![6
The design constraints are presented in table 1. The table presents the specifications given
to us by the Solar Impulse team, as well as the values we calculated and chose for our final
gearbox in bold red.
Powertrain Specification Gearbox Specifications
Motor
Brushless +
Sensor less
Gear Ratio 7.84
Maximum [rpm] 5000 Total Weight [kg] 14.4
Fuel Consumed [L] 0 L of Fossil Fuel
Endurance Life-Gears and
Output Shaft [hrs]
2000
MaximumMotor
Temperature [°C]
135 Temperature Range [°C]
-40 to
+40
Propeller
Twin Blade
Composite
Size limitations [cm]
30 X45
X 45
Propeller Thrust [lbf] 1500 Safety Factor 1.5
Propeller Weight [kg] 160
Propeller-Max RPM [rpm] 525
Table 1- Design Requirments
It should also be noted that the Gears considered in this design are AGMA spur gears of
course diametral pitch, due to their simplicity and versatility. The gears are to be lubricated
with SAE 30W. The team’s target was a fictional loss of less than 5%, and this was successfully
achieved. Finally, the operating conditions considered in our design are shown below.](https://image.slidesharecdn.com/project-report-393-170426201242/85/Gearbox-design-MECH-393-final-project-6-320.jpg)
![7
Operating Conditions
Operation
Electric power to Motor Driver
[hp]
Electric motor Shaft Torque
[Nm]
RPM
Take- Off 40 70.2 4000
Slow Climb 7 16.4 3000
Steep Climb 13.4 29.6 3180
Descent Glide 0.7 2.2 2225
Horizontal
Flight
4.7 14.8 2225
Table 2 Operating Conditions](https://image.slidesharecdn.com/project-report-393-170426201242/85/Gearbox-design-MECH-393-final-project-7-320.jpg)
![25
𝑑bO =
cd∗ fgh
i
∗
jg∗kl
m
no.pq∗ jgr∗sl
m
tu
+
jgw∗kw
m
no.pq∗ jgrw∗sw
m
txy
− 𝑑bS
c
z
{
For our loading, Ta and Mm are zero. This formula was then solved for Nab, our safety fatigue
safety factor for point B.
𝑁}~ =
i∗(•h€
{
••h‚
{
)
cd
∗
jg∗kl
m
tu
+
o.pq∗ jgrw∗sw
m
txy
•ƒ
Note, both these equations consider din . The first formula provides us with an estimate for dB ,
based off NfBDESIRED , which is an initial target safety factor used to provide us with our first
guess. This was set to a conservative 2.25 initially. Next, this value was chosen, rounded to allow
for ease of manufacturability, and then corresponding din was chosen based off the process
outlined in the first section above. These choices caused equation 2 to produce our new NfB . At
this point we would repeat the process again iteratively until the results were satisfying, and we
have a NfB = [1.3,1.5]. Excel was an excellent tool for this iteration.
A sample calculation is presented here detailing all the parameters and factors chosen for design.
These results were the ones chosen for our Shaft 1. For further reference, please see the appendix
and the excel file.
Plug in with Desired Safety Factor of 2.25
𝑑bO =
cd∗(d.dq)
i
∗
ƒ.„p∗cc.p[f††] m
ƒqˆ[k‰Š]
+
o.pq∗ d.dˆ∗podcq.ƒ [f††]
m
ˆ„‹ [k‰Š]
− 10[𝑚𝑚]c
z
{
𝑑bO = 17.8325 [𝑚𝑚]
Choose a 𝑑bO that is appropriate for machining:
𝑑bO = 17 𝑚𝑚
Plug into Safety Factor equation:
𝑁}~ =
𝜋 ∗ (17c
− 10c
)
32
∗
1.67 ∗ 33.7[𝑁𝑚𝑚] d
154[𝑀𝑃𝑎]
+
0.75 ∗ 2.24 ∗ 70235.1 [𝑁𝑚𝑚]
d
469 [𝑀𝑃𝑎]
•ƒ
𝑁}~ = 1.32
which is acceptable as a target safety factor. Note: The development shown above contained
many iterations, but brevity only the final stage is presented.](https://image.slidesharecdn.com/project-report-393-170426201242/85/Gearbox-design-MECH-393-final-project-25-320.jpg)
Gearbox design - MECH 393 final project
- 1. Final Project Report Winter 2017 MECH 393 – Machine Element Design April 11th, 2017 Group Number: 2 Group Members: Georges Matta 260608769 Riad Haissam El Charif 260631084 Stanislav Nemirovsky 260660024
- 2. 2 Table of Contents Executive Summary ................................................................................................................ 4 Individual Contribution ......................................................................................................... 4 Introduction to the Design Problem ................................................................................. 5 Detailed Design Solution ....................................................................................................... 8 I. Gear Development ................................................................................................................... 9 i. Gearbox Layout ..................................................................................................................................................... 9 ii. Determining the Gear Ratio ............................................................................................................................. 9 iii. Power and Torque Requirements .............................................................................................................. 11 iv. Gear 1 Sample Calculations for Safety Factors ..................................................................................... 11 v. Calculations for Gears 2, 3, 4, 5, 6 ............................................................................................................... 18 II. Shaft Development ............................................................................................................... 19 i. Free Body Diagrams of the Shafts with Gears and Bearings: ................................. 21 ii. Shaft 1- Input Shaft ............................................................................................................... 24 iii. Shafts 2-4 - Reduction Shafts ............................................................................................ 27 iv. Shaft 3 - Output Shafts ......................................................................................................... 28 Bearings ................................................................................................................................... 