3.2 Stead state model
3.2.3 Results conversion
𝐹𝑌𝑠= 𝑀𝑎𝑦 (3.24)
𝐹𝑍𝑠= 𝑀𝑔 + 𝐹𝑑𝑜𝑤𝑛𝑓𝑜𝑟𝑐𝑒 (3.25)
𝑀𝜓𝑠 = 𝐼𝜓𝜓̈ (3.26)
The model as a point-mass doesn’t consider any steering input. The lateral friction coefficient used in the simulation was obtained with the assumption that each tire is always operating at a slip angle 𝛼 that corresponds to the highest lateral load. Knowing the value of 𝛼, equation (3.15) can be used to estimate the steering angles, with beta being calculated with equation (3.27).
𝛽1= tan−1(𝑎 + 𝑅𝑐𝑜𝑟𝑛𝑒𝑟sin 𝛽 𝑅𝑐𝑜𝑟𝑛𝑒𝑟cos 𝛽 − 𝑐) 𝛽2= tan−1(𝑎 + 𝑅𝑐𝑜𝑟𝑛𝑒𝑟sin 𝛽
𝑅𝑐𝑜𝑟𝑛𝑒𝑟cos 𝛽 + 𝑑) 𝛽3= tan−1(𝑏 − 𝑅𝑐𝑜𝑟𝑛𝑒𝑟sin 𝛽
𝑅𝑐𝑜𝑟𝑛𝑒𝑟cos 𝛽 − 𝑐) 𝛽4= tan−1(𝑏 − 𝑅𝑐𝑜𝑟𝑛𝑒𝑟sin 𝛽
𝑅𝑐𝑜𝑟𝑛𝑒𝑟cos 𝛽 + 𝑑)
(3.27)
To be able to calculate the longitudinal and lateral forces on the tires, first the vertical forces must be calculated.
𝐹𝑧1= 𝐹𝑍𝑠 𝑑 (𝑐 + 𝑑)
𝑏
(𝑎 + 𝑏)− 𝐹𝑋𝑠 ℎ𝐶𝑜𝐺 (𝑎 + 𝑏)
𝑑
(𝑐 + 𝑑)− 𝐹𝑌𝑠 ℎ𝐶𝑜𝐺 (𝑐 + 𝑑)
𝑏 (𝑎 + 𝑏) 𝐹𝑧2= 𝐹𝑍𝑠 𝑐
(𝑐 + 𝑑) 𝑏
(𝑎 + 𝑏)− 𝐹𝑋𝑠 ℎ𝐶𝑜𝐺
(𝑎 + 𝑏) 𝑐
(𝑐 + 𝑑)+ 𝐹𝑌𝑠 ℎ𝐶𝑜𝐺
(𝑐 + 𝑑) 𝑏 (𝑎 + 𝑏) 𝐹𝑧3= 𝐹𝑍𝑠 𝑑
(𝑐 + 𝑑) 𝑎
(𝑎 + 𝑏)+ 𝐹𝑋𝑠 ℎ𝐶𝑜𝐺 (𝑎 + 𝑏)
𝑑
(𝑐 + 𝑑)− 𝐹𝑌𝑠 ℎ𝐶𝑜𝐺 (𝑐 + 𝑑)
𝑎 (𝑎 + 𝑏) 𝐹𝑧4= 𝐹𝑍𝑠 𝑐
(𝑐 + 𝑑) 𝑎
(𝑎 + 𝑏)+ 𝐹𝑋𝑠 ℎ𝐶𝑜𝐺
(𝑎 + 𝑏) 𝑐
(𝑐 + 𝑑)+ 𝐹𝑌𝑠 ℎ𝐶𝑜𝐺
(𝑐 + 𝑑) 𝑎 (𝑎 + 𝑏)
(3.28)
The lateral forces 𝐹𝑦𝑖 can be calculated with the vertical forces 𝐹𝑧𝑖 and the tire’s lateral friction coefficient and load sensitivity. Those lateral forces, together with the diagram presented in Figure 3-8, the results from equations (3.23), (3.24) and (3.26), and the 50% limit on the front wheels, presented in section 3.2.2, can be used to calculate the longitudinal forces on each wheel.
