The vibration of a solid elastic cylinder is one of the classical applied problems of elastodynamics. Many fundamental forced-vibration problems involving solid elastic cylinders have not yet been studied or solved using the full three-dimensional (3D) theory of linear elasticity. One such problem is the steady-state forced-vibration response of a simply-supported isotropic solid elastic circular cylinder subjected to two-dimensional harmonic standing-wave excitations on its curved surface. In this paper, we exploit certain recently-obtained particular solutions to the Navier–Lamé equation and exact matrix algebra to construct an exact closed-form 3D elastodynamic solution to the problem. The method of solution is direct and demonstrates a general approach that can be applied to solve other similar forced-vibration problems involving elastic cylinders. The obtained analytical solution is then applied to a specific numerical example, where it is used to determine the frequency response of the displacement field in some low wave number excitation cases. In each case, the solution generates a series of resonances that are in exact correspondence with a subset of the natural frequencies of the simply-supported cylinder. The analytical solution is also used to compute the resonant mode shapes in some selected asymmetric excitation cases. The studied problem is of general interest both as an exactly-solvable 3D elastodynamics problem and as a benchmark forced-vibration problem involving a solid elastic cylinder.

中文翻译：

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简支各向同性实心弹性圆柱的谐波驻波激励：精确的 3D 线性弹性动力学响应

固体弹性圆柱体的振动是弹性动力学的经典应用问题之一。许多涉及固体弹性圆柱体的基本受迫振动问题尚未使用全三维 (3D) 线性弹性理论进行研究或解决。其中一个问题是简支各向同性固体弹性圆柱在其曲面上受到二维谐波驻波激励的稳态受迫振动响应。在本文中，我们利用某些最近获得的 Navier-Lamé 方程和精确矩阵代数的特定解来构造该问题的精确闭合形式 3D 弹性动力学解。求解方法是直接的，并展示了一种通用方法，可用于解决涉及弹性圆柱体的其他类似受迫振动问题。然后将获得的解析解应用于特定的数值示​​例，用于确定在某些低波数激励情况下位移场的频率响应。在每种情况下，该解都会产生一系列与简支圆柱体的固有频率子集完全对应的共振。该解析解还用于计算某些选定的非对称激励情况下的谐振模式形状。所研究的问题作为可精确求解的 3D 弹性动力学问题和涉及固体弹性圆柱体的基准强制振动问题都具有普遍意义。然后将获得的解析解应用于特定的数值示​​例，用于确定在某些低波数激励情况下位移场的频率响应。在每种情况下，该解都会产生一系列与简支圆柱体的固有频率子集完全对应的共振。该解析解还用于计算某些选定的非对称激励情况下的谐振模式形状。所研究的问题作为可精确求解的 3D 弹性动力学问题和涉及固体弹性圆柱体的基准强制振动问题都具有普遍意义。然后将获得的解析解应用于特定的数值示​​例，用于确定在某些低波数激励情况下位移场的频率响应。在每种情况下，该解都会产生一系列与简支圆柱体的固有频率子集完全对应的共振。该解析解还用于计算某些选定的非对称激励情况下的谐振模式形状。所研究的问题作为可精确求解的 3D 弹性动力学问题和涉及固体弹性圆柱体的基准强制振动问题都具有普遍意义。在每种情况下，该解都会产生一系列与简支圆柱体的固有频率子集完全对应的共振。该解析解还用于计算某些选定的非对称激励情况下的谐振模式形状。所研究的问题作为可精确求解的 3D 弹性动力学问题和涉及固体弹性圆柱体的基准强制振动问题都具有普遍意义。在每种情况下，该解都会产生一系列与简支圆柱体的固有频率子集完全对应的共振。该解析解还用于计算某些选定的非对称激励情况下的谐振模式形状。所研究的问题作为可精确求解的 3D 弹性动力学问题和涉及固体弹性圆柱体的基准强制振动问题都具有普遍意义。