Mechanics of Serpentine Belt Drives with Tensioner

Lingyuan Kong Robert G. Parker1 e-mail: [email protected] Department of Mechanical Engineering, The Ohio State University, 206 W. 18th Avenue, Columbus, OH 43210

1

Mechanics of Serpentine Belt Drives with Tensioner Assemblies and Belt Bending Stiffness Steady state analysis is conducted on a multipulley serpentine belt drive with a springloaded tensioner assembly. Classical creep theory is extended to incorporate belt bending stiffness as well as the belt stretching and centripetal accelerations. The belt is modeled as an axially moving Euler–Bernoulli beam with nonuniform speed due to belt extensibility and variation of belt tension. The geometry of the belt-pulley contact zones and the corresponding belt tension and friction distributions are the main factors affecting belt slip. Bending stiffness introduces nontrivial span deflections, reduces the wrap angles, and makes the belt-pulley contact points unknown a priori. The free span boundary value problems (BVP) with undetermined boundaries are transformed to a fixed boundary form. A two-loop iteration method, necessitated by the tensioner assembly, is developed to find the system steady state. The effects of system parameters on serpentine drive behavior are explored in the context of an actual automotive belt drive. 关DOI: 10.1115/1.1903002兴

Introduction

Serpentine belt drives are the automotive and truck industry standard for front-end accessory drives. Through a long belt, accessories such as the alternator, air conditioner, power steering, and water pump are powered simultaneously from the crankshaft 共Fig. 1兲. Keeping proper belt tensions to provide adequate torque without belt slip is important for the durable, quiet operation of serpentine drives. For this reason, a tensioner assembly is introduced to automatically adjust to changing conditions. Accurate prediction of the system steady state under different operating conditions is helpful to designers to address competing engineering considerations subject to a variety of practical constraints on geometry and accessory loads. Engineers must balance factors like belt tension, potential for belt slip, bearing loads, belt fatigue, and noise in designing belt drives. The present analysis sets out a two-loop iteration procedure to determine the steady mechanics of general n-pulley serpentine drives with tensioner assemblies. The model considers both the belt spans and pulleys coupled by the friction and normal forces in the contact arcs. The belt model is an axially moving beam that is not restricted to small deflection and curvature. Bending stiffness is an important theoretical and practical consideration because it leads to unknown belt-pulley contact points, curved free span deflections, and nonuniform tension and speed distributions. Belt inertia is also modeled. Solution of the nonlinear equations yields the important quantities required in practice, including the nonuniform tension and speed distributions in the free spans and contact arcs, the friction forces generated in the sliding and adhesion contact zones, the belt-pulley contact points 共wrap angles兲, and the tensioner orientation angle. Other useful quantities such as maximum transmissible moment, power efficiency, and a measure of belt-pulley vibration coupling can be computed from the derived solution. For a simplified three-pulley serpentine belt drive, Beikmann et al. 关1兴 introduce a numerical method to calculate the system steady state where the belt is modeled as an axially moving string without bending stiffness and the belt speed is assumed constant everywhere. Kong and Parker 关2兴 improve this model by consid1 Corresponding author. Contributed by the Mechanisms and Robotics Committee for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received May 18, 2004; revised October 29, 2004. Associate Editor: J. S. Rastegar.

Journal of Mechanical Design

ering belt bending stiffness, but for each free span the belt-pulley contact points are the same as the string model, which are fixed at the points of common tangency of the two bounding pulleys. From the calculated steady state, a belt-pulley coupling indicator is defined to quantify the dynamic coupling between the pulley rotations and the belt transverse vibrations. Both of these models assume that belt tensions are uniform in each free span. There is no analysis of the belt-pulley contact zones. Calculation of the system steady state with these restricted models is mainly for subsequent free vibration analysis. Belt-pulley mechanics has been investigated extensively, especially for a string model without bending stiffness 关3,4兴. In spirit, these analyses focus on belt slip and tension/friction distribution. Creep theory 关3–8兴 is typically used to describe the belt-pulley interaction in the contact zones where the belt is longitudinally extensible and the pulley is modeled as a rigid body. Friction from relative sliding between belt and pulley generates the accessory driving torque governed by a Coulomb law. The belt adheres to or slides over the pulley surfaces in distinct adhesion and sliding zones. For a symmetric two pulley drive 共symmetric means the driver and driven pulleys have the same radius兲, Gerbert 关6兴 uses a string model and obtains the closed-form solution of the sliding/ adhesion zones as well as the tension and friction distributions along the belt. Bechtel et al. 关7兴 and Rubin 关8兴 improve this string model by incorporating belt stretching and centripetal accelerations. Bechtel et al. 关7兴 investigate a symmetric two-pulley drive and give the difference between this model and common engineering models that neglect belt inertia/acceleration. Rubin 关8兴 investigates an asymmetric two-pulley drive and highlights the influence of stretching acceleration on drive performance criteria such as maximum transmissible moment and power efficiency. By incorporating belt bending stiffness while retaining belt inertia and other essential modeling features, Kong and Parker 关9兴 build on the model developed by Bechtel et al. and Rubin. It is shown that bending stiffness has important influences on the system steady states. Neglecting it can result in significant errors in circumstances of importance in applications. The steady state analysis in 关9兴 is conducted on fixed-center, two-pulley drives and cannot be directly applied to an n-pulley serpentine belt drive with a tensioner assembly. In this paper, the axially moving beam-pulley model is extended to n-pulley serpentine belt drives with tensioners such as that shown in Fig. 1. The tensioner assembly is a distinguishing feature that requires a new solution procedure. An iteration

Copyright © 2005 by ASME

SEPTEMBER 2005, Vol. 127 / 957

M = EI␬

Q = dM/ds.

