The effect of angular misalignment on the running torques of tapered roller bearings

Tribology International 95 (2016) 76–85

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Tribology International journal homepage: www.elsevier.com/locate/triboint

The effect of angular misalignment on the running torques of tapered roller bearings Van-Canh Tong, Seong-Wook Hong n Department of Mechanical System Engineering, Kumoh National Institute of Technology, Daehak-ro 61, Yangho-dong, Gumi, Gyeongbuk 730-701, Republic of Korea

art ic l e i nf o

a b s t r a c t

Article history: Received 13 September 2015 Received in revised form 31 October 2015 Accepted 3 November 2015 Available online 11 November 2015

The occurrence of angular misalignment can considerably change bearing characteristics. However, the angular misalignment effect on the running torques for tapered roller bearings (TRBs) has not been thoroughly investigated. This paper presents a comprehensive formula to provide the running torques for TRBs with angular misalignment between inner and outer races. Simulations were performed to investigate the effects of angular misalignment on TRB torques for two representative preloading methods, constant force preload and constant displacement preload. Angular misalignment under the constant displacement preload was found to increase the running torques under mild preload and to slightly decrease the running torques under heavy preload. Under the constant force preload, the running toques consistently decrease with an increase in angular misalignment. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Rolling element bearing Rolling contact Friction Simulation

1. Introduction Tapered roller bearings (TRBs), which have tapered rollers and races, can withstand a combination of high radial and axial loads. Therefore, TRBs have been widely used in heavy-load applications, for example, heavy-duty spindles, train and automobile wheel hubs, aircraft and gearboxes. The running torques for TRBs are usually high due to their long roller–race contact lines and the large sliding friction between the roller end and guide flange. To reduce the TRB running torques, various issues may need addressed, including bearing geometry, surface quality of rolling elements, cooling, lubrication, misalignment, and preloading condition. This paper investigates the running torque of TRBs with angular misalignment. Bearing running torque implies a rotational resistance of a bearing. Starting torque is often discriminated from the running torque. The starting torque is defined as the frictional moment that must be overcome to cause a bearing to rotate. The running torque is the moment required to maintain a constant rotational speed of a bearing. Since the running torque of a bearing relates directly to the power loss of a rotating system, it is crucial to study the bearing running torque in both design and operation to increase the system efficiency. Early studies on bearing torque [1,2] considered only Coulomb friction from the sliding contacts. Palmgren [3] first included the n

Corresponding author. Tel.: þ 82 54 478 7344; fax: þ 82 54 478 7319. E-mail address: [email protected] (S.-W. Hong).

http://dx.doi.org/10.1016/j.triboint.2015.11.005 0301-679X/& 2015 Elsevier Ltd. All rights reserved.

effect of lubricant viscosity in a torque formula based on experimental measurements of torque for ball and roller bearings. Reichenbach [4] introduced the concept of spinning action of the ball with respect to the race, as a major component of overall resistance in ball bearings. Later, with the development of the elasto-hydrodynamic lubrication (EHL) theory, the effect of lubricant film thickness on the friction torque in rolling element bearings was taken into account. Pioneering studies that addressed the effect of EHL-based lubricant film thickness on the bearing friction torque were performed by Harris [5] and Zaretsky et al. [6]. Because of the increasing demand for higher speed and higher loading of rolling element bearings, studies on bearing torque under various lubricating and loading conditions have been conducted. Schuller et al. [7] investigated the effect of rotational speed on the power loss of a high-speed, jet-lubricated ball bearing subjected to axial and radial loads by measuring the running torque. Nishimura and Suzuki [8] presented the friction torque responses of solid-lubricated ball bearings for use in vacuum conditions. Kanatsu and Ohta [9,10] analyzed the running torque of an axially loaded deep groove ball bearing lubricated with a polymer lubricant. The effect of lubrication oil and grease on the friction torque of thrust ball and cylindrical roller bearings was investigated by Cousseau et al. [11,12] and Fernandes et al. [13,14]. Recently, Olaru et al. [15] presented a theoretical model to predict the friction torque of thrust ball bearings. The model accuracy was confirmed by experimental testing of a modified thrust ball bearing with three balls and no cage. Regarding the running torque of TRBs, there are several analytical models, which, however, have little association with angular

