Mechanics and Sliding Friction in Belt Drives With Pulley Grooves

Transcription

1 See discussions, stats, and author profiles for this publication at: Mechanics and Sliding Friction in Belt Drives With Pulley Grooves Article in Journal of Mechanical Design March 2006 Impact Factor: 1.25 DOI: / CITATIONS 23 READS 2,246 1 author: Robert G. Parker Virginia Polytechnic Institute and State Univ 137 PUBLICATIONS 2,852 CITATIONS SEE PROFILE All in-text references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. Available from: Robert G. Parker Retrieved on: 09 May 2016

2 Mechanics and Sliding Friction in Belt Drives With Pulley Grooves Lingyuan Kong Robert G. Parker 1 Professor parker.242@osu.edu Department of Mechanical Engineering, Ohio State University, 650 Ackerman Rd., Columbus, OH The steady mechanics of a two-pulley belt drive system are examined where the pulley grooves, belt extension and wedging in the grooves, and the associated friction are considered. The belt is modeled as an axially moving string with the tangential and normal accelerations incorporated. The pulley grooves generate two-dimensional radial and tangential friction forces whose undetermined direction depends on the relative speed between belt and pulley along the contact arc. Different from single-pulley analyses, the entry and exit points between the belt spans and pulleys must be determined in the analysis due to the belt radial penetration into the pulley grooves and the coupling of the driver and driven pulley solutions. A new computational technique is developed to find the steady mechanics of a V-belt drive. This allows system analysis, such as speed/ torque loss and maximum tension ratio. The governing boundary value problem (BVP) with undetermined boundaries is converted to a fixed boundary form solvable by a general-purpose BVP solver. Compared to flat belt drives or models that neglect radial friction, significant differences in the steady belt-pulley mechanics arise in terms of belt radial penetration, free span contact points, tension, friction, and speed variations. DOI: / Introduction 1 Corresponding author. Contributed by the Power Transmission and Gearing Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received December 15, 2004; final manuscript received June 23, Review conducted by Teik C. Lim. The mechanics between belt and pulley in their contact zones has attracted attention since Euler 1 published on it in Belt-pulley mechanics impact the important industrial considerations of belt tension and life, power transmission efficiency, maximum transmissible moment, and noise. For example, for serpentine belt drives used in the automotive industry, belt tensions are desired to be as small as possible to reduce belt fatigue and prolong bearing life, yet power loss from belt slip is unacceptable. This requires understanding of belt-pulley interactions. Current practically observed behaviors still differ considerably from theoretical prediction for certain belt drives, as communicated by belt drive manufacturers. Belt-pulley friction modeling and interactions with the grooves appear to be major sources of the error and perhaps the least understood aspects of the mechanics. Different theories have been established for the belt-pulley interaction. Comprehensive reviews of belt mechanics can be found in the works of Fawcett 2 and Johnson 3. Although some models were developed on the basis of belt shear deformation theory 4 6, belt creep theory is still the most widely adopted. In this theory, the belt is assumed to be elastically extensible, friction develops due to the relative slip between the belt and pulley, and a Coulomb law describes the belt-pulley friction. For a two-pulley belt drive where the driver and driven pulleys have the same radius, Gerbert 7 used this theory and established that the contact zones for a flat belt are divided into slip and adhesion zones. Bechtel et al. 8 and Rubin 9 incorporated belt inertia effects into this creep theory and presented improved solutions for twopulley belt drives. Kong and Parker further extended this model by incorporating belt bending stiffness and applied it to twopulley belt drives 10 and multiple-pulley serpentine belt drives with tensioner assemblies 11. All of the above models are for flat belt drives without consideration of the pulley grooves. Fewer researchers have studied grooved pulley drives such as V-belt systems. Hornung 12 considered the interaction between a V-belt and the pulley grooves. Due to computational constraints at the time, only qualitative discussion and rough approximate solutions are obtained. Gerbert and Sorge 13 established an effective model to examine sliding of the V-belt in the grooves. They analyzed individual driver or driven pulleys isolated from the rest of the system. The governing equations of the belt on a single pulley are solved by a shooting technique where the boundary value problem BVP is cast as an initial value problem IVP and the boundary conditions are specified at only one point. The equations are then integrated until another point is found that satisfies certain conditions