Non-linear dynamical analysis for an axially moving beam with finite deformation

• The non-linear equations of a simply supported translating beam are derived.
• Analysis reveals the necessity of using geometrically non-linear theory for the translating beam.
• Coupling responses between transverse and longitudinal vibrations are numerically explored.

Non-linear dynamical analysis of a simply supported translating beam considering the interactions between beam translation and flexible deformation is presented. The extended Hamilton's principle is employed to derive the equations of the longitudinal and transverse vibration of the beam under finite deformation theory which are non-linearly coupled. Runge–Kutta method is utilized to solve the non-linear governing equations. The numerical results describe the coupling responses between the beam extension and flexible deformation where the higher the axial velocity is, the stronger the interactions will be. Furthermore, numerical analysis also demonstrates that the resulting responses are distinctly different under small deformation theory and finite deformation theory, and there exists a critical speed under small deformation theory. When the speed exceeds the critical speed, the system becomes unstable because of the divergence or flutter instability. However, under finite deformation theory the system is always stable. Further analysis reveals that, for appropriately considering the influence of axial movement on the beam deformation, a geometrically non-linear beam theory should be utilized even if the deflection is very small. Coupling responses between the transverse and longitudinal vibrations are also numerically explored.