In the Equations of Motion Mentioned Above in Section A

A.6.2.1.4 Wind Model1

A.6.2.1.4 Wind Model

In order to accurately predict the trajectory of a launch the simulations need an atmospheric and wind profile model. This model predicts the conditions a launch vehicle is exposed to when traveling through Earth’s atmosphere. These conditions determine the various aerodynamic forces imparted on a moving vehicle. In the initial design we planned to use a ground launch at sea level, an aircraft launch at an altitude of 15km, and a balloon launch at 30km. The final design is a balloon launch and only uses the wind model for ascension.

In the equations of motion mentioned above in section A.6.2.1 there is a force in all three directions called Fwwhich is the force on the launch vehicle due to the constant wind model coupled with the gusting of the wind. Since we are launching from a balloon at a starting altitude of 30km we are above the effects of wind, which has a maximum altitude of 30km, on the launch vehicle.1 The terms in the free body diagram and equations of motion are there for completeness and are 0. The wind only reaches an altitude of 30km because its effects on a vehicle after the altitude are minimal. This is due to the fact that the atmospheric density is small enough that the wind will not affect the launch vehicle enough.1The air density at 30km is 0.018 kg/m3 and the air density at sea level is 1.225 kg/m3 which is 68 times larger than at 30km.2

A.6.2.1.4.1 Standard Atmosphere Profiles

The atmosphere model used to run the simulations is based on the standard atmosphere data based of the 1976 NASA Standard Atmosphere. Our simulations use a MATLAB function called atmosphere4.m.2 Input into the function is geopotential height.2The function outputs the temperature, density, and pressure at the selected height. The model makes these calculations up to an altitude of 89.9 km. Figures A.6.2.1.4.1.1 through A.6.2.1.4.1.3 show plots of the pressure, temperature and density outputs from this function.

Fig A.6.2.1.4.1.1: Standard atmosphere temperature profile.

(Allen Guzik and Kyle Donahue)

Fig A.6.2.1.4.1.2: Standard atmosphere pressure profile.

(Allen Guzik and Kyle Donahue)

Fig A.6.2.1.4.1.3: Standard atmosphere density profile.

(Allen Guzik and Kyle Donahue)

A.6.2.1.4.2 Constant Wind Profile

In creating the wind profile there had to be several assumptions made. This first assumption we made was that the constant wind is linear at different heights. Another assumption we make is that the constant wind will always be traveling in an easterly direction. This means that the wind can never go east to west. For example the constant wind profile is limited to angles greater than 0o and less than 180o on a compass. We also assumed there would be random wind gusts throughout the wind profile. The restriction on the wind gusts is that their direction can only be ±90o from the constant wind profile. Also the wind gusts will be seen as impulsive. The final assumption we made in creating the wind model is that the wind itself is only 2-D. We assume the wind will only be traveling in the north-east-south-west directions and never in the “up” or “down” directions.

We modeled the constant wind profile after one which was used for the launch of Lunar Orbiter 2.2 Figure A.6.2.1.4.2.1 shows the wind profile which we used as a model.

Fig A.6.2.1.4.2.1:Wind profile from NASA CR-942.

(B.A. Appleby and T.E. Reed)

The dark solid line in the figure is the constant wind profile which we used for our model. We did not model the direction from NASA CR-942, we only used fig. A.6.2.1.4.2.1 for the speed of the constant wind. Figure A.6.2.1.4.2.2 is the constant wind profile which we used in the trajectory and D&C models.

Fig A.6.2.1.4.2.2: Constant Wind Profile for Project Bellerphone.

(Kyle Donahue and Allen Guzik)

Our wind model is a simplified version of one of the current models. We assume that the wind speed is related to altitude in a linear fashion. This is not the case but for our model the linear representation works very well.

We found the constant wind using MATLAB code AAE450_Wind_Model.m. For the initial analysis for Project Bellerphone the wind model was not used. Once some of the designs were not used and the amount of designs getting analyzed further decreased the wind model was implemented into the trajectory code. The wind model is broken up into several parts. The first part is computing the constant wind profile. The second part is determining the wind gusts. And the third part is converting the speed of the wind into the wind force (Fw).