𝐹𝑦𝑖 = 𝜇(𝐹𝑧𝑖) × 𝐹𝑧𝑖 (3.29)
𝐹𝑥1cos 𝛿1− 𝐹𝑦1sin 𝛿1+ 𝐹𝑥2cos 𝛿2− 𝐹𝑦2sin 𝛿2+ 𝐹𝑥3+ 𝐹𝑥4− 𝐹𝑑𝑟𝑎𝑔= 𝐹𝑋𝑠 (3.30) 𝐹𝑥1sin 𝛿1+ 𝐹𝑦1cos 𝛿1+ 𝐹𝑥2sin 𝛿2+ 𝐹𝑦2cos 𝛿2+ 𝐹𝑦3+ 𝐹𝑦4 = 𝐹𝑌𝑠
(3.31)
𝐹𝑥1(sin 𝛿1𝑎 − cos 𝛿1𝑐) + 𝐹𝑥2(sin 𝛿2𝑎 + cos 𝛿2𝑑) − 𝐹𝑥3𝑐 + 𝐹𝑥4𝑑
+𝐹𝑦1(cos 𝛿1𝑎 + sin 𝛿1𝑐) + 𝐹𝑦2(cos 𝛿2𝑎 − sin 𝛿2𝑑) − 𝐹𝑦3𝑏 − 𝐹𝑦4𝑏 = 𝑀𝜓𝑠 (3.32)
Wheel travel ℎ𝑖𝑠 can be estimated using the vertical loads 𝐹𝑧𝑖 on each wheel and the respective wheel-rate (𝐾𝑖), as shown in equation (3.33).
ℎ𝑖𝑠=𝐹𝑧𝑖 𝐾𝑖
(3.33)
Knowing the magnitude of 𝐹𝑥𝑖, the torque on each wheel, 𝑇𝑖, can be calculated by multiplying the forces by the tire radius r. The speed of each wheel, 𝑣𝑖, can be estimated with equation (3.35), aided by Figure 3-8.
𝑇𝑖= 𝐹𝑥𝑖× 𝑟 (3.34)
𝑣𝑖= 𝜓̇𝐿𝑖cos 𝛼𝑖 (3.35)
With torque and speed, the output power on each wheel can be calculated. In a braking event, the power will be negative. Using the limitation imposed by FST Lisboa, regarding the battery, a maximum of 15kW can be regenerated under braking. Therefore, when the magnitude of braking power is higher than 15kW, it is considered that the remaining power above those 15kW will come from the mechanical braking system, Figure 3-24.
There’s always power loss in any type of mechanism. The dissipated power in the motors can be calculated applying the efficiency map shown in Figure C-1, to the power response plot. This dissipated power, together with the power dissipated in the drive train parts 𝐻𝐷𝑇 and the power dissipated in the
Figure 3-24 – Plot of total braking power, electrical regeneration and mechanical braking
braking system 𝐻𝑏𝑟𝑎𝑘𝑒, represents the total dissipated power 𝐻𝑡𝑜𝑡𝑎𝑙, which is useful for estimating the temperatures of all parts of the system.
3.2.4 Operating temperature estimation
One of the main factors that influences the mechanical properties of materials is temperature. To assure the proper stiffness and resistance of all components, their operating temperatures must be estimated. A purpose-developed thermal model was used, created in Simulink, where the dissipated power losses are the input and the temperatures of the various components are the output.
The model is made of interconnected thermal resistances and masses, which are used to estimate the temperatures variation over time. These resistances were obtained by making a thermal finite element analysis of the Concept 4, to which heat powers were applied in different places. In that FEA model, the applied heat power 𝐻 and the temperature differences between components ∆𝑇𝑖→𝑗 were used to calculate the resistance 𝑅𝑖→𝑗, with equation (3.36) [26].
