Euler's Equations of Rigid Body Dynamics Derived | Qualitative Analysis | Build Rigid Body Intuition
Euler's Equations of Rigid Body Dynamics Derived | Qualitative Analysis | Build Rigid Body Intuition
Space Vehicle Dynamics 🤓 Lecture 21: Rigid body dynamics, the Newton-Euler approach, is given. Specifically, from the angular momentum rate equation, we derive the most common forms of Euler’s equations of the rotational dynamics of a rigid body. A qualitative analysis of Euler’s equation is also given to build intuition.
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► Dr. Shane Ross ➡️ aerospace engineering professor, Virginia Tech
Background: Caltech PhD | worked at NASA/JPL \u0026 Boeing
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► Lecture notes (PDF)
https://is.gd/SpaceVehicleDynamicsNotes
► References
Schaub \u0026 Junkins📘Analytical Mechanics of Space Systems, 4th edition, 2018
https://arc.aiaa.org/doi/book/10.2514…
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📚3-Body Problem Orbital Dynamics Course
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📚Space Manifolds
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📚Lagrangian and 3D Rigid Body Dynamics
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📚Nonlinear Dynamics and Chaos
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📚Hamiltonian Dynamics
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📚Center Manifolds, Normal Forms, and Bifurcations
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► Chapters
0:00 Summary so far
0:37 Newton-Euler approach to rigid bodies
10:22 Qualitative analysis to build intuition about rigid bodies
11:06 Spinning top analysis
15:36 Spinning bicycle wheel on string
19:06 Fidget spinner analysis
22:01 Landing gear retraction analysis
24:53 Euler’s equations of rigid body motion derived in body-fixed frame
29:09 Euler’s equation written in components
30:56 Euler’s equation in principal axis frame
35:33 Euler’s equation for free rigid body
40:32 Simulations of free rigid body motion
► Topics covered in course https://is.gd/SpaceVehicleDynamics
- Typical reference frames in spacecraft dynamics
- Mission analysis basics: satellite geometry
- Kinematics of a single particle: rotating reference frames, transport theorem
- Dynamics of a single particle
- Multiparticle systems: kinematics and dynamics, definition of center of mass (c.o.m.)
- Multiparticle systems: motion decomposed into translational motion of c.o.m. and motion relative to the c.o.m.
- Multiparticle systems: imposing rigidity implies only motion relative to c.o.m. is rotation
- Rigid body: continuous mass systems and mass moments (total mass, c.o.m., moment of inertia tensor/matrix)
- Rigid body kinematics in 3D (rotation matrix and Euler angles)
- Rigid body dynamics; Newton’s law for the translational motion and Euler’s rigid-body equations for the rotational motion
- Solving the Euler rotational differential equations of motion analytically in special cases
- Constants of motion: quantities conserved during motion, e.g., energy, momentum
- Visualization of a system’s motion
- Solving for motion computationally
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Content
0.08 -> in outline what have we done we talked about
the transport theorem for many weeks just
5.12 -> to get ourselves oriented about how to deal
with changes of frames and then talked about
11.6 -> single particle dynamics multi-particle dynamics
we got to this end where we talked about if you
18.4 -> have a system of multiple particles or even a
continuum you could break down the dynamics into
24.48 -> translational dynamics of the center of
mass and also the rotational dynamics
31.52 -> and it was described by the two different
equations so this is sometimes called the newton
38.16 -> euler approach to modeling rigid bodies the
newton approach gives the translational motion
46.56 -> if you remember from i think it was lecture
nine we talked about something that looked
51.68 -> like newton's equation f equals m a for the motion
of the center of mass so the translational motion
59.52 -> of the center of mass so that means we've
got an inertial frame some kind of origin
65.44 -> and floating yam arbitrary thing but it
is a rigid body and we're tracking the
72.16 -> location of the center of mass so if this has
a full total force we'll just write it that way
79.6 -> and this is the external force that's a total
external form because we don't even consider
83.92 -> internal forces like if it's magnetic or something
who cares so we have a total force on the body
90.8 -> the newton approach is just that
f is equal to the total mass
95.44 -> times two inertial derivatives of the position
right r c double dot is two inertial derivatives
106.24 -> of the position so this is sometimes called
we called it before the superparticle theorem
111.44 -> some people have even called it newton's law but
it's kind of derived from newton's law it looks
116 -> exactly like newton's law but it's not exactly
the same total external force over here some
