Difference between revisions of "Tutorial:Advanced Rocket Design"

Revision as of 12:03, 20 January 2018

By Vincent McConnell and Kosmo-not

Getting to learn basic rocket science for a space game like Kerbal Space Program can be very important to the success of building rockets that can perform a desired job. In this guide, we will be covering things like calculating the full Delta-V of your ship, explaining how to perform transfer maneuvers, getting Thrust to Weight Ratio, calculating the Peak G-force experienced during a particular burn, also calculating Delta-V needed for a full Hohmann transfer and much more.

Delta-V

${\displaystyle \Delta v}$ (change in velocity) is the bread and butter of rocket science. It is probably the most important thing to know about your rocket because it determines what your rocket is capable of achieving. Among the several things we will explain in this basic tutorial, ${\displaystyle \Delta v}$ is most likely the most useful thing you will apply to Kerbal Space Program while building a rocket. To find the ${\displaystyle \Delta v}$ of your rocket for each stage at a time we have to sum up the part masses of every single part of the stage.

• Total mass: ${\displaystyle m_{\text{total}}}$
• Fuel mass: ${\displaystyle m_{\text{fuel}}}$
• Dry Mass: ${\displaystyle m_{\text{dry}}=m_{\text{total}}-m_{\text{fuel}}}$

The equation only needs the total and dry mass, but as it is easier to get the dry mass by subtracting the fuel mass from the total mass. Of course other combinations like calculating the total mass and measuring the fuel and dry mass are also possible.

The next important part of this set of calculations is to find your engine's specific impulse. Specific impulse is a measure of how fuel efficient an engine is (the greater the specific impulse, the more fuel efficient it is). For example, the non-vectoring stock engine LV-T30 has a vacuum specific impulse of 300 s. So here, we must apply the Tsiolkovsky rocket equation. More informally known as "The Rocket Equation".

It states:

${\displaystyle \Delta v=I_{sp}\cdot \ln \left({\frac {m_{\text{total}}}{m_{\text{dry}}}}\right)}$

If the specific impulse is given in seconds it is necessary to multiply this value by ${\displaystyle 9.82{\frac {m}{s^{2}}}}$ (see also Terminology about I_sp).

So go ahead and sum up your stage's total mass with fuel. Then, go ahead and sum up the mass minus the fuel (this can be done by just adding up the 'dry mass' where given). Input these into the equation in the place of ${\displaystyle m_{\text{total}}}$ and ${\displaystyle m_{\text{dry}}}$. Following is a quick example, where the surface gravity of Earth ${\displaystyle 9.81{\frac {m}{s^{2}}}}$ is used:

Example rocket

Stage 3 (TMI, Mun lander, Return)
${\displaystyle m_{\text{total}}}$	${\displaystyle 3.72t}$
${\displaystyle m_{\text{dry}}}$	${\displaystyle 1.72t}$
${\displaystyle I_{\text{sp}}}$	${\displaystyle 400s}$
${\displaystyle \Delta v}$	${\displaystyle 3027.0{\frac {m}{s}}}$
Stage 2 (Kerbin orbit insertion)
${\displaystyle m_{\text{total}}}$	${\displaystyle 7.27t}$
${\displaystyle m_{\text{dry}}}$	${\displaystyle 5.27t}$
${\displaystyle I_{\text{sp}}}$	${\displaystyle 300s}$
${\displaystyle \Delta v}$	${\displaystyle 946.8{\frac {m}{s}}}$
Stage 1 (Ascent):
${\displaystyle m_{\text{total}}}$	${\displaystyle 38.52t}$
${\displaystyle m_{\text{dry}}}$	${\displaystyle 14.52t}$
${\displaystyle I_{\text{sp}}}$	${\displaystyle 350s}$ (estimated due to atmospheric flight)
${\displaystyle \Delta v}$	${\displaystyle 3349.9{\frac {m}{s}}}$
Total
${\displaystyle \Delta v}$	${\displaystyle 7544.6{\frac {m}{s}}}$

Multiple engines

To calculate the I_sp for multiple engines with different I_sp values, you need to find total thrust and mass flow:

${\displaystyle I_{sp_{avg}}={\frac {\sum \limits _{i}^{n}(thrust_{i})}{\sum \limits _{i}^{n}({\dot {m}}_{i}\cdot g_{0})}}={\frac {\sum \limits _{i}^{n}(thrust_{i})}{\sum \limits _{i}^{n}\left({\frac {thrust_{i}}{I_{sp_{i}}}}\right)}}={\frac {thrust_{1}+thrust_{2}+\dots +thrust_{n}}{thrust_{1}\div I_{sp_{1}}+thrust_{2}\div I_{sp_{2}}+\dots +thrust_{n}\div I_{sp_{n}}}}}$

This will give you the correct I_sp to use for your Δv calculation. If all engines are the same, they act as one engine in this calculation so the sums aren't needed.

Calculating transfer maneuvers

The next part of this tutorial is how to perform a transfer maneuver. This kind of action is called a Hohmann Transfer and it requires two burns at opposite points in an orbit. Adding velocity will boost our apoapsis higher. We would then simply wait until we hit our newly established apoapsis and then add more velocity to boost our periapsis to circularize. Or, we could drop our orbit by subtracting velocity by burning retro-grade.

We can also apply some ${\displaystyle \Delta v}$ calculations to find out how much thrust we will need to perform this maneuver. We will break this burn up into impulses. For example purposes, we will start at a 100 km orbit and then boost into a 200 km orbit. Both circularized. The formula for the first burn is the following:

${\displaystyle \Delta v_{1}={\sqrt {\frac {\mu }{r_{1}+R}}}{\Bigg (}{\sqrt {\frac {2(r_{2}+R)}{r_{1}+r_{2}+2R}}}-1{\Bigg )}}$

This is the formula for the final burn in the transfer:

${\displaystyle \Delta v_{2}={\sqrt {\frac {\mu }{r_{2}+R}}}{\Bigg (}1-{\sqrt {\frac {2(r_{1}+R)}{r_{1}+r_{2}+2R}}}{\Bigg )}}$

Where:

• ${\displaystyle \mu }$= Gravitational parameter of parent body (3530.461 km³/s² for Kerbin).
• ${\displaystyle r_{1}}$= The altitude of our first orbit (100 km in this case).
• ${\displaystyle r_{2}}$= The altitude of our second orbit (200 km in this case).
• ${\displaystyle R}$= The radius of parent body (600 km in this case).

This formula will give us our velocity for the burn in km/s (multiply by 1000 to convert it into m/s). It's important to make sure that you will have the ${\displaystyle \Delta v}$ in the stage to make this burn. Again, you can do that by using the ${\displaystyle \Delta v}$ calculations above.

In our case we get a Δv₁ of 73.65 m/s, a Δv₂ of 71.23 m/s and a total Δv of 144.88 m/s.

Calculating fuel flow

Next, we will explain how to calculate fuel flow in mass to see how much fuel a burn uses up in a specific amount of time.

If we know the ${\displaystyle \Delta v}$ needed for the burn and the total mass of the rocket before the burn, we can calculate how much fuel is required to complete the burn.

First, we calculate the mass of the rocket after the burn is complete. To do this, we use the Tsiolkovsky Rocket Equation, inputting the initial mass and ${\displaystyle \Delta v}$ of the burn. We can then solve the equation for the final mass (“dry mass”) after the burn. The difference between these two masses will be used to determine the length of time that is needed to complete the burn.

The equation for mass flow rate of fuel, given I_sp and thrust, is:

${\displaystyle {\dot {m}}={\frac {thrust}{I_{sp}}}}$

where ${\displaystyle {\dot {m}}}$ is the mass flow rate of fuel consumed. Again if the specific impulse is given in seconds it needed to multiplied by 9.81 m·s⁻² (see also Terminology about I_sp).

Dividing the difference between initial mass and final mass for the burn by the mass flow rate of fuel, we can determine how many seconds are required.

Usually, when the thrust is in kN and the specific impulse is in m/s the result is in Mg/s (= t/s). As the density of the liquid fuel/oxidizer mixture is 5 Mg/m³ this gives 1/5 m³/s = 2 dm³/s (= l/s).

