---
title: Gyroscopic Precession
module: Rotation
moduleNumber: 5
lessonNumber: 6
order: 506
summary: >
  A spinning top leans over but does not fall — it swings its axis in a slow
  horizontal circle instead. The paradox dissolves once torque is read as the rate
  of change of a vector: gravity's torque is perpendicular to the spin angular
  momentum, so it turns $\vec L$ rather than toppling it. We derive the steady
  precession rate $\Omega\simeq Mgr/(I_s\omega_s)$ in the fast-top limit, state the
  assumptions it leans on — dominant spin, slow tilt, negligible bearing torque —
  and read nutation, support motion, and a decaying spin as the ways real
  gyroscopes depart from it.
topics: [Rotation]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 10 — Angular Momentum; §§10-1–10-4"
---

## Angular momentum and steady precession

A spinning symmetric body carries angular momentum approximately along its spin
axis. Its leading spin contribution has symmetry-axis moment of inertia $I_s$ and
rapid spin angular speed $\omega_s$:

$$
\vec L\simeq I_s\omega_s\,\hat e_s.
$$

An external torque changes angular momentum according to

$$
\vec\tau=\frac{\d\vec L}{\d t}.
$$

When the torque is perpendicular to $\vec L$, the leading effect is a change in
direction rather than a change in spin magnitude. Gravity acting on a supported
spinning top produces such a torque about the pivot. The top's centre of mass lies a
distance $r$ from the pivot, so gravitational torque magnitude is

$$
\tau=Mgr\sin\theta,
$$

where $\theta$ is the angle between the spin axis and vertical. The torque is
horizontal for an ordinary tilted top, while the spin angular momentum lies along
the tilted axis. The perpendicular torque produces a gradual rotation of the axis
around the vertical, called precession.

$$
% caption: A supported spinning top. Gravity at the centre of mass produces a torque about the pivot, perpendicular to the dominant spin angular momentum $\vec L$; the small change $\d\vec L=\vec\tau\,\d t$ turns the spin axis around the vertical rather than dropping it.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black,thick] (1.1,0.45)--(1.1,3.0) node[above] {vertical};
\draw[fill=white,draw=black] (1.1,0.45) circle (2pt);
\node[below] at (1.1,0.4) {pivot};
\draw[very thick] (1.1,0.45)--(3.55,2.35);
\draw[thick,fill=black!5] (3.55,2.35) circle (0.34);
\node[black] at (2.95,2.72) {CM};
\draw[->,black,very thick] (3.55,2.35)--(3.55,0.82) node[midway,right] {$Mg$};
\draw[->,acc,very thick] (2.72,1.7)--(4.15,2.82) node[right] {$L$};
\end{tikzpicture}
$$

The vector equation fixes the direction. Draw $\vec L$, construct
$\vec\tau$ from the applied force and lever arm, then add
$\d\vec L=\vec\tau\,\d t$. The new angular-momentum vector is
$\vec L+\d\vec L$, whose tip moves in the torque direction. Spin reversal
reverses $\vec L$ and therefore reverses the observed precession direction under
the same gravitational torque.

**Steady precession of a fast symmetric top.**

In steady precession, the spin axis maintains a nearly constant tilt angle while it
rotates around the vertical with precession rate $\Omega$. The tip of
$\vec L$ traces a horizontal circle of radius $L\sin\theta$, so its rate of
change has magnitude

$$
\left|\frac{\d\vec L}{\d t}\right|=\Omega L\sin\theta.
$$

Equating this to the gravitational torque gives, for the rapid-spin approximation,

$$
\Omega=\frac{Mgr}{I_s\omega_s}.
$$

The tilt factor cancels only after both torque and angular-momentum turning rate
have been written for the same geometry. Faster spin therefore gives slower steady
precession, while a larger centre-of-mass lever arm or larger mass gives faster
precession. The formula requires a dominant spin angular momentum and a nearly fixed
tilt angle. It does not apply unchanged to a top released from rest, a rotor with
substantial transverse angular velocity, or a support that exerts additional torque.