共2兲

Terms involving GV in Eq. 共1兲 correspond to the stretching and centripetal accelerations. To complete the system, a constitutive law relating tension and speed is specified, T = EA共m0V/G − 1兲,

Fig. 1 Seven pulley serpentine belt drive example defined in Table 1

method is proposed to find the steady motion given the system geometric parameters, material properties, and operating conditions. The steady motions of an example automotive belt drive are examined to demonstrate design implications and underlying mechanics of practical systems.

2

Steady Solution for an Axially Moving Curved Beam

The steady motion of a single-span axially moving beam wrapping on two pulleys is a critical component of the iterative serpentine drive solution. Change in the thickness of the beam is neglected resulting in constant beam bending and longitudinal stiffness. Application of Newton’s second law to an infinitesimal segment of an axially moving beam and elimination of higher order terms in infinitesimal quantities leads to the governing equations 关9兴: d共T − GV兲 d␬ + EI␬ = f ds ds

共T − GV兲␬ − EI

d 2␬ = n, ds2

共1兲

where s is the arc coordinate along the beam, T共s兲 is the beam tension, G is the mass flow rate 共which is constant for steady state兲, V共s兲 is the belt speed, EI is the beam bending rigidity, ␬ is the beam curvature, f共s兲 and n共s兲 are tangential and normal contact forces per unit length, respectively. The moment and shearing force are obtained from

共3兲

where EA is the belt longitudinal stiffness and m0 is the belt mass per unit length in the rest state 关7–10兴. Figure 2 presents a general single span stretched between two pulleys. One can view this as the ith span connecting the pulleys i and i + 1 in a serpentine belt drive, in which the pulleys and spans are numbered sequentially in the counterclockwise direction with the crankshaft being pulley 1. The beam in the ith free span is modeled as an Euler elastica. A local coordinate system is adopted where the local xi axis starts from the center of pulley i and goes through the center of pulley i + 1, and the y i axis points to the inside of the belt loop. Li is the distance between the centers of these two pulleys. Once the steady state is obtained in this local coordinate system, it is converted to the global coordinate system whose origin is the center of pulley 1 共Fig. 1兲. Note if i = N, where N is the total number of pulleys, then this Nth span is between pulleys N and 1. Because there is no contact force in the free span, the governing equations in 共1兲 become 共Ti − GVi兲⬘ + EI␬i␬i⬘ = 0

共Ti − GVi兲␬i − EI␬i⬙ = 0,

共4兲

where the subscript i represents the ith span, and 共兲⬘ denotes differentiation with respect to the local arclength si. From the constitutive law 共3兲, Ti − GVi can be treated as one unknown field variable pi共si兲 = Ti − GVi = 共1 − G2 / m0EA兲Ti − G2 / m0. Three boundary conditions can be written for the span in Fig. 2 Ti共0兲 = Tiគ0

␬i共0兲 =

1 Ri

兲= 1 . ␬i共si = L i Ri+1

共5兲

Note that G and Tiគ0 are not initially specified. For reasons to be revealed later, here we assume that these two values are known. The problem has undetermined boundaries where the span end 兲 are calculated from points 共and the total arclength of the beam, L i the geometric requirements that the beam contacts and is tangent with both bounding pulleys. To solve the undetermined boundary problem, the set of governing equations in 共4兲 is expanded by introducing three more field variables ␪i共si兲, xi共si兲, and y i共si兲, where ␪i共si兲 is the inclination angle measured from the positive xi direction and xi共si兲, y i共si兲 are rectangular coordinates of the belt particles in the local coor-

Fig. 2 The ith span connecting the ith and „i + 1…th pulleys in a serpentine belt drive

958 / Vol. 127, SEPTEMBER 2005

Transactions of the ASME

dinate system 共Fig. 2兲. xi共si兲 and y i共si兲 are governed by xi⬘ = cos ␪i and y i⬘ = sin ␪i, as evident from the infinitesimal belt segment in Fig. 2. ␪i共si兲 is governed by ␪i⬘ = ␬i according to the conventional definition of the curvature. The contact and tangency conditions are 关xi共0兲兴2 + 关y i共0兲兴2 = Ri2 xi共0兲 = Ri sin ␪i共0兲

2 兲 − L 兴2 + 关y 共L 2 关xi共L i i i i兲兴 = Ri+1 ,

共6兲

兲 − L = R sin ␪ 共L xi共L i i i+1 i i兲.

共7兲

The two equations in 共6兲 guarantee that the two endpoints contact the surface of pulley i and i + 1, respectively. The first equation in 共7兲 further makes the belt tangent to pulley i, while the second equation in 共7兲 enforces the belt tangency with pulley i + 1. Tangency is imposed by the angles on the pulleys to the contact points 兲, which are also the span endpoint being equal to −␪i共0兲 and ␪i共L i inclination angles. , xˆ Introduction of the nondimensional variables sî = si / L i i 2 = xi / Li, yˆ i = y i / Li, ␬ˆ i = Li␬i , pî = piLi / EI into the governing equations and boundary conditions yields dpî/dsî + ␬ˆ id␬ˆ i/dsî = 0,

d2␬ˆ i/共dsî兲2 − pî␬ˆ i = 0

0 ⬍ sî ⬍ 1, 共8兲

d␪i/dsî = ␬ˆ i,

dxî/dsî = cos ␪i,

dyˆ i/dsî = sin ␪i

0 ⬍ sî ⬍ 1, 共9兲

2/EI, pî共0兲 = P0L i

/R , ␬ˆ i共0兲 = L i i

/R , ␬ˆ i共1兲 = L i i+1

共10兲

共0兲xˆ 共0兲兴2 + 关L 共0兲yˆ 共0兲兴2 = R2 , 关L i i i i i 共1兲xˆ 共1兲 − L 兴2 + 关L 共1兲yˆ 共1兲兴2 = R2 , 关L i i i i i i+1 共0兲xˆ 共0兲 = R sin ␪ 共0兲, L i i i i