and can serve as the other boundary. The disadvantage of this method is that it is difficult to systematically obtain solutions for the physical inputs that are typically specified. The limitation to singlepulley analysis, where one cannot naturally link the driver and driven pulley solutions, prevents straightforward application-tosystem analysis where multiple pulleys always exist and their solutions are coupled. Accordingly, this method cannot be applied directly to study system behavior nor calculate system outputs, such as power efficiency and maximum transmitted moment. In this paper, Gerbert and Sorge s model is adopted and applied to a two-pulley system where the belts sliding in the driver and driven pulley grooves are coupled by the two free spans. The BVP for the entire drive is solved for specified span tensions. Belt radial penetration into the grooved pulleys i.e., wedging leads to initially unknown contact points between the belt spans and pulleys. Consequently, the steady motion is governed by a BVP with unknown boundaries. This is different from many studies in the literature where the boundaries of the belt-pulley contact arcs are assumed to be fixed at the points of common tangency of the driver and driven pulleys The tangential friction that transmits the power and radial friction from seating and unseating of the belt are modeled. Belt inertia in the tangential and normal directions is fully considered. A computational approach is developed to solve the BVP for the entire drive for specified span tensions. Based on this model, the steady mechanics of a twopulley drive are analyzed and some important design criteria, including power efficiency and maximum tension ratio, are examined. 2 Governing Equations of a Belt Sliding in Pulley Grooves Figure 1 a shows cross sections of the belt and the pulley groove. The groove wedge angle is. Friction between the belt and pulley develops in the sliding plane, where the belt edge 494 / Vol. 128, MARCH 2006 Copyright 2006 by ASME Transactions of the ASME

3 Fig. 1 Belt sliding in pulley grooves: a cross section and acting forces, and b velocities contacts the pulley groove, because the contacting material particles and associated relative sliding velocity vector Ṽ s exist in this sliding plane Fig. 1 b. The friction force is projected into the normal plane, which bisects the belt and is perpendicular to the pulley axis, to establish the equations of motion. The angle s is the angle between Ṽ s and the normal plane. is the sliding angle defining the direction of the belt relative sliding velocity vector projected in the normal plane. Figure 1 b shows that the Z component of the belt relative sliding velocity vector Ṽ s is V s sin s. On the other hand, this velocity component can also be written as V s cos s cos tan. Equivalence of these two expressions leads to the relationship between these angles 13 tan s = tan cos 1 where /2 s /2. Figure 2 shows the free body diagram of a segment of an extensible belt in the pulley grooves. The belt is modeled as an axially moving string. An Eulerian formulation is adopted for the control volume. The model is based on that in 13 except that belt inertia through longitudinal and centripetal accelerations is considered here while neglected in 13. Note that Figs. 1 and 2 are similar to those in 13 but with additional vectors GV due to the consideration of belt inertia. For steady motions, conservation of mass requires that G = m s V s = const 2 where G is the mass flow rate, m s is the belt mass density per unit length, V s is the belt speed, and s is the arclength coordinate along the belt. Balance of linear momentum projected along the belt tangential and normal directions in the normal plane leads to d F GV =2p sin sin + cos s sin + 3 ds Fig. 2 a Free body diagram of a moving curved string including belt inertia effect and b pulley velocity r, belt segment velocity V s, and relative speed V s s F GV =2p sin cos cos s cos + 4 where F is the belt tension, p is the normal compressive pressure between the belt and the pulley groove surfaces, is the inclination angle between the belt velocity and the velocity of the overlapping point on the pulley Fig. 2, is the Coulomb friction coefficient, =ds/d is the belt radius of curvature, and is the natural angular coordinate Fig. 2. The belt radial penetration is governed by 13 x = R r s = 2p z k = 2p k cos + sin s 5 where R is the constant belt pitch radius, r s is the belt radius coordinate, k is the radial spring stiffness, and p z is the pressure load component exerted on the belt along the pulley axial direction Fig. 1 a. k is determined mainly by the belt cross-sectional geometry and material properties. Gerbert 7 gives an approximate estimation k=12 H/B E z tan, where H is the belt height, B is the belt width top side of V-belt, and E z is the belt modulus of elasticity in the transverse direction. r and are polar coordinates with origin at the pulley center Fig. 2. Substitution of the geometric relations ds=rd /cos, r =R x, =ds/d, and = d =d d into 3 and 4 leads to the polar coordinate equations T = F GV =2p sin sin + cos s