D:/Dottorato/Pubblicazioni/IN REVISIONE/ASME04/Articoli_Revisionati/Leo/LeoAsme04_REV.dvi

This investigation concerns with the mechanical efficiency of Cardan joints. The model includes also the effects due to manufacturing and mounting errors and the influence of rotation speed on the efficiency. The joint has been modeled as an RCCC spatial linkage and the full dynamic analysis performed by means of dual vectors algebra. NOMENCLATURE [ Â ]i+1 i Dual transform matrix. ai Minimum distance between zi and zi+1 axes. Ci Origin of the reference frame of joint i. D Distance between the bearings of the cross. di Diameter of shaft i. F̂ (ik) C(i) Dual vector of the external dynamic loads (forces and moments) acting on body i, evaluated w.r.t. a point C and expressed in the reference ik. Fi Joint forces at the ith joint. f Friction coefficient. Gi Center of mass of body i. { ĤC j( j−1) }( j) Dual momentum of body ( j− 1), evaluated at pointC j and expressed in the joint reference j. ∗Address all correspondence to this author. [ J (ik) C(i) ] Matrix of inertia of body i expressed in C− xikyikzik reference components. mi Mass of body i. ~R(ik) = ~ CGi vector expressed inC− xikyikzik . { R̂C j( j) }( j) Dual reaction forces acting on link j, evaluated inC j, and expressed in the reference of the joint j. si Relative linear displacement of the links measured according to the Denavit-Hartenberg convention. v̂ (ik) C(i, j) Dual velocity of body i with respect to body j evaluated inC and expressed in the ref. ik. xiyizi Moving cartesian system attached to the ith body. αi Angle between zi and zi+1 axes. β Angle between input and output axes. ε Dual unity ( ε = 0 ) . η Mechanical efficiency of the Cardan jojnt. θi Relative angular displacement of the links measured according to the Denavit-Hartenberg convention (see Fig.1). ωi = θ̇i Relative angular velocity of the ith body w.r.t. body i− 1, measured in the Cartesian system o− xiyizi. τ (i) f Frictional torque at the ith revolute joint. ̂ Denote dual quantities. · Denote differentiation w.r.t. time. ̃ Denote the skew matrix of a vector. 1 Copyright c © 2004 by ASME INTRODUCTION Cardan joints are common devices for transmitting the motion between misaligned intersecting axes. Although their structure has been known for centuries, only recently a complete dynamic analysis has been presented in a series of papers authored by F. Freudenstein and his coworkers [1–4]. In the mentioned references friction is not included and the influence of rotation speed on the efficiency is not investigated. Likely the first scientific contribution on the mechanical efficiency of Cardan joints is due to Morecki [5]. This model, based on a simplified static analysis and including the losses in the yoke bearings only, has been verified with a different analytical approach and refined by including also the losses in the fixed bearings. The results have been plotted in a design chart which allows to compute the efficiency as a function of the angle between input-output shaft axes [6]. In this paper the kinematic and dynamic analysis of the RCCC mechanism by means of the dual numbers algebra is carried out first. Then, the effects of friction and manufacturing errors are included. For this purpose, the following hypotheses are adopted: Coulomb friction; absence of stiction; absence of backlash in the kinematic pairs; rigid bodies. The modeling of manufacturing errors in Cardan joints is introduced by considering a kinematically equivalent RCCC mechanism. It should be acknowledged that the ideal kinematic structure of a Cardan joint is a spherical four-bar RRRR linkage with all the moving links having αi angles equal to 90 degrees. However, the RRRR is an overconstrainedmechanism whose mobility depends on the fulfillment of geometric conditions, such as the convergence of all revolute axes in the same point. For this reason, in technical literature e.g. [1, 2], the effects of manufacturing errors on the kinematics and dynamics of this particular joint are usually monitored through the analysis of an RCCC linkage whose mobility does not depend on the previously mentioned condition. Thus one can study the influence of the missed intersection of kinematic pairs axes or of link dimensional variations on the kinematics of a Cardan joint through the analysis of an RCCC linkage with prescribed axes misalignments. KINEMATIC AND STATIC ANALYSIS OF THE RCCC LINKAGE The relative position between adjacent links and the geometry of the ith link is defined by means of the dual numbers θ̂i = θi+ εsi , (1) α̂i = αi+ εai . (2) a x y z