Goddard optimisation and multistage

Or

Into space for less than $600?

What is Goddard Optimisation?

Start with the conservation of momentum equation

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”.

Three phases of upward flight

So for every rocket there is an optimum velocity at given altitude,

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

For thrust phase by rocket, the most easily solved general form of the equations is:

where and

A useful approximation:

It’s relatively simple to calculate that the thrust phase needs to use about 30% of the total rocket fuel mass.

The ideal rapid acceleration for first stage is from a cannon.

Optimum Velocity Phase

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Optimal Velocity

Coast Phase

When air drag is constant this reduces to

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.

How to get long burn times

1. Multi-staging of off-the-shelf motors2. Home-built motors3. Both together4. Approximate using multi-staging with delays.

To space?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.

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Altitude vs price for multistage G80 rocket motors

1 motor in Stage 1

2 motors in Stage 1

3 motors in Stage 1

5 motors in Stage 1

7 motors in Stage 1

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 12-stage 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?

SummaryI 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.