轴对称薄壁结构自由振动的边界元分析
|
摘要
采用双互易法分析薄壁轴对称结构自由振动的特征频率以及特征模态.首先,采用径向基函数插值域积分里的位移,利用双互易法将域积分转化为子午面边界的积分.然后,将边界物理量、基本解和特解展开为傅里叶级数,沿环向积分后得到的边界积分方程可用于轴对称结构带体积力问题和受非对称载荷的动力学分析,其积分域为轴对称结构子午面边界上的线积分,进一步降低了问题的维度和离散的难度.文章详细探讨了源点处于对称轴的特殊情况,根据基本解和特解的退化形式,针对无体积力和有体积力分别给出了处理奇异矩阵的方案.对于薄壁结构,采用双曲正弦变换处理近奇异积分有效提高积分精度.最后将双互易法和双曲正弦变化应用于薄壁轴对称结构带体积力的静力学和自由振动分析.数值结果表明,文章提出的处理奇异矩阵的方法能够有效处理源点处于对称轴的情况;当圆筒厚高比为10~(-3),边界元计算的特征频率的相对误差为10~(-3),且优于有限元的结果.
The dual reciprocity method(DRM) is extended to study the eigenvalue and eigenmode of thin-walled axisymmetric structures. First the displacement in the domain integral can be approximated by a set of radial basis functions and the domain integral can be converted to the boundary using DRM. Then the displacement and the traction can be expanded as Fourier series and integrate along the circumferential direction. The obtained boundary integral equation can be used for analysis of elastostatics of axisymmetric structure distributed body force and elastodynamics subject to asymmetric loading. The special case of the source point on the axis of symmetry is discussed in detail. New schemes are suggested for dealing with singular matrices for cases with and without body force respectively according to the degenerate form of the fundamental solution and the particular solution. For the thin walled structure, the sinh transformation is applied to improve the accuracy of evaluation of the nearly singular integrals. The developed project has been used to analyze elastostatics with body force and the free vibration of the thin axisymmetric structures. Numerical results indicate that the proposed method for dealing with singular matrices can effectively deal with the situation where the source point is on theaxis of symmetry. and when the thickness ratio reaches 10~(-3), the relative error of the results can approach 10~(-3), which is better than those of FEM.
The dual reciprocity method(DRM) is extended to study the eigenvalue and eigenmode of thin-walled axisymmetric structures. First the displacement in the domain integral can be approximated by a set of radial basis functions and the domain integral can be converted to the boundary using DRM. Then the displacement and the traction can be expanded as Fourier series and integrate along the circumferential direction. The obtained boundary integral equation can be used for analysis of elastostatics of axisymmetric structure distributed body force and elastodynamics subject to asymmetric loading. The special case of the source point on the axis of symmetry is discussed in detail. New schemes are suggested for dealing with singular matrices for cases with and without body force respectively according to the degenerate form of the fundamental solution and the particular solution. For the thin walled structure, the sinh transformation is applied to improve the accuracy of evaluation of the nearly singular integrals. The developed project has been used to analyze elastostatics with body force and the free vibration of the thin axisymmetric structures. Numerical results indicate that the proposed method for dealing with singular matrices can effectively deal with the situation where the source point is on theaxis of symmetry. and when the thickness ratio reaches 10~(-3), the relative error of the results can approach 10~(-3), which is better than those of FEM.
