移动荷载作用下非线性基础梁内共振分析

严巧赟; Yan Qiaoyun

引 en

移动荷载作用下非线性基础梁内共振分析

严巧赟
机构：
上海大学期刊社,上海 200444
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上海大学期刊社,上海 200444；

Internal Resonance Analysis of Nonlinear Foundation Beams under Moving Loads

Yan Qiaoyun
Affiliation：
Periodicals Agency of Shanghai University, Shanghai 200444 , China
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Periodicals Agency of Shanghai University, Shanghai 200444 , China；

通讯作者:

严巧赟,E-mail:qyyan@shu.edu.cn

中图分类号:O32

文献标识码:A

文章编号:1672-6553-2023-21(9)-041-009

DOI:10.6052/1672-6553-2023-061

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目录contents

摘要 Abstract
关键词 Keywords
1 非线性粘弹性基础梁模型
2 3:1内共振条件下的稳定性分析
2.1 3:1内共振下的自由振动
2.2 3:1内共振下的受迫振动分析
3 结论
参考文献

摘要

研究了有限长弹性基础上梁在移动载荷作用下的内共振响应.建立了移动集中力激励的非线性粘弹性基础支承的有限长Euler-Bernoulli梁模型,并对非线性偏微分方程进行离散,在第三阶固有频率与第一阶固有频率成三倍关系时,采用多尺度方法导出了3:1内共振的可解性条件,研究了有无移动载荷时基础阻尼和非线性刚度对梁内共振条件下自由振动响应和受迫振动响应的影响规律.在此基础上,应用Lyapunov第一方法确定了系统的稳定性条件.

Abstract

The internal external resonance response of a beam on a finite length elastic foundation under moving loads is studied. A finite length Euler-Bernoulli beam model supported on a nonlinear viscoelastic foundation excited by a moving concentrated force is established, and the nonlinear partial differential equation is discretized. When the third-order natural frequency and the first-order natural frequency are three times related, the solvability condition for 3:1 internal resonance is derived by using a multi-scale method. The effects of foundation damping and nonlinear stiffness on the free vibration response and forced vibration response of a beam under internal resonance conditions with or without moving loads are studied. On this basis, the stability conditions of the system are determined by using the first Lyapunov method.

关键词

非线性基础梁；移动载荷；内共振；受迫振动；可解性条件

Keywords

nonlinear basic beam ； moving load ； internal resonance ； forced vibration ； solvability condition

引言
移动荷载下基础梁能够作为很多工程结构的力学模型，例如，受到来自高速、重载车辆的作用的非线性基础路面结构.研究移动载荷作用下基础梁的振动分析是结构动力学中的一个重要课题，也是机械、桥梁和铁路工程等不同学科的研究热点.
Ghayesh等^[1]以3:1内共振为例，采用数值的方法研究了梁轴向运动引起的纵向和横向耦合非线性振动的动力学特性，根据不同的参数分析其表现出的动态特性.Hu等^[2]利用多尺度法，得到了空间轨道系留系统中两弹簧悬挂的空间柔性梁的内共振条件，研究了内共振对姿态稳定性的影响以及构件间能量传递趋势的影响.Ding等^[3]在3:1内共振条件下，研究了粘弹性移动梁受迫振动的稳态周期响应，采用数值算例分析了粘弹性行为对稳态周期响应的影响.Wang等^[4]采用输送流体线性梁和非线性弹簧来模拟在弹性基础上放置的流体输送管的振动，用多尺度法求出了各模态在稳态下的频率响应，并用各模态的振幅来检验内共振，通过改变流体的流速来分析系统的稳定性.Wang和Ding^[5]研究了悬臂梁的非线性自由振动、1/3超谐共振以及重力引起的3:1内共振，采用时间多尺度方法，获得其对应振动的模态响应，运用谐波平衡法求解超谐波共振中的激励分量和模型的挠度，研究了重力引起的内部共振对垂直悬臂梁非线性振动的动态影响.Guillot等^[6]对压电片梁中1:3内共振现象，以实验和理论的方法进行了研究.Ding等^[7]采用直接多尺度方法研究了非线性振动对轴向运动梁的应力分布和疲劳寿命的影响，揭示了内共振对V型带稳态响应和疲劳寿命的影响.Younesian等^[8]采用Galerkin方法建立了微梁的无量纲运动控制方程，结合多尺度方法求解非线性方程组，并确定了主共振和次共振条件，研究了不同谐振情况下的谐波响应.顾伟等^[9]研究了预变形叶片在变速条件下的非线性动力学行为，详细研究了温度梯度、阻尼、转速扰动幅度等系统参数对叶片动态响应的影响，在2:1内共振条件下，研究了立方项对方程的影响.毕勤胜和陈予恕^[10]分析了复摆自治系统在1:1内共振时的混沌和分岔特性.魏明海等^[11]研究了内共振和外共振联合作用下的索-梁组合结构非线性振动问题.在众多关于飞线振动的研究成果中，与道路相关的非线性系统内-外联合共振的研究鲜有报道.
目前对于连续体的共振问题多是将连续体以梁或板的形式进行研究^[12-14].杨予等^[15]将桥简化为简支梁，研究其在均布移动载荷作用下的振动响应，指出梁的动态响应与其自振频率、载荷的行驶频率和荷载与梁质量的振动相关.刘涛等^[16]分析了轴向运动速度和材料的非均匀性对轴向运动功能梯度梁发生共振时的影响.Martínez-Castro^[17]以时变模态方程得到了移动荷载作用下非均匀多跨Euler-Bernoulli梁的时域半解析解，并证实了该方法具有很高的精度和鲁棒性.张弛等^[18]针对轴向移动梁结构，讨论了边界支撑参数对振动的影响.Younesian等^[19]采用Galerkin方法结合多尺度方法求解非线性运动控制方程，研究了由非线性粘弹性基础支承的裂纹梁的频率响应，分析了不同参数对频率响应解的影响.Ding等^[20]着眼于车辆与路面的耦合非线性振动问题，将路面建模为非线性基础上的Timoshenko梁，研究了汽车-路面耦合系统的动力学响应.
本文建立了在移动载荷作用下的非线性粘弹性基础上的有限长Euler-Bernoulli梁，且系统的第三阶固有频率与第一阶固有频率成三倍关系，用多尺度法得到内共振条件下非线性系统的自由振动和强迫振动响应.通过参数分析，研究了不同参数对系统动态响应的影响规律.
1 非线性粘弹性基础梁模型
考虑在移动载荷作用下基于非线性粘弹性Kelvin基础的有限长Euler-Bernoulli梁模型，如图1所示.基础梁密度为ρ，梁的宽度和厚度分别为b和h，惯性矩为I，路面长度为L，弹性模量是E.非线性Kelvin基础模型可以如下表示^[21，22]：
图1 非线性粘弹性基础上的有限Euler-Bernoulli梁示意图
Fig.1 Finite Euler-Bernoulli beam on nonlinear viscoelastic foundation

P = k_{1} Y + k_{3} Y^{3} + μ \frac{\partial Y}{\partial T}

(1)

其中，P是在梁上任意点发生位移的力，k₁和k₃分别是基础的线性刚度和非线性刚度系数，μ是基础的阻尼系数.根据Euler-Bernoulli梁理论，Kelvin基础系统的控制微分方程可以表示为
$E I \frac{\partial^{4} w}{\partial X^{4}} + ρ A \frac{\partial^{2} w}{\partial T^{2}} + k_{1} w + k_{3} w^{3} + μ \frac{\partial w}{\partial T} =$

