At Tripoli Central California's January 1998 launch, two rockets (a LOC/Precision Hi-Tech H45 and a Binder Design Thug), built from kits and supposedly stable, went unstable when launched, looping downwind and crashing. An Estes Phoenix and R2D2 surprised us by cruising downwind rather than upwind. Wind velocity at the time was between 10 and 20 miles per hour.

Some 32 other rockets were launched in wind velocities between 10 and 20 miles per hour, and flew normally.

Why did four rockets go unstable, even though they were built from kits and presumably had normal margins of stability? And why did the 32 other rockets fly normally on the same day?

Understanding what the margin of stability actually means, and its limits, is the key to understanding what happened to these four rockets, and preventing the same problem from occurring at future launches.

First of all, it is important to remember that a rocket's center of pressure is not fixed. The Barrowman formula (and software based on it) gives the center of pressure at zero angle of attack; the Center of Lateral Area method gives a rough, conservative estimate of the center of pressure at 90 degrees angle of attack. At angles between zero and 90 degrees, the actual center of pressure will be somewhere between these two locations.

As the angle of attack increases from zero toward 90 degrees, the center of pressure moves forward (Figure 1). If the center of pressure moves forward beyond the center of gravity, the rocket becomes unstable and veers off.

The stability margin, as traditionally defined in High Power rocketry, is the distance between the center of gravity and the center of pressure at zero angle of attack. It is the distance through which the center of pressure can move forward before the rocket goes unstable as the angle of attack increases. This margin is customarily measured in multiples of the body tube diameter, or calibers.

For each rocket, there is a corresponding maximum angle through which the angle of attack can vary without the rocket going unstable. This maximum angle is the angle of attack at which the center of pressure coincides with the center of gravity. Thus, the stability margin can also be specified as the number of degrees that the angle of attack can vary to the right or left before the rocket goes unstable. (I'll call this the angular stability margin.)

The angle of attack is the same as the angle of the wind relative to the rocket's axis. On a dead calm day, with a perfect rocket, the relative wind is produced by the motion of the rocket alone, and is parallel to the rocket's axis; the angle of attack is zero.

When there is a crosswind, it combines with the rocket's velocity by vector addition to produce a resultant relative wind. The velocity of the crosswind and the velocity of the relative wind caused by the rocket's motion can each be represented by an arrow, of a length proportional to the velocity it represents.

To add the vectors graphically, the arrows are drawn at angles corresponding to their respective wind directions, with the head of one arrow touching the tail end of the next. The resultant, or vector sum, is then represented by a new arrow drawn from the tail end of the first vector to the head end of the last vector (Figure 2).

The resultant can also be calculated through the use of trigonometry or analytical geometry.

If the launcher is not tilted, and the rocket is parallel to the launch rod, the trigonometry is very simple, since the crosswind is at right angles to the rocket's velocity. Any scientific calculator can calculate the angle of relative wind directly.

First calculate the ratio of velocities, and then two more keystrokes give you the angle.

For example, if the rocket velocity is equal to the crosswind velocity, the ratio is 1, and the angle of attack is 45 degrees.

If the rocket velocity is twice the crosswind velocity, the ratio is 0.5, and the angle of attack is 26.6 degrees (rounded off to the nearest tenth).

where

v_w = Crosswind velocity, ft./sec.

v_r = Rocket velocity, ft./sec.

As the rocket leaves the top of the launch rod, if the angle of attack is greater than the rocket's angular stability margin, the rocket will go unstable.

If it has a large polar moment of inertia and good acceleration, the rocket may develop enough speed to bring the relative wind back within its stability margin before veering off very much.

If it has a small polar moment of inertia, high damping, and low acceleration, the rocket is likely to turn straight downwind and then fly level like a cruise missile. (When such a rocket's angular stability margin is greater than the relative wind angle, it is likely to weathercock and cruise upwind.)

