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Modeling of material removal rate and surface roughness in finishing of bevel gears by electrochemical honing process
Journal of Materials Processing Technology 214 (2014) 200–209
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Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec
Modeling of material removal rate and surface roughness in finishing of bevel gears by electrochemical honing process Javed Habib Shaikh, Neelesh Kumar Jain ∗ Discipline of Mechanical Engineering, Indian Institute of Technology Indore (MP), India
a r t i c l e
i n f o
Article history: Received 22 March 2013 Received in revised form 26 July 2013 Accepted 20 August 2013 Available online 7 September 2013 Keywords: Modeling Material removal rate Surface roughness Electrochemical dissolution Electrochemical honing Bevel gears
a b s t r a c t This paper describes mathematical modeling of material removal rate (MRR) and surface roughness of the bevel gears finished by the electro-chemical honing (ECH) process. Since, ECH hybridizes electrochemical dissolution (ECD) and mechanical honing therefore, contribution of ECD in MRR and surface roughness has been modeled using Faraday’s law of electrolysis while contribution of mechanical honing has been modeled considering material removal as a phenomenon of uniform wear and using Archard’s wear model. Formulations are also proposed for computing the surface area, required by these two models, along the inter-electrode gap (IEG) based on the geometry of the straight bevel gear tooth surfaces. The developed models were experimentally validated using an indigenously developed experimental setup for finishing of bevel gears by ECH based on an envisaged novel concept of twin complementary cathode gears. An aqueous solution containing 25% NaCl + 75% NaNO3 was used as the electrolyte. The predicted values of MRR and surface roughness have shown close agreement with the experimental values. The experimental results, SEM images and bearing area curve have shown appreciable improvement in the surface roughness and surface integrity ensuring better operating performance of the gears finished by ECH within an optimized finishing time of 2 min. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Gear is one of the essential machine elements used to transmit motion and power by the successive engagements of teeth on their periphery. Application areas for gears are diverse and include machine tools, automotives, aerospace, marine, oil and gas industry equipments, cement and other processing mills, steel processing units, etc. (Davis, 2005). The operating performance such as power transmission efficiency, noise, vibration and durability of the gears, depends on its surface finish and quality. To prevent failure of gears requires careful consideration of gear form accuracy, gear materials, gear tooth forces, lubrication, operating conditions, and surface quality which in turn depend on the gear finishing method. Form errors and surface roughness of gears can be reduced significantly by finishing processes (Jain et al., 2009). Generally used gear finishing processes for bevel gears namely gear grinding and gear lapping, have some limitations such as gear grinding is expensive, complicated and may produce undesirable effects such as grinding burn. Grinding burn damages the surface integrity of the ground workpiece and if it is not detected then loaded gear might fail with severe consequences like tooth breakage. Lapping is the oldest but
∗ Corresponding author. Tel.: +91 732 4240 702; fax: +91 732 4240 700. E-mail addresses: [email protected] (J.H. Shaikh), [email protected], [email protected] (N.K. Jain). 0924-0136/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmatprotec.2013.08.010
still widely used process to finish the surface of the bevel gear tooth flanks and thus reducing the operating noise. It is a time consuming process and can correct only minor errors of the gear tooth profile. If lapping is used for longer duration, it may affect form of the gear teeth profile (Karpuschewski et al., 2008). Electrochemical honing (ECH) hybridizes capabilities of electrochemical dissolution (ECD) and mechanical honing in one operation and is a potential promising alternative to the conventional gear finishing processes. Wei et al. (1986) mentioned that ECH has ability to improve surface finish as well as to correct form errors, smoothening of irregularities on the surface and being productive at the same. ECH can achieve average surface roughness values up to 0.05 m and dimensional tolerances up to ±0.002 m (Benedict, 1987). It offers some unique features such as workpiece material of any hardness can be processed and less cycle time as compared to the existing processes. Misra et al. (2013) studied the effect of electrolyte composition on percentage improvement in bearing ratio by experimental runs conducted using the mixture D-Optimal design approach and found an aqueous solution containing 70% NaCl and 30% NaNO3 as an optimum electrolyte composition. Very few references are available on modeling of material removal rate (MRR) and surface roughness in the finishing of gears by the ECD based processes. Ning et al. (2011) developed a mathematical model for total thickness of the material removed and surface roughness in pulse electrochemical finishing (PECF) of spiral bevel gears and experimentally validated their models. But, they
J.H. Shaikh, N.K. Jain / Journal of Materials Processing Technology 214 (2014) 200–209
Nomenclature Ab Af Ar As
At Dw E f
F Fn Fw H he hh hp
hv
hp hv
I J K k
Ke Li Lc Lr MRRECH Ns P r Rzi
RzECH RzECD1
RzECH1
rb
Area of the bottom land (mm2 ) Area of the tooth flank of the gear (mm2 ) Area of the fillet surface at root of the gear (mm2 ) Surface area of the bevel gear tooth where the electrolytic dissolution takes place through flow of current (mm2 ) Area of the top land (mm2 ) Working depth of gear tooth (mm) Electrochemical equivalent of the workpiece material (g) Factor used to convert the height of rectangle into height of a triangle with same area and the base length (=2) Faraday’s constant (=96,500 C) Total normal load acting along the line of action (N) Facewidth of the bevel gear (mm) Brinell hardness number (BHN) of the workpiece material (N/mm2 ) Total thickness of material removed from the workpiece surface in one cycle of ECD (m) Total thickness of material removed from the workpiece surface in one cycle of honing (m) Distance between center line of the cathode surface roughness and the peaks on the workpiece surface before ECD (m) Distance between center line of the cathode surface roughness and the valleys on the workpiece surface before ECD (m) Distance between center line of the cathode surface roughness and the peaks on the workpiece surface after ECD (m) Distance between center line of the cathode surface roughness and the valleys on the workpiece surface after ECD (m) Amount of the current passed in the IEG (A) Current density in the IEG (A/mm2 ) Wear coefficient Factor that indicates proportion of total thickness of material removed from the valleys in one cycle of ECD and honing Electrical conductivity of the electrolyte (−1 mm−1 ) Length of the involute profile (mm) Length of chord of the involute (mm) Length of arc of fillet at the root (mm) Material removal rate in ECH (mm3 /s) Number of revolutions of the workpiece gear per second (rps) Perimeter of the workpiece gear (mm) Radius of the involute arc (mm) Arithmetic mean of maximum peak-to-valley heights for the UNFINISHED workpiece surface (m) Arithmetic mean of maximum peak-to-valley heights for the ECHed workpiece surface (m) Arithmetic mean of maximum peak-to-valley heights for the workpiece surface AFTER one cycle of ECD (m) Arithmetic mean of maximum peak-to-valley heights for the workpiece surface AFTER one cycle of ECH (m) Radius of the base circle (mm)
S t T V Ve Vh W Wt Wb Y ˛ V
201
Total sliding distance (mm) Finishing time (s) Number of teeth of the workpiece gear Applied voltage (V) Volumetric material removal rate (MRR) due to ECD (mm3 /s) Volumetric material removal rate (MRR) due to honing action (mm3 /s) Width at the base of the tooth (mm) Width of top land (mm) Width of bottom land (mm) Inter-electrode gap (mm) Pressure angle of the involute profile Current efficiency Density of the anodic work material (g/mm3 ) Angle subtended by the involute at its centre (deg.) Total voltage loss in the IEG (V)
finished one tooth at a time and reported improvement in the surface roughness and form errors. Ruszaj and Zybura-Skrabalak (2001) developed a mathematical model for MRR in the ECD with flat ended electrode and compared the theoretical results with the actual results. Park and Kahraman (2009) combined Archard’s wear model with finite-element based contact model for simulation of surface wear of face-milled or face-hobbed hypoid gear pairs. Pavlov (2011) proposed computational methods for estimation of the wear, service life, and efficiency of the straight bevel gears having involute profile while Chernets and Bereza (2009) investigated on kinetics of wear of gear tooth and proposed a method for wear and durability of straight bevel gears. From the review of the past work it is evident that no work has been reported on modeling of MRR and surface roughness of bevel gears finished by the ECH process. The objective of the present work is to bridge this gap. This paper describes development of the models of MRR and arithmetic mean of maximum peak-tovalley heights (Rz ) for the flank surface of bevel gears finished by the ECH process and their experimental validation. These models have been developed as function of rotary speed of the workpiece gear and current in the inter-electrode gap (IEG), which indirectly takes into account the effects of other electrolytic parameters such as concentration, temperature and flow rate. Since, ECH hybridizes ECD and mechanical honing therefore, contribution of ECD in MRR and surface roughness has been modeled using Faraday’s law of electrolysis while, contribution of mechanical honing has been modeled considering material removal by honing as a process of uniform wear and using wear model proposed by Archard (1953). Formulations are also proposed to compute the surface area along the inter-electrode gap (IEG) based on the geometry of the gear tooth surfaces of the straight bevel gears as required by these models. Comparison of the theoretical values with the experimental results has been done. These models and their experimental validation optimized the applied voltage and rotary speed which helps in performance optimization of finishing of gears by ECH. 2. Finishing of bevel gears by ECH 2.1. Working principle Bevel gear finishing by ECH is much more difficult than finishing of cylindrical gears (i.e. spur or helical gears) due to their complex geometry which restricts feeding motion of the workpiece gear required for finishing its entire face width. This problem was resolved by envisaging a novel concept of twin complementary
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Table 1 Details of various ECH parameters used in the experimental validation. Composition (by wt.%) of the workpiece material (i.e. 20MnCr5 alloy steel): Cr (0.8–1.1); Mn (1–1.3); C (0.14–0.19); P (0.035 max.); S (0.035 max.); Si (0.15–0.40); and balance Fe Fixed input parameters
Variable input parameters
1. Electrolyte composition (C): 75% NaNO3 + 25% NaCl 2. Finishing time (t): 2 min 3. Inter electrode gap: 1 mm
1. Voltage (V): 3 levels (8 V; 12 V; 16 V) 2. Rotary speed of workpiece gear (R): 3 levels (40 rpm; 60 rpm; 80 rpm) 3. Electrolyte concentration (C): 3 levels (5%; 7.5%; 10%) (by wt.) 4. Electrolyte temperature (T): 3 levels (27 ◦ C; 32 ◦ C; 37 ◦ C) 5. Electrolyte flow rate (F): 3 levels (10 lpm; 20 lpm; 30 lpm)
cathodic bevel gears ‘3’ and ‘4’ meshing with the anodic workpiece bevel gear ‘1’ as shown in Fig. 1(a). The cathode gear ‘3’ has an insulating layer of Metalon sandwiched between two undercut (by 1 mm) conducting layers of copper while, the other complimentary cathode gear ‘4’ has an undercut (by 1 mm) layer of copper sandwiched between two insulating layers of Metalon. This arrangement ensures finishing of the entire face width of the workpiece gear by the ECD action without its reciprocation and at the same time IEG avoids the short circuiting between the cathodic and anodic gears. A DC voltage is applied across electrodes. The IEG is flooded with a full stream of electrolyte ‘5’ so that cathodic gears can finish the workpiece gear by electrolytic dissolution. The electrolysis action forms a passivating layer of metal oxide on the tooth surfaces of the workpiece gear due to evolution of oxygen. It prevents further material removal by ECD process. This passivating layer is scraped by the honing action between the workpiece gear and a honing gear ‘2’ which is mounted perpendicular to the workpiece and cathode gears. This honing gear is in tight mesh as compared to the cathode gears with the workpiece gear so as to remove the passivating oxide layer. This cyclic sequence of ECD and honing action improves the form accuracy and surface finish of the workpiece gear. The workpiece, cathode and honing gears were cut on bevel
gear generator based on the Gleason method in a gear manufacturing industry. Both honing gear and workpiece gears were made of case hardened 20MnCr5 alloy steel (composition shown in Table 1) having similar surface finish but, having different surface hardness values. The workpiece gears had hardness in the range of 50–54 HRC while, the honing gear had hardness in the range of 58–62 HRC. This difference in the hardness values was found to be sufficient to remove the passivating metal oxide layer and no wear of the honing gear was observed during the finishing time used in the present work. Fig. 1(b) shows the photograph of the machining chamber for finishing the bevel gears by ECH based on this principle. 2.2. Process mechanism Mechanism for finishing of bevel gears by ECH is proposed in which cyclic sequence of finishing by electrolytic dissolution and mechanical honing improves the geometric accuracy and surface finish of the workpiece gear. Fig. 2(a) shows the initial surface profiles of the anode and cathode. It can be seen that the distance between peaks on the workpiece and cathode surfaces is less as compared to the corresponding distance between valleys. Consequently, electrolytic dissolution will remove more material from the peaks of workpiece surface as compared to its valleys thus truncating the highest peaks and giving rapid improvement in the workpiece surface finish. At the end of electrolytic dissolution process, a passivating metal oxide layer is formed on the workpiece surface as shown in Fig. 2(b), (d) and (f) which is removed by the honing action so that further electrolytic dissolution can continue. At the end of honing action some valleys still might be covered with the passivating layer as shown in Fig. 2(c), (e) and (g). In the next cycle of electrolytic dissolution, these peaks will be exposed for further smoothening. Consequently, these peaks are truncated as well as some material is also removed from valleys giving a smoother surface as shown in Fig. 2(g). As the ECH process continues, the improvement in the surface finish and geometry accuracy continues at a diminishing rate due to increasing IEG between the surfaces of cathode and workpiece gears and consequent decreasing MRR. As shown in Fig. 2(g), the roughness which remains after finishing will take very long time to be completely removed by the ECH. Therefore, whenever the desired surface finish or geometric accuracy or both are achieved the process can be stopped. 3. Modeling of material removal rate (MRR) in ECH of gears ECH is a hybrid finishing process of ECD and mechanical honing therefore these processes contribute in the process of material removal and surface roughness generation. Volumetric MRR in ECH is summation of volumetric MRR due to ECD and mechanical honing, i.e. MRRECH = Ve + Vh
Fig. 1. Proposed working principle for finishing of bevel gears by ECH: (a) arrangement of the different gears and (b) photograph of the developed machining chamber based on this principle.
