en
×

分享给微信好友或者朋友圈

使用微信“扫一扫”功能。
通讯作者:

韩勤锴,E-mail:hanqinkai@mail.tsinghua.edu.cn

中图分类号:TB123

文献标识码:A

文章编号:1672-6553-2022-20(4)-040-08

DOI:10.6052/1672-6553-2021-050

参考文献 1
Sahu S K,Datta P K.Research advances in the dynamic stability behavior of plates and shells:1987-2005-Part Ⅰ:Conservative systems.Applied Mechanics Reviews,2007,60(2):65~75
参考文献 2
乔丕忠,王艳丽,陆林军.圆柱壳稳定性问题的研究进展.力学季刊,2018(2):223~236(Qiao P Z,Wang Y L,Lu L J.Advances in stability study of cylindrical shells.Chinese Quarterly of Mechanics,2018(2):223~236(in Chinese))
参考文献 3
赵振,刘才山,陈滨,等.薄壁加筋肋圆柱壳稳定性分析的参数化研究.力学与实践,2004,26(2):17~21(Zhao Z,Liu C S,Chen B,et al.Parameterization study of orthogrid stiffened cylinder shells.Mechanics in Engineering,2004,26(2):17~21(in Chinese))
参考文献 4
王登峰,曹平周.大型薄壁圆柱壳在局部轴向压力作用下的稳定性能研究.工程力学,2009,26(4):38~45(Wang D F,Cao Z Z.Study on stability of large-scale thin-walled cylindrical shells subjected to partial axial compression.Engineering Mechanics,2009,26(4):38~45(in Chinese))
参考文献 5
杜长城,李映辉.功能梯度薄壁圆柱壳的自由振动.动力学与控制学报,2010,8(3):219~223(Du C C,Li Y H.Free vibration of functionally graded cylindrical thin shells.Journal of Dynamics and Control,2010,8(3):219~223(in Chinese))
参考文献 6
王宇,罗忠,李昌.薄壁圆柱壳的高阶模态振动特性研究.动力学与控制学报,2016,14(2):131~137(Wang Y,Luo Z,Li C.Study on vibration characteristics of high-order modal for a thin cylindrical shell.Journal of Dynamics and Control,2016,14(2):131~137(in Chinese))
参考文献 7
郭翔鹰,刘大猛,张伟.压电纤维复合材料层合壳的非线性动力学研究.动力学与控制学报,2017,15(5):430~438(Guo X,Liu D,Zhang W.Nonlinear dynamic analysis of piezoelectric macro-fiber composite laminated shells.Journal of Dynamics and Control,2017,15(5):430~438(in Chinese))
参考文献 8
Ye Z M.The non-linear vibration and dynamic instability of thin shallow shells.Journal of Sound and Vibration,1997,202(3):303~311
参考文献 9
Ng T Y,Hua L,Lam K Y,et al.Parametric instability of conical shells by the generalized differential quadrature method.International Journal for Numerical Methods in Engineering,1999,44:819~837
参考文献 10
Sofiyev A H.The stability of functionally graded truncated conical shells subjected to aperiodic impulsive loading.International Journal of Solids and Structures,2004,41(13):3411~3424
参考文献 11
Sofiyev A H.The buckling of functionally graded truncated conical shells under dynamic axial loading.Journal of Sound and Vibration,2007,305(4-5):808~826
参考文献 12
Sofiyev A H.The vibration and stability behavior of freely supported FGM conical shells subjected to external pressure.Composite Structures,2009,89(3):356~366
参考文献 13
Qin B,Zhong R,Wang T,et al.A unified Fourier series solution for vibration analysis of FG-CNTRC cylindrical,conical shells and annular plates with arbitrary boundary conditions.Composite Structures,2020,232:111549
参考文献 14
Fu T,Wu X,Xiao Z,et al.2021-Analysis of vibration characteristics of FGM sandwich joined conical-conical shells surrounded by elastic foundations.Thin-Walled Structures,2021,165:107979
参考文献 15
Li H,Hao Y X,Zhang W,et al.Vibration analysis of porous metal foam truncated conical shells with general boundary conditions using GDQ.Composite Structures,2021,269:114036
参考文献 16
Ng T Y,Lam K Y,Reddy J N.Parametric resonance of a rotating cylindrical shell subjected to periodic axial loads.Journal of Sound and Vibration,1998,214(3):513~529
参考文献 17
Liew K M,Hu Y G,Ng T Y.Dynamic stability of rotating cylindrical shells subjected to periodic axial loads.International Journal of Solids and Structures,2006,43:7553~7570
参考文献 18
Lam K Y,Hua L.Vibration analysis of a rotating truncated circular conical shell.International Journal of Solids and Structures,1997,34(17):2183~2197
参考文献 19
Shu C,Richards B E.Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations.International Journal for Numerical Methods in Fluids,1992,15:791~798
参考文献 20
Loy C T,Lam K Y,Shu C.Analysis of cylindrical shells using generalized quadrature.Shock and Vibration,1997,4:193~198
参考文献 21
Hua L,Lam K Y.The generalized differential quadrature method for frequency analysis of a rotating conical shell with initial pressure.International Journal for Numerical Methods in Engineering,1997,48:1703~1722
参考文献 22
Lam K Y,Hua L.Generalized differential quadrature for frequency of rotating multilayered conical shell.Journal of Engineering Mechanics,2000,126:1156~1162
参考文献 23
Pei Y C,Tan Q C.Parametric instability of flexible disk rotating at perodically varying angular speed.Meccanica,2009,44:711~720
参考文献 24
Pei Y C,Chatwin C.Elastic connection disk subjected to periodically fluctuating transmitted torque and rotational speed.Journal of Sound and Vibration,2012,331:3083~3095
参考文献 25
Friedmann P,Hammond E,Woo T H.Efficient numerical treatment of periodic systems with application to stability problems.International Journal of Numerical Methods in Engineering,1977,11:1117~1136
参考文献 26
Soedel W.Vibrations of shells and plates.New York:Marcel Dekker,1993
参考文献 27
Irie T,Yamada G,Tanaka K.Natural frequencies of truncated conical shells.Journal of Sound and Vibration,1984,92:447~453
参考文献 28
Sun S,Chu S,Cao D.Vibration characteristics of thin rotating cylindrical shells with various boundary conditions.Journal of Sound and Vibration,2012,331(18):4170~4186
目录contents

