Response to arbitrary excitation: Duhamel integral

Impulse response, the Duhamel convolution integral, step and pulse responses, dynamic load factor and the link to indicial aerodynamics.

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Why it matters

Many loads on an aircraft are neither harmonic nor steady: a landing impact, a discrete gust, a store ejection, a bird strike, a hard control input. The structure's response to such a load depends on how its duration compares with the natural period, and the peak can be up to twice the static deflection. The Duhamel (convolution) integral gives the response of a linear system to any force history, and it is the basis of shock-response spectra and of gust response through indicial aerodynamic functions.

Key ideas

Impulse response. A short impulse Î = ∫F·dt (N·s) applied to a mass at rest changes its momentum without moving it, so the motion starts with x(0) = 0 and ẋ(0) = Î/m. The resulting free vibration of a damped SDOF system is ηh(t), where the unit impulse response function is

h(t) = (1/(m·ω_d))·e^(−ζ·ω_n·t)·sin(ω_d·t), for t ≥ 0 (and h = 0 for t < 0)

For an undamped system h(t) = sin(ω_n·t)/(m·ω_n). Note h has units of m/(N·s).

Building any load from impulses. Split an arbitrary force F(τ) into a sequence of impulses F(τ)·dτ. Each produces F(τ)·dτ·h(t − τ) at a later time t. For a linear system the responses superpose, so

x(t) = ∫₀ᵗ F(τ)·h(t − τ)·dτ

This is the Duhamel integral, a convolution of the force with the impulse response. It assumes the system starts at rest; non-zero initial conditions are handled by adding the free-vibration response to x₀ and v₀ (previous topics) to the integral.

Requirements. Linearity (superposition) and time invariance (constant m, c, k). It is exact for these, but it cannot be used directly for non-linear systems such as a gear strut with a non-linear oleo; those are integrated numerically step by step (Newmark, Runge–Kutta).

Classical results (undamped).

  • Suddenly applied constant force F₀ (a step): x(t) = (F₀/k)·(1 − cos ω_n·t). The peak is 2F₀/k: a suddenly applied load doubles the static deflection, the dynamic load factor (DLF) is 2.
  • Rectangular pulse of height F₀ and duration t_d: during the pulse, the step response; after it, free vibration with amplitude 2·(F₀/k)·|sin(ω_n·t_d/2)|. Short pulses (t_d much less than T_n) act like an impulse F₀·t_d; long pulses (t_d > T_n/2) give DLF = 2.
  • Ramp force rising over a long time compared with T_n gives a nearly static response (DLF close to 1). A load is "quasi-static" if it rises over several natural periods.

Shock response spectrum. Plotting the peak response (or DLF) against t_d/T_n, or against the natural frequency for a given pulse, gives the shock spectrum used to qualify equipment.

Link to aeroelasticity. Unsteady aerodynamic lift after a step change in angle of attack (Wagner function) or on entering a sharp-edged gust (Küssner function) builds up over a few chord lengths of travel. The lift for an arbitrary motion or gust is obtained by the same superposition, a Duhamel integral over the indicial function, and then fed into the structural equations. This is the basis of the gust-response topic.

Frequency-domain equivalent. Convolution in time is multiplication in frequency: X(ω) = H(ω)·F(ω), where H(ω) is the frequency-response function. Both routes give the same answer for linear systems.

Formulas

h(t) = e^(−ζ·ω_n·t)·sin(ω_d·t)/(m·ω_d) — unit impulse response (m/(N·s)), t ≥ 0.

x(t) = ∫₀ᵗ F(τ)·h(t − τ)·dτ — Duhamel integral; F force (N), τ time of each impulse (s); system initially at rest.

x(t) = (1/(m·ω_n))·∫₀ᵗ F(τ)·sin(ω_n·(t − τ))·dτ — undamped form.

x(t) = (Î/(m·ω_n))·sin(ω_n·t) — response to an impulse Î (N·s), undamped.

x(t) = (F₀/k)·(1 − cos ω_n·t) — suddenly applied constant force F₀ (N), undamped; peak 2F₀/k.