121.52 -> people have even gone so far as to call it euler's
first law but that might confuse you just think of
128.64 -> it as newton's law and if anybody asks for details
you could look them up so this gives the dynamics
133.2 -> of the center of mass so if we're talking about
spacecraft maybe that the location of that center
138.56 -> of mass is given with respect to the center of
the earth so we're not saying anything about how
144.24 -> i would rotate we're just saying how the center
of mass moves so the center of mass is moving and
149.52 -> it might also be rotating and of course our focus
is on that rotational part this was translational
156.16 -> motion the rotational motion also harkens back to
lecture nine when we talked about euler's equation
165.44 -> this is the one usually reserved for the word
euler's equation and sometimes people will weirdly
172.24 -> call it euler's second law just letting you know
you're going to hear a variety of terminology out
177.68 -> there in the big bad world so the rotational
motion let's um let's draw the potato again
186.56 -> it's easier than other things it can be drawn
so there it is and it's got a center of mass i
195.44 -> guess we could label that up there and down here
which is a body fixed frame attached to this so
202.32 -> we write the frame as you know b and maybe we
write the body as fancy b and the main thing
207.68 -> that we're going to care about here is the angular
momentum so here's the angular momentum typically
216.24 -> it's angular momentum with respect to the center
of mass b frame is attached to the body and right
223.84 -> now we're not specifying if this is a principal
axis frame it's just a frame attached to the body
230.72 -> then the governing equation the equation of motion
is going to be it's this just one derivative
239.2 -> of i mean maybe i'll write it in analogy to what
i have up there it would be h c over dot equals
249.36 -> l c using the book's notation for what the
moment is the moment or the torque so here is a
261.36 -> torque it's a torque vector about
the center of mass moment or torque
269.52 -> of course hc dot this is just shorthand for an
a derivative with respect to the inertial frame
279.36 -> of the angular momentum of the body about the
center of mass and sometimes this is called
288.64 -> euler's second law but again that's kind of weird
and confusing so we will just call it euler's
294.56 -> equation this is euler's equation and we're going
to write euler's equation in a bunch of different
300.64 -> ways what does lc the total moment total external
moment and we could write a formula for it i'll
307.12 -> draw the shape again all right the moment with
respect to some reference point it doesn't have
313.36 -> to be the point c so in general how do we get
the total moment and about because we might not
319.68 -> always just wonder about the center of mass about
some point p on the body so here's a point p here
327.36 -> is the center of mass and here is a volume so
we're going to have to integrate i'll write the
333.6 -> location of this this is the location of that
volume element this is the volume element dv
341.44 -> it's got a density in it this might not be
uniform density dv with respect to the point
346.24 -> p the total moment with respect to the point
p this is defined to be integral over the body
352.56 -> so over the volume of the location of that volume
with respect to the point p cross and now it's
359.2 -> like the force acting in that little box i'll use
a different color here's a force this is little f
367.76 -> it's a force density f dv so it's at that location
this is a force density which means force per
378.32 -> unit mass we're summing over all of the little
mass elements dm which is rho db so i'll put rho
386.72 -> db we're allowing for the fact that rho the
density might be different throughout the body
391.76 -> if it's not then you could pull it out as
a constant this is how you would write the
395.2 -> moment about an arbitrary point including
the center of mass for example if we're in
400.08 -> uniform gravity like near the surface of
the earth later we won't be in a situation
405.28 -> where we would consider the gravity uniform when
we talk about gravity gradient satellites they
410.16 -> actually use the fact that gravity pulling at
the point closer to the earth is stronger than
414.96 -> the point furthest but for now let's just assume
uniform gravity uniform gravity means that the
420.8 -> force density this is the force per unit mass
that has units of acceleration so this is the
427.28 -> direction of the gravitational acceleration so
uh let's just calculate this for uniform gravity
433.44 -> it is got a triple integral of the
body r dv p cross product g rho dv
445.36 -> this will end up giving you location of the
center of mass cross m g so it looks like
451.92 -> location of the center of mass what's the location
of the center of mass if i were to sketch it up up
456.32 -> here this is the location of the center of mass
with respect to our point p so r c is the same as
