Decomposition of noise sources of synchronous belt drives

Journal of Sound and Vibration 332 (2013) 2239–2252

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Decomposition of noise sources of synchronous belt drives Gang (Sheng) Chen a,n, Hui Zheng b, Mohamad Qatu c a

College of IT and Engineering, Marshall University, Huntington, WV 25755, USA School of Mechanical Engineering, Shanghai Jiaotong University, Shanghai 200240, PR China c School of Engineering and Technology, Central Michigan University, Mount Pleasant, MI 48859, USA b

a r t i c l e i n f o

abstract

Article history: Received 27 November 2011 Received in revised form 11 November 2012 Accepted 27 November 2012 Handling Editor: I. Lopez Arteaga Available online 29 January 2013

In this paper, the noise sources of synchronous belt are decomposed and formulated based on the analysis of the impact dynamics of belt-sprocket tooth interface. The impact/contact of belt-sprocket tooth and the vibration of belt span are modeled. The friction–vibrations interaction of belt tooth and the airflow-induced acoustic wave during belt-sprocket tooth engagement are comprehensively formulated. The structureborne noise consists of structural impact noise and friction-induced noise. The airborne noise is due to airflow-induced acoustic wave during belt-sprocket tooth engaging. The spectral signatures of the varied noise are quantified, and the case studies are given to illustrate the influences of the tooth parameters and operation conditions on noise. The noise due to belt span vibration under impact ranges from hundreds to several thousand Hz. The impact noise, friction-induced noise and airflow-induced noise of belt tooth ranges from 3 kHz to 10 kHz. & 2012 Elsevier Ltd. All rights reserved.

1. Introduction Synchronous belts are widely used for motion and power transmissions in industrial machines, automotive engines and other devices such as photocopy machines. Synchronous belts are drive elements which transmit power between shafts through the action of moulded teeth in a performed sprocket. The belts are usually composite in nature. A belt has tensile cords to carry the extensional loads and rubber compound forming the backing of the belt and the bulk of the belt teeth. A belt also has a facing fabric covering the belt face in contact with the sprocket so as to reinforce the belt teeth and protect the belt wear. The application of synchronous belts has grown over last decades due to their potential advantages of light weight and unlubricated operations. The noise of these applications has been an engineering concern even though there have been improvements through optimal design, material and manufacturing quality. In addition to the improvement to the useful life and reliability of these belts, noise control measures have been needed for synchronous belts system due to customer’s quality perceptions. There are many articles in the literature dedicated to the experimental characterization and analysis of vibrations and noise of synchronous belts [1–21]. However, the quantification of belt noise in design phase remains an important issue. The previous experiments and analysis suggest that the possible root cause of running noise in synchronous belt drives includes the meshing impact noise, noise due to transversal vibrations of belts, friction-induced noise, flow-induced noise, sprocket vibrations and others. They can also be due to collision of the tooth tip of the pulley with the bottom land of the belt at the beginning of the engagement, slight eccentricities of the drive or idlers and sprockets. They can also be attributed to the material non-homogeneity, the polygonal effect at the interface between

n

Corresponding author. Tel.: þ1 907 328 8525. E-mail addresses: [email protected] (G.S. Chen), [email protected] (H. Zheng), [email protected] (M. Qatu).

0022-460X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2012.11.030

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Nomenclature

S

a

S0 St t T0 T u u

A c0 C D En f f0 F(t) F~ d Fd Ff(t) FN(y) F N ðtÞ FNi H K K(St) l LH(y) m M MC me Me pm R Rb Rs R*

parameter characterizing tooth contact given by Eq. (9) area of cross section of belt speed of sound (m/s) viscous damping matrices of multi-degree-offreedom system of belt segment distance between contact point and sprocket center effective elastic modulus frequency (Hz) natural frequency of the first transverse mode of belt span (Hz) force vector between two belt teeth and sprocket tooth fluctuating drag force (N) steady-state drag force (N) friction force in interface tooth static load acting on the side of belt tooth transient contact load coefficient of Fourier series development of FN(t) The height of contact point with respect to sprocket center stiffness matrices of multi-degree-of-freedom system of belt segment proportional coefficient related to Strouhal number length of belt span Hertz contact length of belt tooth tooth number of sprocket mass matrices of multi-degree-of-freedom system of belt segment contraction Mach number masses of the elastic cylinders of impact model maximum pressure distance between centers of sprocket and curvature of the contact sprocket tooth radius of curvature of the contact belt tooth radius of curvature of the contact sprocket tooth effective radius of curvature of the contact teeth.

v vc VF VF0 w W Z

b b0

g d dm d

e y y0

m m0 mn r s ti o O

od o0

channel between belt tooth-sprocket tooth during meshing clearance between belt tooth and sprocket tooth Strouhal number time mean tension of belt dynamic tension of belt speed of sprocket tooth at meshing point displacement vector of multi-degree-of-freedom system of belt segment speed of belt tooth at meshing point channel flow velocity (m/s) relative sliding velocity at meshing point maximum relative sliding velocity at meshing point belt width transverse displacement of belt in vibration substitute value for W specific angle in Fig. 2 initial value of the specific angle in Fig. 2 belt damping coefficient compression or relative displacement of contact teeth of belt and sprocket maximum compression of interface Dirac’s delta function modification factor of theoretical model based on FEM model rotating angle of sprocket initial reference angle of the sprocket tooth coefficient of friction (COF) of belt tooth static coefficient of friction, random portion in COF. mass density of belt amplitude of impact response of belt tooth segment under meshing impact duration rotating frequency of sprocket rotating angular speed of sprocket damped natural frequency of simplified meshing model of belt tooth segment meshing frequency