sin + R x cos =1 2p T sin cos cos s cos + R x 7 cos where T=F GV is the belt tractive tension and is the derivative with respect to the angular coordinate. Because tan =dr/ rd =r /r Fig. 2, substitution of x=r r yields x = R x tan 8 To complete the problem, a constitutive law relating belt tension F and velocity V is needed. Following 8 10,19, the constitutive law is F = EA m 0 V/G 1 ÞT = EAm 0 G 2 V/G EA 9 where EA is the belt longitudinal stiffness and m 0 s is the belt mass density per unit length in the stress-free state, which can be measured. The mass flow rate G is not known initially and is determined in the analysis. Comparison of 9 and Eq. 4 in 13 shows that the constitutive laws are consistent with each other. Velocity analysis from Fig. 2 b reveals that V cos = r + V s sin V sin = V s cos 10 Elimination of the sliding velocity V s and use of r=r x and 9 lead to tan = cos R x EAm 0 G 2 / G T + EA 11 sin In summary, the motion of the belt sliding in the grooves is governed by the three differential equations 6 8 and the four algebraic equations 1, 5, 9, and 11. These equations apply to the entire belt-pulley contact zone on a pulley. The governing equations seem complicated at first sight because they involve coupled differential and algebraic equations with many variables. The primary variables are T,, and x, whose behavior is governed by 6 8. All other variables such as V, p, s,, etc. are intermediate variables that can be explicitly expressed in terms of the three basic variables T,, and x based on the four algebraic equations 1, 5, 9, and 11. In other words, the steady motion of the belt in the belt-pulley contact zone could 6 Journal of Mechanical Design MARCH 2006, Vol. 128 / 495

4 a solution, the governing equations for the steady motion of the whole system, including the two belt-pulley contact zones with undetermined boundaries, are transformed into a standard boundary value problem form on a fixed domain, namely, u t = F t,u t a t b Fig. 3 Two-pulley belt drive with belt penetration into pulley grooves be cast as a boundary value problem for T,, and x, governed solely by three differential equations. Realization of this point aids understanding of the subsequent solution procedure for the full two-pulley system. Nevertheless, the formulation 6 8, 1, 5, 9, and 11 is retained for clarity of equations and convenience of numerical solution. Within a contact zone there is no adhesion zone where the belt penetration, speed, and tension remain constant, as exists in a flat-belt model Gerbert and Sorge 13 gave a mathematical proof of the nonexistence of an adhesion zone. An alternative explanation based on physical insight is given here that clearly shows that an adhesion zone cannot exist in the grooved pulley model. Taking the driven pulley as an example, suppose there is an adhesion zone BC in the belt-pulley contact zone Fig. 3.The only possibility is that it exists in the middle part of the contact zone because belt penetration varies in the entry and exit zones. For this assumed adhesion zone BC, the belt penetration and tension must be constant and the belt speed including that at B must be the same as the linear velocity of the overlapping point B on the pulley, i.e., V B =r B 1, where r B is the belt radius at B and 1 is the rotation speed of the driven pulley in this paper, the subscripts 1 and 2 represent the driven and driver pulley, respectively. At an arbitrary point A in the entry zone outside BC, the belt tension is less than that at point B because the driven pulley entry zone connects with the slack span. According to the constitutive law 9, the belt velocity at A is also smaller than that at B, i.e., V A V B. Because the belt velocity component along the corresponding pulley tangential direction is always less than or equal to its absolute speed, we have V A cos A V A V B. Furthermore, the speed of the overlapping point B on the pulley is less than that at A, r B 1 r A 1, due to the lesser belt penetration in the entry zone. Thus we have V A cos A r A 1, and the belt tangential speed is less than that of the pulley of the same point. This contradicts the requirement that the tangential friction must be opposite the direction of belt travel on the driven pulley. Consequently, the existence of an adhesion zone on the driven pulley is not possible. There is, however, a single point where the belt moves purely in the pulley tangential direction =0 at the transition from seating to unseating. Similar reasoning applies to the driver pulley to rule out the existence of an adhesion zone there. 