4. General Excitation and Response Methods

Introduction and Context

Mechanical systems are rarely subjected only to idealized static or single-frequency loads. In practice, machines experience start-up transients, impacts, rotating unbalance, gear-mesh forces, road irregularities, seismic motion, pressure fluctuations, and loads measured as irregular time histories. The central problem of this chapter is therefore not merely to determine the natural frequencies of a system, but to calculate its response when the excitation is arbitrary in time.

For a linear single-degree-of-freedom system, the governing equation is

\[

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

\]

where \(m\), \(c\), and \(k\) are mass, damping, and stiffness, respectively, and \(p(t)\) is the applied force. Defining

\[

\omega_n=\sqrt{k/m},\qquad \zeta=\frac{c}{2m\omega_n},\qquad \omega_d=\omega_n\sqrt{1-\zeta^2},

\]

for an underdamped system, the response may be obtained in the time domain, frequency domain, or through numerical integration. These methods are mathematically related, but each has particular advantages in engineering practice.

The same ideas extend to multi-degree-of-freedom systems. If \(\mathbf{x}(t)\) is the displacement vector,

\[

\mathbf{M}\ddot{\mathbf{x}}+\mathbf{C}\dot{\mathbf{x}}+\mathbf{K}\mathbf{x}=\mathbf{p}(t).

\]

The methods developed here provide the foundation for transient analysis, vibration qualification, earthquake engineering, machine diagnostics, structural design, and computational dynamics.

Learning Objectives

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

1. Explain impulse response and its physical significance.

2. Derive and use convolution and Duhamel integral expressions.

3. Represent periodic excitation using Fourier series.

4. Use Fourier transforms and frequency-response functions for arbitrary signals.

5. Apply Laplace-transform methods to transient and discontinuous loading.

6. Interpret response spectra and distinguish them from Fourier spectra.

7. Select and implement suitable numerical time-integration methods.

8. Identify modeling, sampling, damping, and numerical errors in vibration analysis.

9. Apply these methods to practical mechanical and structural systems.

4.1 The Impulse Response as the Fundamental Building Block

Consider a unit impulse applied at \(t=0\). The Dirac impulse \(\delta(t)\) has zero duration, infinite idealized amplitude, and unit area:

\[

\int_{-\infty}^{\infty}\delta(t)\,dt=1.

\]

For an impulse of magnitude \(I\), the equation is

\[

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

\]

Integration through the instant of application gives

\[

m[\dot{x}(0^+)-\dot{x}(0^-)]=I.

\]

Thus displacement remains continuous, while velocity undergoes a jump of \(I/m\). For zero initial conditions and an underdamped system, the displacement response is

\[

h(t)=\frac{1}{m\omega_d}e^{-\zeta\omega_n t}\sin(\omega_d t),\qquad t\geq 0.

\]

This function is the impulse response per unit impulse. If the input is an impulse of magnitude \(I\), the response is \(Ih(t)\).

The impulse response contains the complete dynamic signature of a linear time-invariant system. Once it is known, the response to any force can be constructed by treating the force as a collection of infinitesimal impulses. This principle is called superposition.

For a system with nonzero initial conditions, the total response consists of the free response caused by the initial state and the forced response caused by the applied excitation. In state-space form, with \(\mathbf{z}=[x\ \dot{x}]^T\),

\[

\dot{\mathbf{z}}=\mathbf{A}\mathbf{z}+\mathbf{B}p(t),

\]

and the solution is

\[

\mathbf{z}(t)=e^{\mathbf{A}t}\mathbf{z}(0)+\int_0^t e^{\mathbf{A}(t-\tau)}\mathbf{B}p(\tau)d\tau.

\]

The first term is the homogeneous response; the second is the forced response.

4.2 Convolution and the Duhamel Integral

Divide the applied force into small impulses. The force acting during a small interval \(d\tau\) contributes an impulse \(p(\tau)d\tau\). At a later time \(t\), its contribution is

\[

p(\tau)d\tau\,h(t-\tau).

