Differential calc

[XfireZ51]

OK guys. I'd like you to check my math. I am considering changing rear gearing due to the fact that my Z06 wheels are about 1" taller than stock and that my current speedo gear appears to have been there due to the car previously having 4.10s. It now has 3.45s. Using GPS, I find that the speedo to actual error is 10mph at 64 indicated/74 actual. This tells me that my error is ~15.6% tires and everything included. So it seems that the 3.45s are performing actually as 2.91s. If I want to get the performance of 3.45s, I calculate that:

If 3.45 = X*.844 (or 1-.156), then X = 3.45/.844 = 4.0876

Therefore, I need 4.10s:dancing in order for the speedo to be correct and for the acceleration to be what it would be with stock gears.

Guess what I'm looking for now?

[Paul Workman]

OK guys. I'd like you to check my math. I am considering changing rear gearing due to the fact that my Z06 wheels are about 1" taller than stock and that my current speedo gear appears to have been there due to the car previously having 4.10s. It now has 3.45s. Using GPS, I find that the speedo to actual error is 10mph at 64 indicated/74 actual. This tells me that my error is ~15.6% tires and everything included. So it seems that the 3.45s are performing actually as 2.91s. If I want to get the performance of 3.45s, I calculate that:

If 3.45 = X*.844 (or 1-.156), then X = 3.45/.844 = 4.0876

Therefore, I need 4.10s:dancing in order for the speedo to be correct and for the acceleration to be what it would be with stock gears.

Guess what I'm looking for now?

Simple proportion, if I'm not mistaken...

64/74 : 3.45/ X

Solving for "X"

X = 74 x 3.45/64
X = 3.989

Checking:

3.45 is to 3.989 as 64 is to 74

3.45/3.989 = 64/74

Confirmed...the new ratio is 3.989

Or so it seems to me...:p

P.

[XfireZ51]

Paul,

I think the diff here is that I am looking for a gear that will perform LIKE a 3.45 given the current error I have. I am calc'ing the error as actual/indicated which gives me 15.6%. You're calc'ing the error as indicated v actual which is 13.6%.
The error is 10mph at an indicated 64 or 15.6%.
Your proof works because you substituted the answer for X.

[Gunny]

I created an Excel file that calculates gear ratios ... it should provide the calculations you need.

Click HERE (http://tbf.me/a/pdnza) to download. When you open this link you should see a file "GearRatio.xls ... click to download the file. Please note that this link expires in 5 days and is limited to 20 downloads. If you have difficulty with the download please IM me with your email address and I'll send it to you.

George Jones

[Paul Workman]

Paul,

I think the diff here is that I am looking for a gear that will perform LIKE a 3.45 given the current error I have. I am calc'ing the error as actual/indicated which gives me 15.6%. You're calc'ing the error as indicated v actual which is 13.6%.
The error is 10mph at an indicated 64 or 15.6%.
Your proof works because you substituted the answer for X.

The ratios beteen speed difference an gear ratios has to be preserved.

Yours (.844) is the result of a common error (when using percentages) stemming from the corruption of the root ratios by first conveting them to percentages and then mulitplying by their antithesis.:thumbsdo: Apples to oranges in a logical, albeit erroneous manner.

For example, 74/64 = 1.156 or 74 is = 115.6% of 64.

Checking: Let's stick with the ratios... Dividing 74 by 1.156x = 64 = check.

Second check: Inversely, 74 x 1/1.156x = 64. We're good.

But here's the thread of a common mistake: When converting ratios to %, one first multiplies by 100, and then to derive the result as an increase (in this case) we subtract 100 to isolate the change. the antithesis of 15.6% is (in this case) 100% - 15.6% or 84.4%. BUT the reciprical (100%/84.4%) is 118.463% NOT 115.6% which was the orginal ratio 100 x 74/64 or 115.6%

So, keeping with ratio (inv) proportions, the ratio (proportionally speaking) between 64 and 74 is the same ratio that exists between the current 3.45 gear and that of the new gear ratio required to equate to the same ratio in order to get the speedo to agree with the acutal.