If it has a large polar moment of inertia combined with poor acceleration and poor damping, the rocket may loop and crash before ever gaining enough speed to restore stability. Once it starts turning downwind, its rotational momentum tends to keep it turning.

To prevent dangerous flights such as these, we need a way to determine, for any given crosswind, how much thrust a rocket must have so that it can reach the velocity needed for stability.

All three safety codes in effect in California require the launcher to maintain control of the rocket until it has reached this velocity.

The velocity needed is that velocity which results in an angle of relative wind, or angle of attack, that is within the angular stability margin of the rocket.

Thus, to be able to calculate the required velocity, we must first either measure or estimate the angular stability margin of the particular rocket.

If the stability margin of a particular rocket can not be determined easily, then a worst-case conversion factor can be used, based on tests of numerous other rockets. For example, based on the test results shown in Figure 1, you could multiply the stability margin in calibers by 13 to get a worst-case stability margin in degrees. (This conversion factor should be considered preliminary, as it is based on tests of only 4 rockets.) Note that some rockets will have more than 13 degrees per caliber of stability; for example, the Alpha II has 26 degrees at one caliber, and 39 degrees at 1.5 caliber stability margin.

In a wind tunnel or in a large, open field (such as a launch site) on a windy day, a rocket can be mounted on a pivot at its balance point, and its actual angular stability margin can be measured directly if the rocket is not too large.

Or, with the proper mathematical model, it should be possible to predict the angular stability margin before even building the rocket.

Once we know the stability margin in degrees, the velocity needed can be calculated by solving equation 1 (above) for Vr:

(Equation 2)

The acceleration required to produce a given velocity depends on the length of the launch rod, rail, or tower.

(Equation 3)

Where

a = acceleration required, in ft/sec.2

v_r = required velocity for stability, in ft/sec.

h = height of launch rod, rail, or tower, in feet., measured from the bottom of the launch lug (or the forward launch lug if there are two) (or the center of mass if there are no launch lugs) to the top of the launcher.

(Note that if the rocket has two launch lugs spaced far apart, this fact alone can cause the nose to be blown downwind on a windy day when the upper lug clears the launcher, freeing the front end, while the tail end is still constrained.)

Once we know the required acceleration, it is easily converted to a thrust-to-weight ratio by dividing by 32.17 and then adding 1.

Table 1 gives thrust-to-weight ratios required for complete stability at various wind speeds, and 1 or 2 calibers of stability margin, assuming a worst-case conversion factor of 13 degrees per caliber.

Comparison of our January flight card data with Table 1 indicated that many of the 32 rockets that had more or less normal flight paths must have had more than 13 degrees of angular stability margin, and/or sufficient moment of inertia on their yaw axes to prevent them bearing off too far while accelerating to the minimum velocity.

All rockets that had at least a 10:1 thrust-to-weight ratio flew normally despite the wind. But keep in mind that most of the rockets that day were flown in winds of 16 mph or less.

Three of the rockets that went unstable or cruised downwind had thrust-to-weight ratios between 5.7:1 and 6.5:1; less than nearly all the other rockets launched that day, according to flight card data. The Hi-Tech H45 was flown with a G40 motor, the Thug with an F25, and the Phoenix with a D12. The R2D2 had a thrust-to-weight ratio of about 8:1, but has a very low polar moment of inertia and probably a low stability margin also, because it's so short.

The acceleration of the three low-thrust rockets was only about 5 g's. A rocket's velocity at the top of a 3 foot launch rod, with uniform 5 g's acceleration, would be about 28 feet per second. With a crosswind velocity of 15 miles per hour (22 feet per second), the angle of relative wind would be about 38 degrees. Just before we closed down the launch, the wind velocity was about 20 miles per hour (29.33 ft./sec.) and the relative wind angle on these rockets would be about 46 degrees. I doubt that most high power rockets would be stable at such a large angle of attack.