\]

Summing all contributions gives the convolution integral

\[

x(t)=\int_0^t h(t-\tau)p(\tau)d\tau,

\]

or, explicitly,

\[

x(t)=\frac{1}{m\omega_d}\int_0^t p(\tau)e^{-\zeta\omega_n(t-\tau)}

\sin[\omega_d(t-\tau)]d\tau.

\]

This is the Duhamel integral for force excitation. It is valid for zero initial conditions; the free response must be added when initial displacement or velocity is nonzero.

An equivalent form follows by changing the integration variable to \(s=t-\tau\):

\[

x(t)=\frac{1}{m\omega_d}\int_0^t p(t-s)e^{-\zeta\omega_n s}\sin(\omega_d s)ds.

\]

The two forms emphasize different interpretations. The first views the response as the accumulation of past loads. The second views the response as a weighted average of recent and earlier values of the input. Because the kernel decays with time, heavily damped systems have a shorter effective memory than lightly damped systems.

For base excitation, let \(y(t)\) denote the base displacement and \(z=x-y\) the relative displacement. The equation becomes

\[

m\ddot{z}+c\dot{z}+kz=-m\ddot{y}(t).

\]

Thus ground acceleration acts as an effective force \(-m\ddot{y}\), and the Duhamel integral may be applied directly. In relative coordinates,

\[

z(t)=-\frac{1}{\omega_d}\int_0^t \ddot{y}(\tau)e^{-\zeta\omega_n(t-\tau)}

\sin[\omega_d(t-\tau)]d\tau.

\]

Duhamel integration is particularly useful for recorded earthquake acceleration, measured machine forces, and arbitrary shock pulses. It also provides an exact analytical benchmark for testing numerical algorithms.

Example: Rectangular Pulse

Suppose a force \(P_0\) acts from \(t=0\) to \(t=T\). During the pulse,

\[

x(t)=\frac{P_0}{k}\left[1-e^{-\zeta\omega_n t}

\left(\cos\omega_d t+\frac{\zeta\omega_n}{\omega_d}\sin\omega_d t\right)\right].

\]

After the force is removed, the system vibrates freely from the displacement and velocity reached at \(T\). This example illustrates an important engineering fact: the maximum response may occur after the load has disappeared. A short pulse can therefore produce a large dynamic response even when its static effect is small.

4.3 Periodic Loading and Fourier Series

If \(p(t)\) is periodic with period \(T\), it can be represented, under suitable regularity conditions, by a Fourier series:

\[

p(t)=\frac{a_0}{2}+\sum_{n=1}^{\infty}[a_n\cos(n\Omega t)+b_n\sin(n\Omega t)],

\]

where \(\Omega=2\pi/T\),

\[

a_n=\frac{2}{T}\int_0^T p(t)\cos(n\Omega t)dt,

\qquad

b_n=\frac{2}{T}\int_0^T p(t)\sin(n\Omega t)dt.

\]

A linear system responds to each harmonic independently. For harmonic force \(P_n\cos(\omega t)\), the steady-state response is

\[

x_n(t)=X_n\cos(\omega t-\phi_n),

\]

where

\[

X_n=\frac{P_n/k}{\sqrt{(1-r^2)^2+(2\zeta r)^2}},

\qquad

r=\frac{\omega}{\omega_n},

\]

and

\[

\phi_n=\tan^{-1}\left(\frac{2\zeta r}{1-r^2}\right),

\]

with the quadrant selected correctly. The total steady response is the sum of the responses to all harmonics, including the static component.

Fourier analysis reveals why a non-sinusoidal periodic force may excite resonance even when its fundamental frequency is not close to the natural frequency. A square wave, for example, contains odd harmonics; a pulse train contains many harmonics. Resonance can occur when any significant harmonic approaches a natural frequency.

The transient response is not represented by the steady Fourier series alone. It depends on initial conditions and decays according to system damping. In design, it is important to distinguish the periodic steady state from the start-up transient.

4.4 Fourier Transforms and Frequency-Domain Response

For a nonperiodic signal, the Fourier transform is

\[

P(\omega)=\int_{-\infty}^{\infty}p(t)e^{-i\omega t}dt,

\]

with inverse transform

\[

p(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}P(\omega)e^{i\omega t}d\omega.

