Influence of Open Differential Design on the Mass Reduction Function
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThis paper describes an interesting approach to weight optimisation of an open differential transmission. The paper is well elaborated, and thus I recommend it for publication after minor corrections:
- I would suggest to add a couple more of references from the last five years, to demonstrate the state-of-the art knowledge from the field of gears development.
- Figure 1 shall be slightly increased to enhance the readability. Please clearly indicate the starting point in the diagram. Please move the lines more apart, especially the flow lines inside and left of the the "optimisation process" borders, to make it clearer to follow.
- in part of the paper you explain, that you did analytical calculation according to ISO 23509:2006 standard. Please explain: did you use any specific software to do that, such as KISSsoft, or manually?
Author Response
Dear Reviewer,
Thank You for taking the time to read our manuscript. We appreciate your constructive suggestions. With great respect for your expertise in the field, please allow us to address your objections. Based on your suggestions and those of the other reviewers, the manuscript has been updated and improved. All changes are highlighted (red color), and we also include the response to your review. Thank You again for taking the time to read the manuscript and provide us with your feedback.
First comment:
I would suggest to add a couple more of references from the last five years, to demonstrate the state-of-the art knowledge from the field of gears development.
Answer to the first comment:
Thank You for this proposal! We add additional three references. These references are under the numbers [14], [27] and [28].
Second comment:
Figure 1 shall be slightly increased to enhance the readability. Please clearly indicate the starting point in the diagram. Please move the lines more apart, especially the flow lines inside and left of the the "optimisation process" borders, to make it clearer to follow.
Answer to the second comment:
Thank You for this proposal that aims to contribute to the quality of the image structure, better visibility and clarity of the presentation of the proposed algorithm. We made new figure, according to your proposals, and placed figure in the paper. This change is highlighted in the text of the paper.
Third comment:
In part of the paper you explain, that you did analytical calculation according to ISO 23509:2006 standard. Please explain: did you use any specific software to do that, such as KISSsoft, or manually?
Answer to the third comment:
Thank You for this question! KISSsoft is one of the quality software solutions, available on the market, in the part of carrying out calculations of gear geometry, in accordance with available standards (norms). Specifically, we used the MITCalc software solution for geometry creation and inspection, but we also created our own software solution in the part of calculating the geometry of the gears in MatLab, according to the standard ISO 23509:2006.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsThe paper develops and proposes an algorithm that combines the design of the car open differential according to ISO 23509:2006.
Since there is no nomenclature of the notations used in the formulas it is difficult to trace in the formulas what each term represents.
Figure 3 does not correspond to a dynamic model of the vehicle in motion. A motor vehicle is subject to resistance forces (rolling, slope, air, inertia) and the traction force at the wheel generated by the engine!
The wheel force must overcome the sum of the driving resistances and not only the frictional force (as mentioned in line 214-215).
Why is the data mentioned in Table 1 necessary for your calculations?
Table 2, for what are the data mentioned in your calculations necessary? On the basis of which relationships and parameter values did you determine them?
If the engine of the vehicle has 110kW, how does the differential arrive at 53kW (table 1), or 51.457 (table 2)? Which engine operating mode do these values of the input and output power of the transmission ratio correspond to?
The maximum engine torque is 340 Nm, at the wheel it reaches 3546.31 Nm. With what relation have you calculated? What is the transmission efficiency? You did not mention this in any of the input data, although this parameter is important.
What do you mean by "rising resistance force"?
What does "sigma" represent in relation 6?
The FEM analysis in static conditions is not correct, the calculation moments vary continuously.
Figure 7, pinion and bevel gear, deformation 0.194 mm! Such a gear does not work correctly!
Figure 12b you have a deformation of 0.411 mm on the drive shaft teeth! The gear is destroyed!
Figure 13b, on the splines of the shaft there is a torsional moment and not a force on a single tooth!
