Unified energy-flow framework for crack-induced modulation and multi-frequency damage identification in beams and plates

Cracks fundamentally alter the pathways and intensity of vibration energy transfer within load-bearing structures, making accurate modeling of their modulation effects essential for structural health monitoring and dynamic energy analysis. This study develops a unified energy-flow framework that quantitatively links crack-induced local flexibility to energy transmission in beams and plates. Based on an extended Timoshenko theory and a two-dimensional line-spring representation, the model explicitly incorporates shear deformation, rotary inertia, and crack compliance, providing a comprehensive physical description across one- and two-dimensional structures. Numerical and experimental investigations reveal that crack depth primarily governs the amplitude of energy-flow oscillations, while crack position controls their periodicity, forming the characteristic pattern of “location-governing rhythm, depth-governing magnitude.” A normalized input energy-flow criterion is proposed, and a multi-frequency iso-value intersection method is established for simultaneous identification of crack depth and location. Comparative results show that neglecting shear and rotational effects leads to up to 6 dB deviation in energy-flow predictions at high frequencies (Ω = 2.5). Experimental validation using Q235 steel beams and plates confirms that the proposed method achieves unique and robust inversion of crack parameters, with maximum relative errors below 3.75% in depth and 1% in position under ±1% measurement uncertainty. The unified approach provides a scalable, physically interpretable, and error-tolerant foundation for energy-flow–based damage diagnostics and high-frequency health monitoring of complex beam–plate systems.