(1)
Following assumptions are made while developing the mathematical model:
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203
Fig. 2. Cyclic sequence of finishing the workpiece surface in ECH process showing the surface profiles of anode and cathode: (a) Initial, i.e. before ECH; (b) after 1st phase of electrolytic dissolution; (c) after 1st phase of honing action; (d) after 2nd phase of electrolytic dissolution; (e) after 2nd phase of honing action; (f) after 3rd phase of electrolytic dissolution; and (g) after 3rd phase of honing action.
1. Macroscopically, IEG remains constant during the ECH process even though microscopically the workpiece gear surface is smoothened due to finishing action by ECD and mechanical honing. 2. Conductivity of anodic and cathodic gear materials is very large as compared to that of electrolyte. 3. The conductivity of the electrolyte remains constant during the ECH process. 4. The process of material removal by honing action can be considered as the process of uniform wear of workpiece gear by the honing gear and can be modeled using Archard’s (1953) law of wear. 5. The line passing through the end points of the involute profile is parallel to the line tangent to the tooth profile at the pitch point.
3.1. Model of MRR due to ECD Based on Faraday’s laws of electrolysis, volumetric MRR in the ECD process Ve is given by (Jain, 2002):
Ve (mm3 /s) =
EJAs F
takes place through flow of current (mm2 ), and J is current density in this area (A/mm2 ) and is given by (Jain, 2002), J=
Ke (V − V ) Y
in which, Ke is the electrical conductivity of the electrolyte (−1 mm−1 ), Y is the inter-electrode gap (mm), V is the applied voltage and V is the total voltage loss in the IEG. Calculation of MRR due to ECD using Eq. (2) requires computation of the surface area of the workpiece gear As where the electrolytic dissolution takes place through flow of current. It is given by the following equation: As = T [2(Af + Ar ) + At + Ab ]
where is the current efficiency, E is the electrochemical equivalent of the workpiece material (g), F is Faraday’s constant (=96,500 C), is density of the anodic workpiece material (g/mm3 ); As is the surface area of the bevel gear tooth where the electrolytic dissolution
(4)
where T is the number of teeth of the workpiece gear, Af is the area of tooth flank (=Li × Fw ), Ar is the area of fillet surface at root (=Lr × Fw ), At is the area of top land (=Wt × Fw ), Ab is the area of bottom land (=Wb × Fw ), and Li is the length of involute profile; Fw is the face width of the bevel gear; Lr is the length of arc of fillet at the root; Wt is the width of top land and Wb is the width of bottom land. Using these relations the expression for As becomes As = TFw [2(Li + Lr ) + Wt + Wb ]
(2)
(3)
(5)
The values of Fw , Lr , Wt , and Wb can be calculated using the standard design data of the gear, but to the best knowledge of the authors, there is no geometric relation available to calculate the length of the involute profile. Therefore, a method to compute the length of involute profile Li is derived as given below, Li = r ×
Fig. 3. Involute profile: (a) construction and (b) measurement of its different parameters.
(6)
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where r is the radius of the involute arc (refer to Fig. 3(b)) and is the angle subtended by the involute at its centre. Referring to Ocb in Fig. 3(a) to calculate r, tan ˛ =
r yields bc = −→ r = rb tan ˛ Oc rb
(7)
where ˛ is the pressure angle and rb is radius of the base circle. Referring to the efg and ehg shown in Fig. 3(b) to calculate = \gef + \geh = arcsin
gf gh + arcsin ef eh
(8)
Using gf = gh = Lc /2; and ef = eh = r; from Fig. 3(b) gives = 2 arcsin
Lc 2r
(9)
In which, Lc is the length of chord of the involute arc given by
2 + Dw
Lc =
W − W 2 t
(10)
2
Here, Dw is the working depth, W is the width at the base of the tooth and Wt is the width of top land. Using Eq. (7) and Eq. (10) in Eq. (9) gives
= 2 arcsin
2 + (W − W )2 4Dw t 4rb tan ˛
(11)
Using Eq. (7) and Eq. (11) in Eq. (6) to get the expression for the length of involute profile Li
Li = 2rb arcsin
2 4Dw
+ (W − Wt ) 4rb tan ˛
2
tan ˛
(12)
As (mm2 ) = TFw
4rb
arcsin
2 4Dw
where Ns is the number of revolutions per second of the workpiece gear and P is the perimeter of the workpiece gear along the pitch circle at the middle of the face width and is given by,
P = 2T (Li + Lr )
W + W t b
+ (W − Wt )
2
4rb tan ˛
tan ˛ + 2Lr + Wt + Wb
(13)
Using Eq. (17) in Eq. (16) gives
Using Eq. (13) and Eq. (3) in Eq. (2) gives final expression for Ve Ve (mm /s) =
ETF K (V − V ) w e
4rb arcsin
F
Y
2 + (W − W )2 4Dw t 4rb tan ˛
Vh (mm3 /s) =
(KFn Ns T )(Li + Lr )(Wt + Wb ) H
(19)
Using Eq. (14) for Ve and Eq. (19) for Vh in Eq. (1) gives the expression for total MRR in ECH process, i.e.
4rb arcsin
+
tan ˛ + 2Lr + Wt + Wb
(18)
Therefore, volumetric MRR due to honing is given by,
MRRECH (mm3 /s) =
3
(17)
2
S = Ns T [(Li + Lr )(Wt + Wb )]
Substituting for Li from Eq. (12) in Eq. (5) gives final expression for As
Fig. 4. Microscopic view of finishing of surface irregularities on workpiece gear tooth surface along its profile by the ECD process.
ETF K (V − V ) w e F
2 + (W − W )2 4Dw t 4rb tan ˛
(KFn Ns T )(Li + Lr )(Wt + Wb ) H
Y
tan ˛ + 2Lr + Wt + Wb
(20)
(14) 4. Modeling of surface roughness 3.2. Model of MRR due to mechanical honing
Following assumptions were made in modeling the surface roughness produced after ECH of gears:
Using Archard’s (1953) law of wear to calculate the volumetric MRR by the mechanical honing action, volume of the material removed due to wear ‘V’ is given by, Vh (mm3 /s) =
KFn S H
(15)
Here, K is wear coefficient of the workpiece material, Fn is total normal load along the line of action (N), S is total sliding distance (mm); and H is Brinell hardness number of the workpiece material (N/mm2 ). In this expression, the values of K and H can be found from the standard design data for the given workpiece material, value of Fn can be calculated from the nomograph using the gear specifications while, the total sliding distance is given by, S = P × Ns
(16)
1. The actual non-uniform distribution of workpiece surface irregularities are approximated by the uniformly distributed surface irregularities having identical triangular peaks and valleys of the same height as considered by Jain et al. (2006). For such uniform distribution, the ‘Rz ’ (as shown in Fig. 4) becomes the distance between any peak and its adjacent valley on the workpiece surface. 2. Each cycle of ECD and mechanical honing removes workpiece material by an equal amount of ‘he ’ and ‘hh ’ respectively, i.e. the volumetric MRR due to ECD (Ve ) and mechanical honing (Vh ) is same for each cycle of gear finishing by ECH. After each cycle of finishing by ECD and mechanical honing, the value of Rz is recalculated with respect to the new peaks and valleys.