共11兲

xˆ 共1兲 − L , 共12兲 Ri+1 sin ␪i共1兲 = L i i i

is defined as a field funcwhere the unknown final span length L i tion Li = Li共sî兲 governed by /dsˆ = 0, dL i i

0 ⬍ sî ⬍ 1.

共13兲

The reason to introduce the nondimensional variables and define as an unknown function in Eq. 共13兲 is to the unknown constant L i incorporate them into the standard form required for most generalpurpose BVP solvers, namely, u⬘共t兲 = F关t,u共t兲兴, g关u共a兲,u共b兲兴 = 0,

a⬍t⬍b

boundary conditions,

共14兲

where F, u, and g are n-dimensional vectors and F and g may be nonlinear. The standard form involves only coupled differential 共not algebraic兲 equations for unknown functions u共t兲 defined on fixed domain 共a,b兲. After these transformations, the problem is cast entirely into the form 共14兲 on the interval sî 苸共0 , 1兲. The seven boundary conditions 共10兲–共12兲 equal the total order of the six differential Eqs. 共8兲, 共9兲, and 共13兲. For an infinitesimal segment of the belt with arclength dsi in Fig. 2, its unstretched length is dsi / 共1 + ␧i兲, where ␧i = Ti共si兲 / EA is the longitudinal strain. The unstretched length of this whole span, required in subsequent steps, is Li共0兲 =

冕

L i

0

1 dsi . 1 + Ti共si兲EA

共15兲

Direct integration can be used because the tension distribution Ti共si兲 is known from pi共si兲 = Ti共si兲 − GVi共si兲 and the constitutive law 共3兲 or, one may define Ii共si兲 = 兰s0i1 / 1 + Ti共si兲 / EAdsi and add the Journal of Mechanical Design

Fig. 3

Free body diagram of tensioner assembly

additional differential equation and boundary condition 共suitably nondimensionlized兲: 1 dIi共si兲 = , dsi 1 + Ti共si兲EA

0 ⬍ si ⬍ L i

with

Ii共0兲 = 0, 共16兲

兲 is equivalent to the to the previous BVP formulation. Then, Ii共L i desired integral term and is a natural product of the BVP solution.

3 Two-Loop Iteration Method for the System Steady State A prominent characteristic of serpentine belt drives is the tensioner assembly, which consists of an idler pulley on a springloaded tensioner arm that rotates about the pivot O⬘ 共Figs. 1 and 3兲. Its purpose is to maintain proper belt tensions as the accessories are activated/deactivated or engine load/speed changes. The tensioner orientation is not specified a priori, instead its position is determined in the analysis. Because the above BVP formulation can only address spans between two fixed-center pulleys or, more accurately, two pulleys with their center positions known, analysis of the spans adjacent to the tensioner pulley, such as spans 6 and 7 in Fig. 1, requires a specified tensioner angle ␾. This necessitates an inner iteration loop to find the tensioner angle within an outer iteration loop based on span tension. The outline follows the flowchart in Fig. 4. For an initial guess of the span tension T1共0兲关see comment following 共5兲兴, the steady solution for all spans is calculated for an initial guess of the tensioner orientation angle ␾. Iteration on the angle ␾ for fixed T1共0兲 continues until the moment balance for the tensioner and its adjacent spans is satisfied. Then, a compatibility condition requiring the back-calculated unstretched belt length to equal the specified input value is evaluated. If the error exceeds a tolerance, the span tension T1共0兲 is adjusted and the process repeats. 3.1 Outer Loop Iteration for a Multiple Pulley, FixedCenter Drive. Supposing the tensioner orientation angle ␾ is specified, then all the pulleys have fixed centers. Two kinds of segments exist along the beam: free spans and contact zones. For the free spans, the BVP-solver based method presented in the previous section cannot be directly applied because neither G nor any of the boundary tensions Ti共0兲 in 共5兲 are specified or readily measured. Iteration is required. For the contact zones, the governing equation is the same as for the string model because bending stiffness only causes a constant moment uniformly distributed in the contact zones. For this assumed multiple pulley, fixed-center drive, an iteration loop 共outer loop兲 is used to find the system steady state. This loop iterates on T1共0兲, the starting boundary tension for the first span. For an assumed value of T1共0兲, the steady state of each segment is calculated sequentially around the drive along the counterclockwise direction as follows. Because the corresponding belt speed for the assumed T1共0兲 is given by V1共0兲 = R1␻1, where ␻1 is the specified crankshaft 共pulley 1兲 rotational speed, G is obtained from the constitutive law 共3兲. Thus, the steady state of the first span is completely solvable SEPTEMBER 2005, Vol. 127 / 959

˜ − GV ˜ 兲/R = n , 共T i i i i

˜ − GV ˜兲 d共T i i dsi*

M i/Ri =

冕

= − f i = − ␮ in i,

␤iRi

共17兲

f idsi* ,

0

where ˜Ti and ˜Vi are the belt tension and speed in the sliding zone of pulley i, ␮i is the friction coefficient of pulley i, s*i is the local arclength along the sliding zone, and the applied torque M i is positive counterclockwise. The boundary tension ˜T2共0兲 equals 兲, and the belt tension and speed distribution are calculated T1共L 1 using 共17兲. Closed-form solution is readily available, with the tension and speed varying exponentially. The sliding angle ␤2 on pulley 2 is 共Fig. 2兲:

␤i =

−1 Ti共0兲 − GVi共0兲 , ln ␮i 兲 − GV 共L Ti−1共L i−1 i−1 i−1兲

共18兲

i = 2.

