8. Response to Harmonic Excitation

Try these first

1. 1.

   Identify the components of the equation mx¨(t) + cx˙(t) + kx(t) = F sin(ωt).

   Hint

   What do each of the letters stand for in the equation?

2. 2.

   What happens when r equals 1 in relation to resonance?

   Hint

   Think about the natural frequency matching the forcing frequency.

3. 3.

   What occurs at resonance?

   • Response is minimized
   • Response is maximized
   • No response occurs

   Hint

   Recall the nature of resonance in oscillatory systems.

4. 4.

   True or False: The damping ratio increases the amplitude of vibrations.

   • True
   • False

   Hint

   Think about how damping affects motion.

5. 5.

   Given a system with a mass of 10 kg, a damping coefficient of 5 Ns/m, and a stiffness of 200 N/m, calculate the natural frequency and the change in amplitude at resonance.

   Hint

   Think about how to derive natural frequency from fundamental formulas.

6. 6.

   If a building's damping ratio is found to be 0.05, determine how this might affect the building’s response to earthquake-induced harmonic excitation.

   Hint

   Consider the implications of damping on responsiveness during dynamic loading.