F δ (X - V T)

(2)

假定梁两端均为简支，因此，边界条件如下：
${w (X, T)|}_{X = 0} = {w (X, T)|}_{X = L} = 0,$

{E I \frac{\partial ψ}{\partial X} (X, T)|}_{X = 0} = {E I \frac{\partial ψ}{\partial X} (X, T)|}_{X = L} = 0

(3)

根据边界条件，梁的位移函数的表达式如下：

w (X, T) = \sum_{i = 1}^{\infty} q_{i} (T) φ_{i} (X)

(4)

其中，φ_i（X）表示第i阶模态函数，q_i（T）表示关于时间T的第i阶广义位移.梁的模态函数表示为

φ_{i} (X) = s i n \frac{i π X}{L}

(5)

将式（4）代入式（2）得，
$\sum_{i = 1}^{n} [{\ddot{q}}_{i} (T) + 2 ε \hat{μ} {\dot{q}}_{i} (T) + ω_{i}^{2} q_{i} (T)] s i n \frac{i π X}{L} +$

{\hat{k}}_{3} {[\sum_{i = 1}^{n} q_{i} (T) s i n \frac{i π X}{L}]}^{3} = ε L f δ (X - V T)

(6)

其中， $\frac{μ}{ρ A} = 2 ε \hat{μ} ， \frac{k_{1}}{ρ A} + \frac{i^{4} π^{4} E I}{ρ A L^{4}} = ω_{i}^{2} ， \frac{k_{3}}{ρ A} = {\hat{k}}_{3} ， \frac{F}{ρ A} = \frac{L}{2} ε f$ ，ε为小参数（ $0 < ε ≪ 1$ ）.
根据三角函数的正交性，上式等号两边同时乘以 $φ_{m} （ X ） = s i n \frac{m π X}{L}$ ，然后在区间长度[0，L]上进行积分，取Galerkin截断前三阶来求解，可得：
${\ddot{q}}_{1} (T) + ω_{1}^{2} q_{1} (T) = - 2 ε \hat{μ} {\dot{q}}_{1} (T) + ε (- α_{1} q_{1}^{3} -$
$α_{2} q_{1}^{2} q_{3} - α_{3} q_{1} q_{2}^{2} - α_{4} q_{1} q_{3}^{2} - α_{5} q_{2}^{2} q_{3}) +$

ε f s i n (Ω_{1} T)

(7)

${\ddot{q}}_{2} (T) + ω_{2}^{2} q_{2} (T) = - 2 ε \hat{μ} {\dot{q}}_{2} (T) +$
$ε (- α_{6} q_{1}^{2} q_{2} - α_{7} q_{1} q_{2} q_{3} - α_{8} q_{2}^{3} -$

α_{9} q_{2} q_{3}^{2}) + ε f s i n (Ω_{2} T)

(8)

${\ddot{q}}_{3} (T) + ω_{3}^{2} q_{3} (T) = - 2 ε \hat{μ} {\dot{q}}_{3} (T) +$
$ε (- α_{10} q_{1}^{3} - α_{11} q_{1}^{2} q_{3} - α_{12} q_{1} q_{2}^{2} - α_{13} q_{2}^{2} q_{3} -$

α_{14} {\hat{k}}_{3} q_{3}^{3}) + ε f s i n (Ω_{3} T)

(9)

其中， $Ω_{i} = （ i π V ） / L ， α_{2} = - 3 / 4 {\hat{k}}_{3} ， α_{1} = α_{5} = α_{8} = α_{12} = α_{14} = 3 / 4 {\hat{k}}_{3} ， α_{3} = α_{4} = α_{6} = α_{7} = α_{9} = α_{11} = α_{13} = 3 / 2 {\hat{k}}_{3} ， α_{10} = - 1 / 4 {\hat{k}}_{3}$ .
运用多尺度法，将方程的解设为

q_{i} = q_{i 0} (T_{0}, T_{1}) + ε q_{i 1} (T_{0}, T_{1}) + \dots

(10)

其中， $T_{n} = ε^{n} t （ n = 1，2 ， 3 ， \dots ）$ ，将式（10）代入式（7）~（9）可得：
ε⁰阶：

D_{0}^{2} q_{10} + ω_{1}^{2} q_{10} = 0

(11)

D_{0}^{2} q_{20} + ω_{2}^{2} q_{20} = 0

(12)

D_{0}^{2} q_{30} + ω_{3}^{2} q_{30} = 0

(13)

ε¹阶：
$D_{0}^{2} q_{11} + ω_{1}^{2} q_{11} = - 2 D_{0} D_{1} q_{10} - 2 \hat{μ} D_{0} q_{10} -$
$α_{1} q_{10}^{3} - α_{2} q_{10}^{2} q_{30} - α_{3} q_{10} q_{20}^{2} - α_{4} q_{10} q_{30}^{2} -$

α_{5} q_{20}^{2} q_{30} + f s i n (Ω_{1} T)

(14)

$D_{0}^{2} q_{21} + ω_{2}^{2} q_{21} = - 2 D_{0} D_{1} q_{20} - 2 \hat{μ} D_{0} q_{20} -$
$α_{6} q_{10}^{2} q_{20} - α_{7} q_{10} q_{20} q_{30} - α_{8} q_{20}^{3} - α_{9} q_{20} q_{30}^{2} +$

2 f s i n (Ω_{2} T)

(15)

$D_{0}^{2} q_{31} + ω_{3}^{2} q_{31} = - 2 D_{0} D_{1} q_{30} - 2 \hat{μ} D_{0} q_{30} -$
$α_{10} {\hat{k}}_{3} q_{10}^{3} - α_{11} q_{10}^{2} q_{30} - α_{12} q_{10} q_{20}^{2} -$

α_{13} q_{20}^{2} q_{30} - α_{14} q_{30}^{3} + 2 f s i n (Ω_{3} T)

(16)

由式（11）~（13）可以得到下列解的形式.

q_{10} = A_{1} (T_{1}) e^{j ω_{1} T_{0}} + c c

(17)

q_{20} = A_{2} (T_{1}) e^{j ω 2 T_{0}} + c c

(18)

q_{30} = A_{3} (T_{1}) e^{j ω 2 T_{0}} + c c

(19)

