含支撑阶梯梁自由振动的解析型梁段叠加法

鲍四元，吴佳丽，沈峰; Bao Siyuan; Wu Jiali; Shen Feng

引 en

含支撑阶梯梁自由振动的解析型梁段叠加法^*

鲍四元
机构：
苏州科技大学土木工程学院工程力学系,苏州 215011
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，吴佳丽
机构：
苏州科技大学土木工程学院工程力学系,苏州 215011
×
，沈峰
机构：
苏州科技大学土木工程学院工程力学系,苏州 215011
×

苏州科技大学土木工程学院工程力学系,苏州 215011；

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

Bao Siyuan
Affiliation：
School of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011 ,China
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，Wu Jiali
Affiliation：
School of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011 ,China
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，Shen Feng
Affiliation：
School of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011 ,China
×

School of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011 ,China；

通讯作者:

鲍四元,E-mail:bsiyuan@126.com

中图分类号:O327

文献标识码:A

文章编号:1672-6553-2023-21(3)-085-011

DOI:10.6052/1672-6553-2022-017

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出版信息

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目录contents

摘要 Abstract
关键词 Keywords
1 单跨弹性梁的动力分析
1.1 几何模型
1.2 弹性支撑梁的频率特征方程
1.3 不同边界条件下梁的频率方程
1.4 逐段叠加时节点处的平衡方程
2 含支撑梁的动力学模型
2.1 求解思路
2.2 两类振动模型分析
2.2.1 两跨阶梯截面梁
2.2.2 内部含支撑多跨阶梯截面梁
3 数值分析
4 结论
参考文献

摘要

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.

Abstract

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.

关键词

阶梯梁；自由振动；固有频率；解析型叠加法

Keywords

stepped beam ； free vibration ； natural frequency ； analytical superposition method

