Mechanics of Roller Chain Sprocket Contact Mahn Shik Kii Glen E. Johnson Design Laboratory, Mechanical Engineering, and Applied Mechanics University of Michigan, Ann Arbor, MI 48109-2125

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1 Mechanics of Roller Chain-Sprocket Contact1 Mahn Shik Kim2 Glen E. Johnson Design Laboratory, Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, MI 48109-2125 Abstract The contact phenomenon between a roller chain and sprocket was investigated using a new mathematical model which, unlike previous models, made it possible to predict the load distribution without resorting to assumptions about the distribution of the contact points. It was found that the distribution of contact points on the sprocket is affected by the external loading condition, and this relationship was determined. Effects of friction and the dimensional variations in the chain pitch and the sprocket pitch were also investigated. Link tension predicted by the simulation showed excellent agreement with experimental data reported by Naji and Marshek. 1This article is based on the Ph. D. dissertation submitted by the first author to the Rackham Graduate School at the University of Michigan, 10/90. 2Currently employed as a researcher at the Technical Center, Daewoo Motor Company, Inchon, Korea.

1 INTRODUCTION An extensive analysis of the contact phenomenon occurring between the roller chain and the sprocket can be found in Binder's classical textbook [1] published in 1956. In the early 1980's, Naji and Marshek [2, 3, 4] complemented Binder's work by theoretical and experimental investigation of the contact phenomenon under quasi-static conditions. The kinematics of the roller chain drive are quite complicated due to non-trivial geometry of the sprocket tooth, manufacturing tolerance, and the possible change in clearances between components caused by wear. Several simplifying assumptions on the geometry of the roller chain drive were made in previous studies. The geometry of the sprocket tooth standardized by ANSI [5] is defined by four different curvatures as shown in Fig. 1.1. The curve which is at the bottom of the tooth form is often called the seating curve while the curves next to the seating curve are called working curves. The radius of the seating curve is designed, in the ANSI standard, to be slightly bigger than that of the roller. Therefore, the trajectory of the center of the roller on the seating curve is still a circular arc, although the radius of the arc is very small relative to that of the roller. The curvature difference between the roller and the seating curve was neglected in previous models and, therefore, the trajectory of the center of the roller on the seating curve shrank to a point instead of a circular arc as shown in Fig. 1.2 (a). Consequently, the angle of the resultant normal contact force applied on the roller by the sprocket could not been determined from physical reasoning by most models. One exception to this is the "stiff link - soft tooth" model devised by Gerbert [8]. Binder assumed that the angle of the contact force on each tooth with respect to the tooth is constant. But he made no comment on how to determine the value. In the model

of Naji and Marshek a number of rollers on the high tension side contact the tight side tooth flanks and the rest contact the slack tooth flanks. It was further assumed in their model that, if the roller contacts the seating curve, the angle of the normal contact force is constant with respect to axis of the symmetry of the tooth form. The value of the pressure angle used in the theoretical prediction of Naji and Marshek on the chain tension was selected to provide good agreement with their experimental data. The angles of normal contact forces on the sprocket strongly affect the distribution of contact forces and chain link, tensions. It will be shown that in a reliable roller chain drive the rollers contact not only the working curve but also the seating curve. Therefore, the angle of the normal contact force not only on the working curve but also on the seating curve should be precisely estimated to correctly predict the load distribution of the chain and the sprocket. A new model which enables us to determine the load distribution without resorting to any assumption on the distribution of the contact points is developed. It is shown that the distribution of the contact points on the sprocket is in fact determined by the loading condition. In the new model the curvature difference is included and the trajectory of the center of the roller on the seating curve composes a circular arc as shown in Fig. 1.2 (b). Refined modeling of the friction forces is also incorporated. 2 DISTRIBUTION OF CONTACT POINTS In this section, the geometric relationship between contact points on the sprocket surfaces is derived. Only steel roller chains and steel sprockets standardized by ANSI are considered. The chain and the sprocket are regarded as two-dimensional bodies and any varation in the direction parallel to the axis of the rotation of the sprocket is neglected. Furthermore, point contact is assumed and the effects of elastic deformation of the chain and the sprocket are neglected.

The tolerances on the chain pitch and the sprocket pitch are not given by the standard explicitly. However, they can be computed from the tolerances on the chain length and the sprocket diameter given by the standard. New chain should have an over-length error, and the error in the length increases in the course of operation because of wear between the pin and the bushing. On the other hand, the error in the sprocket pitch can be either negative or positive. The effect of a negative error in the sprocket pitch is similar in many aspects to that of a positive error of the same magnitude in the chain pitch. The tolerance on the sprocket pitch turns out to be, in general, larger than the tolerance on the chain pitch. (Detailed treatment of the tolerances on the chain pitch and the sprocket pitch can be found in Kim [7].) Therefore, the error in the sprocket pitch should be also considered whenever the effects of the error in the chain pitch are discussed. The system to be handled here consists of a sprocket and a segment of endless chain wrapping the sprocket as shown in Fig. 2. 1. Without loss of generality it can be assumed that the sprocket rotates counter-clockwise. Each roller and each link is given an integer index for convenient mathematical manipulation. Roller 1 is defined as the roller that has most recently come in contact with the sprocket. Roller 1 is located at the end of the high tension side for the driver sprocket, whereas it is at the end of the low tension side for the driven sprocket. The roller j follows roller j+ 1 in the direction of rotation of the sprocket. m is the instantaneous number of rollers in contact with the sprocket. The link between rollerj and roller j+ 1 is denoted as linkj+ 1. To locate the contact point, coordinate 5 is defined here along the curve of each tooth of the sprocket as shown in Fig. 1.1. Each point at which two different curves meet is assigned an integer from - 4 to 4 from the left end to the right end of each tooth. The coordinate has a linear scale within each segment. When 5 is used with a subscript, it denotes the 5 coordinate of the corresponding roller. The typical relationship between i+l the 5 coordinate of the contact point of roller j+l,

