Impulse calculation

[MaxPayne]

Hello all, I am a n00b.

A question that arises in my mind is : How is impulse measured? I am aware of the method of measurement of thrust, utilized in making thrust =-time curves.

Thank you.

 

[cjl]

Well, if you are familiar with calculus, the total impulse is the time-integral of the thrust curve.

(If you're not familiar with calculus, this just means that the impulse is the average thrust multiplied by the burntime)

 

[Larry Curcio]

Hello all, I am a n00b.

A question that arises in my mind is : How is impulse measured? I am aware of the method of measurement of thrust, utilized in making thrust =-time curves.

Thank you.

You don't reveal your background. Forgive me if I'm aiming too low or too high. Just trying to get a couple of methods across.

Impulse is the area under the thrust/time curve. As cjl pointed out, this is the integral of thrust with respect to time for the burning time of the motor, but it's still just an area, and you don't really need a course in calculus to compute it.

Normally, thrust/time curves are sampled at a number of points. The space between any two adjacent points is called a sampling interval. Let the points in your curve be indexed from 0 to N and let the intervals between those points be indexed from 1 to N. In such a case,

Impulse = SUM{TimeInterval(i)*AverageThrust(i)}

where TimeInterval(i) is the amount of time elapsed in the ith interval (the interval between points i-1 and i). AverageThrust(i) is the average thrust within the same interval.

(This formula reduces the area under the curve to the sum of areas within the various sampling intervals. The area in each interval is represented as that of a rectangle with a height equal to the average thrust in the interval, and a width equal to the length of time in the interval.)

If the sampling points are closely spaced, you can use the approximation

AverageThrust(i) ~= [Thrust(i-1) + Thrust(i)]/2

If the intervals are more widely spaced, you need more sophisticated numerical methods, and you also need faith in the applicability of assumptions that underly them. (Google Runge-Kutta and Richardson Extrapolation if you like.)

Example thrust/time curve (Thrust is in newtons and time is in seconds):

Thrust/Time
0/0
100/0.05
200/0.1
100/0.15
0/0.2

(Note that every time interval in this curve is .05 seconds)

Impulse ~=
.05*[100+0]/2 +
.05*[100+200]/2 +
.05*[200+100]/2+
.05*[100+0]/2

= .05*[50+150+150+50] = .05*400 = 20 newton seconds

That's after-the-fact impluse. If you want to estimate impulse in a motor you are designing, a back-of-the-envelope method is just to take a tyrpical historical specific impulse for your chosen propellant, and multiply it by the weight of propellant you plan to use.

If you want to do more than that, you're into higher math. In that case, you don't need a glib hobbyist and a forum reply; you need a professor and a text book.

Luck and Regards,
-LarryC

 

[powderburner]

I think the NAR S&T guys might use a "threshold" minimum thrust value before they begin measuring impulse from the main portion of the burn. (I could be mis-remembering?) Or it might be a cut-off at some small level of thrust during the end of the burn.

Something to consider, before you get all worried about detailed modeling of a BP motor time-thrust curve down to the gnat's posterior, is that the blackpowder propellant often has variable properties. It can have higher or lower levels of chemical energy due to variations in ingredients, processing, storage, etc. The motors themselves can have small variations in loaded propellant weight from batch to batch. The quirks and foibles of blackpowder performance are probably the main reason that Estes loads its motors to remain well below the theoretical A-B-C impulse class limits, so as not to "go over the line" into the next class.

I get the impression that manufacturing BP motors is as much art as it is science.

 

[GregGleason]

The official definition for total impulse is:

7.8.5 Total impulse shall be measured between the point
when the thrust rises to 5 percent of the motor’s peak thrust to
the point of last measurable thrust prior to ejection or blow
through, or if it is a plugged motor, to the point where all
action has ceased.

-Source, NFPA 1125, 2007 Edition

This is the US government directive given to NAR and Tripoli to use for total impulse.

I made a spreadsheet utility to that performs these calculations and provide graphs based on .eng file data (wRASP format file).

https://www.rocketryforum.com/showpost.php?p=155698&postcount=13

Greg

 

[MaxPayne]

I= F∫dt
Here, we integrate time between limits and take F as average thrust because we do not have equation of F with respect to t ?


Let's talk about instantaneous impulse.

Impulse(i) = {TimeInterval(i)*AverageThrust(i)}

Now, in this curve :

Thrust(N)/time(s)
0/0
100/0.05
200/0.1
250/0/15
250/0.2
250/0.25
250/0.3
200/0.35
250/0.4
250/0.45
200/0.5
100/0.55
0/0.6

Impulse(at 0.2 s)=0.05*250 = 12.5 N s


but if sampling points were not so close, the same data would be :

Thrust(N)/Time(s)
0/0
250/0.2
250/0.4
0/0.6

Impulse(at 0.2 s)=0.2*250 = 50 N s

How to clear this anomaly?

 

[cjl]

I= F∫dt
Here, we integrate time between limits and take F as average thrust because we do not have equation of F with respect to t ?

We do have F wrt t, actually, since that's what is measured, so I = ∫F dt

Let's talk about instantaneous impulse.

Impulse(i) = {TimeInterval(i)*AverageThrust(i)}

Now, in this curve :

Thrust(N)/time(s)
0/0
100/0.05
200/0.1
250/0/15
250/0.2
250/0.25
250/0.3
200/0.35
250/0.4
250/0.45
200/0.5
100/0.55
0/0.6

Impulse(at 0.2 s)=0.05*250 = 12.5 N s


but if sampling points were not so close, the same data would be :

Thrust(N)/Time(s)
0/0
250/0.2
250/0.4
0/0.6

Impulse(at 0.2 s)=0.2*250 = 50 N s

How to clear this anomaly?

