17.9. Special Case: Undamped Systems

Try these first

1. 1.

   What equation represents the motion of undamped systems?

   Hint

   Think about how the system behaves without damping.

2. 2.

   What is Duhamel’s Integral used for?

   Hint

   Consider its application in dynamics.

3. 3.

   What is the characteristic form of modal equations for undamped systems?

   • q¨(t) + ζω²q(t) = F∗(t)
   • q¨(t) + ω²q(t) = F∗(t)
   • q¨(t) = F∗(t)

   Hint

   Consider how damping changes equation behavior.

4. 4.

   True or False: Duhamel’s Integral can be applied only to damped systems.

   • True
   • False

   Hint

   Think about where it is typically applied.

5. 5.

   Given a harmonic oscillator represented by the equation q¨(t) + 25q(t) = 5, solve for q(t) using Duhamel’s Integral given an initial condition of q(0)=0 and q˙(0)=0.

   Hint

   Map out the steps of Duhamel’s process carefully, ensuring initial conditions are incorporated.

6. 6.

   For an undamped spring-mass system with a loading force F(t) = 10sin(ωt), derive the expression for the modal response q(t) using the Laplace Transform.

   Hint

   Remember to handle 's' carefully and recall the properties of inverse transforms.