9.9. Convolution Integral and General Force Response

Try these first

1. 1.

   What is the convolution integral used for in structural analysis?

   Hint

   Think about how different inputs affect the output response.

2. 2.

   Define impulse response function.

   Hint

   What happens when you apply an instantaneous force to a system?

3. 3.

   What does the convolution integral signify?

   • It calculates only the impulse response
   • It combines responses to multiple forces
   • It measures the speed of a building

   Hint

   Focus on the definition and purpose of convolution.

4. 4.

   True or False: The convolution integral can be used for non-linear systems.

   • True
   • False

   Hint

   Remember the system types that allow the use of convolution.

5. 5.

   Consider a simple SDOF system with an impulse response function defined as h(t) = e^(-ζω_nt)sin(ω_dt). If subjected to a constant force F(t)=F_0 for a time window from 0 to T, how would you compute the resultant displacement x(t)?

   Hint

   Think about the shape of the impulse response and how it behaves with constants over time.

6. 6.

   Given an external force F(t) = F_0 * cos(ωt) applied to an LTI system with a known impulse response h(t), derive the total response x(t) using the convolution integral.

   Hint

   Remember the properties of convolution with sinusoidal inputs can lead to specific forms in the output.