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Time varying mesh stiffness calculation of spur gear pair considering sliding friction and spalling defects
Time varying mesh stiffness calculation of spur gear pair considering sliding friction and spalling defects Ankur Saxena, Anand Parey, Manoj Chouksey PII: DOI: Reference:
S1350-6307(16)30119-4 doi: 10.1016/j.engfailanal.2016.09.003 EFA 2961
To appear in: Received date: Revised date: Accepted date:
30 March 2016 5 August 2016 2 September 2016
Please cite this article as: Saxena Ankur, Parey Anand, Chouksey Manoj, Time varying mesh stiffness calculation of spur gear pair considering sliding friction and spalling defects, (2016), doi: 10.1016/j.engfailanal.2016.09.003
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ACCEPTED MANUSCRIPT Time varying mesh stiffness calculation of spur gear pair considering sliding friction and spalling defects
Mechanical Engineering Department, School of Engineering, Indian Institute of Technology, Indore, India b
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a
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Ankur Saxenaa, Anand Pareya,*, Manoj Choukseyb
Mechanical Engineering Department, SGSITS, Indore, India
* Corresponding author. Tel.: +917324240716; Fax: +917324240700.
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E-mail addresses: [email protected] (A. Saxena), [email protected] (A. Parey), [email protected] (M. Chouksey).
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Abstract
Spalling is one of the common tooth surface failures of gear teeth and is defined as the formation of
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deeper cavities that are mainly developed from subsurface defects. The time varying mesh stiffness
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(TVMS)of gear pairs, gives significant information about the health of the system.The change indirection of time varying friction on both sides of the pitch linecauses the change of gear mesh
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stiffness. This article proposes a computer simulation based approach to study the effect of time varying friction coefficient on the total effective mesh stiffness for the spur gear pair. An analytical
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method to calculate the TVMS of the spur gear for different spall shapes, size and location considering sliding friction is also proposed in this study. The results show that spall shape, size and location are very important parametersthat need to be considered for calculation of TVMS and subsequently to know the dynamic response of the gear pair in the presence of a spall. Keywords: Spur gear, mesh stiffness, spalling fault, friction coefficient.
1. Introduction Gearboxes are one of the most critical and widely used power transmission components in industrial applications. The performance of the gear depends mainly on the durability of the gear tooth surface. Spalling is one of the common tooth surface failures and is defined as the formation of deeper cavities that are mainly developed from subsurface defects. Meshari 1
ACCEPTED MANUSCRIPT et al. [1] performed an extensive investigation of gearbox failures and found thatspalled teeth are one of the major causes of gearbox failures. Ding et al. [2] reported that spalling is the
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most destructive surface failure mode of the gear and proposed a hypothesis of material
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ligament collapse as a fundamental mechanism of subsurface defects which are responsible
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for spalling formation and supported the evidence by performing experiments on gears. Many researchers are using dynamic models of gear pairs to study the effect of gear tooth
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surface faults like spalling, pitting, tooth breakage, etc. on gearbox vibrations. Parey et al. [3] developed a 6 DOF gear dynamic model including localized tooth defects like spalling and
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pitting. Jia and Howard [4] presented a 26 DOF gear dynamic model by including the effect of localized tooth spalling and tooth damage. Chaari et al. [5] studied the dynamic responses
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of gear systems by incorporating the effect of spalling and tooth breakage on time varying
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mesh stiffness (TVMS) of single stage spur gear pairs. Stiffness reduction due to change in spalling length and width is compared and the computed values of gear mesh stiffness are
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injected in dynamic model of gear system. Ma et al. [6] calculated the TVMS of the gear pair by using analytical methods proposed in Ref. [5] and incorporated spalling defects by
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introducing a time-periodic defect excitation term in their dynamic model to study dynamic responses. Ma and Chen [7] developed a nonlinear gear model by incorporating spalling defects. The results show that due to spalling defects the stability of the gear system gets affected. Effect of spalling defects on the mesh stiffness was also studied and was included while developing the dynamic model. Jiang et al. [8] developed a 6 DOF helical gear model by incorporating time varying sliding friction and mesh stiffness for simulating dynamic characteristics for helical gear pairs considering spalling defects. The results show that sliding friction influences the helical gears with spalling defect and increases the peak to peak value of the dynamic response. The spalling defects of helical gears under sliding friction also influence the frequency domain of the dynamic response.The above literature [5-8] 2
ACCEPTED MANUSCRIPT shows that TVMS plays a major role in evaluating the vibration response of the gear pair in the presence of spalling. TVMS of a gear pair was also calculated by using the analytical
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technique proposed in Ref. [5].
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Sliding friction has been reported to be an important source of gear vibration and noise. A
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number of researchers have studied the effects of friction on gear dynamics. A detailed review of friction prediction in gear teeth was conducted by Martin [9] who concluded that the values of coefficients of friction can be predicted reasonably accurately based upon
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various lubrication theories. Iida et al. [10] studied the shaft vibration response in the tooth
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sliding direction. Vaishya and Singh [11] proposed an alternate strategy to incorporate friction in dynamic analysis of gear pair and have shown a typical non-linear variation of the
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coefficient of friction. He et al. [12] used several sliding friction formulations in spur gear
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dynamics for the predictions of friction forces. Effects of frictional force on TVMS of gear pair was also studied by Saxena et al. [13] for healthy and cracked gear teeth. Apart from
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these studies load/torque variation also affects the TVMS of the spur gear. Ericson and Parker [14] used finite element program to study the stiffness of the gear pair at various
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experimental torques. Time invariant mesh stiffness is used in their analytical model simulations at the various torque levels. Ma et al. [15] compared the effect of torque variation on the mesh stiffness of the spur gear pair calculated using finite element method and proposed analytical method. Existing studies have some of the following shortcomings which need to be addressed to get accurate results of TVMS. 1) Effect of friction on TVMS in the presence of spall was not done for spur gear pairs. 2) The effect of axial compression and shear deformation was neglected in existing methods [5] for TVMS calculation in spalled tooth pair but Tian [16] showed in his work that axial
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ACCEPTED MANUSCRIPT compression and shear stiffness are important factors and cannot be neglected while calculating TVMS of gear pair.
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3) The shape of the spall was considered as rectangular indentation in Ref. [5-7] but there
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are other possible shapes of spalls which have been discussed in literature. In Ref. [4] a 5
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mm circular spall was used to study torsional stiffness. Hannes and Alfredsson [17] investigated the spall opening angle by assuming a V-shaped spall to study rolling contact fatigue damage. Murakami et al. [18] carried a crack growth analysis by calculating the
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stress intensity factor for a small, arrow headed surface crack which finally forms a V-
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shaped pit. Klein [19] in his Master’s thesis studied the modes of gear contact fatigue failure and found that V-shaped spall occurred on tooth surface. So it can be said that a V-
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shaped spall is more likely to occur in practical engineering and the study related to
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various spall shapes will be more helpful in diagnosis of the gear tooth spalling failure. 4) The effect of symmetry of the spall was also neglected in previous literature and it was
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assumed that the spall was located about the vertical axis of symmetry but the spall may not be symmetric or more than one spall may be present at the line of contact of the gear
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pair and their effect on TVMS needs to be studied for better understanding of vibration responses which was not found in previous work. This literature survey concludes that due to the presence of spalls at the surface of gear teeth, stiffness reduction is observed which directly affects the vibration responses of the gear system. However there are various other parameters which were neglected in previous studies and they need to be considered for better understanding of spalling defects and their effect on gear mesh stiffness. In this paper TVMS has been calculated for a spalled tooth pair using the modified potential energy method. The effect of time varying coefficient of friction, spall shape, size and unsymmetrical spall is considered and stiffness reduction due to these parameters has been simulated.