3 Solution for a Symmetric Two-Pulley Belt Drive The steady motion analysis is presented for a two-pulley belt drive. The driver and driven pulleys are assumed to have the same radius, wedge angle, and friction coefficient. The method presented, however, extends naturally to a general belt drive with different pulleys. The specified parameters are: driver and driven pulley pitch radius R, center distance between the two fixed pulleys L, belt longitudinal stiffness EA, constant rotation speed 2 of the driver pulley, friction coefficient, pulley wedge angle, belt mass density per unit length m 0 s in the stress-free state, radial spring stiffness k, and belt tractive tensions in the slack and tight spans T s and T t, respectively. Figure 3 shows the belt drive. The belt-pulley contact points C 1 C 4 are not known a priori and must be determined. To permit g u a,u b =0 12 where F, u, and g are n-dimensional vectors and F and g may be nonlinear. The undefined boundary requires special treatment. The wrap angles of the belt-pulley contact zones Fig. 3 for the driver and driven pulley are 1 and 2, respectively. They are not known at this point. Nevertheless, they are used to define the following nondimensional variables ˆ 1 = 1 ˆ 2 = 2 0 ˆ 1, ˆ Correspondingly, the governing differential equations for the belt on the driven pulley 0 ˆ 1 1 are dt 1 =2p 1 sin tan 1 + cos s1 tan 1 cos 1 + sin 1 d ˆ 1 R x d 1 d ˆ 1 = 1 2p 1 T 1 sin cos s1 cos 1 sin 1 tan 1 R x dx 1 = R x 1 tan d ˆ 1 To incorporate the unknown constant 1 in the standard BVP form 12, it is defined as the unknown function 1 ˆ 1 governed by d 1 ˆ 1 =0, 0 ˆ d ˆ 1 Similarly, the governing equations for the driver pulley on 0 ˆ 2 1 are dt 2 =2p 2 sin tan 2 + cos s2 tan 2 cos 2 + sin 2 d ˆ 2 R x d 2 d ˆ 2 = 1 2p 2 F 2 sin cos s2 cos 2 sin 2 tan 2 R x dx 2 = R x 2 tan d ˆ 2 d 2 =0 21 d ˆ 2 The intermediate variables, such as p 1, p 2, s1, s2, 1, 2, etc., are still governed by the four algebraic equations 1, 5, 9, and 11 with the subscript 1 or 2 attached for the driven and driver pulleys, respectively. The following boundary conditions are evident for the driven and driver pulleys x 1 0 =0, x 1 1 =0, T 1 0 = T s, T 1 1 = T t / Vol. 128, MARCH 2006 Transactions of the ASME

5 x 2 0 =0, x 2 1 =0, T 2 0 = T t, T 2 1 = T s 23 Additional conditions come from the belt in the pulley grooves being tangent to the free spans at the four belt-pulley contact points C 1 C 4 Fig. 3. Suppose the global coordinate origin is located at the midpoint of the slack span Fig. 3, and the as yet unknown slack span length is. Both spans are straight for a string model of the belt no bending stiffness. The coordinates of the two pulley centers are then x o2 = 2 R cos 2 2 1, y o2 = R sin x o1 = 2 + R cos , 24 y o1 = R sin The pulley centers have fixed distance L x o1 x o2 2 + y o1 y o2 2 = L 2 26 The coordinates of the two belt-pulley contact points for the tight span are determined geometrically as x C2 = R cos 2 + R cos y C2 = R sin R sin x C1 = 2 + R cos R cos y C1 = R sin R sin The tight span goes through point C 2, and it is tangent to the belt in the driver pulley groove. Its slope can be calculated from the three angles 2 0, 2 1, and 2 on the driver pulley as z 2 =tan The line of the tight span can then be written as y y c2 =z 2 x x c2. Similarly, working from the driven pulley, the tight span goes through point C 1 and its slope is z 1 =tan The tight span line is also y y c1 =z 1 x x c1. These two lines must be the same, which requires z 1 z 2 =0 y 1 x c1 z 1 y 2 x c2 z 2 =0 31 In the above analysis, the slack span length, the mass flow rate G, and the driven pulley rotation speed 1 are unknown. Analogous to 17, these unknown constants are incorporated into the standard BVP form 12 by adding three trivial ODEs d dg d 1 =0 =0 =0, 0 ˆ d ˆ 1 d ˆ 1 d ˆ 1 The standard BVP form 12 involves only coupled differential equations. The algebraic equations 26 and 31 are naturally incorporated into the form 12 by treating them as boundary conditions where the unknown constants in 26 and 31 can be written as the values at either boundary for example, 2 can be written as either 2 0 or 2 1. The total order of the 11 differential equations and 32 that define F in 12 equals the number of boundary conditions 22, 23, 26, and 31 that define g in 12. The algebraic equations 22, 23, 26, and 31 are incorporated in the definition of F from the above differential equations and require no special processing. Although the original problem has unknown boundaries, it is now defined entirely on the interval 0,1. This standard BVP form 12 can be solved by general-purpose two-point BVP solvers. This procedure is straightforward to implement, and the accuracy of the results are ensured with use of state-of-the-art solver codes. The pulley torques are M i = 0 i 2 p i cos si sin i R x i 2 /cos i d i i =1,2 33 They are useful for subsequent calculation of the system power efficiency. The torques can be obtained through direct integration of 33 once the distributions of belt tension, speed, and radial penetration have been obtained. Alternatively, by integrating these terms into the standard BVP form, they are a natural product of the BVP solution without additional effort. For example, for the torque on the driven pulley, one defines I 1 1 = p 1 cos s1 sin 1 R x 1 2 /cos 1 d and adds the following ODE and boundary condition to the above BVP formulation di 1 1 = 2 p 1 cos s1 sin 1 R x 1 2, d 1 cos with I 1 0 =0 I 1 1 is the desired torque M 1 on the driven pulley and is a direct output of the solution. Although the added ODE and boundary condition 34 are written in dimensional form over the range 0, 1, use of 13 transforms them into the necessary form on 0,1. The torque on the driver pulley can be similarly obtained. 