\]

Taking the transform of the equation of motion gives

\[

(-m\omega^2+i c\omega+k)X(\omega)=P(\omega).

\]

Therefore,

\[

X(\omega)=H(\omega)P(\omega),

\]

where

\[

H(\omega)=\frac{1}{k-m\omega^2+i c\omega}

\]

is the displacement frequency-response function. Its magnitude and phase are

\[

|H(\omega)|=\frac{1}{\sqrt{(k-m\omega^2)^2+(c\omega)^2}},

\]

and

\[

\angle H(\omega)=-\tan^{-1}\left(\frac{c\omega}{k-m\omega^2}\right),

\]

subject to quadrant correction.

The frequency-domain approach is efficient when the input spectrum is known and the system is linear. It is central to vibration testing, spectral analysis, transfer-path analysis, and operational modal analysis. For random excitation, power spectral densities may be used. If the input force has power spectral density \(S_{pp}(\omega)\), the output displacement spectrum is

\[

S_{xx}(\omega)=|H(\omega)|^2S_{pp}(\omega).

\]

Cross-spectra and coherence functions help determine whether an observed response is causally related to a measured input. However, frequency-domain results do not directly preserve the time sequence of a transient event, and phase information must be retained when reconstructing time histories.

In digital work, the discrete Fourier transform is commonly evaluated using the fast Fourier transform (FFT). Sampling frequency, record length, windowing, leakage, aliasing, and frequency resolution must be considered. The Nyquist criterion requires the sampling frequency to exceed twice the highest frequency of interest, but practical measurements often require a margin and anti-aliasing filters.

4.5 Laplace-Transform Methods

The Laplace transform is particularly convenient for initial-value problems and discontinuous functions. For a function \(f(t)\),

\[

\mathcal{L}\{f(t)\}=F(s)=\int_0^{\infty}f(t)e^{-st}dt.

\]

Using zero initial conditions,

\[

(ms^2+cs+k)X(s)=P(s),

\]

so that

\[

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

\]

The denominator is the characteristic polynomial. Its roots determine the transient behavior. For an underdamped system, the roots are

\[

s_{1,2}=-\zeta\omega_n\pm i\omega_d.

\]

The transfer function is

\[

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

\]

Step, ramp, pulse, and delayed loads can be expressed using standard transforms and the Heaviside function. For example, a force switched on at \(t=a\) is represented by \(P_0u(t-a)\), whose transform is \(P_0e^{-as}/s\). Partial fractions and inverse-transform tables then yield the response.

The Laplace method makes initial conditions explicit and handles discontinuities systematically. It is also the mathematical basis of transfer-function models used in control and simulation software. Its limitations arise when the system is strongly nonlinear, time-varying, or described by data rather than an analytical expression.

4.6 Response Spectra

A response spectrum is a plot of the maximum response of a family of single-degree-of-freedom oscillators subjected to the same transient excitation, plotted against natural period or frequency for a specified damping ratio. For a ground-acceleration record, common spectra include maximum relative displacement, relative velocity, and absolute acceleration.

If \(z(t;T,\zeta)\) is the relative displacement response, the displacement spectrum is

\[

S_d(T,\zeta)=\max_t|z(t;T,\zeta)|.

\]

The pseudo-velocity and pseudo-acceleration spectra are often defined as

\[

S_v=\omega_nS_d,

\qquad

S_a=\omega_n^2S_d.

\]

These quantities are called pseudo-values because they are derived from displacement rather than necessarily being the exact maximum velocity or acceleration. Response spectra are widely used in earthquake-resistant design, shock qualification, and equipment certification. They condense a complete time record into maximum demands, making them useful for design envelopes.

A response spectrum is not the same as a Fourier amplitude spectrum. The Fourier spectrum describes how signal amplitude is distributed over frequency, whereas the response spectrum describes the maximum response of oscillators with different dynamic properties. Two records may have similar Fourier spectra but different response spectra because phase relationships affect the time-domain maxima.

