Duality between the kinematics of gear trains and the statics of beam systems (2).doc

/locate/mechmtduality between the kinematics of gear trains and the statics of beam systemsgordon r. pennock a,*, jeremiah j. alwerdt ba school of mechanical engineering, purdue university, west lafayette, in 47907, usa b allison transmission, indianapolis, in 46206, usareceived 24 april 2006; received in revised form 26 october 2006; accepted 14 december 2006 available online 5 february 2007abstractthis paper provides geometric insight into the duality between the first-order kinematics of gear trains and the statics of beam systems. the two devices have inherent geometrical relationships that will allow the angular velocities of the gears in a gear train to be investigated from a knowledge of the forces acting on the beams of the dual beam system, and vice versa. the primary contribution of the paper is the application of this duality to obtain the dual beam system for a given compound planetary gear train, and vice versa. the paper develops a systematic procedure to transform between the first-order kinematics of a gear train and the statics of the dual beam system. this procedure provides a simple and intuitive approach to study the speed ratios of a planetary gear train and the force ratios of the dual beam system. the speed ratios are expressed in terms of kinematic coefficients, which are a function of the position of the input gear and provide insight into the geometry of the gear train. two numerical examples of simple and compound planetary gear trains are presented to demonstrate the simplicity of the proposed approach. a third example is included to illustrate the systematic procedure to transform from a given beam system to a planetary gear train. the examples take advantage of the principle that the speed ratios for gearing are dual to the force ratios for beam systems, and vice versa. 2007 elsevier ltd. all rights reserved.keywords: kinematics; statics; ordinary gear trains; massless binary link; statically determinate beams; planetary gear trains; compound gearsets; dual beam systems1. introductionit is interesting to note that planetary gear trains (commonly referred to as epicyclic gear trains) were known, and in use, at least 2000 years ago 1. despite the antiquity and widespread applications in machinery, however, the principles of operation of planetary gear trains are not generally understood 2. also, the literature devoted to planetary gear trains is scarce at best 3 although a comprehensive treatise on the theory of epicyclic gears and epicyclic change-speed gears was written by levai 4. planetary gear trains offercorresponding author.e-mail address: (g.r. pennock).0094-114x/s - see front matter 2007 elsevier ltd. all rights reserved. doi:10.1016/j.mechmachtheory.2006.12.0061528g.r. pennock, j.j. alwerdt / mechanism and machine theory 42 (2007) 15271546advantages over ordinary gear trains, for example, for the same speed ratio they can be smaller in size and have less weight 5. there are several techniques that are commonly applied to the kinematic analysis of planetary gear trains; for example, the instant center method, the principle of superposition using a tabular method, and identifying the fundamental circuits of the train 6,7. also, an analogy between planetary gear trains and beam systems using one-dimensional vectors was presented by kerr 8,9. the available methods, however, do not provide geometrical insight into the gear train in a direct manner that is suitable for a specific application. the work presented in this paper attempts to rectify this situation by taking advantage of the duality that exits between gear trains and beam systems.many theories and methods in diverse fields of science and technology can be applied by way of analogies. the analogies exist by virtue of the similarity of the mathematical models. too often, however, the mathematical models do not receive sufficient attention by engineers. it is well known, for example, that an angular velocity vector lying along a specific axis is instantaneously reciprocal to a force vector along some line of action 10. the two vectors are line vectors and, as such, they must both obey the same laws of vector algebra, and can therefore be treated identically. this principle of reciprocity, or duality, is the basis of screw theory 11 and can be used to relate important properties in kinematics and statics by analogy. an in-depth study of first-order instantaneous kinematics and statics by duffy 12 has contributed to a better understanding of both serial robot manipulators and parallel, or platform-type, robot manipulators 1316. the principles underlying the kinematics and the statics of these two types of robot manipulators are the same which makes them dual to each other. davidson and hunt 17 extended this work to the kinetostatics of spatial robots and presented relationships between kinematically equivalent serial and parallel manipulators.shai and pennock 18 introduced the equimomental line in static analysis and showed that this line is dual to the aronholdkennedy theorem of instant centers 3,6. the same work provided new insight into the concept of a face force to allow equimomental lines to be used in a direct manner in static analysis. the duality between the statics of a structure and the kinematics of a mechanism has also been investigated using two new graph representations 19; namely, the flow line graph representation and the potential line graph representation. the duality between a static pillar system and a planar linkage was investigated by using the flow line graph representation for the pillar system and the potential line graph representation for the linkage. this work, and two companion papers 20,21, clearly illustrate the duality that exists between the kinematics of