完备化ANCF薄板单元及在钢板弹簧动力学建模中的应用
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摘要
绝对节点坐标方法已在多体系统动力学研究中广泛应用,但常用来描述板壳类结构的薄板单元,由于梯度不完备而无法直接用于带有初始弯曲参考构型的柔性体变形描述.为避免全参数板单元建立车辆钢板弹簧模型时存在的严重截面闭锁问题,拟采用薄板单元用于板簧建模.为此,探索了将现有绝对节点坐标薄板单元纳入一般连续介质力学弹性力表达的方法,采用中面上单位法向量作为单元厚度方向的梯度向量,从而得到了完备化的薄板单元及其描述初始弯曲构型时消除初应变的方法.在此基础上通过定义簧片的未变形构型,在钢板弹簧中引入可控的预应力,实现对钢板弹簧装配过程的准确模拟.通过数值算例验证了本方法的正确性.建立了车辆钢板弹簧模型,通过建立在簧片上的局部坐标系实现接触点的跨单元搜索,并采用惩罚函数法和平滑化的库伦摩擦模型施加簧片间的接触力.引入参考节点的概念建立了整合车身与吊耳及其机构运动关系的刚柔耦合模型.
The absolute nodal coordinate formulation(ANCF) was widely used in the research of multi-body dynamics.But the thin plate element which designed for modeling thin shell structure couldn't be adopted for initial curved structure directly because of its deficiency of position vector gradient. Thin plate element was chosen for modelling the leaf spring to avoid the troublesome locking problem when using fully parameterized ANCF element. One explored the way to express the elastic force of existing thin plate element via common continuum mechanics system by adopting the unitized normal vector on the centroid plane as the added position vector gradient for thickness direction. So as to modify its elastic force formulation. The completed form of the element was given which could eliminated the introduced initial stress when describing initially curved leaf using the standard procedure which could only be performed on fully parameterized ANCF element. The modification for elastic force would not affect the element property. By setting undeformed structure whichdiffers from initial one, the controlled initial stress could be introduced into the model for describing the assemble process.Numerical results were given to verify the validity of this modification. An initial curved double-leaf spring model was built. In order to specify the exact contact region, a cross element search strategy using a local material coordinate system built on the entire leaf was developed. Introduced a more precise calculation procedure for contact and friction force using penalty method and smooth Coulomb law. The rigid-flexible coupled leaf spring model combined with ANCF reference nodes was built, integrated with shackle, chassis and their mechanism movement.
The absolute nodal coordinate formulation(ANCF) was widely used in the research of multi-body dynamics.But the thin plate element which designed for modeling thin shell structure couldn't be adopted for initial curved structure directly because of its deficiency of position vector gradient. Thin plate element was chosen for modelling the leaf spring to avoid the troublesome locking problem when using fully parameterized ANCF element. One explored the way to express the elastic force of existing thin plate element via common continuum mechanics system by adopting the unitized normal vector on the centroid plane as the added position vector gradient for thickness direction. So as to modify its elastic force formulation. The completed form of the element was given which could eliminated the introduced initial stress when describing initially curved leaf using the standard procedure which could only be performed on fully parameterized ANCF element. The modification for elastic force would not affect the element property. By setting undeformed structure whichdiffers from initial one, the controlled initial stress could be introduced into the model for describing the assemble process.Numerical results were given to verify the validity of this modification. An initial curved double-leaf spring model was built. In order to specify the exact contact region, a cross element search strategy using a local material coordinate system built on the entire leaf was developed. Introduced a more precise calculation procedure for contact and friction force using penalty method and smooth Coulomb law. The rigid-flexible coupled leaf spring model combined with ANCF reference nodes was built, integrated with shackle, chassis and their mechanism movement.