sprocket and belt, oscillation of the camshaft, torque pulses and long spans. The measured noise consists of the low frequency component occurring at the meshing order and the high frequency component of above 5 kHz. The high frequency noise occurs once per a pitch of a meshing and mainly consists of the impact noise, friction-induced noise and airflow-induced noise. They occur at the same time. Various methods have been proposed to reduce the noise associated with these mechanisms. Meshing impact noise has been considered to be due to impact generated by collision of the tooth tip of the sprocket against bottom land of timing belt at the beginning of engagement. Improving the meshing smoothness can reduce the noise. The impact sound of meshing is the most common source of noise. This impact sound occurs as the bottom land of the belt and the tooth tip cylinder collides at the beginning of the meshing of the tight side. Noise due to belt transverse vibration is known as goose noise or whine. External excitation is caused by belt drive itself, like eccentric running of the sprocket. Inherent system excitations are caused by oscillating torque or torsional vibrations from the oscillation of camshaft, which is due to inherent combustion signature and inertial engine imbalances [22]. Particularly, resonance could be excited if the frequency of mesh impact equals the belt natural frequency. Parametric resonance and nonlinear oscillations could occur as well. Theoretical studies show that the resonance could occur if the belt transverse natural frequency is identical to the meshing frequency, ½ of such frequency, or some other fraction (subharmonic) or

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times (harmonic) of meshing frequencies, due to parametric or nonlinear excitations. Friction-induced noise is generated by belt slippage against sprocket, and it occurs almost at the same time as the impact noise. Friction-induced noise is in high frequency range. It is independent of belt speed, tension, belt width and length, and is independent of transverse vibrations. Friction-induced noise is mainly dependent on the belt materials. Particularly, friction-induced noise is dependent on surface topography of contacting parts in mainly dry contact. Airflow-induced noise is due to air pumping and ejection between belt and pulley at meshing. Airborne noise is mainly dependent on the belt profile and geometry. It was illustrated that the high frequency noise is mainly from friction-induced noise and or air pumping noise. However, the sources of belt noise have not been comprehensively quantified, and the features of noise parameter dependence have not been clarified. For instance, [4] showed that the impact sound is caused by the collision of tooth tips of the driving pulley against bottom lands of the toothed belt. However, this is valid only for specific kind of belts. Although the impact sound has become evident as a noise source, the mechanism of belt-sprocket impact-induced vibration, which generates impact sound, is not fully characterized [4,15]. The distinction between impact, friction-induced and airflow-induced noises, despite their co-existence in meshing process, remains to be clarified. A considerable amount of literature on synchronous noise is available, but most of the literature is dedicated to experimental characterization and numerical analysis. There are limited theoretical studies that have been published. Hardly any of these studies focuses on the distinction of high frequency noise components. This paper investigates the noise sources of synchronous belt by quantifying the impact dynamics, friction-induced noise and airflow-induced noise in the context of belt-sprocket engagement and the co-presence of multiple noise sources in the meshing process. The decomposition and formulations of the noise source are based on the impact dynamics of belt-sprocket tooth contact, the vibration of belt span, the frictioninduced tooth vibrations as well as airflow-induced noise during tooth engaging. An attempt is made to develop a comprehensive noise prediction equation in terms of the dimensions, operational parameters and mechanical properties of belt. The study is conducted to decompose the noise source and to elucidate the differences of the sources under different load and operational speeds. The results from different noise patterns and different sources are also compared in terms of the specific characteristics. A large number of parameters of the system are taken into account to describe the impact and vibration characteristics of contact surfaces. The tooth impact process is modeled by considering the geometry and motion, contact and tooth deformations. Rolling and sliding process effects of tooth are considered to extract useful equations for the interaction. A theoretical model for the sliding and impact of the belt tooth is presented based on the dynamics and mechanisms associated with impact-sliding process in belt tooth engaging sprocket. Based on the impact dynamics results, the impact response of belt span structures are estimated based on finite element method (FEM) analysis. It is illustrated that a series of belt transversal natural modes could be excited by the meshing. Moreover, based on the contact dynamics results, a belt tooth segment vibration model with friction is used to illustrate the mechanism of friction-induced vibration and noise. The belt segment is modeled as a three-degree-offreedom dynamic system and the natural modes are estimated by using FEM. The frequency and time domain analyses are conducted for the model system. The analytical and simulation results are qualitatively correlated with the basic features of meshing impact friction-induced noise in conventional publications. To quantify impulsive airflow-induced noise, a noise model of air-pumping and flow ejection during tooth engaging is presented based on Lighthill’s equation. The spectral signatures of the noise are quantified to illustrate the influence of the tooth parameters and operating conditions. The proposed model allows for distinguishing the structural impact noise, friction-induced noise and air flow-induced noise with their respective parameter dependent properties.

2. Modeling of the impact between belt tooth and sprocket tooth Fig. 1 shows the section of one segment of a synchronous belt-sprocket interface. Most synchronous belts have a composite structure, in which a tensile cord carries the extensional loads on the belt, and a polyamide facing fabric covers the belt face in contact with the sprocket. The bulk of the belt teeth are filled up with rubber compounds which also form the backing of the belt. In meshing process, the belt tooth fits in the matching groove of the sprocket. Belt tooth contact involves the combined rolling and sliding motion between belt tooth and sprocket tooth surfaces. The contact per meshing can be quantified locally along the line of action during one single engagement as schematically

Fig. 1. Schematic of belt tooth-sprocket tooth interface.