30 The C parameter is obtained from the manufacturer specification for each bearing, and so to allow for successful iteration, a large amount bearing tables were input into our excel sheets to allow for easy selection. The tables would “iterate” these formulas for many different types of bearings, until a suitable life limit was found. We should note that was not compared directly with our , but rather it was multiplied by a reliability factor of Kr = 0.33 (for a reliability of 98%) for most of our bearing choices. We now used this new to compare with our ideal life. 31 After a number of bearings of acceptable life were found, the particular bearing was selected based off geometrical constraints of our shaft. Since we had a very low stress state, we could afford to add a large shoulder on the bearing portion of our shaft, and so dout was not a factor. However, since we used a hollow shaft, din was the limiting parameter. The selected bearings are presented in later in this report. ................................ 31 Note: Due to our very low loading, the life expectancies of the bearings on shaft 1 are significantly higher than necessary, however these bearings were found to be suitable for our geometry , and satisfied our needs, and
- 4. 4 Executive Summary NikolaDrive Team The design report presented here was commissioned by the Solar Impulse initiative for the design of a gearbox for the titular aircraft. NikolaDrive is a collective of highly motivated innovative aeronautical engineers, who form a vital subdivision within the Solar Impulse family. Headed by our chief engineer Mark Driscoll, the team embarked on the proposed design for a double branch double reduction gearbox, intended for use on the final aircraft. The team had 3 main design goals: Minimize weight, maximize efficiency and endure the aircraft’s lifetime. Integrated design principles were used for the design of Gears, Shafts and Bearings in our system. A targeted safety factor of 1.5 was chosen for our general design allowing for static, dynamic, and fatigue failure analysis to be performed on each component. The designs were iterated until satisfactory results were obtained. All components fall within a safety factor of 1.5. The whole system operates at power losses below 5% as desired. The weight of our system is 14.4 kg, which did not meet the target criterion of 5.5 kg. Further refinement in future designs could be considered. The NikolaDrive Team is proud to present its first and most innovative design: Solar Impulse Double Branch Double Reduction Gearbox. Individual Contribution The NikolaDrive team consists of three engineers, Georges Matta, Stanislav Nemirovsky and Riad Haissam El Charif, working under the supervision of Mr. Driscoll. With such a tightly knit and well-functioning unit, all members had significant contributions on each aspect of design. However, a rough division of individual contribution can nonetheless be made. Georges Matta, a U3 Mechanical Engineering Student at McGill University, oversaw spreadsheet production, gear design, as well as material and bearing selection. Riad Haissam El Charif, also a U3 Mechanical Engineering Student at McGill University worked on shaft and bearing design, layout, and optimization. Finally, Stanislav Nemirovsky, U3 Mechanical Engineering, optimized and realized the design for the bearings, shafts, as well as the gear parameters. The apparent division of labor provided a rough structure of the team’s organization; however, it should be reemphasized that NikolaDrive operates on a highly collaborative structure, in which team members share a significant amount of responsibilities
- 5. 5 Introduction to the Design Problem Solar Impulse’s design objective is simple, yet awe inspiring: Fly around the globe with no onboard fuel. To achieve this unique challenge, every engineering sub team involved in the aircraft design highly optimized parts for maximum efficiency and minimum weight, while maintaining reasonable reliability constraints. Conversely, the uniqueness of this project has given rise to some unusual liberties. First, as an experimental aircraft, no specific certifications are mandated upon our designs, which gave us incredible freedom to pursue our design goals. Second, as a well-funded experimental aircraft team, Solar Impulse has managed to secure enough funds that cost is never taken as a constraint. Such freedom allowed for more open-ended innovation. The gearbox to be designed is of the Double Branched Double Reduction Gearbox type, a sketch of which is shown in Figure 1. Figure 1- Double Branched Design The design allows for the reduction on speed, while increasing torque along two stages. The double branch design allows for the distribution of loads from the input shaft, to reduce stresses on the reduction shaft. The input shaft is directly connected to a 5000-rpm brushless motor, which gets stepdown to a maximum of 525 rpm imparted to the propeller. The propeller shaft, being the heart of our propulsion system, withstands a 1500lb axial load produced by the propeller rotation.
- 6. 6 The design constraints are presented in table 1. The table presents the specifications given to us by the Solar Impulse team, as well as the values we calculated and chose for our final gearbox in bold red. Powertrain Specification Gearbox Specifications Motor Brushless + Sensor less Gear Ratio 7.84 Maximum [rpm] 5000 Total Weight [kg] 14.4 Fuel Consumed [L] 0 L of Fossil Fuel Endurance Life-Gears and Output Shaft [hrs] 2000 MaximumMotor Temperature [°C] 135 Temperature Range [°C] -40 to +40 Propeller Twin Blade Composite Size limitations [cm] 30 X45 X 45 Propeller Thrust [lbf] 1500 Safety Factor 1.5 Propeller Weight [kg] 160 Propeller-Max RPM [rpm] 525 Table 1- Design Requirments It should also be noted that the Gears considered in this design are AGMA spur gears of course diametral pitch, due to their simplicity and versatility. The gears are to be lubricated with SAE 30W. The team’s target was a fictional loss of less than 5%, and this was successfully achieved. Finally, the operating conditions considered in our design are shown below.