Axel Shaft- Figure 14b - maximum deformation 0.14 mm, figure 22b - maximum deformation 0.298 mm for the optimized geometry! By decreasing the mass of the shaft, its deformation has increased, therefore its functional capacity has decreased!
The values of the deformations on the parts subjected to FEM analysis exceed the usual values to ensure a good functioning of the differential mechanism.
Author Response
Dear Reviewer,
Thank You for taking the time to read our manuscript. We appreciate your constructive suggestions. With great respect for your expertise in the field, please allow us to address your objections. Based on your suggestions and those of the other reviewers, the manuscript has been updated and improved. All changes are highlighted (red color), and we also include the response to your review. Thank You again for taking the time to read the manuscript and provide us with your feedback.
Also, thank You very much for the observed and pointed out omissions! We are very embarrassed that it happened! Please accept our apology!
Power of engine Pmax= 110 kW we delete from the Table 1 as and the number of revolutions at which Pmax is reached (nPmax). Please accept our apologies for this mistake we made unintentionally. This was "Copy - Paste" mistake, which we made, because we parallel with this manuscript also are working on the calculation on other product families (variants) of the open differential transmission with different variants of input powers.
Because of the same reason, calculated values of the rising resistance force (FRI), vehicle's inertial force (FIN) and total resistance force (FTRD) in Table 2, are also added incorrectly. Now we correct it!
Comment 1: Since there is no nomenclature of the notations used in the formulas it is difficult to trace in the formulas what each term represents
Answer to the comment:
Thank You very much on this comment! We made correction and we add Nomenclature section at the end of the paper.
Comment 2: Figure 3 does not correspond to a dynamic model of the vehicle in motion. A motor vehicle is subject to resistance forces (rolling, slope, air, inertia) and the traction force at the wheel generated by the engine!
The wheel force must overcome the sum of the driving resistances and not only the frictional force (as mentioned in line 214-215)
Answer to the comment:
Thank You very much on this comment! This was our omission in writing the manuscript, especially in the desire to shorten the paper and present the part that would be more related to obtaining the proposed mathematical model for optimizing the design elements of the differential transmission. Please try to accept our apologies for that!
We expand our manuscript with the calculation of the reaction force (Fw) in expression (4), friction force (Ff) in expression (5), traction force (FTR) in expression (6), rolling resistance force (FRO) in expression (13), rising resistance force (FRI) in expression (14), vehicle's inertial force (FIN) in expression (15). We explain that we did not take in consideration air resistance because the geometry of the vehicle body is not defined. Therefore, the total resistance force (FTRD) is calculated as a sum of the FRO, FRI and FIN. Also we explain this in the manuscript as follows in manuscript lines 269-277:
"Rolling resistance force, rising resistance force and vehicle's inertial force, together determine the total resistance force of the driven machine. These forces represent loads on the output side of the differential transmission. Sum of this forces represent total resistance force of driven machine (FTRD), whose calculation value is presented in Table 2. In the calculation of the total resistance force, the geometry of the vehicle body is not defined, and therefore air resistance is not included in this calculation [38]. In order to achieve the movement of the vehicle, the amount of the traction force (FTR) should be greater than the amount of the total resistance force (FTRD). The values of these forces are presented in Table 2."
Comment 3: Why is the data mentioned in Table 1 necessary for your calculations?
Answer to the comment:
In category Design requirements are data which we used for the following calculations:
Maximum vehicle mass (mv, max) - we used for calculation of the reaction force in the wheel (Fw) in expression (4). Also, we used reaction force for the calculation of the friction force (Ff) in expression (5) and this force we used for the calculation of the traction force (FTR) in expression (6).
Dynamic wheel radius (rd) - we used for torque calculation on the drive wheels (T’w = T’L = T’R) in expression (7). Since the dimension of the tire is determined by the request (205/55 R16), the dynamic wheel radius is 97 % of the length of the actual wheel radius. We mentioned this requirement (dimension of the tire) in manuscript lines 203.