X_res = 2·(F₀/k)·|sin(ω_n·t_d/2)| — amplitude after a rectangular pulse of duration t_d (s), undamped.

DLF = x_max/(F₀/k) — dynamic load factor (dimensionless).

Worked examples

Example 1 (standard). A 2 kg mass on a spring k = 800 N/m (undamped) is struck by a hammer with impulse Î = 4 N·s. Find the maximum displacement.

  1. ω_n = √(k/m) = √400 = 20 rad/s.
  2. Initial velocity Î/m = 4/2 = 2 m/s, x(0) = 0.
  3. x(t) = (Î/(m·ω_n))·sin(ω_n·t) = (4/(2 × 20))·sin(20t) = 0.10·sin(20t) m.

Answer: x_max = 0.10 m (100 mm), reached at t = π/(2ω_n) = 0.0785 s.

Example 2 (GATE level). The same system (m = 2 kg, k = 800 N/m, undamped) is loaded by a rectangular pulse F₀ = 10 N lasting t_d = 0.10 s. Find the maximum displacement and the DLF.

  1. X_st = F₀/k = 10/800 = 12.5 mm. T_n = 2π/20 = 0.314 s, so t_d/T_n = 0.318.
  2. During the pulse x = X_st·(1 − cos ω_n·t). Its first peak would be at ω_n·t = π, t = 0.157 s, which is after the pulse ends; so during the pulse x rises to x(t_d) = 12.5 × (1 − cos 2.0) = 12.5 × 1.416 = 17.7 mm.
  3. After the pulse: X_res = 2·X_st·|sin(ω_n·t_d/2)| = 2 × 12.5 × sin(1.0 rad) = 25 × 0.8415 = 21.0 mm.
  4. The largest of the two is the residual free vibration amplitude.

Answer: x_max ≈ 21.0 mm, DLF = 21.0/12.5 ≈ 1.68. (A direct numerical evaluation of the Duhamel integral gives the same 21.0 mm.)

Common mistakes

  • Writing h(τ)·F(τ) instead of h(t − τ)·F(τ): the impulse response must be evaluated at the time elapsed since each impulse. (The swapped form ∫₀ᵗ F(t − τ)·h(τ)·dτ is also valid.)
  • Forgetting the 1/(m·ω_d) factor, which gives the wrong units and magnitude.
  • Treating a suddenly applied load as static: the peak is twice the static value for an undamped system.
  • Looking for the peak only during the pulse; for short pulses it occurs afterwards, in free vibration.
  • Applying the Duhamel integral to non-linear systems or adding non-zero initial conditions incorrectly.
  • Working in degrees inside sin(ω_n·t): ω_n·t is in radians.

For GATE AE

Expect response to an impulse (initial velocity Î/m), the step response and DLF = 2, residual vibration after a rectangular pulse, and simple convolution integrals that can be done by hand (exponential or ramp forces). Concept questions test linearity and time invariance, the meaning of h(t), and the link between convolution and frequency-response multiplication. Practise evaluating ∫ e^(a·τ)·sin(b·(t − τ)) type integrals cleanly.

Quick check

  1. What is the DLF of an undamped SDOF for a suddenly applied constant load?
  2. An impulse of 2 N·s hits a 4 kg mass at rest. What is its initial velocity?
  3. What two properties must a system have for the Duhamel integral to apply?
  4. For a rectangular pulse with t_d = T_n/2, what is the residual amplitude in terms of F₀/k?
  5. What is h(t) for an undamped system?

Answers: 1. 2; 2. 0.5 m/s; 3. linearity and time invariance; 4. 2F₀/k (sin(π/2) = 1); 5. sin(ω_n·t)/(m·ω_n).

Interview questions

All Structural Dynamics and Aeroelasticity interview questions

Try answering each one aloud before you open it.