planar, spherical, and spatial mechanisms and the statics of a variety of trusses and platform structures.the focus of this current work is the correlation between basic concepts underlying the kinematics of gear trains and the statics of beams 22. an important contribution of this paper is the application of the duality between the speed ratios of gear trains and the force ratios of static beams. the paper shows that the first-order kinematic coefficients in a gear train 3 are analogous to the force ratios in the dual beam system. the beam analogy permits the application of ordinary equations of motion to a system incorporating a planetary gear train. also, the analogy is a useful aid in understanding the behavior of a planetary gear train and will enhance the ability of an engineer to design better transmission systems. the designers of powertrains for the automotive and the aerospace industries are continually searching for compact, light-weight, and highspeed transmission systems that incorporate efficient and sophisticated planetary gear trains. the principles of duality provide insight into the kinematic analysis of a gear train from a static force analysis of a system of beams, and vice versa. based on these principles the paper presents a systematic procedure to obtain a beam, or a system of beams, for a given gear train, and vice versa.the paper is arranged as follows. section 2 presents a procedure to map an ordinary gear train to a statically determinate beam. a force ratio in a static analysis of the beam, which is dual to a speed ratio in a kinematic analysis of the gear train, is introduced and the associated sign convention is defined. the section then introduces a novel concept, namely a massless binary link, to build a static beam system with a single input force, while maintaining the definition of the force ratio. the important properties of a massless binary link, which can also be regarded as an algebraic link, are clearly explained. section 3 extends the idea of duality to planetary gear trains and presents a procedure to map a planetary gear train to a dual beam system. for purposes of illustration, section 4 presents three practical examples. the first two examples clearly illustrate the procedure to transform from a given gear train to a dual beam system. the first example is a compound gear train used in a transmission system, and the second example is a compound planetary gear train that is commonly used in a drill or an electric screwdriver. the third example illustrates the procedure to transform fromg.r. pennock, j.j. alwerdt / mechanism and machine theory 42 (2007) 152715461529a given beam system to a gear train. all three examples emphasize the simplicity of employing the principles of duality. finally, section 5 presents some important conclusions and suggestions for future research.2. a beam dual to an ordinary gear trainconsider the ordinary gear train as shown in fig. 1. the gears are represented by their pitch circles; i.e., the input gear i, rotating about the fixed center oi, is in rolling contact with the output gear j, rotating about the fixed center oj.the point of contact between the two gears is the instantaneous center of zero velocity (henceforth referred to as the instant center) and is denoted as iij. the velocity of the point on gear i and on gear j that is coincident with the instant center iij can be written asvt. = (oi x rt = (of x rji i i j(1)where ri and rj are the vectors from the center of gears i and j to the instant center iij. eq. (1) can be written in scalar form asiri(djrj(2a)or as0= = -l rj (oi(2b)where ri and rj are the radii of the pitch circles of gears i and j, respectively, and xi and xj are the angular velocity of gears i and j, respectively, and h0j i is referred to as the first-order kinematic coefficient of gear j relative to gear i. the kinematic coefficient is negative for external contact between gears i and j and positive for internal contact.the cantilever beam that is dual to the gear train (henceforth referred to as the dual beam) must satisfy the conditions of static equilibrium; i.e., the sum of the external forces and the sum of the moments about an arbitrary point fixed in the beam can be written, respectively, as/ = k0(3a)fig. 1. the ordinary gear train and the dual beam.1530g.r. pennock, j.j. alwerdt / mechanism and machine theory 42 (2007) 15271546andny = rk x fk = 0(3b)where rt is the vector from the point to the force ft-the dual beam, also shown in fig. 1, is obtained from a systematic procedure; namely:step 2.1. the instant center iy defines a point support for the beam (henceforth denoted as ey). note that a dual beam will exist for each instant center between a unique pair of meshing gears. the line of centers otoj defines the length of the beam where the end ot (i.e., the point coincident with the center of the input gear) must be a free end, and the end (3, (i.e., the point coincident with the center of the output gear) must be a pin. since gear j is the output gear then the pin must be a ground pin support.step 2.2. the distance from (3, to ey is equal to the radius r and is denoted as lt. similarly, the distance from oj to ey is equal to the radius rj and is denoted as lj.step 2.3. in order to satisfy eq. (3), the input force (henceforth denoted as ft) must act perpendicular to the dual beam at the point coincident with the center of the input gear i. similarly, the output force (denoted as fj) must act perpendicular to the dual beam at the point coincident with the center of the output gear j.sign convention: since the input gear is rotating counterclockwise then the direction of the input force must act to create a counterclockwise moment about the point support ey. similarly, since the output gear is rotating clockwise then the direction of the output force must act to create a clockwise