Don't feel bad. Retailers make the same mistake all the time by accident (or intentionally). Mark a $100 product up 20% to $120, and then have a 20% reduced price sale; 100%-20% is 80%, but 80% of $120 IS NOT $100! See where this is going?

Keep the ratios intact, and all is well. I'm just sayin.

P.

PS

3 guys check into a cheap hotel room. The clerk initially charges them $30 or $10 each for the room. Later he discovers the room rate was supposed to be $25. The clerk returns to the room with 5 $1 bills. Explaining the error, the guys decide to each take a dollar and leave $2 with the clerk for being so honest. So, in effect, each pays $9 for the room or 3x $9=$27. The clerk keeps $2 and $27 + $2 is $29 NOT $30!!...Where is the missing $1????

Not exactly the same thing, but then sorta IS.;)

[WVZR-1]

I did this a bit differently but the numbers came closer to PW's. I believe pure math makes PW's correct. The displayed error is .86486xxxxx rounded to .865 so I'd guess if we wanted to correct this at the rear gear we'd need 3.45 x 1.135 or a 3.915xxx. Granted this number required rounding up of the error which resulted in a lower value for the multiplier which makes a result that differs from the 3.989 of PW's.

To get your displayed 64 to the actual 74 requires an equal adjustment?

You could do the 4.10's and correct the speedo error after the install.

With no drive/driven speedo adjustments a 1" plus tire from the 315/35 (25.68) would result in an actual 77 MPH with a 74 displayed I believe.

Wasn't this discussed awhile ago?

[XfireZ51]

Given the display error of 10mph with a display of 64, we know the display error is 15.6%. In order for the display error to be 15.6%, my 3.45 gears are effectively 3.45 x .844 = 2.91

Setting up a proportion,
I can say that
2.91:3.45::3.45:X with X being the new gear
Solving for X
3.45x3.45=2.91X
11.90=2.91X
4.09=X

[ZRapid-1]

OP…… not all your info adds up.

Here’s some scenarios:
1. If you want the same acceleration as you had prior to the Z06 wheel changeover:
• Assuming a stock tire diameter of 25.7 inches and then a changeover to Z06 tire diameter of 26.7 inches
• This is only a change of [(26.7 – 25.7) / 25.7 ] x 100% = 3.89 %.
• Note that this does not correlate to the percentage difference between your speedometer and GPS.
• Assuming you have a 3.45 differential ratio, then a change to 3.45 x 1.0389 = 3.58 will provide equivalent acceleration.
• This change, by itself, will still result in an incorrect speedometer reading.
• This analysis shows that the error is not due to just the Z06 tire changeover. If you put the original 25.7 inch diameter tires back on, you will still see a difference between the speedometer and GPS.

2. If you want the speedometer to match the GPS:
• 64 MPH [Indicated Speed] x TBD = 74 MPH [GPS Actual Speed]
• TBD = 1.156
• Need to increase tire rotation by 15.6%
• 3.45 x 1.156 = 3.989
• A differential ratio of 3.989 (and Z06 tires) will provide the correct speedometer reading. It will also provide greater acceleration than the 3.45 gears and 25.7 inch diameter tire combination.
Here’s my theory,
The car started with 3.45 gears and had the correct matching transmission gear for the speedometer.
The car then was converted to 4.10 gears and matching transmission gear (for 4.10).
Another conversion changed the 4.10 gears back to 3.45 – but left the transmission speedometer gear at 4.10.
You then added larger diameter Z06 tires.
3.989 differential gears and Z06 tires will provide the correct speedometer reading.
25.7 inch diameter tires and 4.10 differential gears will also provide the correct speedometer reading.Your other choice is to change the transmission speedometer gear if you just want to get a correct speedometer reading.

[WVZR-1]

I guess the difference of opinions here is the calculation of the "hypothetical" gear!

