Method for calculating inherent frequency of Euler-Bernoulli beam through improved differential transformation method

A technology of natural frequency and transformation method, which can be used in complex mathematical operations and other directions, and can solve problems such as easy divergence and slow convergence rate.

Inactive Publication Date: 2014-07-02
BEIJING UNIV OF TECH
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But this series solution usually converges in a small area, the convergence rate is slow, and it is easy to diverge when the area is large

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  • Method for calculating inherent frequency of Euler-Bernoulli beam through improved differential transformation method
  • Method for calculating inherent frequency of Euler-Bernoulli beam through improved differential transformation method
  • Method for calculating inherent frequency of Euler-Bernoulli beam through improved differential transformation method

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Embodiment Construction

[0069] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments.

[0070] Embodiments The method of the present invention is used to solve the free vibration problem of a rectangular Euler-Bernoulli beam with free ends.

[0071] The relevant specific parameters of the beam are shown in Table 2.

[0072] Table 2 Basic parameters of the beam

[0073] length / m

Width / m

height / m

Young's Mo / Gpa

Density / Kg / m 3

Material

Poisson's ratio

12

2.12

2.06

174

7200

QT600

0.275

[0074] (1) Bring the known relevant parameters of the actual engineering beam into formula (8), establish the partial differential equation of the free vibration of the beam, and perform related transformations according to formulas (9)-(11).

[0075] (2) Put the known beam parameters into formula (12) to get the corresponding dimensionless value. According to formulas (13)-...

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Abstract

The invention relates to a method for calculating the inherent frequency of a Euler-Bernoulli beam through an improved differential transformation method. The method comprises the steps that the improved differential transformation method is deduced on the basis of a differential transformation method; a corresponding oscillatory differential equation is built, the improved differential transformation method is used for solving the free vibration problem of the homogeneous Euler-Bernoulli beam with two freely-supported ends, a control differential equation is converted into an algebraic equation, and boundary conditions are changed to an algebraic frequency equation convenient to calculate; corresponding algebraic operation is carried out to obtain the inherent frequency and the modal shape of any order of the differential equation. According to the method for calculating the inherent frequency of the Euler-Bernoulli beam through the improved differential transformation method, the improved differential transformation method is used for solving the free vibration problem of the homogeneous Euler-Bernoulli beam, an approximate solution of a nonlinear problem is obtained in the mode that iteration is carried out to converge the series, closed-form solutions such as the four-order inherent frequency and the modal shape are obtained, the solution of a Taylor expansion power series can be converged at a larger time interval, and the calculation process is rapider and more accurate.

Description

technical field [0001] The invention belongs to the field of mechanical vibration, and relates to an analysis method for the free vibration of Euler-Bernoulli beams, in particular to a method for calculating the natural frequency of Euler-Bernoulli beams based on an improved differential transformation method. Background technique [0002] Super-span heavy-duty gantry CNC machine tools are urgently needed manufacturing equipment in my country's electric power, aerospace, and large-scale shipbuilding industries, and are national strategic materials. The super-span heavy-duty beam is an extremely important load-bearing structural part of the heavy-duty gantry CNC machine tool, and its dynamic characteristics directly affect the accuracy and performance of the whole machine. Therefore, at the beginning of the design, it is very important to quickly obtain its dynamic performance. Euler-Bernoulli beams are usually used as the mechanical theoretical calculation model of super-sp...

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Application Information

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Patent Type & Authority Applications(China)
IPC IPC(8): G06F17/13
Inventor 蔡力钢殷云尧刘志峰
Owner BEIJING UNIV OF TECH
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