Correcting a Speedometer for a New Wheel Diameter
Date: 11/28/2000 at 08:22:12 From: Mike Subject: Working out angular/linear velocity Hi, I have a problem: For the speedometer on a motor: Originally the speed is measured when the wheel diameter is 65 cm, and it is correct. But when the wheel diameter is increased to 75 cm the reading is not correct. We are required to use a microcontroller in order to correct this problem. I have also been told that I would need to know the angular and linear velocity when using either wheel. I would be grateful for your help or advice. Thanks.
Date: 11/28/2000 at 13:00:48 From: Doctor Rick Subject: Re: Working out angular/linear velocity Hi, Mike. The situation here is that the speedometer measures the angular velocity of the wheel and scales it (multiplies it by a factor) to get the linear velocity. The scale factor depends on the diameter of the wheel. You can find the scale factor by considering how far the car goes when the wheel rotates one full turn. The car moves forward by the circumference of the tire. This distance is the factor by which you have to multiply the angular velocity in revolutions per minute to get the linear velocity in centimeters per minute. You can then multiply by an additional factor to get the linear velocity in km/hour or whatever you need. Find the factor that the speedometer uses, and the factor that it should use in order to get the speed right with a 75 cm diameter wheel. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
Date: 11/29/2000 at 07:51:59 From: Mike Subject: Re: Working out angular/linear velocity Thanks for your quick reply. Pardon me for being bit thick, but I'm still not sure how to work out the factor. I have the assignment sheet here now, so maybe if I write out the exact words it will be a bit clearer: The speedometer and associated electronics for a vehicle have been designed for a wheel diameter of 65 cm; but for rough terrain an alternative wheel with a diameter of 75 cm is used. This means that the road speed information derived from the road speed sensor is too slow. Why is it too slow, and how must the input frequency be adjusted to obtain the correct road speed data? A typical input frequency has a maximum value of 400 Hz. You are required to design a real time system that will accurately read the input frequency to a resolution of better than 1 Hz, calculate the new output frequency using the factor derived above, and to generate that output frequency. I would greatly appreciate your help on this. Thanks.
Date: 11/29/2000 at 08:29:56 From: Doctor Rick Subject: Re: Working out angular/linear velocity Hi again, Mike. The last paragraph is entirely up to you. The math part is to compute the factor by which the sensor frequency must be multiplied so that the speedometer will display the correct velocity with the new tire size. First you can figure out what the existing system does: given a road speed (linear velocity), what is the frequency coming from the sensor? I will assume that the sensor puts out one pulse per rotation; it really doesn't matter as long as the number of pulses per rotation is constant. In one rotation (the time between pulses), the car moves forward by a distance equal to the circumference of the tire. Do you see this? It's the critical fact in the problem. Calculate this distance from the diameter of the tire (the original tire, for which the system was designed). Then use the rate equation, time = distance/speed, to find the time between pulses for a given speed. Convert this to a frequency (pulses per second). What you have just found is the pulse frequency that your system must produce for a given actual speed. Now go through the same process with the new tire size. You'll get a second equation that represents the relationship between the INPUT to your system and the actual speed. Take the two equations and eliminate the speed. Solve for output frequency in terms of the input frequency. That's the relation that your system must model. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
Date: 11/29/2000 at 09:39:30 From: Mike Subject: Re: Working out angular/linear velocity I've managed to complete most of the calculations, but I am stuck on the very last bit. When you say to take the two equations and eliminate the speed to solve for output frequency in terms of the input frequency, I am having problems in solving this relation. Do I need to use simultaneous equations? How do I eliminate the speed factor? Thanks.
Date: 11/29/2000 at 12:34:09 From: Doctor Rick Subject: Re: Working out angular/linear velocity Hello again, Mike. I'd like to see the equations you have. If you have them in the form f_in = a*v f_out = b*v then you can solve these simultaneous equations as follows. First solve the first equation for v: v = f_in/a Then substitute this expression for v in the second equation: f_out = b*(f_in/a) Then you have what you need: a formula that takes the input frequency f_in and tells you the correct output frequency f_out. Now your job is to design the system that will do this frequency conversion, which is outside the scope of Dr. Math. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
Date: 11/29/2000 at 14:31:18 From: Mike Subject: Re: Working out angular/linear velocity Sorry for bothering you again, but I am still a bit confused. These are the equations I have derived. For the 65 cm wheel: circumference = 2*pi*radius (meters) Then, I assumed a speed of 10 m/s. Next I worked out the time by: time = distance/speed = 0.204 s Then I worked out the frequency: F = 1/time = 4.9 Hz I did the same for the 75 cm wheel. As it stands now, I have a frequency value for each wheel, the 65 cm being 4.9 Hz and the 75 cm wheel being 4.24 Hz. I do not understand your equations: f_in = a*v f_out = b*v What are a and b? Please could you explain the two above equations? Thank you for your time. Much appreciated.
Date: 11/29/2000 at 15:50:37 From: Doctor Rick Subject: Re: Working out angular/linear velocity Hi, Mike. Let's go through what you've done, but replacing your assumed velocity of 10 m/s by the variable v. That way, instead of a single number, we'll derive a formula that works for any velocity. time = distance / speed = 2*pi*r / v = pi*d_1 / v [diameter d_1 = 2*r] The frequency that we want to send to the speedometer when the car is going v meters per second is thus freq_out = 1/time = v / (pi*d_1) If v = 10 m/s and d_1 = 65 cm = 0.65 m, then freq_out = (10 m/s) / (3.14*0.65 m) = 4.90 hertz Our answers agree. But since I left the equation in a general form with variables v and d_1, we can easily write the equation for the input frequency, just by replacing d_1 by d_2 = 75 cm = 0.75 m: freq_in = v / (pi*d_2) I have just written the two equations to which I referred in the last e-mail: freq_in = (1/(pi*d_1)) * v freq_out = (1/(pi*d_2)) * v These are in the form I gave as f_in = a*v, where a is a constant that I worked out to be: a = 1/(pi*d_1) = 0.4897 You can work out the constant b. It's cleaner, though, if you keep the variables d_1 and d_2; then you can make the conversion system fully general, able to cope with other wheel sizes besides 65 cm and 75 cm. I showed you how to do the rest of the problem last time. It's up to you to put the pieces together. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
Date: 12/01/2000 at 10:42:59 From: Mike Subject: Re: Working out angular/linear velocity I just wanted to say thanks a lot for your help. I'm sure you'll be glad to know you won't get any more headaches - I handed the work in today. I was up until 8 a.m. finishing it off. Your help and advice really helped a lot. Much appreciated. Mike
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