# Horsepower, Torque, and all that...

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Frequently we hear riders talk about how many horsepower their motorcycle makes. These claims are often not believed, and sometimes not believable. Also, we hear some riders state that they like torque more than horsepower. In this article, I'll explain exactly what torque and horsepower are, and why some motorcycles are dramatically faster than others.

Your motorcycle moves forward because the tire pushes against the road. This is the basic fact we'll use to understand everything else. Push is a non-technical word, and engineers, like doctors and lawyers, love to use more complicated words. The engineering word for push is Force, and a force applied by something round like a wheel is called torque. So, torque is a fancy word for a push by a wheel. More torque equals more push, which equals more acceleration, which wins races.

In chart 1, there's a graph of torque vs. rpm for two Harley-Davidson big twins. Rpm is, of course, revolutions per minute - the number of times your engine crankshaft turns over each minute, and the number which is reported by your tachometer. One of the Harleys on the chart is a stock California model, and the other is modified "for off-road use only". A surprising fraction of all Harley big twins are apparently purchased for dirt riding and closed course competition. This particular modified bike has a Crane Fireball 310 cam, a Mikuni carburetor, and Cycle Shack exhaust pipes. As you can see, the modified engine produces about 25% to 50% more torque, and therefore 25% to 50% more acceleration. Since velocity is acceleration times time, we would expect the modified machine to take about 1/3 less time to go 0 to 60.

A popular performance test is a 50 mph, 200 yard roll-on: full throttle for 200 yards, starting at 50 mph. Let's see what our modified Harley will do in these circumstances. At 50 mph in 5th gear, a big twin is doing about 2200 rpm. At 70 mph, it's doing about 3000 rpm. Over the range of 2000 to 3000 rpm, the modified engine is only producing about 5-10 more foot- pounds of torque than the stock engine; a gain of about 10%. The final speed in a 200 yard roll-on improves from about 72 to about 73. This is not so impressive as the 30% reduction in 0 to 60 times. The calculations used to figure this out are in the "mathematics" sidebar.

The problem is that the modified engine is producing a lot more power, but at a higher rpm. In 0 to 60 tests, we run the engine up to the red line, but in the roll-on, we never get much beyond 3000 rpm. If we did the same roll on test in 3rd gear, we would expect the stock engine to get to about 74, and the modified engine to get to about 80.

Now, we get our complication: the torque is applied at the rear wheel, but it is not produced by the wheel, it's produced by the engine. If the wheel and the engine were directly connected, as they are in a speedway bike, we'd be done. But, on almost all bikes, there's a transmission. The gear ratios in the transmission divide the engine rpm and the torque, with higher gears producing less rpm and less rear wheel torque at a given speed.

The best way to understand this is to make a picture. We'll graph the torque available at the rear wheel at various road speeds. Of course, this graph is going to look a little complicated, because, for example, at 30 mph you could be in any of the five gears, which means you can choose between five available rear-wheel torques and five engine rpm's.

To make this graph, first we need to know how fast the bike goes in its various gears. Harley Davidson supplies the overall gear ratios in their product literature, so all we need is one calibration point. A motorcycle magazine recently reported than in 5th gear at 60 mph a '94 Harley is doing 2570 rpm. Since all '94 Harley big twins use the same rear tire, transmission, and belt pulleys, they all share the same overall gear ratios. Using this information, I filled in the numbers in table 1. For example, at 4000 rpm in 5th gear, the Harley is going 93 mph (60 mph * 4000 / 2570). At 3000 rpm in 3rd gear, the Harley is going 44 mph (60 mph * 3000 / 2570 * 3.15 for fifth gear / 5.05 for third gear).

If you own an earlier evolution big twin, then here are the rules: in '94 all big twins use 29 tooth and 70 tooth pulleys. In earlier years, some used 27 tooth and 70 tooth pulleys, and some used 27 tooth and 61 tooth pulleys. If you have a 27-70 (FXR's and glides), multiply all speeds in this article by 27/29, about .93. If you have a 27-60 (most softtails), multiply all speeds in this article by 70/61*27/29, about 1.07.

 RPM 1st gear (10.212) 2nd gear (6.96) 3rd gear (5.05) 4th gear (3.86) 5th gear (3.15) 2000 14 mph 21 mph 29 mph 38 mph 47 mph 3000 22 mph 32 mph 44 mph 57 mph 70 mph 4000 29 mph 42 mph 58 mph 75 mph 93 mph 5000 36 mph 52 mph 72 mph 94 mph 117 mph 6000 43 mph 63 mph 87 mph 113 mph 140 mph

Now, this is not to say a Harley will necessarily go 140. This just says that if a '94 big twin were ever in 5th gear doing 6000 rpm, it would be doing 140 mph. In fact, the stock Harley rev limiter kicks in at 5200 rpm, and a California big twin will actually only do about 95-105. Also, different model years of Harleys have different gear ratios.

