10.10. Convolution Integral and System Linearity

Try these first

1. 1.

   Define the convolution integral in your own words.

   Hint

   Think about how functions can be combined in mathematics.

2. 2.

   What is an impulse response function?

   Hint

   Consider the response of a system to a sudden force.

3. 3.

   What does the convolution integral compute in an LTI system?

   • Output response
   • Input force
   • Impulse response

   Hint

   Consider what we are trying to determine about the system.

4. 4.

   True or False: The principle of superposition applies to all systems, including non-linear.

   • True
   • False

   Hint

   Review linear vs. non-linear characteristics.

5. 5.

   Describe the process of applying the Duhamel integral to analyze a building’s response to an earthquake, including mathematical formulation.

   Hint

   Focus on the relationship between the ground motion and the system's impulse response.

6. 6.

   Given a non-linear system with a specific input, explain why the convolution integral cannot be directly applied and suggest an alternative approach.

   Hint

   Think of what characterizes non-linear systems and alternative mathematical solutions.