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ACCEPTED MANUSCRIPT 2. Analytical expression of TVMS for spur gear with spalling defects In this study the potential energy method, proposed by Yang and Lin [20] and refined by
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Tian [16] and Chen et al. [21] is used to calculate TVMS of a spalled gear tooth pair. This
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method considers Hertzian contact energy, bending energy, axial compressive energy, shear
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energy and fillet foundation deflection to calculate the stiffness of the gear tooth. To calculate the individual stiffness component this method uses angular variables instead of linear
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variables. This is helpful in investigating parameter properties at some angular displacement of the pinion/gear.
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While considering the spalling fault the change in area of cross-section (A) and moment of inertia (I) due to the spall has to be considered. These changes are difficult to express by
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angular variables, therefore, a modified expression is needed to include linear variables (A,
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I).
Figure1 Healthy tooth model [16] 5
ACCEPTED MANUSCRIPT Using the values and same notations used in Ref. [16] and shown in Fig. 1, modified expressions of Hertzian stiffness, axial compressive stiffness, shear stiffness and bending
1 kb
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2.4 Rb (1 v)cos 2 1 ( 2 )cos d 1 E. As
2
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1 ks
Rb sin 2 1 ( 2 )cos d 1 E. As 2
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1 ka
ELs 4(1 2 )
Rb3 [1 cos 1{( 2 )sin cos }]3 ( 2 )cos d 1 E.I s
2
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kh
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stiffness components are shown in equations (1) – (4) respectively;
(1)
(2)
(3)
(4)
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In equations (1)-(4) Ls, As and Is are the length of contact, area of tooth cross section and
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L Ls L ls
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are moment of inertia respectively and their values will vary according to equations (5) – (7).
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2.h x .L As (2.h x .L) (hs .ls )
2 3 3 .h x .L Is 1 [(2.h )3 .L (h3 .l ) x s s 12
for healthy region for spalled region
(5)
for healthy region for spalled region
(6)
for healthy region
(7) for spalled region
where hx Rb [(2 )cos sin ] and L, ls, hs are shown in Fig. 2. The definition of remaining parameters can be obtained in Ref. [16] and are shown in Figs. 1 and 2.
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Figure 2 Schematic diagram for spalled gear tooth
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Besides the tooth deformation, the fillet-foundation deflection also influences the stiffness
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of the gear tooth. Sainsot et al. [22] derived the fillet-foundation deflection of the gear based
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on the theory of Muskhelishvili [23]. The stiffness with consideration of gear filletfoundation deflection can be obtained by [21],
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1 cos 2 1 * u f L kf EL S f
2
uf * M Sf
* * 2 P (1 Q tan 1 ) ,
(8)
where, L is the tooth width and α1is acting pressure angle of gear tooth. The definition of the remaining parameters uf and Sf is given in Fig. 3. The coefficients L*, M*, P*, Q*can be calculated by use of the following polynomial functions proposed in Ref. [22],
X i* (h fi , f ) Ai f2 Bi h2fi Ci h fi f Di f Ei h fi Fi
(9)
In eq. (9) Xi*denotes the coefficients L*, M*, P*, Q*; hfi = rf/rint; rf, rint and hf are defined in Fig. 3, while the values of Ai, Bi, Ci, Di, Ei and Fi are given in Table 1.
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ACCEPTED MANUSCRIPT Table 1. Values of the coefficients of Eq. (9) [22]. Bi -1.9986 x 10-3 28.100 x 10-3 185.50 x 10-3 9.0889 x 10-3
Ci -2.3015 x 10-4 -83.431 x 10-4 0.0538 x 10-4 -4.0964 x 10-4
Di 4.7702 x 10-3 -9.9256 x 10-3 53.3 x 10-3 7.8297 x 10-3
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Ai -5.574 x 10-5 60.111 x 10-5 -50.952 x 10-5 -6.2042 x 10-5
Ei 0.0271 0.1624 0.2895 -0.1472
Fi 6.8045 0.9086 0.9236 0.6904
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L* (hfi, θf) M* (hfi, θf) P* (hfi, θf) Q* (hfi, θf)
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Figure 3 Geometrical parameters for the fillet-foundation deflection. [21]
Thus for the single – tooth pair meshing duration, the total effective TVMS can be expressed as,
kt
1 1
kh
1
kb1
1
ks1
1
ka1
1
kf1
1
kb 2
1
ks 2
1
ka 2
1
,
(10)
kf 2
where, kh, kb, ks, ka and kf represent the Hertzian, bending, shear, axial compressive and fillet foundation deflection mesh stiffness, respectively and the subscripts 1, 2 mean the pinion and gear, respectively. For the double-tooth-pair meshing duration, the total effective TVMS is the sum of the two pair’s stiffness, which can be expressed as
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kt i 1
1 1
k h ,i
1
kb1,i
1
ks1,i
1
ka1,i
1
k f 1,i
1
kb 2,i
1
ks 2,i
1
ka 2,i
1
(11)
k f 2,i
T
where, i = 1 represents the first pair of meshing teeth and i = 2 represents the second pair of
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meshing teeth.
In this work the TVMS of the gear pair is calculated usingequations (1) - (11).
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3. Analytical expression of coefficient of friction for spur gear pair The analytical method proposed by Saxena et al. [13] is adopted in this work to calculate
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mesh stiffness of the spur gear pairto study the effect of frictional forces. In Ref. [13] a constant value of coefficient of friction (μ) is considered but, in actual case the coefficient of
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friction changes continuously due to varying mesh properties and lubricant film thickness as
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the gear rotates throughout the complete mesh cycle. This type of variation has been already
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presented in literature [11, 12]. Thus, the variation of coefficient of friction is included in the formulation as the position of each tooth and the direction of friction force changes. Buckingham's [24] formula is used to determine the coefficient of friction which will be
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incorporated in the study.
3.1 Friction Coefficient Model included in this study Friction during the gear mesh has two components: sliding friction and rolling friction. Coefficient of friction μ usually refers to coefficient of sliding friction. There are many empirical formulae in literature [9-12] which have been developed based on different assumptions to calculate the value of the coefficient of friction for the gear pair. These empirical formulae state that μ is a function of a number of parameters such as sliding and rolling velocities, radii of curvature of the surfaces in contact, amount of surface roughness,
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ACCEPTED MANUSCRIPT lubricant viscosity, etc. In this study Buckingham's [24] formula is used to determine the coefficient of friction μ which is:
r
2 ; 3
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0.05 4 0.002 V ; ; s a e0.125Vs 3
(12)
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where, μ = average coefficient of friction, μa= average coefficient of friction of approach, μr= average coefficient of friction of recess and Vs = sliding velocity in m/sec.
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In equation (12) the sliding velocity of the spur gear pair varies as the gear rotates, due to which the coefficient of friction will also vary. This variation is incorporated into the TVMS
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calculation of the spur gear pair to study the effect of variation of µ on TVMS. The value of sliding velocity of the gear pair can be obtained by the following equation
where,
d p np 60
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r12 Rb21 Rp1 sin 1 , (13)
=pitch line velocity of gear in m/sec.