    摘要

    本文开展了周期轴向力作用下旋转圆锥薄壳动力稳定性研究.基于Donnell薄壳理论推导了旋转锥壳的动力学方程,采用广义微分求积法和Hill法分析了系统在周期轴向载荷作用下的参数不稳定性,讨论了多个不稳定区随工况和几何参数的变化规律.结果表明:提高转速会导致不稳定区沿频率轴移动,但对不稳定宽度影响不大.增加恒定拉伸轴向载荷,不仅会显著增加失稳宽度,而且会导致失稳区域向更高的频率范围移动.锥角、厚径比或长径比的变化都会导致不稳定区沿频率轴移动.锥角和厚径比会增大失稳宽度(长径比会减小).随着周向波个数的增加,锥角对失稳区的影响逐渐减弱,而厚径比的影响则基本保持不变.

    Abstract

    The dynamic stability of rotating thin conical shells under periodic axial force is studied in this paper. Based on Donnell’s thin shell theory, the motion equation of rotating conical shell is derived. The parametric instability of the system under periodic axial load is analyzed by using the generalized differential quadrature method and Hill’s method. The variations of several instability regions with working conditions and geometric parameters are discussed. The results show that the instability region moves along the frequency axis with the increase of the rotating speed, but the instability width has little effect. Increasing the constant tensile axial load will not only significantly increase the instability width, but also cause the instability region shift to a higher frequency range. The variation of cone angle, thickness to diameter ratio or length to diameter ratio will cause the instability region to move along the frequency axis. Cone angle and thickness diameter ratio will increase the width of instability (length diameter ratio will decrease). With the increase of the number of circumferential waves, the influence of cone angle on the instability region gradually weakens, while the influence of thickness diameter ratio remains unchanged.

  • 引言

  • 薄壁壳体结构在土木工程、机械工程和航空航天工程中有着广泛的应用.在载荷参数值和横向振动固有频率的特定组合下,壳体结构在面内周期力作用下可能发生不稳定的横向振动,导致参数失稳,危及结构安全.半个世纪以来,具有不同几何、边界条件和荷载类型的壳体结构参数失稳问题得到了广泛研究[1,2].

  • 从研究对象看,现有研究[3-7]多数集中于圆柱壳,而对圆锥壳的参数失稳研究相对较少.基于Marguerre型动力方程,Ye[8]分析了扁锥薄壳在周期性横向和面内载荷作用下的非线性振动和动力失稳.Ng等[9]利用广义微分求积(GDQ)方法研究了边界条件对周期边缘载荷作用下截锥壳参数失稳的影响.Sofiyev[10-12]对复合材料层合锥壳的热致动力失稳和屈曲行为进行了一系列研究.近期,不少学者探索了由碳纳米管增强复合材料[13]、功能梯度材料[14]以及金属泡沫材料[15]制成的截锥壳非线性动力失稳问题.

  • 上述研究并未考虑旋转效应.对于周期性轴向载荷作用下的旋转圆柱壳,Ng等[16]和Liew等[17]报道了旋转引起的科里奥利力和离心力对失稳区域有显著影响.Lam和Hua[18]指出,将研究从旋转圆柱壳扩展到旋转圆锥壳所需的分析量相当大.此外,圆锥壳是更为一般的壳体结构形式.因此,研究旋转对周期轴向载荷作用下旋转锥壳参数失稳的影响具有重要意义.

