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