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V
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V ( Rp1 Rp 2 ) Vs R R p1 p 2
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which is derived in Ref. [25],
Rp1, Rp2 = radius of pitch circle of gear and pinion respectively, in m. Rb1 = radius of base circle of gear, in m. r1 = any radius of gear tooth profile, in m. dp= pitch circle diameter of gear in m. np = speed of gear in rpm. α1= pressure angle The value of the time varying coefficient of friction (µ) obtained in equations (12) and (13) is substituted in the TVMS formula developed by Saxena et al. [13]. Hertzian stiffness and stiffness due to fillet foundation deflection will remain unchanged. Expressions for
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ACCEPTED MANUSCRIPT bending mesh stiffness, shear mesh stiffness and axial compressive stiffness considering coefficient of friction is given as follows [13]:
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Bending stiffness; 2 3[{1 cos {( ) sin cos }} { sin {( ) sin cos }}]2 ( ) cos 1 2 1 2 1 2 2 d , 2 EL[sin ( 2 ) cos ]3 1 1 kb 2 3[{1 cos 1{( 2 ) sin cos }} {1 2 sin 1{( 2 ) sin cos }}]2 ( 2 ) cos d , 2 EL[sin ( 2 ) cos ]3 1
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(15)
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(14)
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Shear stiffness; 2 2 1.2(1 )(cos 1 sin 1 ) ( 2 )cos d , EL[sin ( 2 )cos ] 1 1 ks 2 1.2(1 )(cos 1 sin 1 ) 2 ( 2 )cos d EL[sin ( 2 )cos ] 1
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Axial compressive stiffness;
(16)
(17)
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2 ( ) cos (sin cos ) 2 2 1 1 d , 2 EL [sin ( ) cos ] 2 1 1 ka 2 ( ) cos (sin cos ) 2 2 1 1 d 2 EL [sin ( ) cos ] 2 1
(18)
(19)
In the above expressions equations (14), (16) and (18) are derived for the approach process and equation (15), (17) and (19) are derived for the recess process. Using equations (12) and (13) and following the TVMS expressions developed by Saxena et al. [13] the total meshing stiffness of spur gear pair including varying friction coefficient can be obtained.
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ACCEPTED MANUSCRIPT 4. Spall parameters Various parameters of spalling which influences the TVMS are discussed in this section
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4.1 Spall length
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The length of the spall is considered as a parameter responsible for mesh stiffness variation since tooth contact does not take place in this region. In the literature [5-7] the effect of spall length on TVMS is studied but the use of potential energy methods to
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demonstrate this phenomenon is not used in any previous work. For a rectangular spall of length ‘ls’ as shown in Fig. 2, it is assumed that no tooth contact occurs in the spalled area
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which will directly affect Hertzian contact stiffness and has a slight effect on other stiffness
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components. Using equations (1) – (11) the effect of spall length on TVMS can be studied.
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4.2Spall symmetry about vertical axis
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Spalling may develop either symmetrically or asymmetrically about the vertical plane of symmetry. Fig. 4(a) shows a symmetrical spall and the resultant distribution of forces. Fig. 4 (b) shows an unsymmetrical spall and the resultant distribution of forces. It can be seen in
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Fig. 4(b) that due to eccentricity (t) of the resultant force from the axis of symmetry, a torque T acts on the tooth. Due to the torque T, a torsional stiffness (kτ) comes into existence as proposed in Ref. [13]. For simplicity, the change in polar moment of inertia due to spall depth (= 1mm) is neglected. The value of torsional stiffness (kτ) can be calculated by using Ref. [13] which gives
2 12.t 2 (1 v)( 2 )cos 1 d k EL{sin ( 2 )cos }[4.Rb2{sin ( 2 )cos }2 L2 ]
(20)
1
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T
throughout the length of contact.
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Figure 4 Effect of forces on (a) symmetrical spall, (b) unsymmetrical spall 4.3Spall shape
As mentioned in the literature survey many researchers have suggested that the shape of the spall can be rectangular [5], circular [4] or V-shaped [17-19]. In this section the effect of rectangular, circular and V-shaped spalls is studied. In all the cases, the spall is assumed to be present about the pitch point of the pinion and is in the single point contact region only. Also, the spall is assumed to be symmetric.
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indentation as done in Ref. [5-7] having dimensions ls x ws x hs as shown in Fig. 2. Modeling
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of rectangular spall has already been explained in section 2 in equations (1) – (11).
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4.3.2 Modeling of circular spall
Figure 5 Schematic diagram to represent circular spall
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To model a circular spall, a 2 mm radius spall is assumed whose centre coincides with the pitch line of the gear tooth as shown in Fig. 5 (a). Using the equation of a circle,
x2 y 2 r 2 ,
(21)
and assuming the centre of the circle is at the origin as shown in Fig. 5(b), gives, x r 2 y2
(22)
where, r = radius of the circular spalland y varies from -r to +r. Using equation (22), the length of the spall for the angular rotation of the gear can be calculated as, Length of spall (ls) = 2.x
(23) 14
ACCEPTED MANUSCRIPT Using equations (22) and (23) and putting the value of ls in equation (5), the resulting varying length of contact can be calculated. Using this value the stiffness components can be
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calculated as mentioned in Section 2.
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4.3.3 Modeling of V shaped spall
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Figure 6 Schematic diagram to represent a V- shaped spall
A V-shaped spall as shown in Fig. 6 is assumed near the pitch line as observed in Ref.
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[17]. The V – shaped spall has been modeled as an equilateral triangle with side ‘ls’. The length of the spall will uniformly increase from zero to ls in the rolling direction as the gear rotates as shown in Fig. 6 (b). 5. Result and discussion The specifications of the gear pair used in the computer simulation for calculation of TVMS are given in Table 2. The TVMS of the spalled tooth is plotted considering a varying coefficient of friction as mentioned in Section 3 and various spall parameters as mentioned in Section 4. Analytical expressions explained in Section 2 are used to plot these results using
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ACCEPTED MANUSCRIPT MATLAB as computer simulation software. Spall depth (hs) for all the cases has been taken as 0.1 mm.
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Table 2 Specifications of the gear pair [16] Value
Young’s modulus
E = 2.068 × 1011 Pa
Poisson’s ratio
v =0.3
Pressure angle
α1= 200
Width of teeth
L = 0.016 m
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Parameter
Number of teeth on pinion
19 48
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Number of teeth on gear Base radius of gear
Rb1=0.0716 m Rb2=0.02834 m
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Base radius of pinion
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5.1 TVMS of spur gear pair for spalled tooth with varying friction coefficient
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The TVMS of the spur gear pair with varying friction coefficient has been simulated using expressions derived in section 2 and explained in section 3.1. The variation of the
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coefficient of friction was calculated using equations (12), (13). Computer simulation with and without the spall defect can then be simulated assuming the rectangular shape of the spall. Spall length (ls) and spall width (ws) are both assumed to be 4 mm. The spall is considered symmetrical about the vertical axis of symmetry and is assumed to occur near the pitch point. The typical variation of coefficient of friction (μ) with respect to angular rotation is plotted for various speeds of the spur gear as shown in Fig. 7. This plot demonstrates that there is a significant variation in the value of μ as the point of contact moves over the whole tooth surface for various gear speeds.
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Figure 7 Computer simulation of varying friction coefficient w.r.t. gear speed
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Fig. 8 shows the TVMS values for a healthy gear pair for three cases viz. healthy case and
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two cases considering time varying friction for speeds 2000 rpm and 5000 rpm. The computer simulation shows that due to time varying coefficient of friction the value of TVMS
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varies and this variation increases as the speed increases. It has also been observed that at the start of the double point contact, the change in stiffness value is less which keeps on
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increasing as the double tooth pair contact meshing continues. This variation in change in mesh stiffness is expected because it has been discussed in Section 3.1 that friction forces are acting in two different directions for approach process and recess process and the value of friction coefficient is different for both the approach and recess process as mentioned in equation (12). In the case of double tooth pair contact,the first tooth pair contact recess process takes place whereas in second tooth pair contact approach process takes place. This phenomenon is the reason for the variation in mesh stiffness value considering friction. In the case of the single tooth pair contact effect, the approach process and recess process is clearly visible, where due to the approach process, the drop in mesh stiffness value is less as
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ACCEPTED MANUSCRIPT compared to the healthy case. Also at the pitch point where the sliding velocity changesits
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direction, a sudden drop in mesh stiffness value is noticed.
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Figure 8 Computer simulation of healthy tooth with varying friction coefficient on TVMS
Figure 9 Effect of spalled tooth with and without varying friction coefficient on TVMS
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ACCEPTED MANUSCRIPT Fig.9 compares the TVMS values obtained for the healthy gear pair and spalled gear tooth pair with and without considering time varying friction. As discussed in the previous section,
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it can be seen that time varying friction has changed the stiffness values ofthe spur gear pair.
