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轴向运动三阶剪切变形板自由振动的有限元模型及应用*

  • 朱成秀1

    机 构：

    1. 苏州大学 轨道交通学院,苏州 215131

    ×
  • ，随岁寒2

    机 构：

    2. 商丘工学院 机械工程学院,商丘 476000

    ×
  • ，彭丹华2

    机 构：

    2. 商丘工学院 机械工程学院,商丘 476000

    ×
  • ，李成1

    机 构：

    1. 苏州大学 轨道交通学院,苏州 215131

    ×

1. 苏州大学 轨道交通学院,苏州 215131；
2. 商丘工学院 机械工程学院,商丘 476000；

FINITE ELEMENT MODELING AND APPLICATION ON FREE VIBRATION OF AXIALLY MOVING THIRD-ORDER SHEAR DEFORMATION PLATES*

  • Zhu Chengxiu1

    Affiliation：

    1. School of Rail Transportation,Soochow University, Suzhou 215131 , China

    ×
  • ，Sui Suihan2

    Affiliation：

    2. School of Mechanical Engineering,Shangqiu Institute of Technology,Shangqiu 476000 ,China

    ×
  • ，Peng Danhua2

    Affiliation：

    2. School of Mechanical Engineering,Shangqiu Institute of Technology,Shangqiu 476000 ,China

    ×
  • ，Li Cheng1

    Affiliation：

    1. School of Rail Transportation,Soochow University, Suzhou 215131 , China

    ×

1. School of Rail Transportation,Soochow University, Suzhou 215131 , China；
2. School of Mechanical Engineering,Shangqiu Institute of Technology,Shangqiu 476000 ,China；

通讯作者:

李成,E-mail:licheng@suda.edu.cn

中图分类号:O326;O317

文献标识码:A

文章编号:1672-6553-2022-20(5)-049-10

DOI:10.6052/1672-6553-2022-003

参考文献 1
Wang W,Li C Y,Zhou Y X,et al.Nonlinear dynamic analysis for machine tool table system mounted on linear guides.Nonlinear Dynamics,2018,94(3):2033~2045
查找原文
参考文献 2
Pham P T,Hong K S.Dynamic models of axially moving systems:A review.Nonlinear Dynamics,2020,100(1):315~349
查找原文
参考文献 3
随岁寒,晋会杰,李成.轴向运动二维传输结构动力学有限元建模与仿真.力学季刊,2020,41(3):562~570(Sui S H,Jin H J,Li C.Finite element modeling and simulation of dynamics of two-dimensional transmission structures with axial motion.Chinese Quarterly of Mechanics,2020,41(3):562~570(in Chinese))
查找原文
参考文献 4
Yang X D,Zhang W,Chen L Q,et al.Dynamical analysis of axially moving plate by finite difference method.Nonlinear Dynamics,2012,67(2):997~1006
查找原文
参考文献 5
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(Zhang G C,Ding H,Chen L Q.Complex modal analysis of transversally non-linear vibration for supercritically axially moving beams.Journal of Dynamics and Control,2015,13(4):283~287(in Chinese))
查找原文
参考文献 6
王冬梅,张伟,李慕荣.用微分求积法分析轴向移动粘弹性梁的非平面非线性振动.动力学与控制学报,2015,13(1):23~27(Wang D M,Zhang W,Li M R.Using DQM to analyze the nonplanar nonlinear vibrationa of an axially moving viscoelastic beam.Journal of Dynamics and Control,2015,13(1):23~27(in Chinese))
查找原文
参考文献 7
吕海炜,李映辉,刘启宽,等.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(Lv H W,Li Y H,Liu Q K,et al.Transverse vibration of axially moving viscoelastic sandwich beam.Journal of Dynamics and Control,2013,11(4):314~319(in Chinese))
查找原文
参考文献 8
Yao G,Xie Z,Zhu L,et al.Nonlinear vibrations of an axially moving plate in aero-thermal environment.Nonlinear Dynamics,2021,105(4):2921~2933
查找原文
参考文献 9
Wang Y Q,Wu H,Yang F L,et al.An efficient method for vibration and stability analysis of rectangular plates axially moving in fluid.Applied Mathematics and Mechanics-English Edition,2021,42(2):291~308
查找原文
参考文献 10
Hao Y,Dai H L,Qiao N,et al.Dynamics and stability analysis of an axially moving beam in axial flow.Journal of Mechanics of Materials and Structures,2020,15(1):37~60
查找原文
参考文献 11
Wu J M,Shao M Y,Wang Y,et al.Nonlinear vibration characteristics and stability of the printing moving membrane.Journal of Low Frequency Noise Vibration and Active Control,2017,36(3):306~316
查找原文
参考文献 12
胡宇达,朴江民.磁场中旋转运动圆板的分叉与混沌研究.计算力学学报,2017,34(1):17~26(Hu Y D,Piao J M.Bifurcation and chaos of a rotating circular plate in magnetic field.Chinese Journal of Computational Mechanics,2017,34(1):17~26(in Chinese))
查找原文
参考文献 13
Mustafa A M,Hawwa M A,Hardt D E.Vibration of an axially moving beam supported by a slightly curved elastic foundation.Journal of Vibration and Control,2018,24(17):4000~4009
查找原文
参考文献 14
李炀,谭霞,丁虎,等.两端带有弹簧支撑的轴向运动梁振动分析.动力学与控制学报,2019,17(4):335~340(Li Y,Tan X,Ding H,et al.Nonlinear transverse vibration of an axially moving beam with vertical spring boundary.Journal of Dynamics and Control,2019,17(4):335~340(in Chinese))
查找原文
参考文献 15
Ghayesh M H,Amabili M.Nonlinear vibrations and stability of an axially moving Timoshenko beam with an intermediate spring support.Mechanism and Machine Theory,2013,67:1~16
查找原文
参考文献 16
Farokhi H,Ghayesh M H,Amabili M.In-plane and out-of-plane nonlinear dynamics of an axially moving beam.Chaos Solitons & Fractals,2013,54:101~121
查找原文
参考文献 17
Parker R G.Supercritical speed stability of the trivial equilibrium of an axially-moving string on an elastic foundation.Journal of Sound and Vibration,1999,221(2):205~219
查找原文
参考文献 18
Hatami S,Azhari M,Saadatpour M M.Stability and vibration of elastically supported,axially moving orthotropic plates.Iranian Journal of Science and Technology Transaction B-Engineering,2006,30(B4):427~446
查找原文
参考文献 19
Esmaeilzadeh M,Kadkhodayan M,Mohammadi S,et al.Nonlinear dynamic analysis of moving bilayer plates resting on elastic foundations.Applied Mathematics and Mechanics-English Edition,2020,41(3):439~458
查找原文
参考文献 20
Reissner E.The effect of transverse shear deformation on the bending of elastic plates.Journal of Applied Mechanics-Transactions of the ASME,1945,12(2):A69~A77
查找原文
参考文献 21
Mindlin R D.Influence of rotatory inertia and shear on flexural motions of isotropic,elastic plates.Journal of Applied Mechanics-Transactions of the ASME,1951,18(1):31~38
查找原文
参考文献 22
Reddy J N.A simple higher-order theory for laminated composite plates.Journal of Applied Mechanics-Transactions of the ASME,1984,51(4):745~752
查找原文
参考文献 23
Reddy J N.A refined nonlinear-theory of plates with transverse-shear deformation.International Journal of Solids and Structures,1984,20(9-10):881~896
查找原文
参考文献 24
Thai H T,Kim S E.A size-dependent functionally graded Reddy plate model based on a modified couple stress theory.Composites Part B-Engineering,2013,45(1):1636~1645
查找原文
参考文献 25
陈丽华,孙玥,张伟.三阶剪切变形板的振动特性研究.动力学与控制学报,2014,12(1):50~55(Chen L H,Sun Y,Zhan W.Study on vibration characteristic of third order shear deformation theory of plate.Journal of Dynamics and Control,2014,12(1):50~55(in Chinese))
查找原文
参考文献 26
Zamani H A.Frequency analysis of FG-CNT-reinforced composite doubly curved panels on visco-Pasternak medium.Advanced Composites and Hybrid Materials,2021,4(3):830~844
查找原文
参考文献 27
Dung N T,Minh P V,Hung H M,et al.The third-order shear deformation theory for modeling the static bending and dynamic responses of piezoelectric bidirectional functionally graded plates.Advances in Materials Science and Engineering,2021,doi:https://doi.org/10.1155/2021/5520240
查找原文
参考文献 28
Melosh R J.Basis for derivation of matrices for the direct stiffness method.AIAA Journal,1963,1(7):1631~1637
查找原文
参考文献 29
Ren J G,Hinton E.The finite-element analysis of homogeneous and laminated composite plates using a simple higher-order theory.Communications in Applied Numerical Methods,1986,2(2):217~228
查找原文
参考文献 30
Tang Y Q,Chen L Q.Stability analysis and numerical confirmation in parametric resonance of axially moving viscoelastic plates with time-dependent speed.European Journal of Mechanics A-Solids,2013,37:106~121
查找原文
参考文献 31
Tang Y Q,Chen L Q.Nonlinear free transverse vibrations of in-plane moving plates:Without and with internal resonances.Journal of Sound and Vibration,2011,330(1):110~126
查找原文
目录contents