- 7. 7 Operating Conditions Operation Electric power to Motor Driver [hp] Electric motor Shaft Torque [Nm] RPM Take- Off 40 70.2 4000 Slow Climb 7 16.4 3000 Steep Climb 13.4 29.6 3180 Descent Glide 0.7 2.2 2225 Horizontal Flight 4.7 14.8 2225 Table 2 Operating Conditions
- 8. 8 Detailed Design Solution The general methodology followed in our design process was “assume and iterate”. We started process followed for our given points, began with an assumption and tried out multiple different values until we converged to our desired safety factors and geometry. One fundamental design issue we wanted to avoid was the “Design Paradox”; as we learned more about our design and progressed through it, we became more capable of understanding our design, and its specific requirements. However, as we learn more, we became increasingly incapable of editing any of our values due to a deep design investment. To keep a versatile and dynamic system and avoid this crutch, we parametrized all our values through a master Excel Workbook. This workbook contains all our design, diagrams and results and allowed us to modify our design drastically multiple times with ease. This workbook was provided along with the Project Report and will be subsequently referenced multiple times in the sections below. Another important design consideration for any shaft based system is the shaft deflection. This variable typically pushes designers to minimize shaft length ‘L’. However, as our design requirements do not include a shaft deflection analysis, we decided to use up the full horizontal length of 30 cm for our layout as an initial guess. This layout provided us with acceptable results, and so it was chosen for our design with little modification. Our gearbox layout can be seen in Figure 2, as well as in our design drawings: Figure 2 The design process then progressed with a logical order, we designed the gears based off the given inputs and outputs, used the gear data- Face Width and Diameter- to produce a preliminary design layout for our whole gearbox. The layout was then used along with
- 9. 9 torque data from the Gears to produce Torque and Moment diagrams. These were used to size preliminary diameters for the shaft, which was then completed with the selection of an appropriate Ball/Roller Bearing which fits our load and lifetime requirements. Finally, the design was optimized for weight considerations and suitable Safety Factors. I. Gear Development i. Gearbox Layout As shown in Figure 2, a rough sketch of the gearbox was developed in order to have a general idea about the dimensions of all the components. In particular, estimations were made for the overall length of the gearbox, as well as the distance between intermediate gears, and mounting requirements for the shafts and bearings. All four shafts were represented in the above sketch, with an indication of how the six gears and eight bearings would be mounted in this gearbox. ii. Determining the Gear Ratio As a first step in the design process, an ideal gear ratio must be determined in order to satisfy the conditions and constraints provided. Maximum input angular velocities were provided as well as a maximum output RPM. As a result, a minimum gear ratio will be considered a reference for evaluating the several iterations studied. Now, an accepted combination of gear and pinion teeth must result in a gear ratio that is at least equal to this value. Another alternative to this selection criterion is to compare the output RPM generated by the gear ratio selected; this value cannot exceed the given maximum value of 525 rpm. In addition, the fact that a pressure angle of 20° is given means that there is also a minimum number of pinion teeth that could be used in order to avoid interference, the value of which is 18 teeth, as shown in Table 12-4. One of the final constraints when it comes to selecting the number of pinion and gear teeth is checking that the geometry satisfies the requirements given, as the sum of diameters of two consecutive gears in the first stage must be equal to that in the second stage. And since the modulus is taken as constant for all gears, this means that the sum of teeth between adjacent pinions and gears must be equal
- 10. 10 between the two stages. This is expressed by the equations below and can be seen in more detail in the Excel sheet provided. Now, gear ratios can be calculated for several values of N1 through N6, based on the equation for compound gears: where N2 = N6 and N3 = N5 The table below includes several iterations performed before settling on the values shown on the right, which satisfy all the constraints provided. N1 N2 N3 N4 N5 N6 Actual Gear Ratio Satisfie s Geomet ry? Output RPM 18 55 18 45 18 55 7.638888 889 523.6363 636 36 110 36 90 36 110 7.638888 889 523.6363 636 18 40 18 60 18 40 7.407407 407 540 20 50 21 64 21 50 7.619047 619 525.0000 0 18 56 18 44 18 56 7.604938 272 525.9740 26 18 54 18 46 18 54 7.666666 667 521.7391 304 18 60 18 41 18 60 7.5925925 93 526.829 2683 35 105 36 91 36 105 7.5833333 33 527.472 5275 20 56 20 56 20 56 7.84 510.204 0816 GEAR TEETH N1 20 N2 56 N3 20 N4 56 N5 20 N6 56 RATIO 7.84 Output RPM 510.2040816
