刚—柔耦合系统动力学建模理论与仿真技术研究
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摘要
本文对刚柔耦合系统动力学建模理论与仿真技术进行研究。
柔性多体系统的传统混合坐标方法直接采用了结构动力学中的小变形假设,忽略柔性体变形场中的高阶耦合变形项,故本质上是一种零次近似意义上的刚-柔耦合建模方法(MZOA)。1987年,Kane研究发现柔性悬臂梁高速转动时,采用零次近似建模方法将导致错误发散的结果。洪嘉振课题组从连续介质力学和分析动力学出发,考虑柔性梁变形位移的二阶耦合项,首先提出一种基于轴线积分的刚-柔耦合动力学建模方法(MBAI),它是一次近似意义上的建模方法,并且通过全物理仿真实验验证了这种耦合动力学模型的正确性。然而,由于MBAI采用沿整个轴线积分得到二次耦合项的思想,限制了其通用性。其后,课题组提出基于共旋坐标系的建模方法(MBCR),对柔性体的非线性变形场进行描述,虽然能解决上述问题,但是计算效率较低。
显然,评价一个柔性多体系统动力学模型的优劣的重要标准首先是它的可靠性,要能正确的反映刚柔耦合动力学的本质。其次是它应该具有良好的数值性态和计算效率,便于计算机数值仿真的实现。为此,从工程需要出发,建立一个更加通用高效的刚-柔耦合动力学模型也就成为本文的研究目标。
本文提出了一种基于单元耦合变形的刚-柔耦合系统动力学建模方法(MBECD),采用三类参考坐标系:惯性坐标系、浮动坐标系和单元平移坐标系对柔性体上任意一点的位形进行描述。通过浮动坐标系实现对大范围刚性运动和弹性变形的分离,通过单元平移坐标系来描述柔性体的耦合变形。首先根据连续介质力学的基本原理,推导出单元平移坐标系上笛卡尔变形坐标和非笛卡尔变形之间的关系;单元上任意一点的变形可以通过有限元形函数对非笛卡尔变形节点坐标插值得到;然后把非笛卡尔变形节点坐标转化为笛卡尔变形节点坐标表示,最后转化到浮动坐标系上组集,从而实现了对柔性体非线性耦合变形场的描述。由于本文方法中的耦合变形项只和其所在单元的节点坐标相关,摆脱了MBAI中任意一点的耦合变形项与该点到边界之间所有单元相关的局限性,为应用于一般形状柔性体的刚-柔耦合动力学分析提供了可能。
利用本文提出的建模方法,对柔性体的刚-柔耦合动力学进行研究。基于Hamiltion原理和Jourdan速度变分原理,建立了做大范围运动的柔性梁和薄板的刚-柔耦合动力学方程。通过对大范围运动已知和大范围运动自由的柔性梁和薄板的数值仿真算例,验证了本文提出的方法能够很好的应用于柔性体刚-柔耦合动力学分析,不仅不会出现零次近似模型数值发散的问题,而且具有良好的单元收敛性和较高的计算效率。本文还通过对做大范围运动的平面曲梁和不规则形状薄板的动力学仿真,说明本文建模方法具有通用性,为复杂柔性体刚柔耦合动力学程式化的研究打下基础。
为了进一步提高计算效率,便于工程应用,本文还研究了柔性体的刚-柔耦合动力学模型简化问题。分别讨论了耦合变形项对惯性力与弹性力的贡献,分析了它们对刚-柔耦合动力学行为的影响。通过研究指出当采用笛卡尔变形坐标描述时,如果在计算弹性力的时候考虑了耦合变形影响,无论在计算惯性力时是否考虑耦合变形影响,都可以得到稳定收敛的结果。反之,如果在计算弹性力时忽略了耦合变形影响,无论在计算惯性力时是否考虑耦合变形影响,当大范围运动的速度较高时,将会得到错误的发散的结果。因此,通过忽略耦合变形对质量分布的影响,只保留耦合变形对弹性力的影响,可实现对动力学方程的简化。
In this dissertation, the modeling theory and simulation technique of rigid-flexible coupling systems dynamics are investigated.