The loss of time information is a major limitation. A response spectrum cannot generally reconstruct the original acceleration record, nor can it determine whether maximum responses at different frequencies occur simultaneously. For multi-component or nonlinear systems, additional information may be required.

4.7 Numerical Time Integration

When loads, properties, or boundary conditions are complicated, direct numerical integration is often the most practical method. The equation is rewritten in first-order state form:

\[

\dot{x}=v,

\qquad

\dot{v}=\frac{p(t)-cv-kx}{m}.

\]

Explicit Runge–Kutta Methods

The fourth-order Runge–Kutta method estimates the state at the next time step using four slopes. It is accurate for smooth problems and simple to implement, but explicit methods may require very small time steps when high frequencies are present. They can become unstable in stiff systems.

Newmark Methods

For structural and mechanical dynamics, the Newmark family is widely used:

\[

x_{n+1}=x_n+\Delta t\,v_n+\Delta t^2[(1/2-\beta)a_n+\beta a_{n+1}],

\]

\[

v_{n+1}=v_n+\Delta t[(1-\gamma)a_n+\gamma a_{n+1}].

\]

The average-acceleration choice \(\beta=1/4,\gamma=1/2\) is unconditionally stable for linear systems, although stability does not guarantee accuracy. The time step must still resolve the important modes and loading changes. Other choices introduce numerical damping, which may suppress high-frequency response but can distort physically important transients.

At each step, the acceleration is obtained from equilibrium. For linear systems, an effective stiffness matrix can be formed and factorized efficiently. For nonlinear systems, Newton iteration is usually required within each time step.

Choosing a Time Step

A time step should be small relative to both the shortest important vibration period and the shortest significant feature of the input. A computation may remain numerically stable while being physically inaccurate if the step is too large. Convergence should be checked by repeating the analysis with a smaller step and comparing peak response, phase, and energy measures.

Numerical damping, algorithmic dispersion, force interpolation, and round-off error all influence results. Sudden load changes should be represented carefully; replacing a sharp pulse by overly smooth interpolation may remove the very frequencies responsible for the response.

4.8 An Applied Case: Shock Loading of a Precision Instrument

Background

Consider an illustrative case involving a precision optical sensor mounted on an industrial vehicle. The sensor and bracket are modeled initially as a single-degree-of-freedom system with mass \(m=5\) kg, natural frequency \(f_n=20\) Hz, and damping ratio \(\zeta=0.05\). The vehicle encounters a short floor discontinuity, producing a measured base-acceleration pulse.

Situation and Problem

The qualification requirement is to determine whether the sensor displacement relative to its mount remains below 2 mm and whether the maximum absolute acceleration threatens optical alignment. A static calculation is inappropriate because the input lasts for a fraction of the natural period.

Analysis and Approach

The measured acceleration record is first checked for offset, sampling adequacy, and filtering. The relative-motion equation is written as

\[

\ddot{z}+2\zeta\omega_n\dot{z}+\omega_n^2z=-\ddot{y}(t).

\]

The response is computed in three ways: by Duhamel integration, by an FFT-based frequency-domain calculation, and by Newmark time integration. Agreement among the methods is used as a verification check. The response spectrum of the pulse is also calculated for damping ratios between 2 and 10 percent to assess sensitivity to uncertain damping.

The Duhamel calculation preserves the time history and identifies the instant of peak displacement. The frequency-domain method shows whether the pulse contains significant energy near 20 Hz. Newmark integration is then extended to a two-degree-of-freedom model containing the sensor and a flexible bracket.

Outcome

The single-degree-of-freedom model predicts a peak relative displacement close to the allowable value, while the bracket model produces a larger response because its second mode lies within a significant part of the pulse spectrum. The analysis indicates that increasing damping alone is less effective than shifting the bracket mode and introducing a compliant isolation layer. A redesigned bracket moves the second mode away from the dominant excitation range and reduces the computed peak acceleration.

Learning Points

The case demonstrates that arbitrary shocks should not be represented only by their peak acceleration. Duration, frequency content, damping, modal participation, and phase all affect response. It also illustrates the value of using independent solution methods and of analyzing model uncertainty rather than relying on a single nominal result.