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Fig. 4 shows the microscopic view of finishing of surface irregularities on workpiece gear tooth surface along its profile by the ECD process. As the cathode gear surface scans over the anodic workpiece gear surface, the ECD removes relatively more material from the peaks (having less IEG) as compared to that from the valleys (having more IEG). The shaded portion represents the workpiece material removed in one cycle of ECD clearly depicting how ECD improves the surface finish of the workpiece gear. Ning et al. (2011) calculated the maximum surface roughness in PECF considering the surface irregularities as rectangular peaks and valleys. This concept along with the effect of finishing by mechanical honing has been used to derive the expression for the Rz after ECH. A factor ‘f’ has been used to convert the height of rectangle into height of a triangle with same area and the base length. Based
on this, the expression for the Rz after first cycle of ECD RzECD1 is given by
where ‘hh ’ is the total thickness of material removed from the workpiece surface in one cycle of mechanical honing and is given by,
hh (m) =
10−3 fVh As Ns
(27)
Using RzECD1 = hv − hp as shown in Fig. 4, and substituting for hh from Eq. (27) in Eq. (26) gives, RzECH1 (m) = RzECD1 + (2k − 1)
RzECD1 (m) = [khe + hv ] − [(1 − k)he + hp ]
RzECH1 (m) = Rzi + (2k − 1)
RzECD1 (m) = hv − hp + (2k − 1)he
(21)
(22)
where k is the proportion of the material removed from the valleys out of the total material thickness ‘he ’ (or ‘hh ’) removed by ECD (or honing) in its one cycle. The total thickness of workpiece material removed by ECD in one cycle ‘he ’ is given by, 10−3 fVe (m) = As Ns
(29)
10−3 fVe (m) = Rzi + (2k − 1) As Ns
During the finishing time ‘t’ seconds and with workpiece gear making Ns revolutions per second, the total number of finishing cycles by ECH will be Ns t. Therefore, multiplying the second term in Eq. (29) by Ns t, using Eq. (1), and substituting for As from Eq. (13) to get the expression for RzECH , i.e. the Rz value obtained after finishing for time ‘t’ by ECH.
RzECH (m) = Rzi + TFw
4rb
arcsin
10−3 ft(2k − 1)MRRECH
2 +(W −W )2 4Dw t
4rb tan ˛
tan ˛ + 2Lr + Wt + Wb
(30)
(23)
Using Rzi = hv − hp as shown in Fig. 4, and substituting for he from Eq. (23) in Eq. (22) gives, (24)
where Rzi is the value of Rz of the unfinished workpiece surface. In ECH process, the role of honing is to remove the passivating layer formed on the surface irregularities finished by ECD, i.e. surface finished by ECD is the initial surface for the subsequent finishing by the mechanical honing process therefore, the surface finished by honing is the surface finished by the ECH. Consequently, the expression for Rz of workpiece surface after first cycle of ECH (RzECH1 ) is given by
RzECH1 =
10−3 f (Ve + Vh ) As Ns
Rearranging the terms gives,
RzECD1
(28)
Thickness of material removed from valley of the workpiece surface in one cycle of ECD − +Distance between center line of the cathode surface roughness and the valley on the workpiece surface before ECD Thickness of material removed from peak of the workpiece surface in one cycle of ECD +Distance between center line of the cathode surface roughness and the peaks on the workpiece surface before ECD
Using the terms as shown in Fig. 4,
he
10−3 fVh As Ns
Using Eq. (24) in Eq. (28) gives final expression for Rz of the workpiece surface obtained after first cycle of finishing by ECH process,
RzECD1 =
205
5. Experimental validation and discussion The developed models for MRR (Eq. (20)) and surface roughness parameter ‘Rz ’ (Eq. (30)) were validated experimentally using an innovatively developed experimental setup for finishing of bevel gears by ECH. Fig. 5(a) and (b) depicts the schematic diagram and the photograph of the experimental setup respectively. This setup has four subsystems namely (i) power supply system; (ii) electrolyte supply, cleaning and recirculating system; (iii) machining chamber housing workpiece, cathode and honing gears; and (iv) a machine frame (drilling machine of 38-mm drilling capacity) to support the machining chamber and to provide motion to the workpiece gear. Shaikh and Jain (2013) have discussed this experimental
Thickness of material removed from valley of the workpiece surface in one cycle of honing − +Distance between center line of the cathode surface roughness and the valley on the workpiece surface after ECD Thickness of material removed from peak of the workpiece surface in one cycle of honing +Distance between center line of the cathode surface roughness and the peaks on the workpiece surface after ECD
Using the terms as shown in Fig. 4, RzECH1 (m) = [khh + hv ]–[(1 − k)hh + hp ]
(25)
Rearranging the terms gives, RzECH1 (m) = hv − hp + (2k − 1)hh
(26)
setup in detail. An aqueous solution containing 25% NaCl and 75% NaNO3 (by weight) was used as the electrolyte. Three levels of five ECH parameters namely voltage, rotary speed of workpiece gear, electrolyte concentration, electrolyte temperature and electrolyte flow rate (presented in Table 1) were used in the experiments which were planned using L27 orthogonal array based on Taguchi method of design of experiments. These levels were selected on the basis of
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Fig. 6. Comparison between theoretical and experimental values of (a) material removal rate (MRR) and (b) Rz after ECH ‘RzECH ’ using k = 0.4 for the different experiments. Fig. 5. Experimental setup used in validation of the MRR and surface roughness models for the finishing of the bevel gears by ECH: (a) schematic diagram and (b) photograph.
the pilot experiments and literature review. Shaikh and Jain (2013) found from the pilot experiments for finishing of bevel gears using ECH that, finishing for two minutes duration yielded the best quality finished gears therefore finishing was done for two minutes. Surface roughness for the unfinished and the same gear finished by ECH were measured on one of the gear tooth flanks along its profile and perpendicular to the lay direction using contracer-cumsurface roughness tester of KOSAKA, Japan make. Volumetric MRR was calculated by dividing the weight loss of the workpiece gear by finishing time and density of the workpiece material. The weight of the workpiece gear was measured on a precision weighting balance machine (make Essae-Teraoka Ltd.) having at least count of 0.01 g. Table 2 mentions the values of the different parameters used in calculating the theoretical values of MRR [Eq. (20)] and Rz [Eq. (30)]. The electrochemical equivalent for the workpiece material was calculated using the percentage by weight method (Jain, 2002). Theoretical values of MRR and Rz were computed for different value of the factor k (which indicated the proportion of the material removed from the valleys out of the total material removed from the workpiece surface) in the range of 0–0.5. Value of k = 0.4 yielded the minimum errors with respect to the experimental values. Table 3 presents the comparison of the theoretical and experimental values of MRR and Rz (for k = 0.4) along with the corresponding % error for the different combinations of
ECH parameters used in the experiments. The minimum error for MRR prediction was observed for experiment no. 3 while, for surface roughness it was observed for the experiment no. 20. The experiment no. 6 gave minimum prediction errors for both MRR and Rz values. Fig. 6(a) and (b) depicts the comparison between theoretical and experimental values of MRR and Rz respectively using k = 0.4 for the different experiments. It is clear from these figures while almost all the theoretical values of MRR are in close agreement with their corresponding experimental values but, some of the theoretical values of Rz differ from their corresponding experimental values. This may be due to some of the assumptions made such as values of electrolyte conductivity, IEG, and k factor not changing during the ECH process. Actually, these parameters might change slightly during the ECH process and may also vary with applied voltage and rotary speed and their interaction. It can be observed from the equations of the MRR (Eq. (20)) and Rz (Eq. (30)) that the applied voltage V and rotary speed of workpiece gear (in terms of revolutions per second ‘Ns ’) are two most important parameters affecting the MRRECH and RzECH of the bevel gears finished by ECH. Therefore, detailed study on the effects of these two parameters on MRR and surface finish of bevel gears was done. Fig. 7(a) and (b) show the effect of applied voltage and rotary speed of the workpiece gear on theoretical and experimental values of MRR and Rz respectively. It can be observed from Fig. 7(a) that (i) Both theoretical and experimental values of MRR increase continuously with the applied voltage for all rotary speeds due to increased material removal by
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207
Table 2 Values of the different parameters used in computation of the theoretical values of MRR and Rz for finishing of bevel gears by ECH process. Notation
Name (units)
Value
E f F H I K k Ke t V Y ˛ Dw Fn Fw Lr Ns rb T W Wt Wb
Electrochemical equivalent of the workpiece material (i.e. 20MnCr5 alloy steel) using percentage by weight (g) Factor to convert height of rectangle into height of a triangle with same area and the base length Faraday’s constant (C) Brinell hardness number (BHN) of the workpiece material (N/mm2 ) Amount of the current passed in the IEG (A) Wear coefficient Factor indicating proportion of total thickness of material removed from the valleys in one cycle of ECD and honing Electrical conductivity of the electrolyte (25% NaCl + 75% NaNO3 by wt.) at C = 7.5% and T = 27 ◦ C (−1 mm−1 ) Finishing time (s) Total voltage loss in the IEG (V) Inter-electrode gap (mm) Pressure angle of the involute profile (deg.) Current efficiency Density of the anodic workpiece material (g/mm3 ) Working depth of gear tooth (mm) Total normal load acting along the line of action (N) Facewidth of the bevel gear (mm) Length of arc of fillet at the root (mm) Number of revolutions of the workpiece gear per second Radius of the base circle (mm) Number of teeth of the workpiece gear Width at the base of the tooth (mm) Width of top land (mm) Width of bottom land (mm)
27.7661 2 96,500 523 7 1.2 × 10−4 0.4 0.085 120 0.6 1.00 22.5 0.95 0.00781 9.78 90.945 16 1.37 1 35.72 16 8.25 2.80 2.75
ECD; (ii) Experimental MRR increases with rotary speed except at low values of the applied voltage. This is due to more efficient scrapping of passivating layer at higher rotary speeds; and (iii) There is a slight difference between theoretical and experimental values of MRR for a particular value of voltage and rotary speed. This may be due to increase in the IEG and change in the electrolyte conductivity with the finishing time. From Fig. 7(b), it is evident that there exists an optimum value of the applied voltage (i.e. 12 V in the present case) for which theoretical and experimental values of Rz of the gears finished by ECH attains a minimum value for all the rotary speeds except lower rotary speed. This can be explained
by the facts that (i) at low values of applied voltage, the amount of material removed is very less due to truncation of the highest peaks only by ECD resulting in very less improvement in the surface finish, (ii) with increase in the voltage the MRR increases as more number of smaller peaks from the workpiece surface also gets truncated resulting in more improvement in the surface finish, and (iii) But, further increase in voltage though increases MRR due to material removal from peaks and valleys as well which results in poor surface finish. At lower rotary speed (40 rpm in the present case) the experimental values of Rz increases continuously with the applied voltage. This can be explained by the fact that at lower values of
Table 3 Comparison of the theoretical and experimental values of MRR and Rz using k = 0.4 for the different experiments. Exp. no.
