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<p>Beam structures often carry concentrated masses, and the boundary conditions are usually considered as simply supported or clamped. This study develops a transverse nonlinear vibration model for a beam that incorporates a concentrated mass and axially elastic boundaries all while being subjected to a gravitational field. In this model, the effects of concentrated mass, axially elastic boundaries, and gravitational coupling are considered. This provides a more general theoretical basis for describing complex nonlinear responses in engineering structures. The static equilibrium configuration of the system under gravitational effects is studied. The influence of axially elastic boundary on the equilibrium configuration is discussed. Moreover, the governing equation of the system around the static equilibrium configuration is derived. The effects of the concentrated mass and axially elastic boundaries on the natural vibration characteristics are analyzed. Furthermore, the nonlinear dynamic response of the system is examined. The effects of the concentrated mass, axially elastic boundaries, and external excitation amplitude on the amplitude-frequency curves and saddle-node (SN) bifurcation diagrams are investigated. The results show that an increase in the concentrated mass leads to a reduction in the system's natural frequency and induces nonlinear dynamic behaviors at lower frequencies and smaller excitation amplitudes. The strong axially elastic boundaries shift the entire SN bifurcation curve toward a higher frequency region. A large excitation amplitude induces the system to transition from softening-type responses to complex softening-hardening nonlinear dynamic behaviors. Finally, an experimental platform is established, and the parameters of the experimental structure are identified with the restoring force surface method. A high degree of agreement between theoretical predictions and experimental results strongly supports the effectiveness of the proposed model. The proposed model of the beam structure, which incorporates a concentrated mass and axially elastic boundaries, offers a new theoretical and experimental reference for nonlinear dynamic analyses.</p>

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Softening-hardening nonlinear dynamics of a beam with concentrated mass and axially elastic boundaries

  • 期刊论文

  • 发表日期：

    2025年7月19日

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  • Hai-Ting Zheng,
  • Hu Ding

摘要

Beam structures often carry concentrated masses, and the boundary conditions are usually considered as simply supported or clamped. This study develops a transverse nonlinear vibration model for a beam that incorporates a concentrated mass and axially elastic boundaries all while being subjected to a gravitational field. In this model, the effects of concentrated mass, axially elastic boundaries, and gravitational coupling are considered. This provides a more general theoretical basis for describing complex nonlinear responses in engineering structures. The static equilibrium configuration of the system under gravitational effects is studied. The influence of axially elastic boundary on the equilibrium configuration is discussed. Moreover, the governing equation of the system around the static equilibrium configuration is derived. The effects of the concentrated mass and axially elastic boundaries on the natural vibration characteristics are analyzed. Furthermore, the nonlinear dynamic response of the system is examined. The effects of the concentrated mass, axially elastic boundaries, and external excitation amplitude on the amplitude-frequency curves and saddle-node (SN) bifurcation diagrams are investigated. The results show that an increase in the concentrated mass leads to a reduction in the system's natural frequency and induces nonlinear dynamic behaviors at lower frequencies and smaller excitation amplitudes. The strong axially elastic boundaries shift the entire SN bifurcation curve toward a higher frequency region. A large excitation amplitude induces the system to transition from softening-type responses to complex softening-hardening nonlinear dynamic behaviors. Finally, an experimental platform is established, and the parameters of the experimental structure are identified with the restoring force surface method. A high degree of agreement between theoretical predictions and experimental results strongly supports the effectiveness of the proposed model. The proposed model of the beam structure, which incorporates a concentrated mass and axially elastic boundaries, offers a new theoretical and experimental reference for nonlinear dynamic analyses.