Geometrical Methods To Locate Secondary Instantaneous Poles of Single-Dof Indeterminate Spherical Mechanisms Jo urnal o f Me c hanic al Eng ine e ring , Vo l. ME 41, No . 2, De c e mb e r 2010 Tra nsa c tio n o f the Me c h. Eng . Div., The Institutio n o f Eng ine e rs, Ba ng la de sh 80 G EO M ETRIC A L M ETHO DS TO LO C A TE SEC O NDA RY INSTA NTA NEO US PO LES O F SING LE- DO F INDETERM INA TE SPHERIC A L M EC HA NISM S Soheil Zarkandi* Department of Mechanical Engineering Babol University of Technology, Mazandaran, Iran *Corresponding email: zarkandi@gmail.com Abstract: A single-degree-of-freedom (DOF) indeterminate spherical mechanism is defined as a mechanism for which it is not possible to find all the instantaneous poles by direct application of the Aronhold-Kennedy theorem. This paper shows that a secondary instantaneous pole of a two DOFs spherical mechanism lies on a unique great circle instantaneously. Using this property, two geometric methods are presented to locate secondary instantaneous poles of indeterminate single DOF spherical mechanisms. Common approach of the methods is to convert a single DOF indeterminate spherical mechanism into a two DOFs mechanism and then to find two great circles that the unknown instantaneous pole lies on the point of intersection of them. The presented methods are directly deduced from a work done for indeterminate single DOF planar mechanisms. Key Words: instantaneous poles, geometrical techniques, indeterminate single-dof spherical mechanisms Nomenclature PGC Primary great circle IGC Infinity great circle DGC Declination great circle P ij Instant pole between links i and j INTRODUCTION For two co-spherically moving shells, there exist two instantaneously coincident points, each belonging to the respective shell, the linear velocities of which are identical. The place of these common points is called Instantaneous Pole, henceforth referred to as Instant Pole, of the two shells 1 . Instant poles are spherical counterpart of instant centers in planar mechanisms; however they are not fully exploited to study kinematic behavior of spherical mechanisms as the instant centers are for planar ones, see for instance [2-6]. An instant pole which can be found by direct inspection will be referred to as a primary instant pole and an instant pole which cannot be found by direct inspection will be referred to as a secondary instant pole. Planar counterparts of these poles are primary instant center and secondary instant center, respectively. Some scholars dealt with determining secondary instant centers in planar mechanisms e.g. [7- 9] . Foster and Pennock 7, 8 presented some graphical methods to determine the secondary instant centers of any single DOF planar mechanism with kinematic indeterminacy. Gregorio 9 presented an algorithm that analytically computes the instant centers in single DOF planar mechanisms. All of the above mentioned works can be done for spherical mechanisms using the concept of instant poles. For instance, deducing from his work 9 , Gregorio presented an exhaustive algorithm 10 to determine the instant poles’ positions of single DOF spherical mechanisms. Here, the author extends the methodology presented by Foster and Pennock 8 to locate secondary instant poles of indeterminate single DOF spherical mechanisms. The approach adopted in this paper to locate a secondary instant pole of a single DOF indeterminate mechanism is to convert the mechanism into a two DOFs mechanism 7 . There may be several strategies to accomplish this goal, this paper presents two; namely, (i) remove a binary link, or (ii) replace a single link with a pair of connected links by adding a revolute joint. By following a sequence of geometric constructions on the obtained two DOFs mechanism, the secondary instant poles of the original single DOF indeterminate mechanism can be located at the intersection of two unique two great circles. After using this method to find a secondary instant pole, some, or all, of the remaining secondary instant poles can be located by use of Aronhold-Kennedy theorem. This paper is organized as follows: in next section, it is shown that a secondary instant pole of a two DOFs spherical mechanism locates on a specific great circle, instantaneously; in section 3, two geometric methods are presented to show how to apply this concept to determine secondary instant pole of single DOF indeterminate spherical mechanisms; section 4 presents two illustrative examples to show the methods; finally, section 5 presents some conclusions of this research activity. SECONDARY INSTANT POLES OF A TWO DOFS SPHERICAL MECHANISM Spherical mechanisms can be studied by projecting them through the spherical motion center onto a reference sphere with center at the spherical motion