Input parameters V
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
8V 12 V 16 V 8V 12 V 16 V 8V 12 V 16 V 8V 12 V 16 V 8V 12 V 16 V 8V 12 V 16 V 8V 12 V 16 V 8V 12 V 16 V 8V 12 V 16 V
R
40 rpm 60 rpm 80 rpm 60 rpm 80 rpm 40 rpm 80 rpm 40 rpm 60 rpm 80 rpm 40 rpm 60 rpm 40 rpm 60 rpm 80 rpm 60 rpm 80 rpm 40 rpm 60 rpm 80 rpm 40 rpm 80 rpm 40 rpm 60 rpm 40 rpm 60 rpm 80 rpm
Responses C
T
F
5% 5% 5% 5% 5% 5% 5% 5% 5% 7.5% 7.5% 7.5% 7.5% 7.5% 7.5% 7.5% 7.5% 7.5% 10% 10% 10% 10% 10% 10% 10% 10% 10%
27 ◦ C 27 ◦ C 27 ◦ C 32 ◦ C 32 ◦ C 32 ◦ C 37 ◦ C 37 ◦ C 37 ◦ C 27 ◦ C 27 ◦ C 27 ◦ C 32 ◦ C 32 ◦ C 32 ◦ C 37 ◦ C 37 ◦ C 37 ◦ C 27 ◦ C 27 ◦ C 27 ◦ C 32 ◦ C 32 ◦ C 32 ◦ C 37 ◦ C 37 ◦ C 37 ◦ C
10 lpm 20 lpm 30 lpm 20 lpm 30 lpm 10 lpm 30 lpm 10 lpm 20 lpm 20 lpm 30 lpm 10 lpm 30 lpm 10 lpm 20 lpm 10 lpm 20 lpm 30 lpm 30 lpm 10 lpm 20 lpm 10 lpm 20 lpm 30 lpm 20 lpm 30 lpm 10 lpm
MRR (mm3 /s) Theoretical
Experimental
0.15 0.26 0.36 0.19 0.33 0.36 0.22 0.32 0.47 0.22 0.39 0.50 0.22 0.36 0.64 0.22 0.43 0.63 0.22 0.36 0.60 0.22 0.36 0.68 0.29 0.50 0.61
0.20 0.27 0.36 0.37 0.34 0.37 0.28 0.39 0.38 0.29 0.28 0.61 0.21 0.40 0.56 0.24 0.50 0.54 0.31 0.49 0.55 0.30 0.50 0.66 0.30 0.54 0.77
% Error
Rzi (m)
−33.33 −3.84 0.00 −94.73 −3.03 −2.77 −27.27 −21.87 19.14 −31.81 28.20 −22.00 4.54 −11.11 12.5 −9.09 −16.27 14.28 −40.90 −36.11 8.33 −36.36 −38.88 2.94 −3.44 −8.00 −26.22
13.03 9.36 14.26 13.39 10.44 10.66 9.58 13.18 10.08 14.69 7.27 16.49 20.81 10.08 10.51 10.8 11.45 12.46 15.48 8.06 5.98 11.02 10.87 9.36 11.30 12.89 17.06
RzECH (m)
% Error
Theoretical
Experimental
12.92 9.23 14.12 13.3 10.32 10.39 9.5 12.93 9.84 14.61 6.97 16.23 20.65 9.9 10.26 10.69 11.28 11.97 15.37 7.92 5.51 10.94 10.59 9.01 11.08 12.63 16.82
5.18 7.2 7.34 14.4 11.66 10.15 9.07 8.71 13.54 14.18 8.35 6.77 7.2 10.3 11.02 11.66 7.49 19.66 13.32 7.85 7.27 7.13 9 6.55 12.38 6.34 8.35
59.91 21.99 48.02 −8.27 −12.98 2.31 4.53 32.64 −37.60 2.94 −19.8 58.29 65.13 −4.04 −7.41 −9.07 33.6 −64.24 13.33 0.88 −31.94 34.83 15.01 27.30 −11.73 49.80 50.36
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Fig. 8. Surface roughness profile of bevel gear tooth for the optimum values of applied voltage and rotary speed: (a) unfinished gear (Ra 1.79 m and Rmax 10.0 m) and (b) for the same gear finished by ECH (Ra 1.09 m and Rmax 8.42 m).
Fig. 7. Effect of applied voltage and rotary speed of the workpiece gear on theoretical and experimental values of (a) MRR and (b) Rz after ECH ‘RzECH ’ for k = 0.4.
the applied voltage thinner passivating layer is formed and honing action is capable to remove this layer. But, with increase in the voltage the thickness of the passivating layer increases and honing is not capable to remove this layer thus reducing the smoothening of surface irregularities by ECD giving increased value of Rz . Though, theoretical model could not simulate this realistic phenomenon occurring during gear finishing by ECH and hence shows the reverse trend. Consequently, 12 V and 60 rpm were identified as the optimum values of applied voltage and rotary speed respectively giving minimum value of Rz in the present case. Fig. 8(a) and (b) illustrates the surface roughness profiles of an unfinished gear and the same gear finished by ECH for the optimum values of applied voltage and rotary speed. It is clear from these figures that after finishing by ECH for 2 min, the maximum
Fig. 9. Bearing area curve of bevel gear tooth surface for the optimum values of the applied voltage and rotary speed: (a) unfinished gear and (b) same gear finished by ECH.
surface roughness Rmax value has improved significantly from 10.0 m to 8.42 m while, average surface roughness Ra value has improved from 1.79 m to 1.09 m. The improvement in average surface roughness results in reduction in wear and running noise of the gear, while the improvement in maximum surface roughness results in enhanced durability of the gears. Fig. 9(a) and (b) shows the bearing area curve (BAC) for a 0.5 m depth for the unfinished gear and the same gear finished by ECH. The improvement in material ratio in BAC after ECH results in larger contact area and thus minimizing running noise, vibration, and wear. Fig. 10(a) depicts the SEM micrograph of an unfinished gear in which micro-pits and micro-crack on the tooth flank surface are
Fig. 10. SEM micrographs (1000×) for the optimum values of the applied voltage and rotary speed: (a) unfinished gear and (b) gear finished by ECH.
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clearly visible which, may lead to the fatigue failure of the gears. These surface defects were smoothened while finishing the gears by the ECH process as shown by its SEM micrograph in Fig. 10(b). 6. Conclusions This paper reported about mathematical modeling of MRR and Rz of bevel gears finished by ECH, their experimental validation through 27 experiments at different combinations of five ECH parameters and conducted on an indigenously developed setup for finishing of bevel gears by ECH, and study on the effects of voltage and rotary speed on the MRR and final value of Rz . Following conclusions can be drawn from the present work: 1. Almost all the theoretical values of MRR are in close agreement with their corresponding experimental values but, some of the theoretical values of Rz differ from their corresponding experimental values. This may be due to assuming that electrolyte conductivity, IEG, and k factor are constant during the ECH process. 2. Feasible values of the k factor to ensure that surface finish of the gears finished by ECH will improve, should be in the range of 0–0.5. Higher the value of k (i.e. more material is removed from valleys) lesser will be the improvement in the surface finish. Value of k depends on the uniformity of initial surface roughness profile, applied voltage, electrolyte conductivity, and IEG. Moreover, the value of k may change during the ECH process. In the present work, value of k = 0.4 yielded the minimum errors between the theoretical and experimental values of MRR and Rz . 3. From the developed models of MRR and Rz it was found that the voltage and rotary speed of the workpiece gear are the important ECH parameters influencing the surface finish and MRR of the bevel gears finished by the ECH process. This has been confirmed by the experimental results also. 4. MRR is one of the significant responses in ECH, which determines productivity of the process but on the contrary increased MRR adversely affects the surface finish and form accuracy. 5. Maximization of MRR requires selection of maximum possible value of the applied voltage while, there exists an optimum value to minimize the final value of the surface roughness. Therefore, a trade-off might be required to optimize the values of the ECH process parameters so as to achieve the optimum results in terms of surface finish, form accuracy and productivity of the process. 6. In general, large difference between the theoretical and experimental values of MRR indicates excessive honing action. Though honing can correct the form errors and can generate cross-hatch lay pattern in spur and helical gear which is helpful in lubricating oil retention but, excessive honing action may deteriorate the final surface finish and form accuracy due to abrasion effect. Consequently, role of mechanical honing in finishing of gears by ECH should be restricted only to remove the passivating metal oxide layer formed by ECD on the workpiece gear surfaces and to remove the high spots formed due to the differences in dissolution potential of different alloying elements of the work material. The optimum amount of honing action can be achieved by careful selection of the honing parameters such as honing pressure, surface roughness and hardness of the honing gear.
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7. ECH has the potential to become a preferable alternative to the conventional gear finishing processes due to its distinct features such as independence from the gear material hardness, almost negligible tool wear, low cycle time, and ability to significantly improve the form accuracy and surface finish which enhances the performance and durability of the bevel gears. But, for using it commercially requires an exact amount of finishing stock to be incorporated during the gear manufacturing to achieve desired improvement in the form accuracy and surface finish. This requires a sound mathematical model, predicting the amount of material thickness required to be removed and consequently the finishing allowance on the gears in the prefinished condition. 8. Calculation of finishing stock is also useful for tooth profile modification such as profile crowning, tip relief and root relief. This can be achieved by appropriately modifying the profile of the cathode gear in the tool design stage. Acknowledgements The authors acknowledge (i) CSIR, New Delhi (India) for extending the financial support in the Project No. 22/(0468)/09/EMR-II, (ii) SnH Gears, Dewas, MP (India) for providing their facilities for fabrication of the bevel gears, and (iii) VE Commercial Vehicles, Pithampur, MP (India) for allowing to use their facilities for surface roughness measurements. References Archard, J.F., 1953. Contact and rubbing of flat surfaces. Journal of Applied Physics 24, 981–988. Benedict, G.F., 1987. Nontraditional Manufacturing Processes. Marcel Dekker, New York, pp. 147–151. Chernets, M.V., Bereza, V.V., 2009. Method for the evaluation of wear and durability of involute conical straight-bevel gears. Materials Science 45 (4), 595–604. Davis, J.R., 2005. Gear Materials, Properties and Manufacture. ASM International, Ohio, pp. 1–12. Jain, N.K., Naik, L.R., Dubey, A.K., Shan, H.S., 2009. State-of-art-review of electrochemical honing of internal cylinders and gears. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture 223 (6), 665–681. Jain, N.K., Jain, V.K., Jha, S., 2006. Parametric optimization of advanced fine finishing processes. International Journal of Advanced Manufacturing Technology 34 (11–12), 1191–1213, http://dx.doi.org/10.1007/s00170-006-0682-4. Jain, V.K., 2002. Advanced Machining Processes. Allied Publishers Pvt. Ltd., New Delhi, pp. 251–253. Karpuschewski, B., Knoche, H.-J., Hipke, M., 2008. Gear finishing by abrasive processes. CIRP Annals – Manufacturing Technology 57 (2), 621–640. Misra, J.P., Jain, P.K., Dwivedi, D.K., Mehta, N.K., 2013. Mixture D-Optimal design of electrolyte composition in ECH of bevel gears. Advanced Materials Research 685, 347–351. Ning, M., Wenji, X., Xuyue, W., Zefei, W., 2011. Mathematical modeling for finishing tooth surfaces of spiral bevel gears using pulse electrochemical dissolution. International Journal of Advanced Manufacturing Technology 54, 979–986. Park, D., Kahraman, A., 2009. A surface wear model for hypoid gear pairs. Wear 267, 1595–1604. Pavlov, V.G., 2011. Computational estimation of wear, service life, and efficiency of the straight orthogonal bevel gear. Journal of Machinery Manufacture and Reliability 40 (5), 443–449. Ruszaj, A., Zybura-Skrabalak, M., 2001. The mathematical modeling of electrochemical machining with flat ended universal electrodes. Journal of Materials Processing Technology 109, 333–338. Shaikh, J.H., Jain, N.K., 2013. Enhancement of geometric accuracy and surface finish of bevel gears by electrochemical honing process. International Journal of Advanced Manufacturing Technology (submitted for publication). Wei, G., Wu, M., Chen, C., 1986. An investigation into the ability of correcting error in ECH. CIRP Annals – Manufacturing Technology 35 (1), 125–127.
