张量框架下刚体定点转动的运动学新描述  

New kinematics description of the rigid-body fixed-point rotation in a tensor frame

作　　者：刘占芳[1] 李俊峰[2] LIU ZhanFang;LI JunFeng(College of Aerospace Engineering,Chongqing University,Chongqing 400044,China;School of Aerospace Engineering,Tsinghua University,Beijing 100084,China)

机构地区：[1]重庆大学航空航天学院,重庆400044 [2]清华大学航天航空学院,北京100084

出　　处：《中国科学：物理学、力学、天文学》2022年第11期85-97,共13页Scientia Sinica Physica,Mechanica & Astronomica

基　　金：国家自然科学基金委员会与中国工程物理研究院联合基金(编号:U1830115);国家自然科学基金(编号:11772071);中央高校基本科研业务费(编号:2020CDXYHK6,2020CDJQY-Z004)资助项目。

摘　　要：现有理论力学没有给出完整的刚体定点转动的运动学描述,譬如不能确定多次定点转动合成的转角,只有关于欧拉角的角速度矢量公式,而两次转动合成的角加速度矢量形式也存在问题.本文论证了刚体定点转动的转角、角速度、角加速度本质上都不是矢量,而是二阶反对称张量,但可用对应的伪矢量来表达.建立了连续转动的合成是绝对参考系下相应正交张量的有序内积表达式,提出了角速度张量及角速度矢量、角加速度张量及角加速度矢量合成的一般形式.欧拉角描述刚体姿态的方法是借助三次有序定轴转动来等效于定点转动,利用一般的合成法则建立了关于欧拉角的角速度矢量和角加速度矢量的表达式,并且发现角速度矢量和角加速度矢量在莱沙尔(Resal)坐标系下具有十分简洁的形式.The current theoretical mechanics does not give a complete kinematic description of the fixed-point rotation of a rigid body. For example, the rotation angle comprising multiple fixed-point rotations is underdetermined, only the angular velocity vector about Euler angles is defined, and the angular acceleration vector composed of two rotations is an open problem. This paper demonstrates that the rotation angle, angle velocity, and angular acceleration of the rigid-body fixedpoint rotation are essentially not vectors but second-order antisymmetric tensors. The latter can be expressed by the corresponding pseudovectors. The composition of ordered rotations is established to be the ordered inner product of the corresponding orthogonal tensors in the absolute reference system. General composition rules of the angular velocity and angle acceleration tensors and their vectors are formulated. Euler angles describe the attitude of a rigid body through three orderly fixed-axis rotations that are equivalent to a fixed-point rotation. The angle velocity and angular acceleration vectors in terms of Euler angles are presented on the basis of the general composition rule, and their concise form is given in the Resal coordinate system.

关 键 词：角张量 角速度张量 角加速度张量 欧拉角 合成法则 

分 类 号：O313.3[理学—一般力学与力学基础]