4.9 Practical Limitations, Risks, and Professional Judgment

These methods rely strongly on linearity and time invariance. Large clearances, friction, plastic deformation, contact, nonlinear damping, and changing stiffness can invalidate convolution and ordinary frequency-response calculations. In such cases, numerical integration of the nonlinear equations is required, and superposition no longer applies.

Common errors include confusing force and base excitation, omitting initial conditions, using a steady-state formula for a transient problem, applying an incorrect phase convention, and interpreting a response spectrum as an input spectrum. In experimental work, inadequate sampling, sensor resonance, poor synchronization, drift, and aliasing can produce plausible but incorrect conclusions.

Model validation is therefore essential. Measured natural frequencies, damping, mode shapes, and transfer functions should be compared with predictions. Units and sign conventions should be checked, and peak values should be distinguished from root-mean-square values. For safety-critical equipment, independent review, traceable data processing, and documented acceptance criteria are professional requirements.

Digital tools now automate FFT processing, modal superposition, response-spectrum generation, and time integration. Automation improves repeatability but does not remove the need for engineering judgment. Machine-learning methods may assist in anomaly detection or surrogate modeling, yet extrapolation outside the training data and lack of physical interpretability remain concerns. Physical governing equations, uncertainty bounds, and validation measurements should remain central to engineering decisions.

Best-Practice Workflow

A reliable arbitrary-load analysis usually follows this sequence:

1. Define coordinates, excitation type, damping model, and required response quantity.

2. Establish natural frequencies and modes before applying the load.

3. Inspect and condition the input record without removing physically meaningful content.

4. Select the method: Duhamel or direct integration for transients, Fourier methods for periodic or spectral analysis, and response spectra for maximum oscillator demands.

5. Check time-step, frequency-resolution, and sampling adequacy.

6. Compare with an independent method or limiting analytical case.

7. Perform sensitivity studies for damping, stiffness, mass, and uncertain input.

8. Report assumptions, sign conventions, units, peak definitions, and validation evidence.

Chapter Summary

Arbitrary and periodic excitation can be analyzed through several equivalent descriptions of linear system dynamics. The impulse response is the fundamental kernel from which convolution and Duhamel integrals construct the response to any force history. Fourier series resolve periodic loads into harmonics, while Fourier transforms relate arbitrary input spectra to output spectra through the frequency-response function. Laplace transforms are especially effective for initial-value problems and discontinuous loads. Response spectra summarize the maximum response of oscillator families and are valuable for shock and seismic design, although they do not preserve complete time-history information. Numerical methods such as Runge–Kutta and Newmark integration are indispensable for complicated loading, multi-degree-of-freedom systems, and nonlinear behavior. Correct method selection, signal conditioning, convergence checking, validation, and professional interpretation are as important as the equations themselves.

Key Terms

Impulse response: Response of a system to a unit impulsive input.

Dirac impulse: Idealized zero-duration input having unit area.

Convolution: Integral combining an input history with the impulse response.

Duhamel integral: Time-domain expression for the response of a linear system to arbitrary excitation.

Superposition: Property that permits responses to separate inputs to be added in a linear system.

Fourier series: Expansion of a periodic function into sinusoidal harmonics.

Fourier transform: Representation of a nonperiodic time signal in the frequency domain.

Frequency-response function: Complex ratio of steady harmonic output to harmonic input.

Transfer function: Input-output relation expressed in the Laplace domain.

Laplace transform: Integral transform useful for differential equations and initial-value problems.

Response spectrum: Maximum response of a set of oscillators to a specified excitation.

Pseudo-acceleration: Natural-frequency-squared times maximum relative displacement.

Power spectral density: Distribution of mean-square signal content over frequency.

Aliasing: Misrepresentation of high-frequency content caused by insufficient sampling.

Nyquist frequency: One-half of the sampling frequency.

Numerical damping: Artificial dissipation introduced by a time-integration algorithm.

Newmark method: Family of time-stepping algorithms for dynamic equilibrium equations.

Modal superposition: Construction of system response from individual modal responses.

Transient response: Time-dependent response associated with start-up, interruption, or nonsteady excitation.