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An important point worth mentioning is that the spall usually occurs near the pitch point and
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that the change in the direction of the friction forces also takes place on the pitch line, where both phenomenon combined cause an irregular variation in the single contact region near the
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pitch point which is clearly visible in Fig. 9.
5.2 TVMS of spur gear pair for different spall length
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The TVMS of a spur gear pair considering a spall defect have been simulated using expressions derived in section 2 and explained in section 4.1. The effect of friction forces has
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been neglected in this study and the spall was assumed rectangular in shape for the ease of
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simulation. Spall length (ls) was varied as a percentage of the total width of tooth, i.e. spall
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length was assumed as 10%, 20%, and 30% of the total width of the tooth. The spall width (ws = 4 mm) was kept constant for all the cases. The spall was considered symmetrical about the vertical axis of symmetry and was assumed to occur near the pitch point. Fig. 10 shows
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the comparison of the TVMS for these varying spall lengths. After comparison, it can be observed that the reduction in single and double teeth pair contact mesh stiffness has been increased as the spall length is increased.
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Figure 10 Computer simulation of effect of spall length on TVMS
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5.3 TVMS of spur gear pair for symmetric and asymmetric spall TVMS of the spur gear pair has been simulated to study the effect of symmetric and asymmetric spalls using the expressions derived in section 2 and section 4.2. All the other
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gear parameters are kept constant (i.e. ls= 4mm., ws= 4 mm.) and friction has been neglected for both the cases so that the effect of asymmetry on the total mesh stiffness of the gear pair can be compared. It is clearly visible in Fig. 11 that the asymmetric spall reduces the TVMS more than the symmetric spall.
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Figure 11 Computer simulation of effect of spall symmetry on TVMS
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5.4 TVMS of spur gear pair for various spall shape
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TVMS of spur gear pair has been simulated for various spall shapes as discussed in section 4.3. Table 3 gives the dimensions of different shapes considered. For all the shapes,
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the spall depth (hs) has been kept constant at 0.1 mm andthe effect of friction has been neglected. Spall location was kept near to the pitch point for all three cases. In Fig. 12 the stiffness reduction due to the spall shape is shown and it can be observed that as the contact length remains constant during the rectangular spall the reduction in mesh stiffness remains linear. However in the case of a circular and V shaped spall a nonlinear stiffness reduction is observed as the contact length varies according to the gear angular rotation. In order to display the effect of variation in three shapes clearly, a zoomed image of dashed block part is shown in Fig. 13 which shows the pattern in which stiffness reduces due to the presence of a varying spall shape.
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ACCEPTED MANUSCRIPT Table 3 Dimensions of different spall shape Dimensions
Rectangular
Length (ls)= 4 mm. Width (ws)= 4mm.
Circular
Diameter (d) = 4mm.
V-shaped
Initial length= 0 mm. Final length (ls)= 4mm.
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Shape
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Figure 12 Computer simulation of various spall shape on TVMS
Figure 13 Zooming at the dashed-block part in Fig. 12
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ACCEPTED MANUSCRIPT 6. Conclusion A modified analytical formulation for the calculation of TVMS to study the effect of a
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spalling defect has been proposed. The effect of time varying friction coefficient is studied on
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healthy as well as spalled tooth defects to show the resulting mesh stiffness variations. It has
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been found that the value of friction coefficient depends on the speed of the gear shaft and as the speed increases the effect of coefficient of friction also increases. This increase in sliding
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velocity directly affects the mesh stiffness variation of the spur gear pair. It has also been observed that not only is the size of spall an important parameter but its
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symmetry and shape are also key parameters which can affect the TVMS of the gear system. It has been found that the value of TVMS was reduced as compared to healthy gear
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conditions in each of these mentioned cases due to the presence of the spall. One important
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point worth mentioning in this study is that in all cases of the spalled tooth, the reduction in stiffness is localized, i.e. TVMS reduces when the spall is in mesh. A conclusion can be
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drawn from this study that the increase in length of the spall reduces the TVMS, and the unsymmetrical spall reduces the TVMS more than the symmetrical spall and the circular and
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V shaped spall causes a nonlinear reduction of the TVMS. The results of the spalling defects obtained in this paper can provide a theoretical foundation to the spalling fault diagnosis in gearboxes. The proposed model in this paperis limited to spur gear pairs. The scope of our future research work is to develop analytical models of gear mesh stiffness for other gear surface failureswhich will be suitable for different kinds of gear pairs. References [1]
Al-Meshari, A., Al-Zahrani, E., and Diab, M. (2012). Failure analysis of cooling fan gearbox. Engineering Failure Analysis, 20, 166-172.
[2]
Ding, Y., and Rieger, N. F. (2003). Spalling formation mechanism for gears. Wear, 254(12), 1307-1317.
[3]
Parey, A., El Badaoui, M., Guillet, F., and Tandon, N. (2006). Dynamic modeling of spur gear pair and application of empirical mode decomposition-based statistical analysis for early detection of localized tooth defect. Journal of Sound and Vibration, 294(3), 547-561.
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ACCEPTED MANUSCRIPT [4]
Jia, S., and Howard, I. (2006). Comparison of localized spalling and crack damage from dynamic modeling of spur gear vibrations. Mechanical Systems and Signal Processing, 20(2), 332-349.
[5]
Chaari, F., Baccar, W., Abbes, M. S., and Haddar, M. (2008). Effect of spalling or tooth breakage on gearmesh stiffness and dynamic response of a one-stage spur gear transmission. European Journal of
Ma, R., Chen, Y., and Cao, Q. (2012). Research on dynamics and fault mechanism of spur gear pair with
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[6]
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Mechanics-A/Solids, 27(4), 691-705.
spalling defect. Journal of Sound and Vibration, 331(9), 2097-2109.
Ma, R., and Chen, Y. S. (2013). Bifurcation of multi-freedom gear system with spalling defect. Applied
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[7]
Mathematics and Mechanics, 34, 475-488. [8]
Jiang, H., Shao, Y., and Mechefske, C. K. (2014). Dynamic characteristics of helical gears under sliding friction with spalling defect. Engineering Failure Analysis, 39, 92-107.
Martin, K. F. (1978). A review of friction predictions in gear teeth. Wear, 49(2), 201-238.
[10]
Iida, H., Tamura, A., and Yamada, Y. (1985). Vibrational characteristics of friction between gear teeth.
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[9]
Bulletin of JSME, 28(241), 1512-1519.
Vaishya, M., and Singh, R. (2003). Strategies for modeling friction in gear dynamics. Journal of Mechanical Design, 125(2), 383-393.
[12]
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[11]
He, S., Cho, S., and Singh, R. (2008). Prediction of dynamic friction forces in spur gears using alternate
Saxena, A., Parey, A., and Chouksey, M. (2015) Effect of shaft misalignment and friction force on time
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[13]
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sliding friction formulations. Journal of Sound and Vibration, 309(3), 843-851.
varying mesh stiffness of spur gear pair, Engineering Failure Analysis 49, 79-91. [14]
T. M., and Parker, R. G. (2014). Experimental measurement of the effects of torque on the dynamic
[15]
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behavior and system parameters of planetary gears. Mechanism and Machine Theory, 74, 370-389. Ma, H., Pang, X., Feng, R., Zeng, J., and Wen, B. (2015). Improved time-varying mesh stiffness model of cracked spur gears. Engineering Failure Analysis, 55, 271-287. [16]
Tian X. H. Dynamic simulation for system response of gearbox including localized gear faults. Master’s
[17]
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thesis, University of Alberta; 2004. Hannes, D., and Alfredsson, B. (2014). Numerical investigation of the spall opening angle of surface initiated rolling contact fatigue. Engineering Fracture Mechanics, 131, 538-556. [18]
Murakami, Y., Sakae, C., and Ichimaru, K. (1994). Three-dimensional fracture mechanics analysis of pit ormation mechanism under lubricated rolling-sliding contact loading. Tribology transactions, 37(3), 445454.
