# Using Level Accelerations to Determine Climb Performance

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#### Hot Wings

##### Grumpy Cynic
if that's really necessary. Don't see why it would be.
An Hp-35 is to a Ti-89 as a pile of rocks is to a slide rule.
When we took Calc we were taught how to graph a function using stone age clues like the first and second derivative, vertical and Hz asymptotes, local max/min and axis crosses.
Now that it is all done on the Ti, and you had better have your scale set correctly or you won't see much detail. The Ti will do the integration for you.
Nice tool for a lot of things but not so sure it is worth the time to learn for those of us that remember the gas lines of the 70's. IMHO for us time would be better spent polishing our Excel and Python/C++ skills?

#### Saville

##### Well-Known Member
An Hp-35 is to a Ti-89 as a pile of rocks is to a slide rule.
When we took Calc we were taught how to graph a function using stone age clues like the first and second derivative, vertical and Hz asymptotes, local max/min and axis crosses.
Now that it is all done on the Ti, and you had better have your scale set correctly or you won't see much detail. The Ti will do the integration for you.
Nice tool for a lot of things but not so sure it is worth the time to learn for those of us that remember the gas lines of the 70's. IMHO for us time would be better spent polishing our Excel and Python/C++ skills?

Thanks for the suggestion on setting scales on the TI but I doubt I'll need that suggestion.

I program in Python

Every. Single. Day.

....in my day job and use Matplotlib constantly so I doubt I need to polish up those skills.

I don't have a graphing calculator and wouldn't waste a single dime on one. If they need me to graph something I'll graph it. That Hp-35 I owned was back in '75 so I'm not sure why you are comparing it to a TI-89. Nowadays I have an HP 32-S2 scientific - less a pile of rocks. But no matter as I don't see the need for a calculator in a calculus class. We used to secretly laugh at people who said that a calculator will help in a calculus course.

If I need to work out the final value of a definite integral I'll do the indefinite on paper and take it from there. If I need to check my work I might use something like Wolfram Mathematica.

#### Dana

##### Super Moderator
Staff member
I was lucky to have a couple of really good math and calculus teachers. Still didn't keep me from forgetting most of it...

OTOH, when you go to the first day of class and the calculus professor starts talking about "the beauty of the calculus" you know you're gonna be in for a rough ride...

#### Saville

##### Well-Known Member
I was lucky to have a couple of really good math and calculus teachers.

OTOH, when you go to the first day of class and the calculus professor starts talking about "the beauty of the calculus" you know you're gonna be in for a rough ride...

;^)

My first quarter calculus teacher started out by showing how you could make a minor mistake in your algebra and prove that 0 = 1. ;^)

Then he said that mathematicians are the laziest people on earth - they are constantly trying to find simpler ways to do things.

So in retrospect I think he was probably a good teacher. I just wasn't a good enough student.

#### Vigilant1

##### Well-Known Member
In the "old days," teaching/education was the happy beneficiary of some talent market distortions. My best HS science teacher was a young engineer who switched to teaching to get the draft deferment. Similarly, some fantastically bright and talented women became teachers (at low pay) because it was one of the few professions to which they were admitted.
I sure wouldn't want those conditions reimposed, but there was a silver lining to the cloud. With today's low unemployment rate and the widespread emphasis on credentialing from college education departments, I doubt classroom instruction is improving.

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#### BJC

##### Well-Known Member
The best math teacher that I had was my 8th, 9th, 11th, and 12th grade teacher in a government high school in Georgia. She took a few of us through differential equations. Never saw anything new until the third quarter of my sophomore year in AE. Had to learn lots of new stuff on my own for AE courses. After Dr. Li assigned an aero problem that no one could solve, he finally helped us out in broken English: “Hint; use theory residue.” Had to buy Advanced Engineering Mathematics, a very expensive reference back then, at \$11.50, to learn about the theory of residue.

BJC

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#### Hot Wings

##### Grumpy Cynic
But no matter as I don't see the need for a calculator in a calculus class.
We may not, but the modern instructor will. And the tests may be such that you will need the speed of the calculator.

Never saw anything new until the third quarter
Similar experience here. Kids today can get college credit for college level high school classes. Saves a lot of time not having to repeat the last year of high school just to get the credits.

#### Saville

##### Well-Known Member
We may not, but the modern instructor will. And the tests may be such that you will need the speed of the calculator.

We'll see. I sent email to the instructor.

#### gtae07

##### Well-Known Member
The best math teacher that I had was my 8th, 9th, 11th, and 12th grade teacher in a government high school in Georgia. She took a few of us through differential equations.

Her name wasn't Glenda Stevens by chance, was it? Because she sounds a lot like a math teacher we had at our school who was brilliant and able to explain things nobody else could. Based on her age it could have been possible, anyway...