464.48 -> the vector from p to c it's the location of c with
respect to p across the usual force due to gravity
473.6 -> mass times acceleration so that's just an example
it might be different in different contexts okay
479.84 -> so we've got if i were to highlight them again i
like this one this is the one that we'll focus on
485.6 -> that's with respect to the center of mass
if you have an inertially fixed point
490.24 -> then another variation of euler's equation
would be the inertial derivative of the
495.68 -> angular momentum with respect to that point
p think of a pivot point equals the moment
502.24 -> total moment about p so that's another variation
we could say more you write it that way
510.32 -> but we usually write euler's equation in a
body fixed frame which means we want to take
517.36 -> the derivative with respect to a body fixed
frame so we now use the transport theorem
523.84 -> so the transport theorem would say take the
derivative with respect to the body fixed frame
528.8 -> plus and now we're talking about one frame with
respect to another frame here's the b frame and
535.36 -> the end frame they're just different triads of
unit vectors and what we have called angular
540.88 -> velocity is actually the angular velocity of the
b frame with respect to the end frame so then this
547.76 -> would be plus omega of the b frame with respect to
the end frame but that's just the angular velocity
553.36 -> cross h p equals the total moment about that
point so that's just another way to write it
560.4 -> it uses the transport theorem so if we use the
center of mass as our reference point we would
565.52 -> have something that looks really similar it's
just with respect to the center of mass omega
570.8 -> and omega is just the angular velocity of the body
i'll just give you those two right they they look
578.88 -> the same except for that subscript so this is
also called euler's equation i think in the book
585.52 -> this is what 432 it's also called euler's equation
we're just writing things with respect to the body
591.52 -> fixed frame why are we going to want to do that
we're doing that because later we'll write hc
598.32 -> is the moment of inertia matrix what do we
calculate that in not the inertial frame
603.36 -> we usually calculate that in the body fixed frame
because then we've calculated it once and for all
607.68 -> so that means we've got the angular momentum
written in the body fixed frame and it'll be
612.64 -> the moment of inertia times the angular velocity
written in the body fixed frame so we're going
617.68 -> to eventually use this we're not going to use
it quite yet there's more i want to say at this
622.4 -> level to kind of build some intuition there is
this qualitative analysis of spinning bodies
628.56 -> that we can do just to get some intuitive feel for
things i think this is helpful if you haven't seen
637.04 -> it or thought about it then this will provide at
least some idea of what's going on so we'll look
642.96 -> at this this first one that i wrote up here
because this is mostly useful at least in the
649.2 -> examples with there's some kind of pivot point
so just copying what we have up there writing the
654.72 -> time rate of change of the angular momentum
with respect to a body fixed frame plus angular
660.08 -> velocity okay here we go we have this what's
a good picture to have in mind a spinning top
666.32 -> or a gyroscope this is a still from the a video
okay so we got a gyroscope for the gyroscope
673.28 -> there's the part that spins in order to kind
of highlight it there's this giant flywheel
679.6 -> and it's spinning we don't you know which way it's
spinning but it's spinning a lot and it's attached
684.88 -> to a frame we will call this the body fixed frame
and that frame could be moving with respect to
692 -> an inertial frame which i'll just sort of put
centered at this pivot point i'll call that p
697.12 -> and then this is probably over here the center of
mass the idea is if you have a gyroscope there's
701.76 -> a flywheel kind of like your spinner and it's
spinning very rapidly with respect to the frame
708.8 -> so this flywheel bends rapidly with respect
to this b frame that it's attached to
716.08 -> so we could say that it has a very large angular
momentum that isn't changing much that means
723.44 -> over here with respect to that body fixed frame
this is not going to change much and hp is
731.84 -> really large the magnitude of hp is large let me
actually show you a video of this getting spun up
741.92 -> spinning and the way that it's spun
747.76 -> the angular velocity uh is actually
pointing up if we were to watch it again
757.28 -> the it's being spun i guess the way that we're
looking at it it's counterclockwise just gonna
764.4 -> place it on here and it's spinning rapidly it's
gonna have a fixed pivot point what's it gonna do
771.36 -> it's gonna move in some direction you've probably
seen a gyroscope right he's gonna set it down here
776.8 -> it's gonna move either to the left or to the right
and you can figure that out using this qualitative
