## Abstract

Recent experimental studies have shown that tension transition via detachment waves (rather than via sliding, as commonly accepted) occurs at the belt–pulley interface for systems using simple homogeneous or textured flat belts operating under slow speeds. This raises the question of whether or not such detachment waves are universal—e.g., will they persist in systems incorporating belts with composite cross sections, such as those used in commercial applications? Herein, we experimentally explore the behavior of a belt-drive system incorporating a composite belt with tensile cords, with and without a patterned contact surface, and document the persistence of detachment waves. This then leads to a re-evaluation of the Firbank model of belt shear. We also explore the effect of tensile cords on (i) the amplitude and frequency of the observed detachment waves, (ii) the ensuing oscillations of the pulley, (iii) the evolution of belt shear strain, and (iv) the frictional losses of the system. We find that the detachment frequency increases and the pulley rotates steadier with inclusion of tensile cords. The corded belt undergoes shear deformation starting from the entry point due to a speed differential between the pulley envelope and the tension members, which is consistent with the classical Firbank shear model. However, the Firbank model cannot predict the rapid relaxation of the shear traction via detachment waves at the exit region. Additionally, accounting for shear and detachment events, we find that frictional losses in the belt-drive system decrease with inclusion of tensile cords.

## 1 Introduction

Belt drives find application in manufacturing processes and power transmission at nearly every scale. Belts (including poly-V, V, and flat) are used for mechanical power transmission in manufacturing machines and equipment, typically to connect the machine’s electric motor to various driven shafts such as printing presses, pumps, compressors, mixers, etc. They also find wide usage as front end accessory drives in vehicle applications [1].

Although the study of belt drives is expansive, the recent discovery of detachment waves in belt-drive systems with homogeneous flat belts [2] calls into question the universal validity of previously proposed belt-drive models, among which the most prominent is the elastic creep theory by Grashof [3]. Based on elastic creep, the belt–pulley contact arc is divided into a slip arc at the exit and an adhesion arc over the remaining region. Only the slip arc transmits a tangential force via sliding, which is governed by Coulomb’s friction law. In contrast to this theory, Wu et al. [2] recently explored a simple belt-drive system with homogenous flat belts and found no evidence of sliding. Instead, they observed that the relative displacement at the exit region of belt–pulley interface was achieved by detachment waves. Follow-on studies documented detachment wave-induced fluctuations in the pulley rotational response [4] and frictional losses as well as negligible contribution of the slip arc to the power transmission [5] in the drive system.

Modern commercially available belts typically contain inner cords for higher load capacity and increased integrity of the belt-drive system. Regarding this type of belt, Firbank [6] proposed a belt shear theory. In contrast to the elastic creep theory, the adhesion arc transmits a tangential force in response to evolving shear, which develops in the belt due to the speed differential between the pulley-contacting surface and the tensile cords. If and when the shear stress exceeds the maximum allowed by the Coulomb law, a slip arc appears in which sliding governs the remaining tension transition. The belt shear theory has been subsequently refined by considering the effects of belt flexural rigidity [7], belt inertia [79], radial compliance [10,11], and bending stiffness [10,12,13]. Nevertheless, in all studies, the relative displacement in the slip arc has been assumed to be accommodated by sliding. Whether or not detachment waves persist in belts with tensile cords remains unclear.

In light of the above, herein we present an experimental and theoretical study on the effects of belt cord inclusion on dynamic characteristics of detachment waves and self-oscillation and frictional mechanics in a simple belt-drive system. The presence of detachment waves is first examined, and the related self-oscillation in the system are studied and compared with those for the regular flat belt, a patterned belt and a patterned belt with tensile cords. The shear traction at the belt–pulley interface is then measured and compared with the Firbank shear model. Lastly, frictional losses in the system for the belt with tensile cords are estimated via combination of experimental observations and a theoretical model.

