Purpose

A new expression of closed-form solution is given to visually demonstrate the influence of the nonlinear foundation on the responses of a beam subjected to a moving load.

Methods

Greatly different from previous works, the nonlinear partial differential governing equation of the beam motion is discretized into sets of nonlinear Volterra integral equations by using the Laplace transform. Then the Adomian decomposition method combined with a simple iterative formula are employed to analytically solve the Volterra integral equations.

Results

The closed-form solution for responses of the beam can be expressed in the form of infinite series with mode shapes. A good agreement between analytical and numerical results of the responses of the beam for different parameters is observed. It is clear to visually demonstrate the influence of the nonlinear foundation in the closed-form solution.

Conclusion

Nonlinearity of foundation makes the modal coefficients in the closed-form solution, expressed in the form of infinite series with mode shapes, become more complex. When the damping coefficient of the foundation is very small, the beam has more possibility of resonance for the case of nonlinear foundation.

中文翻译：

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基于Volterra积分方程的非线性基础对梁对移动荷载响应的影响研究。

目的

给出了闭式解的新表达式，以直观地证明非线性基础对梁在移动载荷作用下的响应的影响。

方法

与以前的工作有很大不同，使用拉普拉斯变换将梁运动的非线性偏微分控制方程离散化为非线性Volterra积分方程组。然后，将Adomian分解方法与简单的迭代公式结合起来，对Volterra积分方程进行解析求解。

结果

光束响应的封闭形式解决方案可以以具有模式形状的无穷级数形式表示。观察到光束对不同参数的响应的解析结果和数值结果之间的良好一致性。可以直观地证明非线性基础在封闭形式解中的影响。

结论

基础的非线性使闭合形式的解中的模态系数变得更加复杂，以模态无限级数的形式表示。当基础的阻尼系数非常小时，对于非线性基础，梁具有更大的共振可能性。