Heavy math and a non-ideal rocket equation

[steuben]

My little kerbal computers are complaining about some of the calculations I'm asking them to do. Bunch of whiny little It is not without some justification. So I'm trying it to show them it can be done.

The ideal rocket equation is good for spherical cows. But, they're not working with such creatures. Starting with:

m(t)y''(t)=F_E(t)-F_D(t)-gm(t)

m(t)=M_D+M_F-int(0->t)M_R(t)dt
         =M_D+M_F-M_Rt  assuming the throttle is constant

F_E=(((F_V-F_S)y(t))/y_V+F_S)T(t)
     =((F_V-F_S)y(t))/y_V+F_S assuming full throttle all the way

F_D=C_DAR_S/(2y_V)y'(t)²y(t)

F_E engine force
F_D drag force
M_D dry mass
M_F mass of fuel
M_R fuel burn rate
y_V altitude of vacuum
F_V vacuum force of engine
F_S sea level force of engine
C_D coefficient of drag
R_S atmospheric density at sea level

Running everything in I get the monstrosity of

(M_D+M_F-M_Rt)y''(t)= ((F_V-F_S)y(t))/y_V+F_S + C_DAR_S/(2y_V)y'(t)²y(t)

Which looks to be a good reason why the Good Lord Math! asked the Good Lord Electronics! for the miracle of the electronic analog computer.

But trying to find a solution that doesn't rely on numerical processes, I collide with a couple of sticky points in the math. Though it maybe because the math was long ago, and half forgotten. My reflex is to try and run it through the Laplace transform. The two sticky points I have are:

1. La(ty''(t)) -> d(s²Y(s))/ds = 2sY(s)+s²d(Y(s))/ds 
which looking down the road a bit might give me trouble. I think I may have to invoke some matrix math with this portion in the mix.

2. La(y'(t)²y(t)) -> sY(s)*sY(s)*Y(s)  the convolution of the three functions... I think. Which is just as problematic and ugly. I vaguely recall that the transform of a product is a convolution and vice-verse, but I could be wrong.

The math here one of the two reasons that rocket science is hard. Have I mucked up anywhere obvious? Any possible solutions to those Laplace transforms? Leaving aside the fact that if I run t too long I end up with negative mass. 

[mikegarrison]

I loved MIT perhaps more than anything else I've ever done, but this $%^# is one reason I'm happy to have long since graduated.

By the way, I've used several simplified computer fuel-burn tools that would happily start burning the airplane mass if you exceed max fuel capacity range, so you have to know when your model reaches the limits of its applicability.

[K^2]

6 hours ago, steuben said:

I vaguely recall that the transform of a product is a convolution and vice-verse, but I could be wrong.

It is for Fourier Transforms, but I'm not sure about Laplace. Convolution Theorem. (Edit: Actually, per that page: "This theorem also holds for the Laplace transform." So yeah, this works.)

My general recommendation is to convert it into a pure delta-V problem and then go back to original rocket equation to solve for mass ratios. I was able to work out exact forms for vertical ascent in exponentially decaying atmosphere with constant drag coefficient, quadratic drag, and assumption that throttle is varied to always ascend at speed equal to terminal velocity, which is optimal profile for vertical ascent with quadratic drag. It's a very special case, but it's practical, and the total loss to gravity and drag works out to 4gH/v_t, where H is scale height and v_t is terminal velocity at initial altitude. This does come from integrating over a closed form solution to the equations of motion. The solutions involve some hyper trig functions, but otherwise aren't too bad.

If you are interested in derivation of any of that, I can try to dig up some notes.

Edited January 1, 2021 by K^2

[JoeSchmuckatelli]

6 hours ago, mikegarrison said:

I loved MIT perhaps more than anything else I've ever done, but this $%^# is one reason I'm happy to have long since graduated.

By the way, I've used several simplified computer fuel-burn tools that would happily start burning the airplane mass if you exceed max fuel capacity range, so you have to know when your model reaches the limits of its applicability.

My 10 year old daughter saw the Boston Dynamics dancing robot video 

 

... and now she wants to go there.

 

Thank god she's got a head for math; her dad flunked Calc 3 times

[mikegarrison]

Well, good luck to her.

[JoeSchmuckatelli]

57 minutes ago, mikegarrison said:

Well, good luck to her.

Not to brag, or nuthin... but the kid is frickin brilliant

(She's taught herself coding and piano)

Yeah, Dad's proud!

Edited January 1, 2021 by JoeSchmuckatelli

[benzman]

It's a real bummer when a robot can dance much better than me!

 

[Spacescifi]

7 hours ago, benzman said:

It's a real bummer when a robot can dance much better than me!