and j, that of roller j, is shown in Fig. 2.2. The graph was obtained for a single pitched roller chain with pitch of 0.5 inch and matching sprockets of 12, 24, and 36 teeth. The graph is symmetric about line i+l = - $. An analytical procedure to compute the location of a roller from that of neighboring roller can be found in Kim [7]. Obviously, the contact points of all the rollers on the sprocket can be located if any one of them is somehow known. Fig. 2.3 illustrates graphically how to determine 42, 3, and so on sequentially from the values of 41 in two different cases. It can be noticed that the distributions of contact points of the two cases shown in Fig. 2.3 are quite different from each other. In fact, if 5R and tL are 5 coordinates of points R and L shown in Fig. 2.3 respectively, the series 41, c2,..., tm can be classified by the five cases as follow: Case I If 41 > R then >j > j-1 ( = 2, 3..., m) Case II If, = R, then 2 = 3 =..= m = R Case III If 5R > 41 > L, then ij < j-1 (J = 2, 3,..., m) and converges to L. Case IV If 41 = L, then 42 = ~3 =..= m= 4L Case V If Cl < 5L, then j > j ( = 2, 3,..., m) It can be observed that the series are characterized by the two specific points marked as R and L in Fig. 2.2 and 2.3. They will be called the "transition points" from here on. The 5 coordinates of the transition points, 4R and L, can be computed analytically. The algorithm to compute the transition point 5R is listed in Kim [7]. Note that {L = - R. For a chain and a sprocket which have exact reference dimensions, the transition points stay in the working curves very close to the border between the seating curve and the working curve ( = 1 or - 1). The locations of the transition points can be affected by the changes in the chain pitch, the sprocket pitch, and the radius of the working curve. Chain pitch, sprocket pitch, and the radius of the working curve of a real sprocket can differ from

the reference values due to manufacturing errors and wear. Chain pitch and the radius of the working curve change in the course of operation due to wear; sprocket pitch is not affected by wear. The transition points move further toward the inside of the working curves as the chain pitch increases (equivalently, as the sprocket pitch decreases) and as the radius of the working curve increases. Fig. 2.4 shows the location of transition point R of 12-tooth and 24-tooth sprockets for No. 40 and No. 80 chains. The ranges of tolerance on the chain pitch covered in Fig. 2.4 is the same as those allowed by the standard. Distribution of contact points is not determined solely by the geometrical relationship. It is determined only when the equilibrium equations of the system derived in Section 3 are solved along with the geometrical relationship. Therefore, discussion on the five cases of contact point distribution is continued in Section 4 after the force and moment equilibrium of the system is discussed in Section 3. 3 QUASI-STATIC EQUILIBRIUM OF THE CHAIN AND THE SPROCKET IN CONTACT In this section, quasi-static force and moment equilibrium of thrie chain and the sprocket in contact is derived. The friction forces and torques appearing in the equilibrium equations are modeled in accordance with the laws of Coulomb friction. It is shown that the kinematic variables appearing in the equilibrium equations can be computed if the location of any one of the contact points on the sprocket and the direction of its instantaneous movement are known. It is also shown that the equations can be solved under certain conditions including the case of practical interest where tensions of two chain spans wrapping the sprocket are given. A closed form solution is not available since some of the equations governing the relationship between the kinematic variables are highly non

linear. The equations can be solved numerically by iteration at any instant or, equivalently, at any sprocket angle. The procedure for the solution is also presented. 3.1 Equilibrium Equations Equations governing quasi-static equilibrium of each element in Fig. 3.1 are derived. Both scalar variables and vector variables are used to denote forces and torques. Vector quantities will be represented by bold characters and the scalar quantities by plain characters. Several angles and three sets of unit vectors shown in Fig. 3.2 are used. Definitions of angles are shown in Fig. 3.2. Vectors xi and yj are unit vectors fixed in tooth j and mutually perpendicular to each other. Vector yj is parallel to the axis of symmetry of the tooth. Vector z is a unit vector equal to xj x y.. Vector p is a unit vector normal to the tooth surface directing from outside to inside at the contact point with rollerj. Vector Tj is perpendicular to normal vector pj. Vectors pj and zj are obtained by rotating unit vectors x. and yj respectively by angle /. counter-clockwise. Similarly, unit vectors tj and v. are obtained by rotating unit vectors xj and yj respectively by angle a1 counter-clockwise. All angles are measured with respect to axis xj and defined positive counter-clockwise. The formulas derived in this section may be sensitive to the orientation. From here on, subscripts i-2, i, i+2, and so on will be used to denote pin links and subscripts i-1, i+1, and so on for roller links. The orientation of elements relevant to any term with a subscript which includes i should be in accordance with Fig. 3.1. If the formula is not sensitive to the orientation subscript instead of i is used. Let N) and F1 be respectively the normal and the tangential components of the contact force on the sprocket by rollerj. Rollers i and i+l are connected to roller link i+1 as shown in Fig. 3.1. Let Q and S (] = i, i+1l) be respectively the resultant force vector passing

