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<p>This study proposes a hybrid analytical–numerical methodology for accurately predicting the fundamental resonant frequency of complex stepped structures for microsensor applications. Analytical formulations were developed to determine stiffness characteristics, while numerical eigenfrequency simulations in COMSOL Multiphysics were used to extract mode vibration constants (<i>β</i>ₙ). The integration of analytical stiffness and mass expressions with numerically derived parameters yielded hybrid closed-form frequency relations capable of capturing geometric discontinuities. Parametric optimization of step geometry, defined by step length and thickness ratios, was carried out to enhance frequency response. Limited experimental validation using PZT-integrated steel beams confirmed the model’s accuracy. A close correlation between analytical and numerical results was observed, with deviations of less than 5% for stiffness and 2% for resonant frequency. The optimized step configuration achieved up to a 46.8% increase in frequency compared to base model. The proposed hybrid framework offers a robust and computationally efficient approach for the design and optimization of MEMS-based resonant sensors with complex geometries.</p>

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Experimentally validated hybrid analytical-numerical approach to determine frequency characteristics of complex step cantilever shapes for sensor applications

  • 期刊论文

  • 发表日期：

    2026年4月11日

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  • Shubham Kumar Mishra,
  • Mohd. Zahid Ansari,
  • Afzal Husain

摘要

This study proposes a hybrid analytical–numerical methodology for accurately predicting the fundamental resonant frequency of complex stepped structures for microsensor applications. Analytical formulations were developed to determine stiffness characteristics, while numerical eigenfrequency simulations in COMSOL Multiphysics were used to extract mode vibration constants (βₙ). The integration of analytical stiffness and mass expressions with numerically derived parameters yielded hybrid closed-form frequency relations capable of capturing geometric discontinuities. Parametric optimization of step geometry, defined by step length and thickness ratios, was carried out to enhance frequency response. Limited experimental validation using PZT-integrated steel beams confirmed the model’s accuracy. A close correlation between analytical and numerical results was observed, with deviations of less than 5% for stiffness and 2% for resonant frequency. The optimized step configuration achieved up to a 46.8% increase in frequency compared to base model. The proposed hybrid framework offers a robust and computationally efficient approach for the design and optimization of MEMS-based resonant sensors with complex geometries.