Abstract
The variable section structure could be the physical model of many vibration problems, and its analysis becomes more complicated either. It is very important to know how to obtain the exact solution of the modal function and the natural frequency effectively. In this paper, a general analytical method, based on segmentation view and iteration calculation, is proposed to obtain the modal function and natural frequency of the beam with an arbitrary variable section. In the calculation, the section function of the beam is considered as an arbitrary function directly, and then the result is obtained by the proposed method that could have high precision. In addition, the total amount of calculation caused by high-order Taylor expansion is reduced greatly by comparing with the original Adomian decomposition method (ADM). Several examples of the typical beam with different variable sections are calculated to show the excellent calculation accuracy and convergence of the proposed method. The correctness and effectiveness of the proposed method are verified also by comparing the results of the several kinds of the theoretical method, finite element simulation, and experimental method.
Highlights
It is necessary to solve the modal function and natural frequency for studying the vibration characteristics of the beam [5,6,7,8]
How to find an analytical method that could be applied to the vibration problem of a more general variable section beam has become the focuses of subsequent research
Cui et al [20] proposed a semianalytical method for solving modal function and natural frequency of the variable section beam based on Transfer matrix method (TM)
Summary
Based on Euler–Bernoulli beam theory, the force analysis of transverse vibration of beams is shown in Figure 1(a). e moment and shear force on the cross section are expressed as M(x, t) and Q(x, t), respectively. 3. Discussion of Fitting Error e original ADM is applied to discuss the fitting error of beams with different sections under the same order expansion. To reduce the fitting error, a new expansion base point is set to O′, which is the midpoint of the beam, as shown in Figures 2(a) and 2(b). It could be seen that different section functions have a great influence on the fitting error under the same expansion order. Erefore, in practical application, ADM is limited by the expansion order k of the fitting function, so it cannot be fully applied to solve the arbitrary section beam It is almost impossible to substitute the fitting function T1(x) with k 200 into the equations (8), (9), and (12) for iterative computations. erefore, in practical application, ADM is limited by the expansion order k of the fitting function, so it cannot be fully applied to solve the arbitrary section beam
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