> **Worked example.** A bicycle wheel of moment of inertia
> $I_s=0.20\ \mathrm{kg\,m^2}$ (about its axle) spins at $\omega_s=100\ \mathrm{rad\,s^{-1}}$
> and is suspended from one end of its axle, a distance $r=0.25\ \mathrm m$ from the
> support, with the axle horizontal. Its spin angular momentum is
> $L=I_s\omega_s=20\ \mathrm{kg\,m^2\,s^{-1}}$ and the gravitational torque about the
> support is $\tau=Mgr=(2.0)(9.81)(0.25)=4.91\ \mathrm{N\,m}$ for a
> $2.0\ \mathrm{kg}$ wheel. The steady precession rate is
>
> $$
> \Omega=\frac{\tau}{L}=\frac{Mgr}{I_s\omega_s}=\frac{4.91}{20}=0.245\ \mathrm{rad\,s^{-1}},
> $$
>
> a precession period of $2\pi/\Omega=25.6\ \mathrm s$. With a horizontal axle
> ($\theta=90^\circ$) the tilt factor is one; the wheel circles slowly instead of
> falling because gravity's torque turns $\vec L$ rather than dropping it.

$$
% caption: Angular-momentum geometry for steady precession. As the tip of $\vec L$ sweeps a horizontal circle of radius $L\sin\theta$, its rate of change $\d\vec L/\d t$ points along the gravitational torque with magnitude $\Omega L\sin\theta$.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.55,0.42)--(0.55,3.18) node[above] {vertical};
\draw[->,acc,very thick] (0.55,0.42)--(2.75,2.45) node[above left] {$L$};
\draw[black,dashed] (0.55,2.45)--(2.75,2.45);
\node[black,below] at (1.6,2.45) {$L_h$};
\draw[->,black,thick] (2.75,2.45)--(3.75,2.45) node[right] {$dL$};
\draw[thick] ([shift={(0.55,0.42)}]90:0.85) arc (90:43:0.85);
\node[black] at (1.08,1.42) {tilt};
\end{tikzpicture}
$$

The approximation can be tested dimensionally. $Mgr$ has units of torque and
$I_s\omega_s$ has units of angular momentum, giving $\Omega$ in inverse seconds.
A calculation that produces a precession rate proportional to spin speed has
reversed the angular-momentum dependence. A measured top may precess faster or
slower than this estimate when its spin decays, its pivot friction is appreciable,
or its axis nutates through a large range of angles.

## Nutation, energy, and body symmetry

Nutation is oscillation of the tilt angle superposed on precession. A top released
with an initial axis orientation and angular velocity that do not match steady
precession has angular momentum components that change the tilt as well as its
azimuth. The axis can bob above and below its mean tilt while circling the vertical.
Friction, air drag, and energy dissipation usually reduce this motion over time, but
they also change the spin magnitude and therefore the precession rate.

The angular-momentum equation remains valid through nutation. The fast-top formula
does not describe the full motion because $\vec L$ need not remain aligned with
the body symmetry axis and the torque can change the polar motion substantially.
Full rigid-body treatment uses principal moments of inertia and the components of
angular velocity in the body frame. The steady-precession formula is a controlled
limit within that broader dynamics, not a generic law for every rotating object.

$$
% caption: Trace of a nutating spin-axis tip projected onto a plane normal to the vertical. A steady top follows one circle at fixed tilt; the inner and outer excursions are the tilt oscillation superposed on azimuthal precession.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.35,1.65)--(5.45,1.65) node[right] {$x$};
\draw[->,black] (2.9,0.25)--(2.9,3.1) node[above] {$y$};
\draw[acc,thick] plot[domain=0:360,samples=220]
  ({2.9+(1.58+0.24*sin(6*\x))*cos(\x)},{1.65+(0.98+0.15*sin(6*\x))*sin(\x)});
\draw[fill=white,draw=black] (2.9,1.65) circle (1.8pt);
\node[black,below] at (3.5,0.42) {axis-tip trace};
\end{tikzpicture}
$$

Measurement distinguishes steady precession from a short interval of nearly
constant tilt. Record the spin rate, tilt angle, precession period, pivot geometry,
and the time span over which the rate is fitted. A video-derived axis trace needs a
calibrated camera axis and a stated reference frame; perspective projection can make
a circular trace appear elliptical. Comparing the measured $\Omega$ with
$Mgr/(I_s\omega_s)$ is meaningful only after the rotor's symmetry-axis moment and
centre-of-mass distance have been measured for the same support configuration.

