Primary resonance of lateral vibration of a heated beam with an axial stick-slip-stop boundary

As a first endeavor, the present work deals with the primary resonance of lateral vibration of an Euler-Bernoulli beam with a sliding end and under both uniformly distributed heating and harmonic loads. The sliding end is subject to a pair of adjustable normal force and frictional force such that it is initially at a stick status, but may be slightly slipping due to the thermal expansion of the beam until it contacts a stop, i.e., the bound of the clearance. Moreover, this sliding end may also be slipping during the lateral vibration when the vibration amplitude is larger than a critical value. Firstly, based on the nonlinear relation between strain and displacement, a set of partial differential equations of the beam and the axial boundary condition for the sliding end are derived by utilizing Hamilton's principle, where both frictional force and temperature-dependent properties of material are taken into consideration. Then, Galerkin's approach is employed to simplify the partial differential equations to a set of ordinary differential equations. Subsequently, the average approach is used to determine the steady-state primary resonance. Finally, the analytical solutions are well verified through numerical simulations and the influences of system parameters, such as temperature rise and normal force, on the primary resonance of lateral vibration of the beam are discussed. The study shows that it is possible to adjust the primary resonance of lateral vibration of the heated beam with an axial stick-slip-stop boundary via the analytical solutions, which involve the normal force, the clearance and the temperature rise.