For this sliding zone, the unstretched belt length is L␤共0i兲 =

冕

␤iRi

0

⫻ln

1 1 + ˜Ti共si*兲EA

冉

dsi* = 1 −

冊

G2/m0 Ri EA ␮i

兲 − GV 共L 1 + EA/关Ti−1共L i−1 i−1 i−1兲 − M i/Ri兴 , 兲 − GV 共L 兲兴 1 + EA/关T 共L i−1

i−1

i−1

i = 2.

i−1

共19兲 In the adhesion zone, there is no friction because the belt and pulley have the same speed; the tension and speed are constant. Thus, the span 2 starting boundary tension T2共0兲 equals ˜T2共␤2R2兲, allowing calculation of the motion of the second span using the previously outlined single span analysis. The wrap angle ␺2 for pulley 2 is available from the known span 1 and 2 solutions that include the belt-pulley contact points. From geometry, the adhesion angle on pulley 2 is ␣i = ␺i − ␤i, i = 2. The unstretched length of the adhesion zone is L␣共0i兲 = ␣iRi/关1 + Ti共0兲EA兴, Fig. 4 Flowchart of iteration method involving two loops

by specifying i = 1 in Eqs. 共8兲–共13兲. For the contact zone on pulley 2, in which the belt shape is known and the curvature is constant, one only needs to find the distributions of belt tension and friction. This contact zone can be divided into a sliding zone with belt-pulley slip and an adhesion zone with no relative motion. The adhesion zone precedes the sliding zone along the direction of belt motion 共Fig. 1兲关6兴. This is because for driven pulleys the friction force resists the belt motion, so the belt speed in the sliding zone is faster than that of the pulley surface. Belt tension and speed are constant in the adhesion zone; the tension increase along the contact arc in the direction of belt travel occurs entirely in the sliding zone. The increasing belt tension in the sliding zone 共along the belt motion direction兲 leads to the corresponding increase of belt speed in the sliding zone because of the constitutive law 共3兲. On the other hand, the belt speed in the adhesion zone is the same as that of the pulley surface, so even the slowest belt speed in the sliding zone is faster than that in the adhesion zone. If the sliding zone preceded the adhesion zone along the direction of belt motion for driven pulleys, then the belt speed would increase monotonically starting from a speed higher than that of the pulley surface and then reach the same speed as the pulley surface at the point where adhesion begins, which is an impossible situation. Similar reasoning applies to the driver pulley. For the sliding zone of pulley i, including i = 2, the governing equations are 960 / Vol. 127, SEPTEMBER 2005

i = 2.

共20兲

This process repeats around the entire belt drive. There are several points to note in this solution procedure: 共a兲 For any idler pulleys, such as the tensioner pulley, the torque is zero. The sliding angle is zero, and the wrap angle equals the adhesion angle. 共b兲 The single span analysis outlined in the previous section is only suitable for free spans where both bounding pulleys contact the ribbed side of the belt. If one of the two bounding pulleys for the ith free span contacts the flat side of the belt 共like pulleys 4 and 7 in Fig. 1兲, then some of the boundary conditions need modification while the differential equations remain the same. For example, for span 3 where pulley 4 is the end bounding pulley, the dimensional boundary conditions 共5兲 and 共7兲 must be changed to T3共0兲 = T3គ0

␬3共0兲 =

x3共0兲 = R3 sin ␪3共0兲

1 R3

兲=− 1 , ␬3共s3 = L 3 R4

兲 = x 共L − R4 sin ␪3共L 3 3 3兲 − L 3 .

共21兲共22兲

Span 6 is handled similarly. For spans where the starting bounding pulley contacts the flat side of the belt 共like spans 4 and 7兲, the modifications are the same in spirit and there is no difficulty to find the steady states of such free spans. 共c兲 For the contact zones on pulley i, where i ⫽ 2, the solution is obtained by replacing 2 with the individual pulley number i in Eqs. 共17兲–共20兲. The only exception is for the sliding zone on pulley 1. In this case, the friction force exerted on the belt from the crankshaft is in the same direction as the belt motion, which is different from those of the other pulleys for which the friction Transactions of the ASME

Table 1 Physical properties of a practical serpentine belt drive Center X location

Pulley No.