4 Results and Discussion The belt tension F and tractive tension T=F GV differ by GV, which is nearly constant along the belt 10. In the following analysis, references to belt tension always mean the tractive tension, T. Convergence of the numerical BVP solution is not assured because of its complexity. For such a highly nonlinear problem, an initial solution guess by intuition or insight is not reliable. Instead, the initial guess is found using a trial and error method. First, the driver pulley is arbitrarily specified a wrap angle 2 and two boundary tensions T t DR and T s DR ; it is not hard to find its numerical solution from 6 8 plus G =0. The equation G =0 is added because G is an unknown constant; defining it as the field variable G and enforcing zero derivative enables natural inclusion in the standard form 12. The four boundary conditions are similar to those in 23 and 2 is specified. Next, for the driven pulley, the wrap angle 1 is arbitrarily specified, and the two boundary tensions are the same as those for the driver pulley problem. The governing equations and boundary conditions are similar to those of the driver pulley except that G =0 is replaced by 1 =0. G is specified as that computed from the driver pulley and, unlike the driver pulley, the rotation speed of the driven pulley 1 is not known. Again, the numerical solution can be found for the driven pulley. After computing the solutions for the driver and driven pulleys, the geometry of the two pulleys and the belt in their grooves is plotted with the free spans extending from the two pulleys such that the two slack spans align. In general, the two tight spans are not geometrically compatible, i.e., they do not overlap with each other Fig. 4.The parameters, such as the two wrap angles, are adjusted until the geometric compatibility condition i.e., alignment of the two spans is close to being satisfied. At this stage, the numerical solutions of the two individual pulleys, together with the wrap angles, can be used as the initial guess for the solution of the full two-pulley BVP with the same specified parameters as those in the final step of trial and error. This initial guess is typically sufficient for the numerical solution to converge. A continuation procedure avoids repetition of the above process as parameters change. After a numerical solution is obtained from Journal of Mechanical Design MARCH 2006, Vol. 128 / 497

6 Table 1 Physical properties of the example belt drive with two identical pulleys R 1 =R 2 =0.25 m L= m EA=120 kn k=900 kn/m 3 1 =1000 pm m 0 =0.108 kg/m 1 = 2 =0.4 =18 deg T s =100 N Fig. 4 Search of the initial solution guess by trial and error the above process, the parameters can be changed in small increments where each numerically exact solution obtained in the previous step serves as the initial guess for the current step. Even with such a strategy, not all parameter combinations can be solved. For instance, in the example problem, when the two span tensions are out of the range presented in the following figures, the above procedure fails due to the sharp changes of the inclination and sliding angles in the belt-pulley contact zones. Note, however, the large range of span tensions that can be handled. Even for the simpler single pulley case using an alternate numerical method, finding meaningful solutions involves numerical troubles and requires careful selection of the parameters 13. Inclusion of belt bending stiffness might smooth the sharp changes that can occur in the driver pulley exit zone and improve numerical performance. This paper analyzes two-pulley systems. If the belt mechanics on only a single driver or driven pulley are desired as in 13, the presented BVP-solver method remains a convenient technique. This is because the two free span tensions and the wrap angle, which are the three boundary conditions specified for singlepulley analysis 13, can be directly specified and readily varied as desired. This is cumbersome for the shooting method in 13 that requires trial and error. This section presents steady solution results for a belt drive with two identical pulleys. The data are specified in Table 1. Note that the friction coefficient is adopted from 13. Figure 5 shows the steady solutions with increasing tight span tension while the slack span tension remains constant. The belt penetration features are evident for large tight/slack tension ratio. In particular, note the distinctly different belt shape and penetration properties between the two pulleys entry and exit zones. The two belt free spans couple the driver and driven pulley solutions and need to be tangent to the wedging belt in the entry and exit zones. Correspondingly, the two belt free spans are no longer on the line of common tangency of the two pulleys, as in the corresponding string models of flat belt drives 8,9. Instead, the two free spans are nonparallel and this shows why torque loss exits. Figure 6shows the variations of pulley wrap angles and torques with increasing tight span tension. As