Vehicle acceleration (av) – we used for the calculation of the vehicle's inertial force (FIN) in expression (15).
Time required to reach a speed of 100 km/h (t0-100) and maximum vehicle speed at the highest transmission ratio (vmax) – we did not use for calculation, but we put this data as a guideline that gives a clearer picture to the designer, also and reader, to get a more complete idea of the technical system (vehicle) for which it is designed and optimized the differential transmission.
In category Technical characteristics of the driving machine are data which we used for the following calculations:
Max. engine torque (Tmax) and number of revolutions at which Tmax is reached (nTmax) - we used for the calculation of the power on the motor output shaft (PO max) in expression (1).
Power delivered to the drive bevel gear (P1) – in calculation of power P1 are included losses in the kinematic chain. Power is calculated by the expression (2). Losses are included thorough the next efficiencies: efficiency of the gearbox (ηg) and efficiency of the cardan shaft (ηc). Power P1 is used for calculation of the input torque to the differential transmission (T1), determined by the expression (3)
Input torque to the differential transmission (T1) - this is a real value of the Input torque to the differential transmission (torque on the pinion). In calculation of this torque are included losses in the kinematic chain. Losses are included thorough the next efficiencies: efficiency of the gearbox (ηg) and efficiency of the cardan shaft (ηc).
Power (Pmax) - we did not use for calculation. We deleted this data from Table 1
Number of revolutions at which Pmax is reached (nPmax) - we did not use for calculation. We deleted this data from Table 1
Comment 4: Table 2, for what are the data mentioned in your calculations necessary? On the basis of which relationships and parameter values did you determine them?
Answer to the comment:
Thank You on this comment! Answer on this question is as follows:
Reaction force in the wheel (Fw) – this data we determined by the expression (4). This data is necessary for determination of the friction force (Ff), determined by the expression (5).
Friction force (Ff) - this data we determined by the expression (5). This data is necessary for determination of the traction force (FTR), determined by the expression (6).
Transmission ratio of the differential transmission (id) - this data we determined by the expression (12). This data is necessary for determination of the angular velocity of the hypoid ring bevel gear (ω2), determined by the expression (11).
Output power of the differential transmission (P2) - this data we determined by the expression (9). This calculation includes the efficiency of the open differential (ηd). Power P2 is necessary for determination of the torque of the hypoid ring bevel gear (T2), determined by the expression (10).
Rolling resistance force (FRO) - this data we determined by the expression (13). This data is necessary for determination of the total resistance force of driven machine (FTRD).
Rising resistance force (FRI) - this data we determined by the expression (14). This data is necessary for determination of the total resistance force of driven machine (FTRD).
Vehicle's inertial force (FIN) - this data we determined by the expression (15). This data is necessary for determination of the total resistance force of driven machine (FTRD).
Total resistance force of driven machine (FTRD) is determined as the sum of the FRO, FRI and FIN. In response to Comment 2, we gave an answer to the determination of total resistance force of driven machine, which we implemented in the text of the manuscript.
Torque of the hypoid ring bevel gear (T2) - this data we determined by the expression (10). In the calculation amount of this moment is included the efficiency of the open differential (ηd). This is a real value of the hypoid ring bevel gear torque.
Torque of drive wheel (TW, TR, TL) – we explain in the manuscript determination of this data on the lines 264-266, as follows:
"The final amount of torque T2 is shown in Table 2. Also, the final amounts of the torque on the drive wheels is equally divided between the left and right wheel (Tw = TL = TR = T2/2)."
Comment 5: If the engine of the vehicle has 110kW, how does the differential arrive at 53kW (table 1), or 51.457 (table 2)? Which engine operating mode do these values of the input and output power of the transmission ratio correspond to?