- 11. 11 iii. Power and Torque Requirements By considering our system in an ideal scenario, the power generated by the motor will equal the power provided to the propeller. But because our gearbox is a double-branch double-reduction type, we can suggest that the power provided by the motor, which passes through Gear 1, gets divided equally between Gears 2 and 6, remains constant along Gears 3 and 5 (same shaft as 2 and 6 respectively), only to return to approximately the same initial value through Gear 4, powering the propeller. In reality however, losses in power exist due to the presence of factors such as friction; the fact that ball bearings were selected played a role in the assumption that such losses are considered negligible, as will be explained in coming sections. Now, after finding a suitable gear ratio, we can determine the angular velocities and torques at each of the gears: iv. Gear 1 Sample Calculations for Safety Factors
- 12. 12 The modulus m for all the gears was estimated to be 3mm as it was the most reasonable value that provides relatively good safety factors in bending and contact, as will be shown. Thus, the pitch diameter d, the pitch radius r, as well as the addendum a and dedendum b for Gear 1 can all be found from the value of m: The pitch line velocity VT and the tangential component of load WT can also be found: In addition, the life for input Gear 1 (as well as output Gear 4) is double that of the other gears because it is a single driving pinion driving two independent gears causing two fatigue cycles per revolution: Now, the AGMA approach (SI form) for both bending and contact stress will be applied to determine suitable gear parameters and safety factors. The AGMA bending stress equation is given by: The values of these constants and unknowns will now be calculated. The Application Factor KA is chosen as 1.25 since the transmitted load cannot be considered uniform as it fluctuates with time, at least for the driven machine as opposed to the driving machine (electric motor / turbine). Moderate shock is thus chosen. (Refer to Table 12-17)
- 13. 13 The Rim Thickness Factor KB is 1 since we are analyzing a solid-disk pinion. This is not the case for Gears 2, 4, and 6 since rims with spokes are used. The Idler Factor KI is 1 since a non-idler gear is being analyzed. The Size Factor Ks is 1 since no drastic changes in size are present. The Dynamic factor KV attempts to account for internally generated vibration loads from tooth-tooth impacts induced by non-conjugate meshing of the gear teeth. These vibration loads are called transmission error and will be worse with low-accuracy gears. This factor is given by: where: These values of A and B were determined for a quality index QV = 10. Thus, The AGMA Bending Geometry Factor J is determined from Table 12-9 for a pinion by relating the number of pinion teeth (20) to the number of gear teeth (56) This value is different when analyzing Gears 2, 4, and 6 The Face Width F is chosen as the minimum in the suitable range in order to get the best values for safety factors for this gear.
- 15. 15 Thus, Now, the value of the AGMA Contact Stress is: The AGMA Bending Fatigue Strength is given by: where the uncorrected bending strength was chosen for Steel AISI A1-A5 Through Hardened 330 HB (Figure 12-25) Referring to Figure 12-24, The Life Factor KL is given by: The Temperature Factor KT is 1 for steel material in oil temperatures up to 250°F The Reliability Factor KR is taken from Table 12-19 for a reliability of 99% Thus, the value of the AGMA Bending Fatigue Strength is:
- 16. 16 The AGMA Contact Fatigue Strength is given by: where the uncorrected contact strength was chosen for Steel AISI A1-A5 Through Hardened 330 HB (Figure 12-27) Referring to Figure 12-26, The Surface-Life Factor CL is given by: The Hardness Ratio Factor CF is 1 for the pinion (gear 1) The Temperature Factor CT is identical to KT The Reliability Factor CR is is identical to KR Thus, the value of the AGMA Contact Fatigue Strength is: Now, we can calculate our safety factors:
- 17. 17
- 18. 18 v. Calculations for Gears 2, 3, 4, 5, 6 Gears 2 and 6 are identical, and so are Gears 3 and 5 When compared to Gear 1, the only factors that change when calculating the safety factors for the rest of the gears are: • The life for Gears 2, 3, 5, 6 is half that of Gears 1 and 4 (480000000 cycles instead of 960000000 cycles) • The angular velocities, torques, and power differ as shown previously in Section (iii) • The face width changed from stage 1 to stage 2 (from 24mm to 45mm) • The geometry factor J becomes 0.4 instead of 0.34 when analyzing a gear instead of a pinion. • A rim thickness factor Kb exists for gears 2, 4, and 6 due to the presence of spokes. Its value is determined for a certain assumed rim thickness as follows: • The Hardness Ratio Factor CH is not 1 when analyzing gears because of a change in material. Between Gear 2 and Gear 1, it is calculated as follows: A thorough calculation of safety factors for Gear 2 (and Gear 6 by association) can be found in the Appendix, and values for the other gears can be found in the results section of this report as well as in the Excel sheet accompanying it.