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Fig. 2. Schematic of contact point of belt tooth under meshing.

shown in Fig. 1. The tooth contact at each one of these points can be assumed similar to two meshing cylinders with radii Rb and Rs, and with velocities v and u, respectively. The interaction between the belt and sprocket tooth flanks is transient rolling and sliding as shown in Fig. 2. We simplify the problem by assuming that the contact pressure can be approximated by Hertzian pressure during the meshing cycle. The contact during a cycle can then be simplified by the following formulations. For the determination of the pressure in the contact, we must know the transmitted load by the tooth pair considered. The tooth static load acting on the side of belt tooth was derived by the analysis or experiments proposed in [18–25], denoted as FN(y), which is a decreasing function during the meshing process for the driving sprocket and is an increasing function for the driven sprocket. The tooth pressure can be estimated using Hertz formulation. The Hertz contact length is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4F N ðyÞRn LH ðyÞ ¼ 2 (1) pwEn In which w is belt tooth width, E* the effective elastic modulus which is a function of the elastic modulus of belt tooth and sprocket tooth, R* the effective radius of curvature of the two teeth in terms of classic contact mechanics theory [26]. It is noted that Eq. (1) offers the estimation of contact ‘‘length’’, the contact width is belt width w. The contact ‘‘length’’ used here is the commonly used ‘‘contact width’’ in many literatures. For more accurate analysis, the modified length is obtained by using FEM analysis. The numerical results can be incorporated into the following model by using the modification L(y) ¼ eLH(y) in which e is the modification parameter. Based on Fig. 2, the following relationships can be obtained:

y ¼ y0 Ot, D ¼ H=cos y

(2)

In which H is vertical distance of the observation point to the sprocket center. From Fig. 2, it can be obtained,

bðtÞ ¼ arccos½ðDðtÞ2 þ R2s R2 Þ=2DðtÞRs 

(3)

V F ðtÞ ¼ vcos½bðtÞyucos bðtÞ

(4)

In general, the contact load can be decomposed as Fourier series with the fundamental frequency as meshing frequency o0 F N ðtÞ ¼

1 X

  F Ni sin io0 t þ ji

(5)

i¼0

in which FNi, ji are the coefficients of Fourier series after the development of FN(t). For simplification, the discussion is limited to the meshing frequency. Under this situation, the impact event occurs once for each meshing. The impact force denoting as FN1 is estimated as follows. In contact process, the impact force can be approximated as the following sinusoidal distribution:   p F N1 ðt Þ ¼ F 0 sin t (6)

ti

in which ti is impact duration. Hertz’s solution for elastic impact of the cylinder can be derived from the static theory of elastic contact (quasi-static approximation). Consider the impact of two elastic cylinders of masses me and Me with radii of curvatures Rb and Rs, respectively. During impact, their centers approach each other through a compression displacement d due to elastic deformation, and their relative approaching velocity is V ¼dd/dt. The contact force between them at any instant is FN1(t). Consider the elastic contact of two masses with cylindrical surfaces; the maximum impact force F0 can be

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2

Impact and Slip Speed (m/s)

1.8

Impact speed Sliding speed

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Engaging Time (ms) Fig. 3. Impact and sliding speed of contact points in the engaging period (4000 rpm).

solved from following the coupled equations: 2

F 0 ¼ k1 d þme 



d d

(7)

2

dt







d ¼ 0:638 F N0 =a 1v2 ½1=3 þln 2Rb =a =E: a¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4F N0 Rn =pwEn

(8) (9)

me is the effective mass of belt tooth. k1 is the effective suspension stiffness associated with the mass me in the impact model of belt tooth. It is noted that Eq. (7) is consistent with Eq. (8.2.25) given in [26]. Integrating the above equation with respect to initial condition   dd ¼ v0 sin b0 y0 u0 sinb0 dt

(10)

The relative normal velocity and force at the instant of impact can be estimated. At the maximum compression dm, dd/dt¼0, and therefore the maximum compression and maximum impact force can be obtained. The tangential motion equation is approximated as Z t me x_ ¼ m F N1 ðtÞdt (11) 0

It is noted that synchronous belt operates by friction forces and tooth flank normal forces and is likely associated with belt creep and stick-slip motion. Eq. (11) characterizes pure sliding motion which is one of prerequisites of frictioninduced high frequency noise. The friction-induced high frequency noise is a major concern as it is much stronger than stick-slip noise [12]. Fig. 3 shows the impact and sliding speed of contact points in the engaging period (4000 rpm). 3. Vibrations of belt tooth segment and belt span under meshing effect Fig. 5 shows the calculated natural mode shapes of belt tooth segment. The belt parameters are listed in Table 1. The results show that the belt tooth has many local bending and torsional modes. Some of these modes could be excited by impact. For instance, the first bending mode is likely to exhibit decayed vibrations under impact effects during meshing. The impact response of the transverse displacement of tooth segment under meshing can be represented as Z t 1 W¼ FðtÞ ezoN ðttÞ sinod ðttÞdt (12) me od 0 In which me, oN, od, z are the modal parameters of mode equivalent system of a tooth segment undergoing mesh, F(t) is the corresponding impact force given in Fig. 4. In addition to the belt tooth local vibration, the impact could excite natural mode of the belt span. Fig. 6 shows the variation of the fundamental bending natural frequency of the belt span with respect to belt segments. Fig. 7 shows the several lower modes of a belt span, which include (a): 885.16 Hz, 1st bending; (b) 2222.2 Hz, 2nd bending; (3) 3796.4 Hz, 3rd bending; and (d) 4199.8 Hz, span bending and bending about tooth bottom (anti-symmetric). It is noted that the meshing frequency is equal to sprocket tooth number times speed frequency, o0 ¼mo ¼2pm(rpm/60), which