Now, using this table, we can construct our real-wheel torque vs. road speed graph. To do this, we look at the torque curve in chart 1, and at the road speeds in table 1. At 2000 rpm in 1st gear, the bike is going 14 mph. At 2000 rpm, the engine is making 65 foot-pounds of torque. At the rear wheel, this is 65*10.212 = 664 foot-pounds of torque. So, at 14 mph, we make a dot at 664 foot-pounds for 1st gear. At 4000 rpm in 3rd gear, the bike is going 58 mph and the engine is making 55 foot-pounds, which is 55*5.05 = 278 foot-pounds at the rear wheel, so we make a dot at 58 mph and 278 foot-pounds for 3rd gear. I'm sure you see how this goes now, so, here's the resulting graph:

In chart 2, we see a graph of the rear wheel torque at various road speeds in each gear. If you look at 50 mph, you see that there is 200 foot-pounds of torque available in 5th gear, about 240 ft-lbs in 4th gear, about 270 ft-lbs in 3rd gear, and 320 ft-lbs in 2nd gear. Thus, down-shifting this engine from 5th to 3rd gear at 50 mph only gets us about 35% more torque; similarly, up-shifting from 3rd gear to 5th gear at 50 mph only costs us about 28% of our torque. Thus we see that stock Harleys don't change their performance very much when you shift gears. In fact, at 80 mph, 5th and 3rd are almost identical in performance, and 4th is only about 10% better.

The arrows in the graph point at the places where the rear wheel torque curves cross going from one gear to the next. These are the optimum shift points: at about 40, 56, 73, and 95 mph. However, we also see that with this engine, there is very little difference if we shift directly from 2nd to 4th at about 55, or from 3rd to 5th at about 70. We must remember that these shift points are for a '94 Harley; different years have different gearing and different shift points. Since we put in the dots at 2000, 3000, 4000, 5000, and 6000 rpm, we can estimate the optimum shifting rpm: about 5500, 5500, 5000, and 5000 rpm on a 94 Harley. Actually, these rpm estimates are not too bad for nearly any nearly stock Harley big twin.

In chart 3, we see overlaid on the stock torque curves the torque curves for the modified engine. I've left off the dots for the stock CA Harley torque curves to make the chart more readable. From chart 1, we see that the modified engine torque peak is at 4000 rpm. Thus, if we have the modified engine, and we are willing to run it at 3500 to 5500 rpm, we can see from chart 3 that the performance will be much better - there's more available torque, and therefore more available push and acceleration. On the other hand, if we shift at 3000 rpm, as many Harley riders do, there will be very little improvement. For example, at 3000 rpm in first gear, the bike is doing about 22. At 22, there is very little difference between the stock and modified engines in first gear, and there is also very little difference between them in second gear at 22 mph. So, our modified engine will pull much harder, and go a lot faster, but only at a much higher rpm.

In chart 3, the shift points are indicated for both the stock and the modified engines. We see that they are not very different: 42, 57, 78, and 102 compared to 40, 56, 73, and 95 mph. However, we see that shifting gears in the modified engine produces a large effect, while shifting gears in the stock motor produces relatively little effect. At 50, when we down-shifted the stock motor from 5th to 3rd, the torque increased from 200 to 270 - about 30%. With the modified motor, the torque increases from about 220 to 400 - almost double. However, as long as we stay in 5th gear, there's not much difference between the two motors. Again, these numbers are all for a '94 Harley; different years with different gearing will have different shift points in mph.

The modified engine has shift points we can estimate at 5700, 5500, 5400, and 5300 rpm. These numbers are consistently a bit higher than for the stock engine, and above the stock 5200 rpm rev limiter, but within reasonable limits for the engine. Evolution engines are pretty well engineered and can take a reasonable amount of abuse.

Now we see a general rule about making power. In order to make more power, mostly you have to spin the engine faster. Also, each time you shift gears, the torque gets divided by the same ratio that the rpm gets divided. So, to go fast, you want a motor where you don't have to shift gears as often. In fact, if two bikes make the same peak torque - say, 80 foot pounds - the motor which makes the torque at a higher rpm can stay in lower gears longer and therefore go faster. This makes us think it would be useful to have a single number which reported both the peak torque and the peak rpm.