  • GDQ方法最早由Shu和Richards[19]提出,用于直接求解工程问题的控制方程.目前,GDQ方法已广泛应用于壳体结构的振动分析[20-22].本文将基于该方法建立旋转锥壳的振动模型.考虑旋转效应后,周期轴向载荷作用下薄壁锥壳结构的动力学问题属于一类具有参数激励的陀螺动力系统.由于Floquet乘数假设不满足陀螺系统,经典Bolotin方法不能用于此类系统的参数稳定性分析[23,24].需借助更为普适的Hill方法[25].

  • 因此,本文拟开展周期轴向力作用下旋转圆锥薄壳动力稳定性研究.基于Donnell薄壳理论[26]推导旋转锥壳的动力学方程,采用GDQ法和Hill法分析系统在周期轴向载荷作用下的参数不稳定性.通过与文献结果的对比,验证分析模型的准确性.研究不同转速和边界条件时系统主参数稳定区的变化,讨论轴向力相对变化幅值以及几何设计参数(锥角、长径比和厚径比)对不稳定区的影响规律.

  • 1 分析模型

  • 1.1 周期轴向力

  • 本文考虑各向同性的截断型圆锥薄壳,其弹性模量为E,密度为ρ,泊松比为υ.图1给出了以角速度Ω绕其对称轴旋转的圆锥薄壳的几何结构和坐标系.其中,α为锥角,L为轴向长度,h为厚度,ab是锥壳两端的半径.锥壳的基准面取其厚度中面,其中正交坐标系(x-θ-z)是固定的,r=r(x)是任意坐标点(x,θ,z)的半径.旋转锥壳在经向x、周向θ和法向z方向上的变形分别由u,v,w定义.

  • 图1 旋转圆锥薄壳的几何结构和坐标系

  • Fig.1 Geometry and coordinate system of a rotating conical shell

  • 旋转锥壳在图1所示的子午方向上承受周期性载荷Na(t),其形式可假定为恒定载荷上叠加一个小正弦扰动量,即表示为如下形式

  • Na(t)=ηaNcr1+εcosωat
    (1)
  • 式中,ηaε分别表示恒定轴向力的幅值和相对变化幅值.Ncr表示当锥角α=0时的圆柱薄壳的屈曲载荷,有Ncr=Eh/R21-v21/2,其中R表示柱壳的半径.本研究中,令R=a.

  • 1.2 旋转薄壁锥壳振动模型

  • 根据Donnell薄壳理论[26],图1所示的旋转圆锥壳体结构的动力学方程可表示为

  • L+LT+LI+LNu=0
    (2)
  • 式中,u=[u,v,w]T.L表示非旋转锥壳的微分算子矩阵,可参考文献[18].LT表示初始环向张力引起的微分算子矩阵,可表示为

  • LT=LT11 LT12 LT13LT21 LT22 LT23LT31 LT32 LT33
    (3)
  • 式中

  • LT11=ρhΩ22θ2,LT12=0
    (4a)
  • LT13=-ρhrΩ2cosαx
    (4b)
  • LT21=ρhrΩ2cosα2xθ+ρhΩ2sinαθ
    (4c)
  • LT22=ρhrΩ2sinαx,LT23=0
    (4d)
  • LT31=0,LT32=-ρhΩ2cosαθ
    (4e)
  • LT33=ρhΩ22θ2
    (4f)
  • 式中,r=a+xsinα.LI表示惯性力微分算子矩阵,可表示为

  • LI=LI11 LI12 LI13LI21 LR2 LI23LI31 LB2 LI33
    (5)
  • 式中

  • LI11=ρhΩ2sin2α-ρh2t2
    (6a)
  • LI12=-LR21=2ρhΩsinαt
    (6b)
  • LI13=LI31=ρhΩ2sinαcosα
    (6c)
  • LI22=ρhΩ2-ρh2t2
    (6d)
  • LI23=-LB2=-2ρhΩcosαt
    (6e)
  • LI33=ρhΩ2cos2α-ρh2t2
    (6f)
  • LN表示轴向力微分算子矩阵,可表示为

  • LN=LN11 LN12 LN13LN21 LN22 LN23LN31 LN32 LN33
    (7)
  • 其中

  • LN11=LN12=LN13=LN21=0LN22=LN23=LN31=LN32=0
    (8a)
  • LN33=Na(t)2x2
    (8b)
  • 后续分析中,将考虑四种边界条件:小端固支和大端固支(Cs-Cl)、小端简支和大端固支(Ss-Cl)、小端固支和大端简支(Cs-Sl)以及小端简支和大端简支(Ss-Sl).

