力学学报, 2019, 51(6): 1897-1904 DOI: 10.6052/0459-1879-19-205

动力学与控制

黏弹性阻尼作用下轴向运动Timoshenko梁振动特性的研究 1)

周远, 唐有绮,3), 刘星光

上海应用技术大学机械工程学院, 上海 201418

VIBRATION CHARACTERISTICS OF AXIALLY MOVING TIMOSHENKO BEAM UNDER VISCOELASTIC DAMPING 1)

Zhou Yuan, Tang Youqi,3), Liu Xingguang

School of Mechanical Engineering, Shanghai Institute of Technology, Shanghai 201418, China

通讯作者: 3) 唐有绮, 副教授, 主要研究方向: 非线性动力学与振动控制. E-mail:yqtang@126.com

收稿日期: 2019-07-25   接受日期: 2019-09-17   网络出版日期: 2019-09-17

基金资助: 1) 国家自然科学基金资助项目.  11672186

Received: 2019-07-25   Accepted: 2019-09-17   Online: 2019-09-17

作者简介 About authors

2)周远,硕士研究生,主要研究方向:非线性动力学与振动控制.E-mail:zysuzhou@163.com

摘要

黏弹性阻尼一直是轴向运动系统的研究热点之一.以往研究轴向运动系统大都没有考虑黏弹性阻尼的影响.但在工程实际中, 存在黏弹性阻尼的轴向运动体系更为普遍.本文研究了黏弹性阻尼作用下轴向运动Timoshenko梁的振动特性.首先, 采用广义Hamilton原理给出了轴向运动黏弹性Timoshenko梁的动力学方程组和相应的简支边界条件.其次, 应用直接多尺度法得到了轴速和相关参数的对应关系, 给出了前两阶固有频率和衰减系数在黏弹性作用下的近似解析解.最后, 采用微分求积法分析了在有无黏弹性作用下前两阶固有频率和衰减系数随轴速的变化; 给出了前两阶固有频率和衰减系数在黏弹性作用下的近似数值解, 验证了近似解析解的有效性.结果表明: 随着轴速的增大, 梁的固有频率逐渐减小.梁的固有频率和衰减系数随着黏弹性系数的增大而逐渐减小, 其中衰减系数与黏弹性系数成正比关系, 黏弹性系数对第一阶衰减系数和固有频率的影响很小, 对第二阶衰减系数和固有频率的影响较大.

关键词: Timoshenko梁 ; 直接多尺度法 ; 微分求积法 ; 黏弹性阻尼 ; 固有频率 ; 衰减系数

Abstract

Viscoelastic damping has always been one of the research hotspots of axial motion system. The influence of viscoelastic damping has not been considered in most previous researches on axial motion systems. In the present paper, the vibration characteristics of the axially moving Timoshenko beam with viscoelastic damping are studied. The dynamic equations of Timoshenko beams with axial viscoelastic motion and the corresponding boundary conditions of simply supported beams are obtained using the generalized Hamilton principle. The method of direct multiple scales is used to show the corresponding relationship between axial speed and parameters. The approximate analytical solutions of the first two natural frequency and attenuation coefficient are obtained. The differential quadrature method is applied to analyze the variation of the first two natural frequencies and attenuation coefficients with the axial speed under the presence or absence of viscoelasticity. The approximate numerical solutions of the first two natural frequencies and attenuation coefficients under viscoelastic action are given and the validity of approximate analytic solution is verified. It is shown that the natural frequency of the beam decreases gradually with the increasing axial speed. The natural frequency and attenuation coefficient of the beam decrease with the increasing viscoelastic coefficient. The attenuation coefficient is proportional to the viscoelastic coefficient. The viscoelastic coefficient has little effect on the first order attenuation coefficient and natural frequency. But it has a greater influence on the second-order attenuation coefficient and natural frequency.

Keywords: Timoshenko beam ; method of direct multiple scales ; differential quadrature method ; viscoelastic damping ; natural frequency ; attenuation coefficient

PDF (309KB) 元数据 多维度评价 相关文章 导出 EndNote| Ris| Bibtex  收藏本文

本文引用格式

周远, 唐有绮, 刘星光. 黏弹性阻尼作用下轴向运动Timoshenko梁振动特性的研究 1). 力学学报[J], 2019, 51(6): 1897-1904 DOI:10.6052/0459-1879-19-205

Zhou Yuan, Tang Youqi, Liu Xingguang. VIBRATION CHARACTERISTICS OF AXIALLY MOVING TIMOSHENKO BEAM UNDER VISCOELASTIC DAMPING 1). Chinese Journal of Theoretical and Applied Mechanics[J], 2019, 51(6): 1897-1904 DOI:10.6052/0459-1879-19-205

引言

在工程实际及生活中, 轴向运动系统是工程领域中应用最广泛的结构形式, 如钢带、传动带等.由于轴向速度和黏弹性的存在, 轴向运动系统会产生较大的振动, 从而引起有害的振动和噪声.因此, 对轴向黏弹性系统的研究有着重要的意义.

国内外众多学者[1-4]对轴向运动Timoshenko梁的横向振动问题进行了一系列研究.Tang等[5]利用多尺度法和微分求积法分析了轴向运动Timoshenko梁的复模态和复频率.王乐等[6]采用分离变量法求解给出了Timoshenko梁固有频率的方程.唐有绮等[7]分别研究了径向变张力对轴向运动Euler梁和Timoshenko梁的影响.华洪良等[8]应用Rayleigh-Ritz法分析了轴向移动悬臂梁的频率响应特性.黄悦等[9]研究了功能梯度材料Timoshenko梁的动力屈曲.胡璐等[10]研究了黏性流体环境下V型悬臂梁的流固耦合振动特性.Tang等[11]研究了Timoshenko梁的参数共振.Asghari等[12]分别研究了功能梯度材料悬臂梁和功能梯度材料简支梁的静态和自由振动.An等[13]研究了轴速、轴向张力和外部分布力的幅度对轴向移动的Timoshenko梁振幅的影响.Ghayesh等[14-15]分别研究了轴向运动Euler梁和Timoshenko梁的非线性振动.Farokhi等[16]研究了受轴向简谐激振力作用的Timoshenko梁的非线性超临界参数动力学.Qiang等[17]分析了热载荷作用下热黏弹性Timoshenko梁的准静态和动态响应.Lü等[18]分析了变速度下轴向运动黏弹性夹层梁非线性参数振动.Wang[19]运用渐进法研究了三参数模型轴向运动梁的受迫振动.Liu等[20]应用多尺度方法研究了随机无序周期激励下轴向运动黏弹性梁的动力学响应.Simsek[21]对由于移动谐波载荷引起的固定支撑的功能梯度梁进行了非线性动态分析.Mokhtari等[22]基于波谱的有限元模型用于轴向预张力的轴向运动Timoshenko梁的时域和波域动力学分析.随岁寒[23]分析了轴向运动速度、梯度指数、初应力大小等因素对梁的动力响应的影响.

研究者发现, 以往研究轴向运动体系大都没有考虑黏弹性的影响.但是, 在工程实际中, 黏弹性更为普遍, 而目前关于黏弹性对线性振动的研究相对较少.Tang等[24]运用直接多尺度法研究了非齐次边界条件下轴向运动黏弹性Euler梁的固有频率、稳态响应及其稳定性.Yang等[25]利用平均法研究了轴向运动黏弹性Timoshenko梁的动态稳定性.丁虎等[26]研究了黏弹性对轴向运动梁横向振动固有频率的影响.杨鄂川等[27]针对开口裂纹作用下旋转运动Euler-Bernoulli梁的振动特性进行了研究.张国策等[28]研究了简支边界条件下超临界轴向运动梁横向非线性自由振动的固有频率和模态函数.吕海炜等[29]研究了小扰度下轴向匀速运动黏弹性夹层梁的振动模态和固有频率.

