{'fraction_non_alphanumeric': 0.015292190060076462, 'fraction_numerical': 0.002184598580010923, 'mean_word_length': 5.661818181818182, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Fundamental solution and its validation by numerical inverse Laplace transformation and FEM for a damped Timoshenko beam subjected to impact and moving loads

This paper, taking the clamped boundary condition as an example, develops Su and Ma's fundamental solutions of the dynamic responses of a Timoshenko beam subjected to impact load. Based on that, a further extension regarding the general moving load case is also established. Kelvin–Voigt damping, whether proportionally or nonproportionally damped, is incorporated into the model, making it more comprehensive than the model of Su and Ma. Numerical inverse Laplace transformation is introduced to obtain the time-domain solution, where Durbin's formula and the corresponding convergence criteria are utilized in numerical experiments. Further, the real modal superposition method is applied at an analytical level to validate the numerical results by applying a proportionally damped condition. Total comparisons are made between the methods by sufficient case studies. The dynamic responses with and without damping effect are computed with wider slenderness to verify the correctness and effectiveness of the numerical results. Furthermore, parametric studies regarding the damping coefficients are performed to explore the nonproportional damping effect. The results show that the structural damping has significant influences on the dynamic behaviors and is especially stronger at small slender ratios. As the damping decreases the inherent frequencies and excites the low-frequency modal components more actively, a resonant phenomenon appears in high slenderness case when the beam experiences a low-speed moving load. Additionally, the computations in the moving load case indicate that the algorithm convergence is preferable when the number of grids exceeds 1000.