This paper discusses the solvability of quaternion matrix equation A *X *B *±B X A=D and obtains it's general explicit solutions in terms of A,B,D and their Moore Penrose inverses.
There are many achievements in the field of analytical mechanics, such as Lagrange Equation, Hamilton’s Principle, Kane’s Equation. Compared to Newton–Euler mechanics, analytical mechanics have a wider range of app...There are many achievements in the field of analytical mechanics, such as Lagrange Equation, Hamilton’s Principle, Kane’s Equation. Compared to Newton–Euler mechanics, analytical mechanics have a wider range of applications and the formulation procedures are more mathematical. However, all existing methods of analytical mechanics were proposed based on some auxiliary variables. In this review, a novel analytical mechanics approach without the aid of Lagrange’s multiplier, projection, or any quasi or auxiliary variables is introduced for the central problem of mechanical systems. Since this approach was firstly proposed by Udwadia and Kalaba, it was called Udwadia–Kalaba Equation. It is a representation for the explicit expression of the equations of motion for constrained mechanical systems. It can be derived via the Gauss’ s principle, d’Alembert’s principle or extended d’Alembert’s principle. It is applicable to both holonomic and nonholonomic equality constraints, as long as they are linear with respect to the accelerations or reducible to be that form. As a result, the Udwadia–Kalaba Equation can be applied to a very broad class of mechani?cal systems. This review starts with introducing the background by a brief review of the history of mechanics. After that, the formulation procedure of Udwadia–Kalaba Equation is given. Furthermore, the comparisons of Udwadia–Kalaba Equation with Newton–Euler Equation, Lagrange Equation and Kane’s Equation are made, respectively. At last, three di erent types of examples are given for demonstrations.展开更多
基金the National Natural Science Foundation of China(1980 10 11)
文摘This paper discusses the solvability of quaternion matrix equation A *X *B *±B X A=D and obtains it's general explicit solutions in terms of A,B,D and their Moore Penrose inverses.
基金Supported by National Natural Science Foundation of China(Grant No.51705116)Anhui Provincial Science and Technology Major Project of China(Grant No.17030901036)Fundamental Research Funds for the Central Universities of China(Grant Nos.JZ2018HGBZ0096,JZ2018HGTA0217,JZ2018HGTB0261)
文摘There are many achievements in the field of analytical mechanics, such as Lagrange Equation, Hamilton’s Principle, Kane’s Equation. Compared to Newton–Euler mechanics, analytical mechanics have a wider range of applications and the formulation procedures are more mathematical. However, all existing methods of analytical mechanics were proposed based on some auxiliary variables. In this review, a novel analytical mechanics approach without the aid of Lagrange’s multiplier, projection, or any quasi or auxiliary variables is introduced for the central problem of mechanical systems. Since this approach was firstly proposed by Udwadia and Kalaba, it was called Udwadia–Kalaba Equation. It is a representation for the explicit expression of the equations of motion for constrained mechanical systems. It can be derived via the Gauss’ s principle, d’Alembert’s principle or extended d’Alembert’s principle. It is applicable to both holonomic and nonholonomic equality constraints, as long as they are linear with respect to the accelerations or reducible to be that form. As a result, the Udwadia–Kalaba Equation can be applied to a very broad class of mechani?cal systems. This review starts with introducing the background by a brief review of the history of mechanics. After that, the formulation procedure of Udwadia–Kalaba Equation is given. Furthermore, the comparisons of Udwadia–Kalaba Equation with Newton–Euler Equation, Lagrange Equation and Kane’s Equation are made, respectively. At last, three di erent types of examples are given for demonstrations.