**Angular-velocity components and body symmetry.**

A precessing top has more than one angular-velocity component. In the simplest
description, the body spins rapidly about its symmetry axis at rate $\omega_s$ while
that axis rotates about vertical at rate $\Omega$. The angular velocity can be
written schematically as

$$
\vec\omega=\omega_s\hat e_s+
\Omega\hat z.
$$

Angular momentum is not generally parallel to this total angular velocity. For a
body with principal moments $I_s$ about the symmetry axis and $I_t$ about a
transverse axis, the components have different inertia factors. The rapid-spin
approximation takes $I_s\omega_s$ to dominate. A wheel at large spin rate commonly
satisfies this condition. It must be checked when the precession rate is comparable
with spin or when the body is far from axial symmetry.

A laboratory top therefore needs a component-by-component angular-momentum model.
A disk spinning slowly can show pronounced tilt oscillations because transverse
angular momentum is no longer a small correction. Attached masses can give a rotor
different transverse moments and motion outside the symmetric-top picture. Its
measured axis trace then depends on the inertia tensor, initial angular velocity,
and gravitational torque.

The vector equation $\d\vec L/dt=\vec\tau$ remains the primary check.
Any proposed precession model must predict an angular-momentum change parallel to
the applied torque. A calculation based only on the direction of body-axis motion
can give the wrong result when $\vec L$ and the symmetry axis are appreciably
misaligned.

**Energy, tilt, and the two steady-precession branches.**

Steady precession is selected by both torque balance and angular-momentum geometry.
With a fixed pivot, the gravitational potential energy of a symmetric top is

$$
U=Mgr\cos\theta
$$

when $\theta$ is measured from upward vertical. The rotational kinetic energy
contains spin, precession, and cross terms that depend on the chosen Euler-angle
description. Conservation of energy constrains the allowed tilt motion when the
pivot is ideal, while torque determines how angular momentum moves through that
allowed region.

For specified spin and tilt, the full symmetric-top equations can admit two steady
precession rates. The rapid branch approaches $Mgr/(I_s\omega_s)$ at high spin. A
second branch can have much faster precession and a substantial precession angular
velocity contribution to angular momentum. It is often inaccessible from a gently
released fast top because its initial energy and angular momentum do not match that
branch. Treating every observed precession as the slow branch discards this initial-
condition constraint.

An ideal energy calculation excludes pivot friction and air drag. Those effects
reduce mechanical energy and spin angular momentum, causing a slowly changing tilt
and precession rate. A long video record should therefore be divided into short
intervals and fitted locally instead of assigning one constant $\Omega$ to a rotor
whose spin visibly decays.

## Precession measurement and spin decay

The fast-top prediction gives an inverse relation between precession rate and spin
rate. For fixed $M$, $r$, and $I_s$,

$$
\Omega\omega_s=\frac{Mgr}{I_s}.
$$

Plotting measured $\Omega$ against $1/\omega_s$ should give an approximately
straight relation when the slow-precession assumptions hold. The slope estimates
$Mgr/I_s$. This fit requires synchronized measurements: a precession period averaged
over several turns must be paired with the spin rate over the same time interval.
Using an initial spin rate with a later precession period biases the inferred slope.

> **Worked example.** A top precesses at $\Omega_1=0.60\ \mathrm{rad\,s^{-1}}$
> while spinning at $\omega_1=200\ \mathrm{rad\,s^{-1}}$. Bearing friction lowers
> the spin to $\omega_2=120\ \mathrm{rad\,s^{-1}}$, with $M$, $r$, and $I_s$
> unchanged. Since $\Omega\propto1/\omega_s$,
>
> $$
> \Omega_2=\Omega_1\frac{\omega_1}{\omega_2}
> =0.60\times\frac{200}{120}=1.0\ \mathrm{rad\,s^{-1}}.
> $$
>
> The precession speeds up as the spin decays. This is the visible sign that a
> slowing top eventually leaves the fast-top regime, where nutation and a falling
> tilt take over.