    摘要

    采用三阶剪切变形理论,通过虚功原理建立轴向运动板的有限元方程,利用一种包含节点挠度及其斜率和截面转角的四节点四边形单元离散求解域,随后将离散式弹簧支承这一约束首次通过系统势能的形式引入系统有限元方程中.在算例应用部分,首先考虑不同边界条件以及不同支承刚度,与ANSYS计算结果进行对比,证明方法的有效性.接着在两种边界条件下针对轴向运动薄板分别研究运动速度和弹簧刚度与系统复频率实部和虚部的关系,结果显示高速易使板失稳,而高弹簧刚度会提高系统的振动频率.最后,给出板厚与系统复频率的关系,揭示了板厚对轴向运动三阶剪切变形板稳定性的影响.

    Abstract

    On the basis of the third-order shear deformation theory, finite element equation of the axially moving plate is established via principle of virtual work. The solution domain is discreted by applying a four-node quadrilateral element containing nodal deflection, its slope and cross-sectional rotation angle. Then the constraint of discrete spring supports is introduced into the finite element equation in the form of potential energy for the first time. In order to prove effectiveness, the present results are compared with those from ANSYS under the conditions of different boundary conditions and different support stiffness. Subsequently, the relationships between the axial velocity as well as spring stiffness and the real and imaginary parts of the complex frequencies are obtained for the axially moving thin plate with two kinds of boundary conditions. It shows that the axially moving plate may be instable with a high velocity, and the vibration frequency increases with a high spring stiffness. Finally, the relationship between the plate thickness and the complex frequency is demonstrated, and the effect of plate thickness on stability of the axially moving third-order shear deformation plate is thus revealed.