the tight/slack span tension ratio increases, the wrap angles for both pulleys increase considerably. The wrap angle on the driver pulley increases more quickly than on the driven pulley. When the tight/slack span tension ratio is large, the wrap angle on the driver pulley is much larger than that on the driven pulley, and the two free spans are markedly unparallel to each other. When the tight span tension is close to that of the slack span, the two wrap angles are close. Even for such a case, extrapolation of Fig. 6 a shows that the wrap angles would be around 190 deg, larger than the 180 deg for flat belt drives or when belt wedging is ignored. Only when both span tensions drop to zero do the wrap angles become 180 deg. The torques on the driver and driven pulley differ from each other Fig. 6 b, as compared to the always equivalent driver and driven pulley Fig. 5 Steady solutions for the system specified in Table 1: a T t =700 N, b T t =1200 N, c T t =3000 N, and d T t =5000 N Fig. 6 Variation of pulley a wrap angles and b torques with tight span tractive tension for the system specified in Table / Vol. 128, MARCH 2006 Transactions of the ASME

7 Fig. 7 Variation of belt tractive tensions in belt-pulley contact zones with tight span tractive tension for the system specified in Table 1 Fig. 8 Belt radial penetrations along driver and driven contact arcs for the system specified in Table 1: a driver pulley and b driven pulley torques for flat belt drives The torque difference increases with the free span tension difference. The two torques are nearly equal when the tight/slack tension ratio is comparatively small. Figure 7 shows the tension distributions on the belt-pulley contact zones for the driver and driven pulley, respectively. Although the variation shapes are quite different from each other, they share some common characteristics. In the entry or exit zones, both belt tensions vary slowly. This is because in these zones, the belt radial penetrations are small; correspondingly, the friction force is small and does not offer significant tangential force to change the belt tensions. Figure 8 shows the belt radial penetrations in the belt-pulley contact zones. The penetration patterns on the driver and driven pulleys are quite different. For both cases, rapid changes of the penetrations occur in the entry/exit zones. But in the middle zone, the belt penetration on the driver pulley varies little, which differs from the continuously increasing penetration on the driven pulley also see Fig. 5. The belt inclination angles 2 and 1 in the belt-pulley contact zones are given in Fig. 9. Negative positive belt inclination angle means that belt penetration increases decreases at the corresponding point while the belt penetration reaches the maximum point when the belt inclination angle is zero. In the entry/exit zones, the amplitudes of the belt inclination angles are larger than those in the middle zones because of relatively small pressure between the belt and pulley, which leads to rapid seating/ unseating of the belt into the pulley grooves. The seating and unseating rate x is approximately measured by the belt inclination angle see 8. Seating of the belt in the entry zone is determined mainly by the belt entry tensions given the pulley/groove geometry, friction coefficient, and the belt properties. Because the belt entry tension of the driver pulley is higher than that of the driven pulley, the belt on the driver pulley is more quickly seated than on the driven pulley, resulting in larger amplitude belt inclination angles. This point is most apparent for the driver pulley. For the two extreme cases T t =700 N and T t =5000 N, the belt inclination angles at the entry point differ by more than 10 deg Fig. 9 a. While for the driven pulley, although the tight span tensions are very different, the belt entry tension is the same, i.e., 100 N. Accordingly, the belt inclination angles do not change much in the entry zones Fig. 9 b. Unseating of the belt in the exit zone is different from the seating action in the entry zone. The belt unseating rate in the exit zone depends not only on the belt exit tensions but also on how deeply the belt is wedged in the pulley grooves in the middle zone. For pulleys with the same belt penetration in the middle zones, the smaller the belt exit tension, the larger the belt inclination angle required to overcome the belt wedging and unseat the belt. To visualize this, imagine that the belt in the exit zone pulley grooves is pulled out by tugging on the belt in the free spans with the specified tensions. On the other hand, if the belt exit tensions are the same, the deeper the belt penetration in the middle zones, the larger the belt inclination angles in the exit zones see Fig. 9 a where the exit tension is the same for all curves. Although the belt penetrations of driver and driven pulleys in the middle zones are comparable Fig. 8, the exit tension on the driver pulley is lower than that of the driven pulley. Consequently, the belt Journal of Mechanical Design MARCH 2006, Vol. 128 / 499