through the center of rollerj and the accompanying torque in the z direction on rollerj by roller link i+1. From the free body diagram of rollerj in Fig. 3.1 we have Qj = Nj p + Fj j = i,i+1) (3.1) Sj=RR F (= i,i+1) (3.2) where RR is the radius of the roller. Let the contact force by link j-1 on link j passing through the center of the joint be P1 and the magnitude of the accompanying torque in z direction be U1. Equilibrium for pin link i yields Pi- = Pi (3.3) U - Ui +PCi Pi = (3.4) where PC is chain pitch. And, for roller link i+l, we have Pi + Q + Qi + Qi+ = Pi (3.5) - PC Vi+I' Qi - Si - Ui+l - Si+l =-Pc Vi+l Pi-i (3.6) Supplementary information on the relationship connecting the tangential forces and the torques to the normal forces is necessary for the equilibrium equations derived above to be solved. This information is supplied from the following friction model. 3.2 Friction Model Relative motion, or sliding, among the elements of the chain and between the roller and the sprocket is necessary to satisfy the kinematic requirement for the continued motion of the sprocket and the chain. Sliding can occur on the surfaces between the tooth and the roller, between the roller and the bushing, and between the bushing and the pin. Angle 77, the angle between two consecutive chain links, changes as the location of the

contact point changes. The pin should slip against the bushing to accommodate the change in angle r7. Any contact force passing through the contact point can be replaced by the same force whose line of action passes through the pin center and an additional torque. The magnitude of torque Uj can be expressed as U. = sign (i).PB Rp jPj (=i,i+l) (3.7) 1 l+ B in which Upg is the coefficient of kinetic friction between the pin and the bushing and Rp is the radius of the pin, and the operator sign (.) is defined as 1, ifx> 0 sign (x) =0, ifx = (3.8) - 1,ifx<0 Note that the friction torque Uj is not determined by Eq. 3.7) when there is no relative motion of the pin against the bushing. Although this may happen in a chain drive operating at extremely slow speeds or exercising intermittent motion, it will not be considered here. The roller can roll either with slip or without slip on the tooth surface. It is assumed here that the roller slides, if it does, only on either the surface of the sprocket tooth or that of the bushing. If roller j slips on the tooth surface, the magnitude of the friction force Fj- j on the sprocket by the roller j is equal to MRS Nj where MRS is the coefficient of kinetic friction between the roller and the sprocket. The direction of friction force Fj is determined by the direction of the motion of the contact point on the roller relative to the contact point on the sprocket. Let the instantaneous angular velocity of the roller which is assumed to roll without slip be 6. Then, the direction of the relative motion of roller j, which this time slides on the sprocket, is determined by sign (a>- 0') if link j is a roller link, and by sign (yj - 0j) if linkj is a pin link. And it holds Fj = F's where

FS sign (Q - RS Nj for =i -\sign (\i+ - i+1) IRS Nj forj =i+ in which uRS is the coefficient of kinetic friction between the roller and the sprocket. Torque Sj is determined from Eq. (3.2). If roller j rolls without slip on the tooth surface, sliding occurs between the roller and the respective bushing. In this case, resultant force Qj passing through the roller center and the accompanying torque SJ on the roller by the bushing will have a similar relationship to that between force Pj and torque Uj. That is, it holds Sj = Sj where sign ( 6.- -i). n RB IQil forj=i sJA^ =Il1 +/L2 R (3.10) sign(/+-i+ Oi+l) RBI Qi+l[ forj = i+l where,LBR is the coefficient of kinetic friction between the bushing and the roller and RB is the outer radius of the bushing. Friction force Fj and torque Sj can be expressed in terms of normal force Nj from the simultaneous solution to Sj = SB, Eq. (3.1), (3.2), and (3.10). This leads to Fj = F B where sign (- R ) i, Ni forj = i sin (i- i) /R + IR (RR R) f If the magnitude of friction force FB is less than FS, sliding occurs between the bushing and the roller. Otherwise, sliding occurs between the sprocket tooth and the roller. In other words, friction force F. is given by

FpD -. if BR RB <RS RR.+ BR (R BR N. F= N F= (j = ii+1) (3.12) i- N otherwise And torque Sj becomes RR FNi Sj = KN, = j = RR F (= ii+1) (3.13) The directions of the friction forces and the accompanying torques are determined by sign (ij), sign (i - 0i), and sign (ai+1 - i+l) in Eq. (3.7), (3.9), (3.10) and (3.11). Note that angles cxa, fj, j., 77j, and j (j = 1, 2,..., m) except al and 7m are determined from ~j's which in turn can be determined if any one of them, for example 1, is known. Angle al and angle ym are determined by the angle of the sprocket with respect to the chain spans. Due to the geometry of standard chains and sprockets it holds without exception that aj < Oj unless j = I and that Oj < Oj unless j = m. Thus we have sign (a-0) isign(a- 1), ifj=1 sign (d ^)- 0=) (3.14) - sign (j)= sign (j)= sign (1j), if]j 1( and sign.- 0.), ifj m sign ( ) - )= ) (3.15) - sign (6j) = sign (j) = sign (1), ifj ~ m In other words, the friction forces on the sprocket teeth (except teeth 1 and m) have the same direction as the movement of the contact point of roller 1. The directions of the friction forces on teeth 1 and m are not determined solely from geometrical reasoning. They can also be influenced by instantaneous loading conditions. Computation results obtained for a wide range of conditions indicate that it holds that al > 9, and Ym > m.