其中，为上式中所对应的共轭项.
将式（17）~（19）代入到式（14）~（16）可得：
$D_{0}^{2} q_{11} + ω_{1}^{2} q_{11} = - 2 j ω_{1} A_{1}^{'} e^{j ω_{1} T_{0}} + 2 j ω_{1} {\bar{A}}_{1}^{'} e^{- j ω_{1} T_{0}} -$
$2 j \hat{μ} ω_{1} A_{1} e^{j ω_{1} T_{0}} + 2 j \hat{μ} ω_{1} {\bar{A}}_{1} e^{- j ω_{1} T_{0}} +$
$(- α_{1} A_{1}^{3} e^{3 j ω_{1} T_{0}} - 3 α_{1} A_{1}^{2} {\bar{A}}_{1} e^{j ω_{1} T_{0}} -$
$3 α_{1} A_{1} {\bar{A}}_{1}^{2} e^{- j ω_{1} T_{0}} - α_{1} {\bar{A}}_{1}^{3} e^{- 3 j ω_{1} T_{0}} -$
$α_{2} A_{1}^{2} A_{3} e^{j (2 ω_{1} + ω_{3}) T_{0}} - 2 α_{2} A_{1} {\bar{A}}_{1} A_{3} e^{j ω_{3} T_{0}} -$
$α_{2} {\bar{A}}_{1}^{2} A_{3} e^{- j (2 ω_{1} - ω_{3}) T_{0}} - α_{2} A_{1}^{2} {\bar{A}}_{3} e^{j (2 ω_{1} - ω_{3}) T_{0}} -$
$2 α_{2} A_{1} {\bar{A}}_{1} {\bar{A}}_{3} e^{- j ω_{3} T_{0}} - α_{2} {\bar{A}}_{1}^{2} {\bar{A}}_{3} e^{- j (2 ω_{1} + ω_{3}) T_{0}} -$
$α_{3} A_{1} A_{2}^{2} e^{j (ω_{1} + 2 ω_{2}) T_{0}} - 2 α_{3} A_{1} A_{2} {\bar{A}}_{2} e^{j ω_{1} T_{0}}) -$
$α_{3} A_{1} {\bar{A}}_{1}^{2} e^{j (ω_{1} - 2 ω_{2}) T_{0}} - α_{3} {\bar{A}}_{1} A_{2}^{2} e^{- j (ω_{1} - 2 ω_{2}) T_{0}} -$
$2 α_{3} {\bar{A}}_{1} A_{2} {\bar{A}}_{2} e^{- j ω_{1} T_{0}} - α_{3} {\bar{A}}_{1} {\bar{A}}_{2}^{2} e^{- j (ω_{1} + 2 ω_{2}) T_{0}} -$
$α_{4} A_{1} A_{3}^{2} e^{j (ω_{1} + 2 ω_{3}) T_{0}} - 2 α_{4} A_{1} A_{3} {\bar{A}}_{3} e^{j ω_{1} T_{0}}) -$
$α_{4} A_{1} {\bar{A}}_{3}^{2} e^{j (ω_{1} - 2 ω_{3}) T_{0}} - α_{4} {\bar{A}}_{1} A_{3}^{2} e^{- j (ω_{1} - 2 ω_{3}) T_{0}} -$
$2 α_{4} {\bar{A}}_{1} A_{3} {\bar{A}}_{3} e^{- j ω_{1} T_{0}} - α_{4} {\bar{A}}_{1} {\bar{A}}_{3}^{2} e^{- j (ω_{1} + 2 ω_{3}) T_{0}} -$
$α_{5} A_{2}^{2} A_{3} e^{j (2 ω_{2} + ω_{3}) T_{0}} - 2 α_{5} A_{2} {\bar{A}}_{2} A_{3} e^{j ω_{3} T_{0}}) -$
$α_{5} {\bar{A}}_{2}^{2} A_{3} e^{- j (2 ω_{2} - ω_{3}) T_{0}} - α_{5} A_{2}^{2} {\bar{A}}_{3} e^{j (2 ω_{2} - ω_{3}) T_{0}} -$
$2 α_{5} A_{2} {\bar{A}}_{2} {\bar{A}}_{3} e^{- j ω_{3} T_{0}} - α_{5} {\bar{A}}_{2}^{2} {\bar{A}}_{3} e^{- j (2 ω_{2} + ω_{3}) T_{0}} +$

1 / 2 f (e^{j (Ω_{1} T - \frac{π}{2})} + e^{- j (Ω_{1} T - \frac{π}{2})})

(20)

$D_{0}^{2} q_{21} + ω_{2}^{2} q_{21} = - 2 j ω_{2} A_{2}^{'} e^{j ω_{2} T_{0}} + 2 j ω_{2} {\bar{A}}_{2}^{'} e^{- j ω_{2} T_{0}} -$
$2 j \hat{μ} ω_{2} A_{2} e^{j ω_{2} T_{0}} + 2 j \hat{μ} ω_{2} {\bar{A}}_{2} e^{- j ω_{2} T_{0}} +$
$(- α_{6} {\hat{k}}_{3} A_{1}^{2} A_{2} e^{j (2 ω_{1} + ω_{2}) T_{0}} - 2 α_{6} {\bar{A}}_{1} A_{1} A_{2} e^{j ω_{2} T_{0}} -$
$α_{6} {\bar{A}}_{1}^{2} A_{2} e^{- j (2 ω_{1} - ω_{2}) T_{0}} - α_{6} A_{1}^{2} {\bar{A}}_{2} e^{j (2 ω_{1} - ω_{2}) T_{0}} -$
$2 α_{6} {\bar{A}}_{1} A_{1} {\bar{A}}_{2} e^{- j ω_{2} T_{0}} - α_{6} {\bar{A}}_{1}^{2} {\bar{A}}_{2} e^{- j (2 ω_{1} + ω_{3}) T_{0}}$
$α_{7} A_{1} A_{2} A_{3} e^{j (ω_{1} + ω_{2} + ω_{3}) T_{0}} - α_{7} {\bar{A}}_{1} A_{2} A_{3} e^{- j (ω_{1} - ω_{2} - ω_{3}) T_{0}}) -$
$α_{7} A_{1} {\bar{A}}_{1}^{2} e^{j (ω_{1} - 2 ω_{2}) T_{0}} - α_{3} {\bar{A}}_{1} A_{2}^{2} e^{- j (ω_{1} - 2 ω_{2}) T_{0}} -$
$α_{7} A_{1} A_{2} {\bar{A}}_{3} e^{j (ω_{1} + ω_{2} - ω_{3}) T_{0}} - α_{7} {\bar{A}}_{1} A_{2} {\bar{A}}_{3} e^{- j (ω_{1} - ω_{2} + ω_{3}) T_{0}} -$
$α_{7} A_{1} {\bar{A}}_{2} {\bar{A}}_{3} e^{j (ω_{1} - ω_{2} - ω_{3}) T_{0}} - α_{7} {\bar{A}}_{1} {\bar{A}}_{2} {\bar{A}}_{3} e^{- j (ω_{1} + ω_{2} + ω_{3}) T_{0}}) -$
$α_{8} A_{2}^{3} e^{3 j ω_{2} T_{0}} - 3 α_{8} A_{2}^{2} {\bar{A}}_{2} e^{j ω_{2} T_{0}} -$
$3 α_{8} A_{2} {\bar{A}}_{2}^{2} e^{- j ω_{2} T_{0}} - α_{8} {\bar{A}}_{2}^{3} e^{- 3 j ω_{2} T_{0}} -$
$α_{9} A_{2} A_{3}^{2} e^{j ω_{2} + 2 ω_{3}) T_{0}} - 2 α_{9} A_{2} {\bar{A}}_{3} A_{3} A_{3} e^{j ω_{2} 2 T_{0}}) -$
$α_{9} A_{2} {\bar{A}}_{3}^{2} e^{j (ω_{2} - 2 ω_{3}) T_{0}} - α_{9} {\bar{A}}_{2} A_{3}^{2} e^{- j (ω_{2} - 2 ω_{3}) T_{0}} -$
$2 α_{9} {\bar{A}}_{2} {\bar{A}}_{3} A_{3} e^{- j ω_{2} T_{0}} - α_{9} {\bar{A}}_{2} {\bar{A}}_{3}^{2} e^{- j (ω_{2} + 2 ω_{3}) T_{0}} +$

f (e^{j (Ω_{2} T - \frac{π}{2})} + e^{- j (Ω_{2} T - \frac{π}{2})})