Center Y location

Radius R 共m兲

1 0 0 2 0.2116 m 0.0090 m 3 0.2317 m 0.1898 m 4 0.0796 m 0.2097 m 5 −0.2026 m 0.2699 m 6 −0.2000 m 0.1000 m 7 Crankshaft speed: Tensioner: ␾0 = 1.8997 rad Kt = 38.84 N m / rad Tensioner pivot Xo⬘ = 0.033 m Tensioner pivot Y o⬘ = 0.137 m

— 0.0970 m 24.82 0.0625 m 9.09 0.0291 m 0 0.04075 m 18.908 0.06685 m 2.382 0.08245 m 0 0.03775 m ␻1 = 2000 rpm 共clockwise兲 Belt:

forces are opposite to the belt motion direction 共Fig. 1兲. This is because the crankshaft is the only driver pulley in the system while all other pulleys are driven ones. The torque exerted on pulley 1 is negative while all other torques are positive. For pulley 1, the first of 共17兲 and 共20兲 still hold, but the remaining equations of 共17兲–共19兲 become ˜ − GV ˜ 兲 d共T 1 1 ds*1

= f 1 = ␮ 1n 1,

冉

冕

␤1R1

f 1ds*1,

0

␤1 =

L␤共01兲 = 1 −

M 1/R1 = −

1 T1共0兲 − GV1共0兲 , ln ␮1 T 共L N N兲 − GVN共LN兲

共23兲

冊

G2/m0 R1 1 + EA/关T1共0兲 − GV1共0兲 + M 1/R1兴 ln . EA ␮1 1 + EA/关T1共0兲 − GV1共0兲兴共24兲

共d兲 For the analysis of the contact zone on pulley 1, the crankshaft torque M 1 is not specified 共the speed ␻1 is assumed specified兲. Instead, M 1 is calculated in the analysis. Integration of 共1兲 along the entire belt loop leads to

冖冕

␤1R1

fds =

N

f 1ds*1 −

兺 i=2

0

冕

␤iRi

f idsi* = 0.

共25兲

0

Substitution of 共17兲 and 共23兲 into 共25兲 yields N

M 1/R1 = −

兺 M /R . i

i

共26兲

i=2

A criterion is needed to determine if the solution calculated for given T1共0兲 is the correct one. This iteration loop criterion is the compatibility condition N

兺共L i=1

共0兲 i

+ L␤共0i兲 + L␣共0i兲兲 − L共0兲 = 0,

共27兲共0兲

where L共0兲 is the given total unstretched length of the belt, Li is the unstretched length of span i 共see Section 2 for the calculation 共0兲共0兲共0兲 of Li 兲, and L␤ and L␣ are unstretched lengths corresponding to i i the sliding and adhesion zones on pulley i. Physically, the compatibility Eq. 共27兲 means that the unstretched belt length backcalculated from the steady state must equal the user-specified stress free belt length. For a practical serpentine belt drive, this user-specified value L共0兲 is known by manufacturers or can be obtained by measurement in the stress free state. If the error in Eq. 共27兲 is positive 共negative兲, T1共0兲 is increased 共decreased兲, and the Journal of Mechanical Design

Torque M 共N m兲

Friction coefficient

In/Out property

0.95 0.95 0.95 0.95 0.95 0.95 0.95

Ribbed Ribbed Ribbed Flat Ribbed Ribbed Flat

L共0兲 = 2.07918 m m0 = 0.107 kg/ m EA = 111,200 N

process repeated until convergence is achieved to within a chosen tolerance. 3.2 Inner Loop Iteration for Tensioner Orientation Angle. In the above iteration loop, it is assumed that the tensioner orientation angle ␾ is specified. In the analysis, however, the tensioner orientation angle ␾ is not known a priori because the tensioner assembly angle ␾ adjusts for different tensions and operating conditions. To determine the steady state of the entire system, an additional loop 共inner loop兲 iterating on the tensioner orientation angle ␾ is added into the outer loop. The inner loop iterates on the tensioner and its adjacent spans/pulley to enforce tensioner moment balance. The process is outlined in Fig. 4. The tensioner orientation angle ␾ must satisfy the moment balance for the tensioner assembly about its pivot 共Fig. 3, where the strokes of the belt in the free spans are thickened兲. The balance of angular momentum with respect to the tensioner pivot applied to the control volume requires:

兺M

O⬘

= 关共TB − GVB兲rTB − QBrQB兴 + 关共TA − GVA兲rTA − QArQA兴 − M S = 0.

共28兲

The points A and B at the midpoints of the two adjacent spans are selected 共arbitrarily兲 to define the control volume. For the assumed tensioner orientation angle ␾, the two adjacent spans are treated like those between fixed-center pulleys, and there is no difficulty to calculate the steady motions of these two spans. TA and TB are the tension forces at the midspan points while VA and VB are the corresponding speeds. rTB and rQB are the moment arm lengths 共with respect to the tensioner pivot O⬘兲 for the tension TB and shearing force QB at point B 共Fig. 3兲. rTA and rQA for the point A are not drawn in Fig. 3 for figure clarity. The shearing forces 共QA and QB兲 and the moments 共M A and M B兲 are calculated using 共2兲 and the known free span solutions for the current iteration value of ␾. The moment from the tensioner spring is M s = Ks共␾ − ␾o兲, where Ks is the tensioner pivot rotational stiffness and ␾0 is the specified tensioner orientation for zero moment in the pivot spring. If the error in Eq. 共28兲 is positive 共negative兲, ␾ is increased 共decreased兲 and the process repeated until convergence is achieved to within a chosen tolerance. In the above iteration process, the two loops repeat until both the compatibility condition 共27兲 and the moment balance 共28兲 are satisfied 共Fig. 4兲. One can use values calculated from the string model in Ref. 关1兴 as initial guesses for T1共0兲共outer loop兲 and ␾ 共inner loop兲 to speed convergence. SEPTEMBER 2005, Vol. 127 / 961