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Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec
Modeling of material removal rate and surface roughness in finishing of bevel gears by electrochemical honing process Javed Habib Shaikh, Neelesh Kumar Jain ∗ Discipline of Mechanical Engineering, Indian Institute of Technology Indore (MP), India
a r t i c l e
i n f o
Article history: Received 22 March 2013 Received in revised form 26 July 2013 Accepted 20 August 2013 Available online 7 September 2013 Keywords: Modeling Material removal rate Surface roughness Electrochemical dissolution Electrochemical honing Bevel gears
a b s t r a c t This paper describes mathematical modeling of material removal rate (MRR) and surface roughness of the bevel gears finished by the electro-chemical honing (ECH) process. Since, ECH hybridizes electrochemical dissolution (ECD) and mechanical honing therefore, contribution of ECD in MRR and surface roughness has been modeled using Faraday’s law of electrolysis while contribution of mechanical honing has been modeled considering material removal as a phenomenon of uniform wear and using Archard’s wear model. Formulations are also proposed for computing the surface area, required by these two models, along the inter-electrode gap (IEG) based on the geometry of the straight bevel gear tooth surfaces. The developed models were experimentally validated using an indigenously developed experimental setup for finishing of bevel gears by ECH based on an envisaged novel concept of twin complementary cathode gears. An aqueous solution containing 25% NaCl + 75% NaNO3 was used as the electrolyte. The predicted values of MRR and surface roughness have shown close agreement with the experimental values. The experimental results, SEM images and bearing area curve have shown appreciable improvement in the surface roughness and surface integrity ensuring better operating performance of the gears finished by ECH within an optimized finishing time of 2 min. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Gear is one of the essential machine elements used to transmit motion and power by the successive engagements of teeth on their periphery. Application areas for gears are diverse and include machine tools, automotives, aerospace, marine, oil and gas industry equipments, cement and other processing mills, steel processing units, etc. (Davis, 2005). The operating performance such as power transmission efficiency, noise, vibration and durability of the gears, depends on its surface finish and quality. To prevent failure of gears requires careful consideration of gear form accuracy, gear materials, gear tooth forces, lubrication, operating conditions, and surface quality which in turn depend on the gear finishing method. Form errors and surface roughness of gears can be reduced significantly by finishing processes (Jain et al., 2009). Generally used gear finishing processes for bevel gears namely gear grinding and gear lapping, have some limitations such as gear grinding is expensive, complicated and may produce undesirable effects such as grinding burn. Grinding burn damages the surface integrity of the ground workpiece and if it is not detected then loaded gear might fail with severe consequences like tooth breakage. Lapping is the oldest but
∗ Corresponding author. Tel.: +91 732 4240 702; fax: +91 732 4240 700. E-mail addresses: [email protected] (J.H. Shaikh), [email protected], [email protected] (N.K. Jain). 0924-0136/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmatprotec.2013.08.010
still widely used process to finish the surface of the bevel gear tooth flanks and thus reducing the operating noise. It is a time consuming process and can correct only minor errors of the gear tooth profile. If lapping is used for longer duration, it may affect form of the gear teeth profile (Karpuschewski et al., 2008). Electrochemical honing (ECH) hybridizes capabilities of electrochemical dissolution (ECD) and mechanical honing in one operation and is a potential promising alternative to the conventional gear finishing processes. Wei et al. (1986) mentioned that ECH has ability to improve surface finish as well as to correct form errors, smoothening of irregularities on the surface and being productive at the same. ECH can achieve average surface roughness values up to 0.05 m and dimensional tolerances up to ±0.002 m (Benedict, 1987). It offers some unique features such as workpiece material of any hardness can be processed and less cycle time as compared to the existing processes. Misra et al. (2013) studied the effect of electrolyte composition on percentage improvement in bearing ratio by experimental runs conducted using the mixture D-Optimal design approach and found an aqueous solution containing 70% NaCl and 30% NaNO3 as an optimum electrolyte composition. Very few references are available on modeling of material removal rate (MRR) and surface roughness in the finishing of gears by the ECD based processes. Ning et al. (2011) developed a mathematical model for total thickness of the material removed and surface roughness in pulse electrochemical finishing (PECF) of spiral bevel gears and experimentally validated their models. But, they
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Nomenclature Ab Af Ar As
At Dw E f
F Fn Fw H he hh hp
hv
hp hv
I J K k
Ke Li Lc Lr MRRECH Ns P r Rzi
RzECH RzECD1
RzECH1
rb
Area of the bottom land (mm2 ) Area of the tooth flank of the gear (mm2 ) Area of the fillet surface at root of the gear (mm2 ) Surface area of the bevel gear tooth where the electrolytic dissolution takes place through flow of current (mm2 ) Area of the top land (mm2 ) Working depth of gear tooth (mm) Electrochemical equivalent of the workpiece material (g) Factor used to convert the height of rectangle into height of a triangle with same area and the base length (=2) Faraday’s constant (=96,500 C) Total normal load acting along the line of action (N) Facewidth of the bevel gear (mm) Brinell hardness number (BHN) of the workpiece material (N/mm2 ) Total thickness of material removed from the workpiece surface in one cycle of ECD (m) Total thickness of material removed from the workpiece surface in one cycle of honing (m) Distance between center line of the cathode surface roughness and the peaks on the workpiece surface before ECD (m) Distance between center line of the cathode surface roughness and the valleys on the workpiece surface before ECD (m) Distance between center line of the cathode surface roughness and the peaks on the workpiece surface after ECD (m) Distance between center line of the cathode surface roughness and the valleys on the workpiece surface after ECD (m) Amount of the current passed in the IEG (A) Current density in the IEG (A/mm2 ) Wear coefficient Factor that indicates proportion of total thickness of material removed from the valleys in one cycle of ECD and honing Electrical conductivity of the electrolyte (−1 mm−1 ) Length of the involute profile (mm) Length of chord of the involute (mm) Length of arc of fillet at the root (mm) Material removal rate in ECH (mm3 /s) Number of revolutions of the workpiece gear per second (rps) Perimeter of the workpiece gear (mm) Radius of the involute arc (mm) Arithmetic mean of maximum peak-to-valley heights for the UNFINISHED workpiece surface (m) Arithmetic mean of maximum peak-to-valley heights for the ECHed workpiece surface (m) Arithmetic mean of maximum peak-to-valley heights for the workpiece surface AFTER one cycle of ECD (m) Arithmetic mean of maximum peak-to-valley heights for the workpiece surface AFTER one cycle of ECH (m) Radius of the base circle (mm)
S t T V Ve Vh W Wt Wb Y ˛ V
201
Total sliding distance (mm) Finishing time (s) Number of teeth of the workpiece gear Applied voltage (V) Volumetric material removal rate (MRR) due to ECD (mm3 /s) Volumetric material removal rate (MRR) due to honing action (mm3 /s) Width at the base of the tooth (mm) Width of top land (mm) Width of bottom land (mm) Inter-electrode gap (mm) Pressure angle of the involute profile Current efficiency Density of the anodic work material (g/mm3 ) Angle subtended by the involute at its centre (deg.) Total voltage loss in the IEG (V)
finished one tooth at a time and reported improvement in the surface roughness and form errors. Ruszaj and Zybura-Skrabalak (2001) developed a mathematical model for MRR in the ECD with flat ended electrode and compared the theoretical results with the actual results. Park and Kahraman (2009) combined Archard’s wear model with finite-element based contact model for simulation of surface wear of face-milled or face-hobbed hypoid gear pairs. Pavlov (2011) proposed computational methods for estimation of the wear, service life, and efficiency of the straight bevel gears having involute profile while Chernets and Bereza (2009) investigated on kinetics of wear of gear tooth and proposed a method for wear and durability of straight bevel gears. From the review of the past work it is evident that no work has been reported on modeling of MRR and surface roughness of bevel gears finished by the ECH process. The objective of the present work is to bridge this gap. This paper describes development of the models of MRR and arithmetic mean of maximum peak-tovalley heights (Rz ) for the flank surface of bevel gears finished by the ECH process and their experimental validation. These models have been developed as function of rotary speed of the workpiece gear and current in the inter-electrode gap (IEG), which indirectly takes into account the effects of other electrolytic parameters such as concentration, temperature and flow rate. Since, ECH hybridizes ECD and mechanical honing therefore, contribution of ECD in MRR and surface roughness has been modeled using Faraday’s law of electrolysis while, contribution of mechanical honing has been modeled considering material removal by honing as a process of uniform wear and using wear model proposed by Archard (1953). Formulations are also proposed to compute the surface area along the inter-electrode gap (IEG) based on the geometry of the gear tooth surfaces of the straight bevel gears as required by these models. Comparison of the theoretical values with the experimental results has been done. These models and their experimental validation optimized the applied voltage and rotary speed which helps in performance optimization of finishing of gears by ECH. 2. Finishing of bevel gears by ECH 2.1. Working principle Bevel gear finishing by ECH is much more difficult than finishing of cylindrical gears (i.e. spur or helical gears) due to their complex geometry which restricts feeding motion of the workpiece gear required for finishing its entire face width. This problem was resolved by envisaging a novel concept of twin complementary
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Table 1 Details of various ECH parameters used in the experimental validation. Composition (by wt.%) of the workpiece material (i.e. 20MnCr5 alloy steel): Cr (0.8–1.1); Mn (1–1.3); C (0.14–0.19); P (0.035 max.); S (0.035 max.); Si (0.15–0.40); and balance Fe Fixed input parameters
Variable input parameters
1. Electrolyte composition (C): 75% NaNO3 + 25% NaCl 2. Finishing time (t): 2 min 3. Inter electrode gap: 1 mm
1. Voltage (V): 3 levels (8 V; 12 V; 16 V) 2. Rotary speed of workpiece gear (R): 3 levels (40 rpm; 60 rpm; 80 rpm) 3. Electrolyte concentration (C): 3 levels (5%; 7.5%; 10%) (by wt.) 4. Electrolyte temperature (T): 3 levels (27 ◦ C; 32 ◦ C; 37 ◦ C) 5. Electrolyte flow rate (F): 3 levels (10 lpm; 20 lpm; 30 lpm)
cathodic bevel gears ‘3’ and ‘4’ meshing with the anodic workpiece bevel gear ‘1’ as shown in Fig. 1(a). The cathode gear ‘3’ has an insulating layer of Metalon sandwiched between two undercut (by 1 mm) conducting layers of copper while, the other complimentary cathode gear ‘4’ has an undercut (by 1 mm) layer of copper sandwiched between two insulating layers of Metalon. This arrangement ensures finishing of the entire face width of the workpiece gear by the ECD action without its reciprocation and at the same time IEG avoids the short circuiting between the cathodic and anodic gears. A DC voltage is applied across electrodes. The IEG is flooded with a full stream of electrolyte ‘5’ so that cathodic gears can finish the workpiece gear by electrolytic dissolution. The electrolysis action forms a passivating layer of metal oxide on the tooth surfaces of the workpiece gear due to evolution of oxygen. It prevents further material removal by ECD process. This passivating layer is scraped by the honing action between the workpiece gear and a honing gear ‘2’ which is mounted perpendicular to the workpiece and cathode gears. This honing gear is in tight mesh as compared to the cathode gears with the workpiece gear so as to remove the passivating oxide layer. This cyclic sequence of ECD and honing action improves the form accuracy and surface finish of the workpiece gear. The workpiece, cathode and honing gears were cut on bevel
gear generator based on the Gleason method in a gear manufacturing industry. Both honing gear and workpiece gears were made of case hardened 20MnCr5 alloy steel (composition shown in Table 1) having similar surface finish but, having different surface hardness values. The workpiece gears had hardness in the range of 50–54 HRC while, the honing gear had hardness in the range of 58–62 HRC. This difference in the hardness values was found to be sufficient to remove the passivating metal oxide layer and no wear of the honing gear was observed during the finishing time used in the present work. Fig. 1(b) shows the photograph of the machining chamber for finishing the bevel gears by ECH based on this principle. 2.2. Process mechanism Mechanism for finishing of bevel gears by ECH is proposed in which cyclic sequence of finishing by electrolytic dissolution and mechanical honing improves the geometric accuracy and surface finish of the workpiece gear. Fig. 2(a) shows the initial surface profiles of the anode and cathode. It can be seen that the distance between peaks on the workpiece and cathode surfaces is less as compared to the corresponding distance between valleys. Consequently, electrolytic dissolution will remove more material from the peaks of workpiece surface as compared to its valleys thus truncating the highest peaks and giving rapid improvement in the workpiece surface finish. At the end of electrolytic dissolution process, a passivating metal oxide layer is formed on the workpiece surface as shown in Fig. 2(b), (d) and (f) which is removed by the honing action so that further electrolytic dissolution can continue. At the end of honing action some valleys still might be covered with the passivating layer as shown in Fig. 2(c), (e) and (g). In the next cycle of electrolytic dissolution, these peaks will be exposed for further smoothening. Consequently, these peaks are truncated as well as some material is also removed from valleys giving a smoother surface as shown in Fig. 2(g). As the ECH process continues, the improvement in the surface finish and geometry accuracy continues at a diminishing rate due to increasing IEG between the surfaces of cathode and workpiece gears and consequent decreasing MRR. As shown in Fig. 2(g), the roughness which remains after finishing will take very long time to be completely removed by the ECH. Therefore, whenever the desired surface finish or geometric accuracy or both are achieved the process can be stopped. 3. Modeling of material removal rate (MRR) in ECH of gears ECH is a hybrid finishing process of ECD and mechanical honing therefore these processes contribute in the process of material removal and surface roughness generation. Volumetric MRR in ECH is summation of volumetric MRR due to ECD and mechanical honing, i.e. MRRECH = Ve + Vh
Fig. 1. Proposed working principle for finishing of bevel gears by ECH: (a) arrangement of the different gears and (b) photograph of the developed machining chamber based on this principle.