The traditional hybrid-coordinate method of flexible multi-body systems is essentially a coupling modeling method of zero order approximation(MZOA), in which small elastic deformation assumption in structure dynamics is adopted and the high-order coupling terms of flexible body deformation are ignored. In 1987, Kane et al studied the dynamic performance of a flexible beam with rotation motion and the results indicate that the MZOA fails to describe dynamics of the system when the system is in high rotating velocity. Recently, a modeling method of rigid-flexible dynamics based on axial integral (MBAI) was developed by the group of professor Hong Jia-zhen based on the theory of continuum medium mechanics and the theory of analysis dynamics, which was a first-order approximation coupling model method, and second-order coupling term of deformation of flexible body were taken into count. The validity of the MBAI was verified by physical experiments. However, for the coupling deformation term of an arbitrary point in flexible beam is in integral from the bottom point when using the MBAI, which is a limitation for MBAI to investigate the rigid-flexible coupling dynamics of general flexible bodies. Later, the group proposed a modeling method base on Co-rotational coordinates (MBCR) to describe the non-linear deformation field of flexible bodied, which could break the limitation of MBAI. However, the computational efficiency of MBCR is poor.
The first standard to evaluate a dynamical model is whether it could satisfy the request of reliability to reflect the essence the rigid-flexible coupling dynamics exactly. Moreover, it should have good numerical character and computational efficiency, which makes it easy to realize numerical simulation in computer. Therefore, it become the object of this dissertation to established a more general and efficient dynamic model of rigid-flexible coupling systems for the requirement of the engineering.
In this dissertation, the dynamic modeling method of rigid-flexible coupling system based on element coupling deformation (MBECD) is proposed, in which three reference frames (global frame, floating frame, and element translate frame) are used to describe the configuration of arbitrary point in the flexible body. The floating frame is used to realizes the separation of the rigid motion and elastic deformation, and element translate frame is used to describe nonlinear coupling deformation. Firstly, the relationship of Cartesian deformation coordinate and non-Cartesian deformation coordinate is derived based on principle of continuum mechanics; the deformation of a arbitrary point in the element can be interpolated with non-Cartesian nodal deformation coordinates through element shape functions; non-Cartesian nodal deformation coordinates are transferred into Cartesian nodal deformation coordinates and allocated in floating frame, then the nonlinear coupling deformation field is obtained. In MBAI method, the coupling deformation terms of an arbitrary point are related with all the elements between the local element and the bottom. However, in MBECD prospected in this dissertation, the coupling deformation terms only relate with the located element nodal coordinates, which makes it possible for MBECD to investigate general shape flexible bodies.
The rigid-flexible coupling dynamics of flexible bodies are studied by using MBECD prospected in this dissertation. The dynamic equations of flexible beam and thin plate undergoing overall large motions are derived based on Hamiltion principle and Jourdan velocity variation principle respectively. The simulation examples of flexible beam and thin plate undergoing prescribed motions and free motions are investigated to verify the validity of the MBECD to be applied in rigid-flexible coupling dynamic analysis of flexible bodies, which will not suffer the numerical divergence as ZOAC does. Simulation results also show the MBECD has good element convergence and relative high computational efficiency. Moreover, the simulation examples of curve beam and irregular plate are given to verify that the MBECD can be used to study the rigid-flexible coupling dynamics of complex shape flexible bodies.
To improve the computational efficiency, and be applied in engineering conveniently, the simplification of rigid-flexible coupling dynamic equations of flexible bodies undergoing overall large motions are studied. The effects of coupling deformation term on the dynamic behavior of rigid-flexible systems are investigated. Research results indicate that stable solution can be obtained with Cartesian deformation coordinates when the effect of coupling deformation on calculating elastic forces is considered, regardless of the effect of coupling deformation on calculating inertia forces is ignored or not. On other hand, unstable solution can be obtained with high speed motion when the effect of coupling deformation on calculating elastic forces is ignored, regardless of the effect of coupling deformation on calculating inertia forces is ignored or not. It is possible to reduce the dynamic model of rigid-flexible systems through ignoring the effects of coupling deformation on calculating inertia forces and considering the effects of coupling deformation on calculating elastic forces when the Cartesian deformation coordinates are used.