784 -> argument and that's all we're after now which way
is this thing gonna rotate which way is the frame
791.84 -> going to rotate about that pivot so he he's got
the flywheel spinning it turns out he the flywheel
798.48 -> is spinning very rapidly in a particular direction
let's first figure out what the moments are so
804.72 -> when you have this first part equaling roughly
zero then that means you're just left with this
810 -> over here so you could qualitatively figure out
which way is the moment the moment here the moment
816.48 -> about p for this situation is what moment about
p is due to the only force here is gravity so
825.84 -> from this point p to the point c we've
got that which i think before i call that
830.48 -> r little r sub c and then gravity pointing down
so the moment is using the right hand rule r cross
840.88 -> f so it's actually pointing into the screen we
have to know which way the angular momentum is
847.6 -> going the angular momentum is actually going
this way it's hard to see from the video so
854.96 -> if it's spinning very rapidly going that way then
what direction will the frame go so if we know hp
862.08 -> that direction and we know l then we have
to figure out what does omega need to be for
867.84 -> omega cross h to equal l and using the right
hand rule omega needs to be pointing down
875.84 -> so omega this is omega of the b frame with
respect to the end frame so if that's correct
882.88 -> then our prediction is this will be like coming
at us because following the right-hand rule
889.44 -> because of omega if omega's pointing down my thumb
points down my fingers curl in the direction that
894 -> the whole frame is rotating so let's see if that
holds true we'll watch the gyroscope get placed
902.08 -> okay yeah
906.24 -> so this is omega pointing down
omega pointing down means that
910.08 -> your fingers are curling using your right hand
913.36 -> doing that so this had the the very high angular
momentum is actually pointing towards the pivot
922.08 -> we could do another case just to see what's
going on here that pivot point actually
926.32 -> it doesn't matter if it's on uh a stand if you
start with that thing not moving it doesn't
932.56 -> matter what it is so this could be at the end
of a string so there's this other video i think
938.32 -> this one has sound got a bicycle wheel being
from a rope it's sort of like a giant version
944.96 -> of the gyroscope and he's getting
this spinning pretty rapidly
954.72 -> you think of okay which way
is he getting it spinning
958.8 -> uh he's getting it spinning so that the right
now the angular momentum is pointing down
967.12 -> and you'll be able to watch these
971.28 -> then he's going to set it up sideways
974.96 -> and uh let it go and it's going to spin one way or
the other let's try that over here we'll just show
981.2 -> another image so this is spinning we'll
use some different colors here a little
986.32 -> bit easier to see right we still have gravity
here's that point p here's the center of mass
992.08 -> rc and then gravity is pointing down so what's the
moment we know the moment here it's r cross f so
1001.2 -> it's coming out at us so the moment r plus f is
pointing towards us the angular momentum is very
1010.32 -> large but away from the pivot so the only thing
that's different like the moment is the moment
1017.92 -> angular momentum is pointing away from
the pivot so what does omega have to be
1022.48 -> does it point up or down for this equation up
here to be correct we need that so omega if i
1031.6 -> do my right hand omega needs to be pointing
up omega cross h pointing out of the screen
1041.52 -> so it's kind of along here omega
of the the bicycle you know frame
1047.92 -> basically so with that prediction that means
that the the bicycle wheel should be turning
1054.08 -> in the opposite direction as this gyroscope just
because of how it was initially spun up how do i
1059.44 -> draw this nicely kind of around that way and then
coming into the front you're going behind there
1067.44 -> so that's the prediction based on the
qualitative analysis let's see if it bears out
1076.8 -> okay
1093.12 -> and okay good it's going that way
1101.92 -> and this is kind of remarkable because maybe like
how is this how is that bottom of the rope not
1107.36 -> swinging like crazy yeah as long as it starts out
forces it up so that that pivot point is initially
1113.76 -> not moving and that's why this works it is kind of
amazing i think we could look at other situations
1120.96 -> where maybe we don't know the moment we actually
want to find out the moment and that's related
1125.44 -> to the fidget spinner if you've played with
a fidget spinner long enough you know that it
1131.36 -> moves in weird ways if you
hold it you get it spinning
1135.28 -> and then you kind of twist your wrist you
feel something that's not highly intuitive
1146.32 -> what about the fidget spinner fidget
spinner so here is a fidget spinner and
1152.24 -> what we're imagining here is that you're going