𝑅𝑖→𝑗=∆𝑇𝑖→𝑗
𝐻 (3.36)
Faramarz and Salman [27] analysed the temperature distribution and forced convection effect on brake discs and pads. They concluded that the amount of braking power dissipated through the disc and pads is:
Figure 3-25 – Diagram of the thermal model in Simulink
𝐻𝑑𝑖𝑠𝑐= 𝐻𝑏𝑟𝑎𝑘𝑒× 𝑝ℎ𝑒𝑎𝑡 (3.37)
𝐻𝑝𝑎𝑑𝑠= 𝐻𝑏𝑟𝑎𝑘𝑒× (1 − 𝑝ℎ𝑒𝑎𝑡) (3.38)
𝑝ℎ𝑒𝑎𝑡= √𝑘𝑑𝑖𝑠𝑐𝜌𝑑𝑖𝑠𝑐𝑐𝑑𝑖𝑠𝑐
√𝑘𝑑𝑖𝑠𝑐𝜌𝑑𝑖𝑠𝑐𝑐𝑑𝑖𝑠𝑐+ √𝑘𝑝𝑎𝑑𝜌𝑝𝑎𝑑𝑐𝑝𝑎𝑑 (3.39)
Where pheat is the heat partition coefficient and the term √𝑘𝑖𝜌𝑖𝑐𝑖 is the thermal effusivity of the material.
Vidiya and Singh[28] analysed the typical air flow characteristics in regular Formula Student cars, through CFD analysis, obtaining a linear forced-convection coefficient ℎ𝑓𝑜𝑟𝑐𝑒𝑑, dependent on the car’s speed 𝑣:
ℎ𝑓𝑜𝑟𝑐𝑒𝑑= 17𝑣 + ℎ𝑛𝑎𝑡𝑢𝑟𝑎𝑙 (3.40)
To validate the model, the thermal constants were modified to correspond to the ones of the most recent car from FST Lisboa, using the same approach as before to fin the thermal resistances. For the same test session as the one already presented in section 3.1.9, sensors were mounted on the car to measure the temperature of the brake disk, pads and calliper, and the upright, in positions shown in Figure D-1, in Appendix D.
While the car is running, the brake disc is rotating, an infrared sensor (ZTP-135SR) was used to measure its temperature, as it doesn’t require any contact with the surface. This type of sensor is made of a thermopile (which are several thermocouples connected in series), and an NTC resistor, which is used to calculate the reference temperature 𝑇𝑟𝑒𝑓. Using equations (3.41) and (3.42), the measured surface temperature can be calculated.
𝑇𝑟𝑒𝑓= 1
ln ( 𝑈𝑁𝑇𝐶
𝑈𝑖𝑛𝐼𝑅− 𝑈𝑁𝑇𝐶)
3920 + 1
298.15
(3.41)
𝑇𝑑𝑖𝑠𝑐= (𝑈𝐼𝑅
𝑆 + (𝑇𝑟𝑒𝑓+ 273.15)4)
1ൗ4
− 273.15 (3.42)
where 𝑈𝑖𝑛𝐼𝑅 is the input voltage supplied to the sensor (5V), 𝑈𝑁𝑇𝐶 is the measured voltage of the NTC and 𝑈𝐼𝑅 is the measured voltage of the infra-red sensor, both in volt [V]. Temperature values in Celsius [ºC].
𝑆 is a constant that represents the sensor’s field of view, the emissivity of the disc surface, the Stefan-Boltzman constant and the sensor sensitivity, and was defined experimentally, as shown on Figure 3-26. After heating up a sample brake disc to a temperature of 100ºC, it was placed on 3 supports with minimum contact over the infra-red sensor, which was used to measure the temperature variation over time. 𝑆 = 5x10-11 was obtained by comparing the measured values of the infra-red sensor with those of an external temperature sensor, measuring the surface temperature.