[19]
Klein, M. A. An experimental investigation of materials and surface treatments on gear contact fatigue life. Master’s thesis, The Ohio State University; 2009.
[20]
Yang, D. C. H., and Lin, J. Y. (1987). Hertzian damping, tooth friction and bending elasticity in gear impact dynamics. Journal of Mechanical Design, 109(2), 189-196.
[21]
Chen, Z., and Shao, Y. (2011). Dynamic simulation of spur gear with tooth root crack propagating along tooth width and crack depth. Engineering Failure Analysis, 18(8), 2149-2164.
[22]
Sainsot, P., Velex, P., and Duverger, O. (2004). Contribution of gear body to tooth deflections—a new bidimensional analytical formula. Journal of mechanical design, 126(4), 748-752.
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ACCEPTED MANUSCRIPT [23]
Muskhelishvili, N.L., (1975). Some Basic Problems of the Mathematical Theory of Elasticity, second ed., English ed. P. Noordhoff Limited, The Netherlands.
[24]
Buckingham, E., (1949). Analytical Mechanics of Gears, Dover Publication, Inc., New York.
[25]
Lin, H. H., and Liou, C. H. (1998). A parametric study of spur gear dynamics. The University of
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Memphis, Memphis, Tennessee.
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ACCEPTED MANUSCRIPT Highlights:
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1. Time varying mesh stiffness for different spall shape, size and location. 2. Modified analytical method for calculation of time varying mesh stiffness. 3. Effect of varying friction coefficient on time varying mesh stiffness in presence of spall.
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S1350-6307(16)30119-4 doi: 10.1016/j.engfailanal.2016.09.003 EFA 2961
To appear in: Received date: Revised date: Accepted date:
30 March 2016 5 August 2016 2 September 2016
Please cite this article as: Saxena Ankur, Parey Anand, Chouksey Manoj, Time varying mesh stiffness calculation of spur gear pair considering sliding friction and spalling defects, (2016), doi: 10.1016/j.engfailanal.2016.09.003
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ACCEPTED MANUSCRIPT Time varying mesh stiffness calculation of spur gear pair considering sliding friction and spalling defects
Mechanical Engineering Department, School of Engineering, Indian Institute of Technology, Indore, India b
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Ankur Saxenaa, Anand Pareya,*, Manoj Choukseyb
Mechanical Engineering Department, SGSITS, Indore, India
* Corresponding author. Tel.: +917324240716; Fax: +917324240700.
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E-mail addresses: [email protected] (A. Saxena), [email protected] (A. Parey), [email protected] (M. Chouksey).
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Abstract
Spalling is one of the common tooth surface failures of gear teeth and is defined as the formation of
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deeper cavities that are mainly developed from subsurface defects. The time varying mesh stiffness
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(TVMS)of gear pairs, gives significant information about the health of the system.The change indirection of time varying friction on both sides of the pitch linecauses the change of gear mesh
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stiffness. This article proposes a computer simulation based approach to study the effect of time varying friction coefficient on the total effective mesh stiffness for the spur gear pair. An analytical
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method to calculate the TVMS of the spur gear for different spall shapes, size and location considering sliding friction is also proposed in this study. The results show that spall shape, size and location are very important parametersthat need to be considered for calculation of TVMS and subsequently to know the dynamic response of the gear pair in the presence of a spall. Keywords: Spur gear, mesh stiffness, spalling fault, friction coefficient.
1. Introduction Gearboxes are one of the most critical and widely used power transmission components in industrial applications. The performance of the gear depends mainly on the durability of the gear tooth surface. Spalling is one of the common tooth surface failures and is defined as the formation of deeper cavities that are mainly developed from subsurface defects. Meshari 1
ACCEPTED MANUSCRIPT et al. [1] performed an extensive investigation of gearbox failures and found thatspalled teeth are one of the major causes of gearbox failures. Ding et al. [2] reported that spalling is the
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most destructive surface failure mode of the gear and proposed a hypothesis of material
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ligament collapse as a fundamental mechanism of subsurface defects which are responsible
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for spalling formation and supported the evidence by performing experiments on gears. Many researchers are using dynamic models of gear pairs to study the effect of gear tooth
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surface faults like spalling, pitting, tooth breakage, etc. on gearbox vibrations. Parey et al. [3] developed a 6 DOF gear dynamic model including localized tooth defects like spalling and
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pitting. Jia and Howard [4] presented a 26 DOF gear dynamic model by including the effect of localized tooth spalling and tooth damage. Chaari et al. [5] studied the dynamic responses
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of gear systems by incorporating the effect of spalling and tooth breakage on time varying
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mesh stiffness (TVMS) of single stage spur gear pairs. Stiffness reduction due to change in spalling length and width is compared and the computed values of gear mesh stiffness are
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injected in dynamic model of gear system. Ma et al. [6] calculated the TVMS of the gear pair by using analytical methods proposed in Ref. [5] and incorporated spalling defects by
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introducing a time-periodic defect excitation term in their dynamic model to study dynamic responses. Ma and Chen [7] developed a nonlinear gear model by incorporating spalling defects. The results show that due to spalling defects the stability of the gear system gets affected. Effect of spalling defects on the mesh stiffness was also studied and was included while developing the dynamic model. Jiang et al. [8] developed a 6 DOF helical gear model by incorporating time varying sliding friction and mesh stiffness for simulating dynamic characteristics for helical gear pairs considering spalling defects. The results show that sliding friction influences the helical gears with spalling defect and increases the peak to peak value of the dynamic response. The spalling defects of helical gears under sliding friction also influence the frequency domain of the dynamic response.The above literature [5-8] 2
ACCEPTED MANUSCRIPT shows that TVMS plays a major role in evaluating the vibration response of the gear pair in the presence of spalling. TVMS of a gear pair was also calculated by using the analytical
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technique proposed in Ref. [5].
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Sliding friction has been reported to be an important source of gear vibration and noise. A
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number of researchers have studied the effects of friction on gear dynamics. A detailed review of friction prediction in gear teeth was conducted by Martin [9] who concluded that the values of coefficients of friction can be predicted reasonably accurately based upon
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various lubrication theories. Iida et al. [10] studied the shaft vibration response in the tooth
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sliding direction. Vaishya and Singh [11] proposed an alternate strategy to incorporate friction in dynamic analysis of gear pair and have shown a typical non-linear variation of the
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coefficient of friction. He et al. [12] used several sliding friction formulations in spur gear
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dynamics for the predictions of friction forces. Effects of frictional force on TVMS of gear pair was also studied by Saxena et al. [13] for healthy and cracked gear teeth. Apart from
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these studies load/torque variation also affects the TVMS of the spur gear. Ericson and Parker [14] used finite element program to study the stiffness of the gear pair at various
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experimental torques. Time invariant mesh stiffness is used in their analytical model simulations at the various torque levels. Ma et al. [15] compared the effect of torque variation on the mesh stiffness of the spur gear pair calculated using finite element method and proposed analytical method. Existing studies have some of the following shortcomings which need to be addressed to get accurate results of TVMS. 1) Effect of friction on TVMS in the presence of spall was not done for spur gear pairs. 2) The effect of axial compression and shear deformation was neglected in existing methods [5] for TVMS calculation in spalled tooth pair but Tian [16] showed in his work that axial
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ACCEPTED MANUSCRIPT compression and shear stiffness are important factors and cannot be neglected while calculating TVMS of gear pair.
