Exact Modal Analysis and Its Applications to Transverse Vibrations of Axially Travelling Strings

The Hamiltonian dynamics based on symplectic mathematics is used to solve the transverse vibration of axially traveling strings. The Hamiltonian function is formulated and the canonical equation is developed. The exact modal functions are obtained through a linear eigenvalue analysis, and the symplectic-type orthogonality conditions of modes are derived. The expansion theorem is applied to solve the displacement response to free and forced vibrations. Unlike traditional modes of transporting strings, the exact modal functions do not suffer divergence instability, or singularity, at the critical speed, and can be easily used as the base or trial functions for solving linear and nonlinear vibration.