引文
1 Shabana AA.An absolute nodal coordinates formulation for the large rotation and deformation analysis of flexible bodies.Technical Report.No.MBS96-1-UIC,University of Illinois at Chicago,1996
2章孝顺,章定国,陈思佳等.基于绝对节点坐标法的大变形柔性梁几种动力学模型研究.物理学报,2016,65(9):094501(Zhang Xiaoshun,Zhang Dingguo,Chen Sijia,et al.Several dynamic models of a large deformation flexible beam based on the absolute nodal coordinate formulation.Acta Physica Sinica,2016,65(9):094501(in Chinese))
3 Gerstmayr J,Sugiyama H,Mikkola A.Review on the absolute nodal coordinate formulation for large deformation analysis of multi-body systems.Journal of Computational and Nonlinear Dynamics,2013,8(3):031016
4田强,张云清,陈立平等.柔性多体系统动力学绝对节点坐标方法研究进展.力学进展,2010,40(2):189-202(Tian Qiang,Zhang Yunqing,Chen Liping,et al.Review of the advances in absolute nodal coordinate formulation in flexible multi-body system dynamics.Advances in Mechanics,2010,40(2):189-202(in Chinese))
5张越,赵阳,谭春林等.ANCF索梁单元应变耦合问题与模型解耦.力学学报,2016,48(6):1406-1415(Zhang Yue,Zhao Yang,Tan Chunlin,et al.The strain coupling problem and model decoupling of ANCF cable/beam element.Chinese Journal of Theoretical and Applied Mechanics,2016,48(6):1406-1415(in Chinese))
6岳宝增,于嘉瑞,吴文军.多储液腔航天器刚液耦合动力学与复合控制.力学学报,2017,49(2):390-396(Yue Baozeng,Yu Jiarui,Wu Wenjun.Rigid and liquid coupling dynamics and hybrid control of spacecraft with multiple propellant tanks.Chinese Journal of Theoretical and Applied Mechanics,2017,49(2):390-396(in Chinese))
7胡景晨,王天舒.一种O(n)算法复杂度的递推绝对节点坐标法研究.力学学报,2016,48(5):1172-1183(Hu Jingchen,Wang Tianshu.A recursive absolute nodal coordinate formulation with O(n)algorithm complexity.Chinese Journal of Theoretical and Applied Mechanics,2016,48(5):1172-1183(in Chinese))
8刘铖,田强,胡海岩.基于绝对节点坐标的多柔体系统动力学高效计算方法.力学学报,2010,42(6):1197-1205(Liu Cheng,Tian Qiang,Hu Haiyan.Efficient computational method for dynamics of flexible multibody systems based on absolute nodal coordinate.Chinese Journal of Theoretical and Applied Mechanics,2010,42(6):1197-1205(in Chinese))
9 García Vallejo D,Mayo J,Escalona JL,et al.Efficient evaluation of the elastic forces and the jacobian in the absolute nodal coordinate formulation.Nonlinear Dynamic,2004,35:313-329
10 Shabana AA.ANCF tire assembly model for multibody system applications.Journal of Computational and Nonlinear Dynamics,2015,10(2):024504
11 Omar MA,Shabana AA,Mikkola A,et al.Multibody system modeling of leaf springs.Journal of Vibration and Control,2004,10(11):1601-1638
12 Yu ZQ,Liu YG,Tinsley B,et al.Integration of geometry and analysis for vehicle system applications:Continuum-based leaf spring and tire assembly.Journal of Computational and Nonlinear Dynamics,2015 11(3):031011
13 Dufva K,Shabana AA.Analysis of thin plate structures using the absolute nodal coordinate formulation.Proceedings of the Institution of Mechanical Engineers Part K:Journal of Multibody Dynamics,2005,219(4):345-355
14 Wang JS,Li ZK,Jiang QB.The analysis of composite leaf spring by finite element method and experimental measurements//FISITA2012 World Automotive Congress Proceedings-Volume 7:Vehicle Design and Testing,Beijing,China,2012-11-27,803
