#### Forced Transverse Vibration of String — Lesson 4

This lesson covers the analysis of one-dimensional wave equations, specifically focusing on the forced vibration analysis of a string. The lesson begins with a discussion on the equation of motion for damp force vibration of a string, incorporating the effect of damping. It then moves on to the decoupling of the partial differential equation of motion using the modal superposition technique. The lesson also explains the solution of transverse vibration of a string subjected to different types of forces, such as a step input and harmonic excitation. The lesson concludes with a detailed example of a string subjected to a sinusoidal force, demonstrating how to calculate the displacement response of the string.

### Video Highlights

03:21 - Free body diagram of a string element and how the forces acting on it are balanced.

06:30 - Derivation of equation of motion for the string from the force balance equation.

10:33 - Solving modal superposition technique for the equation of motion for the string.

44:58 - How the generalized force in the string is calculated.

54:51 - How the displacement response of the string is calculated when it is subjected to a sinusoidal force.

### Key Takeaways

- The one-dimensional wave equation is used to study wave propagation and transverse vibration of a string.

- The equation of motion for damp force vibration of a string incorporates the effect of damping.

- The modal superposition technique is used to decouple the partial differential equation of motion.

- The solution of transverse vibration of a string can be found when subjected to different types of forces.

- The displacement response of a string subjected to a sinusoidal force can be calculated using the techniques discussed.