  • 引言

  • 轴向运动系统在实际工程中应用广泛,常见的例子有传动带、数控机床的进给系统、机器人末端执行器等[1, 2].为提高这些工程装备的工作效率,对轴向运动速度提出了要求.研究表明提高轴向运动速度能够降低固有振动频率[3],而对低频的振动控制通常是容易实现的.对于轴向运动系统,由于运动因素的存在,即使是低速情况下也会产生横向振动与轴向速度的耦合,从而影响系统的动态特性,导致系统模态的变化[4].因此,运动结构的横向振动往往是制约系统性能的重要原因,尤其是对高速精密系统而言.为了模拟和优化相关运动工程结构,运动系统的振动问题已成为近年来的研究热点[5-7].文献[8]建立了处于气动热环境下的热轧制造过程轴向运动板模型,考虑了运动速度、温度以及气流速度等因素对系统非线性振动的影响,为热轧工艺设计提供了参考.文献[9]基于基尔霍夫板理论,应用有限元方法,研究了流体中轴向运动板的振动问题,并考虑了速度与流体密度对板的振动稳定性的影响.文献[10]研究了流体中轴向运动梁的动力稳定性,其中考虑了流体附加质量以及非线性偏转轴向力的影响.文献[11]针对印刷设备中的运动薄膜,基于冯·卡门非线性理论,研究了运动薄膜的大变形振动问题.文献[12]基于薄板理论,利用哈密顿原理,研究了磁场中旋转运动圆板的分叉与混沌.

  • 值得注意的是,上述研究均未涉及弹性支承对轴向运动结构的影响.在某些工程中,弹性支承是不可或缺的,其对系统横向振动的影响无法忽略.对于弹性支承-轴向运动系统,以往也有一些研究,但大多数都是一维弦或梁模型.文献[13]考虑了在弹性基础上的轴向运动梁模型的振动,研究了路径曲率、轴向速度以及支承刚度对固有频率的影响.文献[14]研究了外部激励下两端带有弹簧支承的轴向运动梁的固有模态.文献[15]应用铁木辛柯梁理论,研究了具有中间弹簧支承的轴向运动梁的非线性振动.文献[16]研究了具有中间弹簧支承的轴向运动梁在分布谐波激励下的纵向和横向非线性耦合振动和稳定性.文献[17]分析了由离散式弹性基础支承的轴向运动弦的临界速度与不稳定性.考虑弹性支承的轴向运动板的文献较少,文献[18]利用经典板理论研究了弹性支承下运动正交各向异性板的稳定性,文献[19]基于一阶剪切变形理论,研究了弹性基础支承的轴向运动双层板的动力响应.目前尚未见基于高阶剪切变形理论的轴向运动板-弹簧系统的研究.

  • 为了更好地分析厚板的力学性能,研究人员发展了很多高阶剪切变形理论[20, 21].其中Reddy提出的三阶剪切变形理论认为横向剪切应变沿板厚呈抛物线分布[22, 23],能够比较准确地反映剪切板的变形与位移,并且计算量较小,已得到广泛应用[24, 25].文献[26]基于Reddy三阶剪切变形理论,研究了位于黏弹性介质上的碳纳米管增强功能梯度板的自由振动,并与一阶剪切变形理论结果对比揭示其优越性.文献[27]应用Reddy三阶剪切变形理论,分析了压电功能梯度板的弯曲与动力特性.

  • 对于轴向运动板的有限元模型,以往多基于薄板理论,并应用哈密顿原理来推导方程,导致对于中厚板的振动预测存在一定误差[3, 9].本文基于三阶剪切变形理论,并应用虚功原理得到有限元方程,考虑了横向剪切应变沿板厚的分布情况,也能够适应于中厚板的振动分析,并且计算量相对于其他高阶变形理论较小.另外,工程结构常置于弹性基础之上,在多数情况下弹性基础可以看作分布式或离散式的弹簧支承.本文将离散式弹簧这一约束通过势能方程引入有限元模型中,并通过改变支承刚度矩阵中非零值的位置,调整离散弹簧支承作用在板面的位置,简化了模型的计算复杂度并提高了预测精确性.

  • 综上,本文应用Reddy三阶剪切变形理论,研究弹性支承下轴向运动板的自由振动.将离散式弹簧支承引至势能方程,并采用一种四节点四边形单元离散板域,得到轴向运动板自由振动的有限元方程.在数值算例部分,利用ANSYS软件对静止板进行了前三阶模态分析,并将本文数值结果与ANSYS结果进行对比,证明本文方法的有效性.随后,探讨轴向速度、弹簧刚度以及板厚对弹性支承下轴向运动板的振动及其稳定性的影响.研究结果可为由空气轴承、张紧臂等支承的二维运动系统的设计与优化提供理论基础.

  • 1 控制方程

  • 建立图1所示的矩形板模型,长、宽、高分别为a、b、h,沿x轴方向的运动速度为v.板的弹性模量为E,密度为ρ,泊松比为μ.板承受离散线弹簧支承,其刚度为k,不考虑转角刚度.

  • 图1 受离散弹簧支承的轴向运动板

  • Fig.1 Axially moving plate with discrete elastic supports

  • 根据Reddy三阶剪切变形理论[22, 23],板的位移可由三阶多项式表示如下:

  • u(x,y,z)=zϕx-αz3ϕx+∂w∂xv(x,y,z)=zϕy-αz3ϕy+∂w∂yw(x,y,z)=w
    (1)

  • 其中,α=43h2,u(x,y,z)、v(x,y,z)、w(x,y,z)分别表示板上一点在x、y、z方向的位移,w表示板中面上的任一点的横向位移, ϕx、ϕy分别表示板的中面法线相对于y轴与x轴的转角.

  • 由此可以得到应变分量与位移分量的关系:

  • εxx=z∂ϕx∂x-αz3∂φx∂x+∂2w∂x2εyy=z∂ϕy∂y-αz3∂ϕy∂y+∂2w∂y2γxy=z∂ϕx∂y+∂ϕy∂x-αz3∂ϕx∂y+∂ϕy∂x+2∂2w∂x∂yγxz=1-βz2ϕx+∂w∂xγyz=1-βz2ϕy+∂w∂y
    (2)

  • 其中β=4h2.

