Transcript Goddard optimisation and multistage

Goddard optimisation and
multistage
Or
Into space for less than $600?
What is Goddard Optimisation?
Start with the conservation of momentum equation
𝑑𝑀𝑉
𝐶𝑑 2
𝑇=
+ 𝑀𝑔 + 𝜌𝑉 𝐴
𝑑𝑡
2
Goddard optimisation is the mathematical solution
of the question: “what is the minimum amount of
fuel needed to increase the altitude of a rocket by a
given amount”.
It is most simply expressed as: “downward force
due to gravity = downward force due to air drag”.
𝑀𝑔 = 12𝐶𝑑 𝜌𝑉 2 𝐴
Three phases of upward flight
So for every rocket 𝑀/𝐴 there is an optimum
velocity at given altitude, 𝑉 = 𝐶𝑑 𝜌𝐴 𝑀𝑔
For minimum fuel usage a rocket’s ascent has to
have three phases, first a THRUST phase that
accelerates the rocket as quickly as possible to
optimum velocity, second an OPTIMUM
VELOCITY phase that maintains that velocity as
closely as possible, third a COAST phase where
no further fuel is used.
Thrust phase
The ideal rapid acceleration for first stage is
from a cannon.
For thrust phase by rocket, the most easily solved general form of the
equations is:
𝑑𝑉
𝐶𝑇
𝐶𝑑 𝜌𝑉
=
−
𝑑𝐻
𝑀 𝐴 𝑉 2(𝑀 𝐴)
𝑑(𝑀 𝐴)
= −𝐶𝑀 𝐶𝑇 𝑉
𝑑𝐻
where 𝐶𝑇 = 𝑇 𝐴 and 𝐶𝑀 = − (𝑑𝑀 𝑑𝑡) 𝑇
A useful approximation:
𝑀 𝐴 = 𝑀 𝐴 ground exp −𝐶𝑀 𝑉
It’s relatively simple to calculate that the thrust phase needs to use about
30% of the total rocket fuel mass.
Optimum Velocity Phase
𝑉 = 𝐶𝑑 𝜌𝐴 𝑀𝑔
When air drag is constant this reduces to
𝑉 = 𝐶𝑇 𝜌𝐶𝑑
This velocity is slow enough for a trip to the edge of space
to require a long thrust duration, eg. 120 seconds, but at a
relatively small thrust.
250
Thrust (N)
200
150
Thrust phase
100
Optimal Velocity
Coast Phase
50
0
0
20
40
60
Altitude (km)
80
100
How to get long burn times
1.
2.
3.
4.
Multi-staging of off-the-shelf motors
Home-built motors
Both together
Approximate using multi-staging with delays.
To space?
Altitude vs price for multistage G80 rocket motors
450
400
350
Altitude (km)
This is an early calculation of mine.
It relies on being able to reduce the
thrust of a G80 motor without
reducing its total impulse in order to
increase its burn time. Apogees
exceeding 100 km.
300
1 motor in Stage 1
250
2 motors in Stage 1
200
3 motors in Stage 1
150
5 motors in Stage 1
100
7 motors in Stage 1
50
0
0
200
400
600
Motor cost ($)
800
1000
1200
An OpenRocket calculation. The
G80s are straight off-the-shelf,
hence the jagged velocity curve.
Apogee 40 km. Total motor cost
$534.
You can’t do that!
The previous OpenRocket calculation requires a 12stage rocket. No-one has ever successfully exceeded
6-stages. You’ll see three reasons given as to why:
1. Fins required by the later stages destabilise early
stages, so by 6 stages the fin sizes get impossibly
large.
2. Each stage separation gives a sideways kick to
the rocket causing it to arch over and nosedive
3. A significant proportion of rocket motors fail. By
6 stages the probability of a motor failure is
extremely high.
Or can you?
1. Fins required by the later stages destabilise early stages,
so by 6 stages the fin sizes get impossibly large.
This turns out to be no problem at all. Attach the fins to the last stage
but have them sited adjacent to or before an early stage. The stability
then becomes embarrassingly large; eg. for the previous OpenRocket
calculation CM is more than 10 diameters above CP for all stages.
A 4-stage Rack Rocket
A successful 5-stage rocket
Or can you?
2. Each stage separation gives a sideways kick to the
rocket causing it to arch over and nosedive
Trying longer overlap and tighter clearances at stage separations
3. A significant proportion of rocket motors fail. By 6
stages the probability of a motor failure is
extremely high.
Non-destructive testing before use? Improved handling?
Summary
I think the possibility exists of rocket launches into
space (100 km) on a budget of less than $600.
The key to doing this is Goddard Optimisation.
The first stage has to be maximum thrust for a very
small time. All subsequent stages have to be low thrust
for a very long time.
Problems with running massively multi-staged still have
to be overcome.