引文
1 Cruse TA,Rizzo FJ.A direct formulation and numerical solution of the general transient elastodynamic problem.I.Journal of Mathematical Analysis and Applications,1968,22(1):244-259
2 Cruse TA.A direct formulation and numerical solution of the general transient elastodynamic problem.II.Journal of Mathematical Analysis and Applications,1968,22(2):341-355
3 Alielahi H,Kamalian M,Adampira M.A bem investigation on the influence of underground cavities on the seismic response of canyons.Acta Geotechnica,2016,11(2):391-413
4 Thirard C,Parot JM.On a way to save memory when solving time domain boundary integral equations for acoustic and vibroacoustic applications.Journal of Computational Physics,2017,348:744-753
5 Ahmad S,Banerjee PK.Multi-domain bem for two-dimensional problems of elastodynamics.International Journal for Numerical Methods in Engineering,1988,26(4):891-911
6 Wuensche M,Sladek J,Sladek V,et al.Dynamic crack analysis in piezoelectric solids under time-harmonic loadings with a symmetric galerkin boundary element method.Engineering Analysis with Boundary Elements,2017,84:141-153
7 Li J,Khodaei ZS,Aliabadi MH.Spectral bem for the analysis of wave propagation and fracture mechanics.Journal of Multiscale Modeling,2017,8(3-4):1-22
8 Wang HC,Banerjee PK.Axisymmetric free-vibration problems by boundary element method.Journal of Applied MechanicsTransactions of the ASME,1988,55(2):437-442
9 Zheng BJ,Gao XW,Zhang C.Radial integration bem for vibration analysis of two-and three-dimensional elasticity structures.Applied Mathematics and Computation,2016,277:111-126
10 Yao ZH.A new time domain boundary integral equation and efficient time domain boundary element scheme of elastodynamics.Cmes-Computer Modeling in Engineering&Sciences,2009,50(1):21-45
11 Nardini D,Brebbia CA.A new approach to free vibration analysis using boundary elements.Applied Mathematical Modelling,1983,7(3):157-162
12 Park KH.A bem formulation for axisymmetric elasticity with arbitrary body force using particular integrals.Computers&Structures,2002,80(32):2507-2514
13 Javaran SH,Khaji N,Moharrami H.A dual reciprocity bem approach using new fourier radial basis functions applied to 2d elastodynamic transient analysis.Engineering Analysis with Boundary Elements,2011,35(1):85-95
14 Javaran SH,Shojaee S.The solution of elastostatic and dynamic problems using the boundary element method based on spherical hankel element framework.International Journal for Numerical Methods in Engineering,2017,112(13):2067-2086
15 Samaan MF,Rashed YF,Ahmed MA.The dual reciprocity method applied to free vibrations of 2d structures using compact supported radial basis functions.Computational Mechanics,2007,41(1):85-105
16 Fahmy MA.Shape design sensitivity and optimization of anisotropic functionally graded smart structures using bicubic b-splines drbem.Engineering Analysis with Boundary Elements,2018,87:27-35
17陆山,黄其青.复杂载荷三维裂纹分析双重边界元法.力学学报,2002,34(5):715-725((Lu Shan,Huang Qiqing.Dual BEM for sif analyses of 3D structures with complex loads.Acta Mechanica Sinica,2002,34(5):715-725(in Chinese))
18 Galvis A F,Rodriguez R Q,Sollero P.Dynamic analysis of threedimensional polycrystalline materials using the boundary element method.Computers&Structures,2018,200:11-20
19晏飞,冯夏庭,周辉.非线性泊松问题的拟线性化边界积分方法研究.力学学报,2010,42(4):798-803(Yan Fei,Feng Xia-Ting,Zhou Hui.A study of boundary integral method for a type of nonlinear problem based on generalized quasilinearlization theory.Chinese Journal of Theoretical and Applied Mechanics,2010,42(4):798-803(in Chinese))
