R. S. Beikmann Noise and Vibration Center, Nortti American Operations, General IVIotors Corporation, IVIiiford, Mictiigan 48380

N. C. Perkins Associate Professor. iVIem, ASiVIE

A. G. Ulsoy Professor. Fellow ASiVIE Mectianicai Engineering and Applied Mectianics, Tlie University of Mictiigan, Ann Arbor, Mictiigan 48109

Design and Analysis of Automotive Serpentine Belt Drive Systems for Steady State Performance Serpentine belt drive systems with spring-loaded tensioners are now widely used in automotive engine accessory drive design. The steady state tension in each belt span is a major factor affecting belt slip and vibration. These tensions are determined by the accessory loads, the accessory drive geometry, and the tensioner properties. This paper focuses on the design parameters that determine how effectively the tensioner maintains a constant tractive belt tension, despite belt stretch due to accessory loads and belt speed. A nonlinear model predicting the operating state of the belt/tensioner system is derived, and solved using (1) numerical, and (2) approximate, closed-form methods. Inspection of the closed-form solution reveals a single design parameter, referred to as the ' 'tensioner constant,'' that measures the effectiveness of the tensioner. Tension measurements on an experimental drive system confirm the theoretical predictions.

Introduction An automotive accessory drive system must maintain proper belt span tensions in order to function effectively. In particular, the belt span tensions must provide adequate torque to drive the accessories vi'ithout slip, and produce minimal noise and vibration. Proper belt tensions must be maintained despite belt wear, assembly variation, and changes in belt length resulting from changing accessory torques, belt speed, and belt temperature. Traditional V-belt drives are limited in their ability to compensate for these changes (Cassidy et al, 1979), and often require high reference tension (tension at rest) which reduces belt and accessory bearing life. Therefore, since 1979, serpentine belt drives with spring-loaded tensioners have become increasingly common; see Fig. 1. When properly designed, the tensioner maintains near constant tractive tension (in the adjacent spans), despite changes in belt length due to the above mechanisms. Serpentine drives employ a multi-rib belt having a thin cross-section as compared to a V-belt, thereby reducing bending stiffness and structural damping. The reductions in bending stress and heat generation, combined with belt tension compensation from the tensioner, generally lead to increased belt life (Cassidy et al., 1979). An ideal spring-loaded tensioner maintains constant tractive tension in the two belt spans that contact it. A less desirable design may allow a considerable decrease in tractive tension in response to increasing accessory torques, or increasing belt speeds. Other designs may actually increase tension with increasing accessory torques (in itself beneficial), but exhibit other undesirable characteristics, as described in this study. The prediction of belt drive tractive tension begins with the prediction of the accessory drive "operating state". The operating state is defined herein as the equilibrium state of the accessory drive under steady accessory torques and steady belt speed. The operating state may be determined after first specifying the "reference state" which is defined here as the equilibrium state of the accessory drive with zero accessory torques and zero belt speed. Homung et al. (1960) examine belt tensions Contributed by the General and Machine Element Design Committee for pubUcation in the JOURNAL OF MECHANICAL DESIGN. Manuscript received Oct. 1996. Associate Technical Editor: S. A. Velinsky.

and curvatures in the operating state for a two-pulley V-belt system with locked-center pulleys. Attention focuses on the speed-dependent belt centrifugal tensioning, which greatly influences system tractive capability and the dynamics of the belt spans. Tensioning due to applied torques and belt speed are analyzed separately, and then superposed. Houser and Oliver (1975) extend the analysis to a two-belt system containing multiple pulleys of arbitrary diameters. Similar tensioning requirements exist in band saw systems, which frequently employ a spring-loaded tensioner that moves parallel to the band spans. The behavior of this type of tensioner, first analyzed by Chubachi (1958), is governed by the dimensionless pulley support constant r], introduced by Mote (1965). This constant depends on the relative compliances of the band saw blade and the tensioner pulley support. This parameter is generalized herein to define a single tensioner constant that measures the effectiveness of the tensioner design. Our objective is two-fold: (1) to predict the steady response of the tensioner, and (2) to assess the effectiveness of the tensioner in maintaining a constant tractive tension. To this end, a theoretical model is derived that determines the operating state, given the reference state. The resulting nonlinear equilibrium problem is solved using a numerical solution procedure and an approximate closed-form procedure (Beikmann et al, 1991). Inspection of the closed-form solution reveals the key design parameters which control the effectiveness of the tensioner design. Belt tension predictions are validated by experimental measurements. The equilibrium solutions evaluated herein are used as a basis for subsequent analyses of linear (Beikmann et al., 1992, 1996a) and nonlinear (Beikmann et al., 1996b) dynamic responses of the entire operating accessory drive system (belt, pulleys, and tensioner).