  • 为了离散求解域,采用四节点四边形单元,广义坐标如下:

  • {d}=w1∂w∂x1φx1∂w∂y1φy1⋯w4∂w∂x4φx4∂w∂y4φy4T
    (3)

  • 其中

  • w=N{d}φx=P{d}φy=Q{d}
    (4)

  • 其中N、P、Q为形函数[28, 29],单元每个节点有横向挠度、绕x方向的转角、绕y方向的转角等3个自由度.

  • 将式(3)和式(4)代入几何方程(2),并将其整理成矩阵形式:

  • ε=[B]{d}
    (5)

  • 其中

  • [B]=zPx-αz3Px+NxxzQy-αz3Qy+NyyzPy+Qx-αz3Py+Qx+2Nxy1-βz2P+Nx1-βz2Q+Ny

  • 考虑应力应变关系,板的物理方程可以写成:

  • σxxσyyσxyσxzσyz=E1-μ2μE1-μ2000μE1-μ2E1-μ200000G00000G00000Gεxxεyyγxyγxzγyz
    (6)

  • 亦可将其表达为σ=[D]ε.

  • 考虑式(5)与式(6),板的应变能变分如下:

  • δU=∫V σxxδεxx+σyyδεyy+σxyδγxy+σxzδγxz+σyzδγyzdV=∫V σδεdV=δ{d}T∫V [B]T[D][B]dV{d}
    (7)

  • 将式(1)和式(4)代入可以得到惯性力做功的变分表达式:

  • δW=-∫V (ρu¨δu+ρv¨δv+ρw¨δw)dV=-δ{d}Tρ∫V z-αz32PTP-αz4-α2z6PTNx+NxTP+α2z6NxTNxz-αz32QTQ-αz4-α2z6QTNy+NyTQy+α2z6NyTNy+NTNdV{d¨}
    (8)

  • 由于板存在轴向运动速度,其科氏力和离心力做功的变分可表达为:

  • δF=-2ρ∫V v∂2w∂x∂tδwdV-ρ∫V v2∂2w∂x2δwdV=-δ{d}T2ρv∫V NTNxdV{d˙}-δ{d}Tρv2∫V NTNxxdV{d}
    (9)

  • 将式(7)、式(8)和式(9)代入如下虚功原理表达式:

  • δU=δW+δF
    (10)

  • 从而得输流管道系统的有限元平衡方程

  • M{d¨}+C{d¨}+K{d}=0
    (11)

  • 其中

  • M=ρ∫V z-αz32PTP-αz4-α2z6PTNx+NxTP+α2z6NxTNxz-αz32QTQ-αz4-α2z6QTNy+NyTQy+α2z6NyTNy+NTNdVC=2ρv∫V NTNxdVK=∫V [B]T[D]T[B]dV+ρv2∫V NTNxxdV

  • 当板面上一点受到弹簧支承时,弹性势能变分为:

  • δUs={d}Tdiag⁡ann{d}
    (12)

  • 其中, ann为支承刚度矩阵对角线上唯一的非零值,下标代表第n行第n列,且ann=k.划分单元时将单元节点与弹簧支承点重合,从而该节点的横向位移对应的整体节点编号即为n.考虑支承刚度后系统的整体刚度矩阵为:

  • D=K+diag⁡ann
    (13)

  • 将式(11)的Κ替换为D,得到受弹簧支承条件下轴向运动板自由振动的有限元方程:

  • M{d¨}+C{d˙}+D{d}=0
    (14)

  • 2 数值算例

  • 考虑两种边界条件:四边固支、与运动方向垂直的两边固支另两边自由,板的材料参数与几何参数如表1所示.

  • 为了验证本文提出的轴向运动弹性支承板模型及其结果的有效性,令板的速度为0,将弹簧刚度分别取极大值与极小值来模拟经典支承条件,然后进行模态分析,并将数值结果与ANSYS计算结果进行对比.

  • 表1 数值算例中的参数值

  • Table1 Parameter values in numerical examples

  • 在四边固支边界下,取板厚为0.02m,弹簧刚度为0,模拟无支承情况,弹簧刚度取正无穷时模拟刚性支承情况.图2为应用ANSYS有限元软件计算得到的模态图,其中图2(a)、(c)、(d)、(e)表示无支承板的前三阶模态,图2(b)为支承刚度无限大条件下的第一阶模态,第二阶模态分两部分,在速度为零的情况下,二者对应的固有频率相等,这与文献[30]和文献[31]的结论一致.对于无支承情况,随着振动模态阶数的提高,板的变形趋势不同,模态愈加复杂.其中无支承板的第一阶模态主要表现为板中心处的大变形,而刚性支承板的第一阶模态的最大变形量位于绕板中心的环状区域,板中心处的变形量等于0.值得注意的是,弹簧刚度为0的条件下,第二和第三阶模态的几何中心处横向变形为0.因此可以推断,当弹簧支承在该点时,弹簧刚度的变化对第二阶和第三阶没有影响.

  • 图3为两边固定两边自由边界条件下板的前三阶模态.在刚度为0和无穷大两种情况下,最大变形量均位于自由板边的中点附近.在无弹性支承条件下,不同模态阶数下板的变形趋势不同.与四边固支不同的是,第三阶模态中板的几何中心处存在较大变形量,因此推断在两边固支两边自由边界约束下,当弹性支承在几何中心点时,弹簧刚度的变化仅对第二阶固有频率不产生影响.图3(b)表示在刚性支承条件下板的第一阶模态,与图3(a)对比可以看出,不同刚度下板的变形趋势存在较大差异,即刚度对板的模态影响较大.

