Mathematical modelling and validation of twist-free lead crowning of spur gears by pulsed electrochemical flank modification process

Abstract

Spur gears are the most commonly used gears in various applications. Their lead crowning helps in mitigating the adverse effects of shaft misalignment and edge contact of their flank surfaces, and improving their operating performance. Conventional contact type processes induce twist-error during lead crowning of spur gears which adversely affects their operating performance and service life. A non-contact process can overcome these problems. Therefore, the present study is focused on development of mathematical model, and innovative cathode gear and experimental apparatus for imparting twist-free lead crowning to spur gears by a non-contact process referred to as pulsed electrochemical flank modification (PECFM) process. The cathode gear provided rotatory motion and spur gear was provided variable reciprocating velocity to ensure lead crowning of the entire flank surfaces of its all teeth. The mathematical model for twist-free lead crowning is developed by equating volumetric material removal based on the spur gear geometry with that given by Faraday’s law of electrolysis. The developed model is validated experimentally by varying applied voltage, pulse-on time, and pulse-off time at four levels each. It found the minimum prediction error for lead crowning model as 0.5%, and minimum and maximum values of imparted lead crowning as 21.5 μm and 46.4 μm respectively. Comparison of topography, microgeometry, and roughness profiles of spur gear flank surfaces before and after their lead crowning confirmed that the developed cathode gear and apparatus through PECFM process not only imparted them twist-free lead crowning but also reduced their arithmetical average roughness and maximum height by 34.3% and 33.6% respectively. They also improved their surface morphology as revealed by their scanning electron microscopy. The outcome of this study will be highly beneficial to the manufacturers and users of spur gears.

Graphical abstract

Appendices

Appendix A

See Figs 12 and 13

Fig. 12

Effect of lead crowning on different portions of workpiece spur gear tooth by PECFM: a instantaneously affected areas on its flank surface, b chordal thickness of its tooth along addendum circle “w;” c involute profile arc length of one flank surface, and d non-involute profile arc length of one flank surface.

Fig. 13

Graph of variation in “b(t)” with lead crowning time combined with graph of variation in reciprocating velocity with distance traveled by workpiece gear

• Gear tooth thickness “w” along the addendum circle can be computed as follows (Fig. 12)

$$w=2\left({r}_a\angle {QOR}^{\prime}\right)=2{r}_a\left(\angle POR+\angle DOP-\angle DOQ\right)=2{r}_a\left[\frac{w_p}{2{r}_p}+ inv\left({\psi}_p\right)- inv\left({\psi}_q\right)\right]$$

(A1)

$$w={r}_a\left[\frac{w_p}{r_p}+2\left\{ inv\left({\psi}_p\right)- inv\left({\psi}_q\right)\right\}\right]$$

(A2)

where ‘inv(Ѱ)’ is expressed by inv(ψ)=tan⁡(ψ)-ψ [41], and “Ѱ_p” and “Ѱ_q” are the involute pressure angles of point “P” and point “Q” at involute profile (rad) respectively and computed as follows:

From the ∆POB and ∆QOA

$${\psi}_p={\cos}^{-1}\left(\frac{r_b}{r_p}\right);{\psi}_q={\cos}^{-1}\left(\frac{r_b}{r_a}\right)$$

(A3)

where “r_b” is radius of base circle (mm).

• Arc length of involute profile “L_lc” of one flank surface of workpiece gear tooth during its lead crowning “L_lc,” shown as DQ in Fig. 12c, can be expressed by Eq. (A4).

$${L}_{lc}={\int}_0^{\theta_q}\left\{\sqrt{{\left(\frac{d x}{d\theta}\right)}^2+{\left(\frac{d y}{d\theta}\right)}^2}\right\} d\theta$$

(A4)

where “x” and “y” are the coordinates of any point on the involute profile DQ which are expressed by parametric equations expressed in Eqs. (A5a) and (A5b); and “θ_q” is the gear roll angle of point Q on the involute profile (radians).

$$x={r}_b\left( cos\theta +\theta sin\theta \right)$$

(A5a)

$$y={r}_b\left( sin\theta -\theta cos\theta \right)$$

(A5b)

$$\mathit{\tan}\left({\psi}_q\right)=\frac{AQ}{OA}=\frac{Arc\ AD}{r_b}=\frac{\theta_q{r}_b}{r_b}\Rightarrow {\theta}_q=\mathit{\tan}\left({\psi}_q\right)$$

(A6)

