Research on the Dynamic Response Characteristics of Soft Coal Under Impact Disturbance Based on Hamilton

To address the limitations of traditional elasticity theory in analyzing the dynamic response of soft coal under external impact, this study establishes a vibration control equation with an analytical solution based on Hamiltonian mechanics. Key control parameters within the equation were solved to determine the theoretical dominant vibration modes and natural frequencies of the weakest coal layer. Triangular and rectangular waves were transformed via FFT to analyze their harmonic components, and the superposition of the first four harmonics was selected as the input impact signal. The modal and natural frequency changes during the fragmentation of the central weak zone under external impact were simulated, and the dynamic displacement response was analyzed. The results indicate a strong response frequency range of 4.4–5.2 Hz, with the rectangular wave identified as the most effective response waveform. A similarity simulation platform was constructed, and experimental data showed that the velocity and displacement response trend of the coal mass aligned closely with theoretical predictions. Therefore, in actual underground operations, emphasis should be placed on monitoring low-frequency vibrations in mines, minimizing rectangular wave disturbances in the low-frequency range, and implementing pressure relief measures in high-risk zones to reduce the likelihood of coal and gas outbursts. By separately modeling high-risk zones and analyzing their dynamic response under external impact, this study explains the outburst mechanism of the weakest layer in soft coal from a dynamic perspective. Combining theoretical and experimental approaches, it provides a new theoretical basis for understanding and preventing coal and gas outbursts.