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Table 1 Parameters of a belt tooth-spocket interface. Property

Value

P T B D m Rpm E rubber/fabric Cof T

8 mm 2.9 mm 2.3 mm 1.1 mm 32 2,000 490 MPa 0.2–0.4 500 N

Horizontal and vertical force (N)

6 Vertical force Horizontal force

5

4

3

2

1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Engaging Time (ms) Fig. 4. Vertical and horizontal projection of impact force on belt due to normal contact and friction.

Fig. 5. Mode shapes of belt tooth segment (4380 Hz, Yaw; 4667 Hz, Torsion; 5472 Hz, Bending; 60588 Hz, Bending).

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Fig. 6. The variation of fundamental natural frequencies of belt span with respective to belt segment.

Fig. 7. The modes of a belt span ((a) 885.16 Hz, 1st bending; (b) 2222.2 Hz, 2nd bending; (c) 3796.4 Hz, 3rd bending; (d) 4199.8 Hz, span bending þbending about tooth bottom (anti-symmetric)).

could coincide with the lower order of belt span modes. Under this situation, there is coincidence of the belt natural frequency with the meshing frequency, and the corresponding response can be estimated using a formulation similar to that of Eq. (12). The belt span vibration under meshing effect has been attributed to the disturbance of two meshing position [1,2,10,14,15], similar to the vibration of chain under effect of polygonal action or the vibration of string driven by an eccentrically mounted pulleys [27–29]. As the bending rigidity of the synchronous belt is negligible, a moving belt model can be used to predict the dynamics of the system [8,9]. The equations of motion for the transverse vibration of a belt span can be represented as,   ðT þT 0 ÞW xx rA W tt þ 2cW xt þ kc2 W xx þ gðW t þ cW x Þ ¼ P ðx,t Þ

(13)

Where rA is the mass per unit length, W is the transverse displacement, t is the time, x is the spatial coordinate, T0 is the mean tension, T is dynamic tension, k is the pulley support system constant, c is system parameter and g is the belt damping coefficient. P(x,t) is belt disturbance load. Assume tooth meshing engaging results in impact response of belt tooth segment at both ends of belt span, which can be obtained from Eq. (12) and represented as boundary conditions for Eq. (13). This yields x¼0 : W ¼s

1 X

sinðio0 t Þ, o0 ¼ mo,

i¼1

x¼l : W ¼s

1 X

  sin io0 t þ fi

(14)

i¼1

where o is sprocket rotating frequency, s is the amplitude of response due to meshing impact effect and m is number of   tooth. Assume variable tension, TðtÞ ¼ T 0 1þ ecosop t , in which T0 is static tension. op is frequency of tension variation. e is ratio of variable portion to average tension. The belt dynamic disturbance force such as tensioner support is assumed as P P ðx,t Þ ¼ P0 dðxdÞ 1 i ¼ 1 cosðio0 t Þ, d is Dirac’s delta function, d the distance of disturbance from end, o0 the meshing frequency. To solve Eq. (13), we substitute transverse displacement W with 1 1   X     X sinðio0 t Þ þ x=l s sin io0 t þ fi W ¼ Z þ 1x=l s i¼1

(15)

i¼1

in which Z is a new variable. Then we get following forced vibration equations with homogeneous boundary condition   ðT þ T 0 ÞZ xx rA Z tt þ2cZ xt þ kc2 Z xx þ gðZ t þcZ x Þ ¼ P ðx,t Þ þ f ðx,t Þ

(16)

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In which, x ¼0:Z ¼0; x¼l:Z¼0, 1 1 X X       f ðx,t Þ ¼ 1x=l rAsi2 o20 sinðio0 t Þ þ x=l rAsi2 o20 sin io0 t þ fi i¼1

(17)

i¼1

By ignoring the damping effect, tension variation and the flexibility of sprocket support as well as the effect of disturbance force which was addressed in [21], the solution of Eq. (16) is given by Z¼

1 71 X s X i¼1

i

i ¼ 71

sin npx   p2 nlo2n 1io0 =on

fZ1 cosðio0 t þ npvxÞ þ cosðnpÞ½sin io0 tnpvð1xÞÞþ Z1 cosðio0 tnpvð1xÞÞg

(18)

where on ¼np(1  v2), Z1 ¼2v(on io0)/(onio0). Finally the transverse vibration can be written as W¼

1 71 X s X i¼1

i

sin npx   2 nlo2 1io =o p n 0 n i ¼ 71

" # 1   1X 1 x sinðio0 t Þ  Z1 cosðio0 t þ npvxÞ þ cosðnpÞ½sin io0 tnpvð1xÞÞ þ Z1 cosðio0 tnpvð1xÞÞ þ þ s 1 l pi¼1 i

(19)