A taller launcher gives the rocket time to accelerate to a higher velocity before it leaves the launcher; thus the relative wind angle will be smaller at any given crosswind speed. But launch rods can't be made longer without also being thicker, otherwise they will flex too much. Thicker rods require larger launch lugs, which have more drag. And unless the rocket has a built-in roll rate or spin, the unbalanced drag on one side will cause the rocket to curve toward the launch lug in flight.

Rail or tower launchers solve this problem. The rail or tower can be made as long as necessary for velocity, and as wide as necessary for stiffness, without increasing the drag of the rocket.

The length of the rod, rail, or tower required for a given thrust-to-weight ratio and velocity can be calculated by solving Equation 3 for h:

(Equation 4)

where

a = 32.17 [(f_i/w)-1]

f_i/w = Initial thrust to weight ratio

f_i = Average initial thrust while the rocket is accelerating up the launcher (pounds)

w = Weight of the rocket at launch (pounds)

v_r = velocity required for stability, ft./sec.

The -1 is for gravity.

The average initial thrust used with Equations 3 and 4 is defined (by the author) as the thrust averaged over the period of time that the rocket is accelerating up the launch rod, rail, or tower. This period of time is the first 0.1 to 0.2 seconds of burn time, for typical model and high power rocket launchers (see motor thrust curves on page 12).

The average initial thrust is usually not the same as the average thrust (which is averaged over the entire burn time of the motor). The difference is most pronounced with thrust profiles like those of the Estes A10 and C5, the Aerotech/Apogee F10, and the Aerotech K125W. With thrust profiles like these, the average initial thrust is several times higher than the average thrust, to give the rocket good acceleration off the launcher.

There are some motors with initial thrust lower than their average thrust, such as the Aerotech K185W, the Rocketflite Silver Streaks, and the Plasmajet I102.

Most Aerotech rocket motors have an initial thrust somewhere between 1.1 and 2.0 times their average thrust.

Rocketeers intending to fly on windy days should learn more about their rocket's stability than merely whether or not the CG is at least 1 caliber forward of the CP.

RSOs and LCOs on windy days should beware rockets with low thrust to weight ratios, especially when they are unusually wide for the motor used.

High thrust/short burn time motors are preferred in windy conditions.

Divide the average thrust of a typical Aerotech motor (in Newtons) by 3.42 to get a rough estimate of the initial thrust in pounds. In windy conditions, divide by 30, or divide by 100 and then multiply by 3 for a rough, usually conservative estimate of maximum rocket gross weight for a thrust/weight ratio of about 10 or 11 to 1 at launch. (Based on initial thrust 1.3 times average thrust.) Just remember that not all Aerotech motors are typical. It's always more accurate to look at the thrust curve.

On a calm day, you can divide the average thrust by 20, which gives the launch weight for a thrust/weight ratio around 5.85:1. As always, the rocket's liftoff weight should never exceed the maximum liftoff weight specified for the particular motor by its manufacturer.

Another method you can use if you have a computer handy is to run flight simulation software and look at the first few feet of altitude. This method could also be used to generate a table which could be used by the RSO and LCO, and by individual rocketeers who want a first approximation when making a choice of which motor to use. Also, RockSim will compute the velocity at the end of your launch rod, and the effect of crosswind.

In windy weather, a scale on the RSO table should be considered a necessity.

Launch lugs spaced far apart are dangerous in windy weather. Single lugs at the CG are safer. Or put one at the CG and one at the CP.

It is not a good idea to launch rockets in winds between about 15 mph and the 20-mph limit, unless their stability characteristics are particularly favorable; for example, rockets with weighted noses, diameters not much greater than the motor diameter, and high thrust to weight ratios. (Be aware that these characteristics will cause the rocket to fly high, and drift far downwind, unless the burn time is short.)

Towers and rail launchers of proper length are very much to be preferred over launch rods in windy weather.

Aerotech F10 (sold by Apogee Components)
Aerotech F12
Aerotech G40
Aerotech K1100
Plasmajet I102
RocketFlight H330 silver streak

Barrowman Formula: A complicated algebraic formula for calculating the center of pressure of a conventionally shaped rocket, using a moment method. This formula can be found in the back of the Handbook of Model Rocketry, 6th Edition, by George Harry Stine (1994). The Rogers Aeroscience CP program, and one or more others, are based on this formula.