to rotate your wrist you get this thing spinning
1157.84 -> so if i let's say i'm holding it and then
i'm spinning it down and i look kind of like
1165.44 -> what's pictured there if i spin it down then
it's each little part of it's going this way
1177.68 -> so what direction is the the large angular
momentum the large angular momentum
1185.12 -> is here i won't even put a little subscript
it's just a large angular momentum
1191.2 -> using the right hand rule it's going
that way and now if i were to uh rotate
1198 -> my wrist so i um maybe i'll move it this way
so if i'm i've got the fidget spinner and i
1205.52 -> try to turn it i can feel it resisting and i got
i have to exert some moment on it to make it not
1214.64 -> turn weird so you rotate your wrist and i don't
know maybe the pivot point is up here for your
1220.48 -> wrist let's say i'm rotating in the direction of
the arrow so that means i have an angular velocity
1225.92 -> i'm imposing some angular velocity that's coming
out of the screen because according to the right
1231.28 -> hand rule if my wrist is rotating that way then
that's the angular velocity vector it's pointing
1235.52 -> out so what is the moment that that my wrist must
impart to maintain this movement well uh what do
1249.2 -> i have i've got omega omega cross h according to
up here must equal the moment that's applied omega
1258.56 -> cross h moment's pointing this way so this
is l is the moment my wrist must impart to
1269.04 -> maintain this movement and you can feel it
resisting that and that's why your moment
1274.48 -> has to be what it is as you rotate your wrist in
the direction shown what you'll feel is that this
1281.28 -> thing wants to kind of turn that way and the
back side turn that that way and so you have
1287.76 -> to exert a moment that's in that direction to
resist it so you could try it out try it at home
1295.6 -> and i could feel it wanting to do weird things
and i have to uh resist now if this was spinning
1302.96 -> like crazy it would break my wrist i couldn't
maintain the movement without breaking my wrist
1308.96 -> by the way these only have a certain size fun size
not dangerous break your wrist size it's all fun
1315.92 -> and games when it's a fidget spinner but what
if it's landing gear what if it's landing gear
1322.4 -> so there's an analogy with landing gear some
planes have had landing gear this way i don't
1328.96 -> know if all planes have yeah they don't they don't
actually do landing gear this way anymore but old
1333.68 -> timey landing gear this this uh this is the wheel
right the wheel as it's spinning during takeoff is
1341.92 -> going to be moving extremely rapidly and if you
try to turn this up it's exactly like the fidget
1348.96 -> spinner and you're going to have lots of moments
imparted up there where things could break so if
1354.24 -> i give you a sketch of landing gear it's kind
of aligned with sort of the same way that this
1360 -> this fidget spinner we've got the wheel rolling
on the ground right here's the ground let's say
1367.76 -> if this is rolling in the way shown by this
arrow then you have a really large moment
1374.64 -> here's at some later time we've got a really large
moment in that direction and we're imposing by
1381.84 -> let's say some motors up here to make the landing
gear turn up and by the right hand rule that
1387.12 -> means that the angular velocity would be in this
direction and because of that to maintain this
1392.4 -> movement the moment that must be imparted is
it's going to be in this direction omega cross h
1399.52 -> so if i have h very large or omega very large
right l i'm just going to rewrite what i have
1406.56 -> up there it equals omega cross h so if you retract
very rapidly spinning landing gear and you retract
1415.04 -> it quickly this moment could be so large that
it does damage right if the magnitude is l is
1421.2 -> large enough it means you're breaking stuff this
could break the landing gear mechanisms could
1427.76 -> break and that's definitely not what you want
especially on takeoff how are you going to land
1434 -> so most landing gear does not fold up this
way anymore it actually just gets pulled up
1440.88 -> then you don't have this kind of omega cross
each moment showing up if you don't worry if
1447.92 -> you fly on commercial jets most landing gear just
retracts upward it doesn't fold sideways but you
1453.68 -> get the same it's completely analogous to holding
this fidget spinner and then trying to turn it up
1460.64 -> and you can feel that it's it's an increased
moment when you turn it rapidly versus slowly
1466.24 -> so when they did do this they probably
tried to do it as slow as they can zoom
1472.48 -> now we have a great intuitive understanding
about three-dimensional rigid body dynamics
1481.84 -> we'll revisit this later but you might you
might think about it or try it out if you
1488.72 -> don't have a fidget spinner get some landing
gear i guess we're going to get back to the
1494.16 -> the quantitative that was qualitative just what
are the directions of vectors but now we're