With the voltage measured in the NTC and the surface temperature 𝑇𝑠𝑢𝑟𝑓, measured in the external temperature sensor, equation (3.41) can be used to calculate the temperature 𝑇𝑟𝑒𝑓, and equation (3.42) can be manipulated into equation (3.43), to find the value of K. The measured 𝑇𝑠𝑢𝑟𝑓 and 𝑇𝑟𝑒𝑓 are presented in Figure 3-28.
𝑆 = 𝑈𝐼𝑅
(𝑇𝑠𝑢𝑟𝑓+ 273.15)4− (𝑇𝑟𝑒𝑓+ 273.15)4 (3.43)
For the remaining measurement points, NTC resistors (NTCLE100E3) were used. Their temperatures, 𝑇𝑁𝑇𝐶𝑖 can be estimated with equation (3.44), where 𝑈𝑖𝑛 is 3.3V and 𝑈𝑖 is the voltage measured in each sensor.
NTC voltage
Thermopile voltage
External temperature sensor
Infra-red sensor Power supply
Figure 3-26 – Setup to obtain the value of K. The infra-red sensor is underneath the brake disc
𝑇𝑁𝑇𝐶𝑖= 1 ln ( 𝑈𝑖
𝑈𝑖𝑛− 𝑈𝑖)
3977 + 1
298.15
(3.44)
Figure 3-27 shows the acquired data overlaid with the simulation results. Comparison between measured and simulated temperatures for other components are presented in Figure 3-27. It’s clear that the simulation is well designed as the presented result is within an error margin of 6%.
Figure 3-27 – Comparison between measured and simulated temperature of the brake disc
Figure 3-28 – Measured temperature variation of 𝑇𝑠𝑢𝑟𝑓 and 𝑇𝑟𝑒𝑓
3.2.5 Simulation results
3.2.5.1 Converted results
Using equations (3.23) through (3.32), for the track shown in Figure 3-22, the forces on each wheel are obtained.
Figure 3-29 – Forces Fx, Fy and Fz on each wheel, for an autocross lap in Formula Student Spain
3.2.5.2 Temperature
With the thermal constants corresponding to Concept 4 (Appendix E), the following results were obtained from the thermal model, assuming an ambient temperature of 25ºC.
Figure 3-30 – Temperatures at the front-left (top) and rear-left (bottom) wheels, during an autocross lap in Formula Student Spain. It represents the fourth lap in a row, which justifies the high temperatures at the beginning of the plot
4 Load cases for structural analysis
The car will be subjected to different types of loading throughout its life. In the earlier days it will be subjected to mild testing, at low speed and low power, since there are multiple electrical and mechanical systems that must be checked. After that, the car will be tested at racing speeds, to find the ideal suspension setup and tune the control systems. Then the car will be subjected to the same types of loadings, but in the competition environment. Finally, there is a testing phase after the competitions, to help in the development of the following year’s car.
To simplify the simulations, the initial phase of testing was neglected, as the expected loads are very small. For the remaining phases, it was assumed that the performed tests represented the dynamic events of the competitions. For the entire life of the car, the expected number of times each event is tested is:
• Acceleration: 84 times, 4 laps each time;
• Skid-pad: 70 times, 4 laps each time;
• Autocross: 14 times, 4 laps each time;
• Endurance: 28 times, 22 laps each time.
With this information, together with the results from the steady state simulations, it is possible to estimate the total expected travelled distance about 760Km, with a total expected operating time about 55h.
4.1 Drive train load cases
The structural design of the gears was performed in KISSsoft (section 5.1.2). The software accepts a full load spectrum, with relative torque, speed and time, the latter representing the percentage of time the system is experiencing the specific torque and speed combination. Table 4-1 shows the first four entries of the spectrum, as an example.
