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Mercedes-Benz 9G-Tronic transmission

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9G-Tronic
Cutaway model of the transmission with components for hybrid drive
Overview
ManufacturerDaimler AG
Jatco Ltd
Production2013–present
Body and chassis
Class9-speed longitudinal automatic transmission
RelatedZF 8HP · Aisin-Toyota 8-speed · Ford-GM 10-speed
Chronology
Predecessor7G-Tronic

9G-Tronic is Mercedes-Benz's trademark name for its 9-speed automatic transmission, starting off with the W9A 700 (Wandler-9-Gang-Automatik bis 700 N⋅m Eingangsdrehmoment; converter-9-gear-automatic with 700 N⋅m (516 lb⋅ft) maximum input torque; type 725.0[1]) as core model. The transmission debuted on the E 350 BlueTEC in 2013,[1] and successively replaced both the 7-speed 7G-Tronic (PLUS) transmission and the 5-speed 5G-Tronic transmission. It includes versions for a maximum input torque of 1,000 N⋅m (738 lb⋅ft).[2]

After the 5G- and 7G-Tronic, this is the 3rd generation of modern automatic transmissions. It is identified internally as NAG3 (New Automatic Gearbox 3rd generation).[3]

Gear Ratios[a]
Gear
Model
R 1 2 3 4 5 6 7 8 9 Total
Span
Span
Center
Avg.
Step
Compo-
nents
W9A All · 2013 −4.932 5.503 3.333 2.315 1.661 1.211 1.000 0.865 0.717 0.601 9.150 1.819 1.319 4 Gearsets
3 Brakes
3 Clutches
W9A All · 2016 −4.798 5.354 3.243 2.252 1.636 1.211 1.000 0.865 0.717 0.601 8.902 1.795 1.314
9AT All · 2019 −4.799 5.425 3.263 2.250 1.649 1.221 1.000 0.862 0.713 0.597 9.091 1.799 1.318
  1. ^ Differences in gear ratios have a measurable, direct impact on vehicle dynamics, performance, waste emissions as well as fuel mileage

Development, production and licensing to Jatco Ltd

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Development took place at the group's headquarters in Stuttgart-Untertuerkheim.[1] Initially, the transmission was produced only at the Daimler plant not far away in Stuttgart-Hedelfingen.[3] Since April 2016, the transmission has also been produced at Daimler's subsidiary Star Assembly in Sebeș, Romania.[4]

In 2019, the Jatco Ltd, based in Fuji, Shizuoka, Japan, started licensed production for use in Nissan and Infiniti vehicles.[5][6] In this version, input torque is limited to 700 N⋅m (516 lb⋅ft), allowing each of the gearsets 1, 2, and 4 to use only three planetary gears.[7] Slightly modified gear dimensions give it a span of about 9.09:1.

Specifications

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Torque converter

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The other main focus was on increasing shift comfort, which is achieved on the one hand by measures in the control system and on the other hand by designing the torque converter accordingly. The hydrodynamic torque converter was largely taken over from the previous 7G-Tronic transmission. It has a torque converter lock-up clutch and a twin-turbine harmonic damper with centrifugal pendulum absorber technology.[8] The torque converter lock-up can operate in all 9 forward gears.

Control system

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The 9G-Tronic is fully electronically controlled. The shift elements are controlled via a new type of hydraulic direct control with electromagnetically actuated valves, which enables fast and smooth gear changes. Compared to the previous transmission, which had a hydraulic pilot control, leakage losses have been reduced by 80%.[8]

Oil supply

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The transmission is equipped with two oil pumps to ensure an energy-efficient supply of long-life synthetic fuel-economy low-friction oil: a mechanical rotary vane pump with chain drive, which is significantly smaller than its predecessor and located next to the main shaft, and a pump driven by a brushless electric DC motor.[8] The mechanically driven pump is responsible for the basic supply of the transmission, with the flow rate depending on the speed of the drive motor. The additional pump is switched on by the electronic transmission control unit as required. This design enables the lubricating and cooling oil volume flow to be regulated as required and makes the 9G-Tronic start/stop-capable.[1] When the drive motor is at a standstill, the transmission remains ready to start solely due to the supply from the electric auxiliary pump.

Filter elements for the two pumps are integrated in the plastic oil pan.