## 2 Experimental Details

### 2.1 Setup.

The setup used herein is custom-built based on a one-pass-slide tribometer, configured with either speed control on the exiting span of the belt (Fig. 1(a)) or the pulley (Fig. 1(b)), respectively. The former configuration enables study of pulley rotational oscillations excited by detachment events at the belt–pulley interface; the latter configuration eliminates pulley oscillations, clarifying study of detachment waves. Both configurations consist of driving, loading, and measuring systems. The driving system, connecting to either the exit span of the belt (Fig. 1(a)) or the pulley (Fig. 1(b)), consists of a dead weight released by an electric motor at a constant prescribed speed. The loading system consists of two adjustable dead weights, one applied to the entering span of the belt and the other applied to either the pulley (torque weight in Fig. 1(a)) or the exiting span of the belt (Fig. 1(b)), so that the tensions in the two spans of the belt can be controlled. Notably, in Fig. 1(a), reversing the direction of the torque allows us to switch between the driver (solid lines) and driven cases (dashed line), and in Fig. 1(b), the switch between the two cases can be achieved via adjusting the two tension weights. The measuring systems consists of two load cells, measuring the tension in the two spans of the belt, a rotary encoder, measuring the angular velocity of the pulley, and two cameras, capturing the contact status at the belt–pulley interface and the shear deformation at the sidewall of the belt, respectively. The data are sampled and processed with a custom application written using the labview software package. The real contact area is seen as a darker region due to optical interference at the belt/pulley interface. We obtain the shear strain by comparing tick marks created on the belt sidewall with a transparent reference—more details can be found in Ref. [5].

Fig. 1
Fig. 1
Close modal

### 2.2 Specimen Preparation and Experimental Conditions.

We fabricate the belt specimens using a transparent elastomer, polydimethylsiloxane (PDMS, Sylgard 184, Dow Corning, Midland, MI). The Sylgard 184 prepolymer and its cross-linker are mixed at a 10:1 ratio and cured for 14 h at 65 °C in light vacuum. We also mold evenly distributed tick marks with ∼0.125 mm intervals to the belt sidewall using a custom template for measuring the shear deformation [5]. Finally, we cut a reference flat belt specimen of 400 mm in length, 8 mm in width, and 3 mm in thickness from the molded form.

We fabricate flat belts with tension members by following the same process except that four cotton cords are placed on the template with 2.4 mm intervals under 0.1 N pretension before molding. The cured belts with tensile cords are then cut as designed in Fig. 2 with 400 mm in length. Additionally, a patterned belt with tensile cords (right one in Fig. 2(c)) is produced via a four-step molding technique replicating the topography of regularly textured example surfaces (details can be found in Ref. [14]). The patterned surface has an area density of AD = (D/a)2 ≈ 70% where a = 1 mm denotes the lattice constant and D denotes the diameter of the inscribed circle, and the aspect ratio, AR = h/D = 0.2, where h denotes the height of the projections (Fig. 2(b)). The four cotton cords are placed 2.75 mm above the patterned mold before the last molding.

Fig. 2
Fig. 2
Close modal

Limited by the strength of the belt without cords (also referred as the reference belt herein), we choose to work with a maximum allowable belt tension of 6 N. The tension difference between the two spans of the belts is adjusted to 4 N. The driving speed and the travel distance of the belt are 3 mm/s and 300 mm (∼5 revolutions of the pulley), respectively. Each test is repeated at least ten times. The temperature and relative humidity in the laboratory were 20 °C and 10%, respectively.

## 3 Detachment Waves and Self-Oscillation

There are two instabilities that have been observed to date using the belt-drive test rig depicted in Fig. 1 [2,4,5]. The first involves local detachment events at the belt–pulley interface, and the other is the appearance of growing pulley oscillations, which are related to the former instability but occur at a differing frequency. The detachment events at the exit region can be visualized using the optical camera where contact and noncontact regions can be differentiated by light interference at the belt–pulley interface. Changes in belt strain (or equivalently, tension) due to detachment events at the exit region also appear in the tension difference signal found using the two load cells (see Fig. 1). In addition, this difference signal encodes a tension changes arising from the pulley oscillations, which generally occur at a different frequency than the detachment events. The pulley oscillations can also be measured directly using the rotary encoder.

Figure 3 plots the time history of the tension difference between the two spans of the belt, and the pulley angular velocity, both measured using a corded belt for the driver and driven cases. The figure also provides images capturing time evolution of the contact area. Similar to previous studies using reference flat belts [2], the relative displacement between the corded belt and the pulley is achieved primarily by means of cyclic detachment vice sliding in both the driver and driven case. As in Ref. [2], we observe more pronounced detachment events in the driver case than the driven case. This is typified by an isolated fold forming in image 3 of Fig. 3(a), where the black region denotes contact area; such folds are missing in the driven case. It is believed that the solitary fold in the driver case is caused by the moment due to the accumulated shear traction at belt–pulley interface (see the schematic in Fig. 3(a)). This important finding confirms the persistence of detachment waves for a belt with a composite cross section and adds to the evidence that such events are a ubiquitous feature of tension transition in belt drives operating at slow speed.