引言
在机械系统中，截面不等的阶梯形杆在实际中应用比较广泛，如石油工程中的阶梯形抽油杆、机械传力的阶梯轴、钻井中的钻柱等.另外，船舶推进轴系的横向弯曲振动往往由不平衡的旋转件或推进器产生周期性弯曲变形引起，该轴系振动问题属于含多弹性支撑梁的振动问题.
在梁的动力分析中，准确计算结构的固有频率和振型具有重要意义.现有振动理论教材^[1，2]一般只提供均质等截面杆件弯曲自由振动的解答.而阶梯梁构件横向振动对应固有频率的求解较为复杂.近年来，学者们对多跨梁结构的振动问题进行了一定的分析和研究.Johansson等^[3]对梁振动的控制微分方程进行拉普拉斯变换，在频域内求解各阶振型，导出了多跨Euler-Bernoulli梁动力问题的一种封闭解.Zhao等^[4]应用Hamilton原理建立结构的运动方程，得到多跨梁的固有频率和自由振动响应，利用插值函数修正了多跨Timoshenko梁的模态振型，模拟了多跨Timoshenko梁的振动模态.Shi等^[5]将模态摄动法推广应用于求解具有任意附加质量和弹性支承的连续非均匀简支梁的振动方程.吴琛等^[6]采用双变量无单元法以广义移动最小二乘法为理论基础，同时考虑挠度和转角双变量，建立了欧拉梁的质量矩阵和刚度矩阵，并进行自由振动的计算.Goranka等^[7]采用格林函数法计算了由任意数目的常刚度平动弹簧支撑的Euler-Bernoulli梁固有频率和振型.周海军等^[8]基于欧拉梁理论和改进傅里叶级数法研究了带弹性支撑多跨距变轴径的轴系振动特性.周渤等^[9]基于改进傅里叶级数给出连续多跨梁结构振动特性的分析.鲍四元等^[10，11]基于改进傅里叶级数法给出弹性支撑多跨梁振动特性分析的两种方法.Yan等^[12]利用含辅助项的傅里叶级数展开法研究了核工程中含弹性支撑燃料棒的振动特性问题.谢瑾荣等^[13，14]基于复合单元法研究阶梯悬臂梁的自由振动并对其进行了实验验证.但上述文献对截面呈现阶梯变化的多跨梁自由振动的理论研究相对较少.
数值解法上，Reddy等^[15]提出适合求解Euler-Bernoulli梁和剪切变形梁理论的位移有限元模型，以Timoshenko梁理论的精确解形式建立了无剪切闭锁有限元模型，该程序可扩展到建立简化三阶梁理论的有限元模型.Moallemi-Oreh等^[16]建议了一种2节点单元用于分析Timoshenko梁动力与屈曲问题，其中挠度采用3次多项式插值，转角采用2次多项式插值，所建立单元在分析振动和屈曲问题时能避免剪切自锁现象，且具有精度好的优点.Lai等^[17]基于Adomian分解方法研究了各种支承条件下Euler-Bernoulli梁的自由振动问题，Mao等^[18]继续应用Adomian分解法研究了阶梯形Euler-Bernoulli梁的自由振动. $\dot{I}$ brahim ^[19]采用切比雪夫小波配置法研究了不同支承条件下非均匀Euler-Bernoulli梁的自由振动问题.Wang等^[20]提出了一种微分积分单元法，用于任意组合边界条件下混合非局部Euler-Bernoulli梁的自由振动分析.Zhang等^[21-22]对杆系结构离散为若干微段，且微段间通过带旋转弹簧的铰链连接，提出基于Hencky杆链划分的一种差分法求解，并成功应用于含内部弹簧的杆结构^[21]、地基上的变截面梁结构^[22]的振动和屈曲问题.而对Euler阶梯梁和Timoshenko阶梯梁的振动问题，Lee等^[23]对控制方程和边界条件、连续性条件分别应用Chebyshev谱展开法进行了研究，所得低阶和高阶频率都与理论解和实验结果吻合.吴兵^[24]提出组合动力系统中的一种精确梁单元并基于该精确单元进行了刚架结构自由振动的分析.Zhu等^[25]对具有一般端部边界条件的功能梯度多段梁结构的振动特性提出一种基于傅里叶级数展开的分析方法.
然而，以上对梁结构自由振动问题的研究大多具有一定的局限性，例如，瑞利-里兹法和有限元法计算梁结构的自振频率时属于近似方法，且部分方法存在数值不稳定或漏根现象.解析解法作为各种数值方法或近似方法的重要验证手段，受到广泛的关注.周盛林等^[26]基于Euler-Bernoulli梁理论建立了双跨梁结构的动力学模型，采用模态分析方法直接求解动力学方程得到简支双跨梁的固有频率及梁上点的时间响应历程曲线.
Banerjee等^[27]推导出单轴对称开口截面薄壁梁的动力刚度矩阵，应用Wittrick-Williams算法求出频率方程的根，从而得到薄壁梁的各阶频率.叶康生等^[28]考虑双向弯扭耦合振动下薄壁杆件的自由振动问题，用动力刚度法求解了薄壁梁和刚架的频率和振型.叶康生等^[29]还据此提出一种有限元新型超收敛算法，并以杆件轴向振动为例进行验证研究.近年来，叶茂等^[30]根据弹性支承点的协调性，基于转换矩阵法，给出不同边界条件且中间带弹性支承的欧拉连续梁的特征方程并求解频率; 谢秋林等^[31]采用类似方法研究了Rayleigh梁的情况.Zhang等^[32]对于非连续元件和复杂边界条件下多跨连续梁的自由振动问题，利用分离变量法和传递矩阵法推导得到梁的特征方程，并做数值计算.刘向尧等^[33]运用参数变易法研究了多跨的三种梁（Timoshenko梁、Rayleigh梁和剪切梁）的横向自由振动模型.Luo等^[34]基于由三角函数、双曲函数和Heaviside阶跃函数组成的通用函数求解中间含弹性支撑的Euler-Bernoulli梁和Timoshenko梁自由振动问题的封闭解.
但这些方法的求解过程相对较为复杂，且不易得到闭合解形式表示的梁结构频率方程，或所得方程不够简洁，实际应用上还有待简化或改进.
为检验各种数值方法的正确性和精度，本文提出一种新型模态分析方法来研究多跨梁的自由振动.其步骤如下:首先，将多跨梁拆为多个单跨梁结构，其连接处需添加人工弹簧，在局部坐标下研究这些多跨梁的频率方程，然后进行梁段叠加，综合单跨梁的频率方程和连接处的动平衡方程，化简得到频率特征方程式; 最后求解多跨梁结构的各阶频率和振型.所提方法基于梁段拆分和复合的原理，得到固有频率解析方程组，其表达形式一般较为简洁，其中所提连接处的动平衡方程对各段独立的固有频率方程起到联系作用，联立后形成固有频率方程组.所提方法为弹性支撑多跨梁提供新颖的求解思路，且得到基于Euler-Bernoulli梁理论多支撑阶梯梁固有频率方程的精确解析形式.
1 单跨弹性梁的动力分析
1.1 几何模型
设长度为L的均质直梁.在梁左右两端分别设置竖直方向的线位移弹簧和旋转弹簧来模拟各种边界条件.左端边界中竖直方向的线弹簧和旋转弹簧的刚度分别为c₁和α，右端边界中竖直方向的线弹簧和旋转弹簧的刚度分别为c₂和β.当边界条件中存在某个方向的位移为固定时，需把对应方向的线弹簧或旋转弹簧的刚度值设为无穷大; 而某方向上弹簧的刚度值设为零则代表不限制该方向位移.此类弹簧称为人工弹簧，弹簧刚度取一般值时可模拟弹性边界条件.
图1 梁端弹性约束的Euler-Bernoulli梁
Fig.1 Euler-Bernoulli beam with elastic supports at the ends
1.2 弹性支撑梁的频率特征方程
取其轴线为x轴，建立如图1所示的坐标系.记坐标为x处梁的横截面积为A，材料的弹性模量为E，质量密度为ρ，截面关于中性轴的惯性矩为I.用w（x，t）表示坐标为x的截面轴线上的点在时刻t的横向位移.考虑Euler-Bernoulli梁的自由振动，等截面均质直梁的弯曲自由振动微分方程^[1]如下:

ρ A \frac{\partial^{2} w}{\partial t^{2}} + E I \frac{\partial^{4} w}{\partial x^{4}} = 0

(1)

设梁具有谐函数形式的横向位移，即

w (x, t) = W (x) s i n (ω t + φ)

(2)

其中固有振型函数
$\begin{matrix} W (x) = a_{1} c o s s x + a_{2} s i n s x + \\ a_{3} c o s h s x + a_{4} s i n h s x \end{matrix}$
ω为梁的自振频率，参数 $s = \sqrt[4]{\frac{ρ A ω^{2}}{E I}}$ .
对图1所示的两端弹性支撑梁，其边界条件为

\{\begin{matrix} {M|}_{x = 0} = {\frac{d^{2} W}{d x^{2}}|}_{x = 0} = {α \frac{d W}{d x}|}_{x = 0} \\ {Q|}_{x = 0} = {E I \frac{d^{3} W}{d x^{3}}|}_{x = 0} = - c_{1} W (0) \\ {M|}_{x = L} = {\frac{d^{2} W}{d x^{2}}|}_{x = L} = - {β \frac{d W}{d x}|}_{x = L} \\ {Q|}_{x = L} = {E I \frac{d^{3} W}{d x^{3}}|}_{x = L} = c_{2} W (L) \end{matrix}

(3)

引入如下参数

\begin{matrix} C_{a} = \frac{c_{1} L^{3}}{E I}, C_{b} = \frac{c_{2} L^{3}}{E I}, \\ R_{α} = \frac{α L}{E I}, R_{β} = \frac{β L}{E I}, \\ ξ = \frac{x}{L}, u = s L \end{matrix}

(4)

则有
$\begin{matrix} \bar{W} (ξ) = a_{1} c o s u ξ + a_{2} s i n u ξ + \\ a_{3} c o s h u ξ + a_{4} s i n h u ξ \end{matrix}$
且式（3）化为:

\{\begin{matrix} {\frac{d^{2} \bar{W}}{d ξ^{2}}|}_{ξ = 0} = {R_{α} \frac{d \bar{W}}{d ξ}|}_{x = 0} \\ {\frac{d^{3} \bar{W}}{d ξ^{3}}|}_{ξ = 0} = - C_{a} \bar{W} (0) \\ {\frac{d^{2} \bar{W}}{d ξ^{2}}|}_{ξ = 1} = - {R_{β} \frac{d \bar{W}}{d ξ}|}_{ξ = 1} \\ {\frac{d^{3} \bar{W}}{d ξ^{3}}|}_{ξ = 1} = C_{b} \bar{W} (1) \end{matrix}

(5)

由式（5）可得关于a₁，a₂，a₃，a₄的齐次方程组.而系数有非零解的条件是其系数行列式为零，化简得到两端弹性支撑梁的频率特征方程为

\begin{matrix} F (u) = u^{8} (1 - H_{1}) - u^{7} (R_{α} + R_{β}) P_{2} - \\ 2 u^{6} R_{a} R_{β} H_{4} + u^{4} [(R_{a} C_{a} + C_{b} R_{β}) (1 + H_{1}) + \\ 2 (R_{β} C_{a} + C_{b} R_{a}) H_{1}] + u^{5} (C_{a} + C_{b}) Q_{2} + \\ u^{3} R_{α} R_{β} (C_{a} + C_{b}) P_{2} + 2 u^{2} C_{a} C_{b} H_{4} - \\ u C_{a} C_{b} (R_{α} + R_{β}) Q_{2} + \\ R_{a} C_{a} C_{b} R_{β} (1 - H_{1}) = 0 \end{matrix}

(6)

其中，P₂，Q₂，H₁，H₂，H₃和H₄分别如下:

\begin{matrix} P_{2} = H_{2} + H_{3} \\ Q_{2} = H_{2} - H_{3} \\ H_{1} = c o s u c o s h u \\ H_{2} = c o s u s i n h u \\ H_{3} = s i n u c o s h u \\ H_{4} = s i n u s i n h u \end{matrix}

(7)

解出系数a₁，a₂，a₃，a₄后，易得梁的模态函数.
1.3 不同边界条件下梁的频率方程
在以下叙述中，将简支边界记为S，自由边界记为F，固支边界记为C，弹性支撑边界记为E.另外刚度c₁、α、c₂和β的含义参见图1中所示的弹簧刚度，而刚度因子C_a、C_b、R_α、R_β的定义按式（4）规定，其中下标与左端或右端对应.
以左端固支、右端弹性约束梁为例，示意图如图2所示.
图2 固支-弹性约束梁的示意图
Fig.2 A clamped-elastically constrained beam
根据图2中左端边界形式，频率方程可基于式（6）用极限形式表示:

\underset{C_{a}, R_{α} \to \infty}{l i m} \frac{F (u)}{C_{a} R_{α}} = 0

(8)

化简得:

\begin{matrix} u^{4} (1 + H_{1}) + u^{3} R_{β} P_{2} - u C_{b} Q_{2} + \\ C_{b} R_{β} (1 - H_{1}) = 0 \end{matrix}

(9)

若图2中右端的弹性支撑改为刚性支撑时，C_b也趋近于无穷大，类似地通过极限运算得频率方程为

R_{β} = \frac{u_{a} Q_{2} (u_{a})}{1 - H_{1} (u_{a})}

(10)

对于不同边界的若干种单跨梁，相应的固有频率方程及特例情况下的固有频率方程归纳为表1.
表1 不同边界条件下梁的频率方程
Table1 Frequency equations of beams under different boundary conditions
注:1. Sym指变形左右对称时所取半结构的中间截面边界，这里是定向支座; AntiSym对应变形左右反对称时的情况，取半结构时中间支座为链杆支座.
2.H₁，P₂，Q₂和H₄的表达式见式（7）.
1.4 逐段叠加时节点处的平衡方程
对多跨梁的自由振动问题，基于如下思路研究:将多跨连续梁等效为若干段梁叠加，一般n个内部支座可把原梁分为（n+1）个梁段.每个梁段在连接处的边界设为弹性约束，边界刚度值的记法同图1，即竖向刚度c₁、c₂以及旋转刚度α、β.根据式（7）由边界条件可得到各段梁的频率方程.另外，分析原系统的动力学特性还需要建立各跨连接处的动平衡条件.
图3 两跨阶梯悬臂梁
Fig.3 A two-span stepped cantilever
图4 两段拆分梁的叠加
Fig.4 The superposition of the two separated beams
各跨连接处的动平衡条件通过图3所示阶梯截面悬臂梁为例说明.可把双跨悬臂梁等效为两段单跨梁（AB^-和B⁺C）的叠加，如图4所示.其中，B^-，B⁺上标中的“-”“+”对应该微段的“左侧”“右侧”.
图4（c）为双跨梁拆分后的节点微元段，微元段B^-B⁺做平面运动，对该微元段的动力学问题按照动静法分析，受力图中 $d m \cdot {\ddot{y}}_{A}$ 是虚加的惯性力，且虚加惯性矩 $Δ M = d J \ddot{θ}$ .由达朗贝尔原理有:

\{\begin{matrix} c_{0} y_{A} + d m \cdot {\ddot{y}}_{A} - c_{1} y_{A} - c_{2} y_{A} = 0 \\ \sum_{i = 1}^{2} M_{i} + Δ M = β θ + α θ + d J \ddot{θ} = 0 \end{matrix}

(11)

式（11）中微元段的质量dm和转动惯量dJ均趋近于零，可以忽略.化简式（11）得连接节点两侧的旋转刚度和为零，竖向刚度和为c₀，即有:

\{\begin{matrix} c_{1} + c_{2} = c_{0} \\ α + β = 0 \end{matrix}

(12)

或转换为无量纲因子形式，即

\{\begin{matrix} \frac{I_{A B}}{I_{B C}} \frac{C_{2}}{a^{3}} + \frac{C_{1}}{(1 - a)^{3}} = \frac{c_{0}}{E I_{B C}} \\ \frac{I_{A B}}{I_{B C}} \frac{R_{β}}{a} + \frac{R_{α}}{1 - a} = 0 \end{matrix}

(13)

其中AB^-段中的C₂、R_β和B⁺C段中的C₁和R_α分别基于式（4）定义.式（12）或式（13）称为结点B的动平衡方程.
2 含支撑梁的动力学模型
2.1 求解思路
多跨连续梁作动力学分析时，可按支座自然拆分得到若干梁段的自由振动问题，其连接处需添加相关的人工弹簧（如图4中的横向弹簧和旋转弹簧）.具体求解分为两步，首先，由这些梁段的频率方程结合各跨连接处的动平衡方程，可得出整个结构固有频率所满足的频率方程，解超越方程得到各阶固有频率.
然后，基于表1中的频率方程，可得各拆分段中所添加人工弹簧的刚度值.最后对每段拆分梁得出对应的振型函数，结合连接处物理变量的连续性可得多跨连续梁的模态.下面通过两跨、多跨阶梯截面梁为例分别说明.
2.2 两类振动模型分析
2.2.1 两跨阶梯截面梁
本节考虑一个两跨悬臂梁自由振动的例子.如图3所示，但跨中支撑改为链杆支撑.
对于左跨拆分梁，如图4（a）所示，左端固支，右端竖向刚度为无穷大，且旋转刚度对应的无量纲因子为 $R_{β}^{（ a ）}$ ，由表1知与固有频率对应的参量u_a所表示的旋转弹簧的表达式为:

R_{β}^{(a)} = \frac{u_{a} Q_{2} (u_{a})}{1 - H_{1} (u_{a})}

(14)

H₁（u_a）、Q₂（u_a）按式（7）定义，且 $u_{a} = s_{a} a ， s_{a} = \sqrt[4]{\frac{ρ A ω_{n}^{2}}{E I_{A B}^{（ a ）}}} ， R_{β}^{（ a ）} = \frac{β a}{E I_{A B}^{（ a ）}}$ .
对于右跨拆分梁，左端竖向刚度为无穷，旋转刚度对应的无量纲因子为 $R_{α}^{（ b ）}$ ; 右端自由，由表1知频率方程为:

R_{α}^{(b)} = \frac{- Q_{2} (u_{b}) u_{b}}{1 + H_{1} (u_{b})}

(15)

其中
$u_{b} = s_{b} - a s_{b}, R_{α}^{(b)} = \frac{α (1 - a)}{E I_{B C}^{(b)}}$
由节点处的动平衡方程可知

\frac{R_{α}^{(b)}}{u_{b}} + p_{a b} \frac{R_{β}^{(a)}}{u_{a}} = 0

(16)

其中p_ab表示AB^-段和B⁺C段截面的惯性矩比，即 $p_{a b} = I_{A B}^{（ a ）} / I_{B C}^{（ b ）}$ .将式（14）、式（15）代入式（16），化简得固有频率因子s_a和s_b满足:

p_{a b} \frac{Q_{2} (a s_{a})}{1 - H_{1} (a s_{a})} - \frac{Q_{2} (s_{b} - a s_{b})}{1 + H_{1} (s_{b} - a s_{b})} = 0

(17)

表2 两跨悬臂梁前2阶固有频率随链杆位置的变化
Table2 The first two order frequencies for a 2-span cantilever with varying location of the chain rod
注:表中括号内的数字表示以百分比表示的相对偏差.
梁长度为1m，考虑等截面梁的情况.均质矩形截面中宽b=0.1m，高h=0.1m.材料密度ρ=7850kg/m³，弹性模量E=2.06×10¹¹Pa.当a=0.1，0.2，0.5，0.6，0.8，1时按式（17）计算得到含中间链杆支撑梁的前2阶固有频率如表2所示.与文献^[10]中的数值解对比，结果显示固有频率的相对偏差在1.0%之内，从而验证本文方法的正确性.
特殊地，对跨正中设置链杆的悬臂梁，其特征方程为cosu（cotu-cothu）=0，其中u=1/2 $\sqrt[4]{\frac{ρ A ω_{n}^{2}}{E I}}$ .解得梁前两阶关于无量纲频率因子u₁=1.5708，u₂=3.9266.代入表1中的频率方程，可得拆分段中所添加人工弹簧的刚度值.跨正中设置链杆时，用分段函数表示梁的模态如下:

W (x) = \{\begin{matrix} λ_{1} [c o s (n x) - c o s h (n x)] + s i n (n x) - \\ s i n h (n x), x \in [0,0.5] \\ λ_{2} [c o s (n - x n) + c o s h (n - x n)] + λ_{3} [s i n (n - \\ x n) + s i n h (n - x n)], x \in (0.5,1] \end{matrix}

(18)

其中n=u_i/L，L=0.5m.参数λ₁、λ₂和λ₃根据2.1节中的方法求得.图5给出a=0.5m时双跨悬臂梁的前2阶振型.
图5 a=0.5m时两跨梁的前2阶振型
Fig.5 The first two modes of a two-span cantilever beam with a=0.5m
2.2.2 内部含支撑多跨阶梯截面梁
本节选取3跨阶梯截面梁自由振动的问题，梁中间分别含有刚性支撑和弹性支撑.
（一）梁内部含刚性支撑的情况
首先研究图6所示含中间刚性支撑三跨圆截面梁的固有频率.边界为左端简支，右端刚性支撑，三个跨的长度分别为L_i（i=1，2，3），各跨截面的面积和惯性矩分别为A_i和I_i（i=1，2，3），三跨梁中间的支撑为刚性支撑.
图6 含刚性支撑的三跨梁示意图
Fig.6 A three-span beam with rigid supports
图7 三段单跨梁的叠加图
Fig.7 The superposition of the three segments of beam
对该三跨梁进行逐段叠加法计算固有频率时，如图7，将第一段拆分梁弹性支撑处的旋转刚度记为β^（1），第二段拆分梁左、右端弹性支撑处的旋转刚度记为α^（2）、β^（2），第三段梁弹性支撑处的旋转刚度记为α^（3）.由结点动平衡条件中的“刚度关系”知:

α^{(2)} + β^{(1)} = 0, β^{(2)} + α^{(3)} = 0

(19)

与图7中三段梁对应的固有频率无量纲因子分别记为:

u_{k} = s_{k} L_{k} = L_{k} \sqrt[4]{\frac{ρ}{E} \frac{ω^{2}}{i_{k}^{2}}} (k = 1,2, 3)

(20)

其中i_k（其中k=1，2，3）为第k段梁截面的惯性半径，其定义为 $i_{k} = \sqrt{I_{k} / A_{k}}$ ，从而有 $u_{2} / u_{1} = \sqrt{i_{1} / i_{2}} L_{2} / L_{1}$ 。为方便计算，设A₁=A₃，I₁=I₃，L₁=L₂=L₃=1m，则有 $u_{1} = u_{3} = \sqrt{i_{2} / i_{1}} u_{2}$ .
图7中，对拆分的第一段梁，左端竖向刚度为无穷，转动刚度为0，右端竖向刚度为无穷，由表1得 $R_{β}^{（ 1 ）}$ 满足:

\frac{β^{(1)} L_{1}}{E I_{1}} = R_{β}^{(1)} = \frac{2 u_{1} H_{4} (u_{1})}{Q_{2} (u_{1})}

(21)

对拆分的第三段梁，右端竖向刚度为无穷，转动刚度为0，右端考虑竖向刚度为无穷，由表1得到 $R_{α}^{（ 3 ）}$ 的表达式为:

\frac{α^{(3)} L_{3}}{E I_{3}} = R_{α}^{(3)} = \frac{2 u_{3} H_{4} (u_{3})}{Q_{2} (u_{3})}

(22)

对拆分的第二段梁，左端和右端的竖向刚度为无穷，两端的旋转刚度设为 $R_{α}^{（ 2 ）}$ ， $R_{β}^{（ 2 ）}$ ，由表1得其频率方程为

\begin{matrix} R_{α}^{(2)} R_{β}^{(2)} [1 - H_{1} (u_{2})] + 2 u_{2}^{2} H_{4} (u_{2}) - \\ u_{2} (R_{α}^{(2)} + R_{β}^{(2)}) Q_{2} (u_{2}) = 0 \end{matrix}

(23)

由式（13）得

\{\begin{matrix} p_{21} R_{α}^{(2)} = - R_{β}^{(1)} \\ p_{23} R_{β}^{(2)} = - R_{α}^{(3)} \end{matrix}

(24)

其中p₂₁=I₂/I₁.将式（24）代入式（21）~式（23），化简得到三跨梁的固有频率方程:

\begin{matrix} 2 u_{1}^{2} H_{4}^{2} (1 - {\tilde{H}}_{1}) + u_{2}^{2} {\tilde{H}}_{4} p_{21}^{2} Q_{2}^{2} + \\ 2 p_{21} u_{1} u_{2} H_{4} Q_{2} {\tilde{Q}}_{2} = 0 \end{matrix}

(25)

其中， ${\tilde{H}}_{1}$ 、 ${\tilde{H}}_{4}$ 和 ${\tilde{Q}}_{2}$ 分别表示H₁（u₂）、H₄（u₂）和Q₂（u₂）.特别地，当整个梁为等截面时，频率方程可化为:

s i n u [3 Q_{2}^{2} + 2 H_{4} (1 - H_{1})] = 0

(26)

其中u₁=u₂，并记为u，而H₁、Q₂、H₄见式（7）.
（二）梁内部含弹性支撑的情况
三跨等截面梁如图8所示.跨长AB=CB=L₁，BC=2L₂.各跨截面的面积和惯性矩分别为A_i和I_i（i=1，2，3），且AB段截面尺寸同CD段的截面尺寸.边界条件为两端简支，B、C处竖向弹性支撑的刚度值c_B=c_C=k.
图8 中间含竖向弹性支撑的三跨梁
Fig.8 A three-span beam with vertical elastic supports in the middle
首先把带弹性支撑三跨梁分别等效为正对称模态、反对称模态下的半结构，如图9（a）、（b）.研究半结构AE时（其中E为BC中点），其自由振动的计算采用两跨截面梁的计算方法，其中节点B处微元段的受力如图9（c）所示.
图9 半结构及拆分后节点B微元段
Fig.9 Half structure and force diagram of node B after separation
考虑图9（c）中的节点B，由节点B的动平衡方程得

c_{1} + c_{2} = c_{B} = k

(27)

α + β = 0 即 p_{21} R_{α}^{(2)} + R_{β}^{(1)} = 0

(28)

其中c₁、c₂分别为结构拆分后AB^-段、B⁺E段在B节点的竖向弹簧刚度值，p₂₁=I_BE₊/I_AB-，β、α分别为拆分后AB^-段、B⁺E段在B节点（实际上指B^-、B⁺位置，且上标中的“-”“+”对应“左侧”“右侧”）的旋转弹簧刚度值.转动刚度β和α可分别转化为参数 $R_{β}^{（ 1 ）}$ 和 $R_{α}^{（ 2 ）}$ .
选取AB段为对象，由表1知其频率方程为:

R_{β}^{(1)} = \frac{2 u_{1} c_{1} m_{1} H_{4} (u_{1}) + u_{1}^{4} Q_{2} (u_{1})}{c_{1} m_{1} Q_{2} (u_{1}) - 2 u_{1}^{3} H_{1} (u_{1})}

(29)

式中 $u_{1} = L_{1} \sqrt[4]{\frac{ρ A_{1}}{E I_{1}} ω_{n}^{2}}$ ，参数 $m_{1} = \frac{L_{1}^{3}}{E I_{1}}$ .
再选取B⁺E段为对象（需按梁的正、反对称变形分类），由表1得其频率方程分别为:

\{\begin{matrix} 正 对 称 : R_{α}^{(2)} = \frac{u_{2}^{4} P_{2} (u_{2}) - 2 u_{2} c_{2} m_{2} H_{1} (u_{2})}{c_{2} m_{2} P_{2} (u_{2}) - 2 u_{2}^{3} H_{4} (u_{2})} \\ 反 对 称 : R_{α}^{(2)} = \frac{u_{2}^{4} Q_{2} (u_{2}) + 2 u_{2} c_{2} m H_{4} (u_{2})}{c_{2} m_{2} Q_{2} (u_{2}) - 2 u_{2}^{3} H_{1} (u_{2})} \end{matrix}

(30)

其中 $u_{2} = L_{2} \sqrt[4]{ρ A_{2} / E I_{2} ω_{n}^{2}}$ ，参数 $m_{2} = L_{2}^{3} / E I_{2}$ 。定义 $i_{k} = \sqrt{I_{k} / A_{k}}$ ，从而有 $u_{2} / u_{1} = \sqrt{i_{1} / i_{2}}$ .
由式（29）和式（30）分别解出对称和反对称模态下弹簧线刚度c₁、c₂的关系式如下（其中参数 $R_{β}^{（ 1 ）}$ 和 $R_{α}^{（ 2 ）}$ 直接记为R_β和R_α）:

正对称: \{\begin{matrix} c_{1} = \frac{2 u_{1}^{3} H_{1} R_{β} + u_{1}^{4} Q_{2}}{m_{1} R_{β} Q_{2} - 2 m_{1} u_{1} H_{4}} \\ c_{2} = \frac{2 u_{2}^{3} {\tilde{H}}_{4} R_{α} + u_{2}^{4} {\tilde{P}}_{2}}{m_{2} R_{α} {\tilde{P}}_{2} + 2 m_{2} u_{2} {\tilde{H}}_{1}} \end{matrix}

(31)

反对称 : \{\begin{matrix} c_{1} = \frac{2 u_{1}^{3} H_{1} R_{β} + u_{1}^{4} Q_{2}}{m_{1} R_{β} Q_{2} - 2 m_{1} u_{1} H_{4}} \\ c_{2} = \frac{2 u_{2}^{3} {\tilde{H}}_{1} R_{α} + u_{2}^{4} {\tilde{Q}}_{2}}{m_{2} R_{α} {\tilde{Q}}_{2} - 2 m_{2} u_{2} {\tilde{H}}_{4}} \end{matrix}

(32)

其中 ${\tilde{H}}_{1}$ 、 ${\tilde{H}}_{4}$ 和 ${\tilde{Q}}_{2}$ 分别表示H₁（u₂）、H₄（u₂）和Q₂（u₂）.为建立频率方程的解析式，当结构具有对称变形时，将式（31）代入式（12），可得

\frac{2 u_{1}^{3} H_{1} R_{β} + u_{1}^{4} Q_{2}}{m_{1} R_{β} Q_{2} - 2 m_{1} u_{1} H_{4}} + \frac{2 u_{2}^{3} H_{4} R_{α} + u_{2}^{4} P_{2}}{m_{2} R_{α} P_{2} + 2 m_{2} u_{2} H_{1}} = k

(33)

注意到 $u_{2} / u_{1} = \sqrt{i_{1} / i_{2}}$ ，且由式（13）得R_β=-p₂₁R_α，代入式（33）后通过解二次方程可得R_α=f₁（u₁）.
类似地，由式（13）得R_α=-R_β/p₂₁，代入式（33）可解出R_β=f₂（u），从而得到奇数阶频率的方程式:

p_{21} f_{1} (u) + f_{2} (u) = 0

(34)