本文研究了黏弹性阻尼作用下轴向运动Timoshenko梁的振动特性, 给出了其对应的控制方程组和简支边界条件.适当取定参数, 发展多尺度法得到了速度和参数的对应关系, 得到了前两阶固有频率和衰减系数的近似解析解, 通过微分求积法对比了有无黏弹性时固有频率和衰减系数随轴速变化的差别, 并对近似解析解进行了数值验证.

1 控制方程

考虑横截面面积为$A$, 密度为$\rho$, 弹性模量为$E$, 惯性矩为$I$,支撑两端间距为$L$, 剪切模量为$G$, 弹性模量为$E$, 黏弹性系数为$\alpha$,初始轴力为$P$, 支撑刚度参数为$\kappa\in [0,1]$. 横向位移为$v(x,t)$的均匀Timoshenko梁以匀速 $\gamma $沿轴向运动. $\varphi$ ($x$, $t)$为弯矩而产生的Timoshenko梁轴线的转角.采用广义Hamilton原理可以得到其无量纲形式的控制方程组[5]和相应的简支边界条件

$\begin{eqnarray}&&v,_{tt} + 2\gamma v,_{xt} + ( {\kappa \gamma ^2 - 1} )v,_{xx} + k_1 ( {\varphi ,_x - v,_{xx} } ) = \\&&\qquad- \varepsilon k_1 \alpha ( {\varphi ,_{xt} - v,_{xxt} } )\end{eqnarray}$
$\begin{eqnarray}&&k_2 ( {\varphi ,_{tt} + 2\gamma \varphi ,_{xt} + \gamma ^2\varphi ,_{xx} } ) - k_f^2 \varphi ,_{xx} + k_1 ( {\varphi - v,_x } ) = \\&&\qquad\varepsilon \alpha [ {k_f^2 \varphi,_{xxt} - k_1 ( {\varphi ,_t - v,_{xt} } )} ]\end{eqnarray}$
$\begin{eqnarray}&&{[ {( {k_f^2 - k_2 \gamma ^2} )\varphi ,_x - k_2 \gamma \varphi ,_t } ]} |_0^1 = 0, v |_0^1 = 0 \end{eqnarray}$

其中$\varepsilon$为无量纲参数, 表征梁的横向位移和黏弹性系数均为小量. 其他无量纲参数为

$\begin{eqnarray}&& v\leftrightarrow \frac{v}{\sqrt \varepsilon L},\ x\leftrightarrow \frac{x}{L},\ t\leftrightarrow \frac{t}{L}\sqrt {\frac{P}{\rho A}},\ \gamma\leftrightarrow \gamma \sqrt {\frac{\rho A}{P}}\\ &&k_f = \sqrt {\frac{EI}{PL^2}},\ \alpha\leftrightarrow \frac{\alpha }{\varepsilon L}\sqrt {\frac{P}{\rho A}},\ k_{1} = \frac{kAG}{P},\ k_2 = \frac{I}{AL^2}\qquad \end{eqnarray}$

2 直接多尺度分析

设式(1)和式(2)的二阶近似解为

$\begin{eqnarray}&&v( {x,t;\varepsilon } ) = v_0 ( {x,T_0 ,T_1 ,T_2 } )+ \varepsilon v_{_1 } ( {x,T_0 ,T_1 ,T_2 } )+\\&&\qquad\varepsilon ^2v_2 ( {x,T_0 ,T_1 ,T_2} ) + O( {\varepsilon ^3} )\end{eqnarray}$
$\begin{eqnarray}\\&& \varphi ( {x,t;\varepsilon } ) = \varphi _0 ( {x,T_0 ,T_1 ,T_2 } ) + \varepsilon \varphi _{_1 } ( {x,T_0 ,T_1 ,T_2 } ) +\\&&\qquad \varepsilon ^2\varphi _2 ( {x,T_0 ,T_1,T_2 } ) + O( {\varepsilon ^3} )\end{eqnarray}$

式中, $T_{0}=t$, $T_{1}=\varepsilon T_{0}$和$T_{2}=\varepsilon ^{2}T_{0}$.将式(5)和式(6)代入式(1) $\sim\!$式(3), 分离$\varepsilon ^{0}$, $\varepsilon ^{1}$和$\varepsilon ^{2}$阶量得

$\varepsilon ^{0}$:

$\begin{eqnarray}&&v_{0,T_0 T_0 } + 2\gamma v_{0,xT_0 } + ( {\kappa \gamma ^2 - 1})v_{0,xx} +\\&&\qquad k_1 ( {\varphi _0 ,_x - v_{0,xx}}) = 0\end{eqnarray}$
$\begin{eqnarray} k_2 ( {\varphi _{0,T_0 T_0 } + 2\gamma \varphi _{0,xT_0 } + \gamma ^2\varphi _{0,xx} } ) - k_f^2 \varphi _{0,xx} + k_1 ( \varphi _0 - v_{0,x} ) = 0\\ \end{eqnarray}$
$\begin{eqnarray}&&[(k_f^2 - k_2 \gamma ^2 )\varphi _{0,x} - k_2 \gamma \varphi _{0,T_0}]|_0^1 = 0,\ {v_0}|_0^1 = 0\end{eqnarray}$

$\varepsilon ^{1}$:

$\begin{eqnarray}&& v_{1,T_0 T_0 } + 2\gamma v_{1,xT_0 } + ( {\kappa \gamma ^2 - 1})v_{1,xx} + k_1 ( {\varphi _{1,x} - v_{1,xx} } ) +\\&&\qquad 2v_{0,T_0 T_1 } + 2\gamma v_{0,xT_1} + k_1 \alpha ( {\varphi _{0,xT_0} - v_{0,xxT_0}}) = 0 \\\end{eqnarray}$
$\begin{eqnarray}&& k_2 ( {\varphi _{1,T_0 T_0} + 2\gamma \varphi _{1,xT_0 } + \gamma^2\varphi _{1,xx} + 2\varphi _{0,T_0 T_1 }+ 2\gamma \varphi _{0,xT_1 } } ) -\\&&\qquad k_f^2 \varphi _{1,xx} - \alpha [ {k_f^2 \varphi _{0,xxT_0 } - k_1( {\varphi _{0,T_0 } - v_{0,xT_0 } } )} ] +\\&&\qquad k_1 ( \varphi _1 - v_{1,x}) = 0\\\end{eqnarray}$
$\begin{eqnarray}&&[ {( {k_f^2 - k_2 \gamma ^2} )\varphi _1 ,_x - k_2 \gamma ( {\varphi _1 ,_{T_0 } + \varphi _0 ,_{T_1 } } )} ] |_0^1 = 0,\\ &&\qquad{v_1 } |_0^1 = 0\end{eqnarray}$

$\varepsilon ^{2}$:

$\begin{eqnarray}&&v_{2,T_0 T_0 } + ( {\kappa \gamma ^2 - 1} )v_{2,xx} + k_1 ( {\varphi _{2,x} - v_{2,xx} } ) + 2v_{0,T_0 T_2 } +\\&&\qquad v_{0,T_1 T_1 } + 2\gamma v_{2,xT_0} + 2\gamma v_{0,xT_2 } + 2v_{1,T_0 T_1 } + 2\gamma v_{1,xT_1} + \\&&\qquad k_1 \alpha ( {\varphi _{0,xT_1} - v_{0,xxT_1} + \varphi _{1,xT_0} - v_{1,xxT_0 } } ) = 0\\\end{eqnarray}$
$\begin{eqnarray}&&\label{eq14} k_2 ( \varphi _{2,T_0 T_0} + 2\gamma \varphi _{2,xT_0 } + \gamma ^2\varphi _{2,xx} + 2\varphi _{0,T_0 T_2} + 2\gamma \varphi _{0,xT_2 } +\\&&\qquad\varphi _{0,T_1 T_1 } + {2\varphi _{1,T_0 T_1 } + 2\gamma \varphi _{1,xT_1 } } ) + k_1 ( \varphi _2 - v_{2,x} ) -\\&&\qquad k_f^2 \varphi _{2,xx} + \alpha k_1 ( {\varphi _{1,T_0} + \varphi _{0,T_1} - v_{0,xT_1} - v_{1,xT_0} } ) -\\&&\qquad \alpha k_f^2 ( {\varphi _{0,xxT_1 } + \varphi _{1,xxT_0 } } ) =0\end{eqnarray}$
$\begin{eqnarray}\\&&\label{eq15}{[ {( {k_f^2 - k_2 \gamma ^2} )\varphi _{2,x} - k_2 \gamma ( {\varphi _{2,T_0 } + \varphi _{1,T_1 } + \varphi _{0,T_2 } } )} ]} |_0^1 = 0, \\ &&\qquad{v_2 }|_0^1 = 0 \end{eqnarray}$

解耦线性派生系统(7)和(8), 整理得到线性派生系统的频率方程[5]

$\begin{eqnarray}&& [ {{\rm e}^{{\rm i}( {\beta _{1n} + \beta _{2n} } )} + {\rm e}^{{\rm i}({\beta _{3n} + \beta _{4n} } )}} ]( {D_{1n} - D_{2n} } )( {D_{3n} } - {D_{4n} } ) -\\&&\qquad [ {{\rm e}^{{\rm i}( {\beta _{1n} + \beta _{3n} } )} + {\rm e}^{{\rm i}( {\beta _{2n} + \beta _{4n} } )}} ]( {D_{1n} } { - D_{3n} } )( {D_{2n} - D_{4n} } ) +\\&&\qquad [ {{\rm e}^{{\rm i}( {\beta _{2n} + \beta _{3n} } )} + {\rm e}^{{\rm i}( {\beta _{1n} + \beta _{4n} } )}} ]( {D_{2n} - D_{3n} })\cdot\\&&\qquad( {D_{1n} - D_{4n} } ) = 0\end{eqnarray}$

从而导出线性派生系统的模态函数[5]

$\begin{eqnarray}&&\phi _n = a_1 \left\{ {{\rm e}^{{\rm i}\beta _{1n} x} - \frac{( {D_{4n} -D_{1n} } )( {{\rm e}^{{\rm i}\beta _{3n} } - {\rm e}^{{\rm i}\beta _{1n} }})}{( {D_{4n} - D_{2n} } )( {{\rm e}^{{\rm i}\beta _{3n} } -{\rm e}^{{\rm i}\beta _{2n} }} )}{\rm e}^{{\rm i}\beta _{2n} x}} \right. -\\&&\qquad \frac{( {D_{4n} - D_{1n} } )( {{\rm e}^{{\rm i}\beta _{2n} } -{\rm e}^{{\rm i}\beta _{1n} }} )}{( {D_{4n} - D_{3n} } )({{\rm e}^{{\rm i}\beta _{2n} } - {\rm e}^{{\rm i}\beta _{3n} }} )}{\rm e}^{{\rm i}\beta _{3n} x} - \\&&\qquad \left[ {1 - \frac{( {D_{4n} - D_{1n} } )( {{\rm e}^{{\rm i}\beta _{3n}} - {\rm e}^{{\rm i}\beta _{1n} }} )}{( {D_{4n} - D_{2n} } )({{\rm e}^{{\rm i}\beta _{3n} } - {\rm e}^{{\rm i}\beta _{2n} }} )}} \right.-\\&&\qquad \left. { \left. {\frac{( {D_{4n} - D_{1n} } )( {{\rm e}^{{\rm i}\beta_{2n} } - {\rm e}^{{\rm i}\beta _{1n} }} )}{( {D_{4n} - D_{3n} })( {{\rm e}^{{\rm i}\beta _{2n} } - {\rm e}^{{\rm i}\beta _{3n} }} )}}\right]{\rm e}^{{\rm i}\beta _{4n} x}} \right\}\end{eqnarray}$
$\begin{eqnarray}\\&& \vartheta _n = a_1 \left\{ {C_{1n} {\rm e}^{{\rm i}\beta _{1n} x} \!-\! \frac{C_{2n} ({D_{4n} - D_{1n} } )( {{\rm e}^{{\rm i}\beta _{3n} } - {\rm e}^{{\rm i}\beta _{1n} }})}{( {D_{4n} - D_{2n} } )( {{\rm e}^{{\rm i}\beta _{3n} } -{\rm e}^{{\rm i}\beta _{2n} }} )}} \right.{\rm e}^{{\rm i}\beta _{2n} x} -\\&&\qquad \frac{C_{3n} ( {D_{4n} - D_{1n} } )( {{\rm e}^{{\rm i}\beta _{2n} } -{\rm e}^{{\rm i}\beta _{1n} }} )}{( {D_{4n} - D_{3n} } )({{\rm e}^{{\rm i}\beta _{2n} } - {\rm e}^{{\rm i}\beta _{3n} }} )}{\rm e}^{{\rm i}\beta _{3n} x}\! - \\&&\qquad C_{4n} \left[ {1 - \frac{( {D_{4n} - D_{1n} } )({{\rm e}^{{\rm i}\beta _{3n} } - {\rm e}^{{\rm i}\beta _{1n} }} )}{( {D_{4n} - D_{2n} })( {{\rm e}^{{\rm i}\beta _{3n} } - {\rm e}^{{\rm i}\beta _{2n} }} )}} \right. -\\&&\qquad \left. {\left. { \frac{( {D_{4n} - D_{1n} } )( {{\rm e}^{{\rm i}\beta_{2n} } - {\rm e}^{{\rm i}\beta _{1n} }} )}{( {D_{4n} - D_{3n} })( {{\rm e}^{{\rm i}\beta _{2n} } - {\rm e}^{{\rm i}\beta _{3n} }} )}}\right]{\rm e}^{{\rm i}\beta _{4n} x}} \right\}\end{eqnarray}$

其中

$\begin{eqnarray}&&D_{jn} = [ {( {k_f^2 - k_2 \gamma ^2} )\beta _{jn} - \omega_n k_2 \gamma } ][ {2\gamma \omega _n \beta _{jn} }+ \\&&\qquad {\omega _n^2 + ( {\kappa \gamma ^2 - k_1 - 1} ) {\beta_{jn}^2 } ]}/{\beta _{jn} },\ \ {j = 1,2,3,4} \qquad\end{eqnarray}$

假定

$\begin{eqnarray}&&v_0 \left( {x,T_0 ,T_1 ,T_{2} } \right) = \sum\limits_{n = 1}^\infty{A_n \left( {T_1 ,T_{2} } \right)\phi _n \left( x \right){\rm e}^{{\rm i}\omega_n T_0 }} + \mbox{cc}\qquad\end{eqnarray}$
$\begin{eqnarray}\\&&\varphi _0 \left( {x,T_0 ,T_1 ,T_{2} } \right) = \sum\limits_{n =1}^\infty {A_n \left( {T_1 ,T_{2} } \right)\vartheta _n \left( x \right){\rm e}^{{\rm i}\omega _n T_0 }} + \mbox{cc}\end{eqnarray}$