One significant source is from the impact between the belt and the sprocket tooth during the meshing process. Figs. 1 and 2 show the schematic of the belt and sprocket configuration at the infinitesimal meshing time. The belt tooth segment sliding range A–B, before entering the sprocket, is moving in a velocity in the belt-span direction but not tangent to the pitch circle of the sprocket. Once this segment sits on the sprocket, it will attempt to follow the circular pitch line of the sprocket, i.e., move in the direction tangent to the sprocket pitch circle. Therefore, the entering segment poses a velocity change at the infinitesimal meshing time, implying that an impact occurs between the entering belt segment and the rotating sprocket. The relative impact velocity of the belt segment to the sprocket causes the vibration of both sprocket and belt that radiate noise into surrounding air. The impact noise of a toothed belt drive is closely related to the system pitch, operational speed, belt tension, and overall dynamic behavior of the belt span. However, the noise peaks always appear at the meshing orders and harmonics in the order spectrum [1–8]. The transverse vibration of belt may amplify the impact force between belt and sprocket, hence resulting in higher sound pressure of the impact noise from the drive. The impact effect could excite the modal vibrations, particularly, when meshing frequency is identical to the natural frequencies of the belt span, resonance occurs. It can be seen that the response includes the meshing frequency and harmonics. The magnitude of response is proportional to the speed due to impact effect. In fact, the experimentally observed 6 dB increase in power which occurs when the belt speed doubles can be related to the vibration velocity amplitude of the belt spans, which also increases proportionally to the belt speed (this could be seen easily in the vibration spectra). This may be due to the increase of the meshing impact force, which is the most likely cause of excitation of transverse vibration of belt spans (in general, not only under resonance conditions). The impact force may change due to variation of the impact velocity (defined as the velocity of the belt at the meshing point) of the tooth tip of the sprocket against the bottom land of the belt with the belt speed. 4. Friction-induced noise Fig. 5 shows the calculated natural mode shapes of belt tooth segment. The results show that the belt tooth has multiple local modes (bending and torsional). Some of these modes have very small separations. For instance, the first bending and first bending-torsioanal modes only have 7percnt difference. This proximity is likely to get the modes coupled under high friction. To elaborate the properties of the friction-induced vibrations, a three degree of freedom model is used in the following analysis. A synchronous belt drive, like any other types of belt drive, has an unavoidable slip motion of belt relative to sprocket during the whole course of belt-sprocket engagement. The slip motion produces the friction force between the belt groove and sprocket tooth, generating friction noise from the drive. Thus, the belt-sprocket friction is responsible for the friction noise generating mechanism in a toothed belt drive. The friction noise constitutes another significant noise source of a toothed belt drive. The peak frequencies of friction-induced noise and its sound pressure level of a synchronous belt drive depend highly on the belt material and geometry, although the speed and system pitch do have effects on friction noise frequency and noise levels. This is different from those of its impact noise where the speed and pitch are more dominant in determining the impact noise frequency and noise levels. Compared to the impact structural noise, friction-induced noise has higher peak frequencies. There have been some studies on synchronous belt noise with belt unit testing machines and engines. It was found that the high frequency noise occurs mainly from the friction at the beginning and end of meshing between the belt and sprocket. It was identified that the portion of high frequency noise is generated by the discontinuous slips between belt and sprocket. Improve the meshing smoothness can reduce this type of noise.

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Fig. 8. Three-degree-of-freedom system model of belt tooth segment.

The fundamental frequency of friction noise are very high and is above 4000 Hz for rubber belt, which does change its value when the belt speed, belt tension, belt width, length of belt was changed. It depends mainly on belt materials and geometry. Most practical belt meshes encompass complex friction characteristics in the tooth contact zone. [30–33] reported the coefficient of friction ranging above 0.3. To give an equation for prediction of the coefficient of friction, the friction law of a friction–velocity trigonometric curve is used (   m0 tan1 V F =V F0 þ mn ðtÞ V F 4 0 m¼ (20) m0 VF ¼ 0 in which m0 is the static coefficient of friction, VF0 is maximum slip speed, mn(t) is the random term. It is noted that for nonslip (VF ¼0) case, there is no friction force. The horizontal reactive force in interface equals to driving force which can be described by static coefficient of friction [26]. It is well established that in a synchronous belt system the power is transmitted both by friction forces and tooth flank normal forces. In this study, the conventional spring-mass models are further extended to include the friction elements. The belt sprocket interaction is made discrete by rigid bodies, springs and mass, a multi-body system with friction as shown in Fig. 8. Each pitch consists of one mass body representing the tooth and another land area. A tooth segment has three degrees of freedom, while a land area segment is assumed to be fixed. A rotational degree of freedom for the tooth segment is believed to have a minor effect on load distribution for belts in complete mesh, therefore it is neglected. Each contact, where translation in the normal direction is present, is modeled with a linear spring for this purpose. In addition to this, there is a spring representing the shearing at each contact. There is rotating spring connecting friction plates that utilizes the above friction model and represents the contact between the belt and pulley. Here, a spring based model capable of handling dynamic load conditions is presented. The model also covers partial meshing effects and utilizes the proposed friction model. In Fig. 8, a single mesh belt tooth segment is used and represented by their base circles. Mesh stiffness is derived from the Hertzian deformation as well as the cantilever bending of the teeth. The displacements of the segment are referred to as u, v, y respectively. Sliding on the belt tooth surface causes a frictional force Ff along the off-line of action direction. During the tooth meshing action, the tooth contact point moves along the line of action direction. The sprocket rotates with mean angular speed O. The magnitude and direction of the speed govern the friction at the surface. In this study, the beginning of the mesh cycle is defined to be coincident with the initiation of contact for the second tooth at point A. The second critical point occurs at the zone of contact passes through point B. Consider the three-degree-of-freedom system shown in Fig. 8. The three-degree-of-freedom model of tooth segment is connected by a contact sliding interface with coefficient of friction m. The equation of motion of the system can be expressed Mu€ þ C u_ þ Ku ¼ Ff

