TWR and Delta-V maths

[tbridges42]

I'm normally really good at math, but I'm having a poor day and can't seem to wrap my head around this.

I use VOID which gives me delta-v and TWR readings on my ship. Let's say I build a reusable launch vehicle, and VOID tells me it has a delta-V of 8000m/s and a TWR of 2.75.

Now lets say I want to introduce an arbitrary payload of x tons, and for some reason I don't want to just put the launch vehicle on there and see what VOID says. Instead I want to be able to predict what the delta-V and TWR of the complete rocket would be if carrying a given payload. How would I go about doing that?

Edited April 27, 2014 by tbridges42

[UmbralRaptor]

TWR is just thrust to weight ratio. That is, thrust / weight or thrust / (mass * local_gravity). (9.81 m/s² for Kerbin's surface, other values findable on the wiki) Given that KSP rockets operate under radically varying gravity, I tend to prefer thinking in terms of thrust / mass, but that's less common.

For ÃŽâ€V, you want the rocket equation. For KSP purposes, it's Isp * 9.82 * ln(Initial_mass/final_mass). Note that you'll need to calculate ÃŽâ€V for each stage separately, and then add them together.

[Streetwind]

In practical application:

- Find your launch vehicle's total thrust by adding up all of the engine thrust values

- Find your launch vehicle's wet mass (can be seen in map view while sitting on the launchpad, but remember to subtract launch clamps)

- Find your launch vehicle's dry mass by removing all fuel from the bottom stage (and only the bottom stage) and then determining its weight again

- Calculate: <normalized Isp> = Sum[<thrust * Isp for each engine>] / <total thrust> <--- this step is not required if you only have engines with exactly the same Isp values

- Calculate: <first stage dV> = <normalized Isp> * 9.82 * ln((<launcher wet mass> + <payload weight>) / (<launcher dry mass> + <payload weight>))

- Calculate: <first stage TWR> = <total thrust> / (<launcher wet mass> + <payload weight>) / 9.81

As UmbralRaptor said, as soon as your launch vehicle has more than one stage (and with that kind of dV I'm fairly sure that is the case), things get more complicated. You basically need to remove the bottom stage in the VAB and then redo the entire six-step list to get the numbers for the second stage. And then you remove that stage in the VAB and do it all again for the third stage, if it exists. And so on.

And because that wasn't confusing enough, you then need to think about "what defines a stage anyway". Because while that is a fairly easy question to answer if you think of real life rockets like the Saturn V, in KSP we tend to do more exotic staging concepts. For example, in the Saturn V, the first stage engines burn only during the first stage, the second stage engines only burn during the second stage, and so on. But in contrast, the space shuttle: The solid boosters burn during the first stage, and the three liquid engines also burn during the first stage... AND the second stage. Stages change when the solid boosters run out, but the liquid engines never even shut down, they just keep burning. Asparagus staging pushes this to the extreme - really heavy lifters may have an onion shell of 18 tanks around the central core, which are dropped in pairs every couple moments. And every dropping action is actually a stage change, even if you're not igniting any new engines but rather have 17 out of 19 first stage engines continue burning. This can easily lead to lifters that require you to calculate over a dozen stages individually just to get out of the atmosphere, some of which are active only for a few seconds and supply only a small few hundred m/s of dV.

Oh, and on the topic of atmospheres: All Isp and dV numbers calculated herein are vacuum numbers. If you want to determine how much actual, atmospheric dV your lifter stage gives you, you need to plot a curve of the Isp profile during a typical ascent from ignition to flameout of that particular stage, with that specific payload on top, and then integrate that over time in order to find an average Isp value to use for the rocket equation. And if you're mixing engines, then you need to of course do this for each type of engine individually.

And if your payload changes? You do ALL of this all over again. I suppose you now see why people prefer using VOID or KER And even those tools are not able to show you actual atmospheric dV, because that depends on the exact ascent path chosen, and would be a pain in the behind even if you knew the exact path. So they only show you the worst-case, sea level atmospheric dV as a comparison point.

Edited April 24, 2014 by Streetwind

[Pecan]

Now lets say I want to introduce an arbitrary payload of x tons, and for some reason I don't want to just put the launch vehicle on there and see what VOID says.

It can be a pain making test loads of a specific mass - stupid_chris has done it for us: NRAP adjustable test weight. Stick that on your rocket and ask KER or MJ what the figures are.

Instead I want to be able to predict what the delta-V and TWR of the complete rocket would be if carrying a given payload. How would I go about doing that?

I predict VOID will agree with KER/MJ within a predictable margin. I further predict they'll all be wrong in practice, partly because of the reasons Streetwind gives and partly because you'll never be able to execute manoeuvres perfectly and instantly, the way they are calculated.