[To understand the material of these documents, I assume the reader have a firm understanding of Newtonian Kinematics.]
Usually the gear ratios of a car is already fine tuned by the car manufacturer. There is no need to adjust the ratios to fully utilize the performance of the car. However, if you happen to be a car enthusiast that loves the black art of modifying your car, then you might well read on. attain the best performance. For people driving cars on the public road, maximum speed of a car probably doesn't mean much for them unless they enjoy the feeling of getting caught speeding. People live in Montana probably don't have this concern because there are freeways that don't have speed limit. So now suppose you are shopping for a car for a trip to one of those freeways or a race track somewhere to get the thrill of your life, you definitely want to find the best deal for the speed. Then you need to know how to calculate the maximum speed of a car.
| Γ(ω)Ggk | 1 | ||
| ------------ | = | -- | cdAρv2 |
| r | 2 |
Now multiply both sides by v where v also equals 2πrω/Ggk. Then we have:
| Γ(ω)Ggk | 2πrω | 1 | |||
| ------------- | × | ------- | = | -- | cdAρv2 |
| r | Ggk | 2 |
Note thath Power P = 2πΓ(ω)ω, so
| ( | 2P | ) | ^(1/3) | ||
| v | = | ( | ------- | ) | |
| ( | cdAρ | ) |
Note that when P is at its peak, v is also at its peak. So when we put in peak power for P, we get the top speed. For our Skyline, this is 262.33km/h.
In reality, the maximum speed is limited by the gearing. Here is why: For each gear, there is a correspondence between engine speed and the speed of the car. Given the top speed occurs only at the peak power that only occurs at a particular engine speed. If at that engine speed, the corresponding car speed is not the maximum speed, that means
This happens because we enter the declining part of our torque curve such that the force generated by the engine is equal to the air resistance. But since our torque curve is declining, there is no way we can overcome the air resistance to attain higher speed. This holds true in every gear. But as we know when we shift up the gears, we get substantially lower torque from our engine and hence it is more likely that the force generated by the engine can be canceled out by the air resistance. To see whether we can attain the ideal top speed for each gear, we need to look at the force generated by engine at red line and compare it with the air resistance.
| Gear | 1st | 2nd | 3rd | 4th | 5th | 6th | reverse |
| Gear Ratio | 3.827 | 2.36 | 1.685 | 1.312 | 1 | 0.793 | 3.28 |
| Peak Engine Force (N) | 16283.38 | 10041.49 | 7169.45 | 5582.39 | 4254.87 | 3374.11 | 13955.97 |
| Air Resistance at Torque Peak (N) | 65.5 | 172.25 | 337.85 | 557.33 | 959.36 | 1525.58 | 89.17 |
| Engine Force at Red Line (N) | 8994.56 | 5546.69 | 3960.24 | 3083.58 | 2350.29 | 1863.78 | 7708.95 |
| Air Resistance at Top Speed (N) | 244.45 | 642.82 | 1260.99 | 2079.91 | 3580.25 | 5693.33 | 332.79 |
As we can see, it is impossible to attain top speed in the 5th and the 6th gear for our Skyline. To see the top speed attainable in these two gears, we need to see at which point the engine force cancels out the air resistance.
| Γ(ω)Ggk | 1 | ||
| ------------ | = | -- | cdAρv2 |
| r | 2 |
For our Skyline, this condition is equivalent to:
| (-2.568ω+580.32)Ggk | 1 | ||
| ------------------------- | = | -- | cdAρv2 |
| r | 2 |
Recall that v=2πrω/Ggk, we can transform the condition into a quadratic equation in terms of v.
| 1 | 2.568G2gk2 | 580.32Ggk | ||||||
| -- | cdAρv2 | + | ------------ | v | − | ------------ | = | 0 |
| 2 | 2πr2 | r |
The quadratic equation for the 5th gear is:
The quadratic equation for the 6th gear is:
To summarize:
| Gear | 5th | 6th |
| Gear Ratio | 1 | 0.793 |
| Drag Limited Top Speed (km/h) | 261.28 | 261.05 |
| Drag Limited Engine Speed (RPM) | 7522.69 | 5960.25 |
| Corresponding Engine Force (N) | 2804.28 | 2799.35 |
| Corresponding Drag Force (N) | 2804.28 | 2799.35 |
As we can see, in reality, our Skyline can never achieve the theoretical top speed 262.33km/h under the current gearing. The top speed 261.28km/h is attained at the 5th gear. This is not too far off the bat.
The result of this section also invalidates the optimal shift point we found in the last section for the 5th gear. In fact, we can never attain that shift point, we reach the maximum speed right before it.
A friend of mine once told me that "You are not really driving when you drive an automatic". I didn't get it before but now I see some merits in his comment. I had many opportunities to climb into automatic cars. I observed how the computer shifts the gear. I found that the shifting strategy is to keep the engine speed between 1500rpm and 2000rpm. This ensures a smooth ride because the car is moving at the weak part of its torque curve. For these automatic cars, the only way you can see their true color is usually at the top gear when the computer doesn't have any way to shift up. But then they lost all the fun in the low gears... ~~;
In contrast, driving a manual car you can choose to accelerate slowly just like the automatics by abiding to the rule that shift up when you feel the engine is noisy. You can also put your car at full throttle and shift only at the red line to get your adrenline rushing until you approach the speed limit. Remember that our body can only feel acceleration because of inertia. We can feel the speed of a car only when we open the window. The force of wind-blowing at the car is constantly available at any speed. And it quadrapules as you double your speed.
Another thing you can do with your manual car is down-shifting in purpose. Consider the case when I am cruising behind a slow vehicle at 50 mph at the 5th gear on a 65mph freeway. When that slow vehicle changes lane and I find that the lane suddenly becomes wide open, I will downshift to the 3rd gear such that I can get to 65mph in the shortest possible time for my Protege. Another useful application of low gear acceleration is merging into an expressway or a freeway. It is so much easier and safer to merge when the speed of your car is about the same as that of the cars in the freeway.
So the rule of public road gear-shifting is pretty straight forward: go to the high gear when you want to cruise, go to the low gear when you want to accelerate.