequency resolution and the trade-offs involved in digital vibration measurement.

Analytical / Application Questions

1. A machine shaft rotates at 2,400 rpm. Determine its rotational frequency in hertz and identify the frequencies of its second and third harmonics. Explain how these components might appear in an acceleration spectrum.
2. For a system with transfer function H(s)=10/(s²+4s+25), determine the natural frequency and damping ratio, and classify the system as underdamped, critically damped or overdamped.
3. A vibration sensor records 4,096 samples at 8,192 samples/s. Calculate the record duration and frequency-bin spacing. Discuss one advantage and one limitation of increasing the record length while keeping the sampling rate fixed.

Multiple Choice Questions with Answers

Question 1

For a causal stable continuous-time system, where must all poles lie?

A. In the open left half-plane

B. On the imaginary axis only

C. In the open right half-plane

D. At the origin only

Correct Answer: A. In the open left half-plane

Question 2

What is the Laplace transform of e^(at), under its usual convergence condition?

A. 1/(s+a)

B. 1/(s−a)

C. s/(s²+a²)

D. a/(s²+a²)

Correct Answer: B. 1/(s−a)

Question 3

A mass–spring–damper system has m = 1 kg, c = 4 N·s/m and k = 3 N/m. What is its static displacement under a 1 N constant force?

A. 1/4 m

B. 1/3 m

C. 3 m

D. 4 m

Correct Answer: B. 1/3 m

Question 4

For an odd periodic function, which Fourier coefficients are necessarily zero?

A. Only b_n

B. a₀ and a_n

C. Only a₀

D. All coefficients

Correct Answer: B. a₀ and a_n

Question 5

The fundamental frequency of a periodic signal with period 0.02 s is:

A. 0.02 Hz

B. 20 Hz

C. 50 Hz

D. 100 Hz

Correct Answer: C. 50 Hz

Question 6

What does the magnitude of a frequency response principally describe?

A. The system’s initial displacement

B. The amplitude ratio at each frequency

C. The sampling interval

D. The number of poles only

Correct Answer: B. The amplitude ratio at each frequency

Question 7

A shaft rotates at 1,800 rpm. Its rotational frequency is:

A. 18 Hz

B. 30 Hz

C. 60 Hz

D. 1,800 Hz

Correct Answer: B. 30 Hz

Question 8

If a signal is sampled at 10 kHz, its Nyquist frequency is:

A. 2.5 kHz

B. 5 kHz

C. 10 kHz

D. 20 kHz

Correct Answer: B. 5 kHz

Question 9

For N = 4,096 samples acquired at f_s = 8,192 samples/s, the DFT frequency-bin spacing is:

A. 0.5 Hz

B. 2 Hz

C. 4 Hz

D. 8,192 Hz

Correct Answer: B. 2 Hz

Question 10

The FFT is best described as:

A. A different physical law from Fourier analysis

B. An efficient algorithm for computing the DFT

C. A method that eliminates aliasing automatically

D. A replacement for sensor calibration

Correct Answer: B. An efficient algorithm for computing the DFT