20 Ghorbani M,Watson D.A collocation and least squares p-singular boundary method without fictitious boundary.Engineering Analysis with Boundary Elements,2015,61:27-32
21 Wang HC,Banerjee PK.Free-vibration of axisymmetric solids by bem using particular integrals.International Journal for Numerical Methods in Engineering,1990,29(5):985-1001
22 Wang HC,Banerjee PK.Generalized Axisymmetric Elastodynamic Analysis by Boundary Element Method.Springer-Verlag,1990:115-131
23姚振汉,王海涛.边界元法.北京:高等教育出版社,2010(Yao Zhenhan,Wang Haitao.Boundary Element Method.Beijing:Higher Education Press,2010(in Chinese))
24 Nardini D,Brebbia CA.A new approach to free-vibration analysis using boundary elements.Applied Mathematical Modelling,1983,7(3):157-162
25 Cruse TA.The Boundary Integral Equation Method in Axisymmetric Stress Analysis Problems.Springer-Verlag,1986
26 Gu Y,Chen W,Zhang C.The sinh transformation for evaluating nearly singular boundary element integrals over high-order geometry elements.Engineering Analysis with Boundary Elements,2013,37(2):301-308
27 Johnston PR,Elliott D.A sinh transformation for evaluating nearly singular boundary element integrals.International Journal for Numerical Methods in Engineering,2005,62(4):564-578
28 Bakr AA.The boundary integral equation method in axisymmetric stress analysis problems.Springer-Verlag,1986
29孙宗光,王本深.匀速旋转实心圆球的弹性力学解.沈阳建筑工程学院学报,1996,12(2):138-143(Sun Zongguang,Wang Benshen.Elastic mechanics solution to rotative medicine globe.Journal of Shenyang Architectural and Civil Engineering Institute,1996,12(2):138-143(in Chinese))
30王大钧.结构力学中的定性理论.北京:北京大学出版社,2014(Wang Dajun.Qualitative Theory in Structural Mechanics.Beijing:Peking University Press,2014(in Chinese))
31 Gladwell GML,Vijay DK.Natural frequencies of free finite-length circular-cylinders.J Sound Vib,1975,42(3):387-397
32 Hutchinson JR,El-Azhari SA.Vibrations of free hollow circular cylinders.Journal of Applied Mechanics,1986,53(3):641-646
2 Cruse TA.A direct formulation and numerical solution of the general transient elastodynamic problem.II.Journal of Mathematical Analysis and Applications,1968,22(2):341-355
3 Alielahi H,Kamalian M,Adampira M.A bem investigation on the influence of underground cavities on the seismic response of canyons.Acta Geotechnica,2016,11(2):391-413
4 Thirard C,Parot JM.On a way to save memory when solving time domain boundary integral equations for acoustic and vibroacoustic applications.Journal of Computational Physics,2017,348:744-753
5 Ahmad S,Banerjee PK.Multi-domain bem for two-dimensional problems of elastodynamics.International Journal for Numerical Methods in Engineering,1988,26(4):891-911
6 Wuensche M,Sladek J,Sladek V,et al.Dynamic crack analysis in piezoelectric solids under time-harmonic loadings with a symmetric galerkin boundary element method.Engineering Analysis with Boundary Elements,2017,84:141-153
7 Li J,Khodaei ZS,Aliabadi MH.Spectral bem for the analysis of wave propagation and fracture mechanics.Journal of Multiscale Modeling,2017,8(3-4):1-22
8 Wang HC,Banerjee PK.Axisymmetric free-vibration problems by boundary element method.Journal of Applied MechanicsTransactions of the ASME,1988,55(2):437-442
9 Zheng BJ,Gao XW,Zhang C.Radial integration bem for vibration analysis of two-and three-dimensional elasticity structures.Applied Mathematics and Computation,2016,277:111-126
10 Yao ZH.A new time domain boundary integral equation and efficient time domain boundary element scheme of elastodynamics.Cmes-Computer Modeling in Engineering&Sciences,2009,50(1):21-45
11 Nardini D,Brebbia CA.A new approach to free vibration analysis using boundary elements.Applied Mathematical Modelling,1983,7(3):157-162
