Vibrations of an Axially Moving Beam with Deployment or Retraction

The longitudinal and transverse vibrations of an axially moving beam are analyzed when the beam has deploying or retracting motion. Comparing the previous studies in which only the transverse displacement is considered, the present study simultaneously considers the longitudinal and transverse displacements when deriving the equations of motion. To include a nonlinear effect, the nonlinear von Karman strain theory is adopted. The derived equation of longitudinal motion is linear and uncoupled with the transverse motion, whereas the equation of transverse motion is nonlinear and coupled with the longitudinal motion. After computing numerical solutions from the equations, the longitudinal and transverse vibrations are analyzed with regard to the moving speed and acceleration of the beam. In addition, the present nonlinear analysis of a moving beam is compared with a previous linear analysis for the variations of the moving acceleration, moving speed, and Young's modulus. The linear and nonlinear vibration analyses result in considerably different dynamic responses when the moving acceleration is large or when Young's modulus is small.