Table 4-1 – First four entries of the drive train load spectrum for a rear wheel
Ner [%] Time [%] Torque [%] Speed
1 0.00241 80.95 45
2 0.00011 80.95 40
3 0.02804 76.19 50
4 0.20479 76.19 45
The load spectrum is the same between the front wheels and between the rear wheels, as
which apart from time, are different in each stage of the transmission system. In the first stage, the reference torque and speed are 21 Nm and 20000rpm, corresponding to the limits of the motor. The references for the second stage depend on the transmission ratio of the first stage. The software uses this spectrum to apply the Palmgren-Miner rule during the analysis.
4.2 Suspension load cases
All components, except for gears, were analysed using FEA (Finite Element Analysis). Those analyses require the definition of load cases and constraints applied to the components. From the calculations made and the results obtained in section 3.2, a load spectrum can be defined, with all the information regarding the car’s performance throughout its life. This spectrum has 877080 load cases, each representing a single set of longitudinal, lateral and vertical forces and accelerations, wheel positions, steering angles and temperatures for each wheel. Naturally this amount of information is too high to be inserted into a FEA software, therefore only the most significant points of the spectrum were considered in those analyses.
To find those reference points, the G-G diagram of the car, shown in Figure 4-1, was observed.
Each grey dot represents a point of the spectrum, with a certain longitudinal and lateral acceleration.
Figure 4-1 – G-G diagram of the car (with Concept 4), with concentric inner and outer rings
The diagram is divided into 2 concentric rings, each divided into 16 areas. These areas represent specific states of the car, such as pure accelerating or braking (areas 1 and 9), pure cornering (areas 5 and 13) and combinations of the two (remaining areas). The outer ring (Oi) represents the absolute peak performance of the car, so from a structural point of view they correspond to the critical points in terms of stress, whereas the inner circle (Ii) represents the load cases that the system will experience more often. Each yellow dot is the selected reference point the respective area, corresponding to a single set of reference loads. The car is considered to be symmetrical, so both wheels on each axle must have
O1
O2
O3
O4
O5
O6
O7
O8
O9
O10
O11
O12
O13
O14
O15
O16
I15
I1
I2
I3
I4
I5
I6
I7
I8
I9
I10
I11
I12
I13
I14
I16
the axle. For the initial finite element analyses, presented in sections 5.1, 5.2 and 5.3, only the maximum load cases from sectors O1, O3, O5, O7, O9, O11, O13 and O15 are considered, which are shown in Table 4-2 and Table 4-3.
Table 4-2 – Maximum reference load cases for each wheel of the front axle Maximum load cases on the front axle
ID. 𝑭𝒙 [N] 𝑭𝒚 [N] 𝑭𝒛 [N] 𝑴𝝋 [Nm] 𝜹 [º] 𝒉𝒔 [mm]
O1 832 0 264 -25 0.0 -8.7
O3 360 442 705 10 1.3 -0.1
O5 1 507 741 23 -1.7 0.6
O7 -915 609 1252 65 1.3 10.5
O9 -1755 0 1823 53 0.0 21.6
O11 -1863 -1332 1915 133 1.9 23.3
O13 22 -2750 2156 111 1.9 28.0
O15 557 -477 756 -2 -1.7 0.9
Table 4-3 – Maximum reference load cases for each wheel of the rear axle Maximum load cases on the rear axle
ID. 𝑭𝒙 [N] 𝑭𝒚 [N] 𝑭𝒛 [N] 𝑴𝝋 [Nm] 𝜹 [º] 𝒉𝒔 [mm]
O1 1359 0 1159 -41
0.0
9.4
O3 712 513 755 -17 1.3
O5 2 424 560 4 -2.6
O7 -401 323 448 15 -4.8
O9 -715 0 742 21 1.1
O11 -890 -1350 1095 15 8.1
O13 43 -2860 1960 -27 25.5
O15 1158 -1629 1606 -49 18.4
Even though the thermal model of section 3.2.4 estimates the temperature variation through time on each component, the temperatures considered for the structural design are the maximum experienced by each part, which are the ones presented in Table 4-4.