其中f₁（u₁）和f₂（u）的表达式基于二次方程的根得到，需要做一定的选取; 类似地，结构存在反对称变形时，联立式（12）和式（32），可得偶数阶频率方程式.然后可分别解出奇数阶固有频率（对应于对称模态）和偶数阶固有频率（对应反对称模态）.从而可求出图9中节点B两侧弹簧刚度c₁和c₂的值，并得到梁的模态.
3 数值分析
本节考虑等截面或阶梯截面多跨梁自由振动的2个算例，跨中含有若干刚性支撑或弹性支撑.具体分别如下:
算例1. 图6所示三跨梁中每段梁长度为1m，各段圆截面直径中，d₁=d₃=0.01m，d₂选为0.01m和0.02m两种情况.质量密度ρA为0.5kg/m，弹性模量E=2.02×10¹¹Pa; 其中跨中弹性支撑的弹簧刚度值取无穷大.
表3 两跨悬臂梁前2阶固有频率随链杆位置的变化
Table3 The first five natural frequencies of a three-span beam
按照2.2.1节方法计算得到含刚性支撑三跨梁的固有频率如表3所示.当d₂=0.01m时所得结果与文献^[10]一致，这说明三跨梁的固有频率方程式（25）的正确性.
算例2. 三跨梁如图8所示.为简便计，整个梁采用矩形等截面，其宽度和高度为:b×h=0.5m×1.3m; 密度ρ=2400kg/m³; 弹性模量E=30GPa; 梁跨度AB=CD=18m，BC=36m; B、C处竖向弹性支撑的刚度值c_B=c_C=k，且k取9个不同值，即k=10ⁱN/m，（i=1，2，···，9）.
按照2.2.2节方法计算，表4给出中间含不同竖向弹性支撑刚度时三跨梁固有频率的计算值，与有限元结果进行对比，所得数值结果一致，相对偏差最大为0.43%.由表中最后2行数据可发现，当k≥10⁸N/m时，弹性支撑趋近于刚性支撑，且梁的各阶频率基本不变.
表4 不同竖向弹性支撑刚度时三跨梁的前3阶固有频率
Table4 The first three order natural frequencies of a three-span beam with different vertical elastic rigidity
4 结论
本文提出阶梯多段梁结构自由振动分析的逐段叠加法.对含弹性支撑多跨梁的振动问题进行分析，求解了这类多跨梁的固有频率.结论如下:
（1）对多跨梁拆分为多段弹性梁的组合; 每段梁建立其振动频率方程; 结合中间连接节点处弹簧刚度应满足的动平衡方程，得到多跨梁固有频率满足的解析方程组.针对两端为经典边界条件、内部含刚性或弹性支撑的阶梯截面多跨梁建立自由振动的解析模型，且解析形式的频率方程组形式相对较为简洁.另外，节点处所添加弹簧刚度满足的动平衡方程是本文的特色，这类方程起着重要作用，即把各段离散梁的自由振动方程联合起来形成频率方程组.还可求出拆分时所添加弹性弹簧的刚度，然后得到梁的模态.
（2）数值算例表明，在不同的边界和跨中含刚性或弹性支撑的条件下，本文能较准确地计算梁的固有频率，并与已有的文献结果或有限元结果进行对比，其误相对偏差均在较小范围之内（＜1.0%），表明文中方法具有良好的精度与求解稳定性.
（3）所提方法属于解析公式法，便于应用.在研究多跨阶梯梁的自由振动时，所提方法得到的精确解可评价其他方法的计算精度.
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基本信息

中图分类号: O327
文献标识码: A
DOI: 10.6052/1672-6553-2022-017
文章编号: 1672-6553-2023-21(3)-085-011

基金信息

国家自然科学基金资助项目(51709194)；
National Natural Science Foundation of China(51709194)；

引用信息

稿件历史

收稿日期: 2022-03-08

参考文献

[1] 胡海岩.振动力学:研究性教程[M].北京:科学出版社,2020.HU H Y.Mechanics of vibration:an investigative course[M].Beijing:Science Press,2020.(in Chinese)
[2] 刘延柱.振动力学[M].北京:高等教育出版社,2019.LIU Y Z.Mechanics of vibration[M].Beijing:Higher Education Press,2019.(in Chinese)
[3] JOHANSSON C,PACOSTE C,KAROUMI R.Closed-form solution for the mode superposition analysis of the vibration in multi-span beam bridges caused by concentrated moving loads [J].Computers and Structures,2013,119(5):85-94.
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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*

摘要

Abstract

关键词

Keywords

1 单跨弹性梁的动力分析

1.1 几何模型

1.2 弹性支撑梁的频率特征方程

(1)

(2)

(3)

(4)

(5)

(6)

(7)

1.3 不同边界条件下梁的频率方程

(8)

(9)

(10)

1.4 逐段叠加时节点处的平衡方程

(11)

(12)

(13)

2 含支撑梁的动力学模型

2.1 求解思路

2.2 两类振动模型分析

2.2.1 两跨阶梯截面梁

(14)

(15)

(16)

(17)

(18)

2.2.2 内部含支撑多跨阶梯截面梁

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

3 数值分析

4 结论

参考文献

基本信息

基金信息

引用信息

稿件历史

参考文献

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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^*