  • 图2 四边固支板前三阶模态的ANAYS结果

  • Fig.2 ANAYS results of the first three modes of the plate fixed at four edges

  • 表2给出了弹簧刚度取0时,厚度分别为0.02m和0.2m的四边固支无运动板的前三阶固有频率,并将本文结果与ANSYS结果进行了对比.可见本文数值解与有限元解基本吻合,最大误差不超过3.97%,证明了本文模型的有效性.另外,本文结果相对于ANSYS结果固有阶频率均偏大,这是由于本文所采用的单元每个节点只有3个自由度,造成本文结果相比ANSYS结果偏大.由于本文基于三阶剪切变形理论,因而对厚板也有良好的适用性,这一点也可由表2中厚度0.2m的对比结果佐证.表3为厚度为0.02m的无运动板在两边固支两边自由边界条件下的前三阶固有频率与ANSYS对比结果,同样具有较好的一致性,证明了本文模型在不同边界条件与不同刚度下均可以取得理想的结果.

  • 图3 两边固支两边自由板前三阶模态的ANAYS结果

  • Fig.3 ANAYS results of the first three modes of the plate fixed on two opposite edges and the others free

  • 表2 四边固支无运动板的前三阶固有频率(k=0,v=0)

  • Table2 The natural frequencies of a stationary plate fixed at four edges

  • 为了分析弹簧刚度与运动速度对模型稳定性的影响,作为对比,图4描述了四边固支条件下,弹簧刚度为0即无支承情况下轴向运动板前三阶固有频率随速度的变化.在速度为0时,前三阶频率实部均为0,随着速度增加,固有频率的虚部逐渐降低,这表明速度的增加使板的刚度降低.当速度增加到320m/s,第一阶固有频率的虚部减小到0,实部由0变为两个分支,此时速度为第一阶模态速度的发散临界值,在临界值到达前,轴向运动板始终保持稳定.在320~340m/s的速度区间内,第一阶振动模态不稳定,固有频率保持实数形式.当速度增加到350m/s时,第一阶固有频率实部回归到0,并且虚部呈上升趋势.第二阶振动模态中包含的两个固有频率在速度较小时保持相等,随着速度增加,二者逐渐分离.另外,第二和第三阶固有频率在0~390m/s区间内保持实部为0,模型在第三阶模态始终具有稳定性.

  • 表3 两边固支两边自由无运动板的前三阶固有频率

  • Table3 The natural frequencies of a stationary plate fixed on two opposite edges and the others free

  • 图4 四边固支弹性支承轴向运动板前三阶固有频率 (k=0)

  • Fig.4 The complex frequencies of the elastically supported axially moving plate fixed at four edges (k=0)

  • 令弹簧刚度k=40×106 N/m,图5为弹性支承下四边固支轴向运动板随速度的变化.与图4类似,在速度较小时,前三阶固有频率实部均为0且虚部随速度增大而降低,板保持稳定状态.第一阶固有频率的值在相同速度下大于无支承时的情况,这表明弹性支承有效增加了板的刚度.第一阶模态的速度临界值为350m/s,相比无支承时变大,在350~400m/s区间内,第一阶固有频率为不稳定状态,不稳定区间的长度相比无支承时增加.第二阶模态包含的两个固有频率分离时的速度相对于无支承时变小,但在速度较大时二者又回到几乎相等的状态.另外,在0~400m/s区间内,第二、三阶固有频率实部始终为0,板在第二、三阶模态保持稳定.第二和第三阶固有频率与弹簧刚度为0时相同,从而验证了前文预测.由此可推知,根据振动控制的需要,可以通过采用不同的弹簧布置方式来控制特定模态的振动.

  • 图5 四边固支弹性支承轴向运动板前三阶固有频率 (k=40×106 N/m)

  • Fig.5 The complex frequencies of the elastically supported axially moving plate fixed at four edges (k=40×106 N/m)

  • 令弹簧刚度趋于无穷大,由于仅仅限制了弹簧位移,截面转角仍保持自由状态,经本文算法验证,离散点刚性支承等价于该点简支的情况.图6给出了局部简支支承下轴向运动板的前三阶固有频率与速度的关系.在速度较小时,板的前三阶频率仍然保持实部为0,且虚部随着速度增大而减小.但在0~400m/s的速度区间内,前三阶固有频率不再出现发散情况,轴向运动板始终保持稳定,这一点与无支承及弹性支承时存在显著差异.另外,第二阶模态的两个固有频率分离时的速度大幅减小,几乎在速度为0时就分离,并且分离程度更加显著.局部简支支承板的第一阶固有频率在相同速度下均大于弹性支承情况,这证实了弹簧刚度增大能够有效增加板的刚度.但值得注意的是,第二、三阶固有频率几乎没有变化,进一步验证了基于模态的预测.另外与图4和图5不同,离散点简支时轴向运动板的第一阶固有频率大于第二阶,这是由于弹簧刚度对第一阶频率的影响远大于第二、三阶,使得第一阶模态发生了扭曲(如图2(b)),从而造成了第一阶固有频率超越无支承条件下的第二阶情形.

  • 图6 四边固支弹性支承轴向运动板前三阶固有频率 (k=+∞)

  • Fig.6 The complex frequencies of the elastically supported axially moving plate fixed at four edges (k=+∞)

  • 图7分析了弹簧刚度对两边固支两边自由板固有频率的影响,与四边固支时相似,在不同弹簧刚度下,前三阶固有频率均随速度增加而降低,证明了轴向运动速度的增加会降低板的总体刚度.由图7(a)可以看出,在弹簧刚度为0时,板的一阶速度发散临界值约为200m/s,远低于四边固支边界情形.由图7(b)可见,随着速度增加,一阶固有频率与二阶固有频率逐渐趋于重合,这表明板的一阶与二阶固有频率可能出现耦合,板将发生颤振不稳定现象.另外,在不同弹簧刚度下,第二阶固有频率没有发生变化,证明了前文模态分析时的推断.注意到与四边固支的情况类似,弹簧刚度对第一阶固有频率的影响大于第二阶,从而使第一阶模态扭曲[如图3(b)],导致了图7(c)中第一阶固有频率超越了第二阶.