Table 4-4 – Temperature of components considered in the structural analyses Temperature [ºC]
Brake disc Calliper Drive train Hub Upright
Front axle 500 100 80 80 70
Rear axle 250 80 80 80 60
4.3 Fatigue
Structural components are frequently subjected to repeated loads, and the resulting cyclic stresses create microscopic physical damage in their materials. Even at stresses significantly below the ultimate strength of the material, the microscopic damage can accumulate with the continued cyclic loading, developing cracks or other macroscopic damages that lead to failure of the component [9].
To account for the variance of stress amplitude during the life of components, Palmgren in 1924 and Miner in 1945, introduced the hypothesis of linear damage accumulation, which is a simple tool for service-life assessment. The resulting Palmgren-Miner rule, characterized by equation (4.1), is used to estimate the cumulative damage on each component. If the estimated total damage is greater than 1, then the rule states a failure is highly likely to occur [8][5].
∑𝑁𝑖 𝑁𝑓𝑖
= 𝐷 (4.1)
where 𝑁𝑖 is the number of cycles the component will experience at a certain stress, 𝑁𝑓𝑖 is the maximum number of cycles the component can withstand at the same stress, without failing, and D is the total accumulated damage, that must be lower than 1.
To use this rule, it’s necessary to know the applied stresses and the corresponding number of cycles. Three random points of each sector of Figure 4-1 are selected, each corresponding to a set of loads that are applied to the FEA model, as presented in section 5.5, resulting in a certain stress distribution in each component. While analysing the stress result of the several components, the location of the highest stress (critical) in each component can be found.
The Von-Mises stress corresponding to the 3 random load cases (𝜎1, 𝜎2, 𝜎3), in the most critical location of each component are inserted into equation (4.2), together with the respective loads of the set, to find constants 𝑠1, 𝑠2 and 𝑠3, which characterize each sector of Figure 4-1. After applying this process to every sector, the complete stress spectrum for every component is obtained, to which the Rainflow method can be applied.
{ 𝑠1 𝑠2 𝑠3} = [
𝐹𝑥1 𝐹𝑦1 𝐹𝑧1 𝐹𝑥2 𝐹𝑦2 𝐹𝑧2
𝐹𝑥3 𝐹𝑦3 𝐹𝑧3 ]
−1
{ 𝜎1 𝜎2
𝜎3} (4.2)
where the indices 1, 2 and 3 correspond to the three selected points per sector.
The Rainflow method is a cycle counting algorithm, suitable for fatigue analysis of structures, that reduces the length of a varying stress spectrum into a set of stress reversals. Originally developed by Tatsuo Endo and M.Matsuishi in 1689, it was improved by Downing and Socie in 1982, becoming one
from the traditional pagoda roofs. This method was applied using a dedicated Matlab function called
“rainflow”, resulting in the number of cycles 𝑁𝑖, and mean and amplitude stresses (𝜎𝑚 and 𝜎𝑎).
Mean and amplitude stresses are necessary to calculate 𝑁𝑓𝑖. Equation (4.3) corresponds to the modified Goodman fatigue criterion, which was developed by Smith in 1942, based on an early proposal by Goodman [9][5]. This criterion is represented by a straight line, as shown in Figure 4-2, which is a conservative approach, so the consequence of its error is an increase in the factor of safety.
Figure 4-2 – Fatigue diagram for various criteria of failure. For each criterion, points “above” the respective line indicate failure [5]
𝜎𝑎 𝑆𝑒+𝜎𝑚
𝑆𝑢 =1 𝑛
(4.3)
where 𝑛 is the factor of safety, 𝑆𝑢 is the tensile strength of the material and 𝑆𝑒 is the endurance limit of the component. If the maximum applied stress is smaller than 𝑆𝑒, it is considered that the component will have an infinite life, so fatigue won’t be the cause of any failure. For the case being presented in this document, 𝑆𝑒 was considered as the stress value at 𝑁𝑓𝑖 = 107 cycles, in the S-N curve of each material.
When the maximum applied stress is greater than 𝑆𝑒, fatigue will have an impact, which can be evaluated with equation (4.4).