The first part of the wind model is coming up with the constant wind profile. The constant wind profile itself is made up of 2 parts: the direction and the speed. The first part of the code determines the speed of the wind. The code uses logic and information which is based on fig. A.6.2.1.4.2.1 The next step is determining the direction in which the constant wind profile is directed. Since the constant wind profile is launch dependent it will change every time the code is used. This works best as it represents the real wind profile more accurately.

A.6.2.1.4.3 Wind Gusting Algorithm

Another aspect of the wind and atmospheric model is the unpredictable nature of wind gusts. Within the wind model architecture there is a wing gusting generator. The concept behind the wind gusting generator is to be able to generate random wind gusts then add them to the constant wind profile.

There are many parameters designed into the gusting algorithm to allow for random gusting. First, unique to each launch, we choose the number of gusts a vehicle can experience. This number is random and can vary from 0 to 20 gusts per launch. Another unique property to each launch is the altitude in the atmosphere where each gust occurs. Each gust is assigned a random altitude where the gust happens. The possible altitude range is from 0 km to 30 km.

The direction of each gust is also random for each gust. This allows for the unpredictable nature of the wind changing direction. The direction is selected to always be parallel to the ground. This means the model does not generate any up or down drafts. Each gust direction is in any direction that is plus or minus 90˚ from the direction of the constant wind profile.

The last component of the wind gust algorithm is the magnitude of each gust. Each gust has a random magnitude that can be anywhere between 0 m/s to 12 m/s. This magnitude is added to the constant wind profile. For example, if a gust of 9 m/s is assigned to the altitude of 10 km and is going in exactly the same direction of the constant wind the resulting wind,at that altitude, is the constant wind, which is 40 m/s, plus the gust resulting in the total wind velocity of 49 m/s. Lastly, concerning the magnitude of the wind gust, is the duration of each gust. The gust is modeled such that the gust is an impulse. This decision is based off of assuming the length of altitude each gust is small enough that when considering the velocity of the rocket traveling through that altitude range, the gust acts as an impulse.

Fig A.6.2.1.4.3.1: Example of a wind profile with gusting in the north/south direction.

(Kyle Donahue and Allen Guzik)

An example of a possible wind profile generated from AAE450_wind_model.m with gusts for a ground rocket launch is provided in Figures A.6.2.1.4.3.1 and A.6.2.1.4.3.2.

Fig A.6.2.1.4.3.2: Example of a wind profile with gusting in the East/West direction.

(Kyle Donahue and Allen Guzik)

A.6.2.1.4.4 Wind Force

The third and final part of the wind model is converting the wind velocity, constant wind profile plus wind gusts, into the wind force which is used in the equations of motion. The equation we used for converting the wind velocity into wind force is using Eq. A.6.2.1.4.4.1

Eq. A.6.2.1.4.4.1

where Fw is the force of the wind, ρ is the density of the air at the particular altitude, W is the total wind velocity which is the constant plus the gust, and Sw is the area exposed to the wind at the given time.

The force of the wind was modeled this way because we determined the wind would be a drag force. This is correct except at some times the wind will actually be helping but that is taken care of in the unit vector basis which was described earlier.

We analyzed the wind on the launch vehicle was analyzed for all three body-fixed directions, br, bθ, and bφ. Since one of our assumptions was that the wind would never moving “up” or “down” the speed and force in the br is therefore 0. It was also analyzed for all altitudes starting with 0 and up till 30km. Figure A.6.2.1.4.4.1 is the wind force in the north/south direction, and fig. A.6.2.1.4.4.2 is the wind force in the east/west direction.

Fig A.6.2.1.4.4.1: Force of the wind in the North/South compass direction.

(Kyle Donahue and Allen Guzik)

Fig A.6.2.1.4.4.1: Force of the wind in the East/West compass direction.

(Kyle Donahue and Allen Guzik)

The two figures above have the constant wind profile and the wind gusts combined into one to create the wind vector which is used to create the wind force.

Reference:

1Appleby, B.A. and Reed, T.E., “Dynamic Stability of Space Vehicles,” NASA CR-942, November 1967.

2Heister, Dr. Stephen, “atmosphere4.m,” AAE 539 Advanced Rocket Propulsion lectures, January 2008.

Author: Kyle Donahue and Allen Guzik