V.-C. Tong, S.-W. Hong / Tribology International 95 (2016) 76–85

misalignment. Palmgren [3] developed a roller bearing torque formula that consists of torque components arising from the applied load and viscosity. However, this torque formula fails to provide the accurate running torque for TRBs subject to high axial load and high rotational speed [16]. Witte [17] developed a semi-empirical formula for predicting the running torque of TRBs under axial and radial loads. This formula is represented by a single term expression that is the product of several constants and exponential terms, including speed, lubricant viscosity, and external load. The TRB geometric effect on the running torque was reflected in terms of a coefficient, but detailed bearing geometry information was not required. The load, speed, and viscosity exponents were derived using curve fitting on experimental torque measurements. Because bearing designs have been appreciably improved for higher efficiency, the constants in Witte's torque formula are no longer applicable. In addition, with increasing usage of high-speed computers along with abundant bearing databases, it is of great concern to formulate bearing torques reflecting all possible factors. Aihara [16] proposed a running torque formula for TRBs that is based on an experiment-assisted theoretical approach. The total TRB torque was defined as the summation of the EHL rolling resistance in roller–raceway contacts, the sliding friction between the guide flange and roller end contact, and the fluid-film friction. The calculated torque agreed well with that from experimental measurements in the case of axially preloaded TRBs. Later, Zhou and Hoeprich [18] extended Aihara's approach to provide an analytical formula for TRB torque calculations. To reduce the calculation efforts and validate the torque formulation for different TRB designs, they employed a numerical data curve-fitting method and established the relationship for load, material, and speed parameters with rolling resistance forces for isothermal EHL line contacts. However, the running torque of TRBs with angular misalignment has not been studied in the above-mentioned studies. Harris [19] introduced the roller bearing friction torque formula based on calculations of friction heat generation. The slicing technique was used in which the actual contact length was divided into a finite number of laminas. The friction heat generated over each lamina was evaluated to provide the total heat generation from the entire contact area. This method can be applied for torque computations in misaligned roller bearings. Recently, Houpert [20] presented a new formula for the TRB running torque. In this study, the TRB running torque was assumed to consist of hydrodynamic rolling forces in roller–raceways, elastic rolling moments at the contact regions between the roller and raceways, and rib– roller end friction moments. The effect of angular misalignment on the TRB torque, however, has not been addressed. Most studies on TRB torque estimations have confined themselves to properly aligned TRBs under axial or combined radial and axial loading. In realistic applications, angular misalignment is unavoidable due to many factors [19]. The occurrence of misalignment would introduce considerable changes in contact characteristics of rollers and raceways, such as contact length and load distribution in rolling elements. These may consequently alter the bearing friction torque. However, the angular misalignment effect has not been adequately included in the TRB torque formulas available in the literature. This paper aims to improve the existing TRB torque formula by taking into account the effect of angular misalignment. Based on the TRB dynamic model presented in Tong and Hong [21,22], the roller contact characteristics, such as the contact forces between the roller and raceways, the contact forces between the roller and flange, the actual contact length between the roller and raceways, and the distance between the flange and roller end contact, are analyzed. Calculation of a TRB dynamic model is performed with respect to the two most commonly employed preloading methods,

77

constant displacement and constant force preloading methods. Then, a modified torque equation that is applicable to misaligned TRBs is proposed and compared with the existing formulas in the literature. The computational results for TRB running torques are presented with regard to angular misalignment, axial and radial loads, and preload method. The obtained results are rigorously discussed.

2. TRB equilibrium In order to estimate TRB torque, the contact forces between the roller and races should be determined in advance. Calculation of these forces is based on solving the bearing dynamic equations relevant to the equilibrium of the rollers and inner ring. For an aligned TRB under pure axial force Fz, the contact forces can be approximated as described by Aihara [16]. However, such a situation is very rare in actual bearing applications. For a general loading condition, as shown in Fig. 1(a), the inner ring of the TRB is assumed to be loaded by an external load vector
 fF gT ¼ F x ; F y ; F z ; M x ; M y ; ð1Þ and the corresponding inner ring displacement vector is
T
 δ ¼ δx ; δy ; δ z ; γ x ; δy :