AMG SpeedShift 9G

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AMG SpeedShift TCT 9G

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The TCT 9G (Torque Converter Technology) transmission is essentially the 9G-Tronic.

AMG SpeedShift MCT 9G

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Mercedes-AMG developed the MCT 9G (Multi Clutch Technology) transmission. It made its debut in the Mercedes-AMG E 63 4Matic+.

The MCT transmission is essentially the 9G-Tronic with a start-off wet clutch (NAK for German Nass-Anfahrkupplung) replacing the torque converter. This saves weight and optimises the response to the accelerator pedal input. It is a computer-controlled double-clutching.[9] The MCT acronym refers to the planetary transmission's multiple clutches and brakes. Its torque is rated at 900 N⋅m (664 lb⋅ft) and it offers 4 drive modes: “C” (Comfort), “S” (Sport), “S+” (Sport plus) and “M” (Manual) and boasts 0.1 second shifts in “M” and “S+” modes. MCT-equipped cars are also fitted with the new AMG Drive Unit as the central control unit for all driving dynamics functions and an innovative Race Start Function.

The driver can change gears either using the steering-wheel shift paddles or conventionally the selector lever. The new Race Start Function is a launch control system that enables maximum acceleration while ensuring optimum traction of the driven wheels.

Layout (Gearset Concept) Progress

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The main objectives in replacing the previous 7G-Tronic model were to improve fuel consumption by adding gears and increasing the gear span, while at the same time reducing manufacturing costs.

The wide gear span[a] allows the engine speed level to be lowered (downspeeding), which is a decisive factor in improving energy efficiency and thus reducing fuel consumption by 6.5 %.[8] In addition, the lower engine speed level improves the noise-vibration-harshness comfort and the exterior noise is reduced by up to 4 dB(A).[1] A speed of 120 km/h is reached in the Mercedes-Benz E 350 BlueTEC in 9th gear at an engine speed of approx. 1350 rpm.[11] By the end of 2024 unsurpassed ratio span among longitudinal automatic transmissions for passenger cars.

  1. ^ First version with a gear ratio span wider than 9:1.[10] Discontinued 2016

Progress Quality

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As the design of the predecessor was significantly more complex than that of the direct competitor 6HP and even the new 8HP model from ZF with one more gear, the specification sheet also stipulate that at least one shift element must be omitted. This was achieved thanks to high-speed computer-aided design and has resulted in a globally patented gearset concept that requires the same installation space as the previous model and is also 1 kg (2.2 lb) lighter.[2] In the process, 85 billion gearset concepts were examined.[12] Additionally, the unit brings the ability to shift in a non-sequential manner – going from gear 9 to gear 4 in extreme situations simply by changing one shift element (actuating brake C and releasing brake A).

After the 5G- and 7G-Tronic, this transmission is the 3rd generation[3] in which in-line epicyclic gearing have been combined with parallel epicyclic gearing. The resulting progress is reflected in an even better ratio between the number of gears and the number of components used compared to all layouts previously used by Mercedes-Benz.

Innovative Strength Analysis
With
Asessment
Output:
Gear
Ratios
Innovation
Elasticity[a]
Δ Output : Δ Input
Input: Main Components
Total Gearsets Brakes Clutches
W9A
Ref. Object

Topic[a]