Orbital velocity

Rather easy is the formula to calculate the orbital velocity of an orbit. This assumes circular orbit or the velocity of a specific point in an orbit. For this, we simply do this calculation:

${\displaystyle {\sqrt {\frac {\mu }{r}}}}$

Where:
${\displaystyle \mu }$ = Gravitational Parameter of parent body. (km³/s²)
${\displaystyle r}$ = radius of orbit. (km)

If we input the radius of the orbit in Kilometers, our orbital velocity will come out in Kilometers per second. In a 100 km orbit, our radius will be 700 km. Meaning our velocity will be ~2.2458 kilometers per second (km/s), or 2245.8 m/s.

Delta-v map

A ${\displaystyle \Delta v}$ map consists of approximate amounts of ${\displaystyle \Delta v}$ needed to get from one place (whether it is on the ground or in space) to another. The ${\displaystyle \Delta v}$ values we have for our ${\displaystyle \Delta v}$ map are approximate and include a fudge factor (in case we slip up on our piloting). Our map is as follows:

Launch to 100 km Kerbin orbit:	4700 m/s
Trans-Munar Injection:	900 m/s
Landing on the Mun:	1000 m/s
Launch from Mun and return to Kerbin:	1000 m/s
Total ${\displaystyle \Delta v}$:	7600 m/s

If we design our rockets to have 7600 total ${\displaystyle \Delta v}$, and the acceleration of the launch stages are adequate, we can have confidence that our rocket is able to land on the Mun and return to Kerbin. A rocket with a little less ${\displaystyle \Delta v}$ can accomplish this goal, but it is less forgiving of less efficient piloting.

Calculate the acceleration

→ See also: Thrust-to-weight ratio

Calculating the thrust-to-weight ratio is very simple. It is important to know the thrust to weight ratio of your rocket to ensure your rocket will actually liftoff. If your TWR is less than 1, you can bet that you won't make an inch in altitude when starting from the launch pad. The minimum optimal TWR to have for your rocket at launch is 2.2.

To lift off the rocket's thrust need to exceed the gravitational force. The formula for this is simply the thrust of all of your current stage engines divided by the weight of your ship, fully fuelled.

${\displaystyle F_{T}>F_{G}=m\cdot g\implies TWR={\frac {F_{T}}{F_{G}}}={\frac {F_{T}}{m\cdot g}}>1}$

To calculate the acceleration simply use Newton's second law:

${\displaystyle F=m\cdot a=F_{T}-F_{G}=F_{T}-m\cdot g=m\cdot a\implies a={\frac {F_{T}}{m}}-g}$

These calculations only work when counteracting gravity. While coasting on an orbit the gravitational acceleration isn't important and thus the TWR may be below one and still work. The acceleration is at minimum directly after launch when the craft is heavy and at maximum immediately before running out of fuel, when the tanks are dry:

${\displaystyle a_{min}\approx {\frac {F_{T}}{m_{total}}}-g}$ and ${\displaystyle a_{max}\approx {\frac {F_{T}}{m_{dry}}}-g}$

The dry mass also includes the fully fuelled upper stages of the craft. To determine the g-force simply divide achieved acceleration by ${\displaystyle g_{0}=9.81{\frac {m}{s^{2}}}}$. As the craft is in free fall, the gravitational acceleration isn't felt by the crew so the accelerations appear to be higher for the crew leading to cancelling out the factor g:

${\displaystyle {\text{g-force}}_{min}\approx {\frac {F_{T}}{m_{total}\cdot g_{0}}}}$ and ${\displaystyle {\text{g-force}}_{max}\approx {\frac {F_{T}}{m_{dry}\cdot g_{0}}}}$

As the weight of the ship depends on the current gravitation (${\displaystyle g}$) the TWR differs between the celestial bodies.

Conclusion

This guide will hopefully have helped with designing your rockets to allow you to get the job done—whatever it may be—with no test flights first. We hope this guide has been helpful to new and continuing KSP pilots alike.