64/74 = .86486..... error

64 X 1.156 = 73.984 for correction

3.45 X .86486 = 2.98 (hypothetical for correction)

2.98 X 1.156 = 3.448 for present ratio

3.45 X 1.156 = 3.98 required for speedo correction

I originally approximated some things and the adjustment I used to correct was in error. See what you think of this rationale!

[Paul Workman]

Given the display error of 10mph with a display of 64, we know the display error is 15.6%. In order for the display error to be 15.6%, my 3.45 gears are effectively 3.45 x .844 = 2.91

Setting up a proportion,
I can say that
2.91:3.45::3.45:X with X being the new gear
Solving for X
3.45x3.45=2.91X
11.90=2.91X
4.09=X

Dom... As other too have pointed out, your error is rooted in the .844 number

The ratios between speeds has to equal the ratio between gear changes, yes?

64:74::3.45:3.989 are valid.

74/64 = 1.156 and the recip. is 1/1.156 = .856, NOT .844.

You can proove it yourself: try applying .844 or 1/.844 to either the 64 or the 74 and see if the other is derived...NEITHER WILL. 74 X .844 = 62.45, NOT 64. 64 X 1/.844 = 75.829 NOT 74

An engineering friend of mine used to call it the "Bugger Factor". That's where a calcuation starts with an answer in imbedded in the mind, and if by chance the treatment delivers a number close to the expected answer, the "Bugger Factor" takes over the mind and the assumption is made. (It is dangerous to assumes a single calc is correct. Hence, PROOFs are necessary to verify the root numbers can be reproduced. Such is not the case here as WVZR-1 and ZeeAreOne also demonstrated.)

The "Bugger Factor" in the soup is the result of you calc being 4.09 and is so close to the expected 4:10 that the calc must be valid...But, it is flawed.

I say we're splitting hairs, in reality:

I suggest you install a 4.09 or 4.10 gear to get back to the equiv of the stock 3.45*, because, 1)that ratio shouldn't be hard to find, whereas the 3.99??? 2) The Bugger Factor will be satisfied because your error will only be about 2.5%, or approx 1.2 mph at 50 mph indicated - well within the window of acceptable error for a spedo, and 3) your shift light is all that really matters on the strip.

*Or going with a 4.56 would put you pretty close to the same as going from a 3.45 to a 4.09 on stock wheels/tires...Just a thought.
And, as the saying goes...your mileage may vary! (Just don't take a job at JPL. They're calculating the trip to Mars right now, and wouldn't want to land on one of its moons instead!:razz:)


P.

[XfireZ51]

My math teacher sister in law says engineers can't solve anything correctly unless the formula is already set up for them.:p;)
We are splitting hairs and at the end of the day the 4.10 is likely the most practical solution. I'm just going to continue the disagreement because it's fun.:cheers:

Proportionality is the most accurate method of solving this:

Effective gear:Actual Gear::Effective gear:Actual Gear

Paul,

You can't accuse me of the Bugger Factor because I was not "looking" for a particular answer. In fact, I had originally thought of 3.90 gears.
The .156 is the error of the indicated speed not the actual. That's why your proof does not work.

WVZR-1,
The error is 15.6% whether you arrive at it by (74/64)-1 or (74-64)/64. The divisor is the indicated speed because that's what I'm trying to solve for.
I use proportionality because I want to maintain the same relationship (since I'm keeping the wheels and speedo gear) I currently have between the 3.45 gear and what it acts like (2.91) when I swap gears.

ZeeAreOne,

That was my suspicion also. Car had 4.10s and at some point 3.45s were installed but the speedo gear never swapped. Of course that under reports the mileage. HMMMMM! No matter.

[Jagdpanzer]

I found this tool handy when I switch over to 19" wheels and larger diameter tires on the rear:
http://www.rimsntires.com/specs.jsp

Prior I was running stock 17" wheels and tires, 4.10 rear gears and 13/43 speedo correction gears. With this combo the speedometer and GPS more or less matched. However, after changing to 19" wheels and larger diameter 345/30ZR19 tires the speedometer showed 4 mph slower with 60 mph indicated on the GPS. Changing to 13/41 speedo gears narrowed the difference to around 1 mph which I find acceptable.

