Measuring Horsepower and Torque Curves of a Car

Contents

• BODY OF ARTICLE
• System Description
• Experiment
• Analysis
• Results
• Conclusions
• Acknowledgments
• REFERENCES
• About the Author
• FIGURES

We have developed a system to measure the motion of a car with very good precision. It requires a laptop computer and a standard sensor interface. The added parts are relatively simple and inexpensive, and can be connected to a variety of cars easily and quickly. Using this system, we are able to derive the horsepower and torque versus engine speed for any manual transmission car. Students enjoy the lab because it gives them an excuse to floor their car but it also produces remarkably good data.

As an added benefit, students see real-time graphs of position and velocity versus time while they are riding in the car experiencing the very motion being plotted. This becomes a wonderful reinforcement of the popular labs in which students move an object in front of a range finder. In this case they are riding in the moving object and experience all the pseudo-forces that arise in the accelerated reference frame. Back in the lab, they analyze the graphs in a spreadsheet program. Imagine the sheepish looks on students' faces when the professor points out the clear graphical evidence that they broke the speed limit.

Figure 1 shows a diagram of the entire system. The method begins with the wheel sensor (shown in Fig. 2). This sensor works similarly to a bicycle odometer in that it merely registers when the wheel completes a revolution.

Figure 1.

Figure 2.

The valve stem cap on the car wheel is replaced with a special magnetic one. It consists of a tiny neodymium magnet (diameter 0.25 × 0.125 in, Edmund K35-429)placed inside a brass valve stem cap. A small washer put in the cap after the magnet ensures that when the cap is screwed on tightly, the valve pin is not depressed.

A special clamp holds a standard magnetic field sensor (Vernier MG-DIN) near the wheel. The clamp was designed to be able to be connected rapidly to almost any car. The probe tip is positioned so that the magnet passes within about 1 cm of it. One difficulty we have encountered is that the suspension is compressed when the car accelerates. This changes the relative position of the probe to the tire. In some cases the probe moved far enough that the signal was lost during the acceleration. We solve this problem by mounting the probe near the rear of the tire instead of near the top. This makes the relative motion of the probe tangent to the circle described by the moving magnet rather than perpendicular to it. Rather than losing the signal, the relative motion introduces a small timing error that we ignore.

The magnetic field sensor creates a small peak in its output voltage when the magnet passes near it. For our system this was about 1 V in height above the baseline. To make data collection more straightforward, we have designed a simple circuit to convert this signal into a digital signal similar to the type produced by digital pulleys (Fig. 3). The circuit (Fig. 4) employs an analog comparator to change the analog input into a digital output. The threshold is controlled by a potentiometer that the students adjust. There is a small amount of positive feedback to give it some hysteresis. The circuit also has an LED indicator that flashes with the digital output signal so that the students can easily adjust the threshold to the optimum level.

Figure 3.

Figure 4.

The output of the conversion circuit is connected to the laptop computer through a standard laboratory interface (we use the ULI interface from Vernier powered by an external 9-V battery). The interface software (Logger Pro) is set to treat the signal as if it were from a digital pulley. To calibrate the system, we measure the distance traveled during one wheel revolution and enter this as the pulley calibration factor.

We have a simple template to configure Logger Pro to display real-time graphs of position and velocity versus time. The saved data file consists of many rows of time intervals and computed positions and velocities. The time intervals correspond to each revolution of the wheel. We then transfer the data into a spreadsheet program (Excel) for analysis.

After attaching the sensor to the car and calibrating it, students drive to a local gravel company to weigh the car and its contents. While driving along, they see graphs of position and velocity versus time.

Figure 5 is a good pedagogical place to begin to understand torque, acceleration, and gear ratios in a qualitative way. In this particular example the car began from rest, and the driver floored it and shifted through the gears in the natural way. When the throttle is wide open, most car engines deliver a constant torque over a broad range of engine speeds. The gearing of the transmission and final drive make the engine spin faster than the wheels and also make the torque on the wheels larger than the torque of the engine. Both effects are simply the factor of the overall gear ratio. So the constant engine torque becomes a higher but constant torque on the wheels. This torque on the wheels contacting the road causes a constant force to accelerate the car (ignoring drag for the moment). On the graph one can see linear sections within each gear, suggesting a constant acceleration. In the higher gears, the gear ratio is smaller so the torque on the wheels is lower, yielding a lower acceleration.

Figure 5.

For the actual data collection, we cover a much wider range of engine speeds than one gets by the normal shifting pattern of Fig. 5. With the car moving slowly in a particular gear (clutch engaged), the gas pedal is floored and data are collected while the engine winds up to the red line or until the speed limit is reached. Then the car is slowed back down again. Now, crawling along in a different gear, the car is floored again. In this manner, data is collected in all the gears but with overlapping speeds as shown in Fig. 6. In fact it is the highest gears that give the best data at low engine speeds and the lowest gears show the high end.

Figure 6.

Air drag and rolling friction are measured by a coasting run. For this we put the car in neutral and coast from a speed of about 50 mph to a stop. Since the coasting trial is especially sensitive to the wind, we use a handheld anemometer to measure the wind speed.