基于等式两边谐波平衡, 假定$v_1 \left( {x,T_0 ,T_1 } \right)$和$\varphi _1 \left( {x,T_0 ,T_1 } \right)$有形式为

$\begin{eqnarray}&&v_1 \left( {x,T_0 ,T_1 ,T_{2} } \right) = \sum\limits_{n = 1}^\infty {P_n \left( {T_1 ,T_{2} } \right)\phi _n \left( x \right){\rm e}^{{\rm i}\omega _n T_0 }} + \mbox{cc}\qquad\end{eqnarray}$
$\begin{eqnarray}\\&&\varphi _1 \left( {x,T_0 ,T_1 ,T_{2} } \right) = \sum\limits_{n = 1}^\infty {Q_n \left( {T_1 ,T_{2} } \right)\vartheta _n \left( x \right){\rm e}^{{\rm i}\omega _n T_0 }} + \mbox{cc}\end{eqnarray}$

将式(20) $\sim\!$式(23)代入式(11)、式(12)中并令两边${\rm e}^{{\rm i}\omega _n T_0 }$的系数相等, 分别乘以模态函数的复共轭积分处理后, 得到

$\begin{eqnarray}&&\beta _n P_n + \eta _n Q_n = A_n ,_{T_{1} } + \alpha \mu _n A_n\end{eqnarray}$
$\begin{eqnarray}&&\xi _n P_n + \psi _n Q_n = A_n ,_{T_{1} } + \alpha \zeta _n A_n\end{eqnarray}$

其中

$\begin{eqnarray}&&\beta _n = - \left[ ( {\kappa \gamma ^2 - k_1 - 1})\displaystyle\int_0^1 {{\phi }''_n \bar {\phi }_n {\rm d}x +2\mbox{i}\omega _n \gamma \displaystyle\int_0^1 {{\phi }'_n \bar {\phi }_n{\rm d}x} }-\right.\\&&\qquad{\left. {\omega _n^2 \displaystyle\int_0^1 {\phi _n \bar {\phi }_n {\rm d}x} }\right]}\Bigg/\Bigg( \mbox{2i}\omega _n \displaystyle\int_0^1 \phi _n \bar {\phi }_n{\rm d}x +\\&&\qquad \mbox{2}\gamma \displaystyle\int_0^1 {\phi }'_n \bar {\phi }_n {\rm d}x \Bigg)\end{eqnarray}$
$\begin{eqnarray}&&\beta _n = - \frac{k_1 \displaystyle\int_0^1 {{\vartheta }'_n \bar {\phi }_n }{\rm d}x}{2\left( {\mbox{i}\omega _n \displaystyle\int_0^1 {\phi _n \bar {\phi }_n {\rm d}x + \gamma \displaystyle\int_0^1 {{\phi }'_n \bar {\phi }_n {\rm d}x} } } \right)}\end{eqnarray}$
$\begin{eqnarray}&&\mu _n = \frac{k_1 \mbox{i}\omega _n \left( {\displaystyle\int_0^1 {{\vartheta }'_n \bar{\phi }_n } {\rm d}x - \displaystyle\int_0^1 {{\phi }''_n \bar {\phi }_n{\rm d}x} } \right)}{2\left( {\mbox{i}\omega _n \displaystyle\int_0^1 {\phi _n \bar{\phi }_n {\rm d}x + \gamma \displaystyle\int_0^1 {{\phi }'_n \bar {\phi }_n{\rm d}x} } } \right)}\end{eqnarray}$
$\begin{eqnarray}&&\xi _n = \frac{k_1 \displaystyle\int_0^1 {{\phi }'_n \bar {\vartheta }_n }{\rm d}x}{2k_2 \left( {\mbox{i}\omega _n \displaystyle\int_0^1 {\vartheta _n \bar{\vartheta }_n {\rm d}x + \gamma \displaystyle\int_0^1 {{\vartheta }'_n \bar {\vartheta}_n {\rm d}x} } } \right)}\end{eqnarray}$
$\begin{eqnarray}&& \psi _n = - \left[ (k_2 \gamma ^2 - k_f^{2})\displaystyle\int_{0}^{1} {{\vartheta }''_n \bar {\vartheta }_n{\rm d}x\mbox{ + }} \mbox{2i}\omega _n k_2 \gamma \displaystyle\int_0^1 {{\vartheta }'_n\bar {\vartheta }_n {\rm d}x + }\right. \\&&\qquad {\left. {\left( {k_1 - \omega _n^2 k_2 } \right)\displaystyle\int_0^1 {\vartheta _n \bar{\vartheta }_n {\rm d}x} } \right]}\Bigg/{\left( {2\mbox{i}k_2 \omega _n\displaystyle\int_0^1 {\vartheta _n \bar {\vartheta }_n {\rm d}x + } } \right.} \\&&\qquad \left. {2k_2 \gamma \displaystyle\int_0^1 {{\vartheta }'_n \bar {\vartheta }_n{\rm d}x} } \right)\end{eqnarray}$
$\begin{eqnarray}&& \zeta _n = \left[ {\mbox{i}\omega _n k_1 \left( {\displaystyle\int_0^1 {\vartheta _n\bar {\vartheta }_n {\rm d}x - \displaystyle\int_0^1 {{\phi }'_n \bar {\vartheta }_n} {\rm d}x} } \right)} \right. - \mbox{i}k_f^2 \omega _n \cdot\\&&\qquad {\left. {\displaystyle\int_0^1 {{\vartheta }''_n \bar {\vartheta }_n {\rm d}x} }\right]}\Bigg/\Bigg[{2k_2}\Bigg(\mbox{i}\omega _n \displaystyle\int_0^1 \vartheta _n \bar{\vartheta }_n {\rm d}x +\\&&\qquad \gamma \displaystyle\int_0^1 {\vartheta }'_n \bar {\vartheta}_n {\rm d}x\Bigg)\Bigg]\end{eqnarray}$

对于给定的参数, 数值计算均表明$\beta_{n}$和$\xi _{n}$是正虚数, $\eta_{n}$和$\psi_{n}$是负虚数, $\mu_{n}$和$\zeta_{n}$是正实数.