  • 图7 两边固支两边自由弹性支承轴向运动板前三阶固有频率

  • Fig.7 The complex frequencies of the elastically supported axially moving plate fixed on two opposite edges and the others free

  • 下面考虑板厚的影响,为了进行对比,取板厚h=0.2m,考虑四边固支无支承时的轴向运动板,其前三阶固有频率的虚部随速度变化如图8所示.与图4结论一致,随着速度增加,固有频率呈现下降趋势.相比于图4,前三阶固有频率大幅增加,这表明增加板厚可以显著提高轴向运动板的刚度.另外,模型在图示速度区间始保持稳定,可推知第一阶模态的速度临界值远大于板厚为0.02m的薄板,这说明较厚的板在高速下能保持稳定.

  • 图8 四边固支弹性支承轴向运动板前三阶固有频率 (h=0.2m, k=0×106 N/m)

  • Fig.8 The complex frequencies of the elastically supported axially moving plate fixed at four edges (h=0.2m, k=0×106 N/m)

  • 3 结论

  • 基于Reddy三阶剪切变形理论研究了弹簧支承下轴向运动板的自由振动,应用四节点四边形单元进行有限元建模和求解,其中考虑了离散式弹性支承与轴向速度,并通过系统势能的形式直接将弹簧支承条件考虑到有限元方程中,从而求得板的动力稳定性.主要结论如下:

  • 1) 运动速度的增加会降低轴向运动板的总体刚度,当速度增加到发散临界值时,板在一定速度区间内会出现不稳定现象.在两边固支两边自由边界条件下,第一阶与第二阶固有频率会发生耦合,从而导致板出现颤振不稳定.

  • 2) 弹簧刚度增大能有效增强轴向运动板的刚度,并提高振动固有频率,且速度发散临界值随着弹簧刚度的增加而提高.另外,当弹簧在中心点支承时,在四边固支条件下,弹簧刚度对第一阶固有频率影响较大,而对第二和第三阶基本无影响.在两边固定两边自由边界下,弹簧刚度仅对第二阶固有频率影响较小.因此可通过改变弹簧布置方式来控制特定模态的振动.

  • 3) 当板厚变大时,固有频率与速度发散临界值均得以显著提高,证明了厚板具有更高的刚度与更好的振动稳定性.

  • 参考文献

    • [1] Wang W,Li C Y,Zhou Y X,et al.Nonlinear dynamic analysis for machine tool table system mounted on linear guides.Nonlinear Dynamics,2018,94(3):2033~2045

    • [2] Pham P T,Hong K S.Dynamic models of axially moving systems:A review.Nonlinear Dynamics,2020,100(1):315~349

    • [3] 随岁寒,晋会杰,李成.轴向运动二维传输结构动力学有限元建模与仿真.力学季刊,2020,41(3):562~570(Sui S H,Jin H J,Li C.Finite element modeling and simulation of dynamics of two-dimensional transmission structures with axial motion.Chinese Quarterly of Mechanics,2020,41(3):562~570(in Chinese))

    • [4] Yang X D,Zhang W,Chen L Q,et al.Dynamical analysis of axially moving plate by finite difference method.Nonlinear Dynamics,2012,67(2):997~1006

    • [5] 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(Zhang G C,Ding H,Chen L Q.Complex modal analysis of transversally non-linear vibration for supercritically axially moving beams.Journal of Dynamics and Control,2015,13(4):283~287(in Chinese))

    • [6] 王冬梅,张伟,李慕荣.用微分求积法分析轴向移动粘弹性梁的非平面非线性振动.动力学与控制学报,2015,13(1):23~27(Wang D M,Zhang W,Li M R.Using DQM to analyze the nonplanar nonlinear vibrationa of an axially moving viscoelastic beam.Journal of Dynamics and Control,2015,13(1):23~27(in Chinese))

    • [7] 吕海炜,李映辉,刘启宽,等.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(Lv H W,Li Y H,Liu Q K,et al.Transverse vibration of axially moving viscoelastic sandwich beam.Journal of Dynamics and Control,2013,11(4):314~319(in Chinese))

    • [8] Yao G,Xie Z,Zhu L,et al.Nonlinear vibrations of an axially moving plate in aero-thermal environment.Nonlinear Dynamics,2021,105(4):2921~2933

    • [9] Wang Y Q,Wu H,Yang F L,et al.An efficient method for vibration and stability analysis of rectangular plates axially moving in fluid.Applied Mathematics and Mechanics-English Edition,2021,42(2):291~308

    • [10] Hao Y,Dai H L,Qiao N,et al.Dynamics and stability analysis of an axially moving beam in axial flow.Journal of Mechanics of Materials and Structures,2020,15(1):37~60

    • [11] Wu J M,Shao M Y,Wang Y,et al.Nonlinear vibration characteristics and stability of the printing moving membrane.Journal of Low Frequency Noise Vibration and Active Control,2017,36(3):306~316

    • [12] 胡宇达,朴江民.磁场中旋转运动圆板的分叉与混沌研究.计算力学学报,2017,34(1):17~26(Hu Y D,Piao J M.Bifurcation and chaos of a rotating circular plate in magnetic field.Chinese Journal of Computational Mechanics,2017,34(1):17~26(in Chinese))