Using the Eqs. (A5a); (A5b); and (A6) in Eq. (A4) give the following relation for “L_lc”

$${L}_{lc}={r}_b{\int}_0^{\theta_q}\theta d\theta \Longrightarrow {L}_{lc}={r}_b\frac{{\theta_q}^2}{2}=\frac{r_b{\mathit{\tan}}^2\left({\psi}_q\right)}{2}$$

(A7)

• Arc length of non-involute profile “L_r” (i.e., profile length below the base circle) of one flank surface of lead crowned workpiece spur gear is shown as EFHI in Fig. 12d. It can be written as summation of arc lengths EF, FH, and HI which can be computed using Eqs. (A9), (A10), and (A11) respectively.

  $$EF={r}_d\left(\angle EOF\right)={r}_d\left(\angle KOI-\angle JOH\right)={r}_d\left\{\frac{BI}{2{r}_b}-{\sin}^{-1}\left(\frac{r_f}{r_d+{r}_f}\right)\right\}$$

$$EF={r}_d\left\{\frac{AI- AB}{2{r}_b}-{\sin}^{-1}\left(\frac{r_f}{r_d+{r}_f}\right)\right\}\Longrightarrow {r}_d\left\{\frac{\frac{2\pi {r}_b}{Z}-{w}_b}{2{r}_b}-{\sin}^{-1}\left(\frac{r_f}{r_d+{r}_f}\right)\right\}$$

(A8)

where “r_f” is the radius of root fillet of workpiece gear tooth and “w_b” is the chordal thickness of the workpiece gear tooth at base circle given by the following expression (using analogy from Eq. (A2))

$${w}_b={r}_b\left[\frac{w_p}{r_p}+2\left\{ inv\left({\psi}_p\right)\right\}\right]$$

(A9)

where “w_p” is chordal thickness of the gear tooth along pitch circle. Using Eq. (A9) in Eq. (A8) gives the following relation for EF:

$$EF={r}_d\left[\frac{\pi }{Z}-\left\{\frac{w_p}{2{r}_p}+ inv\left({\psi}_p\right)\right\}-{\sin}^{-1}\left(\frac{r_f}{r_d+{r}_f}\right)\right]$$

(A10)

$$FH={r}_f\left(\angle OJH\right)={r}_f{\cos}^{-1}\left(\frac{HJ}{OJ}\right)\Rightarrow FH={r}_f{\cos}^{-1}\left(\frac{r_f}{r_f+{r}_d}\right)$$

(A11)

$$HI= OI- OH={r}_b-\sqrt{(OJ)^2-{(HJ)}^2}\Rightarrow HI={r}_b-\sqrt{{\left({r}_d+{r}_f\right)}^2-{r_f}^2}\Longrightarrow HI={r}_b-\sqrt{r_d^2+2{r}_d{r}_f}$$

(A12)

Substituting for arc lengths EF, FH, HI from Eqs. (A10), (A11), (A12) in L_r = EF + FH + HI gives following expression for “L_r:”

$${L}_r={r}_d\left[\frac{\pi }{Z}-\left\{\frac{w_p}{2{r}_p}+ inv\left({\psi}_p\right)\right\}-{\sin}^{-1}\left(\frac{r_f}{r_d+{r}_f}\right)\right]+{r}_f{\cos}^{-1}\left(\frac{r_f}{r_f+{r}_d}\right)+{r}_b-\sqrt{r_d^2+2{r}_d{r}_f}$$

(A13)

• Expressions for lead crowning time at different positions of the anodic workpiece gear during its downward/upward reciprocating motion

Acceleration “a₁” (mm/s²) during first half of the downward/upward stroke can be obtained by using third equation of motion (v² + u² = 2as):

$${u_3}^2-{u_1}^2=2{a}_1\left(\frac{\ {b}_t+{b}_{ccg}}{2}\right)\Rightarrow {a}_1=\frac{{u_3}^2-{u_1}^2}{b_t+{b}_{ccg}}$$

(A14)

where “u₃” is maximum reciprocating velocity and “u₁” is minimum reciprocating velocity of the workpiece gear (mm/s); “b_t” is total face width of anodic workpiece gear (mm); “b_cc” is face width of conductive portion of developed cathode gear for lead crowning (mm). Similarly, deceleration “a₂” during second half of the downward/upward stroke can be expressed as follows:

$${a}_2=\frac{{u_1}^2-{u_3}^2}{b_t+{b}_{ccg}}$$

(A15)

Expression for time instant “t ₂ ” using second equation of motion [s = ut + (1/2)at ² ]:

Distance “S₁₂” traveled by the anodic workpiece gear from position 1 and time t₁ = 0 to position 2 and time “t₂” is S₁₂ = b_cc (Fig. 13). Using second equation of motion, the following expression is obtained:

$${b}_{cc}={u}_1{t}_2+\frac{1}{2}{a}_1{t_2}^2\Rightarrow {a}_1{t_2}^2+2{u}_1{t}_2-2{b}_{cc}=0$$

(A16)

Solving the equation (A16) for obtaining the expression for “t₂” using Shridhar Acharya formula:

$${t}_2=\frac{-{u}_1+\sqrt{{u_1}^2+2{a}_1{b}_{cc}}}{a_1}$$

(A17)

Expression for time instant “t ₃ ” using first equation of motion (v= u + at):

$${t}_3=\frac{u_3-{u}_1}{a_1}$$

(A18a)

Using expression of “a₁” from Eq. (A14) into Eq. (A18a) gives the following expression for “t₃”:

$$\Rightarrow {t}_3=2\frac{b_t+{b}_{cc}}{u_3+{u}_1}$$

(A18b)

Expression for time instant “t ₄ ” using second equation of motion (s = ut + 0.5at ² ):

Distance “S₃₄” traveled by the anodic workpiece gear form position 3 to position 4 during “t′₃₄” (Fig. 13) is as follows:

$${S}_{34}=\frac{b_t-{b}_{cc}}{2}$$

(A19)

Using second equation of motion with distance “S₃₄,” deceleration “a₂,” and time t′₃₄ taken by anodic workpiece gear to travel the distance “S₃₄” gives the following equation:

$${S}_{34}={u}_3{t}_{34}^{\prime }+\frac{1}{2}{a}_2{t_{34}^{\prime}}^2\Rightarrow \frac{b_t-{b}_{cc}}{2}={u}_3{t}_{34}^{\prime }+\frac{1}{2}{a}_2{t_{34}^{\prime}}^2$$

$$\Rightarrow {a}_2{t_{34}^{\prime}}^2+2{u}_3{t}_{34}^{\prime }-\left({b}_t-{b}_{cc}\right)=0$$

(A20)

Solving the Eq. (A20) using Shridhar Acharya formula gives the following equation for t′₃₄:

$${t}_{34}^{\prime }=\frac{-{u}_3+\sqrt{{u_3}^2+{a}_2\left({b}_t-{b}_{ccg}\right)}}{a_2}$$

(A21)

From Fig. 13 “t₄” can be expressed as ${t}_4={t}_3+{t}_{34}^{\prime }$ and using expression of time “t₃” from Eq. (A18a) and “t′₃₄” form Eq. (A21) gives the following equation for time “t₄”:

$${t}_4={t}_3+{t}_{34}^{\prime}\Rightarrow {t}_4=\frac{u_3-{u}_1}{a_1}+\left(\frac{-{u}_3+\sqrt{{u_3}^2+{a}_2\left({b}_t-{b}_{ccg}\right)}}{a_2}\right)$$

(A22)

Expression for time instant “t ₅ ”

The acceleration and deceleration “a₁” and “a₂” are equal in magnitude hence the time “t′₁₃” taken by workpiece gear to travel from position 1 to position 3 (half of the stroke length in first half of upward/downward stroke) is equal to the time “t′₃₅” taken to travel from position 3 to position 5 (the half of the stroke in second half of upward/downward stroke). Hence t′₁₃ = t′₃₅ = t₃ and using expression of “t₃” from Eq. (A18a) the expression for “t₅” is obtained as follows:

$${t}_5={t}_{13}^{\prime }+{t}_{35}^{\prime }=2{t}_3\Rightarrow {t}_5=2\frac{u_3-{u}_1}{a_1}$$

(A23)

Time for completion of one stroke during downward/upward stroke of the workpiece gear “t_s” is equal to the time “t₅.”

The expression for lead crowning time of first downward/upward stroke further used to obtain the expression for lead crowning time of second upward/downward stroke and n^th stroke as mentioned in Table B1.

Appendix B

Table 2 Summary of expressions for lead crowning time at different positions of the workpiece spur gear during its downward/upward reciprocating motion in its lead crowning by PECFM.

Table 3 Model predicted and experimental values of lead crowning (LC), surface roughness parameters, and volumetric material removal rate (MRR).

Patent

A patent with application number 202221016570 was filed to the Indian Patent Office in March 2022 based on the present research work. Response to its first examination report (FER) has also been submitted in Sep 2022 and presently it is under examination.