方程组(24)和(25)是由两个关于$P_{n}$和$Q_{n}$的非齐次微分方程所构成.可解性条件导出

$\begin{eqnarray}A_{n,T_{1}} + \alpha c_n A_n = 0\end{eqnarray}$

其中

$\begin{eqnarray}c_n = \frac{\beta _n \zeta _n - \xi _n \mu _n }{\beta _n - \xi _n }\end{eqnarray}$

将式(32)写成极坐标的形式

$\begin{eqnarray}A_n = a_n \left( {T_1 ,T_2 } \right)\mbox{e}^{{\rm i}\theta _n \left( {T_1,T_2 } \right)}\end{eqnarray}$

其中$a_{n}$和$\theta_{n}$是$T_{1}$和$T_{2}$的实函数, 对应于第$n$阶模态响应的幅值和相角, 从而导出

$\begin{eqnarray}&&v_0 \left( {x,T_0 ,T_1 ,T_{2} } \right) = \sum\limits_{n = 1}^\infty{\phi _n \left( x \right)B_n \left( {T_2 } \right){\rm e}^{{\rm i}\omega _n T_0 -\alpha c_n T_1 }} + \mbox{cc}\\&&\varphi _0 \left( {x,T_0 ,T_1 ,T_{2} } \right) = \sum\limits_{n =1}^\infty {\vartheta _n \left( x \right)B_n \left( {T_2 } \right){\rm e}^{{\rm i}\omega_n T_0 - \alpha c_n T_1 }}+\mbox{cc}\end{eqnarray}$

式(7)和式(8)的所有特解与式(10)和式(11)的解正交, 因此, 将式(35)、式(10)和式(11)的特解代入

式(13), 导出

$\begin{eqnarray}&&v_{2,T_0 T_0} + 2\gamma v_{2,xT_0} + ( {\kappa \gamma ^2 - 1})v_{2,xx} + k_1 (\varphi _{2,x} - v_{2,xx}) =\\&&\qquad ( { - c_n^2 \alpha ^2B_n } \phi _n - 2{\rm i}\omega _n \dot{B}_n \phi _n { - 2\gamma \dot {B}_n {\phi }'_n })\mbox{e}^{{\rm i}\omega _n T_0- \alpha c_n T_1 } - \\&&\qquad ( {c_n \alpha ^2B_n k_1 } {\phi }''_n { - c_n \alpha^2B_n k_1 {\vartheta }'_n } )\mbox{e}^{{\rm i}\omega _n T_0- \alpha c_n T_1 } +\\&&\qquad \mbox{NST} + \mbox{cc}\end{eqnarray}$

其中NST是由式(10)和式(11)产生的长期项, 根据可解性条件导出

$\begin{eqnarray}B_{n} + \alpha ^2d_n B_n = 0\end{eqnarray}$

其中

$\begin{eqnarray}&& d_n = \Bigg(c_n^2 \displaystyle\int_0^1 \phi _n \bar{\phi }_n {\rm d}x -c_n k_1 \displaystyle\int_0^1 \vartheta'_n \bar {\phi }_n {\rm d}x + \\&&\qquad c_n k_1 \displaystyle\int_0^1 \phi''_n \bar{\phi}_n {\rm d}x\Bigg)\Bigg/\Bigg[2\Bigg(\mbox{i}\omega _n \displaystyle\int_0^1\phi _n \bar{\phi }_n {\rm d}x+\\&&\qquad \gamma \displaystyle\int_0^1\phi'_n \bar{\phi}_n {\rm d}x\Bigg)\Bigg]\end{eqnarray}$

将式(37)写成极坐标的形式

$\begin{eqnarray}B_n = b_n \left( {T_2 } \right)\mbox{e}^{{\rm i}\varsigma _n \left( {T_2 }\right)}\end{eqnarray}$

其中$b_n$和$\varsigma _n $是$T_{2} $的实函数. 导出

$\begin{eqnarray}&&v_0 \left( {x,T_0 ,T_1 ,T_2 } \right) = \\&&\qquad\sum\limits_{n = 1}^\infty {\phi_n \left( x \right)b_n \left( {T_2 } \right)\mbox{e}^{\left[ {\sigma _n +\mbox{i}\left( {\omega _n } \right)_{VE} } \right]T_{0} }} + {\rm cc}\qquad \end{eqnarray}$

其中

$\begin{eqnarray}\sigma _n = - \varepsilon \alpha c_n ,\ \ \left( {\omega _n } \right)_{VE} =\omega _n - \varepsilon ^2\alpha ^2{\rm Im}\left( {k_n } \right)\end{eqnarray}$

实部$\sigma_{n}$代表黏弹性Timoshenko梁线性自由振动的幅值随着时间的衰减率; 虚部($\omega _{n})_{VE}$代表其频率.

给定$L=0.3$ m, $G=66$~GPa, $E=30$~GPa, $A=0.133\,5\times 0.067\,412~$m$^{2}$, $\rho =7850$ kg/m$^{3}$, 和$P=107$ N, 由式(4)可解得相应的无量纲参数$k_{\rm f}=0.8$, $k_{1}=71.28$, $k_{2}=0.004\,2$, 若考虑$\kappa =0.5$, 图1给出了参数$c_{1}$和$d_{1}$随着轴向速度的变化情况.从图中可以看出参数$c_{1}$随着轴向速度的增大先是稍微增大而后快速减小, 参数$d_{1}$随着轴向速度的增大而增大.

图1

图1   参数$c_{1}$和$d_{1}$随着轴向速度的变化情况

Fig.1   The parameters $c_{1}$ and $d_{1}$ vary with axial speed


3 数值验证

本节引入微分求积法[30-32]对以上近似解析解进行数值验证.Timoshenko梁的计算区域为$x \in [0,1]$. $x$方向的网点数为$N$. 令$\varepsilon =1$.通过微分求积法将控制方程(1)和(2)离散为

$\begin{eqnarray}&&\ddot {v}_i + 2\gamma \sum\limits_{k = 1}^N {A_{ik}^{(1)} } \dot {v}_k +( {\kappa \gamma ^2 - 1 - k_1 })\sum\limits_{k = 1}^N {A_{ik}^{(2)} } v_k +\\&&\qquad k_1 \sum\limits_{k = 1}^N {B_{ik}^{(1)} } \varphi _k + k_1 \alpha\sum\limits_{k = 1}^N {B_{ik}^{(1)} } \dot {\varphi }_k - k_1 \alpha \sum\limits_{k = 1}^N {A_{ik}^{(2)} } \dot {v}_k = 0\\&&\qquad( {i = 2,3, \cdots ,N - 1} ) \end{eqnarray}$
$\begin{eqnarray}&& k_2 \left( {\ddot {\varphi }_i + 2\gamma \sum\limits_{k = 1}^{\ \ \ N} {B_{ik}^{(1)} }\dot {\varphi }_k + \gamma ^2\sum\limits_{k = 1}^N {B_{ik}^{(2)} } \varphi _k } \right) - k_1 \sum\limits_{k = 1}^N {A_{ik}^{(1)} } v_k - \\&&\qquad k_f^2 \left( {\sum\limits_{k = 1}^{\ \ \ N} {B_{ik}^{(2)} } \varphi _k + \alpha \sum\limits_{k = 1}^N {B_{ik}^{(2)} } \dot {\varphi }_k + \alpha \gamma \sum\limits_{k = 1}^N {B_{ik}^{(3)} } \varphi _k } \right) +\\&&\qquad k_1 \varphi _i + \alpha k_1 \left( {\dot {\varphi }_i - \sum\limits_{k = 1}^{\ \ \ N} {A_{ik}^{(1)} } \dot {v}_k } \right) = 0\\&&\qquad( {i = 2,3, \cdots ,N - 1})\end{eqnarray}$

应用修正权系数法修正后写为矩阵形式

$\begin{eqnarray}( \lambda _n^2 {{ M}} + \lambda _n {{ G}} + {{ K}}){{ S}} = 0 \end{eqnarray}$

其中, $ M$, $ G$和$ K$分别为质量矩阵、陀螺矩阵和刚度矩阵. 它们的维数均为$2N\times 2N$. $ S$表征广义矩阵, 其维数为$2N\times 1$. $\lambda _{n}=\sigma _{n}+{\rm i}\omega _{n}$, 其中$\sigma _{n}$为衰减系数, $\omega_{n}$为固有频率.