$$
% caption: The fast-top prediction $\Omega=Mgr/(I_s\omega_s)$ makes the steady precession rate proportional to inverse spin speed; departures from the line signal nutation, drag, pivot torque, or a transverse angular-momentum component.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.35,0.35)--(5.7,0.35) node[right] {inverse spin};
\draw[->,black] (0.35,0.35)--(0.35,3.1) node[above] {precession rate};
\draw[acc,thick] (0.55,0.6)--(5.3,2.72);
\foreach \x/\y in {1.1/0.82,2.35/1.38,3.7/2.0,4.85/2.52} {\draw[fill=white,draw=black] (\x,\y) circle (1.8pt);}
\node[black,above left] at (4.85,2.52) {data};
\end{tikzpicture}
$$

Measurement uncertainty enters through all four quantities. The centre-of-mass
distance must be measured from the pivot to the mass centre, not to the visible edge
of the rotor. The symmetry-axis inertia can be calculated from geometry or measured
by a torsional or pendulum method, each with its own axis-alignment error. Tilt angle
is needed to test steady motion even though it cancels from the leading slow-branch
formula. A reported agreement should include the range of spin rates, tilt range,
precession-period fit, and whether nutation amplitude was resolved.

## Applied torques and control systems

Torque direction remains the governing issue in gimballed instruments and vehicle
attitude systems. A free rotor retains an angular-momentum direction in an inertial
frame when external torque is negligible. The housing, support rings, and vehicle
can rotate around that direction; the rotor axis changes only by the angular impulse
transferred through bearings, cables, magnetic fields, or other external
interactions. A gimbal constrains selected mechanical rotation axes. A single ring
permits rotation about one axis, and two orthogonal rings permit two independent
rotations. The remaining inability to represent every orientation with independent
small rotations is a kinematic limitation of the mechanism, often called gimbal
lock in three-axis attitude systems.

In a ring-mounted rotor analysis, draw the wheel angular momentum first and then
add each bearing torque about the wheel centre. A torque parallel to the wheel axis
changes wheel speed. A perpendicular torque turns the angular-momentum vector and
appears as a force demand on the gimbal structure. When the outer ring reaches a
mechanical stop, a new constraint torque can be applied abruptly; the motion then
differs from the free-gimbal approximation. Friction at a bearing produces a torque
opposite the relative rotation and gradually transfers angular momentum between the
wheel and its frame. Cable stiffness and encoder leads can supply additional small
torques that are visible in precision instruments as drift or a shifted precession
rate.

> **Worked example.** A flywheel of moment of inertia $I_s=0.50\ \mathrm{kg\,m^2}$
> spins at $\omega_s=300\ \mathrm{rad\,s^{-1}}$, so its angular momentum is
> $L=I_s\omega_s=150\ \mathrm{kg\,m^2\,s^{-1}}$. Its housing is forced to yaw at
> $\Omega=0.40\ \mathrm{rad\,s^{-1}}$ about an axis perpendicular to the spin. To
> turn $\vec L$ at this rate the bearings must supply
>
> $$
> \tau=\Omega L=(0.40)(150)=60\ \mathrm{N\,m},
> $$
>
> and by reaction the rotor pushes back on the bearings with the same
> $60\ \mathrm{N\,m}$, directed perpendicular to both the spin and yaw axes. This
> gyroscopic reaction is why a fast rotor resists being reoriented and loads its
> mounts whenever the vehicle carrying it turns.

$$
% caption: A two-ring gimbal carries a rotor with a large wheel-axis angular momentum. Bearing torques are resolved about the permitted ring axes; a ring stop or cable tension adds an external torque to the rotor-plus-gimbal assembly.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,thick] (0.55,0.4) rectangle (6.3,3.25);
\draw[black,very thick] (2.55,1.82) ellipse (1.05 and 1.55);
\draw[black,very thick] (2.55,1.82) ellipse (1.78 and 0.8);
\draw[acc,thick,fill=acc!8] (2.55,1.82) circle (0.5);
\draw[->,black] ([shift={(2.55,1.82)}]120:0.3) arc (120:-150:0.3);
\draw[->,acc,very thick] (3.05,1.82)--(4.6,1.82) node[right] {wheel axis};
\draw[->,black,thick] (5.35,2.95)--(5.35,2.3) node[midway,right] {torque};
\node[acc,below] at (2.55,1.15) {rotor};
\node[black] at (1.2,0.68) {outer ring};
\node[black] at (4.1,0.85) {inner ring};
\end{tikzpicture}
$$

