Cantilever with an asymmetrically attached tip mass arises in many engineering applications. Both the traditional method of separation of variables and the method of Laplace transform are employed in the present paper to solve the eigenvalue problem of the free vibration of such structures, and the effect of the eccentric distance along the vertical direction and the length direction of the tip mass is considered here. For the traditional method of separation of variables, tip mass only affects to the boundary conditions, and the eigenvalue problem of the free vibration is solved based on the nonhomogeneous boundary conditions. For the method of Laplace transform, the effect of the tip mass is introduced in the governing equation with the Dirac function, and the eigenvalue problem then can be solved through Laplace transform with homogeneous boundary conditions. The computed results with these two methods are compared well with the numerical solution obtained by finite element method and approximate analytical solutions, and the effect of tip mass dimensions on the natural frequencies and corresponding mode shapes is also given.

中文翻译：

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重新审视具有非对称连接尖端质量的悬臂梁的自由振动

具有不对称连接尖端质量的悬臂出现在许多工程应用中。本文采用传统的变量分离方法和拉普拉斯变换方法来解决此类结构的自由振动特征值问题，以及沿垂直方向和长度方向的偏心距的影响。此处考虑尖端质量。对于传统的变量分离方法，尖端质量只影响边界条件，基于非齐次边界条件求解自由振动的特征值问题。对于拉普拉斯变换方法，在具有狄拉克函数的控制方程中引入了尖端质量的影响，并且特征值问题可以通过具有齐次边界条件的拉普拉斯变换来解决。这两种方法的计算结果与有限元法和近似解析解得到的数值解进行了很好的比较，并给出了尖端质量尺寸对固有频率和相应模态振型的影响。