Δ Number
Relative Δ Δ Output

·
Δ Input
W9A
W7A[b]
9[c]
7[d]
Progress[a] 10
11[13]
4
4[e]
3
4
3
3
Δ Number 2 -1 0 -1 0
Relative Δ 0.286
−3.143[a]
·
−0.091
0.000
−0.250
0.000
W9A
8HP[f]
9[c]
8[c]
Current
Market Position[a]
10
9
4
4
3
2
3
3
Δ Number 1 1 0 1 0
Relative Δ 0.125
1.125[a]
·
0.111
0.000
0.500
0.000
9A
3-Speed[g]
9[c]
3[c]
Historical
Market Position[a]
10
7
4
2
3
3
3
2
Δ Number 6 3 2 0 1
Relative Δ 2.000
4.667[a]
·
0.429
1.000
0.000
0.500
  1. ^ a b c d e f g h Innovation Elasticity Classifies Progress And Market Position
    • Automobile manufacturers drive forward technical developments primarily in order to remain competitive or to achieve or defend technological leadership. This technical progress has therefore always been subject to economic constraints
    • Only innovations whose relative additional benefit is greater than the relative additional resource input, i.e. whose economic elasticity is greater than 1, are considered for realization
    • The required innovation elasticity of an automobile manufacturer depends on its expected return on investment. The basic assumption that the relative additional benefit must be at least twice as high as the relative additional resource input helps with orientation
      • negative, if the output increases and the input decreases, is perfect
      • 2 or above is good
      • 1 or above is acceptable (red)
      • below this is unsatisfactory (bold)
  2. ^ Direct Predecessor
    • To reflect the progress of the specific model change
  3. ^ a b c d e plus 1 reverse gear
  4. ^ plus 2 reverse gears
  5. ^ of which 2 gearsets are combined as a compound Ravigneaux gearset
  6. ^ Current Reference Standard (Benchmark)
    • The 8HP has become the new reference standard (benchmark) for automatic transmissions
  7. ^ Historical Reference Standard (Benchmark)
    • 3-speed transmissions with torque converters have established the modern market for automatic transmissions and thus made it possible in the first place, as this design proved to be a particularly successful compromise between cost and performance
    • It became the archetype and dominated the world market for around 3 decades, setting the standard for automatic transmissions. It was only when fuel consumption became the focus of interest that this design reached its limits, which is why it has now completely disappeared from the market
    • What has remained is the orientation that it offers as a reference standard (point of reference, benchmark) for this market for determining progressiveness and thus the market position of all other, later designs
    • All transmission variants consist of 7 main components
    • Typical examples are

Layout Quality

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The ratios of the 9 gears are better distributed in all versions than in the direct competitors 8HP from ZF and much better than in the 10-speed transmissions from Ford/GM and Aisin/Toyota. The only noticable weakness is the relativly small step between 5th and 6th gear and the too small one between 6th and 7th gear. This cannot be eliminated without affecting all other gears. On the other hand, this weakness is not overly significant.