 

 

Oh it gets worse...much worse.

 

[wumpus]

On 12/31/2020 at 4:18 PM, mikegarrison said:

I loved MIT perhaps more than anything else I've ever done, but this $%^# is one reason I'm happy to have long since graduated.

By the way, I've used several simplified computer fuel-burn tools that would happily start burning the airplane mass if you exceed max fuel capacity range, so you have to know when your model reaches the limits of its applicability.

Sounds like what happens to marathon runners after mile 20 or so.  The human body can only store so much glucose, so after about 20 miles or so you run out and it starts to tear apart the rest of the body for more energy (although even the most serious runner has enough fat to last >>100 miles.

[JoeSchmuckatelli]

2 hours ago, wumpus said:

Sounds like what happens to marathon runners after mile 20 or so.  The human body can only store so much glucose, so after about 20 miles or so you run out and it starts to tear apart the rest of the body for more energy (although even the most serious runner has enough fat to last >>100 miles.

Truth: humans are endurance hunters.  A fit person can hunt a horse, and even walk it to death.  The first 3 days, the horse will stay out of range, but by the fourth its exhausted and the hunter can just walk up to it and catch or kill the horse.

 

For some interesting reading - pull up some of the old 19th Century Cavalry manuals.

[steuben]

On 12/31/2020 at 8:55 PM, K^2 said:

My general recommendation is to convert it into a pure delta-V problem

I had the same thought in the shower the other day. But, for different reasons. I've been reading some old electronic analog computer manuals. They say avoid derivatives like Da' Covid.

 I'll have to try the Laplace using y'' as the "solving" variable. This may solve the first sticking point of having to haul around and work with the first derivative of the Transform to find the solution. My naive check is that I would end up with F(s)/s². But, I'll have to sit down with the raw definition of the Laplace transform to confirm. Unless somebody has a better table than the first thirty odd links in google.

[wumpus]

On 1/2/2021 at 9:35 PM, steuben said:

I had the same thought in the shower the other day. But, for different reasons. I've been reading some old electronic analog computer manuals. They say avoid derivatives like Da' Covid.

 I'll have to try the Laplace using y'' as the "solving" variable. This may solve the first sticking point of having to haul around and work with the first derivative of the Transform to find the solution. My naive check is that I would end up with F(s)/s². But, I'll have to sit down with the raw definition of the Laplace transform to confirm. Unless somebody has a better table than the first thirty odd links in google.

Analog computers work on integrals (which reduce noise) and avoid derivatives (which amplify noise).  With less complicated equations you just integrate them until the derivatives go away, but I'm not sure about yours.  To be honest, the only time I've ever heard about analog electrical computers was as a pre-req for DSP type courses (it introduced the fourier transform and used analog computers as an example of the systems you needed to emulate).  To even have a prayer of getting the analog computer in the same class of a digital one (even a cheap desktop CPU), you'd almost certainly have to built the thing entirely on a chip.   And then have fun matching capacitance values.  Analog computers may be revived in the neural net (AI/machine learning/whatever the latest buzzword is) craze, but I that's the only place I can imagine a comeback.

[K^2]

5 hours ago, wumpus said:

With less complicated equations you just integrate them until the derivatives go away, but I'm not sure about yours. 

So long as you can write out the Lagrangian of the system, there are prescriptions in classical mechanics of how to turn your equations of motion into integral equations. So if you have an analog system that's good at solving integral equations rather than differential ones, there's a way to handle that.

Though, to be honest, the only problems I've seen that get nicer from integrating them are in the realms of field theory (classical or quantum). Numerical methods we have for differential equations on trajectories are just so much better.

[steuben]

Doing some quick calculations. Yes. It would appear working through the Laplace space, the transform of the double integral that takes acceleration to distance is F(s)/s².

It really doesn't do much for that product term however. In Laplace space that turns into  Y(s)/s * Y(s)/s * Y(s)/s². There might not be much for it. The mechanics of analog computers can handle it. But, I'm still rummaging for a method of mathing it away, or into a much more tractable solution.

Part of the reason why I'm doing the work on this tarball is to add some solid background notes for my Greenfields series. Which runs from a tech level equivalent to 1950s to early 1970s. So electronic digital computers like we know, are rare, _expensive_, huge, and slower to solve a problem then a pencil, paper, and three books of look up tables. 

The other part is those who forget their technological history are forever doomed to recreate it.

[JoeSchmuckatelli]

2 hours ago, steuben said:

The other part is those who forget their technological history are forever doomed to recreate it.

I may steal that line from you.

Side note: the US Navy is re-teaching its Sailors how to use a sextant.  

Edited January 14, 2021 by JoeSchmuckatelli