15 Zhou ZF,Guo WF,Shen TJ,et al.Research and application on dynamic stiffness of leaf spring.FISITA 2012 World Automotive Congress Proceedings-Volume 10:Chassis Systems and Integration Technology Beijing,China,2012-11-27,2012,111
16 Nunney MJ.Light and Heavy Vehicle Technology.Oxford:Society for Neuroscience,2006:352
17 Dunaevskii B.Calculation of solid beams and flat springs(leaf springs)of varying cross-section.Soviet Engineering Research,1981,61(4):36-38
18周孔亢,陆建辉,侯永涛等.基于RecurDyn的钢板弹簧动力学模型的建立与参数辨识.机械工程学报,2014,50(4):128-134(Zhou Kongkang,Lu Jianhui,Hou Yongtao,et al.Dynamics modeling and parameter identification of leaf spring based on RecurDyn.Journal of Mechanical Engineering,2014,50(4):128-134(in Chinese))
19段亮,杨树凯,宋传学等.平衡悬架钢板弹簧动态特性的研究.机械工程学报,2016,52(6):153-158(Duan Liang,Yang Shukai,Song Chuanxue,et al.Research on dynamic characteristics of the tandem suspension leaf spring.Journal of Mechanical Engineering,2016,52(6):153-158(in Chinese))
20 Wriggers P,Zavarise G.On contact between three-dimensional beams undergoing large deflections.Communications in Numerical Methods in Engineering,1997,13(6):429-438
21 Zavarise G,Wriggers P.Contact with friction between beams in 3-Dspace.International Journal for Numerical Methods in Engineering,2000,49:977-1006
22 Litewka P,Wriggers P.Contact between 3D beams with rectangular cross-sections.International Journal for Numerical Methods in Engineering,2002,53:2019-2041
23 Litewka P.Hermite polynomial smoothing in beam-tobema frictional contact.Computational Mechanics,2007,40:815-826
24 Litewka P.Smooth frictional contact between beams in 3D//Nackenhorst U,Wriggers P.(eds.)IUTAM Symposium on Computational Contact Mechanics.IUTAM Bookseries.Dordrecht:Springer,2007,157-176
25 Konyukhov A,Schweizerhof K.Geometrically exact covariant approach for contact between curves.Computer Methods in Applied Mechanics and Engineering,2010,199:2510-2531
26 Litewka P.Multiple-point beam-to-beam contact finite element//19th International Conference on Computer Methods in Mechanics,Warsaw,Poland,9-12,May,2011
27 Durville D.Contact-friction modeling within elastic beam assemblies:an application to knot tightening.Computational Mechanics,2012,49(6):687-707
28 Wang QT,Tian Q,Hu HY.Dynamic simulation of frictional contacts of thin beams during large overall motions via absolute nodal coordinate formulation.Nonlinear Dynamics,2014,77(4):1411-1425
29 Wang QT,Tian Q,Hu HY.Dynamic simulation of frictional multizone contacts of thin beams.Nonlinear Dynamics,2016,83(4):1919-1937
30 Zahavi E.Analysis of a contact problem in leaf springs.Mechanics Research Communications,1992,19(1):21-27
31 Shabana AA.Computational Continuum Mechanics.London:Cambridge University Press.2012:46-93
32 Liu C,Tian Q,Hu HY.New spatial curved beam and cylindrical shell elements of gradient-deficient absolute nodal coordinate formulation.Nonlinear Dynamics,2012,70(3):1903-1918
33 Gantoi FM,Brown MA,Shabana AA.Finite element modeling of the contact geometry and deformation in biomechanics.Journal of Computational and Nonlinear Dynamics,2013,8(4):041013
34 Shabana AA.ANCF reference node for multibody system analysis.Proceedings of the Institution of Mechanical Engineers Part K:Journal of Multibody Dynamics,2015,229(1):109-112