[RHanselman]

You guys are killing me sitting in your loungers at 1 G...

Us old Fighter Pilots just try to use the KISS principle...

Keep the 3:45's, put the stock wheels back on and then put the stock trans spedo gear back in. Problem solved... :-D

But its kind of fun to see you guys go out to 10 digits and then cut it with an axe!

(4:10's are more fun...)

[XfireZ51]

Dyno,

Thanks for all the higher math. ;). This was going to be my next step.
But let me clarify again the objective. I don't really care about the speedo accuracy. What got me started on this is the realization that I have gone through my upgrades and I am killing performance by having the equivalent of a
2.90 some odd rear gear. So the real purpose of this exercise is to at come up w gearing that approximates the performance of the stock setup. If the speedo is off, I honestly couldn't care.

The other objective of the thread is to give Ron fun.

[XfireZ51]

This also points to why my trap speeds dropped originally when I installed the 19's.

[RICKYRJ1]

I have WAZOO installed 3.73's. I run both 18" and 19" rears depending on what look I prefer at the time. Where does that put me? We installed the correction gear at the time of install. I have only driven her with the 19's on since the install, car feels good. :cheers:

[XfireZ51]

I have WAZOO installed 3.73's. I run both 18" and 19" rears depending on what look I prefer at the time. Where does that put me? We installed the correction gear at the time of install. I have only driven her with the 19's on since the install, car feels good. :cheers:

Just using Dynos calc, I'd estimate u have an effective 3.22 gear.

[WVZR-1]

I have WAZOO installed 3.73's. I run both 18" and 19" rears depending on what look I prefer at the time. Where does that put me? We installed the correction gear at the time of install. I have only driven her with the 19's on since the install, car feels good. :cheers:

Where does that put you? Your 3.73 is "still" a 3.73 gear

These are 1:1 4th gear calculations:

Using Phil's 345/30/19 (27.1) and your 3.73 gear 65 MPH: 3006 RPM, 100 MPH: 4625 RPM

Using a 315/35/17 and your 3.73 gear 65 MPH: 3170 RPM, 100 MPH: 4877 RPM

Your speedometer error actual/displayed: 60/56.8 - 100/94.6 - 140/132.4 These are calculated using advertised diameters not "rolling radius" but it's very close to what Phil mentions using the GPS.

To regain the RPM if you're "performance minded" I'd think it would require a 3.92 gear 65 MPH: 3159 RPM, 100 MPH: 4860 RPM

If you wanted to relate what the 345's did to your present 3.73 I'd guess you'd do the 3.73/3.92 = .9515 X 3.73 = 3.55

Seems logical to me but then again..........

[ZRapid-1]

Dyno,

Thanks for all the higher math. ;). This was going to be my next step.
But let me clarify again the objective. I don't really care about the speedo accuracy. What got me started on this is the realization that I have gone through my upgrades and I am killing performance by having the equivalent of a
2.90 some odd rear gear. So the real purpose of this exercise is to at come up w gearing that approximates the performance of the stock setup. If the speedo is off, I honestly couldn't care.

The other objective of the thread is to give Ron fun.


Changing from the stock ZR-1 tire diameter to one that is an inch larger doesn't have a huge effect.

25.7 x 3.45 = 26.7 x (Effective Dif Ratio)

Effective Dif Ratio = 3.31

Another way of stating it is that a 26.7 inch tire diameter (Z06) with a rear dif ratio of 3.45 is the equivalent of a 25.7 inch tire diameter (stock ZR-1) with a rear dif ratio of 3.31.

Not sure of the weight difference between rims/tires you are using, but acceleration could suffer. With the larger diameter tire, most of the weight is at the tread and thus further away from the axle centerline. A larger rim diameter also puts more metal at a larger radius from the axle centerline. The added rotational inertia will slow the tire's rate of acceleration.