(21)

where M, C and K are the mass, viscous damping, and stiffness matrices for the non-friction system respectively, u is the displacement vector, and Ff is the friction force vector between the belt tooth and sprocket tooth. The friction system consists of an interface with contact and connected in the normal direction, but not in the tangential direction. The tangential friction is modeled as a force   Ff ¼ m Nstatic þ Ndynamic (22) where Nstatic, Ndynamic are the static and dynamic normal forces, respectively. For solution of the dynamic problem, the static force is removed from the equation of motion. Hence the system equation of motion becomes Mu€ þ C u_ þ Ku ¼ mK s Kf u

(23)

There are several mechanisms of friction-induced instability: the mode lock due to negative slope of friction–velocity curve or stick-slip; the model coupling due to larger coefficient of friction [17]. However, based on the derived engaging

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0.5 0.45 0.4

Frequency

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

cof Fig. 9. System natural frequencies as a function of coefficient of friction.

Periodogram Power Spectral Density Estimate 10

Power/frequency (dB/Hz)

0 -10 -20 -30 -40 -50 -60 -70 -80 -90 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (Hz) Fig. 10. System response spectrum.

process attributes shown in Fig. 3, the tooth experience a deceleration process in engagement, thus the mode lock can be eliminated. Actually, the experimental results [11,12] shows that the friction-induced noise in high frequency range, whereas mode lock-related noise is in low frequency range. The simulation results are illustrated in Figs. 9 and 10. Fig. 9 is the system natural frequency as a function of coefficient of friction and Fig. 10 is the system response and spectrum (cofavg ¼0.34). The response of a linear three-degree-of-freedom system with friction constitutes three independent fundamental frequencies. Depending on the system parameters and the coefficient of friction, the fundamental frequencies of the response may relate to each other, exhibiting the synchronization the fundamental frequencies. The fundamental frequencies may also appear to be totally non-synchronizing. The system has two higher frequency natural modes in proximity. When coefficient of friction increases, the two fundamental frequencies moved closer from the two natural modes as illustrated in Fig. 9. When coefficient of friction reaches a sufficiently high level, the two proximity modes get coupled to be identical and the instability of the system is triggered. The published experiments demonstrated that the friction noise could be reduced by reducing friction by sewing up the tooth or applying lubricant [12]. Fig. 11 shows the variation of natural frequencies of the first two tooth modes with respect to belt width, it can be seen that they are almost independent of belt width. This feature can be used to distinguish the high frequency friction-induced noise from the high frequency airflow-induced noise. The high frequency airflow-induced noise is highly dependent on belt width, which will be detailed in next section. 5. Airborne noise There always exists airflow between belt tooth and sprocket grove while belt tooth and sprocket grove engage and disengage. The air will be pumped and ejected when the belt tooth engages the sprocket groove, and air will inject in when

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Fig. 11. The variation of natural frequencies of the first two tooth modes with respect to belt width.

the belt disengage from the sprocket. Sudden pressure changes in the air media accompanying with the air ejection– injection between the belt and sprocket generate impulsive air borne noise. It was considered that the air ejection noise could be relevant in the case of the resonance of duct (the air gap between belt and sprocket teeth) [1,4,14–16]. There are other mechanisms of noise generation of an aero-acoustic nature, due to the existence of dipoles generated by the interaction between the air and the solid surfaces of the pulley flanges. The noise source due to variation of a mass flow source is the monopole source. It is the pulsating escape of the air from channel. For the monopole source, based on the approach of [34,35], a solution of the Lighthill’s equation can be found by using the convolution integral between the gradient of the volume velocity and the Greens-function. The radiated sound pressure can be described as standing wave solution: qffiffiffiffiffiffiffiffiffiffiffiffi

npx Z t 4r0 c2 X 1 dQ ðtÞ xon ðttÞ 2 qffiffiffiffiffiffiffiffiffiffiffiffi pðx,t Þ ¼ e sin sin 1x on ðttÞdt (24) w dt pwS n ¼ 2k1 2 0 n 1x on in which, on ¼ npx=w, n ¼ 1,2,   , w is the width of belt. The air noise generated from belt meshing is the impulsion of air between the belt and the pulley during the engagement. Touuret, Koyama et al. [4,9] found through various experiments that the characteristic frequency of impact sound is affected by space between belt tooth and sprocket tooth, and affected by the belt width. This means that the impact sound comes from the inside the space between the belt bottomland and the sprocket groove [16]. The power spectrum of sound pressure can be obtained by FFT transformation of Eq. (24). Existing experimental results show that if pulley has flange, the sound pressure level of noise has increment of several dB. For this case, the dipole and quadrupole models are needed to account for the phenomena, as the dipole noise is caused by the flow obstacles and quadrupole noise is caused by flow jet. Let us consider a flow channel with a flow obstacle, which could be the flange of sprocket that used in much synchronous belt system. Assume that disturbances going out from the obstacle go on running without reflection. The external forces on the obstacle’s surface remain as a source of noise [36]. Therefore, the source area can be limited very simply as the field of the channel in which the alternating forces on the surface of the obstacle and the channel wall are not negligibly. A solution of the Lighthill equation can be found by using the convolution integral between the gradient of the alternating forces and the Greens-function, similar to Eq. (24); whereas the power spectrum of sound pressure can be obtained by FFT transformation of the solution [36]. The spectral density of the acoustic power W(o) can be approximated by two portions: the first one is below the cut-on frequency of the first transverse channel mode f0 (plane wave propagation) and the second one applies to frequencies higher than f0 (multimodal propagation), W ðoÞ ¼

