12.10. Practical Examples

Try these first

1. 1.

   What is the mathematical expression for a point load applied at the center of a simply supported beam?

   Hint

   Think about where the load is positioned on the beam.

2. 2.

   What type of function is the Dirac delta function?

   Hint

   Consider its properties discussed earlier.

3. 3.

   What does the Dirac delta function model?

   • Continuous force
   • Idealized point effect
   • Distributed load

   Hint

   Recall the key features of the Dirac delta function.

4. 4.

   True or False: The impulse function can be expressed through the Dirac delta function in the time domain.

   • True
   • False

   Hint

   Think about the properties of impulses in dynamics.

5. 5.

   A simply supported beam of length 10 m is subject to a point load of 500 N at its center. Derive the deflection at the midpoint of the beam using the Dirac delta function and given E = 200 GPa, I = 5000 cm4.

   Hint

   Break down the governing equation and apply integration techniques.

6. 6.

   Analyze the response of a mass-spring-damper system under an impulse F(t) = 1000δ(t). Calculate its system response given m=1 kg, k=100 N/m, and b=10 N·s/m.

   Hint

   Use the characteristic equation of the system and determine the form of the response due to this impulse.