  • 1.3 基于GDQ方法的离散模型

  • 在应用GDQ方法之前,需确定圆锥薄壳自由振动的位移场为

  • u=u(x,θ,t)v(x,θ,t)w(x,θ,t)=U(x)cos(nθ+ω¯t)V(x)sin(nθ+ω¯t)W(x)cos(nθ+ω¯t)
    (9)
  • 式中, ω¯是旋转锥壳的固有圆频率,n表示振动壳的周向波数.将位移场方程(9)代入方程(2),导出一组仅在空间域内沿子午x方向变系数的常微分控制方程:

  • L*u*=0
    (10)
  • 其中, u*=[U(x),V(x),W(x)]T是未知的振型空间函数向量,L*是维度为3×3的微分算子矩阵.GDQ方法的思路[19]:一个足够光滑的函数在一个离散点上相对于坐标方向的导数可以近似地表示为所有离散点上函数值的加权线性和.以函数U(x)为例,GDQ方法的基本做法如下:

  • mU(x)xmx=xi=j=1N CijmUxj,i=1,2,,N
    (11)
  • 其中,Nx方向上离散网格点的总数.Cijm是与第m阶导数相关的加权系数.选取x方向上余弦函数点作为离散网格点,即xi={1-cos[(i-1)/(N-1)π]}/2L,(i=1,2,,N).显然,越靠近锥壳的端点,离散网格点的分布就越密集.值得注意的是,锥壳两个端点的坐标为x 1=0和xN =L.V(x)和W(x)导数的离散表达式可与式(11)中U(x)的表达式类似.

  • 将这些GDQ方法的表达式应用于常微分控制方程组[式(10)],得到一组线性代数方程组:

  • L*u*x=xi=R3×15U15×3*x=xi=0
    (12)
  • 式(12)中R 3×15表示系数矩阵,并随离散点坐标的变化而变化.U*3×15是离散网格点上的振型列向量,由下式给出:

  • U15×3*x=xi=Uxi,U(1)xi,U(2)xi,U(3)xi,U(4)xi,Vxi,V(1)xi,V(2)xi,V(3)xi,V(4)xi,Wxi,W(1)xi,W(2)xi,W(3)xi,W(4)xiT
    (13)
  • 式中U(m)xi,V(m)xiW(m)xi可分别由式(11)确定,且m=1,2,3,4.

  • 旋转锥壳的两端一般有8种边界条件.考虑给定的边界条件,将式(12)施加在所有离散网格点上,然后根据固有频率ω-重新排列,得到如下多项式特征值问题:

  • ω-2H2+ω-H1+H0+ηaNcrHad=0
    (14)
  • 式中H 0,H 1,H 2,H a分别表示维度为系数矩阵J ×J的系数矩阵(J=3N-8).需要说明的是,Ha仅由轴向力作用而引起的.d表示第J阶模态振型列向量,有

  • d=Ux2,Ux3,,UxN-2,UxN-1,Vx2,Vx3,,VxN-2,VxN-1,Wx3,Wx4,,WxN-3,WxN-2T
    (15)
  • 通过求解方程(14)的多项式特征值问题,可以得到给定转速下旋转锥壳的前后行波振动频率.在运动方程的推导中,锥壳的位移场可以视作

  • u=U(x)(cos(nθ)q(t)-sin(nθ)p(t))V(x)(sin(nθ)q(t)+cos(nθ)p(t))W(x)(cos(nθ)q(t)-sin(nθ)p(t))
    (16)
  • 与多项式特征值问题的推导类似,旋转锥壳在周期性轴向载荷作用下的运动方程可表示如下:

  • f¨+Gf˙+K0+εKacosωatf=0
    (17)
  • 式中f=q1(t),q2(t),,qJ(t),p1(t),p2(t),pJ(t)T表示维度为2J ×1广义自由度列向量,而质量、陀螺和刚度系数矩阵可表示为

  • M=H200H2,G=0-H1H10,K0=H0+ηaNcrHa00H0+ηaNcrHa,Ka=ηaNcrHa00ηaNcrHa
    (18)
  • 显然,在考虑周期性轴向载荷的情况下,圆锥薄壳的控制方程具有Mathieu-Hill型的周期性变化系数.这类系统又称为参数激励系统,系统的参数稳定性是人们关注的主要问题.

  • 2 参数稳定性分析方法

  • 公式(17)的解的稳定性将通过Hill方法[25]进行研究.该方法基于Floquet理论,其主要思路是式(17)的解可以写成指数部分和周期部分的乘积.用复Fourier级数展开表示周期部分,这个解可以写成

  • f=eλωatk=- fkeikωateλωatk=-NkNk fkeikωat
    (19)
  • 式中,i=-1,λ表示Floquet(或特征)指数,fk是复Fourier系数向量.引入Φ=Ie-iωatI IeiωatTAn=f-1f0f1T (其中I为单位矩阵),可得

  • f=eλωatΦAn
    (20)
  • f˙=λωaeλωatΦAn+iωaeλωatΦJ1An
    (21)
  • f¨=λ2ωa2eλωatΦAn+2iωa2eλωatΦJ1An-ωa2eλωatΦJ2An
    (22)
  • 式中, J1=diag kI J2=diagk2I]).通过三角函数运算, cosωatΦ可重新表示为

  • cosωatΦ=12e-iωat+eiωatΦ=12ΦJ3
    (23)
  • 式中

  • 将式(20)~式(24)代入式(17),应用谐波平衡条件,得到如下代数方程:

  • λ2ωa2IM+2iλωa2IMJ1-ωa2IMJ2+λωaIG+iωaIGJ1+I¯K+12εI~KJ3An=0
    (25)
  • 式中, IM=diag M,IG=diag([G]),IK=diagK0IK=diag([Ka.为了使式(17)具有式(19)形式的非零解,式(25)的系数矩阵的行列式需为零(ωa≠0):

  • detλ2IM+λ2iIMJ1+1ωaIG+-IMJ2+i1ωaIGJ1+1ωa2I~K+12ωa2εI~KJ3=0
    (26)
  • 式(26)可用于分析给定轴向力的变化频率ωa和相对振幅ε的系统稳定性.如果系统是稳定的,所有特征值λ的实部为负,指数部分随着时间的推移而减小.另一方面,如果至少有一个特征值λ具有正实部,则系统是不稳定的.为了得到稳定分析的近似数值特征值,只需少量的Nk就可以满足精度要求.