𝑆𝑓 = { 𝜎𝑎
1 𝑛 −
𝜎𝑚 𝑆𝑢
, 𝜎𝑎> 𝑆𝑎 𝑆𝑒 , 𝜎𝑎≤ 𝑆𝑎
(4.4)
𝑆𝑎= 𝑟𝑆𝑒𝑆𝑢
𝑟𝑆𝑢+ 𝑆𝑒 (4.5)
Yield (Langer) line
Gerber line
Load line
Modified Goodman line
ASME-elliptic line
Midrange stress 𝜎𝑚 Alternating stress𝜎𝑎
𝑆𝑦
𝑆𝑒
𝑆𝑎
𝑆𝑚 𝑆𝑦 𝑆𝑢
0 0
Soderberg line
𝐴 = 𝜎𝑎
𝜎𝑚 (4.6)
𝜎𝑎=𝜎𝑚𝑎𝑥− 𝜎𝑚𝑖𝑛
2
(4.7)
𝜎𝑚 =𝜎𝑚𝑎𝑥+ 𝜎𝑚𝑖𝑛 2
(4.8)
where 𝑆𝑓 is the fatigue strength, 𝑆𝑎 is the limiting value of 𝜎𝑎,𝐴 is the ratio between amplitude and mean stresses, and 𝜎𝑚𝑎𝑥 and 𝜎𝑚𝑖𝑛 are the maximum and minimum applied stresses, respectively [5].
If 𝜎𝑎 > 𝑆𝑎, there will be reduction in life due to fatigue. That reduction is evaluated through the maximum number of cycles until failure, 𝑁𝑓𝑖 , calculated with equations (4.9), (4.13) and (4.14) [13].
log𝑁𝑓𝑖 = 18.21 − 7.73log(𝑆𝑓∗(1 − 𝑅)0.62− 10) (4.9)
log𝑁𝑓𝑖 = 7.56 − 2.73log(𝑆𝑓∗(1 − 𝑅)0.4− 23.7) (4.10)
log𝑁𝑓𝑖 = 7.77 − 2.15log(𝑆𝑓∗(1 − 𝑅)0.79− 28.32) (4.11)
𝑅 = 𝜎𝑚𝑖𝑛
𝜎𝑚𝑎𝑥 (4.12)
where 𝑆𝑓∗= 𝑆𝑓/6.895 and 𝑅 is the stress ratio. They represent the S-N curve of the materials presented in chapter 5, which are aluminium 7075-T6, magnesium ZK60A and steel AISI H13, respectively.
Based on the past experience of FST Lisboa, the Pugsley safety factor model was used to define the factor of safety, n, for the structural design. A factor of n=1.5 was obtained after applying this method [30].
4.4 Stiffness targets
All bodies deform under load, either elastically or plastically [5] If their deflection isn’t controlled, the system in which they are inserted won’t perform as intended.
The suspension geometry is designed with the assumption that all parts suspension system have infinite stiffness, and therefore no deformation. Santos [4] has shown that the compliance of the
suspension system does in fact affect the overall performance of the car, so a stiffness target must defined. Reaching for the highest possible stiffness is not necessarily the most correct approach, as for two parts made of the same material and with similar shape, higher stiffness means more weight, which is detrimental to the performance of the car. A balance must be found, between stiffness and weight, that provides the best overall result.
In recent competitions, the performance difference between teams has been relatively small, which means small gains in the performance may be translated into significant improvements in the final classification order. The most useful dynamic event to observe for this analysis is the skid-pad event, as the others are more dependent on the skill of the drivers. Table 4-5 shows the skid-pad best results from Formula Student Spain 2018.