However, no presumption about this is necessary to solve the equilibrium equations as will be shown later. Note that sign (6c) = sign (rm) =-1. The friction force on each tooth except tooth 1 and tooth m is toward the direction of decreasing 5 in Case III; it is toward the direction of increasing 5 in Cases I and V. 3.3 Solution Procedure Let the components of force Pj parallel to unit vectors tj and v be T and Vj respectively. That is, Pj = T. tj + Vj v ( = i, i+l) (3.16) From Eq. (3.4), (3.7), and (3.16) we have PR V=(sign ( sign (i1) - sign (P iP VT+ (3.17) Solving Eq. (3.17) for Vi gives KVY= T KYTi = ){sign (^i)- sign (7i-l) g RpP (3.18) I 1 + sigB + {s i-sign (7i_)- i) (P-1 B R/PC)2 Ci+2 (the ratio of Vi+2 to Ti+2) can be obtained by replacing i+2 for i in Eq. (3.18). Then, by use of Eq. (3.3) and (3.16) the ratio of Vi+l to Ti+l, Ki+1, can be computed. K _ V i+1 _ (Ti+2 ti+2 + Ki2 Ti+2 Vi+2) *+1 i sin Vri+ l- sv2 cos 7ri+l (3.19) Tli+l (Ti+2 ti+2+ +2 Ti+2 Vi+2). ti+ -cos 71i+1- KV2 sin 1+ And from Eq. (3.7), (3.16) and (3.18) torque Uj can be expressed as U = T, = sign (j) + i i+) (3.20) Substituting Eq. (3.16) and (3.1) along with Eq. (3.12) and (3.18) into Eq. (3.5) and considering two components parallel to tijl and vijl respectively give the following two scalar equations.

Ti+ - Ni (cos vi - ic sin vi) + Ni+1 (cos Pi+I + ciF+1 sin Pi+1) = - Ti (cos ni- sin /i) (3.21) KiV1 Ti - Ni + (+1 svi +l - i +1 Cos i+ll) = - T (sin 77i + KY cos 7i) (3.22) Similarly, Eq. (3.6) can be rewritten as - Kil T+l + Pc (in vi + F c vi)- i) Ni - iS+1 Ni+ = {c (sin 7i + Icv cos 7,) - Ki Ti (3.23) The three scalar equations, Eq. (3.21) to (3.23), can be solved simultaneously for NA, Ni+1, and Ti+1 if the values of Ti and the kinematic variables are known. From Eq. (3.3) and (3.16) we have Ti+1 = (Ti+2 ti+2 + V+2 Vi+2). ti+1 = (- cos 77i+ - ci+2 sin tr+l) Ti+2 (3.24) Thus we can express Ti+2 in terms of Ti+1 and kinematic variables as = ______________i+_ (3.25) - cos 71i+l - Kci+2 sin Thi+l It is sometimes convenient to use the forces at the mid-points of the chain links instead of the forces at the joints, especially when they are compared with experimental data. Let Tf be the tension of the linkj at its mass center. Then, from Fig. 3.1, we have Ti = Pi' ti = Ti (3.26) (3.26) Til = (Pi+l + Qi+I) ti+l = Ti+l + (cos (i+l + icF+ sin i+l) N+ (3.27) Eq. (3.21) to (3.26) hold for all values ofj except for link 1 and link m+l if they are roller links. If link 1 is a roller link, Fo = No = 0 and r77 = 0. Then, from Eq. (3.26), (3.22), and (3.23) for i+l = 1, we have, assuming Uo = 0, the following simultaneous equations for T1 and N1.

P cos c(pi+ icif sin ](Ti }=7* (3.28) pC K _~ PC (-sin o1 + qF cos 01)- 'sJ tV I }O T1 and N1 can be computed by solving Eq. (3.28). If the link m+ is a roller link, setting i = m and noting that Nm+i = 0 in Eq. (3.21) and (3.23), we have 1 -1 (cos v,,- c sin vm) l T+l( - (cos rm Vsi ) \T o Pc (sin vm + KF cos vm)- f1lmm I +c (in C7m + Cm ms N) TM (3.29) It has been shown that values of Ni, T+ 1, N+, and Ti2 can be determined if we know the value of T* and values of the kinematic variable. A typical problem will be one in which T1 and T*1+ are given. In this case, "searching" for the proper value of l1 which yields the given value of T71 is required, since the positions of the contact points of the rollers are not known. Since the endless roller chain is composed of roller links and pin links alternately assembled, the behavior of the chain drive cycles with a period of two pitch angles. Thus the solution obtained over two pitch angles represents the complete solution. Numerical solutions can be obtained at any finite number of sprocket angles. Assume that the sprocket angle ( is defined so that it increases as the sprocket rotates. An algorithm to obtain the solution at any given value of sprocket angle ( is as follows: Step 1 Assume a value of 1. Step 2 Determine sign (51) by noting that sign (1) < 0 if 41 < CR and sign (41) > 0 otherwise. Step 3 Compute all the angles in Fig. 3.2 at ~ and 51. Step 4 Compute also those angles at 'p + A'p and 1 + AI'. A'p is very small and positive; Ad is very small and positive if sign (<1) > 0 and negative otherwise. Then,

compute sign (ij), sign (~i - 6), and sign (ai+1 - 0i+1) in Eq. (3.7), (3.9), (3.10) and (3.11) by assuming that sign (i)= sign (17 (1 + A1, 4 + AO )- 7r (1 ))} (3.30) Step 5 Compute all the forces including TM+1 by use of equations derived in this section. Step 6 Compare the computed value of T.m+ and the given value. If the difference is greater than the given allowance, guess a value of 41 again and repeat step 2 to 4. Repeat this until the difference between the computed value of T^* and the given value falls within a given limit. Step 4 can be skipped if one uses the facts described by Eq. (3.14) and (3.15) and assumptions that sign (l - 0) = sign (am - N,) = -1 and that sign (i7j) = 0. The first assumption is based on the fact the possible variations in al and ym are much larger that those in 01 and m. However, the first assumption can be checked and corrected, if necessary. On the contrary, the second assumption corresponds to neglecting the friction torque between the pin and the bushing. Although the error introduced by this approximation may not seriously degrade the accuracy of the computation, use of this approximation is basically a matter of user's discretion. The computed value of 7,* monotonically changes as the value of hi changes. Therefore, any conventional numerical scheme for the solution of nonlinear equations, such as the bisection method or Newton's method [6], can be incorporated into step 5 to find a satisfactory value of Tm+1. 4 RESULTS In this section results obtained by the analysis presented in the previous sections are discussed. First, a general discussion of the result is presented. Then, factors affecting the load distribution of the system are discussed. Finally, the chain tension data for the