What anomaly? There's no such thing as the total impulse at t = 0.2 seconds. There is only the impulse from t=0 to t = 0.2 seconds. With your first data set, that would be (0+100)/2*0.05 + (100+200)/2*0.05 + (200+250)/2*0.05 + (250+250)/2*0.05 (using a trapezoidal method to approximate the integral). This gives an estimated total impulse from t = 0 to t= 0.2 of 2.5+7.5+11.25+12.5 = 33.75 Ns. Using the trapezoidal method again, with the second data set, the total impulse from t = 0 to t = 0.2 is simply (0+250)/2*0.2 = 25 Ns. The second value differs from the first because you have lost some data when you lowered the resolution. However, they don't differ by that much. Using the same technique for the full burn of this example motor, the first data set gives 115 newton seconds, while the second data set gives 100 newton seconds. Again, they are similar (though not exactly the same, due to the loss of accuracy in the second set).

Oh, and there isn't any such thing as instantaneous impulse. There's only the total impulse between time t1 and t2.

 

[MaxPayne]

One thing I still don't understand,

I = ∫F dt

What will we do in actual calculations(using integrals, no approximation, and without splitting the integral at certain 'in-between' limits)?
Suppose we consider this :

Thrust/Time
0/0
100/0.05
200/0.1
100/0.15
0/0.2

 

[edwinshap1]

When a motor is tested, they take extremely short steps and calculate based off that. If you could calculate an exact time-thrust curve equation, you could take the integral, but since you're taking more of a reimann sum, its pretty much the best you can get without an infinitely short step thrust meter during testing

 

[Larry Curcio]

One thing I still don't understand,

I = ∫F dt

What will we do in actual calculations(using integrals, no approximation, and without splitting the integral at certain 'in-between' limits)?
Suppose we consider this :

Thrust/Time
0/0
100/0.05
200/0.1
100/0.15
0/0.2

You can take impulses for portions of the burn, but they pertain to the intervals between the points, and not to the points themselves. You need, as cjl points out, more than one point.

The fewer points you have covering your burn, the less information you have about the burn. You fill the empty intervals with assumptions about the shape of the curve there. High order integration methods assume polynomial curves, interpolating from the points bordering and otherwise near the interval in question. This may or may not hold. More data give more accurate results. Welcome to the third rock.

That said, it doesn't follow that thrust curves used in simulations having very few points are grossly inaccurate. These are normally tweaked such that the simple method I outlined previously (Euler's method) yields the correct impulse (and burning time) - at the cost of minor differences in shape. In practice, shape differences (G.H. Stine's incorrect math to the contrary) don't make much difference in trajectories unless the rocket is near terminal velocity. When the rocket goes terminal, it's subject to variability in motor performance anyway, so prediction is inherently less acurate even in the best case.

 

[bobkrech]

The official definition for total impulse is:

This is the US government directive given to NAR and Tripoli to use for total impulse.

Greg

Greg

The US government did not give NAR or Tripoli any directives. This is the way NAR S&T has measured total impule from the beginning. When NFPA codified 1122, they adopted the NAR S&T definitions.

Bob

 

[jsdemar]

An altitude simulation is much more sensitive to the total impulse than it is to the shape of the curve or burn time.

The motor data files are an approximation of the high-resolution motor test data. The original data is usually 200 to 1000 samples per second. The "ENG" files (or the "RSE" files) usually have up to 32 points per motor curve. My algorithm for reducing the data, as implemented in my free Thrust Curve Tool program, does the following: it begins with a 3-point approximation and continues adding the most prominent points until the total impulse is within 1% or it reaches the maximum specified number of points; second step does a correction factor to all points to match the total impulse exactly. I calculate the total impulse using linear interpolation and trapezoidal numerical integration, which is what the simulators do. The higher-order numerical methods are used by the simulators for step-wise acceleration and not the motor thrust curve.

Another way to look at the total impulse calculation: you do N-1 sums for N points. Each interval gives a slice of the area under the curve. The area is the product of the time interval and the average thrust during that interval. The first point begins at 5% of the peak thrust. Final point is where the thrust goes to zero. I did a little sketch. See attached.

 

[GregGleason]

An altitude simulation is much more sensitive to the total impulse than it is to the shape of the curve or burn time.

The motor data files are an approximation of the high-resolution motor test data. The original data is usually 200 to 1000 samples per second. The "ENG" files (or the "RSE" files) usually have up to 32 points per motor curve. My algorithm for reducing the data, as implemented in my free Thrust Curve Tool program, does the following: it begins with a 3-point approximation and continues adding the most prominent points until the total impulse is within 1% or it reaches the maximum specified number of points; second step does a correction factor to all points to match the total impulse exactly. I calculate the total impulse using linear interpolation and trapezoidal numerical integration, which is what the simulators do. The higher-order numerical methods are used by the simulators for step-wise acceleration and not the motor thrust curve.

Another way to look at the total impulse calculation: you do N-1 sums for N points. Each interval gives a slice of the area under the curve. The area is the product of the time interval and the average thrust during that interval. The first point begins at 5% of the peak thrust. Final point is where the thrust goes to zero. I did a little sketch. See attached.

Very nice explanation and graphic John!

Greg