1498.88 -> going to get quantitative again and talk about
the rigid body dynamics we're going to look at
1505.6 -> euler's rotational equations motion in more depth
just following on from what we have up there
1513.04 -> and this is if you're in the book 4.2.3
1518.4 -> so we've got this and i'll be focusing on
the moment about the center of mass because
1523.28 -> that's what matters mostly on the spacecraft
the rotational dynamics with respect to the
1528.96 -> center of mass so we've got this plus omega
cross h c equals l c and now we can substitute
1540.64 -> in what we know for the angular momentum
written in the body fixed frame it equals
1547.68 -> the moment of inertia matrix which we have fussed
about times omega so the angular velocity again
1555.52 -> written in the body fixed frame so what this is if
you want uh we could write it out not particularly
1561.2 -> in lightning but there it is and this is all in
body fixed frame and then omega 1 omega 2 omega 3.
1569.28 -> we're trying to make a connection with the angular
velocity because the rigid body kinematics you
1574.64 -> know all the euler angles and how those change was
related to the angular velocity in the body fixed
1579.52 -> frame so how do omega 1 and omega 2 and omega-3
change with time that's what we're going to get to
1585.44 -> we've got this differential equation over here
so if we just plug in everything i'm not going
1590.72 -> to plug in the full way of writing it in terms of
components if we plug in what we have this becomes
1597.92 -> we've got a derivative with respect to the b frame
of this three by three matrix written in the b
1607.2 -> frame times omega written in the b frame plus
omega cross and now we substitute in again i c
1620.72 -> times omega and then this equals l c and we could
also write that in the body fixed frame so this
1628.64 -> is now this is an equation where everything is
written in b frame components this is a vector
1635.04 -> equation all in b frame components whatever that
body fixed frame is and remember the body fixed
1643.84 -> frame could be moving around with respect to
an inertial frame and that's what makes this
1650 -> seem kind of crazy but it's okay it's all right
we have a derivative here let's deal with the
1656.64 -> derivative a derivative of a matrix times a
vector we could use the product rule for that
1663.28 -> it is a derivative of this matrix times the
vector plus the matrix times the derivative
1671.52 -> of the vector then plus omega cross just repeating
everything from there this doesn't change except
1678.96 -> it's all written in b frame components this
is why we write the moment of inertia matrix
1684.16 -> in the body fixed frame because everything's
constant so this thing here you take the
1691.12 -> derivative of things that are all constant they're
not changing right because this is constant that
1696.4 -> matrix is constant all of its entries are constant
you've calculated it once and for all that's the
1701.92 -> mass distribution with respect to the body fixed
frame so this is zero so it falls out goes away
1709.04 -> the mass distribution doesn't change we're
assuming a rigid body if we assume something
1714.48 -> where they're moving parts then okay we would
have to adjust that but for now we are not
1720.24 -> so that part goes away and then we're left with
just this this is equivalent to equation 4.34
1729.36 -> in the book although i'm being more careful
about what frame things are being written in
1734.32 -> so if we were to write this out these derivatives
of the omega vector individually that's just going
1742.32 -> to be omega 1 dot omega 2 dot omega 3 dot
it's just the time rate of change of those
1748 -> so this will all become all right it is
this way i see times omega one dot make
1756 -> it two dot omega three dot it's just the rates
of change of those body fixed frame components
1763.84 -> plus you could write this in different ways if you
want you could write it as omega 1 omega 2 omega 3
1773.6 -> across this matrix you notice there's lots
of omega ones omega twos and omega three
1779.36 -> flying around and then we'll write the moment i'll
just write it in terms of components say l1 l2
1786.24 -> l3 those are the three components of the moment in
the body fix frame and you could take this further
1792.88 -> and write in different ways so if you write
this out it'll look like some horrendous thing
1799.36 -> if you use just any old frame so this is another
version of euler's rotational equations of motion
1807.2 -> you might say is there a difference between
or there's equation and euler's rotational
1810.64 -> equation of motion in fact no it's
just a different way of writing it
1816.4 -> so for whatever reason yeah the book we'll call
this 4.35 things simplify and this this is why i
1824.24 -> wanted you to know what a principal axis frame
was and how to find it if you write this in a
1829.28 -> principal axis frame so if if we choose as our
body fixed frame a principal axis frame then the
1837.12 -> advantage that that gives us is that this moment