8 Fig. 9 Belt inclination angles 2 and 1 along driver and driven contact arcs for the system specified in Table 1: a driver pulley and b driven pulley Fig. 10 Belt sliding angles 2 and 1 along driver and driven contact arcs for the system specified in Table 1: a driver pulley and b driven pulley inclination angle in the driver pulley exit zone is higher than its counterpart on the driven pulley Figs. 9 a and 9 b. This results in more rapid unseating in the exit zone on the driver pulley than on the driven pulley Figs. 5 and 8. The above differences in the entry/exit zones on the driver and driven pulleys cause the two spans to be nonparallel. This effect becomes more apparent with significant tight/slack tension ratio. The belt sliding angles 2 and 1 in the belt-pulley contact zones are given in Fig. 10. Belt sliding angles indicate the direction of the friction force relative to the pulley radial direction Fig. 2. They are determined by the belt sliding speed in the pulley radial direction and the relative speed between belt and pulley surfaces along the pulley tangential direction. For the driven pulley, where the belt drives the pulley, the belt speed along the pulley tangential direction is faster than that of the overlapping point on the groove surface, so the belt sliding angle is in the range deg. For the situation where the pulley drives the belt on the driver pulley, the belt sliding angle is in the range deg. For the driven pulley, when the belt reaches an extremal of belt penetration and the belt inclination angle is zero, the belt speed along the pulley radial direction is zero; correspondingly the belt sliding angle is 90 deg Fig. 10. At this point, the friction force fully contributes to overcoming the driven pulley torque, like the case of a flat belt. A similar situation exists on the driver pulley; at the maximum penetration point, the belt sliding angle is 270 deg and the friction force fully contributes to resisting the pulley driving torque. When the belt sliding angles are away from 90 deg or 270 deg, the belt moves in both the radial and tangential directions relative to the groove surfaces. Inthe extreme case of =0 deg or 360 deg, the belt moves only radially relative to the groove surfaces with decreasing penetration. A sliding angle of 180 deg corresponds to purely radial belt motion with increasing penetration. For both extreme cases, there is no friction contribution to the pulley torque. When the pulley torque increases decreases, the sliding angles adjust to make greater shares of the contact zones close to away from 90 deg for the driven pulley or 270 deg for the driver pulley, as well as increasing the wrap angles. The abrupt changes of the belt sliding angles in the exit zones are caused by the sharp decreases in belt penetration. This study does not consider belt bending stiffness, which is an important factor in belt-pulley drives 10,11,20. Inclusion of bending stiffness might make the belt penetrations and inclinations vary more smoothly in the entry/exit zones, resulting in more parallel spans even with large tension differences. Reducing these sharp changes may also improve numerical convergence for less accurate initial guesses. The power efficiency is defined as the ratio between the powers of the driven and driver pulleys, = M 1 1 / M 2 2. Figure 11 shows that increasing the tight/slack tension ratio significantly decreases the rotation speed of the driven pulley and the power efficiency. The rotation speed of the driven pulley is always less than that of the driver pulley, which is fixed at 1000 rpm. Efficiency decreases because 1 decreases with tension ratio for fixed 2 while the ratio M 1 /M 2 decreases slightly with tension ratio Fig. 6 b. For drives with appreciable free span tension difference, the rotation speed of the driven pulley and the power efficiency are much less than those for flat belt drives, where the 500 / Vol. 128, MARCH 2006 Transactions of the ASME

9 Fig. 11 Variation of system power efficiency and driven pulley rotational speed with tight span tractive tension for the system specified in Table 1 driven pulley rotation speed is close to that of the driver pulley and the power efficiency is always close to unity even for maximum transmitted moment cases 9,10. In flat belt drives, the maximum transmitted moment, or the maximum span tension ratio, is reached when all of a belt-pulley adhesion zone converts to a sliding zone. For drives with pulley grooves, there are no adhesion zones as discussed earlier, and this criterion for the maximum transmitted moment does not apply. Comparison of Figs. 6 b and 11 shows that the rotation speed of the driven pulley decreases with the driven pulley torque. Theoretically, the maximum transmitted moment is reached when the rotation speed of the driven pulley is zero, although Fig. 11 suggests vanishing driven pulley speed may be reached asymptotically. For such a case, V s =V and 1 =90 1 deg on the entire contact arc Fig. 2. In this state, friction on the driven pulley contributes to the torque as much as possible given the seating/ unseating action. Complete contribution of the friction to the torque is impossible except