刚—柔耦合问题与空间多杆柔性机械臂的动力学建模理论研究
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摘要
本文对柔性多体系统中的刚-柔耦合问题以及空间多杆柔性机械臂的动力学建模理论进行了研究。
在柔性多体系统动力学中,传统的零次近似耦合模型在建模过程中直接套用了结构动力学中的小变形假设,忽略了大位移刚体运动与小位移弹性变形之间的高阶耦合项。当系统的大范围运动为高速时,零次近似耦合模型的计算结果将出现发散。为此,国内外众多学者对刚-柔耦合问题进行了研究,并提出了较为精确的一次近似耦合模型。一次近似耦合模型考虑了柔性体大范围刚体运动与其自身变形之间的耦合,解决了零次近似耦合模型的“动力刚化”问题,然而其在建模过程中采用了小变形假设,对动力学方程进行了简化,因此无法处理柔性体的大变形问题。为此,本文基于一次近似耦合模型的理论基础,提出了更为精确的一次耦合模型。
本文以做大范围旋转运动的中心刚体-柔性悬臂梁系统为例对刚-柔耦合建模理论进行了研究,考虑柔性梁横向弯曲变形以及纵向伸长变形,且在纵向位移中计及由于横向变形而引起的纵向缩短项,即非线性二阶耦合变形项。采用假设模态法描述变形,运用第二类Lagrange方程建立了系统的一次耦合动力学方程。在方程中保留了一次近似耦合模型中所省略的二阶耦合变形项的高阶量,由此得到的动力学方程不仅能适用于柔性梁的小变形问题,也同样适用于大变形问题。在此理论基础上,将模型拓展到中心刚体-变截面梁系统以及梁上任意位置带有附加质量的中心刚体-悬臂梁系统。
在一次近似耦合模型的基础上,对做大范围旋转运动的中心刚体-悬臂梁系统的横向弯曲固有频率进行了分析。对动力学方程进行了无量纲化,从固有频率的角度解释了零次近似耦合模型发散的机理。研究了悬臂梁上附加质量的位置、质量、转动惯量对系统固有频率的影响以及变截面梁的结构变化对系统固有频率的影响。
在一次耦合模型的基础上,对做大范围旋转运动的中心刚体-悬臂梁系统的动力学特性进行了研究,比较了一次耦合模型、一次近似耦合模型、零次近似耦合模型之间的差异,并验证了一次耦合模型在大变形条件下的适用性。
基于一次耦合模型的理论基础,对由n个柔性杆和n个柔性铰组成的空间多杆柔性机械臂的动力学建模理论进行了研究。不但考虑了杆件的拉伸、弯曲以及扭转变形,且在纵向位移中计及由于横向变形而引起的纵向缩短项,即非线性二阶耦合变形项。将柔性铰的柔性简化为线弹性扭簧,考虑柔性铰的质量,运用递推拉格朗日动力学方法得到空间多杆柔性机械臂的刚-柔耦合动力学方程。在此基础上编制了通用的空间柔性机械臂的动力学仿真软件,并对平面单杆柔性机械臂、空间多杆柔性机械臂、平面柔性折杆、平面柔性曲杆进行了动力学仿真计算。通过数值仿真算例验证了理论模型的正确性,说明了在建模过程中考虑杆件的“动力刚化”效应、扭转效应、以及铰的柔性的重要性。
In this dissertation, the rigid-flexible coupling problem of the flexible multibody system and the dynamic modeling theory of multi-link spatial flexible manipulator arms are studied.
In the flexible multibody system dynamics, the traditional zero order approximate coupling model applies mechanically the supposes of small deformation in structure dynamics, but ignores the high order coupling term between the large displacement rigidbody motion and the small displacement elastic deformation. When the large overall motion of the system is at high speed, the result of the zero order approximate coupling model will be divergent. So, scholars pay more attention to the rigid-flexible coupling problem, and a more accurate model, which is called the first order approximate coupling model, is proposed. The coupling effect between the large rigidbody motion and its own deformation is considered in the first order approximate coupling model. But it uses the supposes of small deformation in the modeling to simplify the dynamical equation, and can't deal with the large deformation problems. Thus, a more accurate model, which is called the first order coupling model, is proposed based on the theory of the first order approximate coupling model in this dissertation.
