It is well known that most important musical instruments, including the violin and the guitar, involve strings whose natural frequencies and mode shapes play a significant role in their performance. The characteristics of many engineering systems, such as guy wires, electric transmission lines, ropes and belts used in machinery, and thread manufacture, can be derived from a study of the dynamics of taut strings. The free and forced transverse vibration of strings is considered in this chapter. As will be seen in subsequent chapters, the equation governing the transverse vibration of strings will have the same form as the equations of motion of longitudinal vibration of bars and torsional vibration of shafts.
8.2 EQUATION OF MOTION
8.2.1 Equilibrium Approach
Figure 8.1 shows a tightly stretched elastic string or cable of length l subjected to a distributed transverse force per unit length. The string is assumed to be supported at the ends on elastic springs of stiffness and . By assuming the transverse displacement of the string to be small, Newton's second law of motion can be applied for the motion of an element of the string ...