In this paper, the governing differential equation of a beam with axial force is studied using the Lie symmetry method. Considering the inhomogeneous beam and non-uniform axial load, the governing equation is a fourth-order linear partial differential equation with variable coefficients with no closed-form solution. We search for a favourable coordinate system where the governing equation has a simpler-form or a closed-form solution. A favourable coordinate transformation is found using the Lie transformation group method. The system of determining equations for the governing equation of a beam with non-uniform axial load is derived and then solved to find a favourable coordinate system dependent on the spatially variable stiffness, mass, and axial force. The class of non-uniform axially loaded beams which have a closed-form solution is determined. The fixed-free boundary condition is imposed to find the invariant closed-form solution. A comparison between the analytical solution derived by the Lie symmetry method and the numerical solution is presented. Lie symmetry analysis yields hitherto undiscovered closed-form solutions for non-uniform axially loaded beams.

中文翻译：

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使用李对称分析的非均匀轴向载荷梁的闭合形式解

本文采用李对称方法研究了轴受力梁的控制微分方程。考虑梁不均匀和轴向载荷不均匀，控制方程为变系数的四阶线性偏微分方程，无闭解。我们寻找一个有利的坐标系，其中控制方程具有更简单的形式或封闭形式的解。使用李变换群方法找到了一个有利的坐标变换。推导出具有非均匀轴向载荷的梁的控制方程的确定方程的系统，然后求解以找到取决于空间可变刚度、质量和轴向力的有利坐标系。确定具有封闭形式解的非均匀轴向载荷梁的类别。施加固定自由边界条件以找到不变的封闭形式解。给出了由李对称方法导出的解析解与数值解之间的比较。Lie 对称分析为非均匀轴向加载梁提供了迄今为止尚未发现的封闭形式解。