  1. 1.What is the Duhamel integral and how is it used in structural dynamics?Concept

    The Duhamel integral is a mathematical formulation used to determine the response of a linear time-invariant system to arbitrary external excitations. In structural dynamics, it helps calculate the response of structures to dynamic loads by integrating the product of the impulse response function and the excitation function over time.

  2. 2.Explain the significance of the impulse response function in the context of the Duhamel integral.Concept

    The impulse response function characterizes how a system responds to a unit impulse input. In the context of the Duhamel integral, it serves as a fundamental building block to determine the system's response to any arbitrary excitation by convolution with the excitation function.

  3. 3.How does the Duhamel integral relate to the convolution integral?Concept

    The Duhamel integral is essentially a convolution integral. It represents the convolution of the impulse response function of a system with the external excitation function, allowing the calculation of the system's response to arbitrary inputs.

  4. 4.Why is the Duhamel integral important in aeroelasticity studies?Application

    Unsteady aerodynamic loads are naturally described by indicial functions: the Wagner function gives the lift build-up after a step change in angle of attack and the Küssner function the lift on entering a sharp-edged gust. Because the airfoil aerodynamics are treated as linear, the lift for an arbitrary motion or gust profile is a Duhamel (convolution) integral of the indicial response with the rate of change of the input. The structural response to that load is again a convolution with the structural impulse response, which is how discrete-gust loads and transient aeroelastic responses are computed.

  5. 5.What would happen if the impulse response function is not accurately determined in the Duhamel integral?Application

    If the impulse response function is not accurately determined, the calculated response of the system to arbitrary excitations will be incorrect. This can lead to errors in predicting the dynamic behavior of structures, potentially resulting in unsafe designs or failure to meet performance criteria.

  6. 6.Describe a scenario where the Duhamel integral might be used in the design of an aerospace structure.Application

    The Duhamel integral might be used in designing an aircraft wing to predict its response to gust loads. By modeling the wing's impulse response and integrating it with the expected gust load profile, engineers can assess the wing's dynamic behavior and ensure it meets safety and performance standards.

  7. 7.How does the Duhamel integral handle non-linear systems?Application

    The Duhamel integral is specifically formulated for linear time-invariant systems. For non-linear systems, linearization techniques or numerical methods are typically employed to approximate the response, as the Duhamel integral itself cannot directly handle non-linearities.

  8. 8.Calculate the response of a linear system with impulse response h(t) = e^(−2t) to an excitation f(t) = 10·sin(t), starting from rest, using the Duhamel integral.Numerical

    x(t) = ∫₀ᵗ f(τ)·h(t − τ)·dτ = 10·∫₀ᵗ sin τ·e^(−2(t − τ))·dτ. Using ∫ e^(2τ)·sin τ·dτ = e^(2τ)·(2·sin τ − cos τ)/5, the result is x(t) = 2·(2·sin t − cos t + e^(−2t)). It is zero at t = 0, as it must be, the e^(−2t) term is the transient, and the steady state is 2·(2·sin t − cos t), an oscillation of amplitude 2√5 ≈ 4.47 lagging the input.

  9. 9.For a linear system with impulse response h(t) = e^(−3t) and excitation f(t) = 5t, find the response using the Duhamel integral.Numerical

    x(t) = ∫₀ᵗ 5τ·e^(−3(t − τ))·dτ. Integrating by parts, ∫₀ᵗ τ·e^(3τ)·dτ = e^(3t)·(t/3 − 1/9) + 1/9, so x(t) = 5·(t/3 − 1/9 + e^(−3t)/9). It starts at zero, and for large t it follows the ramp with a lag: x ≈ (5/3)·(t − 1/3). At t = 1 it gives x = 5 × (0.3333 − 0.1111 + 0.0055) ≈ 1.14.

  10. 10.What are the limitations of using the Duhamel integral in practical engineering applications?Application

    The Duhamel integral assumes linearity and time-invariance, which may not hold for all real-world systems. Additionally, accurately determining the impulse response function can be challenging, and computational complexity may arise for systems with complex excitations or responses.

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