- 19. 19 II. Shaft Development The 4 shafts were designed based off the data produced by the gear design process. Since the gears designs have little dependence on their inner diameters, the shafts diameters had much sizing freedom. It should be noted that the input shaft, Shaft 1, was the simplest to design due to its low loading state, and had a similar layout to our output shaft, Shaft 3. The reduction shafts, Shaft 2 and 4, were completely identical (albeit operating at a different direction), and as such were designed only once. We began with the input shaft. As described previously, the shaft layout the first step in shaft design. As for the loading conditions on the shafts, all shafts experienced a fully reversed alternating bending moment, caused by either radial and tangential forces or the weight of the shaft. The final shaft also experienced a significant axial load, which was not transferred over to the other shafts. Finally, the torsional loading, due to the torque, was considered to be a constant mean torsion at a magnitude equal to the Take-Off conditions of 40 HP and 4000 rpm. The reasoning for this design consideration was that this steady torque was the largest torque values that would be applied onto the aircraft under regular flight conditions, and so designing for those operating conditions provides us with a more conservative and encompassing failure criterion. Our team could have alternatively chosen the Take-Off conditions and Descent Glide conditions as a Max and Min for an alternating torsion for much more conservative results, but we felt that this would be an inappropriate assumption for our aircraft. Our aircraft is to be flown under highly controlled flight paths, with a known and limited number of landings and take-offs, which makes assuming a completely cyclic alternating torsion an unnecessary overly conservative estimate. Due to the nature of the symmetric double branch acting upon the inner gears (Gear 1 and Gear 4), radial and tangential forces cancel out and no moments diagrams need be produced in those planes, on those shafts. This is shown in the free body diagram below: Figure 3 Wr Wt Wr Wt Tin Tin = 2(WT)(r)
- 20. 20 Due to the fact that no other radial or transverse loadings exist, we used the gear weight as a force in the analysis of Shaft 1. However, due to the small magnitude, its effect was negligible relative to the much larger torque. This is very evident in the numbers presented in our EXCEL calculations. Weight was subsequently ignored in all other shafts. The final design consideration that we applied to all the shafts was the use of a hollow shaft. This was done primarily to conserve on weight, but also gives us a better stiffness/ mass ratio, which improves our design’s deflection resistance (as deflection was not a primary constraint, weight was the primary purpose). The hollow shaft choice meant all our equations had an outer diameter subtracted by an inner diameter. 𝑑N = 𝑑OPQ N − 𝑑SN N , 𝑛 = 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡 𝑖𝑛 𝑎𝑛𝑦 𝑔𝑖𝑣𝑒𝑛 𝑒𝑢𝑞𝑎𝑡𝑖𝑜𝑛 The inner diameter of every shaft was chosen based on an iterative process on the analysis preformed at the smallest diameter of any given shaft. This insured none of our shaft shoulders would clash with our bored inner shaft diameter. Furthermore, the 𝑑SN was chosen to be a full height H of the key chosen for our shaft. This insured that no significant stress concentration would arise between the end of the keyway 𝑑SN. These heights were chosen based off the standard recommendations provided in Norton-Machine Design-Table 10-2.
- 24. 24 The next section contains an analysis for each shaft. However, as there is significant overlap between the analysis methods, Shaft 1 will contain most of the sample calculations and equations. Further calculated references for each shaft can be found in complete expansive detail in our appendix and Excel file. ii. Shaft 1- Input Shaft Shaft 1 was designed to contain gear 1 as well as bearing 1 and 2. Bearing 2 is to be press fit onto the end, while Bearing 1 is locked onto the shaft using a clamp and spacer mechanism (the spacer extending to the motor). The shaft diameter in the bearings is identical, but thinner than that of the rest of the shaft. 1 mm clearance was added between the shoulder and the bearings to allow for thermal expansion during regular operation. The gear is supported axially by a shoulder, and fixed to the shaft using a key. All this is evident in the shaft layout presented in Figure 4, as well as in our machine drawings in the appendix Our moment diagrams, considering weight and torque are presented: A more detailed calculation rundown is provided in our excel file. We can immediately note that the point of highest Bending Moment and Torque is the same, which is at the center of our gear. Given the fact that this point will contain our keyway, it is safe to assume our highest stress concentration will be occurring there, which we named Point B. The Gear shoulder will have a larger d our gear, and so no further stress calculations are needed there. We took the shoulder to be 3mm larger than our dB . Finally, our shaft was fit into our bearings with a d < dB however, due to the low moment and lack of a significant stress concentration, we do not expect a failure point to exist there. Furthermore, we designed the dB with a safety factor range of 1.3-1.5, insuring that we are comfortable away from any potential failure points at the bearings. The d at bearings, henceforth denoted as dbr was sized based off an appropriate design choice of roller bearing. Point B: Critical Stress location. For this shaft, since we have no axial forces and constant Torsion was assumed, the formula dB was: Figure 8
- 25. 25 𝑑bO = cd∗ fgh i ∗ jg∗kl m no.pq∗ jgr∗sl m tu + jgw∗kw m no.pq∗ jgrw∗sw m txy − 𝑑bS c z { For our loading, Ta and Mm are zero. This formula was then solved for Nab, our safety fatigue safety factor for point B. 𝑁}~ = i∗(•h€ { ••h‚ { ) cd ∗ jg∗kl m tu + o.pq∗ jgrw∗sw m txy •ƒ Note, both these equations consider din . The first formula provides us with an estimate for dB , based off NfBDESIRED , which is an initial target safety factor used to provide us with our first guess. This was set to a conservative 2.25 initially. Next, this value was chosen, rounded to allow for ease of manufacturability, and then corresponding din was chosen based off the process outlined in the first section above. These choices caused equation 2 to produce our new NfB . At this point we would repeat the process again iteratively until the results were satisfying, and we have a NfB = [1.3,1.5]. Excel was an excellent tool for this iteration. A sample calculation is presented here detailing all the parameters and factors chosen for design. These results were the ones chosen for our Shaft 1. For further reference, please see the appendix and the excel file. Plug in with Desired Safety Factor of 2.25 𝑑bO = cd∗(d.dq) i ∗ ƒ.„p∗cc.p[f††] m ƒqˆ[k‰Š] + o.pq∗ d.dˆ∗podcq.ƒ [f††] m ˆ„‹ [k‰Š] − 10[𝑚𝑚]c z { 𝑑bO = 17.8325 [𝑚𝑚] Choose a 𝑑bO that is appropriate for machining: 𝑑bO = 17 𝑚𝑚 Plug into Safety Factor equation: 𝑁}~ = 𝜋 ∗ (17c − 10c ) 32 ∗ 1.67 ∗ 33.7[𝑁𝑚𝑚] d 154[𝑀𝑃𝑎] + 0.75 ∗ 2.24 ∗ 70235.1 [𝑁𝑚𝑚] d 469 [𝑀𝑃𝑎] •ƒ 𝑁}~ = 1.32 which is acceptable as a target safety factor. Note: The development shown above contained many iterations, but brevity only the final stage is presented.