2

W ðoÞ ¼

2 F~ d ðoÞ , f of 0 4Sr0 c0



(25)

o2 F~ d ðoÞ 3pc0 ða þ bÞ , f 4f 0 1þ 4oS 24pr0 c30

(26)

In which o is frequency, a and b the cross-sectional dimensions, S the channel cross section area, r0 the ambient density and c0 the ambient sound speed. The total fluctuating drag force is F~ d . Within 1/3-octave bands, it may be assumed [5] that the fluctuating drag force is a function of Strouhal number. F~ d ðoÞ ¼ KðStÞSrv2c =2

(27)

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vc is flow velocity, St is the Strouhal number (fd/vc), d is characteristic dimension, K(St) is proportionality factor. Then the radiated sound power is W ð oÞ ¼

rS 2 K ðSt Þv4c 16c

(28)

The sound power is proportional to the channel cross section area, the fourth power of the flow velocity and its spectral distribution is described by the function K(St). A solution that the RMS value of pressure fluctuation on the obstacle surface can be written as pðoÞ ¼

F~ d ðoÞ 2S

(29)

This solution is strictly valid only for compact sources smaller then 1/4 wavelength. The radiated sound pressure is proportional to the alternating force at the surface of the obstacle. K(St) stands for the spectral efficacy of the noise production. This is the solution for an infinite channel. However, the space between belt and sprocket are considerably shorter and we need to modify solution. K(St) can be represented through the exponential law, K(St)¼cStb. Using the inverse method, K(St) can be deduced values from the sound radiation outside the channel. Although no single universal curve exists, different works lend support to the important idea that a range of components might produce a similar K(St) curve. It usually decays by tens dB per decade. For applications with the sprocket having flange, the small spacing between belt and flange could behave as convergent nozzle to lead to jet flow. The airstream flows from the channel to the ‘‘nozzle’’ to exit to surroundings through an acceleration process. The Mach number is low for no flange sprocket. If there is flange, the Mach number could be as high as 0.6, and the jet flow could generate in the belt-sprocket channel, which leads to quadrupole source. The frequency spectrum of jet exhibits broadband attributes and is given by [32–35,37,38] Y ðoÞ ¼

1 dW o nc c0 4 ¼ W do 5d c pðoKðStÞ=2p þ 2p=oKðStÞ2

(30)

The spectrum curve has a shallow peak with the location at fp ¼ 0.2vc/d, in which vc is exit velocity and d is the dimension size of nozzle. fp is usually at very high frequency beyond audible frequency range. Therefore its spectrum is an increasing function in interested range. In order to investigate the predictions given by the models proposed above, some experimental results that had already been presented in [4] were analyzed and used for comparison. In [4], these results are related to measurements of flow noise for same kind of belts with different width, and for same belts with different tooth profile and clearance between belt tooth and sprocket tooth. Fig. 12 shows the calculated sound spectrum for belts with different belt widths based on the data in [4]. For comparison, the measured noise peak bands given in [4] are also plotted in Fig. 12 as indicated by line A, B and C. From it, we can see the standing wave frequencies shift with the belt length. The amplitude changes with the length are due to the load pressure change. It is noted that the specific frequency corresponding to the peak is belt width dependent for flow-induced noise, whereas the friction-induced noise is independent of belt width. Based on the drive data in [4], Fig. 13 shows the calculated sound spectrum for belts with different cross sectional clearance areas. Results in [4] shows that the noise level has 12 dB reduction after the belt-sprocket clearance under full engagement is reduced by 6 times. In [4], the experimental results also show that the noise at the beginning of meshing became loud in the case of cutting off the belt tooth crest and the tooth flank toward the anti-running direction; this can be interpreted by the model.

Fig. 12. Sound spectrum of airflow-induced noise of belts with different width.

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Fig. 13. Sound spectrum of airflow-induced noise of belts with different cross sectional areas.

6. Conclusion The comprehensive formulations of the noise of synchronous belt shows that the primary noise sources consist of impact-induced vibrations of belt tooth and belt span, the friction-induced vibrations of tooth as well as airflow-induced sound wave during tooth engaging. The noise due to impact-induced belt span vibration ranges from hundreds to several thousand Hz. The noise due to impact-induced tooth vibration, friction-induced tooth vibrations and airflow-induced effect ranges from 3 to 10 kHz. The belt span resonance is due to impact effect, whereas the tooth resonances could be due to both impact effect and friction interaction. Friction-induced tooth resonances are independent of meshing frequency, and also independent of belt width. The airflow-induced noise is independent of meshing frequency, but dependent on belt width. The spectral signatures of various noises are comprehensively quantified, which are consistent with conventional experimental results. The proposed model is expected to provide designers with insight into the issues and tool to improve the tooth belt vibration and noise performance.