(21)

$D_{0}^{2} q_{31} + ω_{3}^{2} q_{31} = - 2 j ω_{3} A_{3}^{'} e^{j ω_{3} T_{0}} + 2 j ω_{3} {\bar{A}}_{3}^{'} e^{- j ω_{3} T_{0}}$
$\begin{matrix} 2 j \hat{μ} ω_{3} A_{3} e^{j ω_{3} T_{0}} + 2 j \hat{μ} ω_{3} {\bar{A}}_{3} e^{- j ω_{3} T_{0}} + \\ (- α_{10} A_{1}^{3} e^{3 j ω_{1} T_{0}} - 3 α_{10} A_{1}^{2} {\bar{A}}_{1} e^{j ω_{1} T_{0}} - \\ 3 α_{10} A_{1} {\bar{A}}_{1}^{2} e^{- j ω_{1} T_{0}} - α_{10} {\bar{A}}_{1}^{3} e^{- 3 j ω_{1} T_{0}} - \\ α_{11} A_{1}^{2} A_{3} e^{j (2 ω_{1} + ω_{3}) T_{0}} - 2 α_{11} A_{1} {\bar{A}}_{1} A_{3} e^{j ω_{3} T_{0}} - \\ α_{11} {\bar{A}}_{1}^{2} A_{3} e^{- j (2 ω_{1} - ω_{3}) T_{0}} - α_{11} A_{1}^{2} {\bar{A}}_{3} e^{j (2 ω_{1} - ω_{3}) T_{0}} - \\ 2 α_{11} A_{1} {\bar{A}}_{1} {\bar{A}}_{3} e^{- j ω_{3} T_{0}} - α_{11} {\bar{A}}_{1}^{2} {\bar{A}}_{3} e^{- j (2 ω_{1} + ω_{3}) T_{0}} - \\ α_{12} A_{1} A_{2}^{2} e^{j (ω_{1} + 2 ω_{2}) T_{0}} - 2 α_{12} A_{1} A_{2} {\bar{A}}_{2} e^{j ω_{1} T_{0}}) - \\ α_{12} A_{1} {\bar{A}}_{2}^{2} e^{j (ω_{1} - 2 ω_{2}) T_{0}} - α_{12} {\bar{A}}_{1} A_{2}^{2} e^{- j (ω_{1} - 2 ω_{2}) T_{0}} - \\ 2 α_{12} {\bar{A}}_{1} A_{2} {\bar{A}}_{2} e^{- j ω_{1} 1 T_{0}} - α_{12} {\bar{A}}_{1} {\bar{A}}_{2}^{2} e^{- j (ω_{1} + 2 ω_{2}) T_{0}} - \\ α_{13} A_{2}^{2} A_{3} e^{j (2 ω_{2} + ω_{3}) T_{0}} - 2 α_{13} A_{2} {\bar{A}}_{2} A_{3} e^{j ω_{3} T_{0}}) - \\ α_{13} {\bar{A}}_{2}^{2} A_{3} e^{- j (2 ω_{2} - ω_{3}) T_{0}} - α_{13} A_{2}^{2} {\bar{A}}_{3} e^{j (2 ω_{2} - ω_{3}) T_{0}} - \\ 2 α_{13} A_{2} {\bar{A}}_{2} {\bar{A}}_{3} e^{- j ω_{3} T_{0}} - α_{13} {\bar{A}}_{2}^{2} {\bar{A}}_{3} e^{- j (2 ω_{2} + ω_{3}) T_{0}} - \end{matrix}$
$\begin{matrix} α_{14} A_{3}^{3} e^{3 j ω_{3} T_{0}} - 3 α_{14} A_{3}^{2} {\bar{A}}_{3} e^{j ω_{3} T_{0}}) - \\ 3 α_{14} A_{3} {\bar{A}}_{3}^{2} e^{- j ω_{3} T_{0}} - α_{14} {\bar{A}}_{3}^{3} e^{- 3 j ω_{3} T_{0}} - \end{matrix}$

f (e^{j (Ω_{3} T - \frac{π}{2})} + e^{- j (Ω_{3} T - \frac{π}{2})})

(22)

2 3:1内共振条件下的稳定性分析
为研究3:1内共振下基础梁的自由振动和强迫振动响应，将系统第3阶模态的共振频率与第1阶模态共振频率的3倍设为相近，即，进而可以确定基础的线性刚度.其中，以及下文提到的均为引入的调谐因子.
2.1 3:1内共振下的自由振动
分析自由振动，即考虑在无移动载荷作用下的振动响应.消除久期项的条件为：
$\begin{matrix} - 2 j ω_{1} A_{1}^{'} - 2 j \hat{μ} ω_{1} A_{1} - 3 α_{1} A_{1}^{2} {\bar{A}}_{1} - \\ α_{2} {\bar{A}}_{1}^{2} A_{3} e^{j σ_{1} T_{1}} - 2 α_{3} A_{1} A_{2} {\bar{A}}_{2} - \end{matrix}$

2 α_{4} A_{1} A_{3} {\bar{A}}_{3} = 0

(23)

$- 2 j ω_{2} A_{2}^{'} - 2 j \hat{μ} ω_{2} A_{2} - 2 α_{6} {\bar{A}}_{1} A_{1} A_{2} -$

3 α_{8} A_{2}^{2} {\bar{A}}_{2} - 2 α_{9} A_{2} {\bar{A}}_{3} A_{3} = 0

(24)

$- 2 j ω_{3} A_{3}^{'} - 2 j \hat{μ} ω_{3} A_{3} - α_{10} A_{1}^{3} e^{- j σ_{1} T_{1}} -$

2 α_{11} A_{1} {\bar{A}}_{1} A_{3} - 2 α_{13} A_{2} {\bar{A}}_{2} A_{3} - 3 α_{14} A_{3}^{2} {\bar{A}}_{3} = 0

(25)

令 $A_{1} = 1 / 2 a_{1} e^{j β 1} ， A_{2} = 1 / 2 a_{2} e^{j β 2}$ 和 $A_{3} = 1 / 2 a_{3} e^{j β 3}$ ，并代入到上式，再分离实部和虚部，可得：
$ω_{1} β_{1}^{'} a_{1} - 3 / 8 α_{1} a_{1}^{3} - 1 / 8 α_{2} a_{1}^{2} a_{3} c o s γ -$

1 / 4 α_{4} a_{1} a_{2}^{2} - 1 / 4 α_{4} a_{1} a_{3}^{2} = 0

(26)

- ω_{1} a_{1}^{'} - a_{1} \hat{μ} ω_{1} - 1 / 8 α_{2} a_{1}^{2} a_{3} s i n γ = 0

(27)

$ω_{2} β_{2}^{'} a_{2} - 1 / 4 α_{6} a_{1}^{2} a_{2} - 1 / 4 α_{9} a_{2} a_{3}^{2} -$

3 / 8 α_{8} a_{2}^{3} = 0

(28)

- ω_{2} a_{2}^{'} - \hat{μ} ω_{2} a_{2} = 0

(29)