4

Results and Discussion

Results are presented for a seven-pulley serpentine belt drive used in a production automobile 共Fig. 1兲. The physical properties are listed in Table 1. Note that the torque on pulley 1 共crankshaft兲 is calculated from 共26兲 instead of being specified. The Ribbed/Flat property refers to whether the ribbed or flat side of the belt contacts the pulley. Figure 5 depicts the system steady motions for three values of bending stiffness EI = 0.01, 0.1, and 0.3 N m2. Practical bending stiffness estimates for polyribbed belts used in serpentine belt drives fall roughly in the range EI = 0.01– 0.1 N m2. Values may be much higher in other applications or in V belts. For small bending stiffness, the wrap angles are only slightly reduced; the span deflections are close to those of the string model. Radial tickmarks indicate string model contact points on the lines of common tangency. For large bending stiffness, the deflections are significantly away from the straight line of the string model, especially for short spans or those with small tensions. The free span deflections cause dynamic coupling between the pulley rotations and span transverse vibrations 关2兴. Through this coupling mechanism, crankshaft speed fluctuations can cause undesired belt span vibrations and noise. The steady states obtained in this study improve on those in 关11兴共calculated from the fixed belt-pulley contact points model 关2兴兲 to better predict the coupled belt-pulley-tensioner natural frequencies, vibration modes, and dynamic response. The influence of bending stiffness on the sliding angles are less pronounced than on the adhesion angles because the sliding angles are determined mainly by pulley torques independent of bending stiffness. Therefore, the dominant parts of the wrap angle reductions are in the adhesion zones. The adhesion zones function as power transmission reserve capacity. When more torque needs to be transmitted by an accessory pulley 共e.g., when the air conditioner is activated兲, a portion of the adhesion zone converts to sliding zone for that pulley. Bending stiffness greatly reduces the system capacity to transmit more power, or, in other words, bending stiffness makes the pulleys more prone to slip fully 共zero adhesion angle兲. Figures 6共a兲–6共c兲 show the numerical variations of these geometric angles with bending stiffness. The wrap angle reductions of small radius pulleys are much larger than those of large radius pulleys 关Fig. 6共d兲兴. For example, for pulley 3 with the smallest radius, when bending stiffness increases from zero to 0.3 N m2, the wrap angle decreases by about 70 deg 共the adhesion angle is reduced to about 1 deg, at the edge of full slip兲. For pulley 1 with the largest radius, the wrap angle is reduced by about 16 deg. The phenomena observed in Fig. 6 can be explained by an approximate analytical solution. Supposing the bending stiffness effect is small, then the slopes in the free spans are low, and the wrap angle for pulley i is reduced by 关see 共36兲 in the appendix兴

␺i共reduced兲 ⬇

冦

1 Ri 1 Ri

冉冑冑冊冉冑冑冊 EI + ¯pi

EI , i ⫽ 1, ¯pi−1

EI + ¯pi

EI , ¯pN

i = 1,

冧

共29兲

where the approximate ¯pi calculated from the string model is used. The analytical and numerical solutions agree well for small bending stiffness as shown in Fig. 7. Equation 共29兲 also explains why the wrap angle reduction for pulley 2 is less than that for pulley 5 in Fig. 6共d兲 even though pulley 2 has smaller radius. The two neighboring span tensions are higher for pulley 2 than those for pulley 5. To prevent full slip in systems where the bending stiffness effect is meaningful 共i.e., large bending stiffness, low tensions, or high speeds兲, accessories with large torques should have large pulley radii 共in addition to the common design requirement of large wrap angles for such accessories兲. This permits use of lower 962 / Vol. 127, SEPTEMBER 2005

Fig. 5 Steady state for the system properties specified in Table 1. „a… EI = 0.01; „b… EI = 0.1, „c… EI = 0.3 N m2. The fixed beltpulley contact points for the string model are marked by short lines perpendicular to the pulleys. Dashed lines delineate the sliding and adhesion zones, denoted by A and S.

tensions, an important consideration in pulley bearing and belt life. Small pulley radii are sufficient for idler pulleys. From this perspective, an increased alternator 共pulley 3兲 radius in Fig. 5 is preferred because the alternator generally requires large torque. Figure 8 shows that the maximum span tensions increase sigTransactions of the ASME

nificantly, and almost linearly, with bending stiffness. Note tension and speed are not uniform along the belt free spans; the maximum tension appears around the span midpoint 关9兴. Bending stiffness causes the belt to experience the cyclic stresses at a higher level. Neglecting bending stiffness results in overestimation of the belt fatigue life, especially for appreciable bending stiffness, which is contrary to safe design practice. The power efficiency ␩ is defined as the ratio between the total power delivered to the driven pulleys and the power supplied by the driver pulley N

兺M␻ i

␩=

i

i=2

兩M 1␻1兩

共30兲

.

Power efficiency changes little with bending stiffness. For the system specified in Table 1, ␩ varies less than 1% in a range near 99% over the bending stiffness values in Fig. 6. This results because the belt has large longitudinal stiffness EA, so tension variation along the belt causes only very small change in belt speed 关V = 共1 + T / EA兲G / mo from the constitutive law 共3兲兴. The nearly uniform belt speed makes the power efficiency ␩ almost unity according to N

␩=

兺

N

兺

M i␻ i

i=2

兩M 1␻1兩

=

N

M iVi共0兲/Ri

i=2

兩M 1V1共0兲/R1兩

兺 M /R i

⬇

i=2

兩M 1/R1兩

i

= 1.