We carry out all the tests on a one-mile straight stretch of road on prairie land that we estimated to be flat to about 0.5 m. Most of the acceleration runs required only a portion of a mile, but the air-drag tests often require somewhat more than a mile and so must be pieced together from two overlapping runs, as shown in Fig. 7.

Figure 7.

In order to convert the measurement of car speed into engine rotational speed, we need the overall gear ratios. These are surprisingly hard to find for many cars and are not always consistent even within a model year. Instead of trying to look them up, we now measure them by using an engine-speed probe. This was constructed by "disconnecting" the clamp-on probe from a timing light and connecting it to a second circuit virtually identical to the circuit for the magnetic probe. The only change was the addition of a low pass filter to clean up the input. When this probe is clamped onto a spark-plug wire, it gives a signal each time that cylinder fires. Car engines are all four cycle and so a complete cycle involves four strokes (intake, compression, power, and exhaust) or two crankshaft revolutions. Thus the probe signal rate is one-half of the engine speed since each cylinder fires once every two rotations. We were hoping to use this probe together with the wheel probe. Unfortunately the Logger Pro software cannot handle two such inputs simultaneously (unlike most probes, which measure a quantity when prompted, these are sending a time signal). Instead we have a switch to change which probe is being monitored. So we have the students drive at a constant speed and switch back and forth between probes. The ratio of the two signals is exactly one-half of the overall gear ratio. This has to be done for each gear.

In the analysis of the coasting data, we experimentally determine the drag versus speed. This is commonly expressed in terms of a dimensionless drag coefficient that is occasionally quoted by manufacturers who are particularly proud of their aerodynamic design. To put our drag coefficients into standard form for comparison, the students need to measure the cross-sectional area of the car. We find this by taking a head-on picture of the car from a distance and measuring the picture with a compensating polar planimeter (a delightfully simple mechanical device for measuring irregular areas on maps).

For the quantitative analysis, the data are pasted into a spreadsheet. We first determine the air drag and rolling friction coefficients from the coasting run. We fit velocity-versus-time graphs with a calculated curve as shown in Fig. 8.

Figure 8.

We assume the standard forms for the air drag force and rolling frictional force:

where C_d is the drag coefficient, the air density, A is the cross-sectional area, v is the velocity, v_w is the wind velocity, µ_r is the coefficient of rolling force, and mg is the weight of the car.

The sum of these forces divided by the mass gives the acceleration of the car. Because the forces are a nonlinear function of the speed, there is no simple expression for the velocity versus time. So the resulting motion can't be analytically solved but it is easily solved numerically. We use the straightforward Euler method of dividing time into short intervals and assuming constant acceleration within each interval. The velocity at the end of the interval is used to determine the drag forces and acceleration in the next interval. The free parameters are: the starting speed, the coefficient of rolling friction, and the air drag coefficient. If you look at the details, you will see various bumps and wiggles in the data. These are due to humps and dips in the road, and variations in wind speed.

The torque and power of the engine are both determined from the acceleration runs. Using a spreadsheet, we take differences in velocities and divide by the time interval to get the acceleration. So the acceleration is given by

We assume the only forces acting on the car are the forward driving force by the road on the wheels, an air-drag force, and a rolling friction force. Solving Newton's second law for the force on the wheel by the road, we have (omitting the index n):

where the drag and rolling forces are calculated from the velocities using the parameters determined above. Ignoring the mass of the wheel, this force at the perimeter of the wheel is created (and opposed) by the torque on the axle. So the torque on the wheel is just the force on the wheel times the wheel radius:

The engine spins faster and has less output torque than the wheels, both of these changed by a factor of the overall gear ratio G. So (neglecting transmission losses) the engine output torque is:

Since we are interested in how torque and power vary with the speed of the engine, the engine speed is found from:

Once the torque is found, the engine power can be directly calculated using the angular equivalent of force times velocity:

Figure 9 is a graph of the engine output torque of a 1986 Honda Accord LX Sedan. One might have expected the results of the four gears to fall on the same line. After all, the engine torque should merely be a function of the throttle and the engine speed. Indeed engine torque is specified at a particular engine speed but with no reference to a particular gear. But the engine torque listed in manuals is measured on a test bench with the engine running at constant speed or slowly accelerating while working against a load. In our case, the output torque of the engine is not just accelerating the mass of the car (including itself) linearly. It must also use some of its torque to provide angular acceleration of its parts, chiefly the flywheel. In the lower gears, the engine rotational speed and acceleration are large compared to the car's speed and acceleration, so this noticeably reduces the net output torque of the engine. To account for this, we define the total engine torque as a sum of the engine output torque and the torque needed for this angular acceleration of the engine.

where I is the moment of inertia and _E the angular acceleration of the engine, which is given by:

Figure 9.

Figure 10 is a graph of the total engine torque of the same car. The engine moment of inertia I was determined graphically by finding a value that made the first-gear data agree with the higher gears. The value we found for this car was 0.20 kg m². This value is reasonable compared to a few examples taken from car racing information, but we have no other source of data on any cars we have measured. In any case the effect is mainly confined to first gear.

