Free Vibration Energy and Period of a String With Time-Varying Length

We introduce a new technique to analyze free vibration energy and natural frequencies of a string with time-varying length by dealing with traveling waves. One of boundaries is vertically fixed but it moves axially at a constant rate. When the string length is varied, the free vibration energy and period also change with time. String tension and non-zero instantaneous vertical velocity at the moving boundary results in the energy variation. When the string undergoes retraction, the vibration energy increases with time. The new technique using traveling waves gives the exact amount of energy transferred into the vibrating system at the moving boundary. The free vibration energy are compared with previous results using the perturbation method and Kotera’s approach.