System Model Figure 1 depicts a three-pulley system containing the essential elements of a serpentine drive: a driving pulley (pulley 1), a driven pulley (pulley 4), a belt, and a spring-loaded tensioner assembly. A detailed derivation of the equations of motion governing the belt spans (continuous elements), and the pulleys and the tensioner arm (discrete elements) is given by Beikmann

162 / Vol. 119, JUNE 1997

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equilibrium state was referred to previously as the "reference state". In this state, the reference tension, P^, is uniform throughout the belt, and the accessory torques, Moi, are zero. The reference state, along with the known operating accessory torques, Moi, and the operating belt speed, c, are used to predict the "operating state", i.e., the equilibrium state under steady accessory torques and steady belt speed. To this end, the following analysis leads to predictions of span tensions and the tensioner arm position in the operating state. For a given belt span (, the operating belt tension, P„,, has two components:

Driving Pulley

Tensioner Arm

t^oi

Tensioner Pulley

~" ^ii

I

"c

(1)

Here, f„ is the tractive tension component in span i, which produces the belt/pulley normal contact force. P^ is commonly called the centrifugal tension (Mahalingham, 1957; Doyle and Hornung, 1969; Houser and Oliver, 1975), and provides the centripetal acceleration of the belt as it accelerates around the pulley perimeter. The centrifugal tension, which can be shown to be uniform throughout the system (Mote, 1965), is Pc = mc^

(2)

where m is the belt mass per unit length, and c is the belt translation speed. Equilibrium for the pulleys requires Pulley 1: (P„, - P„3)r, + M„, = 0

Driven Pulley Fig. 1 Definition diagram for a tliree-pulley serpentine drive system

(1992). Here, the analysis focuses on the equations of equilibrium. Assumptions made in deriving these equations are (1) Negligible belt bending rigidity (Doyle and Hornung, 1969) (2) Uniform belt properties (3) Negligible belt/pulley "wedging" and slipping (4) Negligible friction in the tensioner pivot and tensioner pulley bearing Note that automotive tensioners possess some friction, and that assumption (4) is an idealization. Relaxing this assumption (i.e., including friction) would lead to a continuous set of possible equilibrium positions in a small range. The "friction-free" position calculated herein lies near the center of this range. The analysis begins with the system in equilibrium at rest, with zero belt speed and zero accessory torques. This initial

Pulley 2:

{P„2 - Po^)r^ + M„^ = Q

Pulley 4:

(P,3 - Pot)^ + M„4 = 0

(3) (4) (5)

where M^i is the constant torque applied to pulley i in the operating state, and r, is the radius of pulley ;. The equation of equilibrium for the tensioner arm is P„'"3 cos i/'i + Ptir^ cos ipi + krd^o = 0

(6)

where rs is the tensioner arm radius, i/'i and i/(2 are the alignment angles defined in Figs. 2a and 2b, kr is the tensioner spring constant, and d^o is the deflection of the tensioner spring in its operating state measured from its unstressed position, as defined in Figure 2a. The equations of equilibrium are solved in the following sections using a numerical method (nonlinear) and an approximate method (linear, closed-form). Both methods predict the operating state from the reference state, the latter being described by the uniform reference belt tension, P,., the pulley center coordinates (X,, Z,) and radii (r/, / = 1, 2, 4), the