Caliber: Diameter of the rocket.

Center of Mass, Center of Gravity, or CG: The point on the rocket at which its mass or weight balances; the balance point.

Center of Pressure (CP): The point on a rocket at which aerodynamic lateral forces (acting at 90 degrees to the rocket axis) balance at any given angle of attack.

Damping: The tendency to reduce the amplitude of any oscillations or overshoots, and return to equilibrium quickly. Rockets with large fins tend to have more damping.

Polar Moment of Inertia, or Rotational Inertia: The rotational equivalent of mass; the tendency of an object to resist any change in its rate of rotation around a particular axis (in this article, the yaw axis). Longer rockets and rockets with weighted noses have more rotational inertia.

Rotational Momentum: Rotational Inertia multiplied by the speed of rotation of an object.

Vector: A quantity that has both magnitude and direction in space; A line or arrow representing such a quantity.

Velocity: The combination of speed and direction.

Using the author's wind tunnel:

1. Calculate the predicted CP for the rocket, using the Barrowman Formula.

2. Mark the predicted CP on the rocket (with pencil). Make additional marks at 1/2 caliber intervals toward the nose, up to 2.5 calibers forward of the predicted CP.

3. Make sure the pivot rod in the top of the wind tunnel is perfectly plumb and turns freely.

4. Attach the rocket to the pivot rod, 1/2 caliber forward of the predicted CP.

5. Before turning on the wind, the rocket should turn freely but not tend to swing toward any particular direction. If it does, the pivot rod is not plumb; return to step 3.

6. Point the rocket straight up the wind tunnel and turn on the wind. The rocket should remain pointed straight into the wind. If it will not, turn off the wind, re-hang the rocket 1/2 caliber farther toward the nose, and start again at step 5.

7. Apply torque to the pivot rod (from outside the tunnel) to slowly turn the rocket. You will feel a reaction torque (resistance to turning). A point will be reached where this torque passes through zero and then reverses. If you let go before this point, the rocket will return to center (zero angle of attack) by itself. Beyond this point, the rocket will accelerate toward the side if you let go.

8. Turn the rocket slowly in the opposite direction, repeating Step 7 on the opposite side.

9. Average the left and right readings from Steps 7 and 8.

10. Shut off the wind. Re-hang the rocket at the next mark, 1/2 caliber closer to the nose.

11. Repeat Steps 4 through 10 until you have reached 2.5 calibers, or until the rocket's nose comes within two calibers of the side of the tunnel before reaching the zero-crossing point.

12. Plot each reading from Step 9 on a graph. Draw a smooth curve connecting the points. It should be nearly straight for the first 25. Where this line, when extended, crosses zero degrees, is the actual zero-angle-of-attack CP. You now have a graph of Angular Stability Margin in degrees vs. Static Stability Margin in calibers for this particular rocket.

For a follow-up article on this subject, see What Barrowman Left Out, in the May/June 1999 issue of Sport Rocketry.

This revised version ©2000 by Bob Dahlquist, TRA # 3032.

All commercial rights reserved.

Notes: Liftoff weight includes the weight of the rocket and engine, plus any tracking powder or payload.

Some model rockets are heavier than the maximum liftoff weight for one or more of the motors recommended on the model rocket package! Do not totally trust the motor recommendations on the model rocket package or in the catalog. Doing so may result in a crash. Go by the recommended maximum liftoff weight for the motor (Column 2) on calm days.

A thrust to weight ratio of at least 10 to 1 is recommended on windy days (See Column 4).

A * next to the max. weight means go with the weight in column 2, to prevent late deployment.

Column 4 is based on thrust averaged over the period from 0.05 to 0.2 seconds after ignition.

Numbers have been rounded off to the nearest half ounce.

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