at the single point where 1 =0 because only the friction component in the pulley tangential direction contributes to the torque while some friction in the pulley radial direction is unavoidable due to belt seating and unseating. For the driven pulley, nonzero rotation speed always keeps the belt sliding angle 1, which gives the direction of the friction, away from 90 deg; that is, for unseating 1 0 or for seating 1 0. Neglecting the pulley grooves can significantly underestimate the maximum transmitted moment. The above results show that although the present model and comparable ones for flat belts 7 10 are based on similar creep theory assumptions where Coulomb friction prevails and its existence depends on belt extensibility and relative slip between belt and pulley surface, the consideration of pulley grooves greatly complicates the model, resulting in a two-dimensional radial and tangential contact problem between belt and pulley surfaces. This two-dimensional model is hardly studied in the literature and poses challenging mathematical obstacles to solve it. On the other hand, mechanical textbooks and handbooks emphasize only flat belt conclusions, which are better known because the models are established and far easier to solve. V-belt mechanics are normally approximated from flat belt theory. A typical example is the widely used textbook by Juvinall and Marshek 21. V-belts are treated only briefly, and the main design equation 19.3a is directly modified from equation 19.3 for flat belts with the remark: The flat-belt equations can be modified by merely replacing the coefficient of friction f with the quantity f /sin. Eq then becomes 19.13a. The present simulation results show that distinctive belt behaviors exist that cannot be inferred from flat belt models such as the belt s qualitatively different interactions with the driver and driven pulleys and no differentiation of adhesion/ sliding zones. To date, no experiments exist in the literature to validate this two-dimensional 2D model, whose validity must be evaluated on the underlying mechanics principles and engineering assumptions. The model itself is relatively new, originating in , and there is scope for incorporation of belt bending stiffness and other refinements. A primary purpose of this paper is to advance numerical solution techniques to generate results for a full twopulley drive that can be compared to experiments 13 analyzes only a single pulley. Subsequent experiments demand careful attention to measuring the belt penetrations, maintaining pulley alignment, and the like. Nevertheless, this 2D model deepens knowledge of belt mechanics and explains phenomena that cannot be explained by classical flat belt models for example, the flat belt model predicts no torque loss, as indicated here in Fig. 5 b. 5 Conclusions A computational method based on general-purpose BVP solvers is proposed to compute the steady mechanics of a two-pulley V-belt drive. Belt sliding in the pulley grooves leads to twodimensional tangential and radial friction. This contrasts sharply with common textbook/handbook simplifications that extrapolate V-belt behavior from flat belt behavior through, for example, use of a modified friction coefficient. The belt is modeled as an axially moving string with belt inertia fully considered. The wedging of the belt in the pulley grooves makes the belt-pulley contact points unknown a priori. The original BVP on unknown domain is transformed to a standard BVP form on fixed domain. The steady solutions include belt-pulley contact points, radial penetration in pulley grooves, the magnitude and direction of the friction forces, tension, and belt speed. The main findings include: 1. Wrap angles increase with tight/slack span tension ratio and are significantly larger than those for comparable flat belt drives. 2. There are no adhesion zones on the driver or driven pulley; the belt slides in the pulley grooves along the entire contact arc. 3. Large tight/slack tension ratio causes the belt to exit the pulley grooves abruptly resulting in significant nonparallelism of the two free spans that leads to torque loss. 4. The driven pulley rotation speed is lower than for flat belt drives, especially for heavy loads with significant span tension differences. 5. The theoretical maximum transmitted moment occurs when the driven pulley rotation speed drops to zero. At this point, the system has the maximum tight/slack tension ratio. 6. Neglecting the pulley grooves underestimates the maximum transmitted moment and overestimates the system power efficiency. Acknowledgment The authors thank Mark IV Automotive/Dayco Corporation and the National Science Foundation for support of this research. References 1 Euler, M. L., 1762, Remarques Sur L effect Du Frottement Dans L equilibre, Mem. Acad. Sci., pp Fawcett, J. N., 1981, Chain and Belt Drives - A Review, Shock Vib. Dig., 13 5, pp Johnson, K. L., 1985, Contact Mechanics, Cambridge University Press, Cambridge, England. 4 Firbank, T. C., 1970, Mechanics of Belt Drives, Int. J. Mech. Sci., 12, pp Gerbert, G. 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