$ω_{3} β_{3}^{'} a_{3} - 1 / 8 α_{10} a_{1}^{3} c o s γ - 1 / 4 α_{11} a_{1}^{2} a_{3} -$

1 / 4 α_{13} a_{2}^{2} a_{3} - 3 / 8 α_{14} a_{3}^{3} = 0

(30)

- ω_{3} a_{3}^{'} - \hat{μ} ω_{3} a_{3} + 1 / 8 α_{10} a_{1}^{3} s i n γ = 0

(31)

其中， $γ = σ_{1} T_{1} + β_{3} - 3 β_{1}$ .
式（29）在稳态条件下的响应为

a_{2} = 0

(32)

由式（27）、式（31）和式（32）可得：

a_{1} a_{1}^{'} + \frac{ω_{3} α_{2}}{ω_{1} α_{10}} a_{3} a_{3}^{'} = - \hat{μ} a_{1}^{2} - \frac{ω_{3} α_{2}}{ω_{1} α_{10}} \hat{μ} a_{3}

(33)

令 $f_{13} (T_{1}) = a_{1}^{2} + \frac{ω_{3} α_{2}}{ω_{1} α_{10}} a_{3}^{2}$ ，将其代入式（33），可得：

\frac{d f_{13} (T_{1})}{d T_{1}} = - 2 \hat{μ} f_{13} (T_{1})

(34)

由式（34）可得：

f_{13} (T_{1}) = C e^{- 2 \hat{μ} T_{1}}

(35)

其中C为大于0的常数.
从式（35）可以看出，当T₁→+∞时，f₁₃（+∞）=0，从而理论分析上可以说明系统是随着时间逐渐衰减为0.同样，在时域下变量a₁和a₃的数值解可由式（26）~式（31）得到，如图2~图7所示.图2通过数值仿真给出了系统在自由振动下的衰减过程.
阻尼和非线性刚度对系统自由振动的影响如图3所示.从图3（a）和3（c）可以看出随着阻尼增加，系统的衰减速度就会越快；图3（b）和3（d）可以看出，非线性刚度会增加系统的振动频率，但对其衰减速度基本无影响.
图2 自由振动下幅值的衰减过程
Fig.2 Attenuation process of amplitude under free vibration
图3 自由振动下参数对系统响应的影响
Fig.3 Influence of parameters on system response under free vibration
2.2 3:1内共振下的受迫振动分析
在3:1内共振（即ω₃≈3ω₁时）下分别分析Ω₁≈ω₁、Ω₂≈ω₂和Ω₃≈ω₃时系统发生强迫共振条件.
（1）情况I：Ω₁≈ω₁
假定

Ω_{1} = ω_{1} + ε σ_{2}

(36)

消除久期项，并分离实部和虚部，可以得到：
$ω_{1} β_{1}^{'} a_{1} - 3 / 8 α_{1} a_{1}^{3} - 1 / 4 / α_{4} a_{1} a_{2}^{2} - 1 / 4 α_{4} a_{1} a_{3}^{2} -$

1 / 8 α_{2} a_{1}^{2} a_{3} c o s γ_{1} + f / 2 c o s γ_{2} = 0

(37)

$- ω_{1} a_{1}^{'} - \hat{μ} ω_{1} a_{1} - 1 / 8 α_{2} a_{1}^{2} a_{3} s i n γ_{1} +$

f / 2 s i n γ_{2} = 0

(38)

$ω_{2} β_{2}^{'} a_{2} - 1 / 4 α_{6} a_{1}^{2} a_{2} - 1 / 4 α_{9} a_{2} a_{3}^{2} -$

3 / 8 α_{8} a_{2}^{3} = 0

(39)

- ω_{2} a_{2}^{'} - \hat{μ} ω_{2} a_{2} = 0

(40)

$ω_{3} β_{3}^{'} a_{3} - 1 / 8 α_{10} a_{1}^{3} c o s γ_{1} - 1 / 4 α_{11} a_{1}^{2} a_{3} -$

1 / 4 α_{13} a_{2}^{2} a_{3} - 3 / 8 α_{14} a_{3}^{3} = 0

(41)

- ω_{3} a_{3}^{'} - \hat{μ} ω_{3} a_{3} + 1 / 8 α_{10} a_{1}^{3} s i n γ_{1} = 0

(42)

其中， $γ_{1} = σ_{1} T_{1} + β_{3} - 3 β_{1} ， γ_{2} = σ_{2} T_{1} - β_{1} - π / 2$ .
（2）情况II：Ω₂≈ω₂
假定

Ω_{2} = ω_{2} + ε σ_{2}

(43)

消除久期项，并分离实部和虚部，可以得到：
$ω_{1} β_{1}^{'} a_{1} - 3 / 8 α_{1} a_{1}^{3} - 1 / 4 α_{4} a_{1} a_{2}^{2} - 1 / 4 α_{4} a_{1} a_{3}^{2} -$

1 / 8 α_{2} a_{1}^{2} a_{3} c o s γ_{1} = 0

(44)

- \hat{μ} ω_{1} a_{1} - 1 / 8 α_{2} a_{1}^{2} a_{3} s i n γ_{1} = 0

(45)

$ω_{2} σ_{2} a_{2} - 1 / 4 / α_{6} a_{1}^{2} a_{2} - 1 / 4 α_{9} a_{2} a_{3}^{2} -$

3 / 8 α_{8} a_{2}^{3} + f / 2 c o s γ_{2} = 0

(46)

- \hat{μ} ω_{2} a_{2} + f / 2 s i n γ_{2} = 0

(47)

$ω_{3} (3 β_{1}^{'} - σ_{1}) a_{3} - 1 / 8 α_{10} a_{1}^{3} c o s γ_{1} - 1 / 4 α_{11} a_{1}^{2} a_{3} -$

1 / 4 α_{13} a_{2}^{2} a_{3} - 3 / 8 α_{14} a_{3}^{3} = 0

(48)

- \hat{μ} ω_{3} a_{3} + 1 / 8 α_{10} a_{1}^{3} s i n γ_{1} = 0

(49)

其中， $γ_{1} = σ_{1} T_{1} + β_{3} - 3 β_{1} ， γ_{2} = σ_{2} T_{1} - β_{2} - π / 2$ .
（3）情况III：Ω₃≈ω₃
假定

Ω_{3} = ω_{3} + ε σ_{3}

(50)

消除久期项，并分离方程的实部和虚部可以得到式（51）~式（56）.
$ω_{1} β_{1}^{'} a_{1} - 3 / 8 α_{1} a_{1}^{3} - 1 / 8 α_{2} a_{1}^{2} a_{3} c o s γ -$

1 / 4 α_{4} a_{1} a_{2}^{2} - 1 / 4 α_{4} a_{1} a_{3}^{2} = 0

(51)

- ω_{1} a_{1}^{'} - a_{1} \hat{μ} ω_{1} - 1 / 8 α_{2} a_{1}^{2} a_{3} s i n γ = 0

(52)

$ω_{2} β_{2}^{'} a_{2} - 1 / 4 α_{6} a_{1}^{2} a_{2} - 1 / 4 α_{9} a_{2} a_{3}^{2} -$

3 / 8 α_{8} a_{2}^{3} = 0

(53)

- ω_{2} a_{2}^{'} - \hat{μ} ω_{2} a_{2} = 0

(54)

