针对非圆弧拱面内线性自由振动没有解析的现状，本文提出了一种变系数平衡微分方程近似解析方法来解决该问题。基于笛卡尔直角坐标系下非圆弧拱线性应变与Hamilton原理，推演了非圆弧拱面内自由振动变系数平衡微分方程；基于陡拱与浅拱面内振型没有显著差异的基本假定，将该变系数平衡微分方程对应的常系数平衡微分方程的通解，代入变系数平衡微分方程，得到该变系数平衡微分方程的不平衡差；当该不平衡差沿全拱积分为零时自振频率误差最小，进而得到非圆弧拱面内自振频率高精度实用解析。基于所提出的变系数平衡微分方程近似解析方法，推演了非圆弧两铰拱与无铰拱面内自振频率实用解析，并阐明了非圆弧拱与同参数直梁面内自振频率的逻辑关系。抛物线、悬索线、悬链线与组合线等常用非圆弧两铰拱与无铰拱自由振动算例结果表明：本文的基本假定得到了严格检验；本文方法自振频率与有限元结果吻合较好，非圆弧拱前十阶自振频率中，两铰拱自振频率最大相对误差为7.71%，无铰拱自振频率最大相对误差为4.34%；非圆弧拱与同参数直梁面内自振频率的比例系数，可为行业规范条文修订提供参考。
Abstract
To derive the closed-form solution for the in-plane free vibration of non-circular arches, an approximate analytical method for the solution derivation of the variable coefficient differential equation is proposed. Based on linear strains of non-circular arches in the Cartesian coordinate system and the Hamilton principle, a variable coefficient equilibrium differential equation for the in-plane free vibration of non-circular arches is derived. The analytical general solution of the corresponding constant coefficient differential equation is substituted into the variable coefficient differential equation to yield the unbalanced deviation following by that the in-plane free vibration modes of non-circular steep arches and shallow arches are almost the same. The practical closed-form solution for the in-plane natural frequency is derived by setting the integration of the unbalanced deviation of the variable coefficient differential equation along the span being zero. The analytical solutions of the pin-ended and fixed arch in the Cartesian coordinate system are derived based on the proposed method, and the theoretical relationship of the in-plane frequency between non-circular steep arches, shallow arches and the straight beams with the same parameter are demonstrated. The numerical results of non-circular pin-ended and fixed arches show that, the basic assumptions has been strictly verified; the proposed method agrees well with the finite element method, the maximum relative error of the former tenth natural frequency are 7.71% and 4.34% for pin-ended arches and fixed arches, respectively; the theoretical relationship of natural frequency between non-circular arches and straight beam with the same parameter can be used in the code revision of arch bridges.
关键词
非圆弧拱 /
自由振动 /
Hamilton原理 /
变系数微分方程 /
实用解析
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Key words
Non-circular arches /
Free vibration /
Hamilton principle /
Variable coefficient differential equation /
Practical closed-form solution.
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