10.3. Derivation of Duhamel’s Integral

Try these first

1. 1.

   What does Duhamel's Integral allow us to calculate?

   Hint

   Think about earthquake analysis.

2. 2.

   Define impulse response function.

   Hint

   What happens when a singular force hits the system?

3. 3.

   What is the primary use of Duhamel's Integral?

   • To determine static loads
   • To analyze dynamic responses of structures
   • Neither

   Hint

   Think about its application in earthquake engineering.

4. 4.

   True or False: Superposition applies only to nonlinear systems.

   • True
   • False

   Hint

   Consider the definitions of linear versus nonlinear.

5. 5.

   Given a linear system with mass m, damping coefficient c, and stiffness k, derive the Duhamel Integral expression for the system's response to a sinusoidal force F(t) = F₀ sin(ωt).

   Hint

   Consider how the sine function influences the system's response over its period.

6. 6.

   Assuming a damping ratio ζ < 1, explain how the impulse response changes compared to critically damped and overdamped systems.

   Hint

   Think about the differences in energy dissipation and speed of return to equilibrium.