ð2Þ

Considering the TRB cross-section at a particular roller of location angle ϕ, as indicated in Fig. 1(b), because the roller is displaced from its initial position by {v}T ¼{vr,vz,ψ}, the roller contact forces Qi, Qe, and Qf are generated, as illustrated in Fig. 1(c). Here, the subscripts i, e, and f denote the inner raceway, outer raceway, and flange, respectively. Qi and Qe can be calculated using the well-known slicing method [19,21–23]. In this method, the roller–raceway contact region is divided into ns slices, and the total contact force is calculated by the summation of the contact forces in the individual slices qk. It should be noted that the slice contact force is not uniformly distributed, but depends on the roller and raceway profiles. Thus, when forces Qi and Qe are moved to the middle of the nominal contact length, as illustrated in Fig. 2, moments Mi and Me are induced consequently. Then, the contact forces and moments are described as ns

ns

k¼1

k¼1

Q a ¼ Σ qk ¼ Σ c  δ k

10=9

ns

ns

k¼1

k¼1

Δl;

M a ¼ Σ qk  lk ¼ Σ c  δk

10=9

a ¼ i; e

Δl  l k ;

ð3Þ

a ¼ i; e

ð4Þ

The force Qf is determined by the classical Hertzian contact theory between the flat flange and spherical roller end, as Q f ¼ cf  δf

1:5

ð5Þ

where c and cf indicate the contact constants for the roller-toraceways and the roller-to-flange, respectively, which depend on the material and geometry at the contact. δk and δf represent the contact compressions between the roller and raceway and the roller and flange, respectively. hk denotes the compression drop at slice k due to a modified roller profile. The width and axial position of slice k are Δl and lk, respectively. Then, the roller equilibrium equations are obtained with inclusion of the inertial effects, such as centrifugal force Fc and gyroscopic moment Mg. The roller equations are then solved to provide the roller displacements and the reactive forces Qi, Qe, and Qf. The resultant inner race contact loads at all rollers and the external load {F} give the global equilibrium equations of the inner ring. The inner ring displacement vector {δ} is finally obtained by solving the global equations. The detailed descriptions for the roller load computation and dynamic equations of TRB, including

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Fig. 1. Loads and displacements of TRBs in (a) a global coordinate system (x, y, z), (b) a local coordinate system (r, ϕ, z), and (c) roller contact forces and displacements.

Fig. 2. Demonstration of the slicing technique for roller contact load calculations.

roller and bearing global equilibrium equations, are given in Tong and Hong [22]. Because the roller and inner ring equilibrium equations are non-linear, an iterative Newton–Raphson method is adopted as a solution method. The overall calculation procedure for the TRB equilibrium equations is shown in Fig. 3. Given a permanent misalignment angle γy, along with bearing load components Fx, Fy, Fz, and Mx, unknowns δx, δy, δz, and γx can be found by solving four of the five global equations containing Fx, Fy, Fz, and Mx. The induced moment load My can be calculated by evaluating the global moment about the y-axis. If the inner ring displacement vector {δ} is given, then the applied load vector {F} can be estimated after the local loop is completed. Fig. 3. Calculation procedure for TRB equilibrium equations.

3. Torque calculation for TRBs subjected to angular misalignment Several analytical formulas have been developed for TRB torque estimation [3,16–20]. However, the effect of angular misalignment between the inner and outer races on the TRB running torque has often been neglected. Furthermore, the actual roller–race contact length and contact load distribution along the roller was not explicitly included in previous TRB torque models. In this section, a torque formula for TRBs with angular misalignment is developed by taking advantage of the formula proposed by Aihara [16].

The torque formula for an aligned TRB under axial load Fz can be written as [16] T ¼z

1 ðRe T i þ Ri T e Þ þ ef μs F z cos ϵ Dm

ð6Þ

where Ta (a ¼i,e) are the rolling resistances of the roller in contact with the inner and outer raceways, expressed by [16] Ta ¼

1:76  102

!

1 ðGU Þ0:658 W 0:31 R2 lw 1 þ 0:29L0:78 po

ð7Þ

V.-C. Tong, S.-W. Hong / Tribology International 95 (2016) 76–85

and W is the dimensionless load parameter written as [16] 2F z W¼ Dm lw z sin αe E0

ð8Þ

where z, Dm, lw and ε represent the total number of rollers, the roller mean diameter, the roller effective length, and the semicone angle of the roller, respectively. αe is the contact angle between the roller and outer raceway. Ra (a ¼i,e) are the radii of the raceway contact points on the roller mean diameter. ef indicates the distance between the roller end and flange contact point, and μs is the friction coefficient. L, G and U represent the thermal loading parameter, the dimensionless material and speed parameters, written as L¼

η0 βu2

U¼

to 0.418 μm as reported by Wang et al. [25], and the minimum film thickness is calculated from Hamrock and Downson's formula [26].

