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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 257 21 "Torque-free gyroscope" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 271 "We consi
der a gyroscope that has a counterweight such that the gravitational t
orque is balanced. Such a gyro will not precess. Now we push on the ax
le either gently to induce precession, or rapidly, e.g., by the blow o
f a hammer. This situation is considered in the text " }{TEXT 256 31 "
Elements of Newtonian Mechanics" }{TEXT -1 301 " by J.M. Knudsen and P
.G. Hjorth (Springer-Verlag 2000, 3rd ed.) in Example 13.7. The torque
-free symmetrical top is often the subject of somewhat abstract analys
is demonstrating how the top's symmetry axis and the angular velocity \+
vector precess with respect to the conserved angular momentum vector.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 327 "Our \+
objective is to implement the equations of motion which follow from mo
dified Euler's equations for rigid body rotation and to demonstrate th
e difference between pulse-shaped torques and gentle downward taps on \+
the gyro axis. The parameters for the gyroscope are taken from a probl
em in the text. The gyro axle is of length " }{TEXT 19 1 "l" }{TEXT 
-1 38 ", the gyro itself is a disk of radius " }{TEXT 19 1 "R" }{TEXT 
-1 11 ", and mass " }{TEXT 19 1 "M" }{TEXT -1 227 ", balanced by a cou
nterweight of equal mass at the same distance. We assume the axle to b
e substantially larger than the disk radius, such that the inertia abo
ut the axes perpendicular to the symmetry axis can be assumed to be " 
}{TEXT 19 7 "2*M*l^2" }{TEXT -1 72 ". We assume constant spin about th
e symmetry axis with angular velocity " }{TEXT 19 2 "w0" }{TEXT -1 1 "
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart; Digits:=15:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(plots): with(plottoo
ls):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords ha
s been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 43 "Warning, the name a
rrow has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "M:=2.4: l:=0.3: R:=0.1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "w0:=
200: T_rot:=6.28/w0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&T_rotG$\"0+
+++++9$!#;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "An omega=200 corres
ponds to about 33 rps, or 1812 rpm." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "I1:=2*M*l^2; I3:=M/2*R^2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#I1G$\"$K%!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
I3G$\"0++++++?\"!#;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 215 "As derive
d in the notes the Euler equations are augmented by an additional term
: the 'almost body frame' is defined by an angular velocity vector tha
t excludes the spin about the symmetry axis. To account for it an " }
{TEXT 19 5 "omega" }{TEXT -1 14 " crossed into " }{TEXT 19 6 "L_spin" 
}{TEXT -1 284 " term had to be added to the rate of change of angular \+
momentum. We will specify the torque about the 1-axis later. In the tr
ue body fram the 1- and 2-axes (perpendicular to the symmetry axis lab
eled as the 3-axis) would be spinning around experiencing periodically
 modulated torques." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "EE1:
=I1*diff(w1(t),t)-(I1-I3)*w2(t)*w3(t)+I3*w0*w2(t)=N1(t);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$EE1G/,(*&$\"$K%!\"$\"\"\"-%%diffG6$-%#w1G6#%
\"tGF2F+F+*($\"0++++++?%!#:F+-%#w2GF1F+-%#w3GF1F+!\"\"*&$\"0++++++S#!#
9F+F7F+F+-%#N1GF1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "EE2:=I
1*diff(w2(t),t)+(I1-I3)*w1(t)*w3(t)-I3*w0*w1(t)=0;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$EE2G/,(*&$\"$K%!\"$\"\"\"-%%diffG6$-%#w2G6#%\"tGF2
F+F+*($\"0++++++?%!#:F+-%#w1GF1F+-%#w3GF1F+F+*&$\"0++++++S#!#9F+F7F+!
\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "The next equations e
stablish the relationship of " }{TEXT 19 12 "[w1, w2, w3]" }{TEXT -1 
78 " to the orientation angles for the symmetry axis. The polar nutati
on angle is " }{TEXT 19 5 "theta" }{TEXT -1 40 ", and the azimuthal pr
ecession angle is " }{TEXT 19 3 "phi" }{TEXT -1 443 ". This relationsh
ip is established by the geometry: the body 1-axis is oriented such th
at rotations about it correspond to nutation in the mathematically pos
itive sense. Depending on the nutational orientation the precession fe
eds partially into rotation about the 2- and 3-axes: for the horizonta
l orientation precession has no contribution to rotation about the sym
metry axis, for the vertical orientation it adds to the spin totally. \+
Thus: " }{TEXT 19 100 "[w1, w2, w3] = [-diff(theta(t),t) , -diff(phi(t
),t)*sin(theta(t) , diff(phi(t),t)*cos(theta(t)) ]   " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "OE1:=diff(theta(t),t)=-w1(t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$OE1G/-%%diffG6$-%&thetaG6#%\"tGF,,$
-%#w1GF+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "OE2:=diff(
phi(t),t)=-w2(t)/sin(theta(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$
OE2G/-%%diffG6$-%$phiG6#%\"tGF,,$*&-%#w2GF+\"\"\"-%$sinG6#-%&thetaGF+!