of inertia matrix is just diagonal we lose all
1843.12 -> those off diagonal elements and when i do that
instead of writing it as i 1 1 i'll just call it
1847.6 -> i1 i2 i3 that's it when we've chosen a principal
axis frame when you do that then this euler's
1856.32 -> rotational equation of motion simplifies to
a point where it's kind of easy to work with
1867.6 -> it becomes i 1 omega 1 dot minus i 2 minus i 3
omega 2 omega 3 equals l1 i might say plus cyclic
1883.68 -> permutations which just means all the indices
just sort of move back so this becomes i2 omega 2
1893.76 -> dot so we've just sort of said okay now
you go here you go here you go here well
1900.8 -> and then this goes back over to there that's kind
of ugly looking so i'm going to get rid of that
1911.68 -> all right so then this is if this was i2
and everything moved back so then this is i3
1916.88 -> and this becomes i1 and this is l2 everybody move
over 1 again i 3 omega 3 dot minus i 1 minus i 2
1933.44 -> omega 1 omega 2 equals l 3. sometimes these are
called euler's rotational equation of motion
1943.6 -> it's the same thing it's just written in a nicer
format that's kind of pretty this is just writing
1949.92 -> it in the principal axis frame so this would be
same thing euler's rotational equation of motion
1959.12 -> in principle axis frame okay this is nice what
does this tell us these are three non-linear odes
1969.68 -> for how the body fixed angular velocity components
change so you could think of it as if i were to
1980.4 -> draw not something arbitrary maybe a something
more engineered then there's our center of mass
1987.52 -> and we have an angular velocity written in the
body fixed frame and it's going to have these
1993.68 -> three components omega 1 omega 2 omega 3 they'll
be changing in time right with respect to our
2001.44 -> if this is a box then if we choose our frames
to be b1 b2 and b3 to be normal to the different
2011.92 -> faces of the box like this choose this this
is a principal axis frame for this box shape
2025.44 -> then this equation it's the most simplified
version of euler's rotational equation of
2031.84 -> motion describes how those components change right
it's it's just three odes the i's are all constant
2039.6 -> i1 i2 and i3 are constant now the moment might
be changing it might be changing with time and
2046.48 -> also position i don't know the moment could be
due to thrusters onboard a spacecraft or due to
2056.48 -> some flywheel some momentum wheel that is being
used to redirect how this is orienting it might
2064.64 -> be due to something else maybe it's due to drag
that's heavier on one side versus the other and
2070.08 -> making this thing turn so you'd have to compensate
for that but this is the most simplified thing
2076.08 -> so that from this we would get if you solve it or
used in general you will do this computationally
2083.84 -> we will look at an analytical case just to
get some insight but if you were to solve this
2090.4 -> you would solve it numerically and this
would give you omega 1 omega 2 and omega 3
2097.76 -> as a function of time so you'd start with some
initial condition of what's omega initially
2103.76 -> and then you could solve it you solve that
numerically then from rigid body kinematics
2110.96 -> you can find how the b frame is rotating
with respect to some inertial frame as
2117.12 -> a function of time so if you want that
could be given by some set of euler angles
2123.92 -> or the other thing we talked about the euler
parameters or quaternions however you want to
2129.44 -> parameterize it but that's sort of how the pieces
fit together now the simplest case that we would
2134.48 -> look at is when there's no moment in general you
know out in space there's no moment and the weird
2141.12 -> thing about rigid body dynamics is that you can
get some interesting looking dynamics when there's
2145.52 -> no applied moment no applied moment means that l1
l2 l3 equals zero and that's called the free rigid
2152.8 -> body so that's a special case free rigid body it
just means it's free of of moments or sometimes
2160.24 -> we say torx torque or moment free just means that
l1 l2 l3 are all equal to zero so the right hand
2167.76 -> side up here is equal to zero and i guess i could
bother to rewrite it we set those equal to zero
2175.04 -> and i'll put this stuff on the right hand
side then you could see okay i1 omega 1 dot
2182.4 -> equals minus i2 omega 2 omega 3 i2 omega 2 dot
equals minus i 1 minus i3 omega 3 omega 1 and
2197.44 -> the last one is i3 omega 3 dot equals minus
i2 minus i1 omega 1 omega 2. they're purely
2207.36 -> non-linear because the right hand side and they're
coupled but it can be solved this is a non-linear
2213.44 -> ode that can be solved maybe you're not thinking
if it's a single od it's three coupled ods that
2219.92 -> can be solved given initial conditions well or
so we think you know seems like that would work
2226.08 -> so at some initial time right this vector is
let's call it omega 1 0 omega 2 0 omega 3 0.