给定$N=15$, $\kappa=0.5$, $k_{\rm f}=0.8$, $k_{1}=71.28$和$k_{2}=0.004\,2$, 图2(a)中实线和虚线表示$\alpha =0$时衰减系数随轴速的变化情况, 圆点和加号表示$\alpha=0.01$时衰减系数随轴速的变化情况, 正方形和星号表示$\alpha=0.02$时衰减系数随轴速的变化情况; 图2(b)中实线、圆点和正方形分别表示$\alpha =0$, $\alpha=0.01$和$\alpha=0.02$时前两阶固有频率随轴速的变化情况; 从图2可以看出, 固有频率随着轴速的增大而减小.衰减系数和固有频率随着黏弹性的增大而减小, 黏弹性系数对低阶衰减系数和固有频率的影响小; 对高阶衰减系数和固有频率的影响大.

图2

图2   不同黏弹性系数下衰减系数和频率随着轴速的变化情况

Fig.2   Attenuation coefficient and frequency with different viscoelastic coefficients as a function of axial speed


给定$N=15$, $\kappa =0.5$, $k_{\rm f}=0.8$, $k_{1}=71.28$, $k_{2}=0.004\,2$和$\gamma =1$, 图3给出了不同方法下频率和衰减系数随着黏弹性系数的变化情况.其中实线表示直接多尺度方法近似解析结果; 圆点表示微分求积法数值结果.固有频率和衰减系数都随着黏弹性系数的增大而减小, 黏弹性系数对低阶衰减系数和固有频率的影响很小, 对高阶衰减系数和固有频率的影响较大.

图3

图3   不同方法下衰减系数和频率随着黏弹性系数变化情况的比较

Fig.3   Comparison of attenuation coefficient and frequency with viscoelastic coefficient under different methods


4 结论

本文运用直接多尺度法和微分求积法研究了黏弹性阻尼作用下轴向运动Timoshenko梁的振动特性.得到了参数$c_{1}$和$d_{1}$随轴速变化的曲线, 考虑了黏弹性对前两阶固有频率和衰减系数的影响.研究发现: 参数$c_{1}$随着轴向速度的增大先是小幅度慢速增大而后快速减小, 参数$d_{1}$随着轴向速度的增大而增大.固有频率随着轴速的增大而减小.固有频率和衰减系数都随着黏弹性的增大而减小, 其中衰减系数与黏弹性系数成正比关系, 黏弹性系数对高阶衰减系数和固有频率的影响较大.

参考文献

Lee U, Kim J, Oh H.

Spectral analysis for the transverse vibration of an axially moving Timoshenko beam

Journal of Sound and Vibration, 2004,271:685-703

DOI      URL     [本文引用: 1]

Yan QY, Ding H, Chen LQ.

Nonlinear dynamics of axially moving viscoelastic Timoshenko beam under parametric and external excitations

Applied Mathematics and Mechanics-English Edition, 2015,36:971-984

DOI      URL    

This investigation focuses on the nonlinear dynamic behaviors in the transverse vibration of an axially accelerating viscoelastic Timoshenko beam with the external harmonic excitation. The parametric excitation is caused by the harmonic fluctuations of the axial moving speed. An integro-partial-differential equation governing the transverse vibration of the Timoshenko beam is established. Many factors are considered, such as viscoelasticity, the finite axial support rigidity, and the longitudinally varying tension due to the axial acceleration. With the Galerkin truncation method, a set of nonlinear ordinary differential equations are derived by discretizing the governing equation. Based on the numerical solutions, the bifurcation diagrams are presented to study the effect of the external transverse excitation. Moreover, the frequencies of the two excitations are assumed to be multiple. Further, five different tools, including the time history, the Poincaré map, and the sensitivity to initial conditions, are used to identify the motion form of the nonlinear vibration. Numerical results also show the characteristics of the quasiperiodic motion of the translating Timoshenko beam under an incommensurable relationship between the dual-frequency excitations.

Ghayesh M, Amabili, HM.

Nonlinear dynamics of an axially moving Timoshenko beam with an internal resonance

Nonlinear Dynamics, 2013,73:39-52

DOI      URL    

高晨彤, 黎亮, 章定国 .

考虑剪切效应的旋转FGM楔形梁刚柔耦合动力学建模与仿真

力学学报, 2018,50(3):654-666

[本文引用: 1]

( Gao Chentong, Li Liang, Zhang Dingguo, et al.

Dynamic modeling and imulation of rotating FGM tapered beams with shear effect

Chinese Journal of heoretical and Applied Mechanics, 2018,50(3):654-666 (in Chinese))

[本文引用: 1]

Tang YQ, Luo EB, Yang XD.

Complex modes and traveling waves in axially moving Timoshenko beams

Applied Mathematics and Mechanics, 2018,39(4):597-608

DOI      URL     [本文引用: 4]

王乐, 余慕春 .

轴力对自由边界Timoshenko梁横向动特性影响研究

兵器装备工程学报, 2018 ( 3):36-37

[本文引用: 1]

( Wang Le, Yu Muchun,

Effect of axial force on the lateral vibration characteristics of Timoshenko beam under free boundary condition

Journal of Ordnance Equipment Engineering, 2018, ( 3):36-37 (in Chinese))

[本文引用: 1]

唐有绮 .

轴向变速黏弹性Timoshenko梁的非线性振动

力学学报, 2013,45(6):965-973

DOI      URL     [本文引用: 1]

研究了轴向加速黏弹性Timoshenko梁的非线性参数振动。参数激励是由径向变化张力和轴向速度波动引起的。引入了取决于轴向加速度的径向变化张力,同时还考虑了有限支撑刚度对张力的影响。应用广义哈密尔顿原理建立了Timoshenko梁耦合平面运动的控制方程和相关的边界条件。黏弹性本构关系采用Kelvin模型并引入物质时间导数。耦合方程简化为具有随时间和空间变化系数的积分-偏微分型非线性方程。采用直接多尺度法分析了Timoshenko梁的组合参数共振。根据可解性条件得到了Timoshenko梁的稳态响应,并应用Routh-Hurvitz判据确定了稳态响应的稳定性。最后通过一系列数值例子描述了黏弹性系数、平均轴向速度、剪切变形系数、转动惯量系数、速度脉动幅值、有限支撑刚度参数以及非线性系数对稳态响应的影响。

( Tang Youqi,

Nonlinear vibrations of axially accelerating viscoelastic Timoshenko beams

Chinese Journal of Theoretical and Applied Mechanics, 2013,45(6):965-973 (in Chinese))

DOI      URL     [本文引用: 1]

研究了轴向加速黏弹性Timoshenko梁的非线性参数振动。参数激励是由径向变化张力和轴向速度波动引起的。引入了取决于轴向加速度的径向变化张力,同时还考虑了有限支撑刚度对张力的影响。应用广义哈密尔顿原理建立了Timoshenko梁耦合平面运动的控制方程和相关的边界条件。黏弹性本构关系采用Kelvin模型并引入物质时间导数。耦合方程简化为具有随时间和空间变化系数的积分-偏微分型非线性方程。采用直接多尺度法分析了Timoshenko梁的组合参数共振。根据可解性条件得到了Timoshenko梁的稳态响应,并应用Routh-Hurvitz判据确定了稳态响应的稳定性。最后通过一系列数值例子描述了黏弹性系数、平均轴向速度、剪切变形系数、转动惯量系数、速度脉动幅值、有限支撑刚度参数以及非线性系数对稳态响应的影响。

华洪良, 廖振强, 张相炎 .

轴向移动悬臂梁高效动力学建模及频率响应分析

力学学报, 2017,49(6):1390-1398

[本文引用: 1]

( Hua hongliang, Liao Zhenqiang, Zhang Xiangyan.