  • 3 分析实例

  • 3.1 模型验证

  • 在开展稳定性研究之前,需要对所提出的薄壁锥壳模型和GDQ方法的结果进行验证.采用ωs表示非旋转锥壳的固有频率.考虑旋转后,特定模态(m, n)的前/后行波频率用ωbωf表示,其中,n=1, 2, ···表示周向波的数量,而m=1, 2, ···则表示相应驻波模式中轴向半波的数量.以ρa21-v2/E]1/2为转速、频率的量纲,则无量纲化后的转速和行波频率分别表示为Ω*,ωa*,ωs*,ωb*,ωf*.

  • 表1给出了具有Cs-Cl和Ss-Sl边界条件的非旋转各向同性圆锥壳的固有频率ωs*的比较.以Irie等[27]的数值积分方法得到的结果为基准值,所分析的圆锥壳参数为m=1, υ=0.3, α=30°, h/a=0.01.结果表明,在增加离散网格点的情况下GDQ方法的结果很快得到收敛,且与文献结果具有较好的一致性.将旋转各向同性圆柱壳在不同转速下的前后行波频率与Sun等[28]的结果进行比较,如表2所示.结构参数为υ=0.3, α=0°, h=a/500, L=5a,边界条件为Ss-Sl.从中也可发现类似的一致性,表明本研究所建立的分析模型是正确的,用GDQ方法得到的固有频率结果是可信的.为了精度要求,在后续分析中将总网格点数设为16.

  • 表1 不同边界条件时非旋转圆锥壳的固有频率比较

  • Table1 Comparison of natural frequencies of non-rotating conical shells with different boundary conditions

  • 表2 旋转圆柱壳在不同转速下的前后行波频率比较

  • Table2 Comparison of traveling wave frequencies of rotating cylindrical shells at different speeds

  • 3.2 不同转速下的参数不稳定区

  • 分析的圆锥薄壳参数为m=1, n=2, υ=0.3, α=20°, h/a=0.008, L/a=10,边界条件为Cs-Cl.计算了不同转速时周期轴向力作用下主参数不稳定区,如图2所示.考虑拉伸轴向力(ηa =0.2),且相对变化幅值ε由0.05增加至0.5.可以看出,随着ε的增加,不稳定区域对称地扩大.与对称轴相对应的激励频率即为前后行波频率之和,即ωa*=ωb*+ωf*.显然,对于不旋转的锥壳,有ωa*=2ωs*.考虑旋转后,不稳定区域向高频率范围整体移动,尤其是当转速大于0.002时,如图4(c)所示.但在一定的转速范围内,不稳定区域的位置受转速的影响不大.当边界条件改为Ss-Sl、Cs-Sl或Ss-Cl时,也有类似现象.由于篇幅的限制,这里不给出结果.在后续分析中,只考虑Cs-Cl边界条件,并讨论系统参数对不稳定区域的影响.

  • 3.3 参数影响分析

  • 3.3.1 定常轴向力幅值

  • 图3绘制了具有不同定常轴向力幅值时对应不同模态的不稳定区域变化.考虑了三种轴向力幅值(ηa=0.1,0.2,0.3),相对幅值固定为ε=0.5,转速为Ω*=0.005,其他参数与上一节模型参数相同.随着ηa的增加,不稳定区向高频区整体移动.这是由于增加恒定轴向载荷会增加固有频率值,进而使失稳区域的起始点向更高的频率范围移动.对于给定的ε,增加恒定轴向载荷ηa也会增加周期轴向载荷的变化幅值,刚度参数激励得到加强,使得不稳定区的范围有所扩大,如图3(a)所示.随着周向波数的增加[模态(1,3)和模态(1,4),如图3(b)和图3(c)所示],不稳定区的变化规律与模态(1,1)类似.

