含支撑阶梯梁自由振动的解析型梁段叠加法

An Analytical Beam-segment Superposition Method for Free Vibration of Braced Stepped Beams

针对多跨阶梯梁尚无简洁解析形式固有频率方程的现状,本文得到含支撑多跨阶梯梁频率方程的解析形式.首先基于Euler-Bernoulli梁理论,获得两端弹性约束单跨梁的频率方程,其表达式由梁坐标的三角函数和双曲三角函数的乘积组成,同时含有两端边界对应的横向刚度、旋转刚度等4个参数.给出至少含一个弹簧刚度约束的6种常见边界条件下梁的频率方程,其形式相对较简洁.然后把阶梯截面多跨梁的自由振动等效为各拆分梁段自由振动叠加的分析模型,结合所提出梁段连接节点处微段满足的动平衡方程,推导出多跨阶梯梁频率方程组的闭合解表达式.对于具有不同边界条件和内部支撑多跨梁的几种情况,算例计算出对应多跨梁的前几阶自振频率和振型图.阶梯形截面多跨梁与等截面多跨梁的频率方程可用统一的形式表示.所得解析结果与已有文献结果比较后发现:所得解析解同有限元结果的相对偏差小于1%,说明本文方法合理有效.阶梯多跨梁的自振频率随支撑刚度值、支撑杆位置和突变截面前后的惯性半径、惯性矩变化而变化.所得解析形式的频率方程在理论上未作近似,因此是精确的,形式上相对简单,具有良好的应用价值,故可用于评价其他数值方法的计算精度.

In order to solve the current situation of multi-span stepped beam without simple analytical form of natural frequency equation,an analytical model of closed solution for free vibration of multi-span stepped beams is established in this paper.Firstly,based on the Euler-Bernoulli beam theory,the frequency equation of a single span beam with elastic boundaries is calculated,which composed of trigonometric functions and hyperbolic trigonometric functions.And also the four parameters of transverse spring stiffness and rotational spring stiffness are included in the obtained frequency equation.The beam’s frequency equations for six cases of different boundary conditions including at least one kind of spring stiffness are presented in table form.Then a free vibration analysis model for multi-span beam is established,in which the multi-span beam with stepped sections is equivalent to the superposition of vibration of split beam segments with elastic boundary.Combined with the dynamic equilibrium equation presented by the infinitesimal section located at the linked node between beam segments,the closed-form expressions for natural frequency of multi-span stepped beam are obtained.For several cases of multi-span beam with different boundaries or internal supports,the first several natural frequencies and mode shapes are obtained in the numerical examples.The frequency characteristic equation of a stepped multi-span beam and a uniform-section multi-span beam can be unified in the same form.Compared with the results by using the finite element simulation,the relative deviation is less than 1%,which indicates that the proposed method is reasonable and effective.The natural vibration frequency of a multi-span stepped beam varies with the position,radius and moment of inertia of the supporting bar.The analytical frequency equation is theoretically accurate with a brief form,which has good potential application value and can be used to validate the precision of other numerical methods.