# Using Level Accelerations to Determine Climb Performance

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#### Toobuilder

##### Well-Known Member
Supporting Member
Well, interesting you bring this up. As it happens today is my last day at NTPS (where the Kitplanes author used to instruct), and we just performed this technique last week. Essentially, the acceleration method sweeps all airspeeds in a single pass and does clearly show the best relationship to drag and excess power at the test altitude. Once you have the best acceleration speed for each altitude band, you stack them all together and plot your target airspeeds for the climb and then go out and validate that data in an actual climb series.

Its just a more efficient way to collect data than the endless climbs and descents of a more traditional sawtooth method.

#### Aesquire

##### Well-Known Member
Interesting, but seems to ignore induced drag. Or bury it in the conversion without adequate explanation?

Induced drag goes down with airspeed, while parasitic drag goes up.

Best climb angle is typically at about best sink speed, best climb rate at about max L/D speed. This doesn't apply well with "overpowered" craft like jet fighters that climb partially on pure thrust at steep angles. ( mostly because jet power increases with airspeed in a non intuitive way. ) I'm not counting zoom climbs, they are far more Energy dependent than on lift.

Anyone know the Min Sink & Max L\D speeds for a Mustang?

#### Toobuilder

##### Well-Known Member
Supporting Member
As the aircraft accelerates, it passes through the ranges of induced and parasite drag and shows you in real time what that particular airplane "likes". Also understand that this acceleration technique is not the final solution - it just gets you very close very fast. The validation is actual climbs. The benefit here is that you can narrow the target climb testing down to a much shorter series (only a few knots either side of optimal - rather than starting in the basement and developing a curve)

#### wsimpso1

##### Super Moderator
Staff member
Kitplanes ran an article,
Using Level Accelerations to Determine Climb Performance

What do the wise ones think about it? I don't think it allows for the increase in drag as airspeed increases.
Sounds really smart to me. It is all just first principles. Just by reading the title, I can tell you how to do it.

F=m×a where m is airplane mass, a is airplane acceleration, and F is the excess thrust after the thrust that is overcoming drag is covered. The excess thrust is exactly what is available to either lift the mass of the airplane or to accelerate it.

Once we know the speed at peak accel and know the excess thrust, we can confirm Vy and rate of climb at any speed. With the accel curve known, we can make the drag curve and get Vx from first principles and confirm it with flight test too.

Bill

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#### rv7charlie

##### Well-Known Member
Supporting Member
Does fixed pitch vs c/s prop affect the usefulness of the method?

#### Toobuilder

##### Well-Known Member
Supporting Member
No, because you are looking for the max acceleration of the airplane as a system. The fact that an RV-8 with 150/fixed prop or an RV-8 with 200/CS behave differently does not invalidate the excess power of each.

#### wsimpso1

##### Super Moderator
Staff member
Does fixed pitch vs c/s prop affect the usefulness of the method?
I would not expect so. A constant speed prop allows us to keep the engine at max power regardless of airspeed. Change to a fixed pitch prop and in many airplanes, the power drops as the airspeed drops, but we still get a curves for excess thrust and excess power vs airspeed, so we can still find Vx and Vy. Those speeds may be at a little higher airspeed for the same airframe, engine, and weight with a fixed pitch prop, but the process still works.

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

#### rv7charlie

##### Well-Known Member
Supporting Member
Thanks, guys; makes sense.

#### Pops

##### Well-Known Member
Supporting Member
To me it sounds like a solution to a non-problem.

#### rdj

##### Well-Known Member
Supporting 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.Billski

I remember reading somewhere that mentioning anything like "calculus" in a general circulation magazine instantly dropped the probability of the article being read to something asymptotically approaching zero.

Bob

#### Voidhawk9

##### Well-Known Member
Supporting Member
I remember reading somewhere that mentioning anything like "calculus" in a general circulation magazine instantly dropped the probability of the article being read to something asymptotically approaching zero.
I'm gonna just skip reading this.

#### trimtab

##### Well-Known Member
Vy at straight and level, calculate power used. Calculate max power at the DA.

Vertical speed equals power divided by (mass times gravity).

It gets me to within several percent of book values.

It works for CS props, and surprisingly well even for a few fixed props I've tried it with.

#### Saville

##### Well-Known Member
To me it sounds like a solution to a non-problem.

The one drawback for sawtooth climbs, for me anyway, is that in order to maintain the precision of keeping a constant speed I have to be staring at the airspeed indicator.

Which means I'm not looking outside the aircraft during a climb. Somewhat scary.

#### wsimpso1

##### Super Moderator
Staff member
I remember reading somewhere that mentioning anything like "calculus" in a general circulation magazine instantly dropped the probability of the article being read to something asymptotically approaching zero.

Bob
Good thing neither this forum nor Kitplanes is a general circulation magazine.

Whether calculus is specifically mentioned or not, we do it all the time. Everytime we move something to a new spot, we have to decide how quickly we want it to get there, figure out how much force to apply, compute acceleration, the accel is integrated once to get our velocity, integrated it again to get distance. As we start it moving, we also do quick checks on if it is moving along the path we want and speed we intended, and apply more forces we have estimated to correct it, requiring more calculus toward our goals.

Our airplane wings have local lifting airloads, which accumulate toward the root. This is integration. A second integration occurs to make the bending moment, a third to make the angular deflection, and a fourth integration is then performed - all done naturally by the structure - to result in those airloads deflecting the wings up and down.

Since we are a technical website, we will discuss calculus. Those souls who do not recognize that they do calculus ALL THE TIME are probably missing other connections between their actions and the results...

