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<p>This paper investigates the nonlinear vibration of thermally induced bistable cantilever symmetric laminates under harmonic excitations using a single-degree-of-freedom (SDOF) model. The model is validated by ABAQUS finite element (FE) package and is able to predict the single-well vibration as well as the double-well snapthrough response. Bistable laminates have two stable equilibrium positions at room temperature due to the thermal mismatch upon cooling. It has been proven that thin unsymmetric laminates lose their bistability when clamped or attached to a larger structure. The hybrid bistable symmetric laminates (HBSL) with initially aluminum plies and later bi-directional (BD) glass–epoxy plies resolved this issue. The later layup is called modified hybrid bistable symmetric laminates (m-HBSL). The laminate is subjected to a tip harmonic excitation whose frequency is tuned at the first bending mode to drive the laminate at resonance. The method of multiple-time scales is used to find an approximate analytical solution and the amplitude and phase modulation equations for the small vibration in the vicinity of one of the equilibrium positions. For the steady-state response, the frequency-response plots are obtained and the laminate’s effective nonlinearity was found to be of softening type. Using the numerical integration of the equation of motion, it is found that at low excitation amplitudes, the laminate possesses a single-well periodic response around either well of the static equilibrium positions. As the forcing amplitude increases, the laminate experiences a period-doubling bifurcation that leads to a chaotic single-well and double-well, snapthrough, response. Upon increasing the load further, a periodic snapthrough response is obtained. The main finding of this work is that the cantilevered bistable laminates dynamically snap at substantially reduced loads compared with their static counterparts when they are harmonically excited. Namely, the laminate snaps at a resonant harmonic load that is about 80% of its static value. Given the softening nonlinear behavior, it was interesting to find that the laminate can snap at a pre-resonant load of about 60% of its static value. The novelty of this work is that it presents the snapthrough response of a cantilevered bistable symmetric laminate under harmonic excitation, which mimics the scenario in many real-life applications, using a SDOF model that is inspired and validated by the FE results.</p>

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Nonlinear vibration of a cantilever bistable symmetric laminate using a SDOF model

  • 期刊论文

  • 发表日期：

    2025年5月18日

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  • Samir A. Emam,
  • Ibrahim K. Diab,
  • Daniel J. Inman

摘要

This paper investigates the nonlinear vibration of thermally induced bistable cantilever symmetric laminates under harmonic excitations using a single-degree-of-freedom (SDOF) model. The model is validated by ABAQUS finite element (FE) package and is able to predict the single-well vibration as well as the double-well snapthrough response. Bistable laminates have two stable equilibrium positions at room temperature due to the thermal mismatch upon cooling. It has been proven that thin unsymmetric laminates lose their bistability when clamped or attached to a larger structure. The hybrid bistable symmetric laminates (HBSL) with initially aluminum plies and later bi-directional (BD) glass–epoxy plies resolved this issue. The later layup is called modified hybrid bistable symmetric laminates (m-HBSL). The laminate is subjected to a tip harmonic excitation whose frequency is tuned at the first bending mode to drive the laminate at resonance. The method of multiple-time scales is used to find an approximate analytical solution and the amplitude and phase modulation equations for the small vibration in the vicinity of one of the equilibrium positions. For the steady-state response, the frequency-response plots are obtained and the laminate’s effective nonlinearity was found to be of softening type. Using the numerical integration of the equation of motion, it is found that at low excitation amplitudes, the laminate possesses a single-well periodic response around either well of the static equilibrium positions. As the forcing amplitude increases, the laminate experiences a period-doubling bifurcation that leads to a chaotic single-well and double-well, snapthrough, response. Upon increasing the load further, a periodic snapthrough response is obtained. The main finding of this work is that the cantilevered bistable laminates dynamically snap at substantially reduced loads compared with their static counterparts when they are harmonically excited. Namely, the laminate snaps at a resonant harmonic load that is about 80% of its static value. Given the softening nonlinear behavior, it was interesting to find that the laminate can snap at a pre-resonant load of about 60% of its static value. The novelty of this work is that it presents the snapthrough response of a cantilevered bistable symmetric laminate under harmonic excitation, which mimics the scenario in many real-life applications, using a SDOF model that is inspired and validated by the FE results.