Figure 10.

A similar but less noticeable effect is the rotational inertia of the wheels.¹ The acceleration of the car also involves the angular acceleration of the wheels. Since the wheels always have the same ratio of rolling speed to linear speed, the net effect is simply to increase the effective mass of the car. In a rolling wheel, the mass at the surface of the wheel is effectively doubled while mass at the center of the wheel has no added effect. The radius of gyration of the wheel is the radius that gives the same moment of inertia if all the mass were concentrated there. The radius of gyration of a uniform disk is 0.707R. Clearly the wheels have mass distributed unevenly, but we estimate the radius of gyration of the wheel and tire combination to be about 0.8R. The tangential speed at this radius is just 0.8v. So the total kinetic energy including this term (but not the engine term that we dealt with in a different manner above) is:

We have found that the mass of wheels and tires on various size cars is very close to 5% of the total car's mass. So the adjustment we make to account for this effect is to increase the mass of the car by 5% × (0.8)² 3%.

Note that the torque is very constant over the range of engine speeds measured. This is true of most cars. The torque ultimately comes from the force that the hot combustion gases exert on the piston. The actual details are more complicated, but let us assume that the combustion is completed at the top of the stroke before the expansion begins. Then the force on the piston is not dependent on the speed of the piston but only on the pressure of the gases (the piston speed is always well below the speed of sound). So as long as each intake stroke supplies the same mix, the pressure and torque will be the same regardless of the speed of the receding piston. At high engine speeds, the cylinder is less completely filled during the intake stroke.² In addition, the combustion is not completed before the piston begins receding.³ Both of these cause the torque to drop off. An example of the summary page from a student report is given in Fig. 11. Other reports are available on our website.⁴

Figure 11.

Note the drag coefficient of the car has been determined to be 0.39. Actual drag coefficients of real cars vary from 0.5 down to about 0.25 for very well designed cars.⁵ Although our results generally fall in this range, we have not been able to find manufacturers' quoted values for the cars we have measured.

The total engine power is graphed on the lower right of this report. Since the torque is nearly constant, it is to be expected that the power would increase linearly with engine speed. The power quoted from the manual is specified at the highest engine speed that allows the manufacturer to list a more impressive number. In this example, as in most cases, the measured values of torque and power are somewhat lower than the quoted specifications. Some of this is due to losses in the drive train and the tires that we don't account for. Our measurement of drag does account for the effects of rolling friction of unpowered tires, but when the wheels are powered, there are additional losses. Total losses from engine to road are reported⁶ to be in the range of 15%, which is consistent with our results.

We have demonstrated a system for carrying out horsepower and torque measurements on cars. The system provides a helpful kinesthetic experiment to reinforce understandings of velocity and acceleration. The resulting data are clear enough to compare with manufacturers' numbers.

The author wishes to thank Ken Horst, who used part of a sabbatical from Goshen High School to construct an earlier version of this system.

References

1. Inclusion of this effect was done at the suggestion of our referee. first citation in article
   
2. K. Newton et al., The Motor Vehicle, 10th ed. (Butterworth, 1983), p. 48. first citation in article
   
3. J. Fenton, Handbook of Vehicle Design Analysis (Mechanical Engineering Publications, London, 1996), p. 580. first citation in article
   
4. http://www.goshen.edu/physics/horsepower/index.html. first citation in article
   
5. R.H. Barnard, Road Vehicle Aerodynamic Design, (Longman, Essex, 1996), p. 21. first citation in article
   
6. http://www.pumaracing.co.uk/power3.htm. first citation in article
   

1. A Treasure Trove of Physics from a Common Source—Automobile Acceleration Data
   Christopher M. Graney, Phys. Teach. 43, 506 (2005)
2. An Alternative Approach to "Measuring Horsepower and Torque Curves of a Car"
   Chris M. Graney, Phys. Teach. 43, 363 (2005)

John R. Buschert is a professor of physics at Goshen College. He earned his Ph.D. in solid state physics from Purdue University. Current research interests include the physics of handbells.Goshen College, 1700 South Main St., Goshen, IN 46526; johnrb@goshen.edu

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Fig. 1. Components of the system shown in photo at left. First citation in article

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Fig. 2. Close-up of the wheel sensor. A small magnet in the valve stem cap passes near the magnetic probe sensor. First citation in article

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Fig. 3. Signal from the magnetic probe is converted from analog to digital form. First citation in article

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Fig. 4. Schematic diagram of the conversion circuit. The output on the right is connected to the computer interface. First citation in article

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Fig. 5. Plot of acceleration from a stop using a normal shifting pattern. First citation in article

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Fig. 6. Example acceleration run in gears 1, 2, and 4. First citation in article

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Fig. 7. A coasting run. The run had to be interrupted for a stop sign after one mile. First citation in article

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Fig. 8. Coasting runs in both directions are fit with a calculated curve. First citation in article

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Fig. 9. Engine output torque without correcting for the engine moment of inertia. First citation in article

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Fig. 10. Total engine torque including the correction for the engine moment of inertia. First citation in article

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Fig. 11. A sample lab report summary page. First citation in article

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