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3) The shape of the spall was considered as rectangular indentation in Ref. [5-7] but there
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are other possible shapes of spalls which have been discussed in literature. In Ref. [4] a 5
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mm circular spall was used to study torsional stiffness. Hannes and Alfredsson [17] investigated the spall opening angle by assuming a V-shaped spall to study rolling contact fatigue damage. Murakami et al. [18] carried a crack growth analysis by calculating the
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stress intensity factor for a small, arrow headed surface crack which finally forms a V-
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shaped pit. Klein [19] in his Master’s thesis studied the modes of gear contact fatigue failure and found that V-shaped spall occurred on tooth surface. So it can be said that a V-
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shaped spall is more likely to occur in practical engineering and the study related to
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various spall shapes will be more helpful in diagnosis of the gear tooth spalling failure. 4) The effect of symmetry of the spall was also neglected in previous literature and it was
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assumed that the spall was located about the vertical axis of symmetry but the spall may not be symmetric or more than one spall may be present at the line of contact of the gear
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pair and their effect on TVMS needs to be studied for better understanding of vibration responses which was not found in previous work. This literature survey concludes that due to the presence of spalls at the surface of gear teeth, stiffness reduction is observed which directly affects the vibration responses of the gear system. However there are various other parameters which were neglected in previous studies and they need to be considered for better understanding of spalling defects and their effect on gear mesh stiffness. In this paper TVMS has been calculated for a spalled tooth pair using the modified potential energy method. The effect of time varying coefficient of friction, spall shape, size and unsymmetrical spall is considered and stiffness reduction due to these parameters has been simulated.
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ACCEPTED MANUSCRIPT 2. Analytical expression of TVMS for spur gear with spalling defects In this study the potential energy method, proposed by Yang and Lin [20] and refined by
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Tian [16] and Chen et al. [21] is used to calculate TVMS of a spalled gear tooth pair. This
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method considers Hertzian contact energy, bending energy, axial compressive energy, shear
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energy and fillet foundation deflection to calculate the stiffness of the gear tooth. To calculate the individual stiffness component this method uses angular variables instead of linear
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variables. This is helpful in investigating parameter properties at some angular displacement of the pinion/gear.
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While considering the spalling fault the change in area of cross-section (A) and moment of inertia (I) due to the spall has to be considered. These changes are difficult to express by
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angular variables, therefore, a modified expression is needed to include linear variables (A,
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I).
Figure1 Healthy tooth model [16] 5
ACCEPTED MANUSCRIPT Using the values and same notations used in Ref. [16] and shown in Fig. 1, modified expressions of Hertzian stiffness, axial compressive stiffness, shear stiffness and bending
1 kb
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2.4 Rb (1 v)cos 2 1 ( 2 )cos d 1 E. As
2
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1 ks
Rb sin 2 1 ( 2 )cos d 1 E. As 2
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1 ka
ELs 4(1 2 )
Rb3 [1 cos 1{( 2 )sin cos }]3 ( 2 )cos d 1 E.I s
2
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kh
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stiffness components are shown in equations (1) – (4) respectively;
(1)
(2)
(3)
(4)
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In equations (1)-(4) Ls, As and Is are the length of contact, area of tooth cross section and
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L Ls L ls
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are moment of inertia respectively and their values will vary according to equations (5) – (7).
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2.h x .L As (2.h x .L) (hs .ls )
2 3 3 .h x .L Is 1 [(2.h )3 .L (h3 .l ) x s s 12
for healthy region for spalled region
(5)
for healthy region for spalled region
(6)
for healthy region
(7) for spalled region
where hx Rb [(2 )cos sin ] and L, ls, hs are shown in Fig. 2. The definition of remaining parameters can be obtained in Ref. [16] and are shown in Figs. 1 and 2.
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Figure 2 Schematic diagram for spalled gear tooth
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Besides the tooth deformation, the fillet-foundation deflection also influences the stiffness
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of the gear tooth. Sainsot et al. [22] derived the fillet-foundation deflection of the gear based
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on the theory of Muskhelishvili [23]. The stiffness with consideration of gear filletfoundation deflection can be obtained by [21],
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1 cos 2 1 * u f L kf EL S f
2
uf * M Sf
* * 2 P (1 Q tan 1 ) ,
(8)
where, L is the tooth width and α1is acting pressure angle of gear tooth. The definition of the remaining parameters uf and Sf is given in Fig. 3. The coefficients L*, M*, P*, Q*can be calculated by use of the following polynomial functions proposed in Ref. [22],
X i* (h fi , f ) Ai f2 Bi h2fi Ci h fi f Di f Ei h fi Fi
(9)
In eq. (9) Xi*denotes the coefficients L*, M*, P*, Q*; hfi = rf/rint; rf, rint and hf are defined in Fig. 3, while the values of Ai, Bi, Ci, Di, Ei and Fi are given in Table 1.
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ACCEPTED MANUSCRIPT Table 1. Values of the coefficients of Eq. (9) [22]. Bi -1.9986 x 10-3 28.100 x 10-3 185.50 x 10-3 9.0889 x 10-3
Ci -2.3015 x 10-4 -83.431 x 10-4 0.0538 x 10-4 -4.0964 x 10-4
Di 4.7702 x 10-3 -9.9256 x 10-3 53.3 x 10-3 7.8297 x 10-3
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Ai -5.574 x 10-5 60.111 x 10-5 -50.952 x 10-5 -6.2042 x 10-5
Ei 0.0271 0.1624 0.2895 -0.1472
Fi 6.8045 0.9086 0.9236 0.6904
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L* (hfi, θf) M* (hfi, θf) P* (hfi, θf) Q* (hfi, θf)
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Figure 3 Geometrical parameters for the fillet-foundation deflection. [21]
Thus for the single – tooth pair meshing duration, the total effective TVMS can be expressed as,
kt
1 1
kh
1
kb1
1
ks1
1
ka1
1
kf1
1
kb 2
1
ks 2
1
ka 2
1
,
(10)
kf 2
where, kh, kb, ks, ka and kf represent the Hertzian, bending, shear, axial compressive and fillet foundation deflection mesh stiffness, respectively and the subscripts 1, 2 mean the pinion and gear, respectively. For the double-tooth-pair meshing duration, the total effective TVMS is the sum of the two pair’s stiffness, which can be expressed as
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kt i 1
1 1
k h ,i
1
kb1,i
1
ks1,i
1
ka1,i
1
k f 1,i
1
kb 2,i
1
ks 2,i
1
ka 2,i
1
(11)
k f 2,i
T
where, i = 1 represents the first pair of meshing teeth and i = 2 represents the second pair of
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meshing teeth.
In this work the TVMS of the gear pair is calculated usingequations (1) - (11).
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3. Analytical expression of coefficient of friction for spur gear pair The analytical method proposed by Saxena et al. [13] is adopted in this work to calculate
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mesh stiffness of the spur gear pairto study the effect of frictional forces. In Ref. [13] a constant value of coefficient of friction (μ) is considered but, in actual case the coefficient of
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friction changes continuously due to varying mesh properties and lubricant film thickness as
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the gear rotates throughout the complete mesh cycle. This type of variation has been already
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presented in literature [11, 12]. Thus, the variation of coefficient of friction is included in the formulation as the position of each tooth and the direction of friction force changes. Buckingham's [24] formula is used to determine the coefficient of friction which will be
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incorporated in the study.
3.1 Friction Coefficient Model included in this study Friction during the gear mesh has two components: sliding friction and rolling friction. Coefficient of friction μ usually refers to coefficient of sliding friction. There are many empirical formulae in literature [9-12] which have been developed based on different assumptions to calculate the value of the coefficient of friction for the gear pair. These empirical formulae state that μ is a function of a number of parameters such as sliding and rolling velocities, radii of curvature of the surfaces in contact, amount of surface roughness,
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ACCEPTED MANUSCRIPT lubricant viscosity, etc. In this study Buckingham's [24] formula is used to determine the coefficient of friction μ which is:
r
2 ; 3
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0.05 4 0.002 V ; ; s a e0.125Vs 3
(12)
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where, μ = average coefficient of friction, μa= average coefficient of friction of approach, μr= average coefficient of friction of recess and Vs = sliding velocity in m/sec.