  • 图2 不同转速时周期轴向力作用下旋转锥壳的主参数不稳定区

  • Fig.2 Primary instability regions of conical shells under periodic axial force at different rotational speeds

  • 图3 定常轴向力幅值对不稳定区的影响

  • Fig.3 Effects of axial force amplitude on the instability regions

  • 3.3.2 锥角

  • 本节讨论锥角对不稳定区的影响.分析的圆锥薄壳参数为υ=0.3, h/a=0.008, L/a=10, Ω*=0.005, ηa=0.2, ε=0.5.考虑三种锥角值(α=20°, 40°, 60°),结果如图4所示.对于模态(1,1),增大α值,失稳区整体向低频段移动,且不稳定区范围有所扩大.对于模态(1,3)和模态(1,4),首先使不稳定区域向高频段移动而后向低频段移动,如图4(b)和图4(c)所示.不稳定区范围也有所增加.与模态(1,1)的情况相比,模态(1,3)和模态(1,4)的不稳定区的移动和范围扩大量相对较小.这表明,随着周向波的增加,锥角对失稳区的影响呈减弱趋势.这一现象也与Ng等[9]的发现一致.

  • 图4 锥角对不稳定区的影响

  • Fig.4 Effects of cone angle on the instability regions

  • 3.3.3 厚径比

  • 本节研究厚径比对不同模态的不稳定区影响,如图5所示.分析的圆锥薄壳参数为υ=0.3, α=20°, L/a=10, Ω*=0.005, ηa =0.2, ε=0.5.分析了三种厚径比值,即h/a=0.005, 0.008, 0.012.结果表明,增大h/a值不仅会使不稳定区向高频段移动,而且会使不稳定区明显扩大.对于周向波较大的情况,如图5(c)所示的模态(1,4),不稳定区随h/a的移动和扩大并未减弱.

  • 3.3.4 长径比

  • 讨论长径比对不稳定区的影响.分析的圆锥薄壳参数为υ=0.3, α=20°, h/a=0.008,Ω*=0.005, ηa=0.2, ε=0.5.当L/a=10, 15和20时,不稳定区的结果如图6所示.可以发现,随着L/a的增加,不稳定区主要出现在低频段.对于周向波数较少的失稳模态,如模态(1,1),除失稳区的整体移动外,失稳区宽度的变化不大.但是,对于周向波数较多的失稳模态,增大L/a值也会减小失稳区宽度,如图6(c)所示.

  • 图5 厚径比对不稳定区的影响

  • Fig.5 Effects of thickness-diameter ratio on unstable zone

  • 图6 长径比对不稳定区的影响

  • Fig.6 Effects of length-to-radius ratio on the instability regions

  • 4 结论

  • 本文首先推导了旋转锥壳的动力学方程,采用GDQ法和Hill法分析了系统在周期轴向载荷作用下的参数不稳定性.计算并讨论了多个不稳定区随工况和几何参数的变化规律.结果表明:提高转速会导致不稳定区沿频率轴移动,但对不稳定宽度影响不大.增加恒定拉伸轴向载荷,不仅会显著增加失稳宽度,而且会导致失稳区域向更高的频率范围移动.锥角、厚径比或长径比的变化都会导致不稳定区沿频率轴移动.锥角和厚径比会增大失稳宽度(长径比会减小).随着周向波个数的增加,锥角对失稳区的影响逐渐减弱,而厚径比的影响则基本保持不变.

  • 上述研究基于线性和各向同性薄壁壳体假设.若壳体是非线性的且各向异性,例如考虑大变形、功能梯度或复合材料层合壳材料.进一步的工作应重点探讨非线性因素的引入对系统动力稳定性的影响.

  • 参考文献

    • [1] Sahu S K,Datta P K.Research advances in the dynamic stability behavior of plates and shells:1987-2005-Part Ⅰ:Conservative systems.Applied Mechanics Reviews,2007,60(2):65~75

    • [2] 乔丕忠,王艳丽,陆林军.圆柱壳稳定性问题的研究进展.力学季刊,2018(2):223~236(Qiao P Z,Wang Y L,Lu L J.Advances in stability study of cylindrical shells.Chinese Quarterly of Mechanics,2018(2):223~236(in Chinese))

    • [3] 赵振,刘才山,陈滨,等.薄壁加筋肋圆柱壳稳定性分析的参数化研究.力学与实践,2004,26(2):17~21(Zhao Z,Liu C S,Chen B,et al.Parameterization study of orthogrid stiffened cylinder shells.Mechanics in Engineering,2004,26(2):17~21(in Chinese))

    • [4] 王登峰,曹平周.大型薄壁圆柱壳在局部轴向压力作用下的稳定性能研究.工程力学,2009,26(4):38~45(Wang D F,Cao Z Z.Study on stability of large-scale thin-walled cylindrical shells subjected to partial axial compression.Engineering Mechanics,2009,26(4):38~45(in Chinese))

    • [5] 杜长城,李映辉.功能梯度薄壁圆柱壳的自由振动.动力学与控制学报,2010,8(3):219~223(Du C C,Li Y H.Free vibration of functionally graded cylindrical thin shells.Journal of Dynamics and Control,2010,8(3):219~223(in Chinese))

    • [6] 王宇,罗忠,李昌.薄壁圆柱壳的高阶模态振动特性研究.动力学与控制学报,2016,14(2):131~137(Wang Y,Luo Z,Li C.Study on vibration characteristics of high-order modal for a thin cylindrical shell.Journal of Dynamics and Control,2016,14(2):131~137(in Chinese))