(1)
Following assumptions are made while developing the mathematical model:
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Fig. 2. Cyclic sequence of finishing the workpiece surface in ECH process showing the surface profiles of anode and cathode: (a) Initial, i.e. before ECH; (b) after 1st phase of electrolytic dissolution; (c) after 1st phase of honing action; (d) after 2nd phase of electrolytic dissolution; (e) after 2nd phase of honing action; (f) after 3rd phase of electrolytic dissolution; and (g) after 3rd phase of honing action.
1. Macroscopically, IEG remains constant during the ECH process even though microscopically the workpiece gear surface is smoothened due to finishing action by ECD and mechanical honing. 2. Conductivity of anodic and cathodic gear materials is very large as compared to that of electrolyte. 3. The conductivity of the electrolyte remains constant during the ECH process. 4. The process of material removal by honing action can be considered as the process of uniform wear of workpiece gear by the honing gear and can be modeled using Archard’s (1953) law of wear. 5. The line passing through the end points of the involute profile is parallel to the line tangent to the tooth profile at the pitch point.
3.1. Model of MRR due to ECD Based on Faraday’s laws of electrolysis, volumetric MRR in the ECD process Ve is given by (Jain, 2002):
Ve (mm3 /s) =
EJAs F
takes place through flow of current (mm2 ), and J is current density in this area (A/mm2 ) and is given by (Jain, 2002), J=
Ke (V − V ) Y
in which, Ke is the electrical conductivity of the electrolyte (−1 mm−1 ), Y is the inter-electrode gap (mm), V is the applied voltage and V is the total voltage loss in the IEG. Calculation of MRR due to ECD using Eq. (2) requires computation of the surface area of the workpiece gear As where the electrolytic dissolution takes place through flow of current. It is given by the following equation: As = T [2(Af + Ar ) + At + Ab ]
where is the current efficiency, E is the electrochemical equivalent of the workpiece material (g), F is Faraday’s constant (=96,500 C), is density of the anodic workpiece material (g/mm3 ); As is the surface area of the bevel gear tooth where the electrolytic dissolution
(4)
where T is the number of teeth of the workpiece gear, Af is the area of tooth flank (=Li × Fw ), Ar is the area of fillet surface at root (=Lr × Fw ), At is the area of top land (=Wt × Fw ), Ab is the area of bottom land (=Wb × Fw ), and Li is the length of involute profile; Fw is the face width of the bevel gear; Lr is the length of arc of fillet at the root; Wt is the width of top land and Wb is the width of bottom land. Using these relations the expression for As becomes As = TFw [2(Li + Lr ) + Wt + Wb ]
(2)
(3)
(5)
The values of Fw , Lr , Wt , and Wb can be calculated using the standard design data of the gear, but to the best knowledge of the authors, there is no geometric relation available to calculate the length of the involute profile. Therefore, a method to compute the length of involute profile Li is derived as given below, Li = r ×
Fig. 3. Involute profile: (a) construction and (b) measurement of its different parameters.
(6)
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where r is the radius of the involute arc (refer to Fig. 3(b)) and is the angle subtended by the involute at its centre. Referring to Ocb in Fig. 3(a) to calculate r, tan ˛ =
r yields bc = −→ r = rb tan ˛ Oc rb
(7)
where ˛ is the pressure angle and rb is radius of the base circle. Referring to the efg and ehg shown in Fig. 3(b) to calculate = \gef + \geh = arcsin
gf gh + arcsin ef eh
(8)
Using gf = gh = Lc /2; and ef = eh = r; from Fig. 3(b) gives = 2 arcsin
Lc 2r
(9)
In which, Lc is the length of chord of the involute arc given by
2 + Dw
Lc =
W − W 2 t
(10)
2
Here, Dw is the working depth, W is the width at the base of the tooth and Wt is the width of top land. Using Eq. (7) and Eq. (10) in Eq. (9) gives
= 2 arcsin
2 + (W − W )2 4Dw t 4rb tan ˛
(11)
Using Eq. (7) and Eq. (11) in Eq. (6) to get the expression for the length of involute profile Li
Li = 2rb arcsin
2 4Dw
+ (W − Wt ) 4rb tan ˛
2
tan ˛
(12)
As (mm2 ) = TFw
4rb
arcsin
2 4Dw
where Ns is the number of revolutions per second of the workpiece gear and P is the perimeter of the workpiece gear along the pitch circle at the middle of the face width and is given by,
P = 2T (Li + Lr )
W + W t b
+ (W − Wt )
2
4rb tan ˛
tan ˛ + 2Lr + Wt + Wb
(13)
Using Eq. (17) in Eq. (16) gives
Using Eq. (13) and Eq. (3) in Eq. (2) gives final expression for Ve Ve (mm /s) =
ETF K (V − V ) w e
4rb arcsin
F
Y
2 + (W − W )2 4Dw t 4rb tan ˛
Vh (mm3 /s) =
(KFn Ns T )(Li + Lr )(Wt + Wb ) H
(19)
Using Eq. (14) for Ve and Eq. (19) for Vh in Eq. (1) gives the expression for total MRR in ECH process, i.e.
4rb arcsin
+
tan ˛ + 2Lr + Wt + Wb
(18)
Therefore, volumetric MRR due to honing is given by,
MRRECH (mm3 /s) =
3
(17)
2
S = Ns T [(Li + Lr )(Wt + Wb )]
Substituting for Li from Eq. (12) in Eq. (5) gives final expression for As
Fig. 4. Microscopic view of finishing of surface irregularities on workpiece gear tooth surface along its profile by the ECD process.
ETF K (V − V ) w e F
2 + (W − W )2 4Dw t 4rb tan ˛
(KFn Ns T )(Li + Lr )(Wt + Wb ) H
Y
tan ˛ + 2Lr + Wt + Wb
(20)
(14) 4. Modeling of surface roughness 3.2. Model of MRR due to mechanical honing
Following assumptions were made in modeling the surface roughness produced after ECH of gears:
Using Archard’s (1953) law of wear to calculate the volumetric MRR by the mechanical honing action, volume of the material removed due to wear ‘V’ is given by, Vh (mm3 /s) =
KFn S H
(15)
Here, K is wear coefficient of the workpiece material, Fn is total normal load along the line of action (N), S is total sliding distance (mm); and H is Brinell hardness number of the workpiece material (N/mm2 ). In this expression, the values of K and H can be found from the standard design data for the given workpiece material, value of Fn can be calculated from the nomograph using the gear specifications while, the total sliding distance is given by, S = P × Ns
(16)
1. The actual non-uniform distribution of workpiece surface irregularities are approximated by the uniformly distributed surface irregularities having identical triangular peaks and valleys of the same height as considered by Jain et al. (2006). For such uniform distribution, the ‘Rz ’ (as shown in Fig. 4) becomes the distance between any peak and its adjacent valley on the workpiece surface. 2. Each cycle of ECD and mechanical honing removes workpiece material by an equal amount of ‘he ’ and ‘hh ’ respectively, i.e. the volumetric MRR due to ECD (Ve ) and mechanical honing (Vh ) is same for each cycle of gear finishing by ECH. After each cycle of finishing by ECD and mechanical honing, the value of Rz is recalculated with respect to the new peaks and valleys.
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Fig. 4 shows the microscopic view of finishing of surface irregularities on workpiece gear tooth surface along its profile by the ECD process. As the cathode gear surface scans over the anodic workpiece gear surface, the ECD removes relatively more material from the peaks (having less IEG) as compared to that from the valleys (having more IEG). The shaded portion represents the workpiece material removed in one cycle of ECD clearly depicting how ECD improves the surface finish of the workpiece gear. Ning et al. (2011) calculated the maximum surface roughness in PECF considering the surface irregularities as rectangular peaks and valleys. This concept along with the effect of finishing by mechanical honing has been used to derive the expression for the Rz after ECH. A factor ‘f’ has been used to convert the height of rectangle into height of a triangle with same area and the base length. Based
on this, the expression for the Rz after first cycle of ECD RzECD1 is given by
where ‘hh ’ is the total thickness of material removed from the workpiece surface in one cycle of mechanical honing and is given by,
hh (m) =
10−3 fVh As Ns
(27)
Using RzECD1 = hv − hp as shown in Fig. 4, and substituting for hh from Eq. (27) in Eq. (26) gives, RzECH1 (m) = RzECD1 + (2k − 1)
RzECD1 (m) = [khe + hv ] − [(1 − k)he + hp ]
RzECH1 (m) = Rzi + (2k − 1)
RzECD1 (m) = hv − hp + (2k − 1)he
(21)
(22)
where k is the proportion of the material removed from the valleys out of the total material thickness ‘he ’ (or ‘hh ’) removed by ECD (or honing) in its one cycle. The total thickness of workpiece material removed by ECD in one cycle ‘he ’ is given by, 10−3 fVe (m) = As Ns
(29)
10−3 fVe (m) = Rzi + (2k − 1) As Ns
During the finishing time ‘t’ seconds and with workpiece gear making Ns revolutions per second, the total number of finishing cycles by ECH will be Ns t. Therefore, multiplying the second term in Eq. (29) by Ns t, using Eq. (1), and substituting for As from Eq. (13) to get the expression for RzECH , i.e. the Rz value obtained after finishing for time ‘t’ by ECH.