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In equation (12) the sliding velocity of the spur gear pair varies as the gear rotates, due to which the coefficient of friction will also vary. This variation is incorporated into the TVMS
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calculation of the spur gear pair to study the effect of variation of µ on TVMS. The value of sliding velocity of the gear pair can be obtained by the following equation
where,
d p np 60
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r12 Rb21 Rp1 sin 1 , (13)
=pitch line velocity of gear in m/sec.
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V
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V ( Rp1 Rp 2 ) Vs R R p1 p 2
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which is derived in Ref. [25],
Rp1, Rp2 = radius of pitch circle of gear and pinion respectively, in m. Rb1 = radius of base circle of gear, in m. r1 = any radius of gear tooth profile, in m. dp= pitch circle diameter of gear in m. np = speed of gear in rpm. α1= pressure angle The value of the time varying coefficient of friction (µ) obtained in equations (12) and (13) is substituted in the TVMS formula developed by Saxena et al. [13]. Hertzian stiffness and stiffness due to fillet foundation deflection will remain unchanged. Expressions for
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ACCEPTED MANUSCRIPT bending mesh stiffness, shear mesh stiffness and axial compressive stiffness considering coefficient of friction is given as follows [13]:
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Bending stiffness; 2 3[{1 cos {( ) sin cos }} { sin {( ) sin cos }}]2 ( ) cos 1 2 1 2 1 2 2 d , 2 EL[sin ( 2 ) cos ]3 1 1 kb 2 3[{1 cos 1{( 2 ) sin cos }} {1 2 sin 1{( 2 ) sin cos }}]2 ( 2 ) cos d , 2 EL[sin ( 2 ) cos ]3 1
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(15)
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(14)
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Shear stiffness; 2 2 1.2(1 )(cos 1 sin 1 ) ( 2 )cos d , EL[sin ( 2 )cos ] 1 1 ks 2 1.2(1 )(cos 1 sin 1 ) 2 ( 2 )cos d EL[sin ( 2 )cos ] 1
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Axial compressive stiffness;
(16)
(17)
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2 ( ) cos (sin cos ) 2 2 1 1 d , 2 EL [sin ( ) cos ] 2 1 1 ka 2 ( ) cos (sin cos ) 2 2 1 1 d 2 EL [sin ( ) cos ] 2 1
(18)
(19)
In the above expressions equations (14), (16) and (18) are derived for the approach process and equation (15), (17) and (19) are derived for the recess process. Using equations (12) and (13) and following the TVMS expressions developed by Saxena et al. [13] the total meshing stiffness of spur gear pair including varying friction coefficient can be obtained.
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ACCEPTED MANUSCRIPT 4. Spall parameters Various parameters of spalling which influences the TVMS are discussed in this section
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4.1 Spall length
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The length of the spall is considered as a parameter responsible for mesh stiffness variation since tooth contact does not take place in this region. In the literature [5-7] the effect of spall length on TVMS is studied but the use of potential energy methods to
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demonstrate this phenomenon is not used in any previous work. For a rectangular spall of length ‘ls’ as shown in Fig. 2, it is assumed that no tooth contact occurs in the spalled area
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which will directly affect Hertzian contact stiffness and has a slight effect on other stiffness
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components. Using equations (1) – (11) the effect of spall length on TVMS can be studied.
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4.2Spall symmetry about vertical axis
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Spalling may develop either symmetrically or asymmetrically about the vertical plane of symmetry. Fig. 4(a) shows a symmetrical spall and the resultant distribution of forces. Fig. 4 (b) shows an unsymmetrical spall and the resultant distribution of forces. It can be seen in
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Fig. 4(b) that due to eccentricity (t) of the resultant force from the axis of symmetry, a torque T acts on the tooth. Due to the torque T, a torsional stiffness (kτ) comes into existence as proposed in Ref. [13]. For simplicity, the change in polar moment of inertia due to spall depth (= 1mm) is neglected. The value of torsional stiffness (kτ) can be calculated by using Ref. [13] which gives
2 12.t 2 (1 v)( 2 )cos 1 d k EL{sin ( 2 )cos }[4.Rb2{sin ( 2 )cos }2 L2 ]
(20)
1
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T
throughout the length of contact.
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Figure 4 Effect of forces on (a) symmetrical spall, (b) unsymmetrical spall 4.3Spall shape
As mentioned in the literature survey many researchers have suggested that the shape of the spall can be rectangular [5], circular [4] or V-shaped [17-19]. In this section the effect of rectangular, circular and V-shaped spalls is studied. In all the cases, the spall is assumed to be present about the pitch point of the pinion and is in the single point contact region only. Also, the spall is assumed to be symmetric.
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indentation as done in Ref. [5-7] having dimensions ls x ws x hs as shown in Fig. 2. Modeling
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of rectangular spall has already been explained in section 2 in equations (1) – (11).
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4.3.2 Modeling of circular spall
Figure 5 Schematic diagram to represent circular spall
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To model a circular spall, a 2 mm radius spall is assumed whose centre coincides with the pitch line of the gear tooth as shown in Fig. 5 (a). Using the equation of a circle,
x2 y 2 r 2 ,
(21)
and assuming the centre of the circle is at the origin as shown in Fig. 5(b), gives, x r 2 y2
(22)
where, r = radius of the circular spalland y varies from -r to +r. Using equation (22), the length of the spall for the angular rotation of the gear can be calculated as, Length of spall (ls) = 2.x
(23) 14
ACCEPTED MANUSCRIPT Using equations (22) and (23) and putting the value of ls in equation (5), the resulting varying length of contact can be calculated. Using this value the stiffness components can be
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calculated as mentioned in Section 2.
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4.3.3 Modeling of V shaped spall
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Figure 6 Schematic diagram to represent a V- shaped spall
A V-shaped spall as shown in Fig. 6 is assumed near the pitch line as observed in Ref.
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[17]. The V – shaped spall has been modeled as an equilateral triangle with side ‘ls’. The length of the spall will uniformly increase from zero to ls in the rolling direction as the gear rotates as shown in Fig. 6 (b). 5. Result and discussion The specifications of the gear pair used in the computer simulation for calculation of TVMS are given in Table 2. The TVMS of the spalled tooth is plotted considering a varying coefficient of friction as mentioned in Section 3 and various spall parameters as mentioned in Section 4. Analytical expressions explained in Section 2 are used to plot these results using
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ACCEPTED MANUSCRIPT MATLAB as computer simulation software. Spall depth (hs) for all the cases has been taken as 0.1 mm.
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Table 2 Specifications of the gear pair [16] Value
Young’s modulus
E = 2.068 × 1011 Pa
Poisson’s ratio
v =0.3
Pressure angle
α1= 200
Width of teeth
L = 0.016 m
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Parameter
Number of teeth on pinion
19 48
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Number of teeth on gear Base radius of gear
Rb1=0.0716 m Rb2=0.02834 m
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Base radius of pinion
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5.1 TVMS of spur gear pair for spalled tooth with varying friction coefficient
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The TVMS of the spur gear pair with varying friction coefficient has been simulated using expressions derived in section 2 and explained in section 3.1. The variation of the
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coefficient of friction was calculated using equations (12), (13). Computer simulation with and without the spall defect can then be simulated assuming the rectangular shape of the spall. Spall length (ls) and spall width (ws) are both assumed to be 4 mm. The spall is considered symmetrical about the vertical axis of symmetry and is assumed to occur near the pitch point. The typical variation of coefficient of friction (μ) with respect to angular rotation is plotted for various speeds of the spur gear as shown in Fig. 7. This plot demonstrates that there is a significant variation in the value of μ as the point of contact moves over the whole tooth surface for various gear speeds.