    • [13] Mustafa A M,Hawwa M A,Hardt D E.Vibration of an axially moving beam supported by a slightly curved elastic foundation.Journal of Vibration and Control,2018,24(17):4000~4009

    • [14] 李炀,谭霞,丁虎,等.两端带有弹簧支撑的轴向运动梁振动分析.动力学与控制学报,2019,17(4):335~340(Li Y,Tan X,Ding H,et al.Nonlinear transverse vibration of an axially moving beam with vertical spring boundary.Journal of Dynamics and Control,2019,17(4):335~340(in Chinese))

    • [15] Ghayesh M H,Amabili M.Nonlinear vibrations and stability of an axially moving Timoshenko beam with an intermediate spring support.Mechanism and Machine Theory,2013,67:1~16

    • [16] Farokhi H,Ghayesh M H,Amabili M.In-plane and out-of-plane nonlinear dynamics of an axially moving beam.Chaos Solitons & Fractals,2013,54:101~121

    • [17] Parker R G.Supercritical speed stability of the trivial equilibrium of an axially-moving string on an elastic foundation.Journal of Sound and Vibration,1999,221(2):205~219

    • [18] Hatami S,Azhari M,Saadatpour M M.Stability and vibration of elastically supported,axially moving orthotropic plates.Iranian Journal of Science and Technology Transaction B-Engineering,2006,30(B4):427~446

    • [19] Esmaeilzadeh M,Kadkhodayan M,Mohammadi S,et al.Nonlinear dynamic analysis of moving bilayer plates resting on elastic foundations.Applied Mathematics and Mechanics-English Edition,2020,41(3):439~458

    • [20] Reissner E.The effect of transverse shear deformation on the bending of elastic plates.Journal of Applied Mechanics-Transactions of the ASME,1945,12(2):A69~A77

    • [21] Mindlin R D.Influence of rotatory inertia and shear on flexural motions of isotropic,elastic plates.Journal of Applied Mechanics-Transactions of the ASME,1951,18(1):31~38

    • [22] Reddy J N.A simple higher-order theory for laminated composite plates.Journal of Applied Mechanics-Transactions of the ASME,1984,51(4):745~752

    • [23] Reddy J N.A refined nonlinear-theory of plates with transverse-shear deformation.International Journal of Solids and Structures,1984,20(9-10):881~896

    • [24] Thai H T,Kim S E.A size-dependent functionally graded Reddy plate model based on a modified couple stress theory.Composites Part B-Engineering,2013,45(1):1636~1645

    • [25] 陈丽华,孙玥,张伟.三阶剪切变形板的振动特性研究.动力学与控制学报,2014,12(1):50~55(Chen L H,Sun Y,Zhan W.Study on vibration characteristic of third order shear deformation theory of plate.Journal of Dynamics and Control,2014,12(1):50~55(in Chinese))

    • [26] Zamani H A.Frequency analysis of FG-CNT-reinforced composite doubly curved panels on visco-Pasternak medium.Advanced Composites and Hybrid Materials,2021,4(3):830~844

    • [27] Dung N T,Minh P V,Hung H M,et al.The third-order shear deformation theory for modeling the static bending and dynamic responses of piezoelectric bidirectional functionally graded plates.Advances in Materials Science and Engineering,2021,doi:https://doi.org/10.1155/2021/5520240

    • [28] Melosh R J.Basis for derivation of matrices for the direct stiffness method.AIAA Journal,1963,1(7):1631~1637

    • [29] Ren J G,Hinton E.The finite-element analysis of homogeneous and laminated composite plates using a simple higher-order theory.Communications in Applied Numerical Methods,1986,2(2):217~228

    • [30] Tang Y Q,Chen L Q.Stability analysis and numerical confirmation in parametric resonance of axially moving viscoelastic plates with time-dependent speed.European Journal of Mechanics A-Solids,2013,37:106~121

    • [31] Tang Y Q,Chen L Q.Nonlinear free transverse vibrations of in-plane moving plates:Without and with internal resonances.Journal of Sound and Vibration,2011,330(1):110~126

  • 基本信息

    中图分类号: O326;O317
    文献标识码: A
    DOI: 10.6052/1672-6553-2022-003
    文章编号: 1672-6553-2022-20(5)-049-10

    基金信息

    国家自然科学基金资助项目(11972240,12272064)；
    The work is supported by the National Natural Science Foundation of China (11972240, 12272064).；

    引用信息

    稿件历史

    收稿日期: 2021-11-06

  • 参考文献

    • [1] Wang W,Li C Y,Zhou Y X,et al.Nonlinear dynamic analysis for machine tool table system mounted on linear guides.Nonlinear Dynamics,2018,94(3):2033~2045

    • [2] Pham P T,Hong K S.Dynamic models of axially moving systems:A review.Nonlinear Dynamics,2020,100(1):315~349

    • [3] 随岁寒,晋会杰,李成.轴向运动二维传输结构动力学有限元建模与仿真.力学季刊,2020,41(3):562~570(Sui S H,Jin H J,Li C.Finite element modeling and simulation of dynamics of two-dimensional transmission structures with axial motion.Chinese Quarterly of Mechanics,2020,41(3):562~570(in Chinese))

    • [4] Yang X D,Zhang W,Chen L Q,et al.Dynamical analysis of axially moving plate by finite difference method.Nonlinear Dynamics,2012,67(2):997~1006

    • [5] 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(Zhang G C,Ding H,Chen L Q.Complex modal analysis of transversally non-linear vibration for supercritically axially moving beams.Journal of Dynamics and Control,2015,13(4):283~287(in Chinese))