RzECH (m) = Rzi + TFw
4rb
arcsin
10−3 ft(2k − 1)MRRECH
2 +(W −W )2 4Dw t
4rb tan ˛
tan ˛ + 2Lr + Wt + Wb
(30)
(23)
Using Rzi = hv − hp as shown in Fig. 4, and substituting for he from Eq. (23) in Eq. (22) gives, (24)
where Rzi is the value of Rz of the unfinished workpiece surface. In ECH process, the role of honing is to remove the passivating layer formed on the surface irregularities finished by ECD, i.e. surface finished by ECD is the initial surface for the subsequent finishing by the mechanical honing process therefore, the surface finished by honing is the surface finished by the ECH. Consequently, the expression for Rz of workpiece surface after first cycle of ECH (RzECH1 ) is given by
RzECH1 =
10−3 f (Ve + Vh ) As Ns
Rearranging the terms gives,
RzECD1
(28)
Thickness of material removed from valley of the workpiece surface in one cycle of ECD − +Distance between center line of the cathode surface roughness and the valley on the workpiece surface before ECD Thickness of material removed from peak of the workpiece surface in one cycle of ECD +Distance between center line of the cathode surface roughness and the peaks on the workpiece surface before ECD
Using the terms as shown in Fig. 4,
he
10−3 fVh As Ns
Using Eq. (24) in Eq. (28) gives final expression for Rz of the workpiece surface obtained after first cycle of finishing by ECH process,
RzECD1 =
205
5. Experimental validation and discussion The developed models for MRR (Eq. (20)) and surface roughness parameter ‘Rz ’ (Eq. (30)) were validated experimentally using an innovatively developed experimental setup for finishing of bevel gears by ECH. Fig. 5(a) and (b) depicts the schematic diagram and the photograph of the experimental setup respectively. This setup has four subsystems namely (i) power supply system; (ii) electrolyte supply, cleaning and recirculating system; (iii) machining chamber housing workpiece, cathode and honing gears; and (iv) a machine frame (drilling machine of 38-mm drilling capacity) to support the machining chamber and to provide motion to the workpiece gear. Shaikh and Jain (2013) have discussed this experimental
Thickness of material removed from valley of the workpiece surface in one cycle of honing − +Distance between center line of the cathode surface roughness and the valley on the workpiece surface after ECD Thickness of material removed from peak of the workpiece surface in one cycle of honing +Distance between center line of the cathode surface roughness and the peaks on the workpiece surface after ECD
Using the terms as shown in Fig. 4, RzECH1 (m) = [khh + hv ]–[(1 − k)hh + hp ]
(25)
Rearranging the terms gives, RzECH1 (m) = hv − hp + (2k − 1)hh
(26)
setup in detail. An aqueous solution containing 25% NaCl and 75% NaNO3 (by weight) was used as the electrolyte. Three levels of five ECH parameters namely voltage, rotary speed of workpiece gear, electrolyte concentration, electrolyte temperature and electrolyte flow rate (presented in Table 1) were used in the experiments which were planned using L27 orthogonal array based on Taguchi method of design of experiments. These levels were selected on the basis of
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Fig. 6. Comparison between theoretical and experimental values of (a) material removal rate (MRR) and (b) Rz after ECH ‘RzECH ’ using k = 0.4 for the different experiments. Fig. 5. Experimental setup used in validation of the MRR and surface roughness models for the finishing of the bevel gears by ECH: (a) schematic diagram and (b) photograph.
the pilot experiments and literature review. Shaikh and Jain (2013) found from the pilot experiments for finishing of bevel gears using ECH that, finishing for two minutes duration yielded the best quality finished gears therefore finishing was done for two minutes. Surface roughness for the unfinished and the same gear finished by ECH were measured on one of the gear tooth flanks along its profile and perpendicular to the lay direction using contracer-cumsurface roughness tester of KOSAKA, Japan make. Volumetric MRR was calculated by dividing the weight loss of the workpiece gear by finishing time and density of the workpiece material. The weight of the workpiece gear was measured on a precision weighting balance machine (make Essae-Teraoka Ltd.) having at least count of 0.01 g. Table 2 mentions the values of the different parameters used in calculating the theoretical values of MRR [Eq. (20)] and Rz [Eq. (30)]. The electrochemical equivalent for the workpiece material was calculated using the percentage by weight method (Jain, 2002). Theoretical values of MRR and Rz were computed for different value of the factor k (which indicated the proportion of the material removed from the valleys out of the total material removed from the workpiece surface) in the range of 0–0.5. Value of k = 0.4 yielded the minimum errors with respect to the experimental values. Table 3 presents the comparison of the theoretical and experimental values of MRR and Rz (for k = 0.4) along with the corresponding % error for the different combinations of
ECH parameters used in the experiments. The minimum error for MRR prediction was observed for experiment no. 3 while, for surface roughness it was observed for the experiment no. 20. The experiment no. 6 gave minimum prediction errors for both MRR and Rz values. Fig. 6(a) and (b) depicts the comparison between theoretical and experimental values of MRR and Rz respectively using k = 0.4 for the different experiments. It is clear from these figures while almost all the theoretical values of MRR are in close agreement with their corresponding experimental values but, some of the theoretical values of Rz differ from their corresponding experimental values. This may be due to some of the assumptions made such as values of electrolyte conductivity, IEG, and k factor not changing during the ECH process. Actually, these parameters might change slightly during the ECH process and may also vary with applied voltage and rotary speed and their interaction. It can be observed from the equations of the MRR (Eq. (20)) and Rz (Eq. (30)) that the applied voltage V and rotary speed of workpiece gear (in terms of revolutions per second ‘Ns ’) are two most important parameters affecting the MRRECH and RzECH of the bevel gears finished by ECH. Therefore, detailed study on the effects of these two parameters on MRR and surface finish of bevel gears was done. Fig. 7(a) and (b) show the effect of applied voltage and rotary speed of the workpiece gear on theoretical and experimental values of MRR and Rz respectively. It can be observed from Fig. 7(a) that (i) Both theoretical and experimental values of MRR increase continuously with the applied voltage for all rotary speeds due to increased material removal by
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Table 2 Values of the different parameters used in computation of the theoretical values of MRR and Rz for finishing of bevel gears by ECH process. Notation
Name (units)
Value
E f F H I K k Ke t V Y ˛ Dw Fn Fw Lr Ns rb T W Wt Wb
Electrochemical equivalent of the workpiece material (i.e. 20MnCr5 alloy steel) using percentage by weight (g) Factor to convert height of rectangle into height of a triangle with same area and the base length Faraday’s constant (C) Brinell hardness number (BHN) of the workpiece material (N/mm2 ) Amount of the current passed in the IEG (A) Wear coefficient Factor indicating proportion of total thickness of material removed from the valleys in one cycle of ECD and honing Electrical conductivity of the electrolyte (25% NaCl + 75% NaNO3 by wt.) at C = 7.5% and T = 27 ◦ C (−1 mm−1 ) Finishing time (s) Total voltage loss in the IEG (V) Inter-electrode gap (mm) Pressure angle of the involute profile (deg.) Current efficiency Density of the anodic workpiece material (g/mm3 ) Working depth of gear tooth (mm) Total normal load acting along the line of action (N) Facewidth of the bevel gear (mm) Length of arc of fillet at the root (mm) Number of revolutions of the workpiece gear per second Radius of the base circle (mm) Number of teeth of the workpiece gear Width at the base of the tooth (mm) Width of top land (mm) Width of bottom land (mm)
27.7661 2 96,500 523 7 1.2 × 10−4 0.4 0.085 120 0.6 1.00 22.5 0.95 0.00781 9.78 90.945 16 1.37 1 35.72 16 8.25 2.80 2.75
ECD; (ii) Experimental MRR increases with rotary speed except at low values of the applied voltage. This is due to more efficient scrapping of passivating layer at higher rotary speeds; and (iii) There is a slight difference between theoretical and experimental values of MRR for a particular value of voltage and rotary speed. This may be due to increase in the IEG and change in the electrolyte conductivity with the finishing time. From Fig. 7(b), it is evident that there exists an optimum value of the applied voltage (i.e. 12 V in the present case) for which theoretical and experimental values of Rz of the gears finished by ECH attains a minimum value for all the rotary speeds except lower rotary speed. This can be explained
by the facts that (i) at low values of applied voltage, the amount of material removed is very less due to truncation of the highest peaks only by ECD resulting in very less improvement in the surface finish, (ii) with increase in the voltage the MRR increases as more number of smaller peaks from the workpiece surface also gets truncated resulting in more improvement in the surface finish, and (iii) But, further increase in voltage though increases MRR due to material removal from peaks and valleys as well which results in poor surface finish. At lower rotary speed (40 rpm in the present case) the experimental values of Rz increases continuously with the applied voltage. This can be explained by the fact that at lower values of
Table 3 Comparison of the theoretical and experimental values of MRR and Rz using k = 0.4 for the different experiments. Exp. no.