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Figure 7 Computer simulation of varying friction coefficient w.r.t. gear speed
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Fig. 8 shows the TVMS values for a healthy gear pair for three cases viz. healthy case and
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two cases considering time varying friction for speeds 2000 rpm and 5000 rpm. The computer simulation shows that due to time varying coefficient of friction the value of TVMS
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varies and this variation increases as the speed increases. It has also been observed that at the start of the double point contact, the change in stiffness value is less which keeps on
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increasing as the double tooth pair contact meshing continues. This variation in change in mesh stiffness is expected because it has been discussed in Section 3.1 that friction forces are acting in two different directions for approach process and recess process and the value of friction coefficient is different for both the approach and recess process as mentioned in equation (12). In the case of double tooth pair contact,the first tooth pair contact recess process takes place whereas in second tooth pair contact approach process takes place. This phenomenon is the reason for the variation in mesh stiffness value considering friction. In the case of the single tooth pair contact effect, the approach process and recess process is clearly visible, where due to the approach process, the drop in mesh stiffness value is less as
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direction, a sudden drop in mesh stiffness value is noticed.
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Figure 8 Computer simulation of healthy tooth with varying friction coefficient on TVMS
Figure 9 Effect of spalled tooth with and without varying friction coefficient on TVMS
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ACCEPTED MANUSCRIPT Fig.9 compares the TVMS values obtained for the healthy gear pair and spalled gear tooth pair with and without considering time varying friction. As discussed in the previous section,
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it can be seen that time varying friction has changed the stiffness values ofthe spur gear pair.
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An important point worth mentioning is that the spall usually occurs near the pitch point and
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that the change in the direction of the friction forces also takes place on the pitch line, where both phenomenon combined cause an irregular variation in the single contact region near the
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pitch point which is clearly visible in Fig. 9.
5.2 TVMS of spur gear pair for different spall length
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The TVMS of a spur gear pair considering a spall defect have been simulated using expressions derived in section 2 and explained in section 4.1. The effect of friction forces has
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been neglected in this study and the spall was assumed rectangular in shape for the ease of
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simulation. Spall length (ls) was varied as a percentage of the total width of tooth, i.e. spall
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length was assumed as 10%, 20%, and 30% of the total width of the tooth. The spall width (ws = 4 mm) was kept constant for all the cases. The spall was considered symmetrical about the vertical axis of symmetry and was assumed to occur near the pitch point. Fig. 10 shows
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the comparison of the TVMS for these varying spall lengths. After comparison, it can be observed that the reduction in single and double teeth pair contact mesh stiffness has been increased as the spall length is increased.
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Figure 10 Computer simulation of effect of spall length on TVMS
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5.3 TVMS of spur gear pair for symmetric and asymmetric spall TVMS of the spur gear pair has been simulated to study the effect of symmetric and asymmetric spalls using the expressions derived in section 2 and section 4.2. All the other
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gear parameters are kept constant (i.e. ls= 4mm., ws= 4 mm.) and friction has been neglected for both the cases so that the effect of asymmetry on the total mesh stiffness of the gear pair can be compared. It is clearly visible in Fig. 11 that the asymmetric spall reduces the TVMS more than the symmetric spall.
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Figure 11 Computer simulation of effect of spall symmetry on TVMS
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5.4 TVMS of spur gear pair for various spall shape
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TVMS of spur gear pair has been simulated for various spall shapes as discussed in section 4.3. Table 3 gives the dimensions of different shapes considered. For all the shapes,
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the spall depth (hs) has been kept constant at 0.1 mm andthe effect of friction has been neglected. Spall location was kept near to the pitch point for all three cases. In Fig. 12 the stiffness reduction due to the spall shape is shown and it can be observed that as the contact length remains constant during the rectangular spall the reduction in mesh stiffness remains linear. However in the case of a circular and V shaped spall a nonlinear stiffness reduction is observed as the contact length varies according to the gear angular rotation. In order to display the effect of variation in three shapes clearly, a zoomed image of dashed block part is shown in Fig. 13 which shows the pattern in which stiffness reduces due to the presence of a varying spall shape.
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ACCEPTED MANUSCRIPT Table 3 Dimensions of different spall shape Dimensions
Rectangular
Length (ls)= 4 mm. Width (ws)= 4mm.
Circular
Diameter (d) = 4mm.
V-shaped
Initial length= 0 mm. Final length (ls)= 4mm.
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Shape
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Figure 12 Computer simulation of various spall shape on TVMS
Figure 13 Zooming at the dashed-block part in Fig. 12
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ACCEPTED MANUSCRIPT 6. Conclusion A modified analytical formulation for the calculation of TVMS to study the effect of a
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spalling defect has been proposed. The effect of time varying friction coefficient is studied on
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healthy as well as spalled tooth defects to show the resulting mesh stiffness variations. It has
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been found that the value of friction coefficient depends on the speed of the gear shaft and as the speed increases the effect of coefficient of friction also increases. This increase in sliding
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velocity directly affects the mesh stiffness variation of the spur gear pair. It has also been observed that not only is the size of spall an important parameter but its
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symmetry and shape are also key parameters which can affect the TVMS of the gear system. It has been found that the value of TVMS was reduced as compared to healthy gear
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conditions in each of these mentioned cases due to the presence of the spall. One important
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point worth mentioning in this study is that in all cases of the spalled tooth, the reduction in stiffness is localized, i.e. TVMS reduces when the spall is in mesh. A conclusion can be
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drawn from this study that the increase in length of the spall reduces the TVMS, and the unsymmetrical spall reduces the TVMS more than the symmetrical spall and the circular and
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V shaped spall causes a nonlinear reduction of the TVMS. The results of the spalling defects obtained in this paper can provide a theoretical foundation to the spalling fault diagnosis in gearboxes. The proposed model in this paperis limited to spur gear pairs. The scope of our future research work is to develop analytical models of gear mesh stiffness for other gear surface failureswhich will be suitable for different kinds of gear pairs. References [1]
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Jia, S., and Howard, I. (2006). Comparison of localized spalling and crack damage from dynamic modeling of spur gear vibrations. Mechanical Systems and Signal Processing, 20(2), 332-349.
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sliding friction formulations. Journal of Sound and Vibration, 309(3), 843-851.
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T. M., and Parker, R. G. (2014). Experimental measurement of the effects of torque on the dynamic
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behavior and system parameters of planetary gears. Mechanism and Machine Theory, 74, 370-389. Ma, H., Pang, X., Feng, R., Zeng, J., and Wen, B. (2015). Improved time-varying mesh stiffness model of cracked spur gears. Engineering Failure Analysis, 55, 271-287. [16]
Tian X. H. Dynamic simulation for system response of gearbox including localized gear faults. Master’s
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thesis, University of Alberta; 2004. Hannes, D., and Alfredsson, B. (2014). Numerical investigation of the spall opening angle of surface initiated rolling contact fatigue. Engineering Fracture Mechanics, 131, 538-556. [18]
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Klein, M. A. An experimental investigation of materials and surface treatments on gear contact fatigue life. Master’s thesis, The Ohio State University; 2009.
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Chen, Z., and Shao, Y. (2011). Dynamic simulation of spur gear with tooth root crack propagating along tooth width and crack depth. Engineering Failure Analysis, 18(8), 2149-2164.
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Muskhelishvili, N.L., (1975). Some Basic Problems of the Mathematical Theory of Elasticity, second ed., English ed. P. Noordhoff Limited, The Netherlands.
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Memphis, Memphis, Tennessee.
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ACCEPTED MANUSCRIPT Highlights:
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1. Time varying mesh stiffness for different spall shape, size and location. 2. Modified analytical method for calculation of time varying mesh stiffness. 3. Effect of varying friction coefficient on time varying mesh stiffness in presence of spall.
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