- 26. 26 Material: For our shaft, we arrived at our chosen 𝑁}~ by selecting SAE 1020 Cold Rolled Steel (Table A-9, Norton). Subsequently, due to our Lifetime of 2000 hour ,our number of cycles for shaft 1 is in the range of 𝑁 = 10• 𝐶𝑦𝑐𝑙𝑒𝑠. Based on this, we shall design for an endurance limit of 𝑆– using: 𝑆– = 𝐶—S˜– 𝐶™OŠ• 𝐶—Pš} 𝐶Q–†› 𝐶š–™SŠ~ 𝑆–œ where 𝑆–œ ≅ 0.5𝑆PQ for steels. • 𝐶—S˜– was calculated using 𝐶—S˜– = 1.189𝑑•o.o‹p , and remained consistent with excel parametrization. • 𝐶™OŠ•=1 for loading in bending loads, as this shaft as no axial loads. Shaft 3 is the only shaft in which 𝐶™OŠ• = 0.70. • 𝐶—Pš} was obtained using the relationships 𝐶—Pš} ≅ 𝐴 𝑆PQ ~ , with our parameters A and b obtained from Norton Table 6-3, for Machined steel. 𝐶—Pš} ≅ 0.798. • 𝐶Q–†› =1, as our operating conditions for our gearbox is to be −40℃ ≤ 𝑇 ≤ 40℃. • 𝐶š–™SŠ~=0.814 for a reliability of 99% which seemed like an acceptable reliability range for the critical application needed for Solar Impulse. It should be noted that all parts of the aircraft undergo extensive quality testing, and so a higher reliability is unnecessary. Finally, our 𝑘} and 𝑘}—† values for our analysis at point B (keyway) were obtained from this development: Obtain 𝐾Qand 𝐾Q— from Norton Figure 10-16, an estimate of 2.2 was taken for an r/d ratio of 0.021 for the first iteration. This was later corrected as d was obtained. 𝐾Q = 2.2 𝐾Q— = 3.0 These was converted a fatigue safety factor using the Neuber equation: 𝐾} = 1 + 𝑞 𝐾Q − 1 𝑞 = 1 1 + 𝑎 𝑟 where 𝑎 is obtained from Table 6-6. Finally, Test 𝐾} 𝜎¦§¨ NO† < 𝑆ª if true then 𝐾}† = 𝐾} and 𝐾}†— = 𝐾}—. This was the case for all the shafts.