4. Computational results The TRB torque calculation is performed for a sample TRB (30206-A) whose geometrical properties are given in Table 1. The TRB misalignment angle (γ ¼ γy) is selected from 0 to 5 mrad. The TRB is lubricated with synthetic oil SAE20 using the circulating oil method, which is described in detail in [16]. It is also assumed that the temperature for all the TRB elements and lubricant oil remains constant at 50 °C.

ð9Þ

0

k

G ¼ p0 E

4.1. Model comparison 0

ð10Þ

η0 u

ð11Þ

E0 R

where p0 and β are the pressure and temperature coefficients of viscosity, respectively. η0 and k0 are the dynamic viscosity and thermal conductivity of the lubricant, respectively. R is the equivalent radius at the roller–raceway contact in the rolling direction. The equivalent Young's modulus E0 and the rolling speed at the roller–race contact u are given in Harris [19]. It should be noted that the torque formulas (6) and (7) are applicable only for properly aligned and axially loaded TRBs, in which all rollers have identical loading conditions. Moreover, these torque formulas are valid only when the load distributions along rollers and the roller–race contact lengths are uniform. However, roller loading is changed with regard to roller location in the case of misaligned and/or radially loaded TRBs. Radial loading and misalignment angle can lead to non-uniform load distributions in rollers and the roller–race contact lengths in TRBs. In addition, the curved profiles in the roller and/or raceways in TRBs make the load distributions non-uniform. Therefore, in order to estimate the TRB torque accurately, the torque contributed by an individual roller should be estimated separately by evaluating that of each contact slice. To this end, the torque formula shown in Eq. (6) is improved, summing the individual torques, as   z 1 Re T¼ Σ ðRe T i þ Ri T e Þ þ ef μs Q f ð12Þ Dm j ¼ 1 Dm j

Fig. 4 shows the dependence of the TRB torque on the rotational speed evaluated under a constant force preload of Fz ¼1000 N. The running torque of the aligned TRB is calculated using the current model and using the torque formulas developed by Palmgren [3], Aihara [16], Witte [17], and Houpert [20]. When the TRB impends to rotate, or is at a very low rotational speed, the sliding friction between the inner ring guide flange and the roller end are relatively high and predominantly raises the TRB torque [27]. The roller and flange friction decreases rapidly when the rotational speed increases due to the formation of a fluid film at the contact length [16]. Thus, the TRB running torque at high speeds is mainly due to the rolling resistance between the roller and raceways. Fig. 4 shows that Palmgren's and Witte's formulas are likely to underestimate the TRB starting torque. Palmgren's formula underestimates the TRB running torque, regardless of the rotational speed, in comparison with other formulas. The TRB torque calculated by Aihara's and the current Table 1 TRB 30206-A basic geometrical parameters. Parameter

Symbol (mm)

Value Parameter

Symbol

Bore diameter Outside diameter Total width

d D

30 62

Pitch diameter Number of rollers

dm (mm) 45.92 z 17

B

17.25

Roller effective length

lw (mm)

where j is the roller index number. Because the roller and raceway contact regions are divided into ns slices, the rolling resistances of the roller in contact with the inner and outer raceways can be calculated by using the rolling resistance of each slice as  n  s T a ¼ Σ T k ; ða ¼ i; eÞ ð13Þ

and w is the load parameter represented by qk RΔlE0

ð15Þ

where qk, the contact force between the roller and raceway for slice k, is obtained from the TRB model in the previous section. The roller end and flange contact distance ef and the contact force Qf can be obtained from the TRB model [22]. The roller end and flange contact friction coefficient μs depends on the composite surface roughness and the minimum film thickness as described by Aihara [16]. In this study, the composite surface roughness is set

10.54

Palmgren [3] Witte [17] Aihara [16] Houpert [20] Current study

450 400

a

where Tk is the rolling resistances of slice k, expressed as [24], ! 1:76  102 1 Tk ¼ ðGU Þ0:658 w0:31 R2 Δl ð14Þ 1 þ 0:29L0:78 po

Value

500

350 Torque (Nmm)

k¼1

w¼

79

300 250 200 150 100 50 0 0

500

1000

1500

2000

2500

3000

Rotational speed (rpm)

Fig. 4. Comparison of torques for the aligned TRB using the current method and four other formulas (Fz ¼ 1000 N).