In fact, there is just such as number: horsepower. Horsepower equals torque times rpm; thus an engine which makes 80 foot pounds of torque at 8000 rpm makes twice as many horsepower as another engine which makes the same 80 foot pounds at 4000 rpm. Although the two engines deliver the same amount of torque, and therefore the same amount of push, the first engine can go twice as fast in each gear. While I'm doing 60 topped out in second gear on my 80 foot-pound Harley, some other guy can be doing 60 not quite topped out in first gear on his 80 foot-pound Kawasaki ZX-11. Of course, he's winning the race.

It's about time we considered some different bikes. In chart 4, we see the torque curves for several different engines overlaid. The bikes are all about the same in terms of engine size - a bit over one liter. Interestingly, with the exception of the California Harley and the BMW, we see that they all produce about the same amount of peak torque - about 80 foot- pounds. We all know that Harley engines can make significantly more power than what the factory supplies. Based on these curves, we suspect the same is true for the BMW.

In chart 5, we see the horsepower curves for the same five bikes. In spite of the similarity of the peak torque, the horsepower curves are very different.

Notice that the modified Harley, which makes the most torque up to almost 5000 rpm, also makes the most horsepower up to 5000 rpm. The Honda ST1100 makes the most torque and horsepower from 5000 to 6500 rpm, and the Kawasaki makes the most from there up.

Let's see how to calculate horsepower. There's no big trick at all. From chart 1, we see that a stock California Harley is making 65 foot-pounds at 2000 rpm, 64 ft-lbs at 3000, 55 at 4000, and about 45 at 5000. Horsepower is torque times rpm divided by 5252; so this Harley is making 25 horsepower at 2000 rpm, 37 hp at 3000 rpm, 42 at 4000, and 43 at 5000. (Yes, it's true: stock California big twins make about 43 horsepower. 49 state big twins make about 50 horsepower.)

Here's something interesting: the torque peaks at about 2000 rpm, but the horsepower peaks at about 5000 rpm. Even though the torque is going down, the rpm is going up faster than the torque is going down, and therefore the horsepower is increasing.

Also, we see from the formula that at 5252 rpm, horsepower is equal to torque. This is always true. If your dyno tells you that the horsepower and torque curves cross anywhere other than 5252 rpm, your dyno is broken.

Finally, we see that at any given rpm, the engine which makes the most torque at that rpm also makes the most horsepower at that rpm. To make more horsepower, an engine has to either make more torque or run at higher rpm. The ZX-11 makes the same 80 foot-pounds of torque as the modified Harley, but it makes this torque at 8000 rpm - exactly twice the 4000 rpm peak of the modified Harley. Because of this, the ZX-11 makes twice the horsepower - 120 hp - at 8000 rpm that the modified Harley makes at 4000 rpm - 60 hp - even though both are making 80 foot-pounds of torque.

Since the Harley torque curve drops off a bit gradually from 4000 to 5000 rpm, and the ZX-11 torque curve drops more quickly from 8000 to 10000 rpm, the ZX-11 doesn't quite make twice as much horsepower as the modified Harley - "only" 130 hp compared to 71 hp. Kawasaki riders need not totally despair - the ZX-11 ram air system is not active on a dyno. It is believed that at 150 mph, the Kawasaki is making about 145 hp due to the ram air effect.

We see that if all you care about is engine peak torque, you would be equally happy with the Harley and the ZX-11. If all you care about is Horsepower, you definitely want the ZX-11. If what you're really interested in is power at low rpm, now you definitely want the Harley. So, those riders who claim to prefer torque over horsepower really mean they prefer engines with power at low rpm.

Let's see how this horsepower advantage translates into a torque advantage. We'll make the same picture for the ZX-11 rear wheel torque that we made for the Harleys. In table 2 below, there are the gear ratios and road speed worked out for the ZX-11 (the gear ratio data was supplied by U.S.Kawasaki). The ZX-11 has a six speed transmission. I've worked out the road speed in 2,000 rpm increments from 2,000 rpm to 10,000 rpm. The ZX-11 actually has a red line of 11,000 rpm, so each gear actually goes a little faster than the table shows.

For those of you who aren't familiar with ZX-11's, this bike has been clocked by radar guns at 172 mph, stock. It does 135 mph in the quarter mile in about 10.3 seconds. These days, the ZX-11 pretty much defines fast.

The ZX-11 engine torque curve is illustrated in chart 4 above. We do the same cross-reference look up exercise as before, and we get the rear wheel torque curves. Interestingly, these curves seem to indicate that, for racing purposes, fourth gear could be eliminated with no effects, and fifth gear is only useful from 120 mph to 135 mph. It looks like removing fourth and fifth gear and replacing them with one gear with the ratio 5.372 (31/25 internally) would be an improvement.