[Paul Workman]

Oh, hell...Going all the way around the barn here; rpm, inches per second... :rolleyes:

Cutting to the chase:

Let the (effective reference) radius from the center of the wheel (A) to the contact patch = 1
Let the (effective) radius from the center of then new wheel (B) to the contact patch = 1.2
Let the current differential drive shaft to axel ratio (C) =1:1
Let "X" = the new differential ratio.

X = Bx2xPi/Ax2xPi x C
simplifying...
X = B/A(reference) x C
substituting values...
X = 1.2/1 x 1 = 1.2

Checking:

The reference case: The (effective) circumferance of (A) = DxPi, or radius x 2 x Pi = 1 x 2 x 3.14 = 6.28
The (effective) circumferance of (B) = 1.2(radius) x 2 x 3.14 = 7.536

The new driveshaft axel ratio: X = B/A x C = 7.536/6.28 x 1:1
The new ratio = 1.2:1 (check)

B (7.536) / 1.2:1 = 6.28 = A...Check!!

Proof #2:

Using a proportional computation: A is to B as 3.45 is to X expressed as-

1:1.2 :: 3.45:X

solving the proportion-

X = 1.2x3.45/1 = 4.14 (in this hypothetical case)

4.14/1.2 = 3.45 Check (again)

Cliff (Dynomite) is right, far as using the speedo reading and GPS doesn't address the gear ratio directly, leaving a lot to chance, unless you assume 100% accuracy of the speedo and GPS values. Direct measuring is always less likely to introduce hidden factors.

Dom, you measure the effective radius of your wheel (center to patch), and I'll measure mine (currently a F1 GS D3 315x35x17) and then we'll have a pretty good basis for predicting final ratio to equate to the 3.45:1 on stock wheels. K?

My radius = 12.25"

X = (your radius)/12.25 x 3.45 = your new ratio to exactly equate to the 3.45

I'm outta here. Oh, wait!...I'm outta here except to trade friendly jousts with your sister: Being a math teacher in public school is no glowing recommendation . I've had to teach logarithms to high school grads that couldn't balance their checkbook. I made a good living for several years teaching math to HS grads that didn't have a grasp of the 4 basic principles of arithmetic. (Did your sister teach you math too??:sign10:)

P.

[WVZR-1]

Interesting!

I measured my BFG G-force some time ago (years) doing a single rotation for circumference of the tire and I really don't recall the answer BUT a good while ago I was told to use this for "rolling radius" and it's usually been very close.

Advertised diameter less 1/2 of the tread depth x .96/2 but I've never passed that on anywhere. I can't presently measure mine but it seems like it's actually a good tool.

Your F1 315/35R17 25.7 - .16 (1/2 - 10/32) X .96 = 24.52/2 = 12.26

It seems a match or very close! Air pressure!

[XfireZ51]

I'm outta here. Oh, wait!...I'm outta here except to trade friendly jousts with your sister: Being a math teacher in public school is no glowing recommendation . I've had to teach logarithms to high school grads that couldn't balance their checkbook. I made a good living for several years teaching math to HS grads that didn't have a grasp of the 4 basic principles of arithmetic. (Did your sister teach you math too??:sign10:)

P.

Paul,

I missed this last part. First of all, Kathy is my sister-in-law and she teaches at a top girls' Catholic High School in Chicago, Trinity. Its run by the Dominican nuns. She teaches the math honors classes and Kathy has her math major and physics minor degree from UIC. Not too shabby, wouldn't you say? If Kathy was incorrect, its because of how I set up the problem. And honestly, the snark on teachers in public schools is unnecessary. I have a good friend who taught math and physics in public school. Pretty good guy who knows his stuff.
Now that that's cleared up, turns out I was completely blind and the
Oracle of Addison :worship:(not you Paul) straightened me out.:redface: I had same wheel/tire combo on Azzuro-1 and measured ODO v GPS at that time. Rear gears were 3.45s. ODO was off by 5% with correct trans gear so at 60mph, ODO read 3 mph low and at the time I thought that wasn't enough to bother with re: correct trans gear. If I replace the trans gear with the correct 3.45 gear, I will be off by same amount now as well. Marc agreed I have 4.10 trans speedo gear in there now. A 5% diff means the car runs like a 3.28:1. To get it back to a 3.45 would take something like a 3.66. Closest would be a 3.73. Marc however still encouraged me to go with the 4.10's (which BTW I found a killer price on for OEM Spicer gears) and they effectively would run like a ~ 3.90 with the same wheel/tire combo.
Now all I need is $$$$. :-x:dancing