    • [7] 郭翔鹰,刘大猛,张伟.压电纤维复合材料层合壳的非线性动力学研究.动力学与控制学报,2017,15(5):430~438(Guo X,Liu D,Zhang W.Nonlinear dynamic analysis of piezoelectric macro-fiber composite laminated shells.Journal of Dynamics and Control,2017,15(5):430~438(in Chinese))

    • [8] Ye Z M.The non-linear vibration and dynamic instability of thin shallow shells.Journal of Sound and Vibration,1997,202(3):303~311

    • [9] Ng T Y,Hua L,Lam K Y,et al.Parametric instability of conical shells by the generalized differential quadrature method.International Journal for Numerical Methods in Engineering,1999,44:819~837

    • [10] Sofiyev A H.The stability of functionally graded truncated conical shells subjected to aperiodic impulsive loading.International Journal of Solids and Structures,2004,41(13):3411~3424

    • [11] Sofiyev A H.The buckling of functionally graded truncated conical shells under dynamic axial loading.Journal of Sound and Vibration,2007,305(4-5):808~826

    • [12] Sofiyev A H.The vibration and stability behavior of freely supported FGM conical shells subjected to external pressure.Composite Structures,2009,89(3):356~366

    • [13] Qin B,Zhong R,Wang T,et al.A unified Fourier series solution for vibration analysis of FG-CNTRC cylindrical,conical shells and annular plates with arbitrary boundary conditions.Composite Structures,2020,232:111549

    • [14] Fu T,Wu X,Xiao Z,et al.2021-Analysis of vibration characteristics of FGM sandwich joined conical-conical shells surrounded by elastic foundations.Thin-Walled Structures,2021,165:107979

    • [15] Li H,Hao Y X,Zhang W,et al.Vibration analysis of porous metal foam truncated conical shells with general boundary conditions using GDQ.Composite Structures,2021,269:114036

    • [16] Ng T Y,Lam K Y,Reddy J N.Parametric resonance of a rotating cylindrical shell subjected to periodic axial loads.Journal of Sound and Vibration,1998,214(3):513~529

    • [17] Liew K M,Hu Y G,Ng T Y.Dynamic stability of rotating cylindrical shells subjected to periodic axial loads.International Journal of Solids and Structures,2006,43:7553~7570

    • [18] Lam K Y,Hua L.Vibration analysis of a rotating truncated circular conical shell.International Journal of Solids and Structures,1997,34(17):2183~2197

    • [19] Shu C,Richards B E.Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations.International Journal for Numerical Methods in Fluids,1992,15:791~798

    • [20] Loy C T,Lam K Y,Shu C.Analysis of cylindrical shells using generalized quadrature.Shock and Vibration,1997,4:193~198

    • [21] Hua L,Lam K Y.The generalized differential quadrature method for frequency analysis of a rotating conical shell with initial pressure.International Journal for Numerical Methods in Engineering,1997,48:1703~1722

    • [22] Lam K Y,Hua L.Generalized differential quadrature for frequency of rotating multilayered conical shell.Journal of Engineering Mechanics,2000,126:1156~1162

    • [23] Pei Y C,Tan Q C.Parametric instability of flexible disk rotating at perodically varying angular speed.Meccanica,2009,44:711~720

    • [24] Pei Y C,Chatwin C.Elastic connection disk subjected to periodically fluctuating transmitted torque and rotational speed.Journal of Sound and Vibration,2012,331:3083~3095

    • [25] Friedmann P,Hammond E,Woo T H.Efficient numerical treatment of periodic systems with application to stability problems.International Journal of Numerical Methods in Engineering,1977,11:1117~1136

    • [26] Soedel W.Vibrations of shells and plates.New York:Marcel Dekker,1993

    • [27] Irie T,Yamada G,Tanaka K.Natural frequencies of truncated conical shells.Journal of Sound and Vibration,1984,92:447~453

    • [28] Sun S,Chu S,Cao D.Vibration characteristics of thin rotating cylindrical shells with various boundary conditions.Journal of Sound and Vibration,2012,331(18):4170~4186

  • 参考文献

    • [1] Sahu S K,Datta P K.Research advances in the dynamic stability behavior of plates and shells:1987-2005-Part Ⅰ:Conservative systems.Applied Mechanics Reviews,2007,60(2):65~75

    • [2] 乔丕忠,王艳丽,陆林军.圆柱壳稳定性问题的研究进展.力学季刊,2018(2):223~236(Qiao P Z,Wang Y L,Lu L J.Advances in stability study of cylindrical shells.Chinese Quarterly of Mechanics,2018(2):223~236(in Chinese))

    • [3] 赵振,刘才山,陈滨,等.薄壁加筋肋圆柱壳稳定性分析的参数化研究.力学与实践,2004,26(2):17~21(Zhao Z,Liu C S,Chen B,et al.Parameterization study of orthogrid stiffened cylinder shells.Mechanics in Engineering,2004,26(2):17~21(in Chinese))