experimental system studied by Naji and Marshek in [2,3] are compared with the results obtained by this simulation. 4.1 General If friction is neglected, Eq. (3.21) to (3.27) are reduced to * sin pj T* (4 T;r 1=-Tsin (4.1) sin vj sinv ( =sinv= (+). T (4.2) J sin vj The ratio between tensions of two adjacent links and the ratio between the normal contact force on the sprocket and the link tension are determined basically by the angles tp1 and vj (see Fig. 3.2). Friction changes the ratio slightly. Note that angle (pj is the angle between link and the normal vector at the contact point and angle vj is the angle between link j+l and the normal vector. The angles of links j and j+l with respect to tooth j (angle aj and yj in Fig. 3.2 respectively) are little affected by the change in the location of the contact point. Thus, angles vj and Tj are affected primarily by the location of the contact point of roller j. This implies that locating contact points correctly is very important for accurate prediction of link tensions and contact forces. The ratio between tensions of two chain spans wrapping the sprocket can be expressed, from Eq. (4.1), as Tm+i = sin hi sin T2 sin m4.3) *i_ (4.3) T* sin v sin v2 sin vm All the angles except ql and vm shown in the right hand side of Eq. (4.3) can be computed from X. Angles (p and vm are determined from the instantaneous angle of the sprocket with respect to the two chain spans. For fixed qp and v, the tension ratio is a monotonic

function of ~. In Case I, it holds that 0 < p <r - v < (j = 1, 2,..., m). More specifically, it holds that sin (p1 ~ sin vl, sin oT < sin vi (' = 2, 3,..., m-1), and sin pm < sin v,. The equalities in the first and the third equations hold at the instants of engagement and disengagement respectively. This yields, from Eq. (4.3), that Trn < T7. Due to the numbering convention for chain links and the direction of rotation of the sprocket we assumed (as shown in Fig. 2.2) it should be satisfied that T* > T for the driven sprocket. Therefore, Case I cannot occur in the driven sprocket. Similarly, it holds that Tm7+ > T1 in Case V and cannot occur in the drive sprocket. Cases II and IV would occur if the tension ratio T*+l/T were to fluctuate according to the periodic changes in the angle of the sprocket, or, angles Ad and vm. This condition is not met by a normal chain drive in motion. Case III can occur for both the drive sprocket and the driven sprocket. In Case I, the tension ratio T*+,/IT is smaller than in Case III. Case I occurs in the drive sprocket if the slack tension is extremely low relative to the tight tension and/or only a small fraction of the total number of teeth are in contact with the chain. In this case, the roller moves outside the tooth as the sprocket rotates and derailment can possibly occur (see Fig. 2.3). A similar argument can be made for the driven sprocket in Case V. Obviously Cases I and V do not represent desirable operational conditions. It is possible to compute the value of the tension ratio at which the distribution of contact points transitions from Case I or V to Case III. Case III is the most desirable case and it occurs in most chain drives designed in accordance with conventional design practice. This line of arguments provides the theoretical basis for the conventional practice of selecting relative shaft position. It is said that the most favorable operating conditions are obtained when the drive approximates the horizontal [9]. It is also recommended that the larger sprocket should be placed in the lower position if the drive is not horizontal. When

the slack span is horizontal, the slack tensions of the two sprockets are positive due to gravity. As the inclination angle of the drive increases, the slack tension of the sprocket in the lower position decreases and it becomes more probable that it will exhibit the contact point distribution of Case I (for a drive sprocket) or Case V (for a driven sprocket) rather than Case III. Since the possibility of having a Case III distribution increases as the number of teeth in contact increases, the larger sprocket should be placed in the lower position. If the condition for a Case III distribution is not satisfied, use of a tensioner sprocket should be considered. Note that a symmetric distribution of the contact points on the drive sprocket in Case I can be found from a distribution of contact points on the driven sprocket in Case V, in which Sj of the drive sprocket is replaced by -~mj Similar arguments can be made for distributions of the contact points of both the drive sprocket and the driven sprocket in Case III. Furthermore, the link tensions and the contact forces on the sprocket can be exactly symmetric (or T* of the drive sprocket is equal to T* + of the driven sprocket, etc.), if J m-j+ 1 there were no friction. As was discussed in Subsection 3.2, the directions of the friction forces on the sprocket are not affected by whether the sprocket is the drive sprocket or the driven sprocket. They are determined by the direction of the movement of the contact points on the sprocket and the direction of the rotation of the sprocket. It is assumed that the sprocket rotates counter-clockwise. The contact points on the drive sprocket in Case I move in the same direction as those on the driven sprocket in Case V. In Case III, the roller comes in contact with the sprocket at a point very close to transition point R as it engages with the sprocket. Then it moves toward the other transition point L until it disengages from the sprocket. Thus if friction is involved, symmetry between the drive sprocket and the driven sprocket no longer exists.