Steady-state response: Response remaining after transients have decayed under persistent excitation.

Review Questions

Short Answer Questions

1. What physical change occurs in velocity when a finite impulse is applied to a mass?

2. State the Duhamel integral for a damped single-degree-of-freedom system.

3. Why can a harmonic of a periodic load cause resonance?

4. Distinguish a Fourier amplitude spectrum from a response spectrum.

5. What is the purpose of an anti-aliasing filter?

Long Answer Questions

1. Derive the impulse response of an underdamped single-degree-of-freedom system and explain its physical meaning.

2. Explain how convolution is developed by representing an arbitrary force as a sequence of infinitesimal impulses.

3. Describe the Fourier-series method for calculating the steady response to a periodic nonsinusoidal force.

4. Explain the use of Laplace transforms for step, pulse, and delayed loading.

5. Discuss the assumptions, advantages, and limitations of Newmark time integration.

Analytical / Application Questions

1. A lightly damped machine is subjected to a rectangular force pulse whose duration is close to one-half of its natural period. Explain why the maximum displacement may occur after the pulse ends and propose a suitable analysis method.

2. An acceleration record is sampled at a rate only slightly above twice the highest expected frequency. Identify the risks and recommend a defensible signal-processing procedure.

3. Two shock records have similar Fourier amplitude spectra but produce different maximum responses in a test oscillator. Explain how this can occur and identify which additional information should be examined.

22. Multiple Choice Questions with Answers

Question 1

For a zero-initial-condition underdamped SDOF system, which function is convolved with the applied force to obtain displacement?

A. The static flexibility only

B. The unit impulse response

C. The natural frequency

D. The damping ratio

Correct Answer: B. The unit impulse response

Question 2

In the Duhamel integral, the contribution of a force element p(τ)dτ at time t is weighted by which quantity?

A. h(t−τ)

B. h(t+τ)

C. p(t−τ) only

D. The static displacement only

Correct Answer: A. h(t−τ)

Question 3

A periodic square-wave load can excite resonance when:

A. Only its mean value equals the natural frequency

B. Any significant harmonic approaches a natural frequency

C. Its peak value is zero

D. Its period is independent of the system

Correct Answer: B. Any significant harmonic approaches a natural frequency

Question 4

For a linear system in the frequency domain, the output spectrum is related to the input by:

A. X(ω)=P(ω)/H(ω)

B. X(ω)=H(ω)P(ω)

C. X(ω)=H(ω)+P(ω)

D. X(ω)=H(ω)−P(ω)

Correct Answer: B. X(ω)=H(ω)P(ω)

Question 5

Which method is especially convenient for delayed step and pulse loads?

A. Laplace transform

B. Fourier series only

C. Static equilibrium alone

D. Dimensional analysis

Correct Answer: A. Laplace transform

Question 6

A response spectrum primarily provides:

A. The complete phase history of an input

B. The maximum response of oscillators over natural periods

C. Only the mean value of an excitation

D. The exact original time record

Correct Answer: B. The maximum response of oscillators over natural periods

Question 7

If the sampling frequency is inadequate, high-frequency components may appear at incorrect lower frequencies. This phenomenon is:

A. Modal coupling

B. Aliasing

C. Numerical damping

D. Hysteresis

Correct Answer: B. Aliasing

Question 8

For a linear system, the average-acceleration Newmark parameters are:

A. β=0, γ=0

B. β=1/4, γ=1/2

C. β=1/2, γ=1

D. β=1, γ=1/4

Correct Answer: B. β=1/4, γ=1/2

Question 9

For base excitation, the effective force in the relative-coordinate equation is:

A. mẍ only

B. −mÿ(t)

C. kẏ(t)

D. cÿ(t)

Correct Answer: B. −mÿ(t)

Question 10

Which statement correctly describes numerical stability?

A. A stable calculation is always accurate

B. Stability prevents unbounded numerical growth but does not ensure adequate accuracy

C. Stability eliminates the need for convergence checks

D. Stability is unrelated to time-step size

Correct Answer: B. Stability prevents unbounded numerical growth but does not ensure adequate accuracy