Nomenclature

'ia > 'i 1 ' ' ( 2

c = Steady state belt speed EA = longitudinal belt stiffness modulus kt = translational (belt) stiffness of tensioner arm kg = translational (geometric) stiffness of tensioner arm kr = rotational spring stiffness of tensioner arm ks = translational (spring) stiffness of tensioner arm /, = length of belt span ; etc. = auxiliary lengths (see Figs. 2a and 2b L = total belt length

Journal of Mechanical Design

m = belt mass per unit length, constant throughout system M„i = applied static torque on accessory ( Pc = centrifugal tension (uniform throughout) in the operating state P„, = total span tension in span i in the operating state Pr = belt tension (uniform throughout) in reference state P„ = tractive tension for span ; in the operating state n = radius of ith discrete element in system U = potential energy in the system X, = pulley center coordinate for pulley / Xs = translational tensioner arm travel: Xs = r^B-i

Z, = pulley center coordinate for pulley (• 7i = angle between tensioner arm and horizontal (see Fig. 3) ^3„ = tensioner arm deflection in the operating state. B-sr = tensioner arm deflection in the reference state. K = tensioner support constant, -dP,ldPc 7] = tensioner support constant, dPJdPc, also tensioner effectiveness if/u ip2 = alignment angles of tensioner (see Figs. 2a and 2b) ^, = rate of change of cos i//,, C,i = d(cos ^i)/dx,, 4>i - total wrap angle for pulley ;' JUNE 1997, Vol. 1 1 9 / 1 6 3

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pulley (' (see Fig. 1). The reference belt length, L^, is defined similarly: (8)

L, = X (/„• + r, and accounting for the difference in length terminology, Eq. (32) recovers the handsaw Eq. (13). Having calculated 77, the centrifugal tensioning problem can be solved approximately by using Eq. (16). The result (31) highlights the design variables that control JUNE 1997, Vol. 1 1 9 / 1 6 5

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tensioner effectiveness. Three principal design variables arise upon re-writing (32) as 77 = hlih

+ kg + K)

Component properties for system used in stability analysis

Description

Variable Name

Value

Units

Pulley Radii

ri, r2, r4

0.05387

Tensioner Arm Radius Pulley 1 Center Coordinates

r3 (X1.Z1)

0,07184 (O.O, 0.0)

meters meters meters

Beit Modulus

(X2,Z2) (X4,Z4) EA

(-0.1524,0,0) (-0.3048. 0.0) 88960

meters meters newtons

Belt Mass/Unit Length

m

Tensioner Spring Constant Reference Tension

kr

0.0893 0.0, 54.4, 544.0, or 5440.0 445.0

kg/meter N-m/rad newtons

(33)

where the three stiffnesses, kb = {EAILr){cos i/'i -t- cos 1/^2)^

(34)

K = P.(Ci + C2)

(35)

and fCy ~" fvf / ' ,

Table 1

(36)

are introduced. The quantities ki„kg, and k^ are components of the tensioner stiffness, deriving from three distinct mechanisms: (1) ^i,, the ' 'belt stiffness", derives from the belt material property, EA, (2) kg, the "geometric stiffness", derives from changes in alignment between the direction of tensioner pulley center motion and the adjacent belt spans, and (3) k^, the "spring stiffness", derives from the tensioner spring. Note that the components kb and k^ are always positive, while kg is signindefinite. Thus, while r; is usually between zero and one (as is always the case for the band saw), 77 may now attain any value, in general. Inspection of Eq. (33) reveals that tensioner designs for which kb is dominant {kb > \kg\, kb> K) have values of 77 » 1, and are therefore effective in maintaining constant tractive tension. Tensioner designs with rj > \ would actually increase tractive tension as the belt stretches. This would appear to be quite beneficial, as increased accessory torques would then induce greater tractive tension. However, designs for which 77 > 1 have a high value of \kg\, accompanied by a low take-up gain | dLI dxs I. Thus, such designs require larger tensioner travel to take up belt stretch (static or dynamic), increasing the likelihood of tensioner wear and dynamic motion of the tensioner and belt. Also, achieving high values of 77 by using high | kg| (low | dL/ dXs\) makes the tensioner angles i/r, more sensitive to build variations in belt length, pulley center location, and pulley radius. This could cause significant variation in 77 due to the high sensitivity (d(cos tpt )ldXs) inherent with the low take-up gain. Moreover, the operating tensioner position may be marginally stable for such systems, as will be shown in the next section.