All in all

  • the extra effort, reflected in the acceptable elasticity compared to the ZF 8HP is more than justified and
  • compared to the 10-speed gearboxes from Ford/GM and Aisin/Toyota, the absence of the 10th gear is more than compensated for by the significantly better distribution.
Gear Ratios
With Assessment Weight Planetary Gearset: Teeth[a] Count Total[b]
Center[c]
Avg.[d]
Simpson Simple[e]
Model
Type
Version
First Delivery
with Con-
verter + Oil
S1[f]
R1[g]
S2[h]
R2[i]
S3[j]
R3[k]
S4[l]
R4[m]
Brakes
Clutches
Ratio
Span
Gear
Step[n]
Gear
Ratio
R
1
2
3
4
5
6
7
8
9
Step[n] [o] [p]
Δ Step[q][r]
Shaft
Speed
Δ Shaft
Speed[s]
Torque
Ratio[t]
Efficiency[t] ηR η1 η2 η3 η4 η5 η6 η7 η8 η9
W9A ALL
725.0
1,000 N⋅m (738 lb⋅ft)[2]
2013[u]
95 kg (209 lb)[2] 46
98
44
100
36
84
34
86
3
3
9.1495
1.8194
1.3188[n]
Gear
Ratio
−4.9316[o]
5.5032
3.3333
2.3148
1.6611[r]
1.2106
1.0000[s]
0.8651[r][s]
0.7167
0.6015
Step 0.8961[o] 1.0000 1.6510 1.4400 1.3935 1.3722 1.2106 1.1559 1.2072 1.1915
Δ Step[q] 1.1465 1.0333 1.0156[r] 1.1335 1.0473 0.9575[r] 1.0131
Speed -1.1159 1.0000 1.6510 2.3774 3.3130 4.5459 5.5032 6.3611 7.6789 9.1495
Δ Speed 1.1159 1.0000 0.6510 0.7264 0.9356 1.2329 0.9573[s] 0.8579[s] 1.3178 1.4706
Torque[t] –4.6393 5.2806 3.2633 2.2450 1.6274 1.1957 1.0000 0.8578 0.7104 0.5902
Efficiency[t] 0.9407 0.9595 0.9790 0.9698 0.9797 0.9877 1.0000 0.9915 0.9913 0.9813
W9A ALL
725.0
1,000 N⋅m (738 lb⋅ft)[2]
2016[v]
95 kg (209 lb)[2] 46
98
44
100
37
83
34
86
3
3
8.9022
1.7946
1.3143[n]
Gear
Ratio
−4.7983[o]
5.3545
3.2432
2.2523
1.6356[r]
1.2106
1.0000[s]
0.8651[r][s]
0.7167
0.6015
Step 0.8961[o] 1.0000 1.6510 1.4400 1.3770 1.3511 1.2106 1.1559 1.2072 1.1915
Δ Step[q] 1.1465 1.0457 1.0192[r] 1.1160 1.0473 0.9575[r] 1.0131
Speed -1.1159 1.0000 1.6510 2.3774 3.2737 4.4231 5.3545 6.1892 7.4714 8.9022
Δ Speed 1.1159 1.0000 0.6510 0.7264 0.8964 1.1493 0.9314[s] 0.8347[s] 1.2822 1.4308
Torque[t] –4.5151 5.1392 3.1759 2.1849 1.6031 1.1957 1.0000 0.8578 0.7104 0.5902
Efficiency[t] 0.9410 0.9598 0.9793 0.9701 0.9802 0.9877 1.0000 0.9915 0.9913 0.9813
Jatco 9AT
JR913E
700 N⋅m (516 lb⋅ft)
2019[w]
99.5 kg (219 lb)[16] 45
96
41
91
38
86
37
92
3
3
9.0910
1.7994
1.3177[n]
Gear
Ratio
−4.7991[o]
5.4254
3.2632
2.2496
1.6491[r]
1.2213
1.0000[s]
0.8619[r][s]
0.7132
0.5968
Step 0.8846[o] 1.0000 1.6626 1.4505 1.3641 1.3503 1.2213 1.1603 1.2085 1.1950
Δ Step[q] 1.1462 1.0634 1.0102[r] 1.1056 1.0526 0.9601[r] 1.0113
Speed -1.1305 1.0000 1.6626 2.4117 3.2899 4.4423 5.4254 6.2950 7.6074 9.0910
Δ Speed 1.1305 1.0000 0.6626 0.7491 0.8782 1.1525 0.9831[s] 0.8696[s] 1.3124 1.4836
Torque[t] –4.5149 5.2061 3.1953 2.1818 1.6161 1.2055 1.0000 0.8544 0.7069 0.5855
Efficiency[t] 0.9408 0.9596 0.9792 0.9699 0.9800 0.9871 1.0000 0.9914 0.9912 0.9810
Ratio
R & Even
[e]
Ratio
Odd
Algebra And Actuated Shift Elements[x]
Brake A[y]
Brake B[z] (❶)[aa]
Brake C[ab] [e]
Clutch D[ac]
Clutch E[ad]
Clutch F[ae] [e]
  1. ^ Layout
    • Input and output are on opposite sides
    • Planetary gearset 1 is on the input (turbine) side
    • Input shafts are S1, C4 (planetary gear carrier of gearset 4), and, if actuated, C1 (planetary gear carrier of gearset 1)
    • Output shaft is C3 (planetary gear carrier of gearset 3)
  2. ^ Total Ratio Span (Total Ratio Spread · Total Gear Ratio)
    • A wider span enables the
      • downspeeding when driving outside the city limits
      • increase the climbing ability
        • when driving over mountain passes or off-road
        • or when towing a trailer
  3. ^ Ratio Span's Center
    • The center indicates the speed level of the transmission
    • Together with the final drive ratio
    • it gives the shaft speed level of the vehicle
  4. ^ Average Gear Step
    • With decreasing step width
      • the gears connect better to each other
      • shifting comfort increases
  5. ^ a b c d Except in 4th gear when used in the Simpson configuration
  6. ^ Sun 1: sun gear of gearset 1
  7. ^ Ring 1: ring gear of gearset 1
  8. ^ Sun 2: sun gear of gearset 2
  9. ^ Ring 2: ring gear of gearset 2
  10. ^ Sun 3: sun gear of gearset 3
  11. ^ Ring 3: ring gear of gearset 3
  12. ^ Sun 4: sun gear of gearset 4
  13. ^ Ring 4: ring gear of gearset 4
  14. ^ a b c d e Standard 50:50
    — 50 % Is Above And 50 % Is Below The Average Gear Step —
    • With steadily decreasing gear steps (yellow highlighted line Step)
    • and a particularly large step from 1st to 2nd gear
      • the lower half of the gear steps (between the small gears; rounded down, here the first 4) is always larger
      • and the upper half of the gear steps (between the large gears; rounded up, here the last 4) is always smaller
    • than the average gear step (cell highlighted yellow two rows above on the far right)
    • lower half: smaller gear steps are a waste of possible ratios (red bold)
    • upper half: larger gear steps are unsatisfactory (red bold)
  15. ^ a b c d e f g Standard R:1
    — Reverse And 1st Gear Have The Same Ratio —
    • The ideal reverse gear has the same transmission ratio as 1st gear
      • no impairment when maneuvering
      • especially when towing a trailer
      • a torque converter can only partially compensate for this deficiency
    • Plus 11.11 % minus 10 % compared to 1st gear is good
    • Plus 25 % minus 20 % is acceptable (red)
    • Above this is unsatisfactory (bold)
  16. ^ Standard 1:2
    — Gear Step 1st To 2nd Gear As Small As Possible —
    • With continuously decreasing gear steps (yellow marked line Step)
    • the largest gear step is the one from 1st to 2nd gear, which
      • for a good speed connection and
      • a smooth gear shift
    • must be as small as possible
      • A gear ratio of up to 1.6667:1 (5:3) is good
      • Up to 1.7500:1 (7:4) is acceptable (red)
      • Above is unsatisfactory (bold)
  17. ^ a b c d From large to small gears (from right to left)
  18. ^ a b c d e f g h i j k l m Standard STEP
    — From Large To Small Gears: Steady And Progressive Increase In Gear Steps —
    • Gear steps should
      • increase: Δ Step (first green highlighted line Δ Step) is always greater than 1
      • As progressive as possible: Δ Step is always greater than the previous step
    • Not progressively increasing is acceptable (red)
    • Not increasing is unsatisfactory (bold)
  19. ^ a b c d e f g h i j k l m Standard SPEED
    — From Small To Large Gears: Steady Increase In Shaft Speed Difference —
    • Shaft speed differences should
      • increase: Δ Shaft Speed (second line marked in green Δ (Shaft) Speed) is always greater than the previous one
    • 1 difference smaller than the previous one is acceptable (red)
    • 2 consecutive ones are a waste of possible ratios (bold)
  20. ^ a b c d e f g h Torque Ratio And Efficiency
    • Ratio of output torque to input torque
    • Assumed efficiency η0 of the combined epicyclic meshing for
    • Assumed efficiency η0 of a single meshing:
  21. ^ First version with a gear ratio span wider than 9:1.[10] Discontinued 2016
  22. ^ 2nd version introduced without notice with the Mercedes-Benz E-Class (W213)
    • to reduce the step between gear 4 and 5 below the one of the 7G-Tronic (1.3684:1 [26:19])[14]
    • AMG SpeedShift MCT 9G is rated at 900 N⋅m (664 lb⋅ft)[9]
  23. ^ Under license from Daimler[15]
  24. ^ Permanently coupled elements
    • R1 and C2
    • R2, S3, and S4
  25. ^ Blocks C1 (carrier 1)
  26. ^ Blocks S2
  27. ^ Not involved. Only serves to maintain the shift logic: only one shift element is changed for step up or down
  28. ^ Blocks R3
  29. ^ Couples S1 with C1 (carrier 1)
  30. ^ Couples C1 (carrier 1) with R2
  31. ^ Couples C3 (carrier 3) with R4