Fig. 3
Fig. 3
Close modal

In order to characterize the effect tensile cords have on the character of the observed detachment events, the same tests as those described above have been conducted on a regular flat belt (Ref), a hexagonally patterned belt (Pattern), and a hexagonally patterned belt with tensile cords (Pattern-Cord). The effects are documented in Fig. 4 based on the frequency and amplitude of oscillations in the tension difference and pulley angular velocity. Although the tension difference fluctuations are affected by both the contact instabilities and the pulley oscillations, as shown in Fig. 3, we can extract frequency and amplitude information of only detachment events (Fig. 4(a)) using a wavelet decomposition. For the pulley rotational oscillations, we simply apply a fast Fourier transformation to the angular velocity signal to obtain the corresponding frequencies and amplitudes (Fig. 4(b)).

Fig. 4
Fig. 4
Close modal

When comparing the belt with tension cords (Fig. 3) and the reference belt, we find that the presence of the belt cords results in steadier pulley rotations with only subtle fluctuations around the prescribed angular velocity. The rotational oscillations in both the driver and driven cases have higher frequencies (Fig. 4(c)), but smaller amplitudes (Fig. 4(d)). These trends can be expected due to the significant increase in the belt stiffness provided by cords, which should act to increase oscillation frequencies and decrease deformations. In addition, the belt contact instabilities affect pulley rotation leading to a self-oscillation [4], i.e., the local contact instabilities serve as an excitation source for the pulley rotational response, which, in turn, stores a periodic tension pattern in the belt, further destabilizing the pulley rotation. The detachment waves, which cause the self-oscillation of the pulley, take place more frequently (Fig. 4(a)) in a corded belt as compared with a regular flat belt. On the other hand, the corded belt, limited by the tension cords, can detach from the pulley to a smaller extent compared with a regular flat belt, leading to a decrease in the corresponding amplitude (Fig. 4(b)). This effect is not pronounced in the driven case since the detachment events happen mainly due to peeling, which is weakly affected by the presence of tension cords.

When we pattern the contact surface of the belt with tension cords, we expect to see additional benefits associated with disruption of the contact surface, as reported in Ref. [14]. An important question then arises: does the combination of surface patterning and inclusion of tension cords eliminate contact instabilities and rotational oscillations? As documented in Fig. 4, the detachment events and pulley oscillations persist on a corded belt with patterns, confirming again their ubiquitous nature. However, the amplitudes of the contact instabilities and rotational oscillations (Figs. 4(b) and 4(d)) are minimized by the combination, while a small decrease in frequency for both can be noted (Figs. 4(a) and 4(c)).

## 4 Frictional Dissipation

The presence of tensile cords in the belt results in a significant difference in the belt-drive mechanics, especially for the shear traction along the belt–pulley interface. Shear traction develops and accumulates in the belt from the start of the contact arc due to the speed gradient between the tension member and belt–pulley interface, as described first by Firbank [6]. Herein, we evaluate the shear strain along with the contact status of the belt as a function of position for the corded belt. The shear strain is obtained by quantifying the deformation using visual observation of the tick marks. The contact status is determined by the light interference at the belt/pulley interface. Unlike our previous study [5], the measurement of the longitudinal stretching strain is not feasible since the stretching strain is significantly smaller due to the larger stiffness of the composite belt.

In the driver case, the distribution of shear strain is correlated with the contact area at the instance when a fold (noncontact region in white separating two contact regions) is formed at the exit zone (Fig. 5(a)). The shear traction is defined as positive when the pulley drives the belt (the driver pulley). Along the direction of belt motion, as shown in the small schematic in Fig. 5(b), the shear strain at the belt–pulley interface develops gradually from 180 deg (entry) to 62 deg, and then grows dramatically up to its peak from 62 deg to 37 deg, and then finally drops down to zero within 10 deg. Compared with a regular flat belt, the gradual growth of shear (180 deg to 62 deg) is unique to the corded belt, which, we believe, is due to the speed gradient in the belt. It is notable that the position of the maximum shear strain coincides with the right edge of the fold; hence, the fold and the region to its left is a relaxation zone, carrying no traction regardless of whether this region is in contact or not.

