In this paper the new semi-analytical solution for the moving mass problem on massless foundation, published by the author of this paper, is extended to account for inertial foundation modelled by a continuous homogeneous finite depth foundation with simplified shear resistance. Derivations are presented for infinite as well as finite homogeneous beams. Mode expansion method is used to solve the problem on finite beams, thus vibration modes, the corresponding orthogonality condition, reengagement of coupled equations to ensure significant calculation time savings are derived. Methods of integral transforms and contour integration are exploited to obtain the solution on infinite beams. Resulting vibrations are derived as a sum of the steady and unsteady harmonic vibrations and a transient contribution. The unsteady harmonic vibration is proven to be a useful indicator of unstable behaviour through the mass induced frequencies. Besides frequency lines also discontinuity lines are determined and their influence on the proximity of harmonic and full solutions is discussed. Even if the differences between these two versions are larger than for the massless foundation, it is shown that the harmonic solution provides a very good estimate of the full solution (in several cases perfect match is achieved) with the advantage to be obtainable by a simple evaluation of the derived closed-form results. Like for the massless foundation, also here, vibrations on infinite beams can be obtained on long finite beams with eliminated effect of its supports. All mentioned approaches are also validated by the finite element method.

中文翻译：

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由具有简化抗剪强度的有限深度基础支撑的横梁的质量引起的振动的半解析分析

在本文中，由本文作者发表的无质量基础上移动质量问题的新半解析解被扩展到考虑由具有简化抗剪力的连续均匀有限深度基础建模的惯性基础。给出了无限和有限均匀梁的推导。模态扩展方法用于解决有限梁上的问题，从而推导出振动模态、相应的正交条件、耦合方程的重新啮合以确保显着的计算时间节省。利用积分变换和轮廓积分的方法来获得无限梁上的解。由此产生的振动是作为稳态和非稳态谐波振动和瞬态贡献的总和得出的。非定常谐波振动被证明是通过质量感应频率的不稳定行为的有用指标。除了频率线外，还确定了不连续线，并讨论了它们对谐波和全解接近度的影响。即使这两个版本之间的差异比无质量基础的差异更大，但表明谐波解决方案提供了对完整解决方案的非常好的估计（在一些情况下实现了完美匹配），其优点是可以通过简单的方法获得对导出的封闭形式结果的评估。与无质量基础一样，这里也可以在长有限梁上获得无限梁上的振动，并消除其支撑的影响。所有提到的方法也通过有限元方法进行了验证。