{"title":"Specific Features of the Dynamics of the Rectilinear Motion of the Darboux Mechanism","authors":"S. N. Burian","doi":"10.1134/s1063454123040179","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The Darboux mechanism is considered. It is proved that this hinge mechanism allows the rotational movement of one link to be converted into (strictly) straight linear movement of its top <i>H</i>. The links of the Darboux mechanism can form geometric shapes such as triangles and squares (with diagonals drawn). In the “square”-shaped configuration of the mechanism, geometrically, branching may occur when the vertex <i>H</i> can move both along a straight line <i>L</i> and along a curve γ. In this case, the rank of the holonomic constraints of the system diminishes by one. For direct linear motion of the vertex <i>H</i>, the Lagrange equation of the second kind in terms of the point <i>H</i> coordinates is derived. The coefficients of this equation can be smoothly continued through a branching point. The “limiting” behavior of the reaction forces in the rods is studied when the mechanism moves to the branching point. An external force that does not do work on point <i>H</i> leads to unlimited reactions in the rods. The kinematics at the branching point is also studied. The inverse problem of dynamics at the point where the rank of the holonomic constraints is not a maximum is solvable. The Lagrange multipliers Λ<sub><i>i</i></sub> at the branching point are not defined in a unique way, but the corresponding forces acting on the mechanism vertices are uniquely defined.</p>","PeriodicalId":43418,"journal":{"name":"Vestnik St Petersburg University-Mathematics","volume":"34 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Vestnik St Petersburg University-Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s1063454123040179","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The Darboux mechanism is considered. It is proved that this hinge mechanism allows the rotational movement of one link to be converted into (strictly) straight linear movement of its top H. The links of the Darboux mechanism can form geometric shapes such as triangles and squares (with diagonals drawn). In the “square”-shaped configuration of the mechanism, geometrically, branching may occur when the vertex H can move both along a straight line L and along a curve γ. In this case, the rank of the holonomic constraints of the system diminishes by one. For direct linear motion of the vertex H, the Lagrange equation of the second kind in terms of the point H coordinates is derived. The coefficients of this equation can be smoothly continued through a branching point. The “limiting” behavior of the reaction forces in the rods is studied when the mechanism moves to the branching point. An external force that does not do work on point H leads to unlimited reactions in the rods. The kinematics at the branching point is also studied. The inverse problem of dynamics at the point where the rank of the holonomic constraints is not a maximum is solvable. The Lagrange multipliers Λi at the branching point are not defined in a unique way, but the corresponding forces acting on the mechanism vertices are uniquely defined.
摘要 本文研究了达尔布机构。达布机构的链节可以形成三角形和正方形（画对角线）等几何形状。在机构的 "正方形 "构型中，当顶点 H 既能沿直线 L 运动，又能沿曲线 γ 运动时，就会出现几何分支。对于顶点 H 的直接直线运动，可以根据点 H 的坐标推导出第二类拉格朗日方程。该方程的系数可通过分支点平滑延续。当机构运动到分支点时，研究了杆中反作用力的 "极限 "行为。不对 H 点做功的外力会导致杆中产生无限的反作用力。同时还研究了分支点的运动学。在整体约束条件的秩不是最大值的点上的动力学逆问题是可解的。分支点上的拉格朗日乘数Λi 并非以唯一方式定义，但作用在机构顶点上的相应力却是唯一定义的。
期刊介绍：
Vestnik St. Petersburg University, Mathematics is a journal that publishes original contributions in all areas of fundamental and applied mathematics. It is the prime outlet for the findings of scientists from the Faculty of Mathematics and Mechanics of St. Petersburg State University. Articles of the journal cover the major areas of fundamental and applied mathematics. The following are the main subject headings: Mathematical Analysis; Higher Algebra and Numbers Theory; Higher Geometry; Differential Equations; Mathematical Physics; Computational Mathematics and Numerical Analysis; Statistical Simulation; Theoretical Cybernetics; Game Theory; Operations Research; Theory of Probability and Mathematical Statistics, and Mathematical Problems of Mechanics and Astronomy.
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