Fig. 5
Fig. 5
Close modal

In the driven case, the belt transitions tension smoother than in the driver case, with no detachment waves at the belt–pulley interface. The contact area in the driven case is larger than that in the driver case, but the magnitude of the shear strain in the driven case is smaller than that in the driver case. The shear strain in the driven case is negative, indicating that the belt drives the pulley. The shear strain develops gradually due to the speed gradient in the belt from entry until relaxation at around 10 deg.

Our experimental results for shear strain distribution are next compared with predictions by the Firbank model, which was proposed for belts with tension members. According to the Firbank model [6], the shear strain γ in the adhesion arc as a function of the angle θ measured from the entry point obeys
$γ(θ)=kRθ+1t∫0θΔFT(θ)EtAtRdθ$
(1)
The first term in the expression represents the shear strain due to the speed differential between the pulley envelope and tension members. R denotes the pulley radius and k the speed differential factor, k = (VcordV)/V. Here, Vcord = (R + t) · ωpulley is the cord speed at the entry point and V = R · ωpulley, where ωpulley denotes the angular velocity of the pulley. The second term represents the shear strain due to the contraction/extension of the belt, with ΔFT(θ) denoting the tension drop along arc θ, which can be estimated by the tension distribution formula developed by Della Pietra and Timpone [15] (driver pulley, Eq. (2a); driven pulley, Eq. (2b)),
$FT(θ)FTS=eμkβ{1+μsλ[coshλ(π−β)−coshλθsinhλ(π−β)]},0≤θ≤π−βFT(θ)FTS=eμk(π−θ),π−β≤θ≤π$
(2a)
$FT(θ)FTS=1+μs(coshλθ−1)λsinhλ(π−β)−μs(coshλ(π−β)−1),0≤θ≤π−βFT(θ)FTS=FT(π−β)FTSeμk(θ−π+β),π−β≤θ≤π$
(2b)
where μs and μk are static and kinetic friction coefficients (both are set to be 1.3 for our system based on Ref. [16]), and $λ=GBR2/tEtAt$ is a parameter related to the geometry and material properties of the belt (provided in Sec. 2.2 and Table 1). Here, the slip arc angle, β, can be computed iteratively: β is varied until the ratio between the tension in the slack and tight spans of the belt, FTS and FTT, respectively, from Eq. (2) matches the ratio as prescribed during the experiment. Then, the shear strain distribution in the adhesion arc can be estimated using Eq. (1). The shear strain in the slip arc is obtained by computing the sliding friction per unit area via Coulomb’s friction law (by multiplying the normal force per unit area, FT(θ)/RB, by the kinetic friction coefficient μk) and then converting the shear stress to shear strain using the shear modulus G (driver pulley, Eq. (3a); driven pulley, Eq. (3b))
$γ(θ)=μkFT(θ)RB1G=μkFTSGRBeμk(π−θ),π−β≤θ≤π$
(3a)
$γ(θ)=μkFT(θ)RB1G=μkFT(π−β)GBR2eμk(θ−π+β),π−β≤θ≤π$
(3b)
Table 1

Model parameters

NomenclatureValue (driver/driven)
SymbolParameter
FTSTension at the slack span, N2
FTTTension at the tight span, N6
GShear modulus of the belt, MPa0.53
νPoisson’s ratio0.499
BWidth of the belt, mm8
tThickness of the belt, mm3
VDriving speed, mm/s3
EtElastic modulus of tension member, GPa3.3
AtTension member cross section area, mm20.196
γa_exMax. shear strain at the exit zone(0.24/0.10)
fdwDetachment wave frequency, Hz(2.28/2.03)
v0Characteristic crack speed, μm/s0.1
αExponent0.33
NomenclatureValue (driver/driven)
SymbolParameter
FTSTension at the slack span, N2
FTTTension at the tight span, N6
GShear modulus of the belt, MPa0.53
νPoisson’s ratio0.499
BWidth of the belt, mm8
tThickness of the belt, mm3
VDriving speed, mm/s3
EtElastic modulus of tension member, GPa3.3
AtTension member cross section area, mm20.196
γa_exMax. shear strain at the exit zone(0.24/0.10)
fdwDetachment wave frequency, Hz(2.28/2.03)
v0Characteristic crack speed, μm/s0.1
αExponent0.33

Note that Eqs. (1) and (3) determine the shear strain from the entry point to exit point, which requires a change of the angular coordinate, θπθ, to compare directly with the experimental results.