Pulley 2 Center Coordinates Pulley 4 Center Coordinates

P-

stressed state. Taking the derivative with respect to tensioner arm position, x^, yields

dll_EA dXs

dL

L

dxg

which, using the take-up gain, dL/dx., = (cos i/'i -I- cos i/'2), and (L - L„) = {LIEA)P„ can be expressed as

d£ dx.

(38)

Pr (cos i/'i -I- cos i/^a) + Kxs

Taking the second derivative, and using the definition d(cos ij/i)/dxs = C,i yields d^U _ dPr - j - j = -3-' (cos iA, + cos t/'a) + Pri^i + Cz) + k, (39) dxl dxs Using dL = {LIEA)dPr, and the chain rule leads to dPrldXs = {EAIL)dLldXs

= {EAIL){cos i/'i -I- cos ^2)

Substituting the above result into (39), and specifying the reference state by setting L = L, yields d^U dxl

EA Lr

(cos i/'i -I- cos lAa)^ + F,(Ci + C2) + h (40)

Thus, the requirement for static stability of the reference state, d'^UIdxl > 0, becomes EA

A Note on Equilibrium Stability

( c o s I/'I + c o s l/Zj)^ -I- P r ( C l + ^ 2 ) + ^ , >

0

As in any physical system, the equilibrium positions found or, equivalently, using Eqs. (34) through (36) here may be either stable or unstable. Maintaining stability of (41) the (equilibrium) operating state is obviously essential to the kb + kg + k, > 0 operation of a serpentine drive design. To determine the stability of the reference state, a classical energy-based stability analysis Thus, the sum of these tensioner stiffness components must be greater than zero for stability. Note that, in Eq. (41), kb and k, is performed. The potential energy of the system is are always positive, while kg is sign-indefinite. Therefore, the stability criterion is met unless kg < —{kb + K). Also note, (37) from (33), that the stability criterion has the alternate form

^=Kf)^^-^"^^4^^^'

77 > 0 (stable equilibrium) where U is the system potential energy, L is the total belt length in an arbitrary state, and L„ is the total belt length in its un-

(42)

Thus, the tensioner constant 77 assesses both the effectiveness (77 « 1) and stability (77 > 0) of the tensioner design.

Results and Discussion Tensioner Rvot -^

Fig. 3

Example three-pulley system with properties listed in Table 1

166 / Vol. 119, JUNE 1997

Example results are presented which highlight key features of the equilibrium solution. The first system to be analyzed is illustrated in Fig. 3, and its physical properties are listed in Table I. Figure 4 shows how the tensioner constant 77 varies with the reference tensioner arm position ji. For convenience, the angle 71 is introduced in Fig. 3 to show the orientation of the (reference) tensioner arm position with respect to the horizontal. The tensioner pulley center coordinates remain constant, while the pivot coordinates vary as ji changes. The four curves shown correspond to the four values of the tensioner Transactions of the ASME

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!

fcr

=0

:

'

'

\

;

1

:

;

Table 2 cation

i

Component properties for system used in experimental verifi-

Pespription

Variable Name

Value

Units

Pulley 1 Radius Pulley 2 Radius Pulley 4 Radius Tensioner Arm Radius

ri

0.0889 0.0452 0.0270 0.0970

meters meters meters meters

Pulley 1 Center Coordinates Pulley 2 Center Coordinates Pulley 4 Center Coordinates Tensioner Pivot Coordinates

(Xi,Zi)

(0.5525. 0.0556) (0.3477, 0.05715) (0.0, 0.0) (0.2508. 0.0635)

Belt Modulus Belt Mass/Unit Length Tensioner Spring Constant

EA m kr

170000 0.1029 20.37

meters meters meters meters newtons kg/meter N-m/rad

PI

127.7

newtons

Reference Tension

n H 13

(X2.Z2) (X4.Z4) (X3.Z3)

/^^=54T^^ 1 y