4.2 Factors Affecting the Load Distribution There are several factors that can affect the load distribution of the chain and the sprocket. The effects of the tension ratio of the two chain spans, friction, and the dimensional variations in the chain pitch and the sprocket pitch are discussed in this section. The link tension shown in the figures of this section represents the tension which a link experiences from the instant of engagement to that of disengagement as the sprocket rotates. The angle of rotation in the figures of this section is defined such that it is zero when the link of interest engages with the drive sprocket or when the link disengages from the driven sprocket.3 Since the two chain spans are assumed to be parallel, the angle of rotation is 180 degrees when the link engages with the driven sprocket and disengages from the drive sprocket. Therefore, the angle of rotation changes from zero to 180 degrees for the drive sprocket and from 180 to zero degrees for the driven sprocket. The ratio of the slack tension to the tight tension has significant effect on the load distribution of the sprocket and the chain. Fig. 4.1 and 4.2 show the tension of a pin link on the drive sprocket and the driven sprocket respectively. They were obtained with No. 2040 standard steel chains and matching 31-tooth steel sprockets under five different ratios of the slack tension to the tight tension. The same drive and the same tension ratios were used in [2 and 3]. The load distributions have almost the same value up to a certain sprocket rotation angle regardless of the tension ratio. This is because of the fact that the contact points of the first several rollers at the side of the tight tension stay very close to the transition points as shown in Fig. 4.3 and 4.4. Existence of friction between the bushing and the roller and between the roller and the sprocket makes the load distribution of the drive sprocket different from that of the driven 3The link tension is maximized when the angle of rotation is zero. The definition of the angle of rotation is the same as used by Naji and Marshek [2,3].

sprocket. Fig. 4.5 shows the tension of a pin link for various values of the coefficient of kinetic friction for the bushing and the roller. The slack tension is equal to the tight tension in Fig. 4.5. If no friction exists, the tension on the drive sprocket is exactly the same as that on the driven sprocket. The tension drops more quickly after the chain link engages with the drive sprocket as the friction coefficient decreases. The friction between the pin and the bushing has little influence on the load distribution. The dimensional variations in chain pitch and sprocket pitch significantly affect the load distribution. A negative error in the sprocket pitch showed almost the same influence on the load distribution as a positive error of the same magnitude of the chain pitch. The net error in chain pitch used in Fig. 4.6 is defined as the error in the chain pitch minus the error in the sprocket pitch. The effect of the net error in the chain pitch is reduced as the chain elongates as shown in Fig. 4.6. Fig. 4.7 and 4.8 show simulated tension of a pin link for one of the chain drives measured by Naji and Marshek in [2 and 3]. The net error in the chain pitch used in the simulation is approximately 0.007 inch per pitch (or 0.084 inch per foot). The coefficient of kinetic friction used in the simulation is 0.08 which is within the typical range for lubricated steel surfaces. Estimated link tension shown in Fig. 4.7 and 4.8 shows an excellent agreement with the measurement by Naji and Marshek (as shown in Fig. 4.9 and 4.10)4 respectively. Estimated values of the coefficients of the kinetic friction and the net error in the chain pitch are used in the new analysis for a quantitative prediction of the contact phenomenon in roller chain drives. Estimated values of the coefficients of the kinetic friction and a constant value for the pressure angle were used in previous analyses. 4Fig. 4.9 and 4.10 are listed here with the permission of Dr. M. R. Naji.

The primary argument in favor of the new analysis is that the net error in the chain pitch can be measured or, at least, related to the physical dimensions of the chain and the sprocket. The effect of the dimensional variations in the chain pitch and the sprocket pitch can be fully appreciated by the new analysis. Finally, a recent experimental investigation [10] found strong spikes in the chain sideplate force as the point where the roller exits the load sprocket and at the point where the roller enters the motor sprocket. Chain speeds were in the moderate to high range in this study, and this suggests that the load distribution problem may be significantly complicated by dynamic factors (impact and inertia) at higher speeds. This is an excellent area for future research. 5 CONCLUSIONS The contact phenomenon between a roller chain and sprocket was investigated using a system model which includes significant geometrical detail as well as friction effects. The distribution of the contact points on the sprocket is derived from the model (rather than assumed), hence a logical prediction of the contact phenomenon is possible. Comparison of the simulated link tension with existing experimental data [2, 3] showed an excellent agreement. It was observed that the distribution of contact points on the sprocket is affected by the external loading condition, and this relationship was determined. The distribution of the contact points can be classified practically into two different cases. In the first case the contact between the chain and the sprocket occurs outside the two points identified as transition points (symmetrically positioned on each tooth). This tends to happen if the ratio of the slack tension to tight tension is extremely low and/or if the number of teeth in contact is small. The load on the sprocket is more concentrated on the first several teeth at the side of the tight span in this case than otherwise. Moreover, the rollers in contact with the

sprocket can be easily stripped out from the sprocket, and the motion of the chain drive can become jerky. This can be avoided by use of an idler sprocket. In the second case, the contact between the chain and the sprocket occurs between the transition points and reliable power transmission is achieved. This condition is satisfied by most chain drives designed in accordance with conventional guidelines on the selection of relative shaft position. The locations of the transition points vary depending on the dimensional variations in the chain pitch, t the sprocket pitch, and the radius of the working curve. Those variations are caused by manufacturing tolerance and wear. The transition points of a roller chain drive with dimensions that correspond to the nominal values given in the ANSI standard [5] reside close to the border with the seating curve. As the chain elongates the transition points move toward the inside of the working curves. But, the transition points of any chain in workable condition will stay within a small portion of the working curve next to the seating curve. The contact points of the first several rollers from the end of the tight span remain very close to the transition point at the side of tight span. Suggestions for correct interpretation of some terminologies used in the relevant ANSI standard [5] can be made. In the standard the pressure angle5 is given for different cases. The "pressure angle for a new chain" is denoted as the pressure angle of the inner end point of the working curve (the border point between the working curve and the seating curve). This seems to be a good approximation for the first several rollers from the tight span, since they remain very close to the transition point and the transition point for a new chain resides very close to the inner end point of the working curve. But, it should be also noticed that some of the rollers may contact the seating curve. The "minimum pressure angle" is denoted in the standard as the pressure angle of the outer end point of the working curve. This should not be interpreted as an implication that a fully elongated, but 5In fact, the definition of pressure angle cannot be found from the standard. It is defined in Ref. 1 as the angle (p in Fig. 1.1.