Answer to the comment:
We started our calculation with the determination of power on the motor output shaft (PO max). We determined this power by the expression (1). Value of this power is 56.94 kW (see expression 1). For this calculation, we use amount of the max. engine torque (Tmax) which was achieved at the number of revolutions nTmax. Tmax and nTmax are input data our calculation of the open differential transmission and their values are given in Table 1.
Power (P1) is a power delivered to the drive bevel gear (power arrived in the open differential transmission). This power is calculated by the expression (2). In calculation of the power P1 are included losses in the kinematic chain over the efficiency of the gearbox (ηg) and the efficiency of the cardan shaft (ηc). Values of ηg and ηc are added in the manuscript in text line 202-203. Because of this losses, amount of the power P1 is less from the amount of the power PO max. Calculated value of the power P1 is 53.05 kW and is given in Table 1.
Power (P2) is the output power of the differential transmission. This power is calculated by the expression (9). In this expression is included efficiency of the open differential (ηd). Value of ηd is added in the manuscript in text line 203. Because of this loss, amount of the power P2 is less from the amount of the power P1. Calculated value of the power P2 is 51.457 kW and is given in Table 2.
For the calculation of this power we used gearbox transmission ratio i1 = 3.5. This transmission ratio and other transmission ratios (i2 = 2.1, i3 = 1.32, i4 = 0.97, i5 = 0.76, iR = 3.55) are added as input requirements in text manuscript lines 200, 201.
Power of engine Pmax= 110 kW we delete from the Table 1 as and the number of revolutions at which Pmax is reached (nPmax). Please accept our apologies for this mistake we made unintentionally. This was "Copy - Paste" mistake, which we made, because we parallel with this manuscript also are working on the calculation on other product families (variants) of the open differential transmission with different variants of input powers.
Comment 6: The maximum engine torque is 340 Nm, at the wheel it reaches 3546.31 Nm. With what relation have you calculated? What is the transmission efficiency? You did not mention this in any of the input data, although this parameter is important.
Answer to the comment:
We expand our manuscript with the calculation presented from equation (1) till the equation (15). For the calculation, torque T2, we used expression (10). This torque is divided on the torque on the left drive wheel and on the torque on the right drive wheel. We add on our manuscript:
"The final amount of torque T2 is shown in Table 2. Also, the final amounts of the torque on the drive wheels is equally divided between the left and right wheel (Tw = TL = TR = T2/2). Their calculated amounts are presented in Table 2."
We add in the manuscript data about the value of the efficiency of the open differential (ηd). This value is ηd = 0.97. In previous comments we mentioned that we expand our manuscript with this data.
Comment 7: What do you mean by "rising resistance force"?
Answer to the comment:
The rising resistance force is a component of weight force vector that acts in the opposite direction to the motion of the vehicle, at the moment when the vehicle climbs the slope.
We add this explanation in the manuscript (lines 282-283)
Comment 8: What does "sigma" represent in relation 6?
Answer to the comment:
Thank You on this comment. "sigma" is shaft angle and value of this angle is 90° (Σ=90°). In the paper, we add under expression 21, clarification of the meaning of this variable. We marked this change in the text of the paper in red color. Also, we add in Nomenclature section, clarification of the meaning of this variable.
Comment 9: The FEM analysis in static conditions is not correct, the calculation moments vary continuously.
Answer to the comment:
The authors would like to thank Reviewer for this comment. The authors are aware that calculation moments are not constant and continuously vary and that the simulation of dynamic problems with static analysis can lead to deviating results in some situations. On the other hand, numerical analysis of continuous moment variations can be quite challenging, computationally resource demanding and time consuming. Therefore, the authors have limited themselves to the static FEM analysis in this phase of the study, but will also consider the variation of moments as well as the effects of dynamic loading in the numerical analyzes in the next phase of the study, where more comprehensive results can be expected. The authors also mention this in the conclusion of this paper.
Comment 10: Figure 7, pinion and bevel gear, deformation 0.194 mm! Such a gear does not work correctly!