共31兲

When the belt longitudinal stiffness is small, because the speed variation along the belt is more pronounced, the system power efficiency is a much stronger function of longitudinal stiffness. For example, in Fig. 9 the belt longitudinal stiffness EA varies across one order of magnitude. The power efficiency drops from about 99% to 94% with this decrease in longitudinal stiffness. Notice the steepness of the curve for the lower values of EA; further decrease dramatically decreases efficiency. The longitudinal stiffness is dictated primarily by thin cords embedded in a rubber belt, so this property is at the designers’ discretion without affecting the belt adhesion properties. In practice, cords with a wide range of longitudinal stiffness are used in vehicle systems. Bending stiffness has little influence on the tensioner orientation angle. The value of the tensioner orientation angle ␾ varies less than 1% about 165 deg over the bending stiffness range in Fig. 6. Tensioner orientation is similarly insensitive to friction coefficient and accessory loads, as shown later. This is also due to the large belt longitudinal stiffness EA. Usually the steady torque in the tensioner rotational spring under operating conditions is less than that in the rest state 共for the system in Fig. 1, it means that the tensioner orientation angle in the rest state is larger than under operating conditions兲. This is because, from the rest state to the operating states, more belt moves into the tensioner assembly part, whose two adjacent spans are the slackest ones of all spans. This reduces the tensions of these two spans, and less torque from the pivot rotational spring is needed to keep the moment balance 共28兲. When EA is large, as is typical, a very small amount of belt moves into the tensioner assembly part under operating conditions, and only small changes in belt orientation angle are needed to accommodate the system requirements for the compatibility condition 共27兲 and the moment balance 共28兲. Therefore, for systems with large belt longitudinal stiffness EA, the tensioner orientation angle is a weak function of design parameters. In contrast, tensioner orientation depends strongly on longitudinal stiffness. Figure 9 shows that tensioner orientation angle decreases for decreasing belt longitudinal stiffness EA with a sharp slope for small EA. For comparison purpose, in Fig. 9, instead of having the same total unstrected belt length over the EA variation range, the tensioner orientation angles in the rest state are the same 167.51 deg. Under Journal of Mechanical Design

the same operating conditions, the variation in tensioner orientation angle from that in the rest state is higher for systems with small belt longitudinal stiffness EA. Figure 10 shows the effect of friction coefficient on the system steady states. The sliding and adhesion angles are sensitive to the friction coefficient, especially for the crankshaft, whose torque is much higher than the other accessories. As the friction coefficient decreases to 0.75, the crankshaft is at the edge of full slip. Belt wear reduces the friction coefficient, which affects the system capacity to function properly. Temperature also changes the friction coefficient. The friction coefficient has little influence on the wrap angles, maximum span tensions, the tensioner orientation, and power efficiency. This is also because of the large belt longitudinal stiffness EA. Conversion of the contact arcs between the sliding and adhesion zones due to the variation of friction coefficient causes only very small changes of belt tension/speed in the contact zones, which have little influence on the rest of the system. Changing the torque of an individual pulley influences only that pulley and any upstream components. For example, in Fig. 11, when the torque on pulley 5 is doubled, the sliding angles of pulleys 5 and 1 are increased 共with corresponding decrease in adhesion angles兲 while the sliding angles of the pulleys between them decrease. The tensions of the spans between pulleys 1 and 5 increase. The rest of the system is almost unchanged. This is because the tensioner orientation angle is a weak function of accessory torques due to the large longitudinal stiffness EA as discussed earlier. Therefore, the tensions of the two spans adjacent to the tensioner remain almost the same from the moment balance about the tensioner assembly 共28兲. This is the main design purpose of the tensioner assembly. Because the torque on pulley 6 is not changed, the sliding/adhesion angles on pulley 6 and the tension of span 5 remain almost the same. Because the accessory torque for pulley 5 is doubled, span 4 has higher tension. Part of the adhesion zone on this pulley converts to an enlarged sliding zone to transmit the increased torque. The increased tension of span 4 leads to higher tensions of the upstream spans 1–3. For middle pulleys like pulley 3, both of the two neighboring span tensions 共spans 3 and 2兲 increase. Thus, the beam tension throughout the pulley 3 sliding zone increases 共it increases exponentially from the tension of the neighboring slack span 3 to the tension of the neighboring tight span 2兲. The normal force between the belt and pulley also increases, and hence so does the friction force. Consequently, a smaller sliding zone is needed to transmit the unchanged torque M 3. For the crankshaft, because the torque is increased according to 共26兲 and the tension of the neighboring span 7 is almost unchanged, the sliding zone increases.

5

Conclusions

Steady state analysis is conducted on a general multipulley serpentine belt drive with a tensioner assembly. The belt is modeled as an axially moving beam with bending stiffness, and the stretching and centripetal accelerations are incorporated. A two-loop iteration method is developed: the inner loop iterates on the tensioner orientation angle to satisfy the moment balance about the tensioner pivot while the outer loop iterates on the belt tension at the entry point of the crankshaft to satisfy a belt length compatibility condition. Key building blocks of the iteration are: 共a兲 the solution for a single span wrapping on two bounding pulleys, a problem with undetermined boundaries solved by reformulating into a fixed domain problem, and 共b兲 the exact solution for the belt on the pulleys. The main conclusions include: 1. Bending stiffness significantly decreases the wrap angles, making the pulleys more prone to slip fully due to reduction of the adhesion angles. 2. The maximum span tensions increase with bending stiffness, accelerating belt fatigue compared to commonly used engineering approximations based on string models. SEPTEMBER 2005, Vol. 127 / 963

Fig. 6 Variations of geometric angles for the system specified in Table 1: „a… wrap angles with bending stiffness; „b… adhesion angles with bending stiffness; „c… sliding angles with bending stiffness; „d… reduction of wrap angles „0 Ï EI Ï 0.3 N m2… with respect to pulley radii

Fig. 7 Comparison of variation of wrap angles with bending stiffness for the system specified in Table 1. „—… numerical solution; „---… approximate solution.