$ω_{3} (3 β_{1}^{'} - σ_{1}) a_{3} - 1 / 8 α_{10} a_{1}^{3} c o s γ_{1} -$
$1 / 4 α_{11} a_{1}^{2} a_{3} - 1 / 4 α_{13} a_{2}^{2} a_{3} -$

3 / 8 α_{14} a_{3}^{3} + 1 / 2 f c o s γ_{2} = 0

(55)

- \hat{μ} ω_{3} a_{3} + 1 / 8 α_{10} a_{1}^{3} s i n γ_{1} + 1 / 2 f s i n γ_{2} = 0

(56)

其中， $γ_{1} = σ_{1} T_{1} + β_{3} - 3 β_{1} ， γ_{2} = σ_{2} T_{1} - β_{3} - π / 2$ .
（4）稳定性分析
有限长梁和非线性基础的物理几何参数如表1所示.以情况I为例分析系统的稳定性，观察方程组式（37）~式（42）可以看出稳态下a′₂=a₂=0，则方程组可以写为
$ω_{1} β_{1}^{'} a_{1} - 3 / 8 α_{1} a_{1}^{3} - 1 / 4 α_{4} a_{1} a_{3}^{2} -$

1 / 8 α_{2} a_{1}^{2} a_{3} c o s γ_{1} + f / 2 c o s γ_{2} = 0

(57)

- ω_{1} a_{1}^{'} - \hat{μ} ω_{1} a_{1} + f / 2 s i n γ_{2} = 0

(58)

$ω_{3} β_{3}^{'} a_{3} - 1 / 8 α_{10} a_{1}^{3} c o s γ_{1} - 1 / 4 α_{11} a_{1}^{2} a_{3} -$

3 / 8 α_{14} a_{3}^{3} = 0

(59)

- ω_{3} a_{3}^{'} - \hat{μ} ω_{3} a_{3} + 1 / 8 α_{10} a_{1}^{3} s i n γ_{1} = 0

(60)

又因为

β_{3}^{'} = γ_{1}^{'} - 3 γ_{2}^{'} + 3 σ_{2} - σ_{1}

(61)

β_{1}^{'} = σ_{2} - γ_{2}^{'}

(62)

表1 基础梁和移动载荷的物理几何参数^[1]
Table1 Physical geometric parameters of foundation beams and moving load
将式（61）和式（62）代入式（57）~式（60）可得，

a_{1}^{'} = - \hat{μ} a_{1} - \frac{α_{2}}{8 ω_{1}} a_{1}^{2} a_{3} s i n γ_{1} + \frac{f}{2 ω_{1}} s i n γ_{2}

(63)

a_{3}^{'} = - \hat{μ} a_{3} + \frac{α_{10}}{8 ω_{3}} {\hat{k}}_{3} a_{1}^{3} s i n γ_{1}

(64)

$γ_{1}^{'} = σ_{1} + (\frac{α_{11}}{4 ω_{3}} - \frac{9 α_{1}}{8 ω_{1}} {\hat{k}}_{3}) a_{1}^{2} + (\frac{3 α_{14}}{8 ω_{3}} -$
$\frac{3 α_{4}}{4 ω_{1}}) a_{3}^{2} + (\frac{α_{10}}{8 ω_{3} a_{3}} a_{1}^{3} - \frac{3 α_{2}}{8 ω_{1}} a_{1} a_{3}) c o s γ_{1} +$

\frac{3 f}{2 a_{1} ω_{1}} c o s γ_{2}

(65)

$γ_{2}^{'} = σ_{2} - \frac{3 α_{1}}{8 ω_{1}} a_{1}^{2} - \frac{α_{4}}{4 ω_{1}} a_{3}^{2} - \frac{α_{2}}{8 ω_{1}} a_{1} a_{3} c o s γ_{1} +$

\frac{f}{2 a_{1} ω_{1}} c o s γ_{2}

(66)

以 ${[a_{1} ， a_{3} ， γ_{1} ， γ_{2}]}^{T}$ 为状态向量，通过线性化式（63）~式（66）可以得到Jacobian矩阵

(67)