\"\"F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 292 "The third component of
 the angular velocity vector needs to be eliminated from the equations
. The Euler equation of motion for the third component expresses conse
rvation of the 3-component of angular momentum, and is not used explic
itly. We solve for the two orientational degrees of freedom " }{TEXT 
19 8 "theta(t)" }{TEXT -1 5 " and " }{TEXT 19 6 "phi(t)" }{TEXT -1 31 
". Redefine the Euler equations:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "EE1:=subs(w3(t)=diff(phi(t),t)*cos(theta(t)),EE1);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$EE1G/,(*&$\"$K%!\"$\"\"\"-%%diffG6
$-%#w1G6#%\"tGF2F+F+**$\"0++++++?%!#:F+-%#w2GF1F+-F-6$-%$phiGF1F2F+-%$
cosG6#-%&thetaGF1F+!\"\"*&$\"0++++++S#!#9F+F7F+F+-%#N1GF1" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "EE2:=subs(w3(t)=diff(phi(t),t)*cos(
theta(t)),EE2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$EE2G/,(*&$\"$K%!
\"$\"\"\"-%%diffG6$-%#w2G6#%\"tGF2F+F+**$\"0++++++?%!#:F+-%#w1GF1F+-F-
6$-%$phiGF1F2F+-%$cosG6#-%&thetaGF1F+F+*&$\"0++++++S#!#9F+F7F+!\"\"\"
\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 280 "Let us first consider the
 problem of the hammer blow treated not by means of a torque, but a mo
mentum transfer. Suppose one SI unit of angular momentum is transferre
d into the nutational motion at time zero. Let us understand by how mu
ch the angular momentum changes in magnitude:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 14 "L_spin:=I3*w0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%'L_spinG$\"0++++++S#!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
10 "L_blow:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'L_blowG\"\"\"" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "theta_dot:=L_blow/I1;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*theta_dotG$\"0\"[\"[\"[\"[J#!#9" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "IC:=phi(0)=0,w2(0)=0,theta(
0)=evalf(Pi/2),w1(0)=-theta_dot;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#ICG6&/-%$phiG6#\"\"!F*/-%#w2GF)F*/-%&thetaGF)$\"0!\\zEjzq:!#9/-%#w1GF
)$!0\"[\"[\"[\"[J#F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "We select
ed no precessional motion at time zero. The gyro is torque-free:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "EE10:=subs(N1(t)=0,EE1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%EE10G/,(*&$\"$K%!\"$\"\"\"-%%diffG6
$-%#w1G6#%\"tGF2F+F+**$\"0++++++?%!#:F+-%#w2GF1F+-F-6$-%$phiGF1F2F+-%$
cosG6#-%&thetaGF1F+!\"\"*&$\"0++++++S#!#9F+F7F+F+\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "sol:=dsolve(\{EE10,EE2,OE1,OE2,IC\}
,numeric,output=listprocedure):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 49 "Theta:=eval(theta(t),sol): Phi:=eval(phi(t),sol):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plot([Theta(t)-Pi/2,Phi(t)],t=0..2,
color=[red,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 1217 458 458 
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WG6$;F(F^^m%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
30 "We have the same amplitude in " }{TEXT 19 5 "theta" }{TEXT -1 5 " \+
and " }{TEXT 19 3 "phi" }{TEXT -1 141 ". We note that the gyroscope ca
rries out simple periodic motion - something we might not have expecte
d at first from the nonlinear equations." }}{PARA 0 "" 0 "" {TEXT -1 
239 "We have chosen a massive change in angular momentum (tremendous b
low compared to the spin). Considerable nutational and precessional os
cillations are the result. What is remarkable is that the nutation is \+
about the original inclination of " }{TEXT 19 4 "Pi/2" }{TEXT -1 120 "
, the precessional oscillation is about a displaced centre angle. The \+
conserved angular momentum vector points downward." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 301 "An interesting question:
 what sets the rather long time scale for the nutational/precessional \+
oscillations? One can derive the equations of motion for the theta and
 phi angles, linearize them, and show that in the limit of small-ampli
tude oscillations harmonic solutions exist with circular frequency " }
{TEXT 19 14 "Omega=I3*w0/I1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Omega:=I3*w0/I1; \+
T_osc:=evalf(2*Pi/Omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&OmegaG
$\"0cbbbbbb&!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&T_oscG$\"0K#HbL(
48\"!#9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 12 "Exercise 1: " }}
{PARA 0 "" 0 "" {TEXT -1 155 "Explore the motion as a function of the \+
strength of the momentum transfer to the gyro axle (imparted angular m
omentum change). Find out for which range of " }{TEXT 19 5 "theta" }
{TEXT -1 206 " oscillations the linearized approximation is valid, i.e
., the period follows the above formula. Also explore initial conditio
ns for which the excursion amplitude in the nutation is larger than gi
ven above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
263 11 "Exercise 2:" }}{PARA 0 "" 0 "" {TEXT -1 189 "Explore the chara
cter of the motion when the gyro is not started in the horizontal orie
ntation. This should result in motion in the nonlinear regime. How wou
ld you characterize this motion?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 121 "Can we depict the relation of the three \+
vectors: angular momentum, angular velocity, symmetry axis as a functi
on of time?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 7 "L_spin;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0++++
++S#!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "N:=300: for i f