workable, chain may have this small value of the pressure angle. REFERENCES 1. Binder, R. C., 1956, Mechanics of the Roller Chain Drive, Prentice-Hall, Englewood Cliffs, New Jersey. 2. Naji, M. R., 1981, On Timing Belt and Roller Chain Load Distribution, Ph. D. Dissertation, University of Houston, Houston, Texas. 3. Naji, M. R. and Marshek, K. M., 1983, "Experimental Determination of the Roller Chain Load Distribution," ASME Journal of Mechanisms, Transmission, and Automation in Design, Vol. 105, pp. 331 - 338. 4. Naji, M. R. and Marshek, K. M., 1984, "Analysis of Roller Chain Sprocket Pressure Angles," Mechanism and Machine Theory, Vol. 19, No. 2, pp. 197 - 203. 5. American National Standards Institute, 1975, "Precision Power Transmission Roller Chains, Attachments, and Sprockets," ANSI Standard B29.1 - 1975, New York. 6. Conte, S. D. and de Boor, Carl, 1972, Elementary Numerical Analysis, McGraw-Hill. 7. Kim, M. S., 1990, Dynamic Behavior of Roller Chain Drives at Moderate and High Speeds, Ph. D. dissertation, University of Michigan. 8. Gerbert, G., 1989, "Tooth Action in Chain and Timing Belts," International Power Transmission and Gearing Conference - Vol. 1, Book No. 10288A - 1989, ASME, NY, pp 81 - 89. 9. Bouillon, G. and Tordion,G. V., 1965, "On Polygonal Action in Roller Chain Drives", ASME J. Engineering for Industry, Vol. 87B, pp 243 - 250. 10. Conwell, J. C., 1989, An Examination of Transient Forces in Roller Chain Drives, Ph.D. dissertation, Vanderbilt University.

W M_ \ 3600 360'A = 35 + A = 35 + Notes: 561 1. Coordinate 5 and the integers on the N tooth surface are not parts of the standard. R = 1.005 RR + 0.0015 (in inches) 2. E and F can be computed from other M = 1.6 R cos A constants shown in the right or found from =2.8 Ros 180 the standard. N Fig. 1.1 Standard Sprocket Tooth Form for Roller Chains

Trajectory of the center of roller j+1 ~~~~~/ ~ Trajectory of the center of roller j A Roller j+1 A'\ / \ HRollerj 'I \F / / 0 \ / * The contact point on the seating curve cannot be determined from physics (a) Previous Models Trajectory of the center of roller j+1 Trajectory of the center of roller] / H' \ X8Roller ]+1\ ~chain pitch Note: the radii difference between the roller and. / / the seating curve is shown exaggerated. / * Radii difference between the roller and the seating curve (curve DE) considered * Contact point on the seating curve can be located (b) New Model Fig. 1.2 Differences in Modeling of the Contact Point on the Seating Curve Seating Curve

Link m+1 Roller m Link m Roller m+1 Roller 0 Link 1 Roller 1 Link 2 Roller 2 Fig. 2.1 Roller Chain Drive

~j+! 4-7-~ 1 24 teeth 3|2 4 teeth...................................................................................... —..-......-''. —.-. —...................................................... Transition point R I Transition point L -4 -, - - -4 -3 -2 -1 0 1 2 3 4 Fig. 2.2 Relationship between the Locations of the Contact Points of Two Adjacent Rollers

~j+l 4 0.................................................................. I, I I, " I.I / R oer2 ller 3 - -- - - - --- 7 Case I / Roller 2 - ----- p i, i i i Roller 1.... FI g. 2.3 D Istr u of Transition point R Roller.,- Roller- 1 t f --- —~- Roller 2. Case III Transition 4; — l — -" — Roller 3 -1 4 ~ ~1 4Roler 3. 45 44 53 52 514R 1 2 3 -1 0 1 2 3 4 Fig. 2.3 Distribution of Contact Points in Two Typical Cases

-0-a- /N=12, e= 0 5R^~~~~~~~~~ 5R^~~~~~ -i^ — N= 12, E = 0.06 inch 1.12 1.12 O N=24,e=O -4-4- N= 24, E= 0.06 inch 1.1 1.1 N: number of teeth e: error in the radius 1.08A^^~~~~~~ 1.08^~~~ ~of the working curve 1.08 t ^ /O 1.08 1.06.3 1.06 1.04 1.04 1.02 `fl 1.02 1.02 NO. 80 chain(pitch: 1 inch) No. 40 chain(pitch: 1/2 inch) 1 -3 5x1 0- 10 0 1- 2x103 Error in the chain pitch - inch Error in the chain pitch - inch Fig. 2.4 Variations in 5 Coordinate of Transition Point R