- 27. 27 iii. Shafts 2-4 - Reduction Shafts Shaft 2 and 4 are the reduction shafts, and they both have identical designs while rotating in opposite directions. Unlike Shaft 1, these two shafts have significant bending loads due to the tangential and radial forces applied at the gears. Bearing 3,4,7 and 8 are to be press fit onto the ends of both shafts. The shaft diameter in the bearings are identical, but thinner than the rest of the shaft. Just as Shaft 1, a 1 mm clearance was added between the shoulder and the bearings to allow for thermal expansion during regular operation. The gears are supported axially by a shoulder, and both are fixed to the shaft using keys. All this is evident in the shaft layout presented in Figure 5, as well as in our machine drawings in the appendix. Due to the 3- dimensional nature of the loading, we produced moment diagrams in two planes, then composed them into a total Moment diagram. Our moment diagrams, considering weight and torque: A more detailed calculation rundown is provided in our excel file. We can immediately note that the point of highest Bending Moment and Torque is the same, which is at the center of our Gear 3. Given the fact that this point will contain our keyway, it is safe to assume our highest stress concentration will be occurring there, which we named Point B. Similar reasoning for why this point will have the highest stress concentration is followed as Shaft 1. Since this diameter will be designed for failure, we shall use an identical diameter for the location at Gear 2. We followed an identical analysis procedure for fatigue failure as Shaft 1. The diametric results are shown in the results section, as well as our Excel file. Note, to size our bearings, we had to choose from a list of standard roller bearings, which forced us to tailor our bearing diameter to this shaft. To make sure these diameters do not cause failure, a safety factor analysis was done using shear stress from shear force as the only loading (as no other loading exists on those points). They were all comfortably within acceptable range. Figure 9
- 28. 28 Material: For our shaft, we arrived at our chosen 𝑁}~ by selecting SAE 1050 Cold Rolled Steel (Table A-9, Norton). Subsequently, due to our Lifetime of 2000 hour ,our number of cycles for shaft 1 is in the range of 𝑁 = 10• 𝐶𝑦𝑐𝑙𝑒𝑠. Based on this, we shall design for an endurance limit of 𝑆– using: 𝑆– = 𝐶—S˜– 𝐶™OŠ• 𝐶—Pš} 𝐶Q–†› 𝐶š–™SŠ~ 𝑆–œ All developments beyond this point are identical to Shaft 1. iv. Shaft 3 - Output Shafts Shaft 3 has slightly different considerations than the last two shafts, since it contains a cantilevered propeller weight, as well as a significant thrust load. The design methodology for the shaft itself was identical; iterate different values of din and dout until an acceptable safety factor is achieved. Bearing 5 and 6 are carefully selected to account for the thrust load. This is further expanded upon in the bearing section. Bearing 5 is to be press fit, while bearing 6 is attached using a clamp and nut arrangement. No clearance was given between the shaft shoulder and bearing 6, due to the axial thrust consideration. The shaft diameter in bearing 5 is thinner than that of the rest of the shaft, while bearing 6’s diameter will be our design diameter. A 1 mm clearance was added between the shoulder and bearing 5 to allow for thermal expansion during regular operation. The gear is supported axially by a shoulder, and fixed to the shaft using a key. Keys are typically non-ideal for attachments were axial loading is present, but our bearing 6 was designed to absorb all axial loads from the propeller. All this is evident in the shaft layout presented in Figure 6, as well as in our machine drawings in the appendix Our moment diagrams, considering weight and torque are presented: Unlike previous shafts, the point of highest stress is not so immediately clear. Two points are likely contenders, the keyway at the gear attachment, due to a high torque and moment, and the second point is at Bearing 6 which has a high torque and maximum moment. Due to this, the design analysis was repeated on those two points. It is safe to assume that the rest of the shaft Figure 10
- 29. 29 will have lower stresses than those two points, just as we did in previous shafts. We started with point B at our bearing. This formula : 𝑑bO = cd∗ fgh i ∗ jg∗kl m no.pq∗ jgr∗sl m tu + jgw∗kw m no.pq∗ jgrw∗sw m txy − 𝑑bS c z { Is not applicable for this shaft, as it was derived from the Case 3 loading of the modified Goodman diagram, with the assumption of no axial load. To remedy this, the original equations were used for the calculations instead. Namely: 1 𝑁𝑓 = 𝜎Š 𝑆– + 𝜎† œ 𝑆PQ where 𝜎Š = 𝑘} 32𝑀Š 𝜋(𝑑bO c − 𝑑bS c ) 𝜎† œ = 𝜎† Š¬SŠ™ d + 3𝜏† d o.q 𝜏† = 𝑘}—† 16𝑇† 𝜋(𝑑bO c − 𝑑bS c ) 𝜎† Š¬SŠ™ = 𝑘}† 34𝐹Š¬SŠ™ 𝜋(𝑑bO d − 𝑑bS d ) We then proceeded in a fashion similar to the previous shafts, iterating diameters until our target safety factor was achieved. We then repeated this process for the keyway point, and found it to be less critical. We chose a diameter 5mm larger than that of the bearing point. Material: For our shaft, we arrived at our chosen 𝑁}~ by selecting SAE 1050 Cold Rolled Steel (Table A-9, Norton). Subsequently, due to our Lifetime of 2000 hour ,our number of cycles for shaft 1 is in the range of 𝑁 = 10• 𝐶𝑦𝑐𝑙𝑒𝑠. Based on this, we shall design for an endurance limit of 𝑆– using: 𝑆– = 𝐶—S˜– 𝐶™OŠ• 𝐶—Pš} 𝐶Q–†› 𝐶š–™SŠ~ 𝑆–œ where 𝐶™OŠ• = 0.70, as opposed to 1 in our other shafts. Note, our 𝐾Q and 𝐾Q— were obtained from Norton Tables C-1 – C-3 for our bearing analysis , as it has a shoulder stress concentration point at that area. Otherwise, 𝑘} and 𝑘}—† were obtained identically to Shaft 1.