柔性多体系统的传统混合坐标方法直接采用了结构动力学中的小变形假设,忽略柔性体变形场中的高阶耦合变形项,故本质上是一种零次近似意义上的刚-柔耦合建模方法(MZOA)。1987年,Kane研究发现柔性悬臂梁高速转动时,采用零次近似建模方法将导致错误发散的结果。洪嘉振课题组从连续介质力学和分析动力学出发,考虑柔性梁变形位移的二阶耦合项,首先提出一种基于轴线积分的刚-柔耦合动力学建模方法(MBAI),它是一次近似意义上的建模方法,并且通过全物理仿真实验验证了这种耦合动力学模型的正确性。然而,由于MBAI采用沿整个轴线积分得到二次耦合项的思想,限制了其通用性。其后,课题组提出基于共旋坐标系的建模方法(MBCR),对柔性体的非线性变形场进行描述,虽然能解决上述问题,但是计算效率较低。
显然,评价一个柔性多体系统动力学模型的优劣的重要标准首先是它的可靠性,要能正确的反映刚柔耦合动力学的本质。其次是它应该具有良好的数值性态和计算效率,便于计算机数值仿真的实现。为此,从工程需要出发,建立一个更加通用高效的刚-柔耦合动力学模型也就成为本文的研究目标。
本文提出了一种基于单元耦合变形的刚-柔耦合系统动力学建模方法(MBECD),采用三类参考坐标系:惯性坐标系、浮动坐标系和单元平移坐标系对柔性体上任意一点的位形进行描述。通过浮动坐标系实现对大范围刚性运动和弹性变形的分离,通过单元平移坐标系来描述柔性体的耦合变形。首先根据连续介质力学的基本原理,推导出单元平移坐标系上笛卡尔变形坐标和非笛卡尔变形之间的关系;单元上任意一点的变形可以通过有限元形函数对非笛卡尔变形节点坐标插值得到;然后把非笛卡尔变形节点坐标转化为笛卡尔变形节点坐标表示,最后转化到浮动坐标系上组集,从而实现了对柔性体非线性耦合变形场的描述。由于本文方法中的耦合变形项只和其所在单元的节点坐标相关,摆脱了MBAI中任意一点的耦合变形项与该点到边界之间所有单元相关的局限性,为应用于一般形状柔性体的刚-柔耦合动力学分析提供了可能。
利用本文提出的建模方法,对柔性体的刚-柔耦合动力学进行研究。基于Hamiltion原理和Jourdan速度变分原理,建立了做大范围运动的柔性梁和薄板的刚-柔耦合动力学方程。通过对大范围运动已知和大范围运动自由的柔性梁和薄板的数值仿真算例,验证了本文提出的方法能够很好的应用于柔性体刚-柔耦合动力学分析,不仅不会出现零次近似模型数值发散的问题,而且具有良好的单元收敛性和较高的计算效率。本文还通过对做大范围运动的平面曲梁和不规则形状薄板的动力学仿真,说明本文建模方法具有通用性,为复杂柔性体刚柔耦合动力学程式化的研究打下基础。
为了进一步提高计算效率,便于工程应用,本文还研究了柔性体的刚-柔耦合动力学模型简化问题。分别讨论了耦合变形项对惯性力与弹性力的贡献,分析了它们对刚-柔耦合动力学行为的影响。通过研究指出当采用笛卡尔变形坐标描述时,如果在计算弹性力的时候考虑了耦合变形影响,无论在计算惯性力时是否考虑耦合变形影响,都可以得到稳定收敛的结果。反之,如果在计算弹性力时忽略了耦合变形影响,无论在计算惯性力时是否考虑耦合变形影响,当大范围运动的速度较高时,将会得到错误的发散的结果。因此,通过忽略耦合变形对质量分布的影响,只保留耦合变形对弹性力的影响,可实现对动力学方程的简化。
In this dissertation, the modeling theory and simulation technique of rigid-flexible coupling systems dynamics are investigated.
