Revisited on the Free Vibration of a Cantilever Beam with an Asymmetrically Attached Tip Mass

Xiangsheng Lei,Xinghua Wang,Yanfeng Wang,Gang Lin,Shihong Shi,Francesco Pellicano

doi:10.1155/2021/8024879

Abstract

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.

Highlights

In engineering practices, the problem of a beam carrying a concentrated mass at its end or middle may arise
E most widely used approach solves the homogeneous partial differential equation that describes the free vibration with separation of variables to yield a pair of ordinary differential equations, and with the requirement of the nontrivial solution of the linear equations based on the introduction of nonhomogeneous boundary conditions, frequency equation and mode shape can be obtained . is approach is described as the traditional method of separation of variables (MSV), as it mainly depends on the nonhomogeneous boundary conditions
ANSYS is used here to determine the natural frequency and mode shape, and the obtained numerical results are considered as the benchmark solution. e cantilever is modeled as a three dimensional structure and meshed with SOLID186 element. e convergence test is performed in advance to make sure that the numerical solution can be treated as the benchmark solution

Summary

Introduction

The problem of a beam carrying a concentrated mass at its end or middle may arise. Offshore wind turbines [1], mast antenna structures, wind tunnel stings carrying an airplane or a missile model, large aspect ratio wings carrying heavy tip tanks, or launch vehicles with payload at the tip [2], and all these structures can be modeled as a beam carrying a concentrated mass at its end or middle. In these applications, the concept of an ideal concentrated mass or moment of initial is often not applicable, as the attachment point does not coincide with the center of gravity of the mass. Anderson pointed out that due to the importance in airplane and missile design, it is of interest to consider the problem

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