- 30. 30 Bearings Early on in the process of bearing selection the team decided it would be best to choose roller/ball bearings for the entire system rather than journal bearings. Ball bearings have many desirable attributes, primarily an very operating friction, which reduces frictional loses in our overall system, that would have otherwise been generated by the viscosity in a journal bearing. Rolling bearings also have no transient start-up speeds, which makes them ideal for a critical application such as an aircraft. Finally , they can handle axial and radial loads, which is desirable if the aircraft happens to operate under unexpected conditions. Roller bearings are a lot less sensitive to any potential interruptions with their lubrication as well. We used ball bearings for most of our system, where axial loads were not present. The selection of these bearings was fairly straight forward. For our axial load on the output shaft, a spherical roller bearing was chosen to absorb all the axial loads from the propeller. This choice was made so that no stress concentrations may arise in the remainder of Shaft 3, shielding the gear and preventing the transfer of any axial loads to the rest of the gearbox in the case of an unexpected fluctuation in the propeller. The Ball bearing selection process can now be outlined. For every one of our shafts we used the Free Body Diagram analysis outlined in the Shaft section (Figure 4,5, 6 and 7) to obtain the reaction forces necessary for our bearings. This allowed us to obtain reaction forces FR and FA , for radial and axial respectively. We then combined those loads using: 𝑃 = 𝑋𝑉𝐹± + 𝑌𝐹³ where X, V and Y are bearing load factors obtained from Norton Figure 11-24. Now, using 𝐿ƒo = µ ‰ c we may obtain the expected 𝐿ƒo life in millions of revolutions. This 𝐿ƒo life can then be compared to the ideal number of cycles provided as a design constraint, to appropriately select a bearing that will exceed this limit. Our 𝐿S•–Š™ = •ooo ¶š ∗„o∗·¸ ƒo¹ Where 𝜔N is the angular velocity of each shaft.
- 31. 31 The C parameter is obtained from the manufacturer specification for each bearing, and so to allow for successful iteration, a large amount bearing tables were input into our excel sheets to allow for easy selection. The tables would “iterate” these formulas for many different types of bearings, until a suitable life limit was found. We should note that was not compared directly with our , but rather it was multiplied by a reliability factor of Kr = 0.33 (for a reliability of 98%) for most of our bearing choices. We now used this new to compare with our ideal life. After a number of bearings of acceptable life were found, the particular bearing was selected based off geometrical constraints of our shaft. Since we had a very low stress state, we could afford to add a large shoulder on the bearing portion of our shaft, and so dout was not a factor. However, since we used a hollow shaft, din was the limiting parameter. The selected bearings are presented in later in this report. Note: Due to our very low loading, the life expectancies of the bearings on shaft 1 are significantly higher than necessary, however these bearings were found to be suitable for our geometry , and satisfied our needs, and thus they were chosen. The same applies for Shaft 2, 4 , however the life expectancies are not much higher than the ideal requirement. The final spherical thrust bearing was chosen from SKF , and it was chosen for a suitable its suitable geometry and life expectancies. The bearing is a Spherical Roller Bearing.
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- 36. 36 Modified Goodman Diagrams After compiling all the data in the tables above, we can now illustrate the stress and safety factors using modified Goodman Diagrams. The equations below will help us to determine the different values that construct the axis and lines. To obtain the safety factor, we also have to choose for which failure case we are solving. A standard choice would be case 3 where we assume that Ơm and Ơa will increase in a constant ratio.
- 40. 40 Appendix A – All Calculations • Full Calculations for Gear 2 (and Gear 6 by Association)
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- 44. 44 Appendix B – Figures and Tables 0 20000 40000 60000 80000 100000 120000 0 50 100 150 200 250 300 T2 -1200 -1000 -800 -600 -400 -200 0 200 400 600 800 0 50 100 150 200 250 300 V2 yz -2500 -2000 -1500 -1000 -500 0 500 1000 1500 0 50 100 150 200 250 300 V2 xy 0 10000 20000 30000 40000 50000 60000 0 50 100 150 200 250 300 M2 yz -20000 0 20000 40000 60000 80000 100000 120000 140000 0 50 100 150 200 250 300 M2 xy 0 20000 40000 60000 80000 100000 120000 140000 160000 0 50 100 150 200 250 300 M total -180000 -160000 -140000 -120000 -100000 -80000 -60000 -40000 -20000 0 0 50 100 150 200 250 300 M3 yz -2000 -1500 -1000 -500 0 500 1000 1500 2000 0 50 100 150 200 250 300 V3 yz 0 20000 40000 60000 80000 100000 120000 140000 160000 180000 0 50 100 150 200 250 300 M total -1200 -1000 -800 -600 -400 -200 0 200 400 600 800 0 50 100 150 200 250 300 V4 yz 0 20000 40000 60000 80000 100000 120000 0 50 100 150 200 250 300 T4 0 10000 20000 30000 40000 50000 60000 0 50 100 150 200 250 300 M4 yz -2500 -2000 -1500 -1000 -500 0 500 1000 1500 0 50 100 150 200 250 300 V4 xy -20000 0 20000 40000 60000 80000 100000 120000 140000 0 50 100 150 200 250 300 M4 xy 0 20000 40000 60000 80000 100000 120000 140000 160000 0 50 100 150 200 250 300 M total -80000 -70000 -60000 -50000 -40000 -30000 -20000 -10000 0 0 20 40 60 80 100 120 140 T1 Nmm -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 140 V1 (yz plane) 0 5 10 15 20 25 30 35 40 0 20 40 60 80 100 120 140 M1 yz (Nmm) 0 5 10 15 20 25 30 35 40 0 20 40 60 80 100 120 140 M total
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- 49. 49 Top View
- 50. 50 Courtesy of Engineering edge website. http://www.engineersedge.com/mechanical,045tolerances/general_iso_tolerance_.htm