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formulas are similar because they are based on the same conceptual approach. Houpert's formula predicts the starting and running torques at low speeds, but deviates from other formulas as the rotational speed increases.

My (Nmm)

4

4.2. Effect of misalignment and loading on TRB torque

4.2.1. Constant displacement preloaded TRB In the case of constant displacement preload, an axial displacement between the inner and outer races of a TRB (δz) is set by using spacers, shims, lock nuts, etc. Then, the axial distance may be assumed constant during operation. Fig. 5 shows the relationship between the TRB running torque and the misalignment angle varying from 0 to 5 mrad. The TRB is initially preloaded by δz ¼0.002 mm for all loading conditions. Fig. 5 shows that the TRB running torque gradually increases with an increase in the misalignment angle, irrespectively of the rotational speed. This is because the induced moment and axial loads are consistently increased with increasing misalignment angle, as demonstrated in Fig. 6. Fig. 5 also shows that the angular misalignment significantly affects the starting torque of TRBs, as indicated by the torque at the rotational speed n ¼ 0. Fig. 7(a) illustrates the effect of angular misalignment on the TRB running torque calculated at a rotational speed of 3000 rpm. Increasing the displacement preload from 0 to 0.05 mm considerably increases the TRB torque. Moreover, it is interesting to note that, when a displacement preload higher than 0.02 mm is applied, the TRB running torque tends to slightly decrease with an increase in misalignment angle. The effect of radial force and misalignment angle on the TRB torque is shown in Fig. 7(b). The TRB torque is calculated for radial forces varying from 5000 to 10,000 N, while the TRB displacement preload and rotational speed remain constant at δz ¼0.005 mm and n¼3000 rpm, respectively. Unlike the axial preload effect, the radial load causes a local minimum for the TRB running torque when the 800

γy = 0 mrad

700

γy = 1 mrad γy = 2 mrad

600

Torque (Nmm)

γy = 3 mrad 500

γy = 4 mrad γy = 5 mrad

400 300

100 0

3 2 1

0

500

1000

1500

2000

2500

3000

Rotational speed (rpm)

Fig. 5. Effect of misalignment angle on TRB running torque with an axial preload of δz ¼ 0.002 mm.

1

1.5

2

2.5

3

3.5

4

4.5

5

3

3.5

4

4.5

5

4000 3000 2000 1000 0

0

0.5

1

1.5

2

2.5

γy (mrad) Fig. 6. Induced moment load My and axial load Fz (δz ¼ 0.01 mm, n¼ 3000 rpm).

bearing is misaligned by a certain misalignment angle. This is because the aligned TRB, under a radial load, is subjected to additional induced moment load due to so-called back loading, and introducing a certain misalignment angle can relieve the TRB from back loading to consequently reduce the bearing running torque. Fig. 8 shows the contact characteristics between the roller and outer race of the TRB with angular misalignment under an axial displacement preload of δz ¼0.01 mm and δz ¼0.05 mm. The angular misalignment leads to an increase in contact force between the roller and outer race, as shown in Fig. 8(a) and (c). On the other hand, it reduces the contact length between the roller and outer race, as depicted in Fig. 8(b) and (d). In general, the TRB running torque at a constant rotational speed is dependent on the roller–race contact forces, as well as the corresponding contact lengths. When a small axial displacement preload is applied (δz ¼0.01 mm), the effect of the increased contact force due to angular misalignment is greater than that of the decreased contact length. As a result, the TRB torque increases, as shown in Fig. 7(a). However, the reduction of the roller contact length becomes dominant for the TRB torque, especially when a heavy displacement preload is applied (in this case, when δz is greater than 0.02 mm). In practical applications, the TRB may be misaligned in an arbitrary direction, and the misalignment may be represented by a combination of two orthogonal misalignment angles γx and γy. Fig. 9 displays the effect of misalignment and radial loading on the TRB running torque under a constant displacement preload. Fig. 9(a)– (c) shows the torque behavior of TRBs under radial loads: Fx ¼5000 N and Fy ¼0 N, Fx ¼0 N and Fy ¼5000 N, and Fx ¼Fy ¼ 5000 N, respectively. Fig. 9(a) and (b) illustrate that the sign of the misalignment angle shows a negligible effect on the TRB torque, because the graphs are nearly symmetric with respect to the x- and y-axes. If the resultant radial load, load direction angle, and combined misalignment angle are expressed, respectively, by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F r ¼ F 2x þ F 2y ð16Þ

