Nonlinear vibration analysis of axially moving strings based on gyroscopic modes decoupling

A novel idea that applies the multiple scale analysis to a discretized decoupled system of gyroscopic continua is introduced and an axial moving string is treated as an example. First, the invariant manifold method is applied to the discretized ordinary differential equations of the axially moving string. Complex gyroscopic mode functions that agree well with true analytical results are obtained. The gyroscopic modes are subsequently used for the discretized ordinary differential equations with gyroscopic and nonlinear coupling terms that yield a gyroscopically decoupled system. Further the method of multiple scales is used to obtain the equations at a slow scale. This novel procedure is compared to solutions obtained by directly applying the classical multiple scale analysis to the gyroscopically coupled system without decoupling. The modal decoupled system analysis yields better frequency with comparing to the classic method. The proposed methodology provides a novel alternative for nonlinear dynamic analysis of gyroscopic continua.