    • [4] 王登峰,曹平周.大型薄壁圆柱壳在局部轴向压力作用下的稳定性能研究.工程力学,2009,26(4):38~45(Wang D F,Cao Z Z.Study on stability of large-scale thin-walled cylindrical shells subjected to partial axial compression.Engineering Mechanics,2009,26(4):38~45(in Chinese))

    • [5] 杜长城,李映辉.功能梯度薄壁圆柱壳的自由振动.动力学与控制学报,2010,8(3):219~223(Du C C,Li Y H.Free vibration of functionally graded cylindrical thin shells.Journal of Dynamics and Control,2010,8(3):219~223(in Chinese))

    • [6] 王宇,罗忠,李昌.薄壁圆柱壳的高阶模态振动特性研究.动力学与控制学报,2016,14(2):131~137(Wang Y,Luo Z,Li C.Study on vibration characteristics of high-order modal for a thin cylindrical shell.Journal of Dynamics and Control,2016,14(2):131~137(in Chinese))

    • [7] 郭翔鹰,刘大猛,张伟.压电纤维复合材料层合壳的非线性动力学研究.动力学与控制学报,2017,15(5):430~438(Guo X,Liu D,Zhang W.Nonlinear dynamic analysis of piezoelectric macro-fiber composite laminated shells.Journal of Dynamics and Control,2017,15(5):430~438(in Chinese))

    • [8] Ye Z M.The non-linear vibration and dynamic instability of thin shallow shells.Journal of Sound and Vibration,1997,202(3):303~311

    • [9] Ng T Y,Hua L,Lam K Y,et al.Parametric instability of conical shells by the generalized differential quadrature method.International Journal for Numerical Methods in Engineering,1999,44:819~837

    • [10] Sofiyev A H.The stability of functionally graded truncated conical shells subjected to aperiodic impulsive loading.International Journal of Solids and Structures,2004,41(13):3411~3424

    • [11] Sofiyev A H.The buckling of functionally graded truncated conical shells under dynamic axial loading.Journal of Sound and Vibration,2007,305(4-5):808~826

    • [12] Sofiyev A H.The vibration and stability behavior of freely supported FGM conical shells subjected to external pressure.Composite Structures,2009,89(3):356~366

    • [13] Qin B,Zhong R,Wang T,et al.A unified Fourier series solution for vibration analysis of FG-CNTRC cylindrical,conical shells and annular plates with arbitrary boundary conditions.Composite Structures,2020,232:111549

    • [14] Fu T,Wu X,Xiao Z,et al.2021-Analysis of vibration characteristics of FGM sandwich joined conical-conical shells surrounded by elastic foundations.Thin-Walled Structures,2021,165:107979

    • [15] Li H,Hao Y X,Zhang W,et al.Vibration analysis of porous metal foam truncated conical shells with general boundary conditions using GDQ.Composite Structures,2021,269:114036

    • [16] Ng T Y,Lam K Y,Reddy J N.Parametric resonance of a rotating cylindrical shell subjected to periodic axial loads.Journal of Sound and Vibration,1998,214(3):513~529

    • [17] Liew K M,Hu Y G,Ng T Y.Dynamic stability of rotating cylindrical shells subjected to periodic axial loads.International Journal of Solids and Structures,2006,43:7553~7570

    • [18] Lam K Y,Hua L.Vibration analysis of a rotating truncated circular conical shell.International Journal of Solids and Structures,1997,34(17):2183~2197

    • [19] Shu C,Richards B E.Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations.International Journal for Numerical Methods in Fluids,1992,15:791~798

    • [20] Loy C T,Lam K Y,Shu C.Analysis of cylindrical shells using generalized quadrature.Shock and Vibration,1997,4:193~198

    • [21] Hua L,Lam K Y.The generalized differential quadrature method for frequency analysis of a rotating conical shell with initial pressure.International Journal for Numerical Methods in Engineering,1997,48:1703~1722

    • [22] Lam K Y,Hua L.Generalized differential quadrature for frequency of rotating multilayered conical shell.Journal of Engineering Mechanics,2000,126:1156~1162

    • [23] Pei Y C,Tan Q C.Parametric instability of flexible disk rotating at perodically varying angular speed.Meccanica,2009,44:711~720

    • [24] Pei Y C,Chatwin C.Elastic connection disk subjected to periodically fluctuating transmitted torque and rotational speed.Journal of Sound and Vibration,2012,331:3083~3095

    • [25] Friedmann P,Hammond E,Woo T H.Efficient numerical treatment of periodic systems with application to stability problems.International Journal of Numerical Methods in Engineering,1977,11:1117~1136

    • [26] Soedel W.Vibrations of shells and plates.New York:Marcel Dekker,1993

    • [27] Irie T,Yamada G,Tanaka K.Natural frequencies of truncated conical shells.Journal of Sound and Vibration,1984,92:447~453

    • [28] Sun S,Chu S,Cao D.Vibration characteristics of thin rotating cylindrical shells with various boundary conditions.Journal of Sound and Vibration,2012,331(18):4170~4186

  • WeChat

    Mobile website