In chart 6, we see a direct comparison of the rear wheel torque of a modified Harley, and a Kawasaki ZX-11. As you can see, the ZX-11 has a much larger advantage over the modified Harley than the modified Harley has over the stock Harley. Not only does the ZX-11 make more rear wheel torque, it can also go faster in first gear than the Harley can go in second gear, and it can go about as fast in third gear as the Harley can go in fifth. And, in fact, it is now well established that a ZX-11 can pretty much beat any Harley, any where, any time, in any race. So, how come Harleys outsell ZX-11's about a hundred to one?

 RPM 1st gear (12.133) 2nd gear (8.905) 3rd gear (6.890) 4th gear (4.996) 5th gear (3.15) 6th gear (4.485) 2000 12 mph 17 mph 22 mph 26 mph 30 mph 33 mph 4000 25 mph 34 mph 44 mph 52 mph 60 mph 67 mph 6000 37 mph 50 mph 65 mph 78 mph 90 mph 100 mph 8000 49 mph 67 mph 87 mph 104 mph 120 mph 134 mph 10000 62 mph 84 mph 109 mph 130 mph 150 mph 167 mph

In the Fall 1994 issue of American Rider, Joe Minton presents a graph of horsepower for a Harley 1340. His article did not specify the modifications made, but in the context of his article, we can safely assume they include substantial head work. His engine's torque curve can be easily deduced from the horsepower curve: at each rpm, we multiply the horsepower by 5252 and divide by the rpm. The result indicates an engine making between 87 and 93 foot-pounds of torque from 2500 to 6250 rpm. This is an engine with the same torque per cubic inch as a ZX-11, and a power band which is 25% wider. If true, this is quite an accomplishment.

In chart 7, I have graphed the rear wheel torque curves for Mr. Minton's bike. As you can see, the performance would be substantially better than either the stock or the lightly modified Harley. Also, we see that this engine is poorly mated to its gear box: optimum shifting points would occur at 7,500 to 6,500 rpm. This level of rpm severely over stresses an evolution big twin. It appears that a change of camshafts to move the torque curve down 500 to 1000 rpm would give this bike performance better suited to its gear box and engine design parameters.

With Mr. Minton's bike, we would expect the 0 to 60 times to be cut roughly in half compared to a stock CA Harley, and the 50 mph roll-on speed to be about 75 mph in fifth, and 86 mph in third. We would also expect that when shifting from first gear to second at 6000 rpm, the bike's acceleration would drop dramatically.

Mark Lawrence has a degree in Engineering. He has been riding motorcycles since 1969. In this time he has owned 21 bikes, ridden 305,000 miles through 28 states, accumulated 27 traffic tickets, and set a California state record for traffic school attendance. He currently rides a Road King with a modified "closed course competition use only" motor. He has not yet raced his Road King in a Grand Prix, but expects to do well if he ever does.

Sidebar 1:

The Mathematics

For those who are interested, here's a brief summary of the physics and mathematics used in this article.

1. force = mass * acceleration
2. torque = force * moment arm length
3. distance = 1/2 * acceleration * time * time
4. velocity = acceleration * time = force / mass * time
5. time = velocity * mass / force
6. time = square root( 2 * distance / acceleration )
7. velocity = square root( 2 * distance * acceleration )
8. velocity = square root( 2 * distance * force / mass )
9. horsepower = torque * rpm / 5252
10. rear wheel torque = engine torque * primary gear ratio * transmission gear ratio * final gear ratio
11. rear wheel force = rear wheel torque * rear wheel radius
12. rear wheel radius is almost always 1 foot, +/- about 1 inch, for any motorcycle

To calculate the roll-on performance, here's what I did:
published final speed = 72 mph.
Modified bike has 10% greater torque, average, over 50-80 mph.
By equation 5, velocity = K * square root( force )
new velocity / old velocity = square root( new force / old force ) = square root( 1.1 ) = 1.05
old velocity = 72 - 50 = 22 mph. 5% of 22 mph = 1 mph. new velocity = 50 + 22 + 1 = 73 mph.

For third gear modified bike, I noted that the modified bike is making about 370 foot-pounds of rear-wheel torque in third gear from 50 to about 80, and it's making about 220 foot-pounds from 50 to 80 in fifth gear, so the math is the same up to:

new velocity / old velocity = square root( new force / old force ) = square root( 380 / 220 ) = square root( 1.7 ) = 1.3
30% of 23 is 7, so the modified bike hits about 50 + 23 + 7 = 80 mph in third gear.

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