Maintenance

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Compared to the predecessor gearboxes NAG1 (5G-Tronic) and NAG2 (7G-Tronic), the NAG3 gearbox is much more highly integrated, meaning that repairs are only possible by replacing entire assemblies when servicing is required.[1] This applies, for example, to the oil filters permanently integrated in the plastic oil pan.[10] Another example is the fully integrated mechatronic module with sensors, control unit and electrohydraulic shift plate. This module must be replaced as a unit, even if, for example, only one sensor is defective.[10]

Nomogram

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Planetary gearSet 1Planetary gearSet 2Planetary gearSet 3Planetary gearSet 4

▶️ Interactive Nomogram

This nomogram is a real geometric calculator exactly representing the rotational speeds of the transmission's 3x4 = 12 internal shafts for each of its 9 ratios (+ reverse), grouped according to their 4 permanent coupling on 3 joint ordinates and 5 independent ordinates. These ordinates are positioned on the abscissa in strict accordance with the proportions of the sun gears' teeth numbers relative to those of their rings. Consequently, the output ratios on the 3rd ordinate (carrier of the third planetary gearset) follows closely those of the actual transmission. This advantageous geometric construction sets us free from Robert Willis' famous and tedious formula,[17] because all calculations are exclusively determined by lengths ratios, respectively teeth numbers on the abscissa for the 4 epicyclic ratios, and of rotational speeds on the 3rd ordinate for the 10 gear ratios.

