Purpose

The geometrically non-linear free and forced vibrations of a multi-span beam resting on an arbitrary number of supports and subjected to a harmonic excitation force is investigated.

Methods

The theoretical model developed here is based on the Euler–Bernoulli beam theory and the von Kármán geometrical non-linearity assumptions. Assuming a harmonic response, the non-linear beam transverse displacement function is expanded as a series of the linear modes, determined by solving the linear problem. The discretised expressions for the beam total strain and kinetic energies are then derived, and by applying Hamilton’s principle, the problem is reduced to a non-linear algebraic system solved using an approximate method (the so-called second formulation). The basic function contribution coefficients to the structure deflection function and the corresponding backbone curves giving the non-linear amplitude-frequency dependence are determined. Considering the non-linear forced response, an approximate multimode approach has been used in the neighbourhood of the predominant mode, to obtain numerical results, for a wide range of vibration amplitudes.

Results

The effects on the non-linear forced dynamic response of the support number and locations, the excitation frequency and the level of the applied harmonic force (a centered point force or a uniformly distributed force) have been investigated and illustrated by various examples.

中文翻译：

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欧拉-伯努利多跨梁几何非线性强迫振动的多模方法

目的

研究了搁在任意数量的支撑架上并受到谐波激励力的多跨梁的几何非线性自由振动和强迫振动。

方法

这里开发的理论模型是基于Euler–Bernoulli梁理论和vonKármán几何非线性假设的。假定为谐波响应，则非线性光束横向位移函数将扩展为通过解决线性问题而确定的一系列线性模式。然后导出梁总应变和动能的离散表达式，并通过应用汉密尔顿原理，将该问题简化为使用近似方法（所谓的第二种公式）求解的非线性代数系统。确定对结构挠度函数的基本函数贡献系数以及给出非线性幅度-频率依赖性的相应主干曲线。考虑到非线性强制响应，

结果

已经通过各种示例研究和说明了支撑数量和位置，激励频率和施加的谐波力（中心点力或均匀分布的力）对非线性强迫动力响应的影响。