其中
$J_{11} = - \hat{μ} - \frac{α_{2}}{4 ω_{1}} a_{1} a_{3} s i n γ_{1},$
$J_{12} = - \frac{α_{2}}{8 ω_{1}} a_{1}^{2} s i n γ_{1},$
$J_{13} = - \frac{α_{2}}{8 ω_{1}} a_{1}^{2} a_{3} c o s γ_{1}$
$J_{14} = \frac{f}{2 ω_{1}} c o s γ_{2}, J_{21} = \frac{3 α_{10}}{8 ω_{3}} {\hat{k}}_{3} a_{1} s i n γ_{1},$
$J_{22} = - \hat{μ}, J_{23} = \frac{α_{10}}{8 ω_{3}} {\hat{k}}_{3} a_{1}^{3} c o s γ_{1}, J_{24} = 0,$
$J_{31} = (\frac{α_{11}}{2 ω_{3}} - \frac{9 α_{1}}{4 ω_{1}} {\hat{k}}_{3}) a_{1} +$
$(\frac{2 α_{10}}{8 ω_{3} a_{3}} a_{1}^{2} - \frac{3 α_{2}}{8 ω_{1}} a_{3}) c o s γ_{1} - \frac{3 f}{2 a_{1}^{2} ω_{1}} c o s γ_{2},$
$J_{32} = (\frac{3 α_{14}}{4 ω_{3}} - \frac{3 α_{4}}{2 ω_{1}}) a_{3} +$
$(- \frac{α_{10}}{8 ω_{3} a_{3}^{2}} a_{1}^{3} - \frac{3 α_{2}}{8 ω_{1}} a_{1}) c o s γ_{1},$
$J_{33} = - (\frac{α_{10}}{8 ω_{3} a_{3}} a_{1}^{3} - \frac{3 α_{2}}{8 ω_{1}} a_{1} a_{3}) s i n γ_{1},$
$J_{34} = - \frac{3 f}{2 a_{1} ω_{1}} s i n γ_{2}$
$J_{41} = - \frac{3 α_{1}}{4 ω_{1}} a_{1} - \frac{α_{2}}{8 ω_{1}} a_{3} c o s γ_{1} - \frac{f}{2 a_{1}^{2} ω_{1}} c o s γ_{2}$
$J_{42} = - \frac{α_{4}}{2 ω_{1}} a_{3} - \frac{α_{2}}{8 ω_{1}} a_{1} c o s γ_{1},$
$J_{43} = \frac{α_{2}}{8 ω_{1}} a_{1} a_{3} s i n γ_{1}, J_{44} = - \frac{f}{2 a_{1} ω_{1}} s i n γ_{2}$
根据Lyapunov第一方法，可以判定：若矩阵J的特征值的实部全为负则系统稳定，反之系统不稳定.本文根据以上所述的方法，分别给出了在情况I、情况II和情况III下当k₃=8×10^9.9N/m⁴时的稳定性分析图，如图4~图6所示.
图4 k₃=8×10^9.9N/m⁴时，幅频响应及其稳定性（情况Ⅰ）
Fig.4 Amplitude-frequency response and its stability at k₃=8×10^9.9N/m⁴ (Case Ⅰ)
图4（c）画出了情况Ⅰ下特征值实部的变化情况，可以看出在区间（σ_2B，σ_2A），即38.45＜σ₂＜132.15时，系统存在正实特征值，则该部分存在不稳定解，其它部分为1个稳定解.与图4（c）相对应的，图4（a）和图4（b）分别给出了第一阶模态和第三阶模态的幅频相应图.同时，结合表2可知由A和C点计算得到的Jacobian矩阵特征值都具有负实部，可以判定A和C点所在的曲线为稳定解用实线表示；由B点计算得到的Jacobian矩阵特征值实部中有两个具有正实部，可以判定B点所在的曲线为不稳定解用虚线表示.
表2 图4标注各点所对应的Jacobian矩阵特征值
Table2 Figure4 shows the eigenvalues of Jacobian matrix corresponding to each point
运用同样的方法分别判断了系统在情况Ⅱ和情况Ⅲ下的稳定性.从图5（a）和图5（b）可以看出在区间（σ_2B，σ_2A），即38.16＜σ₂＜131.67时，系统存在2个稳定解和1个不稳定解，其它部分为稳定解.从图6（a）和图6（b）可以看出在区间（σ_2A，σ_2C）和（σ_2F，σ_2B），系统存在2个稳定解和1个不稳定解；在区间（σ_2C，σ_2E），系统存在3个稳定解和2个不稳定解；在区间（σ_2E，σ_2D），系统存在3个稳定解和4个不稳定解；在区间（σ_2D，σ_2F），系统存在2个稳定解和3个不稳定解，其它区间只存在1个稳定解.
图5 k₃=8×10^9.9N/m⁴时，幅频响应及其稳定性（情况Ⅱ）
Fig.5 Amplitude-frequency response and its stability at k₃=8×10^9.9N/m⁴ (Case Ⅱ)
图6 k₃=8×10^9.9N/m⁴时，幅频响应及其稳定性（情况Ⅲ）
Fig.6 Amplitude-frequency response and its stability at k₃=8×10^9.9N/m⁴ (Case Ⅲ)
3 结论
研究了在移动荷载作用下非线性粘弹性基础梁的振动问题.采用Galerkin方法对非线性偏微分运动方程进行离散，并利用多尺度法求解了非线性运动控制方程.在此基础上，研究了非线性基础梁在3:1 内共振下的自由振动，以及在内共振作用下梁受外激励作用的第一、第二和第三阶模态主共振的非线性动力学行为及其动力学响应的稳定性问题.研究揭示了基础参数对内共振下弹性梁振动响应的影响规律，并发现在发生第三阶主共振时，幅频响应中会出现7个解支，包括3个稳定解支和4个不稳定解支.
参考文献
- [1] GHAYESH M H,KAZEMIRAD S,AMABILI M.Coupled longitudinal-transverse dynamics of an axially moving beam with an internal resonance [J].Mechanism and Machine Theory,2012,52:18-34.
- [2] HU W P,YE J,DENG Z C.Internal resonance of a flexible beam in a spatial tethered system [J].Journal of Sound and Vibration,2020,475:115286.
- [3] DING H,HUANG L L,MAO X Y,et al.Primary resonance of traveling viscoelastic beam under internal resonance [J].Applied Mathematics and Mechanics-English Edition,2017,38(1):1-14.
- [4] WANG Y R,WEI Y H.Internal resonance analysis of a fluid-conveying tube resting on a nonlinear elastic foundation [J].European Physical Journal Plus,2020,135(4):364.
- [5] WANG G X,DING H,CHEN L Q.Dynamic effect of internal resonance caused by gravity on the nonlinear vibration of vertical cantilever beams [J].Journal of Sound and Vibration,2020,474:115265.
- [6] GUILLOT V,GIVOIS A,COLIN M,et al.Theoretical and experimental investigation of a 1:3 internal resonance in a beam with piezoelectric patches [J].Journal of Vibration and Control,2020,26(13-14):1119-1132.
- [7] DING H,HUANG L L,DOWELL E,et al.Stress distribution and fatigue life of nonlinear vibration of an axially moving beam [J].Science China Technological Sciences,2019,62(7):1123-1133.
- [8] YOUNESIAN D,SADRI M,ESMAILZADEH E.Primary and secondary resonance analyses of clamped-clamped micro-beams [J].Nonlinear Dynamics,2014,76(4):1867-1884.
- [9] 顾伟,张博,丁虎,等.2:1内共振条件下变转速预变形叶片的非线性动力学响应 [J].力学学报,2020,52(7):1131-1142.GU W,ZHANG B,DING HU,et al.Nonlinear dynamic response of pre-deformed blade with variable rotational speed under 2:1 internal resonance [J].Chinese Journal of Theoretical and Applied Mechanics,2020,52(7):1131-1142.(in Chinese)
- [10] 毕勤胜,陈予恕.双摆内共振分岔分析 [J].应用数学和力学,2000,21(3):226-234.BI Q S,CHEN Y S.Bifurcation analysis of a double pendulum with internal resonance [J].Applied Mathematics and Mechanics,2000,21(3):226-234.(in Chinese)
- [11] 魏明海,肖仪清.内外共振联合作用下索-梁组合结构非线性振动分析 [J].振动与冲击,2012,31(7):79-84.WEI H M,XIAO Y Q.Nonlinear vibration analysis for a cable-beam coupled system under simultaneous internal and external resonances [J].Vibration and Shock,2012,31(7):79-84.(in Chinese)
- [12] LI S H,YANG S P,CHEN L Q,et al.Effects of parameters on dynamic responses for a heavy vehicle-pavement-foundation coupled system [J].International Journal of Heavy Vehicle Systems,2012,19(2):207-224.
- [13] DING H,CHEN L Q.Approximate and numerical analysis of nonlinear forced vibration of axially moving viscoelastic beams [J].Acta Mechanica Sinica,2011,27(3):426-437.
- [14] DONG Z J,MA X Y.Analytical solutions of asphalt pavement responses under moving loads with arbitrary non-uniform tire contact pressure and irregular tire imprint [J].Road Materials and Pavement Design,2018,19(8):1887-1903.
- [15] 杨予,滕念管,黄醒春,等.承受移动均布质量的简支梁振动反应分析 [J].振动与冲击,2005,24(3):19-22+6.YANG Y,TENG N G,HUANG X C,et al.Vibration analysis of a simply supported beam traversed by uniform distributed moving mass [J].Vibration and Shock,2005,24(3):19-22+6.(in Chinese)
- [16] 刘涛,周洋忻,胡伟鹏.轴向运动功能梯度梁横向振动问题的保结构分析 [J].动力学与控制学报,2022,20(6):101-105.LIU T,ZHOU Y X,HU W P.Structure-preserving analysis on transverse vibration of functionally graded beam with an axial velocity [J].Journal of Dynamics and Control,2022,20(6):101-105.(in Chinese)
- [17] MARTINEZ-CASTRO A E,MUSEROS P,CASTILLO-LINARES A.Semi-analytic solution in the time domain for non-uniform multi-span Bernoulli-Euler beams traversed by moving loads [J].Journal of Sound and Vibration,2006,294(1-2):278-297.
- [18] 张弛,毛晓晔,丁虎,等.受轴向激励弹性支承梁的稳定性分析 [J].动力学与控制学报,2022,20(3):66-76.ZHANG C,MAO X Y,DING H,et al.Stability analysis of axially excited beam with elastic boundary [J].Journal Dynamics and Control,2022,20(3):66-76.(in Chinese)
- [19] YOUNESIAN D,MARJANI S R,ESMAILZADEH E.Nonlinear vibration analysis of harmonically excited cracked beams on viscoelastic foundations [J].Nonlinear Dynamics,2013,71(1-2):109-120.
- [20] DING H,YANG Y,CHEN L Q,et al.Vibration of vehicle-pavement coupled system based on a Timoshenko beam on a nonlinear foundation [J].Journal of Sound and Vibration,2014,333(24):6623-6636.
- [21] ANSARI M,ESMAILZADEH E,YOUNESIAN D.Frequency analysis of finite beams on nonlinear Kelvin-Voight foundation under moving loads [J].Journal of Sound and Vibration,2011,330(7):1455-1471.
- [22] SAPOUNTZAKIS E J,KAMPITSIS A E.Nonlinear response of shear deformable beams on tensionless nonlinear viscoelastic foundation under moving loads [J].Journal of Sound and Vibration,2011,330(22):5410-5426.

