Solve Counterintuitive Space Vehicle Problem w/Two Thrusters

[D H]

I like to toss a problem at interviewees to see how they think. My new question is so simple if looked at correctly, but so very counterintuitive that I caught a freshly minted PhD with it today.

A space vehicle with two thrusters is depicted below.

+=====v=========+
|...|...|
|...+-----O...|
|...|...|
+=====^=========+


The thrusters (the 'v' and '^' in the diagram) direct their exhaust normal to the long axis of the vehicle. The two thrusters have the same Isp and generate identical force magnitude. The vehicle is perfectly symmetric about the long axis. A single fuel tank (the 'O' in the diagram) feeds the two thrusters and is located some distance from the thrusters along the long axis of the vehicle.

The vehicle starts in a quiescent state at rest wrt some inertial observer. The two thrusters are activated simultaneously. The thrust from each thruster quickly ramps up to a constant value; both thrusters ramp up in an identical manner. After firing for some time, both thrusters are shut down simultaneously. Both thrusters ramp down to null firing quickly and in the same manner.

Describe the behavior of the vehicle.

 

[Danger]

I'm no scientist by any means, but my first thought (and second) is (neutral colour answer--highlight it to read)
I expect that it would travel forward (to the right in your diagram) due to reaction thrust against the fuel tank.

 

[vanesch]

My intuitive answer at first sight:

A slight shift to the right ?

 

[DaveC426913]

Danger's answer was my first thought too. But:
<that force is counteracted by the force directed against the fuel lines where they turn 90 degrees. Thus the rocket will go nowhere. Actually there will be a SLIGHT movement to the right while the centre of mass shifts as the fuel drains. The movement will stop once the engines stop. The craft will have moved less than its own length and will be stationary thereafter.>
Thus, Vanesch is correct. And if I hadn't bothered to be so verbose, I would have beaten him.

Also, here's your rocket with no dots:
+=====v=========+
|. . . . . |. . . . . . . . |
|. . . . . +-----O . . . |
|. . . . . |. . . . . . . . |
+=====^=========+

Copy this code and remove the spaces from the [ color] tags.
+=====v=========+
|[ color="#E9E9E9"]. . . . . [/COLOR]|[ color="#E9E9E9"]. . . . . . . . [/COLOR]|
|[ color="#E9E9E9"]. . . . . [/COLOR]+-----O[ color="#E9E9E9"] . . . [/COLOR]|
|[ color="#E9E9E9"]. . . . . [/COLOR]|[ color="#E9E9E9"]. . . . . . . . [/COLOR]|
+=====^=========+

 

[Danger]

That's a good point, Dave. I was thinking of radiused bends in the fuel lines (I always assume that, because I always design things that way ). It seemed to me that the thrust on the perpendicular pipes would be distributed in various directions and thus not counteract that upon the front of the tank. Is that correct? In any event, they are shown as right-angle junctions which negates that.

 

[George Jones]

Well, I've only thought a bit quantitatively about the question, but I'm getting a completely different answer - at the end of the day, in the original inertial frame, the ship is drifting to the left.

I'll try and think more about this when I get home, but I doubt that my daughter will let me.

 

[Danger]

I'll try and think more about this when I get home

You'd bloody well better; you have me intrigued, and you know how impatient I can be...

 

[D H]

Most of you saw that it drifts to the right while the thrusters are firing. Only George saw that it reverses direction when the thrusters are deactivated.

Way to go, George.


During a design review, one engineer complained that the vehicle design is invalid. His complaint: The thrusters would plume the space station when the design vehicle approached the station for berthing. The designer, who really liked his design because of its very fine control capabilities, thought for a second. He said, "Ahah. My new design avoids plume impingement problems and still has very fine motion control when I fire both jets simultaneously. In fact, it has the exact same equations of motion as the first design!" His new design:


+===============+
|...|
>-----------O---<
|...|
+===============+


How does this vehicle have the same equations of motion as the first?

 

[D H]

The easy way to solve this is to invoke conservation laws. The conservation laws dictate what the answer must be. The causal actions that result in that answer are a bit harder to puzzle out, but that is often the case. For example, I don't have to model contact forces, bending, and flexing to know the final state after one spacecraft docks of one spacecraft with another. I just need to know their initial states and their configuration after the transients die out.

Back to the problem at hand: The fuel flowing from the tank to the thrusters has linear momentum. The momentum resulting from that flow has to be balanced by some other translational motion to keep the total linear momentum zero as seen by the inertial observer who saw the vehicle start at rest. The vehicle has to move to the right while the thrusters are firing.