35 Shabana AA.Dynamics of Multibody Systems.Cambridge:Cambridge University Press,2005.
36田强.基于绝对节点坐标方法的柔性多体系统动力学研究与应用.[博士论文].武汉:华中科技大学,2009(Tian Qiang,Flexible multibody dynamics research and application based on the absolute nodal coordinate method.[PhD Thesis].Wuhan:Huazhong University of Science and Technology,2009(in Chinese))
37 Yu ZQ,Shabana AA.Mixed-coordinate ancf rectangular finite element.Journal of Computational and Nonlinear Dynamics,2015,10(6):061003
38 Shabana AA,Zaazaa KE,Sugiyama H.Railroad Vehicle Dynamics:A Computational Approach.Boca Raton:CRC Press.2007:145-147
2章孝顺,章定国,陈思佳等.基于绝对节点坐标法的大变形柔性梁几种动力学模型研究.物理学报,2016,65(9):094501(Zhang Xiaoshun,Zhang Dingguo,Chen Sijia,et al.Several dynamic models of a large deformation flexible beam based on the absolute nodal coordinate formulation.Acta Physica Sinica,2016,65(9):094501(in Chinese))
3 Gerstmayr J,Sugiyama H,Mikkola A.Review on the absolute nodal coordinate formulation for large deformation analysis of multi-body systems.Journal of Computational and Nonlinear Dynamics,2013,8(3):031016
4田强,张云清,陈立平等.柔性多体系统动力学绝对节点坐标方法研究进展.力学进展,2010,40(2):189-202(Tian Qiang,Zhang Yunqing,Chen Liping,et al.Review of the advances in absolute nodal coordinate formulation in flexible multi-body system dynamics.Advances in Mechanics,2010,40(2):189-202(in Chinese))
5张越,赵阳,谭春林等.ANCF索梁单元应变耦合问题与模型解耦.力学学报,2016,48(6):1406-1415(Zhang Yue,Zhao Yang,Tan Chunlin,et al.The strain coupling problem and model decoupling of ANCF cable/beam element.Chinese Journal of Theoretical and Applied Mechanics,2016,48(6):1406-1415(in Chinese))
6岳宝增,于嘉瑞,吴文军.多储液腔航天器刚液耦合动力学与复合控制.力学学报,2017,49(2):390-396(Yue Baozeng,Yu Jiarui,Wu Wenjun.Rigid and liquid coupling dynamics and hybrid control of spacecraft with multiple propellant tanks.Chinese Journal of Theoretical and Applied Mechanics,2017,49(2):390-396(in Chinese))
7胡景晨,王天舒.一种O(n)算法复杂度的递推绝对节点坐标法研究.力学学报,2016,48(5):1172-1183(Hu Jingchen,Wang Tianshu.A recursive absolute nodal coordinate formulation with O(n)algorithm complexity.Chinese Journal of Theoretical and Applied Mechanics,2016,48(5):1172-1183(in Chinese))
8刘铖,田强,胡海岩.基于绝对节点坐标的多柔体系统动力学高效计算方法.力学学报,2010,42(6):1197-1205(Liu Cheng,Tian Qiang,Hu Haiyan.Efficient computational method for dynamics of flexible multibody systems based on absolute nodal coordinate.Chinese Journal of Theoretical and Applied Mechanics,2010,42(6):1197-1205(in Chinese))
9 García Vallejo D,Mayo J,Escalona JL,et al.Efficient evaluation of the elastic forces and the jacobian in the absolute nodal coordinate formulation.Nonlinear Dynamic,2004,35:313-329
10 Shabana AA.ANCF tire assembly model for multibody system applications.Journal of Computational and Nonlinear Dynamics,2015,10(2):024504
11 Omar MA,Shabana AA,Mikkola A,et al.Multibody system modeling of leaf springs.Journal of Vibration and Control,2004,10(11):1601-1638
12 Yu ZQ,Liu YG,Tinsley B,et al.Integration of geometry and analysis for vehicle system applications:Continuum-based leaf spring and tire assembly.Journal of Computational and Nonlinear Dynamics,2015 11(3):031011
13 Dufva K,Shabana AA.Analysis of thin plate structures using the absolute nodal coordinate formulation.Proceedings of the Institution of Mechanical Engineers Part K:Journal of Multibody Dynamics,2005,219(4):345-355
14 Wang JS,Li ZK,Jiang QB.The analysis of composite leaf spring by finite element method and experimental measurements//FISITA2012 World Automotive Congress Proceedings-Volume 7:Vehicle Design and Testing,Beijing,China,2012-11-27,803
15 Zhou ZF,Guo WF,Shen TJ,et al.Research and application on dynamic stiffness of leaf spring.FISITA 2012 World Automotive Congress Proceedings-Volume 10:Chassis Systems and Integration Technology Beijing,China,2012-11-27,2012,111