12 Park KH.A bem formulation for axisymmetric elasticity with arbitrary body force using particular integrals.Computers&Structures,2002,80(32):2507-2514
13 Javaran SH,Khaji N,Moharrami H.A dual reciprocity bem approach using new fourier radial basis functions applied to 2d elastodynamic transient analysis.Engineering Analysis with Boundary Elements,2011,35(1):85-95
14 Javaran SH,Shojaee S.The solution of elastostatic and dynamic problems using the boundary element method based on spherical hankel element framework.International Journal for Numerical Methods in Engineering,2017,112(13):2067-2086
15 Samaan MF,Rashed YF,Ahmed MA.The dual reciprocity method applied to free vibrations of 2d structures using compact supported radial basis functions.Computational Mechanics,2007,41(1):85-105
16 Fahmy MA.Shape design sensitivity and optimization of anisotropic functionally graded smart structures using bicubic b-splines drbem.Engineering Analysis with Boundary Elements,2018,87:27-35
17陆山,黄其青.复杂载荷三维裂纹分析双重边界元法.力学学报,2002,34(5):715-725((Lu Shan,Huang Qiqing.Dual BEM for sif analyses of 3D structures with complex loads.Acta Mechanica Sinica,2002,34(5):715-725(in Chinese))
18 Galvis A F,Rodriguez R Q,Sollero P.Dynamic analysis of threedimensional polycrystalline materials using the boundary element method.Computers&Structures,2018,200:11-20
19晏飞,冯夏庭,周辉.非线性泊松问题的拟线性化边界积分方法研究.力学学报,2010,42(4):798-803(Yan Fei,Feng Xia-Ting,Zhou Hui.A study of boundary integral method for a type of nonlinear problem based on generalized quasilinearlization theory.Chinese Journal of Theoretical and Applied Mechanics,2010,42(4):798-803(in Chinese))
20 Ghorbani M,Watson D.A collocation and least squares p-singular boundary method without fictitious boundary.Engineering Analysis with Boundary Elements,2015,61:27-32
21 Wang HC,Banerjee PK.Free-vibration of axisymmetric solids by bem using particular integrals.International Journal for Numerical Methods in Engineering,1990,29(5):985-1001
22 Wang HC,Banerjee PK.Generalized Axisymmetric Elastodynamic Analysis by Boundary Element Method.Springer-Verlag,1990:115-131
23姚振汉,王海涛.边界元法.北京:高等教育出版社,2010(Yao Zhenhan,Wang Haitao.Boundary Element Method.Beijing:Higher Education Press,2010(in Chinese))
24 Nardini D,Brebbia CA.A new approach to free-vibration analysis using boundary elements.Applied Mathematical Modelling,1983,7(3):157-162
25 Cruse TA.The Boundary Integral Equation Method in Axisymmetric Stress Analysis Problems.Springer-Verlag,1986
26 Gu Y,Chen W,Zhang C.The sinh transformation for evaluating nearly singular boundary element integrals over high-order geometry elements.Engineering Analysis with Boundary Elements,2013,37(2):301-308
27 Johnston PR,Elliott D.A sinh transformation for evaluating nearly singular boundary element integrals.International Journal for Numerical Methods in Engineering,2005,62(4):564-578
28 Bakr AA.The boundary integral equation method in axisymmetric stress analysis problems.Springer-Verlag,1986
29孙宗光,王本深.匀速旋转实心圆球的弹性力学解.沈阳建筑工程学院学报,1996,12(2):138-143(Sun Zongguang,Wang Benshen.Elastic mechanics solution to rotative medicine globe.Journal of Shenyang Architectural and Civil Engineering Institute,1996,12(2):138-143(in Chinese))
30王大钧.结构力学中的定性理论.北京:北京大学出版社,2014(Wang Dajun.Qualitative Theory in Structural Mechanics.Beijing:Peking University Press,2014(in Chinese))
31 Gladwell GML,Vijay DK.Natural frequencies of free finite-length circular-cylinders.J Sound Vib,1975,42(3):387-397
32 Hutchinson JR,El-Azhari SA.Vibrations of free hollow circular cylinders.Journal of Applied Mechanics,1986,53(3):641-646