Table 4-5 – Skid-pad results for Formula Student Spain 2018 - Electric
Skid-pad results in Formula Student Spain 2018 - Electric
Place Car num. Team Best time [s] Score
1 E77 DHBW Stuttgart 4,871 75,0
2 E31 TU München 4,895 73,1
3 E40 Eindhoven University of Technology 5,029 62,7
4 E26 Universität Stuttgart 5,070 59,7
5 E14 DHBW Ravensburg 5,096 57,8
6 E13 UAS Munich 5,101 57,5
7 E33 ETH Zurich 5,142 54,6
8 E81 Karlsruhe Institute of Technology 5,178 52,1
9 E8 Chalmers University of Technology 5,180 52,0
10 E146 Politecnico di Torino 5,181 51,9
It is expected, based on the results for the simulations of section 3.2, to have a skid-pad time of about 4.9s. On that range, to lose a position the car had to be 0.129s slower, which is equivalent to a lateral acceleration variation of 0.7m/s2, or 5%, as shown in equations (4.13), (4.14) and (4.15). From the stiffness point of view, a goal was to assure the entire system (including wheel rim and suspension links) doesn’t have a toe-angle or camber-angle variation greater those 5%, relative to the angle values that correspond to the maximum lateral performance (±0.25º or camber-angle and ±0.35º for toe-angle).
𝑎𝑦= 𝑣2
𝑅𝑐𝑜𝑟𝑛𝑒𝑟 =(2𝜋𝑅𝑐𝑜𝑟𝑛𝑒𝑟/𝑡)2
𝑅𝑐𝑜𝑟𝑛𝑒𝑟 =4𝜋2𝑅𝑐𝑜𝑟𝑛𝑒𝑟
𝑡2 (4.13)
Δ𝑎𝑦= 𝑎𝑦2− 𝑎𝑦1 = 4𝜋2𝑅𝑐𝑜𝑟𝑛𝑒𝑟(1 𝑡22− 1
𝑡12) (4.14)
𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 =Δ𝑎𝑦
𝑎𝑦1 × 100% (4.15)
Analysing the stiffness of a transmission system is a complex task. The shafts and bearings that hold the gears must have small deflections, otherwise the gears won’t mesh properly. It also must have low distortion (torsion), or there will be a significant delay in the system’s response. During the structural design in section 5.1.2, a centre distance tolerance of +/-0.0105mm (JS7) is set, to assure the proper functioning of the transmission. That tolerance serves two purposes, the first is a machining and assembly tolerance, and the other is a margin for the deflections of the parts. Both purposes have high importance on service, so the tolerance for machining and assembly was reduced to JS5 (+-0.0045mm) and the remaining +/-0.006mm is the maximum acceptable variation due to deflection.
5 Final solution
Iterative design is a methodology based on a cyclic process with the several consecutive steps of the design. Based on the results of the last step of each cycle, changes and refinements are made, and the process is restarted. This type of process is intended to ultimately improve the quality of the design. Figure 5-1 shows the iterative process applied to the system being developed [31].
In chapter 2 it was shown that the best solution to the system is to have the motor inside the upright, followed by a two-stage planetary gear train. Concept 4 is the starting point for the iterative process and is only a representation of that kind of solution, so further development must be performed, to obtain a feasible design. This chapter will only show the final result of the iterative process.
There are two main considerations to be taken in mechanical design, while trying to achieve the performance targets: the systems must be designed to be functional, serviceable and easy to manufacture and assemble. All the components of the system must have enough resistance to withstand the applied loads during their expected life.
Different manufacturing processes are considered differently during design process. Only processes that are applicable to small scale production and accessible to the FST Lisboa team are considered: CNC machining, conventional machining, hobbing and W-EDM. Laser-sintering, or other 3D printing processes, allow the production of highly complex geometries, however, the anisotropic properties of the material (after production) and the distortion due to the high temperatures while fusing layers, means that a deeper analysis must be performed to properly use this process.
For these production processes and the overall temperature and loads applied to the system (section 4.2), only metallic materials are recommended for the components. Aluminium, titanium, magnesium or steel alloys are the most common materials used in motorsport for similar applications to the one being studied [32].
CAD modelling
Machinability
&
serviceability
FEA Fatigue
verification
Figure 5-1 – Illustration of the iterative process used in the design