The vehicle attains a constant velocity as soon as the fuel reaches a constant mass flow rate. Ignoring transients, the exhaust gases in the first configuration have longitudinal velocity equal to the vehicle's velocity. When thrust terminates, the only moving entities in the closed vehicle+fuel+exhaust system are the exhaust and the vehicle. The exhaust has net momentum to the right. The vehicle has to be moving to the left after thrusting terminates.

If you want to know what causes this behavior, you have to look at the transients. Imagine that you just closed the valve to a faucet a bit too quickly. That bang you just heard is momentum transferred from the no-longer-flowing water to the pipes.

 

[DaveC426913]

The exhaust has net momentum to the right.

Why?
10 char

If you want to know what causes this behavior, you have to look at the transients. Imagine that you just closed the valve to a faucet a bit too quickly. That bang you just heard is momentum transferred from the no-longer-flowing water to the pipes.

I'm not convinced of this. It sounds like the rocketry equivalent of attaching a giant fan to the stern of a sailboat.

 

[D H]

The exhaust has net momentum to the right.

Why?

Imagine you and a buddy are driving in a dune buggy at a constant velocity on the moon. Your buddy is firing tennis balls normal to your direction of travel. To you, those tennis balls appear to be moving in a straight line away from the vehicle. The balls maintain their longitudinal velocity (the velocity component in the direction of your travel) after being fired. The exact same concept applies to the streams of exhaust. The vehicle structure is moving; those exhaust streams maintain that longitudinal motion. The thrusters (configuration #1) merely add a normal component to that velocity.

 

[DaveC426913]

Imagine you and a buddy are driving in a dune buggy at a constant velocity on the moon. Your buddy is firing tennis balls normal to your direction of travel. To you, those tennis balls appear to be moving in a straight line away from the vehicle. The balls maintain their longitudinal velocity (the velocity component in the direction of your travel) after being fired. The exact same concept applies to the streams of exhaust. The vehicle structure is moving; those exhaust streams maintain that longitudinal motion. The thrusters (configuration #1) merely add a normal component to that velocity.

You're saying that firing tennis balls normal to our forward motion is imparting a force to slow us down. How so?

Note that, if our dune buggy is floating in free space, you can no longer tell we're moving, and in fact, might not be. Yet I can still fire tennis balls out the sides. By your logic, I would start moving backwards. Or is it forwards?

 

[D H]

I said nothing of the sort. I used this as an analogy. Regarding the tennis balls, I said "you and a buddy are driving in a dune buggy at a constant velocity". The point of this analogy is that the balls are moving along with the vehicle longitudinally while they move away from the vehicle transversely. This is the exact same situation in the first configuration of the vehicle. The stream of exhaust doesn't lose its longitudinal velocity just because it left the vehicle, does it? To think otherwise is to revert to Aristotelian physics.

BTW, the equations of motion for this system are

$$m_v(t)\ddot r_v(t) + \ddot m_v(t) l = 0$$

where $$m_v$$ is the mass of the vehicle, including the fuel remaining on the vehicle, $$r_v$$ is the velocity of the vehicle structure (not the vehicle center of mass), and $$l$$ is the vector from the point midway between the two thrusters and the fuel tank. The transients are the only thing that changes the vehicle's velocity.

 

[DaveC426913]

Question:

Given this arrangment (where I've eliminated the fuel line factor):

+=====v=========+
|. . . . . |. . . . . . . . |
|. . . . . O. . .. . . . . |
|. . . . . |. . . . . . . . |
+=====^=========+

and a slight rightward motion (wrt to what I don't know), which way would the rocket move?

 

[D H]

By eliminating the fuel line factor you just eliminated the sole motive force. The vehicle moves with a constant, slight rightward motion (wrt to the same unknown what). Look at the EOM in post #13.

BTW, your vehicle will look better if you use a fixed-point font. Wrap the whole thing in a [ FONT="Courier New"] ... [ /FONT] construct.

Also, thanks for showing me how to hide the periods in my original drawing.

 

[DaveC426913]

OK, I wasn't sure if the fuel line factored into your leftward force setup. It does.

I'm still not getting where the leftward force comes from.
It's a force that
- occurs after the thrusters stop
- cancels the slight rightward motion caused by the shift in CoG
- adds an additional component that starts it in a leftward direction

 

[DaveC426913]

That's a good point, Dave. I was thinking of radiused bends in the fuel lines (I always assume that, because I always design things that way ). It seemed to me that the thrust on the perpendicular pipes would be distributed in various directions and thus not counteract that upon the front of the tank. Is that correct? In any event, they are shown as right-angle junctions which negates that.

Doesn't matter. No matter how you distribute them, everything cancels out except the exhaust component.

If you could do that, then you could create motion merely by shaping your vehicle like a wedge using the logic that the air would just slide off the slanted end while pushing on the flat end.

