The presented work concerns the kinematically excited transient vibrations of a cantilever beam with a mass element fixed to its free end. The Euler–Bernoulli beam theory and the fractional Zener model of the beam material are assumed. A fractional Caputo derivative is used to formulate a viscoelastic material law. A characteristic equation, modal frequencies, eigenfunction and orthogonality conditions are achieved for the beam considered. The equations of motion of the system are solved numerically. A numerical solution of a multi-term fractional differential equation is obtained by means of a conversion to a mixed system of ordinary and fractional differential equations, each of the order of \(0 < \gamma \le 1\). The transient time histories of the beam vibrations during the passage through resonance are calculated. A comparison between the beam responses obtained with a fractional and an integer viscoelastic material model is presented. The calculations performed reveal that use of the fractional damping affects on the time histories of the system. The calculated beam responses show that for some values of the order of the fractional derivative \(\gamma\), the amplitudes occurring in the area of the second resonance are greater than those obtained in the area of the first resonance, which does not occur in the case of the integer order of the fractional derivative. Moreover, an evaluation is made of the difference between the results obtained for the calculations using the fractional Zener model and the fractional Kelvin model. It is shown that for some physical beam parameters, the calculation results obtained using both models are virtually the same for both models, which means that the the simpler, fractional Kelvin–Voigt material can be used instead of the fractional Zener material model. This simplifies the solution and decreases the time needed to make the numerical calculations.

所提出的工作涉及悬臂梁的运动激发瞬态振动，其中质量元件固定在其自由端。假设 Euler-Bernoulli 梁理论和梁材料的分数齐纳模型。分数 Caputo 导数用于制定粘弹性材料定律。对于所考虑的梁，实现了特征方程、模态频率、特征函数和正交性条件。系统的运动方程被数值求解。多项分数阶微分方程的数值解是通过转换为常微分方程和分数阶微分方程的混合系统而获得的，每个阶数为\(0 < \gamma \le 1\). 计算梁振动通过共振的瞬态时间历程。呈现了使用分数粘弹性材料模型和整数粘弹性材料模型获得的梁响应之间的比较。执行的计算表明，分数阻尼的使用会影响系统的时间历程。计算的光束响应表明，对于分数阶导数\(\gamma\)，出现在第二共振区域中的幅度大于在第一共振区域中获得的幅度，这在分数阶导数的整数阶的情况下不会发生。此外，对使用分数齐纳模型和分数开尔文模型的计算所获得的结果之间的差异进行了评估。结果表明，对于某些物理梁参数，使用两种模型获得的计算结果几乎相同，这意味着可以使用更简单的分数 Kelvin-Voigt 材料代替分数齐纳材料模型。这简化了解决方案并减少了进行数值计算所需的时间。