Exact Solution For Free Vibration Research Articles

Vibration of String with Time-Varying Length. On the Exact Solution.

The simplest exact solution of free vibration of a string with timevarying length is presented. Taking into account virtual sources of waves which can be assigned to reflecting waves, an extended Stokes' formula is obtained. Coordinates of the virtual sources of the waves which can be observed at (t, x) are given by (0, -at+x) and (0, at+x). Using values of the virtual sources, the solution at (t, x) can be obtained.

Vibration of String with Time-Varying Length (Method Using Hypergeometric Differential Equation).

Free vibration of a string with time-varying length is analyzed. An equation of motion cannot be solved by the method of separation of the variables because the length of the string varies with time. New variables of position and time are introduced, and then the equation of motion which can be solved by the method using a hypergeometric differential equation is obtained. Exact solutions of free vibration are obtained by Laplace transformation.