 

[D H]

This problem is an offshoot of our investigations into a discrepancy in our propagated orbit insertion position. We hadn't accounted for the flow momentum in our equations of motion. A vehicle casts off 90% of its mass to get to into orbit; that momentum needs to be taken into account.

Added bonus: It makes for an incredibly nasty interview question.

 

[DaveC426913]

Well, I guess I won't doubt that you're right then...

 

[D H]

I'm still not getting where the leftward force comes from.
It's a force that
- occurs after the thrusters stop
- cancels the slight rightward motion caused by the shift in CoG
- adds an additional component that starts it in a leftward direction

Starting the flow of fuel requires a transfer of momentum from the vehicle to the fuel in the line. It is this momentum transfer that makes the vehicle proper start moving. That momentum is transferred back to the vehicle when the flow is stopped. However, the vehicle mass has been reduced in the interim. Instead of stopping the vehicle, the momentum transfer reverses the vehicle's direction of travel.

 

[vanesch]

My reasoning was simply:
the center of gravity of the entire system must remain at rest (because only internal forces act).
Now, if you look at the center of gravity before the thrusters are activated, it is at a certain point on the symmetry line of the vehicle. At the end of the day, the center of gravity of the fuel has been displaced, from the center of the tank, to the point on the symmetry line between the two thrusters, and it stays there. So in order to compensate, the vehicle (without the fuel) must shift a bit to the right (in the direction of the empty tank).

So I still claim that the overall motion is just a shift to the right:
when the thrusters fire, there will be a rightward velocity, which will drop to 0 when the thrusters will stop firing. The only thing that counts is the fuel flow from the tank to the thrusters.

 

[George Jones]

You'd bloody well better; you have me intrigued, and you know how impatient I can be...

Sorry, I was too exhausted last night to do anything more than lurk. I hope D.H. statisfied your curiosity before your impatience killed you.

D.H. has already explained the puzzle, but here,in my own words, is the reasoning that I used. All motion is with respect to the initial frame of reference.

I used Newton's second law, which says that the (time) rate of change of the momentum of a system equals the total external force on the system. I took the system to be everything, rocket plus fuel plus exhaust. In this case, there is no extrenal force, as all the forces are internal. Since the external force is zero, the momentum of the system is zero (its inital value) throughout the entire process.

Once the fuel starts flowing to the left, the ship has to move to the right in order to keep the momentum of the system zero. The velocity of the exhaust has no horizontal component, i.e., is completely vertical (up and down) with repect to the ship. Because the ship is moving to the right, the exhaust moves (partly) to the right. After the thrusters are turned off, fuel no longer flows, but the exhaust is still moving a bit to the right. In order for the momentum of the system to be zero, the ship has to be drifting to the left.

D.H. - nice puzzle.

 

[vanesch]

The velocity of the exhaust has no horizontal component, i.e., is completely vertical (up and down) with repect to the ship. Because the ship is moving to the right, the exhaust moves (partly) to the right..

Right !

 

[D H]

D.H. - nice puzzle.

Thanks, and congrats.

Nobody has responded to the second part of the puzzle, post #8, repeated herein.

During a design review, one engineer complained that the vehicle design is invalid. His complaint: The thrusters would plume the space station when the design vehicle approached the station for berthing. The designer, who really liked his design because of its very fine control capabilities, thought for a second. He said, "Ahah. My new design avoids plume impingement problems and still has very fine motion control when I fire both jets simultaneously. In fact, it has the exact same equations of motion as the first design!" His new design:


+===============+
|...|
>-----------O---<
|...|
+===============+


How does this vehicle have the same equations of motion as the first?

 

[Gokul43201]

Nobody has responded to the second part of the puzzle, post #8, repeated herein.

Are there flow regulators on the exhaust pipes or does the flow go like the length of exhaust pipe?

 

[D H]

You are reading to much into this. Nobody in their right mind would really use such a design. At least I hope that is the case. I have seen some truly crackpot jet layout schemes over the years with equally crackpot justifications for these layouts. "Very fine control" is one of justifications.

Back to the problem at hand: Just assume the mass flow rates along the two feed lines are equal and that the two thrusters generate equal-but-opposite forces at all times. How that is accomplished I shall leave up to the original designer.

 

[vanesch]

Back to the problem at hand: The fuel flowing from the tank to the thrusters has linear momentum. The momentum resulting from that flow has to be balanced by some other translational motion to keep the total linear momentum zero as seen by the inertial observer who saw the vehicle start at rest. The vehicle has to move to the right while the thrusters are firing.