[XfireZ51]

Interesting!

I measured my BFG G-force some time ago (years) doing a single rotation for circumference of the tire and I really don't recall the answer BUT a good while ago I was told to use this for "rolling radius" and it's usually been very close.

Advertised diameter less 1/2 of the tread depth x .96/2 but I've never passed that on anywhere. I can't presently measure mine but it seems like it's actually a good tool.

Your F1 315/35R17 25.7 - .16 (1/2 - 10/32) X .96 = 24.52/2 = 12.26

It seems a match or very close! Air pressure!

Is there any accomodation for "growth" of the tire due to centrifical force and heating of air at x mph? I'm supposing its negligible or within the margin of error for the speedo itself.

[sammy]

dominic .not sure if you are still interested in the 392s that came out of the callaway car .if you are i gave them to bill boudreau. he will sell them to you for the price we talked about .if you arent they are for sale and are in great shape just contact bill boudreau .

[XfireZ51]

dominic .not sure if you are still interested in the 392s that came out of the callaway car .if you are i gave them to bill boudreau. he will sell them to you for the price we talked about .if you arent they are for sale and are in great shape just contact bill boudreau .

Brad,

I'll contact Bill on this. Thx

[Paul Workman]

Paul,

I missed this last part..

Dom,

Yes, you did (miss it)...AGAIN!

Well, Let me apologize, Dom. I got tickled when you threw your sister in law (I stand corrected) onto your funeral pire. I was tying to inject my sick sense of humor into the conversation, and ... well ... that's the problem with these boards: without facial expressions etc, things can be misunderstood. Actually, I was poking fun AT YOU, Dom, or (more precisely) your damned Italian stubbornness!! :p


Now that that's cleared up, turns out I was completely blind and the
Oracle of Addison :worship:(not you Paul) straightened me out.:redface: I had same wheel/tire combo on Azzuro-1 and measured ODO v GPS at that time. Rear gears were 3.45s. ODO was off by 5% with correct trans gear so at 60mph, ODO read 3 mph low and at the time I thought that wasn't enough to bother with re: correct trans gear. If I replace the trans gear with the correct 3.45 gear, I will be off by same amount now as well. Marc agreed I have 4.10 trans speedo gear in there now. A 5% diff means the car runs like a 3.28:1. To get it back to a 3.45 would take something like a 3.66. Closest would be a 3.73. Marc however still encouraged me to go with the 4.10's (which BTW I found a killer price on for OEM Spicer gears) and they effectively would run like a ~ 3.90 with the same wheel/tire combo.
Now all I need is $$$$.

You may or may not be out of the woods yet, my friend... Cliff (Dynomite) made a very astute observation::thumbsup: The ratios you should be concerned with are the actual wheel and drive train ratios, i.e., direct measurements, and NOT your speedo or GPS readings, or phase of the moon or anything else (read: indirect measurement) unless you have no other choice.

Truth is where you find it. So, wheter it was Marc or Cliff, or me, or some others too that shone the light, in the end it's all good.

I believe my math was correct - provided mulitple proofs - but I'm willing to be proven otherwise. That said, in (one) example you gave, I too arrive at the same "3.28:1" ratio you (or was it Marc?) finally derived. Regardless of how you get there, as long as the wheel change ratio matches the gear change ratio, you're golden.

Sorry if you thought I was taking a shot at your sister in law. I wasn't. But, I was messing with you, Dom!;)

P.