Fig. 8 Variations of span maximum tensions with bending stiffness for the system specified in Table 1


Fig. 9 Influence of belt longitudinal stiffness on power efficiency and tensioner orientation angle. The physical properties are from Table 1 with EI = 0.05 N m2.

3. Tensioner orientation and system power efficiency are generally insensitive to changes of other design parameters because of the large belt longitudinal stiffness. These quantities are sensitive to belt longitudinal stiffness, where reducing longitudinal stiffness decreases the system power efficiency and increases the variation of tensioner orientation angle for different operating conditions. 4. Friction coefficient has little influence on wrap angles and the belt steady states in the free spans. The sliding and adhesion angles, however, are sensitive to changes of friction coefficient. Belt wear decreases friction coefficient, as can temperature, which reduces the system capacity to function properly with sufficient reserve power transmission capacity in the adhesion zones. 5. Changing the torque of an individual pulley influences only the steady states of that pulley and upstream components with little effect on components in the downstream direction.

Fig. 10 Influence of friction coefficient for the system specified in Table 1 with EI = 0.05 N m2

Acknowledgments The authors thank Mark IV Automotive/Dayco Corporation and the National Science Foundation for support of this research.

Appendix Suppose an axially moving belt is bounded between two pulleys with radius R1 and R2. When the bending stiffness is small, then the deflection and curvature of the beam in the free spans are small. Elimination of higher order terms of infinitesimal quantities in 共1兲 leads to linearized equations for the free spans d共T − GV兲 =0 ds

共T − GV兲␬ − EI

d 2␬ = 0. ds2

共32兲

Integrating these gives T − GV = ¯p = constant

␪共s兲 =

D 冑冑 + C1e p/EIs + C2e− p/EIs , ¯p 共33兲

where ¯p is the span tractive tension and D is a constant. For the stated assumptions, ¯p is approximated well using the tension from a serpentine system string model 关1兴 and uniform belt speed. Sub 兲 = ␪⬘共L 兲 = ± 1 / R gives stitution of ␬共0兲 = ␪⬘共0兲 = ± 1 / R1 and ␬共L 2 关共⫾兲 indicates whether the pulley is above/below the horizontal line of common tangency兴 Journal of Mechanical Design

Fig. 11 Influence of accessory torques. The physical properties are from Table 1 with EI = 0.1 N m2 except that the torque on pulley 5 is doubled, as indicated

SEPTEMBER 2005, Vol. 127 / 965

␪共s兲 ⬇

1 D ⫿ ¯p R1

冑

EI −冑¯p/EIs 1 e ± ¯p R2

冑

EI 冑¯p/EI共s−L 兲 e . ¯p

共34兲

When EI → 0, the inclination angle ␪共s兲 should be zero 共recovering the solution for the string model兲, which gives D = 0. Further in Eq. 共34兲 gives the boundary inclinamore, letting s = 0 or s = L tion angles

␪共0兲 ⬇ ⫿

1 R1

冑

EI ¯p

兲 ⬇ ± 1 ␪共L R2

冑

EI . ¯p

共35兲

The total wrap angle around a pulley is reduced by

␺共reduced兲 ⬇

1 R

冉冑冑冊 EI + ¯p1

EI , ¯p2

共36兲

where ¯p1 and ¯p2 are the tractive tensions of the two neighboring spans.

References 关1兴 Beikmann, R. S., Perkins, N. C., and Ulsoy, A. G., 1996, “Design and Analysis of Automotive Serpentine Belt Drive Systems for Steady State Performance,”


ASME J. Mech. Des., 119, pp. 162–168. 关2兴 Kong, L., and Parker, R. G., 2003, “Equilibrium and Belt-Pulley Vibration Coupling in Serpentine Belt Drives,” ASME J. Appl. Mech., 70共5兲, pp. 739– 750. 关3兴 Fawcett, J. N., 1981, “Chain and Belt Drives: A Review,” Shock and Vibration Dig., 13, pp. 5–12. 关4兴 Johnson, K. L., 1985, Contact Mechanics, Cambridge University Press. 关5兴 Gerbert, G., and Sorge, F., 2002, “Full Sliding Adhesive-Like Contact of V-Belts,” ASME J. Mech. Des., 124共4兲, pp. 706–712. 关6兴 Gerbert, G., 1999, Traction Belt Mechanics, Chalmers University of Technology, Sweden. 关7兴 Bechtel, S. E., Vohra, S., Jacob, K. I., and Carlson, C. D., 2000, “The Stretching and Slipping of Belts and Fibers on Pulleys,” ASME J. Appl. Mech., 67, pp. 197–206. 关8兴 Rubin, M. B., 2000, “An Exact Solution for Steady Motion of an Extensible Belt in Multipulley Belt Drive Systems,” ASME J. Mech. Des., 122, pp. 311–316. 关9兴 Kong, L., and Parker, R. G., 2004, “Steady State Mechanics of Belt-Pulley Systems,” ASME J. Appl. Mech., 72共1兲, pp. 25–34. 关10兴 Leamy, M. J., 2004, “On a New Perturbation Method for the Analysis of Unsteady Belt-Drive Operation,” ASME J. Appl. Mech., in press. 关11兴 Kong, L., and Parker, R. G., 2004, “Coupled Belt-Pulley Vibration in Serpentine Drives with Belt Bending Stiffness,” ASME J. Appl. Mech., 71共1兲, pp. 109–119.