The vehicle attains a constant velocity as soon as the fuel reaches a constant mass flow rate. Ignoring transients, the exhaust gases in the first configuration have longitudinal velocity equal to the vehicle's velocity. When thrust terminates, the only moving entities in the closed vehicle+fuel+exhaust system are the exhaust and the vehicle. The exhaust has net momentum to the right. The vehicle has to be moving to the left after thrusting terminates.

Actually, this depends on the exact flow and capacity of the pipes!
If you consider that you first move a certain amount of fuel on the axis to in between the thruster pipes, bring it to a halt there, then you will agree with me that the spaceship only underwent a small shift (an internal motion of a mass). Now, if at that point, we activate the thrusters with that amount of fuel, nothing will happen, as the spaceship is not in motion when the thrusters are activated. If we now wait until the thrusters have finished burning this fuel, before bringing in the next amount of fuel, and repeat the cycle again, at no point the thrusters are having a longitudinal motion when they are in action. So in this pulsed mode, the spaceship will NOT have a final leftwards velocity, right ?
And if we would pump some of the fuel back in the tank while burning another part, the exhaust would get a leftgoing component, and hence, in such a repeated motion, at the end, the spaceship would have a final rightward velocity, or am I wrong ?

Meaning, that the final velocity is depending on exactly how the fluid is moving, and at what moment the thrusters are activated...

 

[D H]

So in this pulsed mode, the spaceship will NOT have a final leftwards velocity, right ?

That is exactly correct. Many spacecraft motion models implicitly invoke a pulsed behavior ("consider a small packet of fuel ..."). As I mentioned in a previous post, this nifty little problem is an off-shoot of an investigation into an error in our propagated orbit insertion position. A model based on continuum physics yields more accurate results for continuous flow rockets.

An APU (which pulses) would indeed result in different physics.

 

[cesiumfrog]

Are there flow regulators on the exhaust pipes or does the flow go like the length of exhaust pipe?

Apparently the former, such that the exhaust never produces thrust in the craft's frame. This requires slightly more impulse to get the fuel to start/stop flowing in the longer pipe.

 

[D H]

Apparently the former, such that the exhaust never produces thrust in the craft's frame. This requires slightly more impulse to get the fuel to start/stop flowing in the longer pipe.

Exhaust pipes? Those are supply lines. If you want specifics, the fuel is hydrazine. The tanks are pressure regulated. The longer fuel line is slightly larger in diameter than the shorter one so that the flow rates in the two are equal. The thrusters contain honeycombed iridium that absorbs any momentum in the incoming fuel and serves as catalyst to decompose the fuel, thereby producing thrust. By some miracle, both thrusters produce the exact same amount of force.

 

[DaveC426913]

Once the fuel starts flowing to the left, the ship has to move to the right in order to keep the momentum of the system zero. The velocity of the exhaust has no horizontal component, i.e., is completely vertical (up and down) with repect to the ship. Because the ship is moving to the right, the exhaust moves (partly) to the right. After the thrusters are turned off, fuel no longer flows, but the exhaust is still moving a bit to the right. In order for the momentum of the system to be zero, the ship has to be drifting to the left.

I hope I'm not beating this to death. (And I don't doubt that D_H and you are correct.) But I really want to understand (I hate not understanding).

I understand the rationale in the above explanation (the overall momentum must be zero). But I want to understand it "from the inside" i.e. from the frame of ref of the occupants. It would have to be an internally consistent cause and effect. Somehow, the exhaust plume must be observed as slightly rightward-of-normal - in order for the ship to ultimately have a slightly leftward movement.

 

[D H]

I understand the rationale in the above explanation (the overall momentum must be zero). But I want to understand it "from the inside" i.e. from the frame of ref of the occupants. It would have to be an internally consistent cause and effect. Somehow, the exhaust plume must be observed as slightly rightward-of-normal - in order for the ship to ultimately have a slightly leftward movement.

At the end of the day, I would expect to see two backwards letter Cs drifting away from each other tranvsersely. The tip and tail of each C will be stationary longitudinally. The Cs will stretch out as time passes.

The equations of motion for this system, assuming continuous thruster firings, are

$$m_v(t)\ddot r_v(t) + \ddot m_v(t) l = 0$$

where l is the vector from the midpoint of the two thrusters and the fuel tank.

The derivation is left to the reader.

The vehicle accelerates only when the flow rate changes. The exhaust plumes will replicate the graph of the flow rate.

 

[DaveC426913]

OOHHHH!

Upon turning on the jets, the craft accelerates to the right.
Upon turning off the jets, the craft accelerates to the left BUT the craft is slightly less massive than before, causing it to have an slightly GREATER acceleration leftwards!

Jeez. Why didn't you just say so!

Izzat right? I hope that's right.

 

[D H]

That is exactly correct.

 

[DaveC426913]

I wonder if I can make a career out of lay-interpreting scientific explanations.