#### FinnFlyer

##### Well-Known Member
Having read the article, I find one thing curious. The author has us extract the velocity curve to 4th power and five digits, then calculates airspeed at one second intervals to then extract acceleration. Anybody ever pass a physics class. Take an airspeed curve and differentiate it once to get the acceleration curve. Then you can work at whatever time base (or airspeed base) you want. Here is how it works:

v(t) = A*t^4 + B*t^3 + C*t^2 + D*t + E

Differentiating once gives

a(t) = 4A*t^3 + 3*B*t^2 + 2*C*t + D

Now you can run the case at 1 knot or 1 mph or 1 ft/s airspeed intervals, as suits you without extra fuss.

Billski

Finally got what you wrote. The thing is that if using a spreadsheet it's a lot easier to copy and paste the trend line equation and add the dV/dt and dH/dt columns than differentiating the tend line equation.

Finn

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#### Hot Wings

##### Grumpy Cynic
We'll see. I sent email to the instructor.

Hope your experience is less 'eye opening' than mine was after 20+ years of mathematical stagnation.

#### BJC

##### Well-Known Member
Her name wasn't Glenda Stevens by chance, was it? Because she sounds a lot like a math teacher we had at our school who was brilliant and able to explain things nobody else could. Based on her age it could have been possible, anyway...
Mrs. Stokes. She would be well over 100 years old today.

BJC

#### Saville

##### Well-Known Member
Hope your experience is less 'eye opening' than mine was after 20+ years of mathematical stagnation.

" My syllabus also included a sentence that it is strongly recommended that each student in this course purchase a graphing calculator comparable to the TI 83 or TI 84. Nevertheless, having a graphing calculator is not required. It seems to me that any scientific calculator with buttons for LN, LOG, SIN, COS and TAN should be sufficient. "

He continues:

"All these functions can also be calculated with an Excel spreadsheet. As a result, using Excel should be sufficient, especially if you learn how to produce graphs with Excel. Actually, it seems to me that it is easier to evaluate functions and to produce usable graphs with an Excel spreadsheet than with a graphing calculator.
....
Another option is to use the calculator on a cell phone. If you turn your cell phone sideways, then you should see a calculator with scientific functions. However, it seems to me to be better to not allow cell phone use on exams, because too many students seem to upload exam questions to services that produce answers a half hour later if that is allowed."

Neither the TI-83 or 84 is a printing calculator. Therefore I get the impression from what he wrote that the graphs are for the student to get a better sense of what's going on and not anything to be turned in for homework or tests. Especially since he suggests using Excel. Excel on a laptop is worse than a cell phone for cheating and he doesn't like cell phones for tests.

So...a scientifc calculator is all I need, plus Excel (or mathematica) or Matplotlib.

#### Hot Wings

##### Grumpy Cynic
Sounds like his students are going to get a better course than they get at my local college.
Here the Ti-89 is 'required' and the instructors didn't like having to deal with the Casio equivalent.
The Ti-89 is?/was actually allowed in the proctored math test labs.

I had to download an HP-48 emulator.....was 1/2 of an engineering course on Matlab I wanted to take. It was the instructors favorite calculator. He had a stash of still working units. The HP was kind of nice for conversions and simultaneous equations with the RPN. He still required showing work using Cramer's rule on some of the tests just to make sure we knew how.

#### wsimpso1

##### Super Moderator
Staff member
Finally got what you wrote. The thing is that if using a spreadsheet it's a lot easier to copy and paste the trend line equation and add the dV/dt and dH/dt columns than differentiating the tend line equation.

Finn
Harramph. 18 minutes total to enter the data, get the coefficients, to write the program, put in the coefficients, and make the equations work. Now you the speed on whatever time base you want. Easier? 18 minutes, and now everyone has the program....

#### Attachments

• Level Accel Sheet.xlsx
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#### Dana

##### Super Moderator
Staff member
Kids these days... we learned calculus just fine making plots with colored pencils, a french curve, and graph paper.

Now, where's my pencil sharpener?

#### FinnFlyer

##### Well-Known Member
Harramph. 18 minutes total to enter the data, get the coefficients, to write the program, put in the coefficients, and make the equations work. Now you the speed on whatever time base you want. Easier? 18 minutes, and now everyone has the program....

Sorry. You're missing the point. If testing at multiple altitudes and weights you may be looking at tens of runs (and thus equations).
Also, I don't limit it to 4th order and just 5 decimal points.

I guess, to each their own.

Finn

#### wsimpso1

##### Super Moderator
Staff member
Sorry. You're missing the point. If testing at multiple altitudes and weights you may be looking at tens of runs (and thus equations).
Also, I don't limit it to 4th order and just 5 decimal points.

I guess, to each their own.

Finn
Actually, I think you are missing the point. The test method IS a tremendous time saver, but the data reduction, as presented, is weak statistically. This engineer trained in statistics knows that good data analyzed poorly can be ... misleading. What is worse than having poor data? Learning something that is far off the mark is worse than poor data.