The predicted shear strain for both the driver and the driven cases obtained using the Firbank model is plotted by dashed blue lines in Fig. 5. The most obvious difference between the experimental and theoretical results lies in the exit region. In the experiment, there is a significant accumulation of the shear traction followed by a rapid relaxation via detachment waves. The Firbank model fails to predict this relaxation. Instead, the exit region in the Firbank model is governed by Coulomb’s friction law when the shear traction exceeds the maximum static friction. In fact, the shear traction at the exit point predicted by the Firbank model is nonzero, which violates compatibility since the shear strain must be zero in the spans immediately next to the exit region. Nevertheless, the Firbank model makes decent predictions of the shear distribution in the adhesion arc for both cases, although there is more apparent conflict with the experimentally obtained results in the driver case. This deviation results from the Firbank model’s inaccurate estimation in the relaxation region, which influences the shear distribution in the adhesion region.

Following the measurement of shear traction at the belt–pulley interface, we can estimate the mechanical losses due to rolling friction for corded belts according to the approach outlined in Ref. [5]. For the belt segment in contact with the pulley, the sum of all moments about the pulley axis is zero due to conservation of angular momentum. These moments include ones due to tension forces (concentrated on tension cords to a first approximation), shear traction at the belt–pulley interface (integration of shear strain over the contact arc multiplied with the shear modulus), and shear and tension forces together with moments applied at the free belt spans. As measured and computed (see Table 1), the moment due to the tangential (shear) traction forces about the pulley axis equals to 57.5 N·mm in the driver case (representing the power source/input when the pulley drives the belt) and to −43.7 N·mm in the driven case (representing the load/useful output when the pulley is driven by the belt). The moment due to the difference in the tension forces are −49.5 N·mm (useful output) and 48.3 N·mm (input) for the driver and driven cases, respectively. Notably, the moment due to the difference in tension forces is larger than that of a regular flat belt under the same loading condition since the tension forces in the composite belt, concentrated at the tension member, possess larger moment arms about the pulley axis. Based on the moment equilibrium, the rolling friction moments are estimated to be −8.0 N·mm and −4.6 N·mm in the driver and driven cases, respectively. The rolling friction moment is much higher in the driver case due to the appearance of detachment waves.

The rolling friction for a corded belt is quite small for both the driver and the driven case (Fig. 6(a)) compared with the rolling friction for a reference belt. A further calculation based on our model predicting the rolling friction can explain this behavior quite well. In the model, the mechanical losses result from both the viscoelastic deformation of the belt material and the detachment events at the belt–pulley interface. The viscoelastic deformation is due to five different types of cyclic loadings, including bending, stretching, radial compression, shear, and stretching fluctuation. The energy dissipation due to detachment events is the difference between the energy spent during detachment and the energy gained during attachment. Here, the detachment/attachment hysteresis is treated as the fracture/healing process between two layers for the calculation of the associated spent/gained energy [17,18]. The details about this model are provided in Ref. [5]. The parameters needed for the calculation can be found either in Table 1 or in Refs. [19,20].

Fig. 6
Fig. 6
Close modal

The rolling friction in the belt with tensile cords is mostly due to cyclic shear and bending, and for the driver case, there is an additional contribution from adhesion (Fig. 6(b)). Compared with the regular flat belt case (Ref), the contribution from cyclic stretching is absent and the rolling friction moment due to shear is much smaller. As a result, the rolling friction in the belt with tensile cords is much smaller. The contribution from shear is small here since the maximum shear strain and the frequency of the cyclic variation of shear deformation are both smaller than those in the reference belt.

## 5 Conclusions

The results obtained in this work allow us to draw the following conclusions:

1. Detachment waves persist in the presence of belt tensile cords; however, the pulley rotates steadier due to the increased stiffness of the composite belt. The detachment events are weaker (i.e., amplitude is smaller) and occur at higher frequencies.

2. Shear strain accumulates in the adhesion arc due to a speed differential between the belt contacting the pulley surface and the tensile cords and is relaxed at the exit region by detachment waves. The new understanding of the shear relaxation by detachment waves requires a re-assessment of the Firbank’s shear model—this is proposed as future work.

3. Compared with a regular flat belt, the frictional losses are smaller in both the driver and driven cases for corded belts.

## Acknowledgment

The authors would like to acknowledge support from the National Science Foundation under Grant No. 1562129.

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