Nonlinear Dynamic Behavior Research of a Simplified Longitudinal-Transversal Vibrating Mechanical Boom Beam System Equipped with Nonlinear Boundaries

As a foundation bearing unit, the beam structure is widely used in construction, aviation, aerospace, and shipbuilding engineering fields. To effectively suppress its vibration response, it is crucial to understand the dynamic characteristics and complex behavior of the beam structure. This paper presents a dynamic model for the longitudinal–transverse coupled vibration of beams with nonlinear boundary conditions. The Galerkin truncation method was used to predict the steady-state response, and the reliability of the process was confirmed by the Lagrangian method. Based on this, the influence of harmonic excitation sweep direction and nonlinear support parameters on the dynamic response of the longitudinal and transverse vibrating beam system with a nonlinear support boundary is studied. Systematic analysis demonstrates that the proposed methodology accurately characterizes coupled dynamics under nonlinear constraints. The system exhibits noticeable initial value sensitivity, and the scanning direction of the excitation has a significant effect on the vibration response curve of the system. With the increase in the stiffness of the nonlinear support, the damping effect of the system becomes increasingly pronounced. Still, excessive nonlinear stiffness will lead to complex responses within the system. The research provides a theoretical basis for optimizing and designing the stability of nonlinear vibration suppression strategies for small deformation beam structures.