This nomogram reflects the version from 2013.

Legend

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A: Brake (blocks S2 sun gear)
B: Brake (blocks R3 ring gear)
C: Brake (blocks C1 carrier gear)
D: Clutch (couples C3 carrier gear with R4 ring gear)
E: Clutch (couples C1 carrier gear with R2 ring gear)
F: Clutch (couples S1 sun gear with C1 carrier gear)

Applications

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Mercedes models

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Mercedes C-Class

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Mercedes E-Class

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Mercedes S-Class

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Mercedes V-Class

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Mercedes GLC-Class

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Mercedes-Benz GLE-Class

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  • 2016–2019 GLE (W166) (except 63 AMG & 350 models)
  • 2020–present GLE (W167) (except 63 models)

Mercedes-Benz GLS-Class

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Mercedes-Benz SLK-Class

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Mercedes-AMG models

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Mercedes-AMG SL

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Jatco Ltd model JR913E

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Nissan

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Infiniti

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Aston Martin

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See also

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References

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  1. ^ a b c d e f "New nine-speed automatic transmission debuts in the Mercedes-Benz E 350 BlueTEC: Premiere of the new 9G-TRONIC – Daimler Global Media Site". media.mercedes-benz.com. 2013-07-24. Retrieved 2024-10-29.
  2. ^ a b c d e f "9G-Tronic · Vertiefende Informationen" (in German). Archived from the original on 2015-11-20. Retrieved 2015-11-20.
  3. ^ a b c "50 years of automatic transmissions from Mercedes-Benz". media.mercedes-benz.com. 2011-04-12. Retrieved 2024-10-29.
  4. ^ "Daimler launches production of nine-speed automatic transmissions in Romania". 2016-04-04. Retrieved 2024-10-29.
  5. ^ "Daimler-Renault-Nissan – The alliance in action".
  6. ^ "Fact Sheet:Press Releases and Project Overview Daimler & Renault-Nissan Alliance" (PDF).
  7. ^ "Jatco Technical Review No. 20 - 2021 - see cutaway model Figure 4 - p. 72" (PDF). Retrieved 2022-11-11.
  8. ^ a b c d Christoph Dörr · Henrik Kalcynksi · Anton Rink · Marcus Sommer: Nine-Speed Automatic Transmission 9G-Tronic By Mercedes-Benz (english version), in: ATZ 116 (2014) · No. 1 · pp. 20–25 · Springer Vieweg · Wiesbaden
  9. ^ a b Harald Naunheimer · Bernd Bertsche · Joachim Ryborz · Wolfgang Novak · Peter Fietkau: Fahrzeuggetriebe (in German) · Berlin/Heidelberg 2019 · pp. 571–572
  10. ^ a b c d "Automatic Transmission 9G-Tronic · 725.0 · System Description" (PDF). documents.epfl.ch. September 2013. Retrieved 2020-01-16. (PDF)
  11. ^ "Thomas Harloff: Neun-Gänge-Menü". 2014-05-27.
  12. ^ "Developed for in-house drive systems: The best out of 85 billion possibilities". 2014-03-06. Retrieved 2024-10-29.
  13. ^ "Archived copy of Mercedes-Benz Automatic Transmission 722.9 Technical Training Materials". Archived from the original on 2019-06-28. Retrieved 2019-06-28.
  14. ^ "Der neue Mercedes-Benz SL: Die Legende – jetzt noch dynamischer – Daimler Global Media Site" (in German). Media.daimler.com. Retrieved 2020-01-16.
  15. ^ "Jatco Technical Review No. 20 · 2021 · pp. 71 – 74". Retrieved 2022-11-11.
  16. ^ "Jatco Technical Review No. 20 · 2021 · see Table 1 · p. 72". Retrieved 2022-11-11.
  17. ^ Robert Willis (1841). "Principles of mechanism" (PDF). Retrieved 2024-11-04.
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