2240.24 -> it can show some interesting dynamics typically
what we want to do first is look at special cases
2248.48 -> so say what's a special case if you stare at
these odes a special case would be if two of the
2256.08 -> moments of inertia are equal so
these are further special cases
2260.48 -> and two of the moments these are principal moments
are equal because if two of them are equal then
2268.88 -> one of these coefficients is going to be zero
so it's commonly taken that i one is equal to i2
2275.28 -> and then we let i3 be the one that that's that's
different so for example if i1 is equal to i2 this
2283.04 -> case and it's the same thing if i three is equal i
want or whatever it's just for ease of explanation
2291.04 -> it's typically taken that i one is equal to
i2 this case of two equal principal moments
2296.88 -> of inertia is called a symmetric rigid body
sometimes it's called axi-symmetric but i think
2306.24 -> this can be deceiving because yes it's true for
if you have a cylinder like a perfect cylinder
2315.6 -> then yes this is an axisymmetric body and
the moment about b1 and about b2 are equal
2323.28 -> maybe we'd call those the the transverse
moments they're equal to each other and then
2330 -> i3 is not equal to that value if it was then
we'd have something that's completely symmetric
2337.68 -> these are called symmetric bodies but it
doesn't have to be a cylinder you could
2342 -> have a rectangle that has a square cross section
so rectangular block with a square cross section
2349.84 -> it's not axisymmetric but it also has i1 equal
to i2 right if we were to write this as b1
2358.16 -> this is b2 and then this third one is b3 these
moments are also equal i1 equals i2 like this
2370.64 -> and then these have kind of two different
cases and we'll just we'll just have to work
2375.04 -> on it next time one of them is we've got the i3
does not equal it if i3 is the largest moment of
2382.72 -> inertia so if it's bigger than i t that means you
have sort of a squished thing kind of like what
2388.96 -> i drew here i think this would this probably
also qualifies this big piece of lead coaster
2398.48 -> so it has a lot of mass that's perpendicular
to that axis and that is called an oblate
2406.32 -> shape that squished if i 3 is less than i t then
we call this prolate if you want it's told skinny
2417.92 -> so this is an example just from how i drew it
and what this is this is prolate so the moment
2422.64 -> of inertia about that direction is the smallest
and you'll just get you'll get different behavior
2429.6 -> and just to give you a preview with some videos
here is an oblate body out in space it's hard to
2436.4 -> see but right the angular momentum vector is that
h and it's not changing it's purple here angular
2441.92 -> velocity is a blue thing that's incredibly hard
to see but you get this idea of in general this
2448.4 -> thing is going to be looking like it's tumbling
it's not tumbling erratically in fact this motion
2453.92 -> is called precession so that is uh one type and
then here's the kind of tall skinny thing prolate
2464.4 -> and it it moves seems to be moving differently
and we will understand all that stuff later all
2472.88 -> right so that's all i have for today and so if i
put this one third of the way up i can balance it
2483.36 -> wow that's cool should make
your own triangular plate
Source: https://www.youtube.com/watch?v=z4-vL6lxR8U