Abstract By introducing relaxation summands in the formula of Hooke’s law, in order to consider the finite propagation velocity of stress and deformations, an equation for damped oscillations of an elastic rod was obtained, including, as opposed to the well-known equation, the third derivative of motion in time as well as the mixed derivative of the spatial variable and time. By using Fourier’s method, its accurate analytical decision was found, the study of which showed that considering relaxation factors results in elimination of step-wise change of stresses and deformations. Comparison of the results of theoretical studies and experimental data applied to longitudinal oscillations of a rod fixed at one end showed that the amplitude and frequency of their oscillation coincided in a satisfactory manner.

中文翻译：

---

双相滞后阻尼弹性杆振荡的数学模型

摘要 通过在胡克定律公式中引入松弛加数，为了考虑应力和变形的有限传播速度，得到了弹性杆阻尼振荡方程，与众所周知的方程相反，第三运动的时间导数以及空间变量和时间的混合导数。通过使用傅立叶方法，找到了其准确的分析决策，研究表明，考虑松弛因素导致应力和变形的逐步变化被消除。将理论研究的结果和应用于固定在一端的杆的纵向振荡的实验数据进行比较表明，它们的振荡幅度和频率以令人满意的方式重合。