moment about the point support ey.the input force ft is dual to the input angular velocity a, and the output force fj is dual to the output angular velocity coj. therefore, the kinematic coefficient of gear/ relative to gear i, defined in eq. (2b), has a corresponding property in the dual beam. this property is the force ratio of beam section/ with respect to beam section i, henceforth, referred to as the force ratio of the beam and defined asfii = (4)fithe force ratio of the beam is positive if the moments about the point support ey created by the forces ft and fj are in the same direction and negative if the moments about the point support ey created by the forces are in opposite directions.step 2.4. if gear/ is an intermediate gear (i.e., a gear between the input and the output gears) then the pin coincident with gear center oj will correspond to a massless binary link (consistently denoted throughout the paper as link m). this massless (or algebraic) link will ensure that the forces acting on the two pins of the link are equal in magnitude but opposite in direction; i.e., a massless binary link can only transmit a force along the line of centers of the link, commonly referred to as a two force member 3.for an ordinary gear train, the input force is applied to a single dual beam, however, for a planetary gear train, the input force must be applied to multiple dual beams (henceforth referred to as a dual beam system). massless binary links will be used to couple the free ends of a beam system to account for the single input force and provide the correct direction of the applied moments. in other words, a massless binary link will load each beam in such a way that the input force is produced at the appropriate end of the beam. the introduction of a massless binary link is a simple intuitive approach similar to the idea of using a linear spring to connect and load two beams and provide an equal and opposite force on each beam.to illustrate the concept of a massless binary link in a dual beam system, consider the ordinary gear train shown in fig. 2. gear 2 is the input gear (rotating counterclockwise with an angular velocity m2), gear 3 is the intermediate gear, and gear 4 is the output gear (rotating counterclockwise with an angular velocity 014). the dual beam system; i.e., the two statically determinate beams denoted as 0203 and o3o4, is also shown in fig. 2.the free end 02 is subjected to the input force f2 and the pin support o4 is subjected to the output force f4 (see step 2.2). the massless binary link m, connecting the free ends o3 and 0, guarantees the correct directions of the moments about the two point supports 23 and 34.the following section will extend the concepts that were introduced here and present a systematic procedure to obtain a system of beams that is dual to a given planetary gear train.g.r. pennock, j.j. alwerdt / mechanism and machine theory 42 (2007) 152715461531fig. 2. the dual beam system for the ordinary gear train.3. a beam system dual to a planetary gear traina simple planetary gear train is a mechanism that has two degrees of freedom; i.e., the mechanism requires two independent input angular velocities in order to obtain a unique output angular velocity 3,5,13. for convenience, and without any loss in generality, this section will assume that the angular velocity of one of the inputs is zero; i.e., one of the inputs is locked. consider the simple planetary gear train shown in fig. 3 in which the ring gear h is locked (i.e., the angular velocity of the ring gear xh = 0) and the planet carrier (also referred to as the arm) is taken to be the input (i.e., the angular velocity xarm = xi).in general, the first-order kinematic coefficient of the planet gear j in rolling contact with a rotating ring gear h can be written asrhc0jrj(oharmarm(5a)where the negative sign denotes external contact between gears j and h and the positive sign denotes internal contact. similarly, the first-order kinematic coefficient of the planet gear j in rolling contact with the sun gear k can be written asrk rj(5b)armuk armnote that if the ring gear is removed and the arm is locked (i.e., xarm 0 then eq. (5b) is identical to eq. (2b) and the planetary gear train reduces to the ordinary gear train that is shown in fig. 1.since there is internal contact between the planet gear and the fixed ring gear (i.e., xh = 0) then the positive sign in eq. (5a) is used and the equation can be written as(6)k arm armrjcoj = (rh rj)(0arm1532g.r. pennock, j.j. alwerdt / mechanism and machine theory 42 (2007) 15271546fig. 3. a simple planetary gear train with the ring gear locked.similarly, since there is external contact between the planet gear and the sun gear then the negative sign in eq. (5b) is used and the equation can be written asrkk = rj0:j + (rk + rj)j)arm(7a)the angular velocity of the sun gear k can be expressed in terms of the input angular velocity of the arm by substituting eq. (6) into eq. (7a) and simplifying; i.e.,rkk = (rk + rh)arm = 27?armwarmc7b)analogous to an ordinary gear train (see step 2.1), a dual beam will exist for every instant center that is coincident with the point of contact between each unique pair of meshing gears. since there are two points of rolling contact in this gear train (i.e., the instant centers ijf, and ijk), see fig. 3, then two dual beams are created in terms of the input force. in this example, there is a dual beam for the planet and ring gears and a dual beam for the planet and the sun gears. note that when multiple dual beams are defined in terms of the input force, then the free ends of each dual beam can be pinned to a massless binary link to create a dual beam system.the dual beam system is obtained from a systematic procedure; namely:step 3.1. the instant center 7 defines a point support for the first dual beam (denoted as /,) and