The traditional hybrid-coordinate method of flexible multi-body systems is essentially a coupling modeling method of zero order approximation(MZOA), in which small elastic deformation assumption in structure dynamics is adopted and the high-order coupling terms of flexible body deformation are ignored. In 1987, Kane et al studied the dynamic performance of a flexible beam with rotation motion and the results indicate that the MZOA fails to describe dynamics of the system when the system is in high rotating velocity. Recently, a modeling method of rigid-flexible dynamics based on axial integral (MBAI) was developed by the group of professor Hong Jia-zhen based on the theory of continuum medium mechanics and the theory of analysis dynamics, which was a first-order approximation coupling model method, and second-order coupling term of deformation of flexible body were taken into count. The validity of the MBAI was verified by physical experiments. However, for the coupling deformation term of an arbitrary point in flexible beam is in integral from the bottom point when using the MBAI, which is a limitation for MBAI to investigate the rigid-flexible coupling dynamics of general flexible bodies. Later, the group proposed a modeling method base on Co-rotational coordinates (MBCR) to describe the non-linear deformation field of flexible bodied, which could break the limitation of MBAI. However, the computational efficiency of MBCR is poor.
The first standard to evaluate a dynamical model is whether it could satisfy the request of reliability to reflect the essence the rigid-flexible coupling dynamics exactly. Moreover, it should have good numerical character and computational efficiency, which makes it easy to realize numerical simulation in computer. Therefore, it become the object of this dissertation to established a more general and efficient dynamic model of rigid-flexible coupling systems for the requirement of the engineering.
In this dissertation, the dynamic modeling method of rigid-flexible coupling system based on element coupling deformation (MBECD) is proposed, in which three reference frames (global frame, floating frame, and element translate frame) are used to describe the configuration of arbitrary point in the flexible body. The floating frame is used to realizes the separation of the rigid motion and elastic deformation, and element translate frame is used to describe nonlinear coupling deformation. Firstly, the relationship of Cartesian deformation coordinate and non-Cartesian deformation coordinate is derived based on principle of continuum mechanics; the deformation of a arbitrary point in the element can be interpolated with non-Cartesian nodal deformation coordinates through element shape functions; non-Cartesian nodal deformation coordinates are transferred into Cartesian nodal deformation coordinates and allocated in floating frame, then the nonlinear coupling deformation field is obtained. In MBAI method, the coupling deformation terms of an arbitrary point are related with all the elements between the local element and the bottom. However, in MBECD prospected in this dissertation, the coupling deformation terms only relate with the located element nodal coordinates, which makes it possible for MBECD to investigate general shape flexible bodies.
The rigid-flexible coupling dynamics of flexible bodies are studied by using MBECD prospected in this dissertation. The dynamic equations of flexible beam and thin plate undergoing overall large motions are derived based on Hamiltion principle and Jourdan velocity variation principle respectively. The simulation examples of flexible beam and thin plate undergoing prescribed motions and free motions are investigated to verify the validity of the MBECD to be applied in rigid-flexible coupling dynamic analysis of flexible bodies, which will not suffer the numerical divergence as ZOAC does. Simulation results also show the MBECD has good element convergence and relative high computational efficiency. Moreover, the simulation examples of curve beam and irregular plate are given to verify that the MBECD can be used to study the rigid-flexible coupling dynamics of complex shape flexible bodies.
To improve the computational efficiency, and be applied in engineering conveniently, the simplification of rigid-flexible coupling dynamic equations of flexible bodies undergoing overall large motions are studied. The effects of coupling deformation term on the dynamic behavior of rigid-flexible systems are investigated. Research results indicate that stable solution can be obtained with Cartesian deformation coordinates when the effect of coupling deformation on calculating elastic forces is considered, regardless of the effect of coupling deformation on calculating inertia forces is ignored or not. On other hand, unstable solution can be obtained with high speed motion when the effect of coupling deformation on calculating elastic forces is ignored, regardless of the effect of coupling deformation on calculating inertia forces is ignored or not. It is possible to reduce the dynamic model of rigid-flexible systems through ignoring the effects of coupling deformation on calculating inertia forces and considering the effects of coupling deformation on calculating elastic forces when the Cartesian deformation coordinates are used.
引文
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