Using a better data reduction should help the process and can be automated so it is very little effort beyond getting the data off the flight logger and installing it in the spreadsheet. Issues include not holding enough significant digits in the curve fit coefficients, no check of significance of terms in the selected equations, no investigation of math that fits better, and enforced granularity. I am merely asking folks to improve the precision of the data reduction, which is intended to reduce the amount of subsequent runs to refine and check your estimates of Vy and Vx ...

Coefficient Precision - Using the fixed decimal point format in the Excel curve fit on the Graph gave us the t^4 term (shown in the Kitplanes article) of 0.00001. That means it is somewhere between 0.0000051 and 0.0000150. The argument the 1E-5 is not important, t=100 = 1e2, t^4 = 1e8 and 0.00001*t^4 = 1000 I would call that a bunch of error if not managed. Using more points right of the decimal point helps but just changing to scientific notation in the format always gives you whatever number of significant digits you feel you need. Four is a good idea.

Coefficient Significance - When I took the 10 seconds of data supplied and plotted it, I ran Linest() on it and got data with more significant digits on the coefficients. My practiced eye led me to try adding terms for t^5 and sqrt(t), then checked the value of each coefficient (1st row) against the error estimator for the term (2nd row). If the coefficient is several times larger than the error estimate, it is probably real, if the error estimator is around the size of the coefficient or bigger, the coefficient is probably garbage. On that data, the first order term, t appears to be crap, but the higher order terms all appeared significant. Most statisticians well tell you if the x^3 term is significant, you keep the x term and the x^2 term, even if they do not appear to be significant, so we keep t. R^2 with the original fit is 99.981% which sounds really good, but with it including t^5 and sqrt(t), it improved to 99.995% - it has a quarter of the error of the base approach. Now you have an improved base to differentiate once for an accel curve.

Granularity - The data and analysis as presented gives you info at 1 second intervals while the airplane is changing speed at 5 knots a second. While I agree that the rate of climb curves are relatively smooth near Vy and near Vx, a little more precision is available, so why not use it. With calculated and fabulously fitted acceleration equations, we plot our reserve power based rate of climb estimate at shorter time steps and get a smoother curve for picking off Vy and Vx, as well as Vz.

Since it can be set up in Excel and just put in from flight logger data, we can just program the data reduction once and run it on each set of data. Why would you skip it if it were easy to do? I assure you guys, writing this note took me WAY longer than doing the math and programming it.

Billski

#### Saville

##### Well-Known Member
Actually, I think you are missing the point. The test method IS a tremendous time saver, but the data reduction, as presented, is weak statistically. This engineer trained in statistics knows that good data analyzed poorly can be ... misleading. What is worse than having poor data? Learning something that is far off the mark is worse than poor data.

Using a better data reduction should help the process and can be automated so it is very little effort beyond getting the data off the flight logger and installing it in the spreadsheet. Issues include not holding enough significant digits in the curve fit coefficients, no check of significance of terms in the selected equations, no investigation of math that fits better, and enforced granularity. I am merely asking folks to improve the precision of the data reduction, which is intended to reduce the amount of subsequent runs to refine and check your estimates of Vy and Vx ...

Coefficient Precision

Coefficient Significance -

Granularity -

Since it can be set up in Excel and just put in from flight logger data, we can just program the data reduction once and run it on each set of data. Why would you skip it if it were easy to do? I assure you guys, writing this note took me WAY longer than doing the math and programming it.

Billski

What I would like to know is:

When you are doing the flying, how precise must your speed and/or rate of climb be? A lot of fliers trying to work out the numbers for their POH are not test pilots and many not even IFR pilots. Holding speed/rate of climb at a constant isn't a skill they necessarily have.

So how bad can that accuracy be and still get useful data?

#### Toobuilder

##### Well-Known Member
The info presented is for education. It shows how real test pilots do real testing and ultimately get the performance charts used by the aircrews in the airliner thats hauling your family to Hawaii or the combat crews to get a slight edge on an advisary. Does it matter much if you are 5 knots off on your climb speed in an RV? No. This endeavor is about getting smarter, even if the knowledge is not directly applicable to your specific situation.

And yes, the flying needs to be precise. Any error will contaminate the data reduction and therefore the final product. In the case of a "real" fest point, the airplane will be equipped with an onboard Data Aquisition System (DAS), and the performance will be recorded in real time. In NTPS, we would sit in the back of the airplane and furiously write down data seen on the needles and stopwatches, but that was really to teach us how and why the accuracy of data was important. In the classroom after the flight we would do the reduction and the instructors first words were usually "did you get the data you expected"? Usually, it was way off, either because the weather was horrible or we just did not collect the data accurately enough. But the lesson learned was that you need to make a call in real time if the data is off. You certainly dont want to burn a whole flight and THEN process garbage data just because the points fell randomly on a graph. You need to be able to say with confidence "we need to re rerun that last test point".

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