Input parameters V
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
8V 12 V 16 V 8V 12 V 16 V 8V 12 V 16 V 8V 12 V 16 V 8V 12 V 16 V 8V 12 V 16 V 8V 12 V 16 V 8V 12 V 16 V 8V 12 V 16 V
R
40 rpm 60 rpm 80 rpm 60 rpm 80 rpm 40 rpm 80 rpm 40 rpm 60 rpm 80 rpm 40 rpm 60 rpm 40 rpm 60 rpm 80 rpm 60 rpm 80 rpm 40 rpm 60 rpm 80 rpm 40 rpm 80 rpm 40 rpm 60 rpm 40 rpm 60 rpm 80 rpm
Responses C
T
F
5% 5% 5% 5% 5% 5% 5% 5% 5% 7.5% 7.5% 7.5% 7.5% 7.5% 7.5% 7.5% 7.5% 7.5% 10% 10% 10% 10% 10% 10% 10% 10% 10%
27 ◦ C 27 ◦ C 27 ◦ C 32 ◦ C 32 ◦ C 32 ◦ C 37 ◦ C 37 ◦ C 37 ◦ C 27 ◦ C 27 ◦ C 27 ◦ C 32 ◦ C 32 ◦ C 32 ◦ C 37 ◦ C 37 ◦ C 37 ◦ C 27 ◦ C 27 ◦ C 27 ◦ C 32 ◦ C 32 ◦ C 32 ◦ C 37 ◦ C 37 ◦ C 37 ◦ C
10 lpm 20 lpm 30 lpm 20 lpm 30 lpm 10 lpm 30 lpm 10 lpm 20 lpm 20 lpm 30 lpm 10 lpm 30 lpm 10 lpm 20 lpm 10 lpm 20 lpm 30 lpm 30 lpm 10 lpm 20 lpm 10 lpm 20 lpm 30 lpm 20 lpm 30 lpm 10 lpm
MRR (mm3 /s) Theoretical
Experimental
0.15 0.26 0.36 0.19 0.33 0.36 0.22 0.32 0.47 0.22 0.39 0.50 0.22 0.36 0.64 0.22 0.43 0.63 0.22 0.36 0.60 0.22 0.36 0.68 0.29 0.50 0.61
0.20 0.27 0.36 0.37 0.34 0.37 0.28 0.39 0.38 0.29 0.28 0.61 0.21 0.40 0.56 0.24 0.50 0.54 0.31 0.49 0.55 0.30 0.50 0.66 0.30 0.54 0.77
% Error
Rzi (m)
−33.33 −3.84 0.00 −94.73 −3.03 −2.77 −27.27 −21.87 19.14 −31.81 28.20 −22.00 4.54 −11.11 12.5 −9.09 −16.27 14.28 −40.90 −36.11 8.33 −36.36 −38.88 2.94 −3.44 −8.00 −26.22
13.03 9.36 14.26 13.39 10.44 10.66 9.58 13.18 10.08 14.69 7.27 16.49 20.81 10.08 10.51 10.8 11.45 12.46 15.48 8.06 5.98 11.02 10.87 9.36 11.30 12.89 17.06
RzECH (m)
% Error
Theoretical
Experimental
12.92 9.23 14.12 13.3 10.32 10.39 9.5 12.93 9.84 14.61 6.97 16.23 20.65 9.9 10.26 10.69 11.28 11.97 15.37 7.92 5.51 10.94 10.59 9.01 11.08 12.63 16.82
5.18 7.2 7.34 14.4 11.66 10.15 9.07 8.71 13.54 14.18 8.35 6.77 7.2 10.3 11.02 11.66 7.49 19.66 13.32 7.85 7.27 7.13 9 6.55 12.38 6.34 8.35
59.91 21.99 48.02 −8.27 −12.98 2.31 4.53 32.64 −37.60 2.94 −19.8 58.29 65.13 −4.04 −7.41 −9.07 33.6 −64.24 13.33 0.88 −31.94 34.83 15.01 27.30 −11.73 49.80 50.36
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Fig. 8. Surface roughness profile of bevel gear tooth for the optimum values of applied voltage and rotary speed: (a) unfinished gear (Ra 1.79 m and Rmax 10.0 m) and (b) for the same gear finished by ECH (Ra 1.09 m and Rmax 8.42 m).
Fig. 7. Effect of applied voltage and rotary speed of the workpiece gear on theoretical and experimental values of (a) MRR and (b) Rz after ECH ‘RzECH ’ for k = 0.4.
the applied voltage thinner passivating layer is formed and honing action is capable to remove this layer. But, with increase in the voltage the thickness of the passivating layer increases and honing is not capable to remove this layer thus reducing the smoothening of surface irregularities by ECD giving increased value of Rz . Though, theoretical model could not simulate this realistic phenomenon occurring during gear finishing by ECH and hence shows the reverse trend. Consequently, 12 V and 60 rpm were identified as the optimum values of applied voltage and rotary speed respectively giving minimum value of Rz in the present case. Fig. 8(a) and (b) illustrates the surface roughness profiles of an unfinished gear and the same gear finished by ECH for the optimum values of applied voltage and rotary speed. It is clear from these figures that after finishing by ECH for 2 min, the maximum
Fig. 9. Bearing area curve of bevel gear tooth surface for the optimum values of the applied voltage and rotary speed: (a) unfinished gear and (b) same gear finished by ECH.
surface roughness Rmax value has improved significantly from 10.0 m to 8.42 m while, average surface roughness Ra value has improved from 1.79 m to 1.09 m. The improvement in average surface roughness results in reduction in wear and running noise of the gear, while the improvement in maximum surface roughness results in enhanced durability of the gears. Fig. 9(a) and (b) shows the bearing area curve (BAC) for a 0.5 m depth for the unfinished gear and the same gear finished by ECH. The improvement in material ratio in BAC after ECH results in larger contact area and thus minimizing running noise, vibration, and wear. Fig. 10(a) depicts the SEM micrograph of an unfinished gear in which micro-pits and micro-crack on the tooth flank surface are
Fig. 10. SEM micrographs (1000×) for the optimum values of the applied voltage and rotary speed: (a) unfinished gear and (b) gear finished by ECH.
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clearly visible which, may lead to the fatigue failure of the gears. These surface defects were smoothened while finishing the gears by the ECH process as shown by its SEM micrograph in Fig. 10(b). 6. Conclusions This paper reported about mathematical modeling of MRR and Rz of bevel gears finished by ECH, their experimental validation through 27 experiments at different combinations of five ECH parameters and conducted on an indigenously developed setup for finishing of bevel gears by ECH, and study on the effects of voltage and rotary speed on the MRR and final value of Rz . Following conclusions can be drawn from the present work: 1. Almost all the theoretical values of MRR are in close agreement with their corresponding experimental values but, some of the theoretical values of Rz differ from their corresponding experimental values. This may be due to assuming that electrolyte conductivity, IEG, and k factor are constant during the ECH process. 2. Feasible values of the k factor to ensure that surface finish of the gears finished by ECH will improve, should be in the range of 0–0.5. Higher the value of k (i.e. more material is removed from valleys) lesser will be the improvement in the surface finish. Value of k depends on the uniformity of initial surface roughness profile, applied voltage, electrolyte conductivity, and IEG. Moreover, the value of k may change during the ECH process. In the present work, value of k = 0.4 yielded the minimum errors between the theoretical and experimental values of MRR and Rz . 3. From the developed models of MRR and Rz it was found that the voltage and rotary speed of the workpiece gear are the important ECH parameters influencing the surface finish and MRR of the bevel gears finished by the ECH process. This has been confirmed by the experimental results also. 4. MRR is one of the significant responses in ECH, which determines productivity of the process but on the contrary increased MRR adversely affects the surface finish and form accuracy. 5. Maximization of MRR requires selection of maximum possible value of the applied voltage while, there exists an optimum value to minimize the final value of the surface roughness. Therefore, a trade-off might be required to optimize the values of the ECH process parameters so as to achieve the optimum results in terms of surface finish, form accuracy and productivity of the process. 6. In general, large difference between the theoretical and experimental values of MRR indicates excessive honing action. Though honing can correct the form errors and can generate cross-hatch lay pattern in spur and helical gear which is helpful in lubricating oil retention but, excessive honing action may deteriorate the final surface finish and form accuracy due to abrasion effect. Consequently, role of mechanical honing in finishing of gears by ECH should be restricted only to remove the passivating metal oxide layer formed by ECD on the workpiece gear surfaces and to remove the high spots formed due to the differences in dissolution potential of different alloying elements of the work material. The optimum amount of honing action can be achieved by careful selection of the honing parameters such as honing pressure, surface roughness and hardness of the honing gear.
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7. ECH has the potential to become a preferable alternative to the conventional gear finishing processes due to its distinct features such as independence from the gear material hardness, almost negligible tool wear, low cycle time, and ability to significantly improve the form accuracy and surface finish which enhances the performance and durability of the bevel gears. But, for using it commercially requires an exact amount of finishing stock to be incorporated during the gear manufacturing to achieve desired improvement in the form accuracy and surface finish. This requires a sound mathematical model, predicting the amount of material thickness required to be removed and consequently the finishing allowance on the gears in the prefinished condition. 8. Calculation of finishing stock is also useful for tooth profile modification such as profile crowning, tip relief and root relief. This can be achieved by appropriately modifying the profile of the cathode gear in the tool design stage. Acknowledgements The authors acknowledge (i) CSIR, New Delhi (India) for extending the financial support in the Project No. 22/(0468)/09/EMR-II, (ii) SnH Gears, Dewas, MP (India) for providing their facilities for fabrication of the bevel gears, and (iii) VE Commercial Vehicles, Pithampur, MP (India) for allowing to use their facilities for surface roughness measurements. References Archard, J.F., 1953. Contact and rubbing of flat surfaces. Journal of Applied Physics 24, 981–988. Benedict, G.F., 1987. Nontraditional Manufacturing Processes. Marcel Dekker, New York, pp. 147–151. Chernets, M.V., Bereza, V.V., 2009. Method for the evaluation of wear and durability of involute conical straight-bevel gears. Materials Science 45 (4), 595–604. Davis, J.R., 2005. Gear Materials, Properties and Manufacture. ASM International, Ohio, pp. 1–12. Jain, N.K., Naik, L.R., Dubey, A.K., Shan, H.S., 2009. State-of-art-review of electrochemical honing of internal cylinders and gears. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture 223 (6), 665–681. Jain, N.K., Jain, V.K., Jha, S., 2006. Parametric optimization of advanced fine finishing processes. International Journal of Advanced Manufacturing Technology 34 (11–12), 1191–1213, http://dx.doi.org/10.1007/s00170-006-0682-4. Jain, V.K., 2002. Advanced Machining Processes. Allied Publishers Pvt. Ltd., New Delhi, pp. 251–253. Karpuschewski, B., Knoche, H.-J., Hipke, M., 2008. Gear finishing by abrasive processes. CIRP Annals – Manufacturing Technology 57 (2), 621–640. Misra, J.P., Jain, P.K., Dwivedi, D.K., Mehta, N.K., 2013. Mixture D-Optimal design of electrolyte composition in ECH of bevel gears. Advanced Materials Research 685, 347–351. Ning, M., Wenji, X., Xuyue, W., Zefei, W., 2011. Mathematical modeling for finishing tooth surfaces of spiral bevel gears using pulse electrochemical dissolution. International Journal of Advanced Manufacturing Technology 54, 979–986. Park, D., Kahraman, A., 2009. A surface wear model for hypoid gear pairs. Wear 267, 1595–1604. Pavlov, V.G., 2011. Computational estimation of wear, service life, and efficiency of the straight orthogonal bevel gear. Journal of Machinery Manufacture and Reliability 40 (5), 443–449. Ruszaj, A., Zybura-Skrabalak, M., 2001. The mathematical modeling of electrochemical machining with flat ended universal electrodes. Journal of Materials Processing Technology 109, 333–338. Shaikh, J.H., Jain, N.K., 2013. Enhancement of geometric accuracy and surface finish of bevel gears by electrochemical honing process. International Journal of Advanced Manufacturing Technology (submitted for publication). Wei, G., Wu, M., Chen, C., 1986. An investigation into the ability of correcting error in ECH. CIRP Annals – Manufacturing Technology 35 (1), 125–127.