15.12.2. Transform of Shifted Functions

Try these first

1. 1.

   Define a shifted function in your own words.

   Hint

   Think about when you add something to a system.

2. 2.

   What is the purpose of the unit step function in shifted functions?

   Hint

   Consider how the function behaves before and after the delay.

3. 3.

   What is the formula for the Laplace transform of a shifted function?

   • L{f(t-a)u(t-a)} = e^{-as}F(s)
   • L{f(t+a)u(t+a)} = e^{as}F(s)
   • L{f(t-u(t-a))} = F(s)

   Hint

   Focus on how the shift modifies the standard Laplace transformation.

4. 4.

   True or False: The unit step function allows a function to begin analysis at t<0.

   • True
   • False

   Hint

   Remember the definition of the unit step function.

5. 5.

   A system experiences a load that starts fluctuating after 4 seconds. If the function describing load fluctuation is f(t) = sin(t), express its Laplace transform and explain the significance of transformations in engineering.

   Hint

   Reflect on how you account for the shift when applying the transform.

6. 6.

   Model a scenario where material is added to a structure 5 seconds after construction begins. Define the function and calculate its Laplace transform.

   Hint

   Break down the process into defining the load before transforming.