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Mechanical Engineering QUESTION #2447
Question 1
The Duhamel integral gives the response of an undamped SDOF system to an arbitrary forcing function \(F(t)\) as:
  • \(x(t) = \frac{1}{m\omega_n}\int_0^t F(\tau)\sin[\omega_n(t-\tau)]\,d\tau\) plus homogeneous solution✔️
  • \(x(t) = \frac{1}{k}\int_0^t F(\tau)\,d\tau\)
  • \(x(t) = \int_0^t F(\tau)\cos[\omega_n(t-\tau)]\,d\tau\)
  • \(x(t) = F(t)/k\)
Correct Answer Explanation
The Duhamel (convolution) integral for an undamped SDOF system: \(x(t)=\frac{1}{m\omega_n}\int_0^t F(\tau)\sin[\omega_n(t-\tau)]d\tau\) plus the free-vibration homogeneous solution from initial conditions.