基本信息

中图分类号: O32
文献标识码: A
DOI: 10.6052/1672-6553-2023-061
文章编号: 1672-6553-2023-21(9)-041-009

基金信息

引用信息

稿件历史

收稿日期: 2023-03-24

参考文献

[1] GHAYESH M H,KAZEMIRAD S,AMABILI M.Coupled longitudinal-transverse dynamics of an axially moving beam with an internal resonance [J].Mechanism and Machine Theory,2012,52:18-34.
[2] HU W P,YE J,DENG Z C.Internal resonance of a flexible beam in a spatial tethered system [J].Journal of Sound and Vibration,2020,475:115286.
[3] DING H,HUANG L L,MAO X Y,et al.Primary resonance of traveling viscoelastic beam under internal resonance [J].Applied Mathematics and Mechanics-English Edition,2017,38(1):1-14.
[4] WANG Y R,WEI Y H.Internal resonance analysis of a fluid-conveying tube resting on a nonlinear elastic foundation [J].European Physical Journal Plus,2020,135(4):364.
[5] WANG G X,DING H,CHEN L Q.Dynamic effect of internal resonance caused by gravity on the nonlinear vibration of vertical cantilever beams [J].Journal of Sound and Vibration,2020,474:115265.
[6] GUILLOT V,GIVOIS A,COLIN M,et al.Theoretical and experimental investigation of a 1:3 internal resonance in a beam with piezoelectric patches [J].Journal of Vibration and Control,2020,26(13-14):1119-1132.
[7] DING H,HUANG L L,DOWELL E,et al.Stress distribution and fatigue life of nonlinear vibration of an axially moving beam [J].Science China Technological Sciences,2019,62(7):1123-1133.
[8] YOUNESIAN D,SADRI M,ESMAILZADEH E.Primary and secondary resonance analyses of clamped-clamped micro-beams [J].Nonlinear Dynamics,2014,76(4):1867-1884.
[9] 顾伟,张博,丁虎,等.2:1内共振条件下变转速预变形叶片的非线性动力学响应 [J].力学学报,2020,52(7):1131-1142.GU W,ZHANG B,DING HU,et al.Nonlinear dynamic response of pre-deformed blade with variable rotational speed under 2:1 internal resonance [J].Chinese Journal of Theoretical and Applied Mechanics,2020,52(7):1131-1142.(in Chinese)
[10] 毕勤胜,陈予恕.双摆内共振分岔分析 [J].应用数学和力学,2000,21(3):226-234.BI Q S,CHEN Y S.Bifurcation analysis of a double pendulum with internal resonance [J].Applied Mathematics and Mechanics,2000,21(3):226-234.(in Chinese)
[11] 魏明海,肖仪清.内外共振联合作用下索-梁组合结构非线性振动分析 [J].振动与冲击,2012,31(7):79-84.WEI H M,XIAO Y Q.Nonlinear vibration analysis for a cable-beam coupled system under simultaneous internal and external resonances [J].Vibration and Shock,2012,31(7):79-84.(in Chinese)
[12] LI S H,YANG S P,CHEN L Q,et al.Effects of parameters on dynamic responses for a heavy vehicle-pavement-foundation coupled system [J].International Journal of Heavy Vehicle Systems,2012,19(2):207-224.
[13] DING H,CHEN L Q.Approximate and numerical analysis of nonlinear forced vibration of axially moving viscoelastic beams [J].Acta Mechanica Sinica,2011,27(3):426-437.
[14] DONG Z J,MA X Y.Analytical solutions of asphalt pavement responses under moving loads with arbitrary non-uniform tire contact pressure and irregular tire imprint [J].Road Materials and Pavement Design,2018,19(8):1887-1903.
[15] 杨予,滕念管,黄醒春,等.承受移动均布质量的简支梁振动反应分析 [J].振动与冲击,2005,24(3):19-22+6.YANG Y,TENG N G,HUANG X C,et al.Vibration analysis of a simply supported beam traversed by uniform distributed moving mass [J].Vibration and Shock,2005,24(3):19-22+6.(in Chinese)
[16] 刘涛,周洋忻,胡伟鹏.轴向运动功能梯度梁横向振动问题的保结构分析 [J].动力学与控制学报,2022,20(6):101-105.LIU T,ZHOU Y X,HU W P.Structure-preserving analysis on transverse vibration of functionally graded beam with an axial velocity [J].Journal of Dynamics and Control,2022,20(6):101-105.(in Chinese)
[17] MARTINEZ-CASTRO A E,MUSEROS P,CASTILLO-LINARES A.Semi-analytic solution in the time domain for non-uniform multi-span Bernoulli-Euler beams traversed by moving loads [J].Journal of Sound and Vibration,2006,294(1-2):278-297.
[18] 张弛,毛晓晔,丁虎,等.受轴向激励弹性支承梁的稳定性分析 [J].动力学与控制学报,2022,20(3):66-76.ZHANG C,MAO X Y,DING H,et al.Stability analysis of axially excited beam with elastic boundary [J].Journal Dynamics and Control,2022,20(3):66-76.(in Chinese)
[19] YOUNESIAN D,MARJANI S R,ESMAILZADEH E.Nonlinear vibration analysis of harmonically excited cracked beams on viscoelastic foundations [J].Nonlinear Dynamics,2013,71(1-2):109-120.
[20] DING H,YANG Y,CHEN L Q,et al.Vibration of vehicle-pavement coupled system based on a Timoshenko beam on a nonlinear foundation [J].Journal of Sound and Vibration,2014,333(24):6623-6636.
[21] ANSARI M,ESMAILZADEH E,YOUNESIAN D.Frequency analysis of finite beams on nonlinear Kelvin-Voight foundation under moving loads [J].Journal of Sound and Vibration,2011,330(7):1455-1471.
[22] SAPOUNTZAKIS E J,KAMPITSIS A E.Nonlinear response of shear deformable beams on tensionless nonlinear viscoelastic foundation under moving loads [J].Journal of Sound and Vibration,2011,330(22):5410-5426.

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移动荷载作用下非线性基础梁内共振分析

Internal Resonance Analysis of Nonlinear Foundation Beams under Moving Loads

摘要

Abstract

关键词

Keywords

1 非线性粘弹性基础梁模型

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2 3:1内共振条件下的稳定性分析

2.1 3:1内共振下的自由振动

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2.2 3:1内共振下的受迫振动分析

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3 结论