Billski

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#### Toobuilder

##### Well-Known Member
Supporting Member
To me it sounds like a solution to a non-problem.

Sort of. In our world, figuring out the best climb rate through the envelope is not hard to pinpoint even from a TLAR standpoint, and then not that expensive to validate through test. But if you have a new jet fighter or airliner, then sawtooth climbs are a very time consuming (expensive) way to build an excess power plot. And if you have ever seen how complex the energy maps are for some of them, it becomes clearer.

These energy maps and various perf charts end up in the -1's and AFM's for all these airplanes as a result of tedious math followed by tedious test. The level accel method is an easy, quick and efficient method to grab a LOT of useful data for later data reduction used to narrow down your validation envelope so that you then reduce the need for the more challenging climb validation.

#### Toobuilder

##### Well-Known Member
Supporting Member
....Since we are a technical website, we will discuss calculus. Those souls who do not recognize that they do calculus ALL THE TIME are probably missing othe connections between ther actions and the results...

Billski

This short course covered about 6 months of material in a few hours. They showed us the math, but only enough to illustrate the effect of variables on the equations. To be honest, the construct of the equations would have gone right over my head even if I took the full course, but honest to God, more than once I thought to myself "Bill would be LOVING this right now..."

#### wsimpso1

##### Super Moderator
Staff member
This short course covered about 6 months of material in a few hours. They showed us the math, but only enough to illustrate the effect of variables on the equations. To be honest, the construct of the equations would have gone right over my head even if I took the full course, but honest to God, more than once I thought to myself "Bill would be LOVING this right now..."
Actually, that part would be teaching me nothing - I have done exactly that sort of programming. I would find it interesting to note how many are just making notes so they can do this later when they need it, and how many are going "Wow, that's how that works!"

One of the cool things about this applied in an airplane is that the separation between the thrust and power needed for straight and level is so cleanly separated from the power available for accel/ climb/ maneuver. If you are pulling a T-6 from the hangar with one flat tire, application of this theory gets tougher because overcoming drag from the flat tire positively dominates the forces required.

Working my first job out of college at a musket mill in Ilion New York, we had gun barrels instrumented for pressure vs time and a chronograph to measure ejecta velocity. When we wanted to check calibrations, we hooked the outputs to the mainframe (1980...) to record and calculate the double integral of the pressure. The calculated velocity was usually less than 1% higher than the actual velocity. Throw in ejecta drag on the bore, it got even closer. Shot shells need almost no correction while centerfire rifle ammo needed something like 100 pounds to match perfect.

This whole topic is how Inertial Navigation works. You know, submarines, ballistic missiles, landing on the Moon, going to Mars, that sort of thing. Set up sensitive accelerometers, know where you start or update position along the route, integrate the accels twice, you know where you are and how fast you are going, Schrodinger not withstanding...

Billski

#### Hot Wings

##### Grumpy Cynic
Supporting Member
Those souls who do not recognize that they do calculus ALL THE TIME are probably missing othe connections between ther actions and the results...

Yes, yes they are. They live among us and their percentage of the population is increasing.

Not intended to be a political statement, just an observation of our society.

I actually find it pretty easy to explain the concept of an integral to the average person, if they can understand the idea of limits and infinity. Most do - or pretend to?
Slope of the tangent line is harder to explain, at least for me, because most don't recognize Y=mX+b. I find that rather sad too.

Edit:
Seems like a good time to inject one of my favorite math jokes

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#### FinnFlyer

##### Well-Known Member
To me it sounds like a solution to a non-problem.

Not at all. I had done numerous climbs (and descents) in my RV-4 testing trying to determine Vx and Vy over many, many flights. Terrible results.

In less than an hour during one flight I had numbers for 1,200', 2000', 3000', 4000', 5000' and 6000'.
Not terribly accurate because I had a hard time keeping the altitude as the plane accelerated from near stall to max speed.
But that's a pilot issue.
I had a conversation with Nigel (I wondered about the need to start at 500' below desired test altitude). Turns out the method was developed for jets (taking a long time to spool up to full power). In my RV-4 it worked much better for me to slow down to near stall at desired test altitude and then advance to full throttle keeping my eyes glued to VSI. Discounting any prop stall, it's up to full power in a second or two. A factor is that RPM increases as airspeed increases, but apparently that's not an issue. Someone here may be able to explain that?

This method does require data logging. Having an Dynon 100 EFIS (10A is fine to0), that is taken care of.

After a flight I download the data log, copy and paste airspeed, altitude and OAT columns into a spreadsheet (I use Excel 97). Identify the rows containing the acceleration, chart the airspeed column for those rows. (select the cells you want to chart, Insert, Chart ..., Line.) Right-click on the curve and Add trend line. Select option to show equation for trend line. Right-click on the equation, Format Data Labels..., Number, Number, increase to 10 or more digits. Click on displayed equation, starting with the "=", copy equation into Notepad. Replace All "x" with "*A2^" (remove last "^"). Copy and paste the result into cell B2 in a new sheet for that test altitude. Click on cell B2 and drag down 30 or so rows. (When you click or select a cell or cells you'll see a solid black square in the lower right hand corner. Simply grab that square with your mouse pointer and drag that down.) Enter 0 in A2, 1 in A3. Select A2 and A3 and drag down 30 or so rows. (Column A is your time column in one-second increments.)

When you have a sheet for one altitude, you can copy that sheet. In the copy you just need to paste the equation for the next test altitude in cell B2 and drag down.

The attached needs do be redone with data from flights where acceleration is flown more accurately.

By displaying the climb vs airspeed columns as a XY (scatter) chart, you can hold up a ruler to the screen from 0,0 to where the ruler touches the curve and see Vx. Vy is of course the top of the curve.

Perhaps I should make a video on how to do this in Excel if anyone are serious about using the method.

Finn

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