7.15. Numerical Solution of Free Vibration (Finite Difference Method)

Try these first

1. 1.

   What is the finite difference method?

   Hint

   Think about how we might break up time into smaller sections.

2. 2.

   Why might we need to discretize time?

   Hint

   Consider why continuous equations can be challenging.

3. 3.

   What does the finite difference method primarily address?

   • Analytical solutions of equations
   • Numerical solutions of dynamic problems
   • Geometric properties analysis

   Hint

   Recall the primary uses of the method.

4. 4.

   True or False: The finite difference method can model continuous systems with variable properties more effectively than analytical methods.

   • True
   • False

   Hint

   Think about the advantages over analytical methods.

5. 5.

   Using the finite difference method, if a system has a natural frequency of 5 rad/s and initial displacements of x(0)=0 and x(1)=0.1, calculate the displacement at the next time step given a Δt of 0.1 seconds.

   Hint

   Start by identifying your constants and then plug into the formula appropriately.

6. 6.

   Discuss the impact of increasing the time step size on the accuracy of results obtained from the finite difference method.

   Hint

   Consider how a larger time step could skip details of the system's motion.