16 Nunney MJ.Light and Heavy Vehicle Technology.Oxford:Society for Neuroscience,2006:352
17 Dunaevskii B.Calculation of solid beams and flat springs(leaf springs)of varying cross-section.Soviet Engineering Research,1981,61(4):36-38
18周孔亢,陆建辉,侯永涛等.基于RecurDyn的钢板弹簧动力学模型的建立与参数辨识.机械工程学报,2014,50(4):128-134(Zhou Kongkang,Lu Jianhui,Hou Yongtao,et al.Dynamics modeling and parameter identification of leaf spring based on RecurDyn.Journal of Mechanical Engineering,2014,50(4):128-134(in Chinese))
19段亮,杨树凯,宋传学等.平衡悬架钢板弹簧动态特性的研究.机械工程学报,2016,52(6):153-158(Duan Liang,Yang Shukai,Song Chuanxue,et al.Research on dynamic characteristics of the tandem suspension leaf spring.Journal of Mechanical Engineering,2016,52(6):153-158(in Chinese))
20 Wriggers P,Zavarise G.On contact between three-dimensional beams undergoing large deflections.Communications in Numerical Methods in Engineering,1997,13(6):429-438
21 Zavarise G,Wriggers P.Contact with friction between beams in 3-Dspace.International Journal for Numerical Methods in Engineering,2000,49:977-1006
22 Litewka P,Wriggers P.Contact between 3D beams with rectangular cross-sections.International Journal for Numerical Methods in Engineering,2002,53:2019-2041
23 Litewka P.Hermite polynomial smoothing in beam-tobema frictional contact.Computational Mechanics,2007,40:815-826
24 Litewka P.Smooth frictional contact between beams in 3D//Nackenhorst U,Wriggers P.(eds.)IUTAM Symposium on Computational Contact Mechanics.IUTAM Bookseries.Dordrecht:Springer,2007,157-176
25 Konyukhov A,Schweizerhof K.Geometrically exact covariant approach for contact between curves.Computer Methods in Applied Mechanics and Engineering,2010,199:2510-2531
26 Litewka P.Multiple-point beam-to-beam contact finite element//19th International Conference on Computer Methods in Mechanics,Warsaw,Poland,9-12,May,2011
27 Durville D.Contact-friction modeling within elastic beam assemblies:an application to knot tightening.Computational Mechanics,2012,49(6):687-707
28 Wang QT,Tian Q,Hu HY.Dynamic simulation of frictional contacts of thin beams during large overall motions via absolute nodal coordinate formulation.Nonlinear Dynamics,2014,77(4):1411-1425
29 Wang QT,Tian Q,Hu HY.Dynamic simulation of frictional multizone contacts of thin beams.Nonlinear Dynamics,2016,83(4):1919-1937
30 Zahavi E.Analysis of a contact problem in leaf springs.Mechanics Research Communications,1992,19(1):21-27
31 Shabana AA.Computational Continuum Mechanics.London:Cambridge University Press.2012:46-93
32 Liu C,Tian Q,Hu HY.New spatial curved beam and cylindrical shell elements of gradient-deficient absolute nodal coordinate formulation.Nonlinear Dynamics,2012,70(3):1903-1918
33 Gantoi FM,Brown MA,Shabana AA.Finite element modeling of the contact geometry and deformation in biomechanics.Journal of Computational and Nonlinear Dynamics,2013,8(4):041013
34 Shabana AA.ANCF reference node for multibody system analysis.Proceedings of the Institution of Mechanical Engineers Part K:Journal of Multibody Dynamics,2015,229(1):109-112
35 Shabana AA.Dynamics of Multibody Systems.Cambridge:Cambridge University Press,2005.
36田强.基于绝对节点坐标方法的柔性多体系统动力学研究与应用.[博士论文].武汉:华中科技大学,2009(Tian Qiang,Flexible multibody dynamics research and application based on the absolute nodal coordinate method.[PhD Thesis].Wuhan:Huazhong University of Science and Technology,2009(in Chinese))
37 Yu ZQ,Shabana AA.Mixed-coordinate ancf rectangular finite element.Journal of Computational and Nonlinear Dynamics,2015,10(6):061003
38 Shabana AA,Zaazaa KE,Sugiyama H.Railroad Vehicle Dynamics:A Computational Approach.Boca Raton:CRC Press.2007:145-147