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This answer explains that the gimbaling actuators on each of the nine Rutherford engines on the Electron rocket are in fact electrically powered DC brushless motors. Currently it's not known if they are linear motors, or rotary motors.

This kind of actuator is able to produce substantial force or torque even at zero linear or angular velocity.

I was wondering, very roughly, how much force or torque is involved here? When the engine is not running, there is just the inertia of the engine to deal with, and that's a very straightforward mechanical engineering problem.

Let's take gimbaling the smaller Rutherford engine as a working example for the moment, since it has the electric gimbaling actuators. I'm not sure if there is enough information around to estimate how much force those blue cylinders would need to accelerate an engine by say 1 degree per second squared, but assuming most of the mass is near the pivot point and I = 100 kg m^2, then you'd only need a few Newtons to push sideways, or perhaps 10 Newtons considering the mechanical disadvantage of the angle at which they are mounted.

I have no idea if gimbaling thrust at 1 degree per second squared is fast enough, responsive enough, or too slow to keep a rocket like this flying stable and smoothly. Any ideas on that?

The other half of the question is what happens when the engine is running and there is thrust? Is it harder to gimbal? There is some more mass in the chamber, pumps and tubing, and if there is flexible propellant feed tubing it will become much stiffer under high pressure. If this dominates, then it would be pretty hard to estimate because of all the specifics.

However, there may be some numbers out there that could be at least scaled for an informal ballpark estimate, and that would be fine in this case!

note: In the title of the question I've asked about a mid-sized nine-engine rocket and I've shown pics of both Merlin and Rutherford engines. Either one could be the focus for an answer.

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above x2: Illustrations of the nine Rutherford engines on Rocket Lab's Electron rocket. From here.

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above: jkavalik's excellent picture of a Merlin engine, from OrganicMarble's question SpaceX Engine Mounting / Gimbaling.

uhoh
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    There are good videos of e.g. the SSMEs gimbaling in prelaunch check; you could make some estimates from that. I suspect typical gimbal rates are a lot faster than 1degree/s^2 -- going to say closer to 20 as a wild guess. – Russell Borogove May 22 '17 at 15:18
  • @RussellBorogove That's half of an answer right there! "What kind of forces and response times/accelerations..." – uhoh May 22 '17 at 15:22
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    (And also that the acceleration is high enough and top rate low enough that it's easier to treat it as rate than acceleration) – Russell Borogove May 22 '17 at 15:26
  • @RussellBorogove I wonder if a lot of the mechanical work done is actually for "noise cancellation" - a lot of fast, small motions responding to turbulence within the nozzle, and the slow vectoring for turns is actually a small part of the work done? – uhoh May 22 '17 at 15:29
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    SSME gimbal max rate was 20 deg/sec https://science.ksc.nasa.gov/shuttle/technology/sts-newsref/sts-mps.html#sts-mps-tvc so excellent guess @RussellBorogove – Organic Marble May 22 '17 at 15:38
  • @uhoh I don't think so on the "noise cancellation" because TVC is part of the attitude control/guidance system. The engines are not going to be commanded to gimbal unless the vehicle needs to turn or needs to correct its attitude. The attitude control/guidance system will know nothing of "noise cancellation" unless it affects the trajectory. – Organic Marble May 22 '17 at 15:43
  • @OrganicMarble - I had the impression there was constant small-scale adjustment correcting for measurement error, crosswind, and/or unstable aerodynamics. – Russell Borogove May 22 '17 at 15:44
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    Yes, those are trajectory effects. He mentioned turbulence within the nozzle. – Organic Marble May 22 '17 at 15:45
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    Uneven thrust (even by a fraction of a percent) would also be a source of instability, so I didn't want to pooh-pooh that. – Russell Borogove May 22 '17 at 15:47
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    A hand-drawn (!) diagram of the F-1 gimbal system indicates Saturn V 1st stage engines had a 5 deg/sec gimbal rate "under load", second stage J-2 engines 9.6 deg/sec, and 3rd stage J-2 8 deg/sec. https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19700009516.pdf -- I also found a 15 deg/sec figure for the never-completed M-1 engine. – Russell Borogove May 22 '17 at 15:49
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    This paper https://www.scribd.com/document/7361853/00581656 says the SSME actuators had to withstand transient forces of 60 klbf ! – Organic Marble May 22 '17 at 15:52

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