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Propellant tanks are a key part of any propulsion system; usually they are the heaviest single parts of a launch vehicle or sounding rocket. They are also one of the more dangerous as the somewhat recent 2007 Scaled Composite purchase clomid has shown. So let’s spend a couple of blog posts discussing the design.

First off, today is the basic stress analysis. We will just focus on the general sizing analysis for tanks today and move to materials and joints in the next post.

In a perfect world, you would just have spherical tanks that were not part of the primary structure load path. In that case, the analysis is very easy: Stress = (Pressure x Radius) / (2 x Thickness). So if we had a 6 inch diameter sphere at 200 psi out of Al-6061 with a FS of 2, that is a (42 ksi / 2) = (220 psi x 3 in) / (2 x thick) which mean we can have a 0.029″ wall thickness. Now, a sphere is pretty hard to package so if we use a cylinder with spherical endcaps, the cylinder max stress is Stress = (Pressure x Radius) / (Thickness). Thus, a cylinder would have to be twice as thick at 0.057″. Of course, if you don’t use a spherical endcap, things get trickier and you need to either do FEA analysis or SP-125 has some rules of thumb for ellipsoidal endcaps on page 338.

And for a flat constant thickness endcap assuming simply supported there is a simple solutions given to us by the trusty where can i purchase clomid

Max Stress = 3/4 * Pressure * radius^2 / (4  * thickness ^2 )

Now that we have general thickness sizing of the tanks, we can check for buckling loads and for bending loads. For buckling loads, we can just assume that if the tank is pressurized it will not buckle, using the same rational as a pressurized soda can cannot be smashed but a empty can is easy to smash. This is called pressure stabilizing and it is really convenient. If you want more information on buckling, you should check out purchase clomid over counter. For most tanks, you can do a first pass for beam bending by assuming Sigma = My / I, or Stress = Moment  / ( pi *  Radius ^2  * Thickness) . For a gimballed engine, Moment can be assumed to be the Distance from the Cg to the gimbal plane * sin (gimbal angle ) * Thrust. For an ungimbaled engine, assume 5 degrees for the gimbal angle for a first pass. So with 200 lbf, set 40 inches for the Cg and a 5 degree angle this gives us a moment of 700 lb*in and a stress of 450 psi for our 6″ diameter 0.057″ thick tank.

As you can see, pressure loads dominate for initial sizing and other flight loads are relatively trivial. In a more in depth analysis, other flight loads can have a large impact, especially at joints.

Since pressure loads are the driving factor, a term called Tank Factor is fairly common for initial tank sizing.  Tank Factor (m) = Tank Volume (m^3) * Pressure ( Pa) / ( Tank Mass (kg) * g (m/s^2)).  This equation is used to initially size tanks and compare tanks made with a common fabrication technique as they should all share Tank Factors. A good first pass for an Aluminum tank with high end amateur construction is 2000-2500 meters.

Stay tuned next time for some more tank analysis.

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Today, let’s cover the basics of writing a trajectory code. There are many different levels of fidelity for an analysis, but today we will start at the basic 1-DOF code, just worrying about altitude. Also, we will assume constant drag coefficient, thrust, and gravity. If you want a more detailed code, the first step is to add thrust and drag coefficients that vary with altitude. The next step would be to add a 2nd degree of freedom with downrange, as well as altitude. At this point, if you are doing just basic sizing, you could stop. Then, a 3-DOF model is made by adding angle of attack with vehicle center of gravity and center of pressure. Then, add wind into the 3D model as well as thrust vector response. This is the most complicated that I have done and it is sufficient for sizing gimbals and fins in a variety of wind loads. From here, you would add the next 3-DOF and then probably add a Monte Carlo analysis on top of that.

If you are just using normal hobby rockets, you could get away with using a code like buy clomid pct and not have to write your own. But, let’s be honest, if you are reading this blog, you probably like doing the code for the fun of it. I still recommend using a well tested code like RASAero to ballpark the first few answers your analysis gives you.

The basics of the code is Newton’s Second Law that the Sum of Forces = Mass x Acceleration and the knowledge that velocity is the integral of acceleration and position is the integral of velocity. So we start with a known position, velocity, and mass and solve for all of the forces (drag and thrust) to find the acceleration. Then we go to the next step and find a new velocity as a function of acceleration and the old velocity, new position as a function of old position, old velocity, and acceleration, new mass as a function of old mass and mass flow rate, and then start the cycle over again.

X_new = X_old + V_old*dt + 1/2 * A_old  * dt^2

V_new = V_old + A_old * dt

A_new = Force_new / Mass_new

For the forces, we set Thrust = Thrust when thrusting, and Thrust = 0 otherwise. Drag is more complicated as it is a function of speed and air density. For air density, we use the old standby: the 1976 Standard Atmosphere. Annoyingly, density does not curve fit well, so I curve fit pressure and temperature (they are both exponential), then calculated density. And gravity is always 9.8 m/s pointing to the earth

Force_new = Thrust – Drag – Gravity

Drag = 1/2 * Density * Velocity * abs(Velocity) * Cd * Area

Density = Pressure * Molar Mass / Temperature / Gas Constant

The velocity is multiplied by the absolute value of velocity instead of just squaring to retain the proper sign; otherwise, drag acts as thrust when the vehicle is coming back down to earth.

So you pretty much just take all of these equations and add them together into one integrated iterator and you are good to go. I have put an example for Earendel done in Openoffice Calc below. It has an OK accuracy, achieving 86 km altitude and a max speed of 1200 m/s vs. a more accurate 2D model with variable thrust and drag and finer stepping that achieves 1299 m/s max speed and an altitude of 106 km. Not too shabby for a simple code whipped up in an hour.

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Thanks to Professor Anderson at Purdue for teaching me this in undergrad at Purdue.