An efficient dynamic modeling method of an axially moving cantilever beam and frequency response analysis

Chinese Journal of Theoretical and Applied Mechanics, 2017,49(6):1390-1398 (in Chinese))

[本文引用: 1]

黄悦, 韩志军, 路国运 .

轴向载荷下功能梯度材料Timoshenko梁动力屈曲分析

高压物理学报, 2018,144(4):96-103

[本文引用: 1]

( Huang Yue, Han Zhijun, Lu Guoyun,

Dynamic buckling of functionally graded timoshenko beam under axial load

Journal of High Pressure Physics, 2018,144(4):96-103 (in Chinese))

[本文引用: 1]

胡璐, 闫寒, 张文明 .

黏性流体环境下V 型悬臂梁结构流固耦合振动特性研究

力学学报, 2018,50(3):643-653

[本文引用: 1]

( Hu Lu, Yan Han, Zhang Wenming, et al.

Analysis of flexural vibration of Vshaped beams immersed in viscous fluids

Chinese Journal of Theoretical and Applied Mechanics, 2018,50(3):643-653 (in Chinese))

[本文引用: 1]

Tang YQ, Chen LQ, Zhang HJ, et al.

Stability of axially accelerating viscoelastic Timoshenko beams: Recognition of longitudinally varying tensions

Mechanism and Machine Theory, 2013,62:31-50

DOI      URL     [本文引用: 1]

Stability of axially accelerating viscoelastic Timoshenko beams is treated. The effects of longitudinally varying tensions due to the axial acceleration are focused in this paper, while the tension was approximatively assumed to be longitudinally uniform in previous works. The dependence of the tension on the finite axial support rigidity is also modeled. The governing equations and the accurate boundary conditions for coupled planar motion of the Timoshenko beam are established based on the generalized Hamilton principle and the Kelvin viscoelastic constitutive relation. The boundary conditions were approximate in previous studies. The method of multiple scales is employed to investigate stability in parametric vibration. The stability boundaries are derived from the solvability conditions and the Routh-Hurwitz criterion for principal and summation parametric resonances. Some numerical examples are presented to demonstrate the effects of the tension variation, the viscosity, the mean axial speed, the shear deformation coefficient, the rotary inertia coefficient, the stiffness parameter, and the pulley support parameter on the stability boundaries. (C) 2012 Elsevier Ltd.

Asghari M, Rahaeifard M, Kahrobaiyan MH, et al.

The modified couple stress functionally graded Timoshenko beam formulation

Materials & Design, 2011,32(3):1435-1443

DOI      URL     PMID      [本文引用: 1]

Nanometer zinc oxide has become a new hotspot in the research of tissue engineering materials due to its excellent antibacterial properties, biocompatibility, and anti-tumor properties. In this paper, the existing research results were summarized, generalized, and analyzed. The antibacterial mechanism of nanometer zinc oxide was discussed in depth. The antibacterial properties and advantages of the latest nanometer zinc oxide composite materials were introduced in detail. In this review, we made prospect of the future application of nanometer zinc oxide.

An C, Su J.

Dynamic response of axially moving Timoshenko beams: integral transform solution

Applied Mathematics and Mechanics, 2014,35(11):1421-1436

DOI      URL     [本文引用: 1]

Ghayesh MH, Ghayesh HA, Reid T.

Sub- and super-critical nonlinear dynamics of a harmonically excited axially moving beam

International Journal of Solids and Structures, 2012,49:227-243

DOI      URL     [本文引用: 1]

The sub- and super-critical dynamics of an axially moving beam subjected to a transverse harmonic excitation force is examined for the cases where the system is tuned to a three-to-one internal resonance as well as for the case where it is not. The governing equation of motion of this gyroscopic system is discretized by employing Galerkin's technique which yields a set of coupled nonlinear differential equations. For the system in the sub-critical speed regime, the periodic solutions are studied using the pseudo-arclength continuation method, while the global dynamics is investigated numerically. In the latter case, bifurcation diagrams of Poincare maps are obtained via direct time integration. Moreover, for a selected set of system parameters, the dynamics of the system is presented in the form of time histories, phase-plane portraits, and Poincare maps. Finally, the effects of different system parameters on the amplitude-frequency responses as well as bifurcation diagrams are presented. (C) 2011 Elsevier Ltd.

Ghayesh MH, Amabili M.

Nonlinear vibrations and stability of an axially moving Timoshenko beam with an intermediate spring support

Mechanism and Machine Theory, 2013,67(67):1-16

DOI      URL     [本文引用: 1]

Farokhi H, Ghayesh MH.

Supercritical nonlinear parametric dynamics of Timoshenko microbeams

Communications in Nonlinear Science & Numerical Simulation, 2018,59:592-605

DOI      URL     PMID      [本文引用: 1]

Measuring temperature and moisture are important in many scenarios. It has been verified that temperature greatly affects the accuracy of moisture sensing. Moisture sensing performance would suffer without temperature calibrations. This paper introduces a nonlinearity compensation technique for temperature-dependent nonlinearity calibration of moisture sensors, which is based on an adaptive nonlinear order regulating model. An adaptive algorithm is designed to automatically find the optimal order number, which was subsequently applied in a nonlinear mathematical model to compensate for the temperature effects and improve the moisture measurement accuracy. The integrated temperature and moisture sensor with the proposed adaptive nonlinear order regulating nonlinearity compensation technique is found to be more effective and yield better sensing performance.

Qiang LY, Li JJ, Zhang NH.

Quasi-static and dynamical analyses of a thermoviscoelastic Timoshenko beam using the differential quadrature method

Applied Mathematics and Mechanics, 2019,40(4):549-562

DOI      URL     [本文引用: 1]

HW, Li L, Li YH.

Non-linearly parametric resonances of an axially moving viscoelastic sandwich beam with time-dependent velocity

Applied Mathematical Modeling, 2018,53:83-105

DOI      URL     [本文引用: 1]

Wang B.

Asymptotic analysis on weakly forced vibration of axially moving viscoelastic beam constituted by standard linear solid model

Applied Mathematics and Mechanics, 2012,33(6):817-828

DOI      URL     [本文引用: 1]

Liu D, Xu W, Xu Y.

Dynamic responses of axially moving viscoelastic beam under a randomly disordered periodic excitation

Journal of Sound and Vibration, 2012,331:4045-4056

DOI      URL     [本文引用: 1]

We investigate dynamic responses of axially moving viscoelastic bean subject to a randomly disordered periodic excitation. The method of multiple scales is used to derive the analytical expression of first-order uniform expansion of the solution. Based on the largest Lyapunov exponent, the almost sure stability of the trivial steady-state solution is examined. Meanwhile, we obtain the first-order and the second-order steady-state moments for the non-trivial steady-state solutions. Specially, we discuss the first mode theoretically and numerically. Results show that under the same conditions of the parameters, as the intensity of the random excitat on increases, non-trivial steady-state solution fluctuation will become strenuous, which will result in the non-trivial steady-state solution lose stability and the trivial steady-state solution can be a possible. In the case of parametric principal resonance, the stochastic jump is observed for the first mode, which indicates that the stationary joint probability density concentrates at the non-trivial solution branch when the random excitation is small, but with the increase of intensity of the random excitation, the probability of the trivial steady-state solution will become larger. This phenomenon of stochastic jump can be defined as a stochastic bifurcation. (C) 2012 Elsevier Ltd.

Simsek M.