-Pi+2 Pin link i+2 \ pi+1 P. Roller link iPi+I -i+i S i+i /i^^ pi+, Roller i+1 Pi- F1i+ qi+l Footh i+g 1 Roller link i+l ti+1 - Ni pi Qi Tooth i Q u Pi "' ' P i jVj+ Pj Pin link i Fig. 3.1 Free Body Diagram of Elements of Roller Chain

Rollerj Rollerj +1 \Y PJ Link j I~Y;r \ Roller-1 V- p ~ /P ig. 3.3 Two Equivalent Contact Loadings on a Cylinder tan-I y

Chain number: 2040 (Double pitch chain, pitch: 1 inch) Number of sprocket teeth: 31 Roller - bushing kinetic friction coefficient: 0.08 Bushing radius/roller radius: 0.75.8 Net error in the chain pitch: 0 C): \ Low tension/High tension ~I _ -o-o-.03 mD -.25 -0-o0 —.50 oX -Q — ~.75 CD \ - 1.0 0. Angle of rotation - degree Fig. 4.1 Pin Link Tension on the Drive Sprocket

Chain number: 2040 (Double pitch chain, pitch: 1 inch) Number of sprocket teeth: 31 Roller - bushing kinetic friction coefficient: 0.08 8 \ ~ Bushing radius/roller radius: 0.75 c.8 -Net error in the chain pitch: 0 Ca^ | \~~ ~Low tension/High tension. -o-o-.03 O.^~~6 ~-6- \ — A-.25 / >' [ i-c.75 ^ \ -~~~~~-0-0.50 {} ^ V -VI 1.0 N -~.2 0 ' ' 0 30 60 90 120 150 180 Angle of rotation - degree Fig. 4.2 Pin Link Tension on the Driven Sprocket

1.2: ~ Hightension \ z\ g -4 -:25'-o-d.03 '-.8 O 5 0 30 60 90 120 150 180 210 Angle of rotation - degree Fig. 4.3 Locations of Contact Points for the System in Fig. 4.1 -1.2 E -4 - --.03 \.) 0 Hgte- s.50 o ~ -.8 4 -.75 0- l 0, ~ 1.0 ~ -1.2 0 30 60 90 120 150 180 210 Angle of rotation - degree Fig. 4.4 Locations of Contact Points for the System in Fig. 4.2

Chain number: 2040 (Double pitch chain, pitch: 1 inch) Number of sprocket teeth: 31 Ratio of the slack tension to the tight tension: 1.00 Q.8 1 Net clearance on the chain pitch: 0.000 inch 0 Cn )c ^\ Sprocket, Friction coefficients* Z -U-Ea - No friction** C.6D\ -6 — Drive, 0.06 I^~~~ ^\\<-0 --- Drive, 0.12 \ -+\ —4~- Driven, 0.06 ^ \\\\\i~~ -*-*- Driven, 0.12 C JI.2 * Constant value of the coefficient of kinetic friction is used for pin/bushing, bushing/roller, and roller/tooth surfaces. ** The tension of the chain on the drive sprocket is exactly the same as that on the driven sprocket in this case. 0 ~0 30 60 90 120 150 180 Angle of Rotation (in degrees) Fig. 4.5 Effects of Friction on Pin Link Tension

~.8 ~ ~ Net error in the chain pitch 8 \ -0-0-.000 I ~ -6=-,.005 inch..6 ~ - -0.010 inch CZ 0.~ 0 I, 0 30 60 90 120 150 180 Angle of rotation - degree (a) Pin Link Tension on the Drive Sprocket o.8 Net errorinthechain pitch: P~ -0-.000 a) -&-A —.005 inch.6 - E - -.010 inch.4 N CZ 0 0 30 60 90 120 150 180 Angle of rotation - degree (b) Pin Link Tension on the Driven Sprocket Fig. 4.6 Effects of the Net Error in the Chain Pitch on Link Tension

Chain number: 2040 (Double pitch chain, pitch: 1 inch) Number of sprocket teeth: 31 Roller - bushing kinetic friction coefficient: 0.08 \8 ~ Bushing radius/roller radius: 0.75.8 \ Net error in the chain pitch: 0.007 inch 0 Cn 1^~~~ \ ~ Low tension/High tension r^ \~tr,~~~ -0-0-.03 o.6 -~^ ---.25 -0-0-.50 n -~ ----.75 CD -* — 1.0 N o.4 0 C 0 Cn C J.2 0 0 30 60 90 120 150 180 Angle of rotation - in degree Fig. 4.7 Simulated Pin Link Tension on the Drive Sprocket

Chain number: 2040 (Double pitch chain, pitch: 1 inch) Number of sprocket teeth: 31 Roller - bushing kinetic friction coefficient: 0.08 Bushing radius/roller radius: 0.75.8 Net error in the chain pitch: 0.007 inch 0 ~S t~~ ^ ~~Low tension/High tension 0,g~~~) ^ ~~~-u-U-.03. 6 -AC.25 c \-0-0-.506 _0 — *- 1.0 N 0.4 0 0 0 30 60 90 120 150 180 Angle of rotation - in degree Fig. 4.8 Simulated Pin Link Tension on the Driven Sprocket

.8 z_ 0.) 0 ~-^~.0 0 30 60 90 120 150 180 Fig. 4.9 Measured Pin Link Tension on the Drive Sprocket (from Ref. 2)

i..8 H z Ir. Q: z 0 4 w z.2 0.0 0 30 60 90 120 150 180 Fig. 4.10 Measured Pin Link Tension on the Driven Sprocket (from Ref. 2)

3 9015 02828 3813 THE UNIVERSITY OF MICHIGAN DATE DUE /o- (~7 ~'4