    • [6] 王冬梅,张伟,李慕荣.用微分求积法分析轴向移动粘弹性梁的非平面非线性振动.动力学与控制学报,2015,13(1):23~27(Wang D M,Zhang W,Li M R.Using DQM to analyze the nonplanar nonlinear vibrationa of an axially moving viscoelastic beam.Journal of Dynamics and Control,2015,13(1):23~27(in Chinese))

    • [7] 吕海炜,李映辉,刘启宽,等.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(Lv H W,Li Y H,Liu Q K,et al.Transverse vibration of axially moving viscoelastic sandwich beam.Journal of Dynamics and Control,2013,11(4):314~319(in Chinese))

    • [8] Yao G,Xie Z,Zhu L,et al.Nonlinear vibrations of an axially moving plate in aero-thermal environment.Nonlinear Dynamics,2021,105(4):2921~2933

    • [9] Wang Y Q,Wu H,Yang F L,et al.An efficient method for vibration and stability analysis of rectangular plates axially moving in fluid.Applied Mathematics and Mechanics-English Edition,2021,42(2):291~308

    • [10] Hao Y,Dai H L,Qiao N,et al.Dynamics and stability analysis of an axially moving beam in axial flow.Journal of Mechanics of Materials and Structures,2020,15(1):37~60

    • [11] Wu J M,Shao M Y,Wang Y,et al.Nonlinear vibration characteristics and stability of the printing moving membrane.Journal of Low Frequency Noise Vibration and Active Control,2017,36(3):306~316

    • [12] 胡宇达,朴江民.磁场中旋转运动圆板的分叉与混沌研究.计算力学学报,2017,34(1):17~26(Hu Y D,Piao J M.Bifurcation and chaos of a rotating circular plate in magnetic field.Chinese Journal of Computational Mechanics,2017,34(1):17~26(in Chinese))

    • [13] Mustafa A M,Hawwa M A,Hardt D E.Vibration of an axially moving beam supported by a slightly curved elastic foundation.Journal of Vibration and Control,2018,24(17):4000~4009

    • [14] 李炀,谭霞,丁虎,等.两端带有弹簧支撑的轴向运动梁振动分析.动力学与控制学报,2019,17(4):335~340(Li Y,Tan X,Ding H,et al.Nonlinear transverse vibration of an axially moving beam with vertical spring boundary.Journal of Dynamics and Control,2019,17(4):335~340(in Chinese))

    • [15] Ghayesh M H,Amabili M.Nonlinear vibrations and stability of an axially moving Timoshenko beam with an intermediate spring support.Mechanism and Machine Theory,2013,67:1~16

    • [16] Farokhi H,Ghayesh M H,Amabili M.In-plane and out-of-plane nonlinear dynamics of an axially moving beam.Chaos Solitons & Fractals,2013,54:101~121

    • [17] Parker R G.Supercritical speed stability of the trivial equilibrium of an axially-moving string on an elastic foundation.Journal of Sound and Vibration,1999,221(2):205~219

    • [18] Hatami S,Azhari M,Saadatpour M M.Stability and vibration of elastically supported,axially moving orthotropic plates.Iranian Journal of Science and Technology Transaction B-Engineering,2006,30(B4):427~446

    • [19] Esmaeilzadeh M,Kadkhodayan M,Mohammadi S,et al.Nonlinear dynamic analysis of moving bilayer plates resting on elastic foundations.Applied Mathematics and Mechanics-English Edition,2020,41(3):439~458

    • [20] Reissner E.The effect of transverse shear deformation on the bending of elastic plates.Journal of Applied Mechanics-Transactions of the ASME,1945,12(2):A69~A77

    • [21] Mindlin R D.Influence of rotatory inertia and shear on flexural motions of isotropic,elastic plates.Journal of Applied Mechanics-Transactions of the ASME,1951,18(1):31~38

    • [22] Reddy J N.A simple higher-order theory for laminated composite plates.Journal of Applied Mechanics-Transactions of the ASME,1984,51(4):745~752

    • [23] Reddy J N.A refined nonlinear-theory of plates with transverse-shear deformation.International Journal of Solids and Structures,1984,20(9-10):881~896

    • [24] Thai H T,Kim S E.A size-dependent functionally graded Reddy plate model based on a modified couple stress theory.Composites Part B-Engineering,2013,45(1):1636~1645

    • [25] 陈丽华,孙玥,张伟.三阶剪切变形板的振动特性研究.动力学与控制学报,2014,12(1):50~55(Chen L H,Sun Y,Zhan W.Study on vibration characteristic of third order shear deformation theory of plate.Journal of Dynamics and Control,2014,12(1):50~55(in Chinese))

    • [26] Zamani H A.Frequency analysis of FG-CNT-reinforced composite doubly curved panels on visco-Pasternak medium.Advanced Composites and Hybrid Materials,2021,4(3):830~844

    • [27] Dung N T,Minh P V,Hung H M,et al.The third-order shear deformation theory for modeling the static bending and dynamic responses of piezoelectric bidirectional functionally graded plates.Advances in Materials Science and Engineering,2021,doi:https://doi.org/10.1155/2021/5520240

    • [28] Melosh R J.Basis for derivation of matrices for the direct stiffness method.AIAA Journal,1963,1(7):1631~1637

    • [29] Ren J G,Hinton E.The finite-element analysis of homogeneous and laminated composite plates using a simple higher-order theory.Communications in Applied Numerical Methods,1986,2(2):217~228

    • [30] Tang Y Q,Chen L Q.Stability analysis and numerical confirmation in parametric resonance of axially moving viscoelastic plates with time-dependent speed.European Journal of Mechanics A-Solids,2013,37:106~121

    • [31] Tang Y Q,Chen L Q.Nonlinear free transverse vibrations of in-plane moving plates:Without and with internal resonances.Journal of Sound and Vibration,2011,330(1):110~126

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