Non-linear vibration analysis of a functionally graded Timoshenko beam under action of a moving harmonic load

Composite Structures, 2010,92(10):2532-2546

DOI      URL     [本文引用: 1]

Abstract

In this paper, non-linear dynamic analysis of a functionally graded (FG) beam with pinned–pinned supports due to a moving harmonic load has been performed by using Timoshenko beam theory with the von-Kármán’s non-linear strain–displacement relationships. Material properties of the beam vary continuously in thickness direction according to a power-law form. The system of equations of motion is derived by using Lagrange’s equations. Trial functions denoting transverse, axial deflections and rotation of the cross-sections of the beam are expressed in polynomial forms. The constraint conditions of supports are taken into account by using Lagrange multipliers. The obtained non-linear equations of motion are solved with aid of Newmark-β method in conjunction with the direct iteration method. In this study, the effects of large deflection, material distribution, velocity of the moving load and excitation frequency on the beam displacements, bending moments and stresses have been examined in detail. Convergence and comparison studies are performed. Results indicate that the above-mentioned effects play a very important role on the dynamic responses of the beam, and it is believed that new results are presented for non-linear dynamics of FG beams under moving loads which are of interest to the scientific and engineering community in the area of FGM structures.

Mokhtari A, Mirdamadi HR, Ghayour M.

Wavelet-based spectral finite element dynamic analysis for an axially moving Timoshenko beam

Mechanical Systems and Signal Processing, 2017,92:124-145

DOI      URL     [本文引用: 1]

随岁寒 .

轴向运动功能梯度梁的动力学分析

[硕士论文]. 苏州: 苏州大学, 2015

[本文引用: 1]

( Sui Suihan .

Dynamic analysis of axial motion functionally graded beams

[Master Thesis]. Suzhou: Suzhou University, 2015 (in Chinese))

[本文引用: 1]

Tang YQ, Zhang DB, Gao JM.

Parametric and internal resonance of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions

Nonlinear Dynamics, 2016,83(1-2):401-418

DOI      URL     [本文引用: 1]

Yang XD, Tang YQ, Chen LQ, et al.

Dynamic Stability of Axially Accelerating Timoshenko beams: Averaging Method

European Journal of Mechanics A/Solids, 2010,29:81-90

DOI      URL     [本文引用: 1]

Ding H, Tang YQ, Chen LQ.

Frequencies of transverse vibration of an axially moving viscoelastic beam

Journal of Vibration and Control, 2017,23(20):3504-3514

DOI      URL     PMID      [本文引用: 1]

A simple microfabrication technique for the preparation of a tapered microchannel for thermally generated pH gradient isoelectric focusing (IEF) has been demonstrated. The tapered channel was cut into a plastic sheet (thickness was 120 microm), and the channel was closed by sandwiching the plastic sheet between two glass microscope slides. The length of the microchannel was 5 cm. The width of the separation channel was 0.4 mm at the narrow end and 4 mm at the wide end. The channel was coated with polyacrylamide to prevent electroosmotic flow (EOF) during focusing. Two electrolyte vials were mounted on top of each end of the channel with the wide end of the channel connected to the cathodic vial and the narrow to the anodic vial. The feasibility of the thermally generated pH gradient in a tapered channel was demonstrated. Important parameters that determined the feasibility of using a thermally generated pH gradient in a tapered channel were analyzed. Parameters to be optimized were control of EOF and hydrodynamic flow, selection of power supply mode and prevention of local overheating and air bubble formation. Tris-HCl buffer, which has a high pK(a) dependence with temperature, was used both to dissolve proteins and as the electrolyte. The thermally generated pH gradient separation of proteins was tested by focusing dog, cat and human hemoglobins with a whole column detection capillary IEF (CIEF) system.

杨鄂川, 李映辉, 赵翔 .

含旋转运动效应裂纹梁横向振动特性的研究

应用力学学报, 2017,34(6):1160-1165

[本文引用: 1]

( Yang Echuan, Li Yinghui, Zhao Xiang, et al.

Investigations of transverse vibration characteristics of a rotating beam with a crack

Journal of applied mechanics, 2017,34(6):1160-1165 (in Chinese))

[本文引用: 1]

张国策, 丁虎, 陈立群 .

复模态分析超临界轴向运动梁横向非线性振动

动力学与控制学报, 2015,13(4):283-287

[本文引用: 1]

( Zhang Guoce, Ding Hu, Chen Liqun,

Complex modal analysis of transversally nonlinear vibration for supercritically axially moving beams

Journal of Dynamics and Control, 2015,13(4):283-287 (in Chinese))

[本文引用: 1]

吕海炜, 李映辉, 刘启宽 .

轴向运动粘弹性夹层梁的横向振动

动力学与控制学报, 2013,11(4):314-319

[本文引用: 1]

( Haiwei, Li Yinghui, Liu Qikuan, et al.

Transverse Vibration of Axially Moving Viscoelastic Sandwich Beam

Journal of Dynamics and Control, 2013,11(4):314-319 (in Chinese))

[本文引用: 1]

陈红永, 李上明 .

轴向运动梁在轴向载荷作用下的动力学特性研究

振动与冲击, 2016,35(19): 75-80

URL     [本文引用: 1]

研究了轴向运动Timoshenko梁在轴向载荷作用下的振动特性。首先通过考虑轴向拉压载荷作用,根据Timoshenko梁理论和Hamilton原理建立了梁的横向振动控制微分方程,推导了简支-简支边界条件下的梁的无量纲频率随轴向载荷的变化关系,采用新的无量纲化形式消除了无载荷作用下控制方程的奇异性。通过微分求积法进行数值求解并对结果进行验证,分析结果表明:无载荷作用下,长细比越大,越易达到失稳状态;在相同运动速度下,受压状态时比受拉状态下更易达到失稳;临界速度随着轴向载荷的绝对值的增大而减小。通过研究探索了影响临界速度和临界载荷的因素以及两者的关系,对于轴向受载运动系统设计具有一定指导意义。
关键词:轴向运动Timoshenko梁;轴向载荷;横向振动;微分求积法

( Chen Hongyong, Li Shangming,

Study on dynamic characteristics of axial moving beam under axial load

Journal of Vibration and Shock, 2016,35(19): 75-80 (in Chinese))

URL     [本文引用: 1]

研究了轴向运动Timoshenko梁在轴向载荷作用下的振动特性。首先通过考虑轴向拉压载荷作用,根据Timoshenko梁理论和Hamilton原理建立了梁的横向振动控制微分方程,推导了简支-简支边界条件下的梁的无量纲频率随轴向载荷的变化关系,采用新的无量纲化形式消除了无载荷作用下控制方程的奇异性。通过微分求积法进行数值求解并对结果进行验证,分析结果表明:无载荷作用下,长细比越大,越易达到失稳状态;在相同运动速度下,受压状态时比受拉状态下更易达到失稳;临界速度随着轴向载荷的绝对值的增大而减小。通过研究探索了影响临界速度和临界载荷的因素以及两者的关系,对于轴向受载运动系统设计具有一定指导意义。
关键词:轴向运动Timoshenko梁;轴向载荷;横向振动;微分求积法

丁虎 .

数值仿真轴向运动黏弹性梁非线性参激振动

计算力学学报, 2012,29(4):545-550

( Ding Hu,

Numerical investigation into nonlinear parametric resonance of axially moving accelerating viscoelastic beams

Chinese Journal of Computational Mechanics, 2012,29(4):545-550 (in Chinese))

Bert CW, Malik M.

The differential quadrature method in computational mechanics: a review

Applied Mechanics Reviews, 1996,49:1-28

DOI      URL     [本文引用: 1]

/