Answer to the comment:
The authors thank the Reviewer for this comment and for pointing out this deformation value. Prior to the final analysis, the authors performed several preliminary analyzes. This figure was erroneously taken from a preliminary analysis where the numerical model was not properly constrained. There was an initial gap between the parts in contact giving results with significantly larger deformations. The final analysis was performed with an initial contact between the parts, resulting in reasonable deformation values, which can now be seen in Figure 7. The deformation of the parts in the contact region is also consistent and the deformation values are significantly lower, as it can see in this close-up.
Comment 11: Figure 12b you have a deformation of 0.411 mm on the drive shaft teeth! The gear is destroyed!
Answer to the comment:
As the Reviewer noted, the deformation of the drive shaft in Figure 12b was too large and is not comparable to the deformations of the optimized drive shaft in Figure 18, because the largest value of the optimized deformation was 0.064774 mm, while the deformations of the non-optimized shaft, which has higher strength and stiffness, were larger with values of 0.411 mm. The authors would like to apologize that the same situation occurs here as in the previous comment. Figures 12a and 12b, which show the stresses and deformations of the drive shaft, are not from the final analysis but from a preliminary analysis in which the numerical model was not properly constrained, resulting in a gap between the drive shaft and ring gear. This has now been corrected and Figures 12a and 12b are from the final analysis. The deformation of the optimized drive shaft now has slightly larger values than in the initial analysis of the drive shaft, as it should be, and is in the same order of magnitude.
Comment 12: Figure 13b, on the splines of the shaft there is a torsional moment and not a force on a single tooth!
Answer to the comment:
In the numerical analysis, the authors used a remote force load that simulates a moment and that lead to an increase in deformation in a part of the axel shaft. The authors agree with the Reviewer that a torsional moment would be more appropriate for this analysis. Therefore, the authors repeated the numerical analysis, which showed a more uniform distribution of displacement with better deformation values, but also showed that there is no scope for reducing the mass of the axel shaft. Therefore, the authors would like to thank the Reviewer for this advice. This prompted the authors to restructure the paper and remove the design parameter inner radius of the axel shaft r2, leaving four design parameters: Vertical chamfer of ring gear a, horizontal chamfer of ring gear b, projection height of ring gear h and inner radius of drive shaft r. The authors recreated the central composite design of the experiment (Table 10) and determined the mathematical functions of the response variables using the response surface method. The change can be seen in Equation 39, where the total mass is affected by a, b, h and r and not by the inner radius of the axel shaft r2. The sensitivity indices for the responses: total mass, mass of ring gear and mass of drive shaft are also changed in Figure 17.
Comment 13: Axel Shaft- Figure 14b - maximum deformation 0.14 mm, figure 22b - maximum deformation 0.298 mm for the optimized geometry! By decreasing the mass of the shaft, its deformation has increased, therefore its functional capacity has decreased!
Answer to the comment:
According to the Reviewer's advice in a previous comment, the author's response in a previous comment, and the restructuring of the investigation to remove the inner radius of the axel shaft r2 from the design parameters, the axel shaft was no longer included in the optimization and mass reduction.
Comment 14: The values of the deformations on the parts subjected to FEM analysis exceed the usual values to ensure a good functioning of the differential mechanism.
Answer to the comment:
According to the Reviewer’s notes, the authors found that figures from the preliminary analysis, in which the numerical model was not properly constrained so that there was an initial gap between the parts in contact, led to results with considerable larger deformation presented in paper. After presenting figures from the final analysis with properly set initial contact, the deformation values presented in paper do not endanger a good functioning of the differential mechanism.
The authors would like to thank the Reviewer for his comments, which made it possible to improve the quality of the work.
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
Comments and Suggestions for Authors The authors reanalyzed the paper and took into account the recommendations made. I propose the acceptance and publication of the paper in its current form.