To study the modeling theory of the rigid-flexible problem, a hub-beam system with large overall motion is researched in the dissertation. Both the transversal deformation and the longitudinal deformation of the flexible beam are considered. And the non-linear coupling term, also known as the longitudinal shortening caused by transversal deformation, is considered in the total longitudinal deformation. The approach of assumed modes is used to describe the deformation of the beam. The first order coupling dynamical equations of the system are established via employing the second kind of Lagrange's equation. The high order quantities of the non-linear coupling term, which are omitted in the first order approximate coupling model, are retained in the equations. Thus, the dynamical equations can be used not only in the small deformation problems, but also in the large deformation problems. Based on this theory, the model can be extended to the hub-variable section beam system and the hub-beam system with a payload on the arbitrary position of the beam.
The natural frequencies of the hub-beam system with large overall motion are studied though the transversal bending vibration analysis based on the first order approximate coupling model. For convenience of discussion, dimensionless parameters are used in the dynamical equations. The divergence mechanism of the zero order approximate coupling model is explained from the angle of the natural frequencies. Then, the influences of the position, mass, rotary inertia of the payload and the structure change of the variable section beam on the natural frequencies of the system are studied.
The dynamic characteristics of the hub-beam system with large overall motion are studied based on the first order coupling model. The differences among the first order coupling model, the first order approximate coupling model and the zero order approximate coupling model are compared. And the applicability of the first order coupling model under large deformation situation is verified.
Based on the theory of the first order coupling model, the dynamic modeling theory of the multi-link spatial flexible-link flexible-joint manipulator arms is investigated. The system considered here is an n-flexible-link manipulator driven by n DC-motors through n revolute flexible-joints. The flexibility of each flexible joint is modeled as a linearly elastic torsional spring, and the mass of the joint is also considered. For the link's flexibility, both the stretching deformation, bending deformation and the torsional deformation are included. The complete governing equations of motion of the system are derived via the Lagrangian equations. In the modeling the nonlinear description of the deformation field of the flexible link is adopted, and thus the dynamic stiffening effects are captured. Based on this model, a general-purpose software package for dynamic simulation of multi-link spatial flexible manipulator arms is developed. Several illustrative examples including a planar single-link flexible manipulator, two multi-link spatial flexible manipulators, a planar flexible fold link, and a flexible curve link are given to validate the algorithm presented and to indicate that the dynamic stiffening effects, the torsional effects, the flexibility of the structure all have significant influence on the dynamics of the manipulator.
在柔性多体系统动力学中,传统的零次近似耦合模型在建模过程中直接套用了结构动力学中的小变形假设,忽略了大位移刚体运动与小位移弹性变形之间的高阶耦合项。当系统的大范围运动为高速时,零次近似耦合模型的计算结果将出现发散。为此,国内外众多学者对刚-柔耦合问题进行了研究,并提出了较为精确的一次近似耦合模型。一次近似耦合模型考虑了柔性体大范围刚体运动与其自身变形之间的耦合,解决了零次近似耦合模型的“动力刚化”问题,然而其在建模过程中采用了小变形假设,对动力学方程进行了简化,因此无法处理柔性体的大变形问题。为此,本文基于一次近似耦合模型的理论基础,提出了更为精确的一次耦合模型。
本文以做大范围旋转运动的中心刚体-柔性悬臂梁系统为例对刚-柔耦合建模理论进行了研究,考虑柔性梁横向弯曲变形以及纵向伸长变形,且在纵向位移中计及由于横向变形而引起的纵向缩短项,即非线性二阶耦合变形项。采用假设模态法描述变形,运用第二类Lagrange方程建立了系统的一次耦合动力学方程。在方程中保留了一次近似耦合模型中所省略的二阶耦合变形项的高阶量,由此得到的动力学方程不仅能适用于柔性梁的小变形问题,也同样适用于大变形问题。在此理论基础上,将模型拓展到中心刚体-变截面梁系统以及梁上任意位置带有附加质量的中心刚体-悬臂梁系统。
在一次近似耦合模型的基础上,对做大范围旋转运动的中心刚体-悬臂梁系统的横向弯曲固有频率进行了分析。对动力学方程进行了无量纲化,从固有频率的角度解释了零次近似耦合模型发散的机理。研究了悬臂梁上附加质量的位置、质量、转动惯量对系统固有频率的影响以及变截面梁的结构变化对系统固有频率的影响。
在一次耦合模型的基础上,对做大范围旋转运动的中心刚体-悬臂梁系统的动力学特性进行了研究,比较了一次耦合模型、一次近似耦合模型、零次近似耦合模型之间的差异,并验证了一次耦合模型在大变形条件下的适用性。
基于一次耦合模型的理论基础,对由n个柔性杆和n个柔性铰组成的空间多杆柔性机械臂的动力学建模理论进行了研究。不但考虑了杆件的拉伸、弯曲以及扭转变形,且在纵向位移中计及由于横向变形而引起的纵向缩短项,即非线性二阶耦合变形项。将柔性铰的柔性简化为线弹性扭簧,考虑柔性铰的质量,运用递推拉格朗日动力学方法得到空间多杆柔性机械臂的刚-柔耦合动力学方程。在此基础上编制了通用的空间柔性机械臂的动力学仿真软件,并对平面单杆柔性机械臂、空间多杆柔性机械臂、平面柔性折杆、平面柔性曲杆进行了动力学仿真计算。通过数值仿真算例验证了理论模型的正确性,说明了在建模过程中考虑杆件的“动力刚化”效应、扭转效应、以及铰的柔性的重要性。
In this dissertation, the rigid-flexible coupling problem of the flexible multibody system and the dynamic modeling theory of multi-link spatial flexible manipulator arms are studied.
