... sum of theproduct i × Li. Hence,iiLi= 2e. (2.7)Let vkdenote the number of vertices of degree k, namely,v2denotes the number of vertices of degree two, v3the number of vertices of ... and chords. The arcs of G consist of all the elements of E that form the spanning tree T , whereas the chordsconsist of all the elements of E that are not in T . The union of the arcs and chordsconstitutes ... detailed descriptions of the theory.2.1 DefinitionsA graph consists of a set of vertices (points) together with a set of edges or lines.The set of vertices is connected by the set of edges. Let the...
... of computer-aided kinematic and dynamic analysis of mechanisms. For example, Freudenstein and Yang [7]applied the theory of fundamental circuits for the kinematic and static forceanalysis of ... 1970, Synthesis ofKinematic Structure of Geared Kinematic Chains and other Mechanisms, Journal of Mechanisms,5, 357–392.[2] Chatterjee, G. and Tsai, L.W., 1994, Enumeration of Epicyclic-Type ... different kinematic representations of four epicyclic geartrains.3.4.1 Advantages of Using Graph RepresentationThe advantages of using the graph representation are:1. Many network properties of...
... SymbolNumber of verticesvNumber of linksnNumber of edgeseNumber of jointsjNumber of vertices of degreeiviNumber of links havingijointsniDegree of vertexidiNumber of joints on ... mechanisms4.3 Degrees of FreedomThe degrees of freedom of a mechanism is perhaps the first concern in the study of kinematics and dynamics of mechanisms. The degrees of freedom of a mechanismrefers ... number of links, number of joints, and type of joints. It is alsopossible to establish an equation that relates the number of independent loops to thenumber of links and number of joints in a kinematic...
... trains, the number of gear pairs is equal to the number of independentloops; the number of turning pairs is equal to the number of links diminished by one;and the number of degrees of freedom is ... 7.1summarizes the number of labeled nonisomorphic graphs in terms of the number of degrees of freedom and the number of independent loops. Appendix F provides anatlas of labeled graphs and typical ... structuralcharacteristics. This results in a family of EGTs that have the same number of degrees of freedom and number of independent loops as that of the parentbar linkages.5. Sketch the corresponding...
... Viewpoint ofKinematic Structure,ASME Journal of Mechanisms, Transmissions, and Automation in Design, 105,2, 259–266.[4] Freudenstein, F. and Maki, E.R., 1984, Kinematic Structure of Mechanismsfor ... the functional requirements. Then, we translatesome of the requirements into structural characteristics for the purpose of enumeration of the kinematic structures. Lastly, we apply the remaining ... evaluation of the kinematic structures. This results in a class of feasible mechanisms or design alternatives.Since we are primarily concerned with the enumeration and qualitative evaluation of various...
... of two-dof joints, and thethird indicates the number of three-dof joints. For example, type 201 limb has twoone-dof, zero two-dof, and one three-dof joints, whereas type 120 limb consists of one ... represents a family of parallel manipulators for which the number of limbsis equal to the number of degrees of freedom of the manipulator, and the total number of joint degrees of freedom in each ... character -istics of a mechanism. For example, the number of degrees of freedom is governedby Equation (4.3); the number of independent loops, number of links, and number of joints are related...
... by a com-puter program using a nested-do loops algorithm to vary the value of each xiandcheck for the validity of the solutions. A more rigorous procedure for solving onelinear equation ... a method for solving one equation in n unknowns. Then, we extendthe method to solving a system of m equations in n unknowns.A.1 Solving One Equation in n UnknownsConsider the following linear ... imposed:xi≤ qi(constant) . (A.2)Since there are n unknowns in one equation, we may choose n − 1 number of unknowns arbitrarily and solve Equation (A.1) for the remaining unknown, providedthat all the...
... precisely by the set of edges GEOMETRY OF THE UNIFORM SPANNING FOREST475We will need the following known lemma, which is a corollary of themain result of von Weizs¨acker (1983). Its proof is included ... allow for future applications.The bulk of the paper is devoted to the proof of the upper bound onmax N(x, y) in (1.1). We now present an overview of this proof. Let U(n)bethe relation N(x, y) ... thendimS(LR) = min{dimS(L) + dimS(R),d} . GEOMETRY OF THE UNIFORM SPANNING FOREST483event has positive probability), is the union of E1with the set of edges of aUST on the graph obtained from...
... multidegree of X equalsthe sum of multidegrees of components of Y that happen also to be compo-nents of X. By hypothesis, the multidegrees of X and Y coincide, so the sum of multidegrees of the ... formulae for the multidegrees of ladder determi-nantal rings.The proofs of the main theorems introduce the technique of “Bruhat in-duction”, consisting of a collection of geometric, algebraic, ... closureBwTB+ of the B × B+orbit on Mnthrough the permutation matrix wT. ThusXwis irreducible of dimension n2− length(w), and wTis a smooth point of it.Proof. The stability of Xwunder...