rom 1 to N do: t:=0.03*i: Prec:=vector([l*sin(Theta(t))*cos(Phi(t)),l*
sin(Theta(t))*sin(Phi(t)),l*cos(Theta(t))]) ; ll[i] := arrow(vector([0
,0,0]), Prec, .1, .4, .1):\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 117 "i:='i': display3d([seq(ll[i],i=1..N)], color=red,orientation=
[66,48],axes=boxed,insequence=true,scaling=constrained);" }}{PARA 13 "
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{PARA 0 "" 0 "" {TEXT -1 70 "We change the initial conditions to imply
 no motion without the pulse:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 50 "IC:=phi(0)=0,w2(0)=0,theta(0)=evalf(Pi/2),w1(0)=0;" }}{PARA 11 "
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63 "sol:=dsolve(\{EE1,EE2,OE1,OE2,IC\},numeric,output=listprocedure):
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l): Phi:=eval(phi(t),sol):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
68 "plot([Theta(t)-Pi/2,Phi(t)],t=0..2,color=[red,blue],numpoints=2000
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1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 162 "The pulse results in action very much along th
e lines of what we did through initial conditions above. The motion se
ts in at a time when the pulse gains strength." }}{PARA 0 "" 0 "" 
{TEXT -1 58 "On a technical note: if we delay the pulse to later time,
 " }{TEXT 19 15 "dsolve[numeric]" }{TEXT -1 85 " has problems to conne
ct the initial condition to a proper solution due to round-off." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "The ampli
tude of oscillation is much less than before. What is the implied angu
lar momentum change?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eva
lf(Int(N1(t),t=0..0.4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!0V#QWK#>
p&!#;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Now repeat the procedur
e for a pulse that is longer-lived, and has a rise time that does not \+
resonate." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "N1:=t->-exp(-(
(t-1)/(20*T_rot))^4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#N1Gf*6#%\"
tG6\"6$%)operatorG%&arrowGF(,$-%$expG6#,$*&#\"\"\"\"'++;F3*&,&9$F3F3!
\"\"\"\"%%&T_rotG!\"%F3F8F8F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "t:='t': plot(N1(t),t=0..2,numpoints=1000);" }}{PARA 
13 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 1177 383 383 
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0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2
" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 489 "The behaviour is qualitativ
ely different now: rather than exciting resonant oscillations in the g
yroscope the rise and fall times of the pulse are so long that we have
 almost adiabatic motion. We can compare the situation to a mass-sprin
g system: excitation on short time scales comparable to the natural fr
equency would lead to oscillations, while slow pushes would just move \+
the mass from one equilibrium position (without force applied) to a ne
w one (with force applied) and vice versa." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "The pulsed torque is of longer \+
duration and leads to more angular momentum change:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 25 "evalf(Int(N1(t),t=0..1));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$!0ZcG$*Q;p&!#:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 200 "In this case the angular momentum vector precessed to a new lo
cation via excursion to nutation angles below the horizontal and back.
 The above graph shows clearly that precession follows the nutation." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 502 "What i
s curious about the above discussion: the natural oscillations of the \+
gyroscope are slow - the period is about 1.02 seconds (it is somewhat \+
shorter than in the linearized limit). What controls whether the gyro \+
responds by strong oscillations on this time scale or by precession wi
th small oscillations superimposed is not resonance with this frequenc
y, but the rise time of the pulse compared to the rotation period mult
iplied by some number. This number involves the ratio of moments of in
ertia " }{TEXT 19 5 "I3/I1" }{TEXT -1 53 ". For our example this numbe
r is small (almost 1/40)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT 259 11 "Exercise 3:" }}{PARA 0 "" 0 "" {TEXT -1 92 "Explo
re the behavior of the tapped gyroscope by changing the parameters of \+
the torque pulse." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 260 22 "Intellectual exercise:" }{TEXT -1 84 " compare the prese
nt approach to the direct solution of the equations of motion for " }
{TEXT 19 8 "theta(t)" }{TEXT -1 5 " and " }{TEXT 19 6 "phi(t)" }{TEXT 
-1 19 " as carried out in " }{TEXT 19 8 "Gyro.mws" }{TEXT -1 1 " " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 185 "N:=150: for i from 1 to N do: t:=0.02*i: Prec:=vec
tor([l*sin(Theta(t))*cos(Phi(t)),l*sin(Theta(t))*sin(Phi(t)),l*cos(The
ta(t))]) ; ll[i] := arrow(vector([0,0,0]), Prec, .1, .4, .1):\nod:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "i:='i': display3d([seq(ll[
i],i=1..N)], color=red,orientation=[35,50],axes=boxed,insequence=true,
scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT3D 1120 424 424 
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35.000000 1 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 13 "Mini-
project:" }{TEXT -1 174 " demonstrate the time evolution of the angula
r momentum vector. Show how close its evolution comes to the naive tre
atment by the angular momentum theorem (steady precession)." }}}
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