Acknowledgment The authors thank Dr Jim Zhu, SAIC Motor Technical Center for his supports in this study. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

A. Kubo, On the running noise of toothed belt drive, Bulletin of the JSME 14 (1971) 991–997. A. Kubo, On the running noise of toothed belt drive, Bulletin of the JSME 14 (1971) 998–1007. N. Mizuno, Analysis of synchronous belt vibration in automotive valve train, SAE (1988) 880077. J. Tourret, Y. Birembaut, M. Deschamps, Noise generation of synchronous belts using acoustic intensity mapping, Noise Control Engineering Journal 30 (1988) 11–14. C.C.J. Wong, Noise control of office copy machine (timing belt application), Noise Control Engineering Journal 46 (1998) 11–14. C.C.J. Wong, Timing belt noise of office machines, Noise Control Engineering Journal 35 (1990) 103–112. C.C.J. Wong, Timing belt noise of office machine, Noise Control Engineering Journal 39 (1992) 77–90. K. Watanabe, T. Koyama, A study on timing belt noise: theoretical analysis for forced transverse vibration of timing belt with parametric excitation, Transaction of ASME, Journal of Mechanical Design 112 (1990) 424–429. T. Koyama, K. Watanabe, K. Nagai, A study on timing belt noise: how to reduce resonant noise, Transaction of ASME, Journal of Mechanical Design 112 (1991) 419–423. H.J. Kaiser, J. Querengasser, G.M. Bundgegs, Special noise problems in automotive timing belts, SAE 931316. H. Tokoro, Analysis of transverse vibration in engine timing belt, JSAE Review 18 (1997) 33–38. H. Tokoro, M. Nakamura, N. Sugiura, H. Tani., K. Yamamoto, T. Shuku, Analysis of high frequency noise in engine timing belt, JSAE Review 19 (1998) 33–38. H. Ueda, M. Kagotani, T. Koyama, M. Nishioka, Noise and life of helical timing belt drives, Transaction of ASME, Journal of Mechanical Design 121 (1999) 274–279. R. Di Sante, G.M. Revel, G.L. Rossi, Measurement techniques for the acoustic analysis of synchronous belts, Measurement Science Technology 11 (2000) 1463–1472. R. Di Sante, G.L. Rossi, A new approach to the measurement of transverse vibration and acoustic radiation of automotive belts using laser Doppler vibrometry and acoustic intensity techniques, Measurement Science Technology 12 (2001) 525–533. W.M. Zhang, T. Koyama, A study on noise in synchronous belt drives (experimental and theoretical analysis of impact sound), ASME Journal of Mechanical Design 125 (2003) 773–778. G. Sheng, H. Zheng, M. Qatu, R.V. Dukkipati, Modelling of friction-induced noise of timing belt, International Journal of Vehicle Noise and Vibration 4 (2008) 285–303. T. Johannesson, M. Distner, Dynamic loading of synchronous belts, ASME Journal of Mechanical Design 124 (2002) 79–91. M. Callegari, F. Cannella, G. Ferri, Multi-body modelling of timing belt dynamics, Proceedings of the Institution of Mechanical Engineers 217 (2003) 63–75. M. Gregg, US Patent No: 5,254,049, Synchronous drive belt, 1993.

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[21] S. Abrate, Vibrations of belts and belt drives, Mechanism and Machine Theory 27 (6) (1992) 645–659. [22] H. Rahnejat, Multi-Body Dynamics: Vehicle, Machine and Mechanisms, PEP(IMechE) and SAE (Joint Publishers), 1998. [23] M. Distner, T. Johannesson, Measurements of forces between a synchronous belt and a pulley, Proceedings of the DETC’00 8th International Power Transmission and Gearing Conference, ASME, 2000, Baltimore, USA. [24] N.A. Karolev, P.W. Gold, Load distribution of timing belt drives transmitting variable torques, Mechanism and Machine Theory 30 (1995) 553–559. [25] T.H.C. Childs, J.K. Parker, Power transmission by flat, V and timing belts tribological design of machine elements, Lees-Lyon 15. TribologySeries, 14, Elsevier, Amsterdam, (1989) 133. [26] B. Bhushan, Principles and Applications of Tribology, John Wiley & Sons, 1999. [27] K. Uehara, T. Nakajima, On the noise of roller chain drives, Fifth Congress on Theory of Machines and Mechanisms (1979) 906–909. [28] H. Zheng, Y.Y. Wang, K.P.Quek, et al., Investigation of meshing noise of roller chain drives for motorcycles, Noise Control Engineering Journal 50 (2002) 5–11. [29] J. Moon, J.A. Wickert, Non-linear vibration of power transmission belts, Journal of Sound and Vibration 886 (1993) 308–320. [30] Y. Murakami, M. Waranabe, N. Ohsawa, Study of belt toughness through an examination of the strain on belt cords, SAE (1988) 880415. [31] K. Shimizu, T. Fujii, A simple modeling for analyzing the load distribution of toothed belts under fluctuating torque load, SAE (1995) 950542. [32] R. Kido, T. Kusano, A new approach for analyzing load distribution of toothed belts at steady states using FEM, SAE (1996) 940690. [33] R. Kido, T. Kusano, T. Fujii, Fatigue life evaluation for toothed belts based on FE analysis, SAE (1996) 960712. [34] A.P. Dowling, J.E.Fowcs Williams, Sound and Sources of Sound, Ellis Horwood Limited, 1983. [35] L.L. Beranek, Noise and Vibration Control, McGraw-Hill Inc., 1971. [36] H.H. Heller, S.E. Widnall, Sound radiation from rigid flow spoilers correlated with fluctuating forces, Journal of the Acoustical Society of America 47 (1970) 924–936. [37] P.A. Nelson, C.L. Morfey, Aerodynamic sound production in low speed flow duct, Journal of Sound and Vibration 79 (1981) 263–289. [38] D.Y. Ma., Noise Control Technology, Scientific Publishing, 1980.