The propagation properties of waves in Timoshenko beams resting on flexible supports and with periodically attached harmonic locally resonant oscillators are studied by the transfer matrix methodology. Through calculating the differential equations of the beam for the flexible vibration and the dynamic equations of the oscillators in series, the matrix of dynamic stiffness and the resulting transfer matrix are derived. Accordingly, the band gap in infinite system characterized by the propagation constant can be verified by comparing to the curve of transmission property, determined with the finite element method for the finite system. The mechanism of each band gap formation is further explored. Numerical results show that different from the single degree-of-freedom mass-spring model, one more locally resonant band gap is generated in the system of two oscillators in series. The introduction of flexible supports, allowing for variable internal coupling between the adjacent cells, produces an extra band gap with a minimum frequency of zero. It is also found that the starting frequencies of the locally resonant gaps are related to the spring stiffness and mass of the oscillator. Therefore, the positions and widths of the band gaps can be tuned by properly adjusting the four parameters of the oscillators and also the stiffness of the flexible supports.

中文翻译：

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周期性 Timoshenko 梁中的弯曲振动带隙，振荡器串联放置在柔性支架上

通过传递矩阵方法研究了基于柔性支撑并具有周期性连接的谐波局部谐振振荡器的 Timoshenko 梁中波的传播特性。通过计算梁的柔性振动微分方程和串联振荡器的动力学方程，推导出动刚度矩阵和由此产生的传递矩阵。因此，以传播常数为特征的无限系统中的带隙可以通过与有限系统的有限元方法确定的传输特性曲线进行比较来验证。进一步探索了每个带隙形成的机制。数值结果表明，与单自由度质量弹簧模型不同的是，在两个串联振荡器的系统中产生了另一个局部谐振带隙。引入柔性支撑，允许相邻单元之间的可变内部耦合，产生最小频率为零的额外带隙。还发现局部谐振间隙的起始频率与振荡器的弹簧刚度和质量有关。因此，可以通过适当调整振荡器的四个参数以及柔性支架的刚度来调整带隙的位置和宽度。还发现局部谐振间隙的起始频率与振荡器的弹簧刚度和质量有关。因此，可以通过适当调整振荡器的四个参数以及柔性支架的刚度来调整带隙的位置和宽度。还发现局部谐振间隙的起始频率与振荡器的弹簧刚度和质量有关。因此，可以通过适当调整振荡器的四个参数以及柔性支架的刚度来调整带隙的位置和宽度。