γ¼ 0

0.5

γy (mrad)

θ ¼ tan  1

200

4

0

Fz (N)

In this section, the TRB torque is calculated using the developed model with regard to misalignment angles and loading conditions. A preload is normally applied to the rolling element bearings to increase their rigidity and proper seating of the rolling elements in the races. This study considers two representative preloading methods, constant displacement preload and constant force preload. The effect of the misalignment angle, along with radial and axial loading, on the TRB torque is then investigated.

x 10



Fy Fx

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ 2x þ γ 2y

 ð17Þ ð18Þ

then, low torque regions occur along the Fr direction with inclined angle θ, as shown in Fig. 9(a)–(c). For comparison, Fig. 9(d) shows the relationship between the misalignment angle and the torque for a TRB subject only to an axial preload.

V.-C. Tong, S.-W. Hong / Tribology International 95 (2016) 76–85

4.2.2. Constant force preloaded TRB The effect of angular misalignment on the running torque of TRB under a constant force preload method is investigated. In the constant force preload method, an axial force, which is assumed to

81

remain constant during operation, is generated by a soft spring, a magnetic field, or pneumatic cylinders. Fig. 10 demonstrates the effect of misalignment angle on the TRB torque. The TRB is subjected to a constant force preload of

Fig. 7. TRB torque as a function of misalignment angle and loading (n¼ 3000 rpm): (a) displacement preload δz and misalignment angle γy effects, (b) radial force Fx and misalignment angle γy effects (with δz ¼ 0.005 mm).

γy = 0 mrad

γy = 1 mrad

γy = 2 mrad

γy = 3 mrad

γy = 4 mrad

γy = 5 mrad

9

180 210

150

240

120

270

90 500 1000 1500 (N) 60

300

Roller-outer race contact length (mm)

8.5

8

7.5

7

6.5

6 330

30

5.5

0

45

90

135

180

225

270

315

360

270

315

360

Position angle ( o)

180 210

0

11 150

240

120

270

90 1000 2000 3000 (N) 300

60

Roller-outer race contact length (mm)

10.5

10

9.5

9

8.5

8

330

30 0

7.5

0

45

90

135

180

225

Position angle ( o)

Fig. 8. Roller-outer race contact characteristics of the TRB as a function of misalignment angle and displacement preload (n¼ 3000 rpm): (a) and (b) δz ¼ 0.01 mm, (c) and (d) δz ¼ 0.05 mm, (a) and (c) contact force distribution, (b) and (d) contact length.

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Fig. 9. TRB torque under general misalignment effects (δz ¼0.05 mm, n¼ 3000 rpm): (a) Fx ¼ 5000 N, (b) Fy ¼5000 N, (c) Fx ¼ Fy ¼5000 N, (d) pure axial loading. 500

γy = 0 mrad 450

γy = 1 mrad γy = 2 mrad

400

γy = 3 mrad

Torque (Nmm)

350

γy = 4 mrad γy = 5 mrad

300 250 200 150 100 50

0

500

1000

1500

2000

2500

3000

Rotational speed (rpm)

Fig. 10. Effect of misalignment angle on the TRB torque under a constant force preload (Fz ¼ 1000 N).

1000 N and a rotational speed varying from 0 to 3000 rpm. The angular misalignment has a negligible effect on the starting torque of the TRB. However, an increase in misalignment angle reduces the running torque of the TRB when the rotational speed is greater than 200 rpm. The variation of the axial displacement as a function of misalignment angle is shown in Fig. 11(a). While the axial force preload remains constant, the axial displacement of the TRB gradually

decreases with increasing misalignment angle to maintain the equilibrium of the TRB in the axial direction. Fig. 11(b)–(c) shows the roller-to-outer-race contact characteristics as a function of misalignment angle. The occurrence of angular misalignment results in uneven load distribution, as well as reduced the contact length between the roller and raceways. The above-mentioned internal changes reflect the TRB running torque reduction as depicted in Fig. 10. Fig. 12 shows the effects of angular misalignment, along with axial and radial loadings on the TRB running torque that are evaluated at a rotational speed of 3000 rpm. Fig. 12 (a) demonstrates that the TRB torque increases with an increase in axial preload. This is similar to the torque trend under the constant displacement preload shown in Fig. 7(a). On the other hand, increasing misalignment angle results in a decrease in the TRB running torque due to the decrease of the roller contact length and/or the uneven contact load distribution. The TRB running torque as a function of misalignment angle and radial force is shown in Fig. 12(b). The bearing is initially preloaded by a constant axial force of 3000 N. The TRB running torque is influenced by both angular misalignment and radial loading in the same manner. It is interesting to note that increasing radial load leads to a decrease in the TRB torque. A similar conclusion was previously reported by Witte [17] based on experimental testing of aligned TRBs subjected to radial loads. Fig. 13 indicates the effects of combined misalignment angles γx and γy on the TRB running torque calculated for the constant force preload case. The angular misalignment about the x-axis γx shows a similar effect on the TRB torque as γy, because the TRB torque

