6.8. Worked Examples with Engineering Applications

Try these first

1. 1.

   What is the general equation for a beam under uniform load?

   Hint

   Think about the governing differential equation for beams.

2. 2.

   Define a Particular Integral.

   Hint

   It contributes to the total solution.

3. 3.

   What is the role of the Particular Integral in a non-homogeneous equation?

   Hint

   Think about how discrimination differs from natural responses.

4. 4.

   True or False: A homogeneous differential equation can have external forcing functions.

   Hint

   Consider the definition of homogeneous terms.

5. 5.

   Given a beam with length L subjected to a uniform load q, derive an expression for the maximum deflection. Assume fixed boundary conditions at both ends.

   Hint

   Pay close attention to boundary conditions for fixed supports.

6. 6.

   For a damped vibrating system represented by m d²y/dt² + c dy/dt + ky = F cos(ωt), solve for the steady-state response when ω = ω0.

   Hint

   Understand resonance behavior clearly to derive correct PID function.