Part 26: The Driving Wheel, Chapter I

**Brian Beckman, PhD**

**©Copyright July 2001**

Imagine 400 ft-lbs of torque measured on a chassis dynamometer like a DynoJet (see references at the end). This is a very nice number to have in any car, street or racing. In dyno-speak, however, the interpretation of this number is a little tricky.

To start a dyno session, you strap your car down with the driving wheels over a big, heavy drum, then you run up the gears gently and smoothly until you're in fourth at the lowest usable engine RPM, then you floor it, let the engine run all the way to redline, then shut down. The dyno continuously measures the time and speed of the drum and the engine RPM through a little remote radio receiver that picks up spark-plug noise. The only things resisting the motion of the driving wheels are the inertia of the driveline in the car and the inertia of the drum. The dyno 'knows' the latter, but not the former. Without these inertias loading the engine, it would run up very quickly and probably blow up. Test-stand dynos, which run engines out of the car, load them in different ways to prevent them from free running to annihilation. Some systems use water resistance, others use electromagnetic; in any case, the resistance must be easy to calibrate and measure. We are only concerned with chassis dynos in this article, however.

What is the equation of motion for the car + dyno system? It is a simple
variation on the theme of the old, familiar second law of Newton. For linear
motion, that law has the form
**F*** = ma*, where

For rotational motion, like that of the driveline and dyno, Newton's second
law takes the form where ** T** is the net torque on an object,

To get the rotational equation of motion, we assume that the dyno drum is strong enough that it will never fly apart, no matter how fast it spins. We model it, therefore, as a bunch of point masses held to the centre of rotation by infinitely strong, massless cables. With enough point masses, we can approximate the smooth (but grippy) surface of the dyno drum as closely as we would like.

Assume each particle receives a force of * F / N* in the
tangential direction. Tangential, of course, means the same as circumferential
or longitudinal, as clarified in recent instalments about slip and grip of
tyres. So, each particle accelerates according to

Since * r* is constant, it has no rate of change. Only has one, measured in radians per second per
second, or radians per second squared, or , and denoted with an overdot: . The equation of motion, so far, looks like . Now, we know that torque is just force
times the lever arm over which the force is applied. So, a force of

This value for * J* only works for this

So, now, the dyno has a known, fixed value for * J*, and it
measures very accurately. This enables it to calculate trivially
just how much torque is being applied by the driving wheels of the car to the
drum. But it does

Everyone knows that the transmission and final-drive on a car *multiply*
the engine torque. The torque at the driving wheels is almost always much larger
than the flywheel torque, and it's larger in lower gears than in higher gears.
So, if you run up the dyno in third gear, it will accelerate faster than if you
run it up in fourth gear. Yet, the dyno reports will be comparable. Somehow,
without knowing any details about the car, not even drastic things like gear
choice, the dyno can figure out flywheel torque. Well, yes and no.

It turns out that all the dyno needs to know is engine RPM. It does not
matter whether the dyno is run up quickly with a relatively large drive-wheel
torque (* DWT*) or run up slowly with a relatively small

Wheel RPM is directly proportional to drum RPM, assuming the longitudinal
slip of the tyres is within a small range. The reason is that at the point of
contact, the drum and wheel have the same circumferential (longitudinal,
tangential) speed, so . Let's write , where . Engine RPM is related to wheel RPM by a factor that
depends on the final-drive gear ratio * f* and the selected gear
ratio . We write . Usually, engine RPM is much larger than wheel RPM, so
we can expect to be larger than 1 most of the time. So, we get

We also know that, by Newton's Third Law, that the force applied to the drum by the tyre is the same as the force applied to the tyre by the drum. Therefore the torques applied are in proportion to the radii of the wheel + tyre and the drum, namely that

or . Recalling that the transmission gear and final drive multiply engine torque, we also know that , so . But we already know : it's the ratio of the RPMs, so , or, more usefully,

Every term on the right-hand side of this equation is measured or known by the dyno, so we can measure engine torque independently of car details! We can even plot versus , effectively taking the run-up time and the drum data out of the report.

Almost. There is a small gotcha. The engine applies torque indirectly to the
drum, spinning it up. But the engine is *also* spinning up the clutch,
transmission, drive shaft, differential, axles, and wheels, which, all together,
have an unknown moment of inertia that varies from car-to-car, though it's
usually considerably smaller than * J*, the moment of inertia of the
drum. But, in the equations of motion, above, we have not accounted for them.
More properly, we should write

This doesn't help us much because we don't know , so we pull a fast one and rearrange the equation:

This is why chassis dyno numbers are always lower than test-stand dyno numbers for the same engine. The chassis dyno measures , and the test-stand measures . Of course, those trying to sell engines often report the best-sounding numbers: the test-stand numbers. So, don't be disappointed when you take your hot, new engine to the chassis dyno after installation and get numbers 15% to 20% lower than the advertised 'at the crankshaft' numbers in the brochure. It's to be expected. Typically, however, you simply do not know : it's a number you take on faith.