In the flexible multibody system dynamics, the traditional zero order approximate coupling model applies mechanically the supposes of small deformation in structure dynamics, but ignores the high order coupling term between the large displacement rigidbody motion and the small displacement elastic deformation. When the large overall motion of the system is at high speed, the result of the zero order approximate coupling model will be divergent. So, scholars pay more attention to the rigid-flexible coupling problem, and a more accurate model, which is called the first order approximate coupling model, is proposed. The coupling effect between the large rigidbody motion and its own deformation is considered in the first order approximate coupling model. But it uses the supposes of small deformation in the modeling to simplify the dynamical equation, and can't deal with the large deformation problems. Thus, a more accurate model, which is called the first order coupling model, is proposed based on the theory of the first order approximate coupling model in this dissertation.
To study the modeling theory of the rigid-flexible problem, a hub-beam system with large overall motion is researched in the dissertation. Both the transversal deformation and the longitudinal deformation of the flexible beam are considered. And the non-linear coupling term, also known as the longitudinal shortening caused by transversal deformation, is considered in the total longitudinal deformation. The approach of assumed modes is used to describe the deformation of the beam. The first order coupling dynamical equations of the system are established via employing the second kind of Lagrange's equation. The high order quantities of the non-linear coupling term, which are omitted in the first order approximate coupling model, are retained in the equations. Thus, the dynamical equations can be used not only in the small deformation problems, but also in the large deformation problems. Based on this theory, the model can be extended to the hub-variable section beam system and the hub-beam system with a payload on the arbitrary position of the beam.
The natural frequencies of the hub-beam system with large overall motion are studied though the transversal bending vibration analysis based on the first order approximate coupling model. For convenience of discussion, dimensionless parameters are used in the dynamical equations. The divergence mechanism of the zero order approximate coupling model is explained from the angle of the natural frequencies. Then, the influences of the position, mass, rotary inertia of the payload and the structure change of the variable section beam on the natural frequencies of the system are studied.
The dynamic characteristics of the hub-beam system with large overall motion are studied based on the first order coupling model. The differences among the first order coupling model, the first order approximate coupling model and the zero order approximate coupling model are compared. And the applicability of the first order coupling model under large deformation situation is verified.
Based on the theory of the first order coupling model, the dynamic modeling theory of the multi-link spatial flexible-link flexible-joint manipulator arms is investigated. The system considered here is an n-flexible-link manipulator driven by n DC-motors through n revolute flexible-joints. The flexibility of each flexible joint is modeled as a linearly elastic torsional spring, and the mass of the joint is also considered. For the link's flexibility, both the stretching deformation, bending deformation and the torsional deformation are included. The complete governing equations of motion of the system are derived via the Lagrangian equations. In the modeling the nonlinear description of the deformation field of the flexible link is adopted, and thus the dynamic stiffening effects are captured. Based on this model, a general-purpose software package for dynamic simulation of multi-link spatial flexible manipulator arms is developed. Several illustrative examples including a planar single-link flexible manipulator, two multi-link spatial flexible manipulators, a planar flexible fold link, and a flexible curve link are given to validate the algorithm presented and to indicate that the dynamic stiffening effects, the torsional effects, the flexibility of the structure all have significant influence on the dynamics of the manipulator.
引文
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