V.-C. Tong, S.-W. Hong / Tribology International 95 (2016) 76–85

83

0.015

Axial displacement (mm)

0.01 0.005 0 -0.005 -0.01 -0.015 -0.02 -0.025

0

1

γy = 0 mrad

2 3 Misalignment angle (mrad)

γy = 1 mrad

γy = 2 mrad

4

γy = 3 mrad

5

γy = 4 mrad

γy = 5 mrad

10 180 210

150

240

120

270

200

90 400

600 (N)

300

60

330

30

Roller-outer race contact length (mm)

9 8 7 6 5 4 3

0

2

0

45

90

135

180

225

270

315

360

Position angle (o)

Fig. 11. TRB characteristics and misalignment angle dependence with a constant force preload (Fz ¼1000 N, n¼ 3000 rpm): (a) axial displacement of TRB, (b) roller-outer race contact force, (c) roller-outer race contact length.

Fig. 12. Effect of misalignment angle and loading on the TRB torque (n ¼3000 rpm): (a) axial preload and misalignment angle effects, (b) radial force and misalignment angle effects (with Fz ¼ 3000 N).

decreases with the increase of either γx or γy independently of the radial load. Moreover, the sign of the misalignment angle has a minor influence on the TRB torque, similarly to the constant

displacement preload case. Despite the fact that angular misalignment manifests a benefit of reducing the TRB torque for the constant axial force preload bearing, angular misalignment should

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V.-C. Tong, S.-W. Hong / Tribology International 95 (2016) 76–85

γx = 1 mrad

γx = 2 mrad

γx = 3 mrad

700

700

650

650

600

600

550

550

Torque (Nmm)

Torque (Nmm)

γx = 0 mrad

500 450

450 400

350

350

-4

-3

-2

-1

0

1

2

3

4

5

γx = 5 mrad

500

400

300 -5

γx = 4 mrad

300 -5

-4

-3

-2

γy (mrad)

-1

0

1

2

3

4

5

γy (mrad)

Fig. 13. TRB torque under combined misalignment effects (Fz ¼ 2000 N, n¼ 3000 rpm): (a) Fy ¼ 5000 N, (b) Fy ¼ 0 N.

be avoided because it would degrade other characteristics of the TRB, e.g., reduce the bearing fatigue life [22], lower the bearing stiffness, and reduce the natural frequencies of the spindle-TRB system [28].

Acknowledgments

5. Conclusions

References

An improved running torque formula was developed that is applicable for misaligned TRBs. The proposed formula takes advantage of the slicing method for contact load estimation of individual rollers. Thus, the proposed formula accounts for more realistic roller–race contact lengths and contact load distributions. The effect of angular misalignment, along with radial and axial loadings, on the TRB running torque was investigated for various rotational speeds. Two representative preloading methods were examined, constant displacement and constant force preloading methods. A comprehensive analysis of TRB torque provided the TRB torque characteristics associated with angular misalignment and loading conditions. From the computational results, the following conclusions are drawn:

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1) Angular misalignment critically alters the contact characteristics of rolling elements, which leads to subsequent changes in the TRB running torque for all loading conditions. 2) Axial preload has a significant effect on the TRB running torque behavior regardless of the preload method. 3) The preloading method can change the TRB running torque behavior with regard to the radial load. 4) Angular misalignment increases the TRB running torques under a mild constant displacement preload, but decreases the TRB running toques under a constant force preload or heavy constant displacement preload. Based on the proposed running torque model, further investigation of the TRB torque behavior may be performed, for example, investigations of lubricant and operating temperature effects, bearing asperity, and bearing internal design. In addition, the developed torque model provides a valuable tool for the design and analysis of TRBs by permissible misalignment control, bearing selection, and preload calculations.

This research was financially supported by the Research Fund of Kumoh National Institute of Technology, Republic of Korea.

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