Let's run a quick sample. The following numbers are pulled out of thin air, so don't hang me on them. Suppose the drum has 3-foot radius, is solid, and weighs 6,400 lbs, which is about 200 slugs (remember slugs? One slug of mass weighs about 32 pounds at the Earth's surface). So, the moment of inertia of the drum is about . Let's say that the engine takes about 15 seconds to run from 1,500 RPM to 6,000 RPM in fourth gear, with a time profile like the following:

t |
e RPM |
V MPH |
v FPS |
drum |
drum |
drum RPM |
RPM ratio |
Torque |

0 | 1,500 | 35 | 51.33 | 17.11 | 0.00 | 163.40 | 0.1089 | 0.00 |

1 | 1,800 | 42 | 61.60 | 20.53 | 3.42 | 196.08 | 0.1089 | 335.51 |

2 | 2,100 | 49 | 71.87 | 23.96 | 3.42 | 228.76 | 0.1089 | 335.51 |

3 | 2,400 | 56 | 82.13 | 27.38 | 3.42 | 261.44 | 0.1089 | 335.51 |

4 | 2,700 | 63 | 92.40 | 30.80 | 3.42 | 294.12 | 0.1089 | 335.51 |

5 | 3,000 | 70 | 102.67 | 34.22 | 3.42 | 326.80 | 0.1089 | 335.51 |

6 | 3,300 | 77 | 112.93 | 37.64 | 3.42 | 359.48 | 0.1089 | 335.51 |

7 | 3,600 | 84 | 123.20 | 41.07 | 3.42 | 392.16 | 0.1089 | 335.51 |

8 | 3,900 | 91 | 133.47 | 44.49 | 3.42 | 424.84 | 0.1089 | 335.51 |

9 | 4,200 | 98 | 143.73 | 47.91 | 3.42 | 457.52 | 0.1089 | 335.51 |

10 | 4,500 | 105 | 154.00 | 51.33 | 3.42 | 490.20 | 0.1089 | 335.51 |

11 | 4,800 | 112 | 164.27 | 54.76 | 3.42 | 522.88 | 0.1089 | 335.51 |

12 | 5,100 | 119 | 174.53 | 58.18 | 3.42 | 555.56 | 0.1089 | 335.51 |

13 | 5,400 | 126 | 184.80 | 61.60 | 3.42 | 588.24 | 0.1089 | 335.51 |

14 | 5,700 | 133 | 195.07 | 65.02 | 3.42 | 620.92 | 0.1089 | 335.51 |

15 | 6,000 | 140 | 205.33 | 68.44 | 3.42 | 653.60 | 0.1089 | 335.51 |

The "v MPH" column is just a straight linear ramp from 35 MPH to 140 MPH,
which are approximately right in my Corvette. The "v FPS" column is just 22 / 15
the v MPH. The drum is in radians per second and is just v FPS divided
by 3 ft, the drum radius. The drum is just the stepwise difference of the drum numbers. It's constant, as we would
expect from a run-up of the dyno at constant acceleration. The drum RPM is times the drum . The RPM ratio is just drum RPM divided by engine
RPM, and it must be strictly constant, so this is a nice sanity check on our
math. Finally, the torque column is the RPM ratio times
* J* = 900 slug - ft

Dyno reports often will be labelled 'Rear-wheel torque' (* RWT*)
or, less prejudicially, 'drive-wheel torque' (

In the next instalment, we relate the equations of motion for the driving wheel to the longitudinal magic formula to compute reaction forces and get equations of motion for the whole car.

**References:**

http://www.c5-corvette.com/DynoJet_Theroy.htm
[sic]

http://www.mustangdyne.com/pdfs/7K%20manualv238.pdf

http://www.revsearch.com/dynamometer/torque_vs_horsepower.html

**Attachments:**

I've included the little spreadsheet I used to simulate the dyno run. It can be downloaded here.

**ERRATA:**

* Part 14, yet again, the numbers for frequency are
actually in radians per second, not in cycles per second. There are cycles per radian, so the 4 Hz natural
suspension frequency I calculated and then tried to rationalize was really 4 /
6.28 Hz, which is quite reasonable and not requiring any rationalization. Oh,
what tangled webs we weave…

* Physical interpretations of slip on page 2 of part 24: "Car (hub) moving
forward, CP moving slowly forward w.r.t. ground, resisting car motion." Should
be "Car (hub) moving forward, CP moving slowly forward w.r.t. * HUB*,
resisting car motion."

* Part 21, in the back-of-the-envelope numerical calculation just before the 3-D plot at the end of the paper, I correctly calculated , but then incorrectly calculated as -0.266. Of course, it's 0.688 - 0.822 = -0.134. One of the hazards of doing math in one's head all the time is the occasional slip up. Normally, I check results with a calculator just to be really sure, but some are so trivial it just seems unnecessary. Naturally, those are the ones that bite me.