卡斯蒂利亚诺定理中积分符号下的微分

J. Rédl
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引用次数: 0

摘要

在这篇文章中,我们讨论了用定参数积分定义的函数的微分的应用。从材料力学的角度对数学问题进行了分析。建立了受载梁的数学模型。应用截面法确定了弯矩的精确函数。忽略了悬臂节点的附加性能。利用积分符号下的莱布尼茨求导规则,给出了修正Castigliano定理的推导。应用修正的卡斯蒂利亚诺定理,得到了梁挠度的精确解。在PTC Mathcad Prime软件(PTC Parametric Technology)中完成了梁挠度的精确求解。在Microsoft Visual c# 2010编写的程序中,对弯矩数据集和定义的挠度函数进行了数值积分。数值积分采用Dormand-Prince数值积分法。通过对精确解和数值解的比较,得出了数值积分解的误差。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Differentiating under integral sign in Castigliano’s theorem
In this contribution we are dealing with application of differentiating of function defined with determined parametrical integral. The mathematical problem was analyzed form point of view of mechanics of materials. The mathematical model of loaded beam was created. Applying the cross section method we defined the exact function of bending moment. Additional properties of cantilever joint were neglected. We showed the derivation of modified Castigliano’s theorem via Leibnitz rule of differentiating under integration sign. Appling the modified Castigliano’s theorem we got the exact solution of the deflection of the beam. The exact solution of beam deflection was finished in PTC Mathcad Prime software (PTC Parametric Technology). The numerical integration of the bending moment dataset and defined deflection function was done in program written in Microsoft Visual C# 2010. The Dormand-Prince numerical integration method was used for the numerical integration. Comparing the exact and numerical solution we got the error of numerical integration solution.
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