A reaction wheel uses the same momentum exchange without a gimbal. An electric
motor accelerates a flywheel fixed to a spacecraft body. The motor applies equal
and-opposite torques: increasing wheel angular momentum along one body axis turns
the spacecraft in the opposite sense. With negligible external torque,
$\Delta\vec L_{\rm wheel}+\Delta\vec L_{\rm body}=0$. The manoeuvre changes
attitude while conserving the total angular momentum of the wheel-spacecraft
system. Desaturation requires an external torque because the wheel has a finite
speed range. Thrusters, magnetic torque rods interacting with a planetary magnetic
field, or gravity-gradient torque can remove accumulated wheel momentum before the
wheel reaches its speed limit. Wheel saturation is a control limit rather than a
loss of angular-momentum conservation.

$$
% caption: A reaction wheel exchanges angular momentum with its spacecraft body. Motor torque increases the wheel momentum one way and rotates the body the other; external actuators unload stored wheel momentum before the wheel reaches its speed limit.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,thick,fill=black!4] (0.6,0.55) rectangle (5.45,3.05);
\draw[acc,very thick] (2.6,1.75) circle (0.6);
\draw[acc,thick] (2.6,1.75) circle (0.2);
\draw[->,acc,very thick] ([shift={(2.6,1.75)}]40:0.82) arc (40:315:0.82);
\draw[->,black,very thick] (3.4,1.75)--(4.8,1.75) node[right] {wheel torque};
\draw[->,black,very thick] (1.6,1.75)--(0.9,1.75);
\node[black,above left] at (1.75,1.78) {body};
\node[acc,above] at (2.6,2.78) {reaction wheel};
\node[black,below] at (3.2,0.5) {spacecraft body};
\end{tikzpicture}
$$

Spacecraft attitude estimates combine wheel-speed data with star trackers, Sun
sensors, or inertial sensors. Wheel speed alone measures stored angular momentum
only after the wheel inertia and spin-axis alignment have been calibrated. A small
misalignment maps wheel torque into more than one body axis. Structural flexibility,
fuel motion, and vibration can also separate the sensor-measured body attitude from
the rigid-body model. Controller commands must account for the inertia tensor, the
available wheel torque, and the time over which the command is applied. Treating the
vehicle as a rigid body with diagonal inertia is appropriate only when cross-axis
coupling and flexible modes remain below the required pointing accuracy.

Measurement resolution becomes restrictive near slow rotations. A gyroscope
reports angular rate relative to its sensing frame, while an attitude estimate is an
orientation relative to a specified inertial or celestial reference. Bias drift
integrates into an attitude error. Finite encoder resolution can mask small gimbal
motion, and sampling too slowly can alias a nutation or wheel imbalance frequency.
Ground tests must separate support torque from the applied control torque; a vehicle
on a suspension, air bearing, or test stand is not torque-free. Comparing measured
body angular momentum with the sum of wheel momentum and known external impulses
directly checks the model's angular-momentum balance.

## Stability, support motion, and impulses

Small nutation is a perturbation of a steady-precession state. Write the tilt as a
steady value plus a small displacement and retain the first-order terms in the
rigid-body equations. The resulting coupled tilt and azimuth equations have
oscillatory solutions when the steady state is dynamically stable. Their frequencies
depend on the spin angular momentum, the transverse and symmetry-axis moments of
inertia, the centre-of-mass offset, and the selected precession branch. A single
formula based only on $Mgr/(I_s\omega_s)$ cannot supply the nutation frequency: that
expression describes the slow steady-precession rate after transverse motion has
been neglected. The measured nutation period is therefore a separate observable,
not an error term to discard when fitting the precession period.

During an ideal nutation cycle, gravitational potential energy and rotational energy
associated with transverse motion exchange while total mechanical energy remains
constant. At a tilt turning point, the instantaneous tilt rate vanishes and the
effective tilt energy is locally potential-like. Between turning points, the tilt
rate is nonzero and the corresponding transverse kinetic contribution increases.
The detailed partition depends on the chosen generalized coordinates, but the
turning-point sequence and bounded tilt excursion are directly observable. A
trajectory with constant mean tilt and a finite repeating excursion indicates a
stable oscillatory mode within the tested range of amplitudes.

Damping changes both the amplitude and the reference state about which the top
oscillates. Pivot friction and air drag remove mechanical energy, while bearing
friction also reduces spin angular momentum. A diminishing nutation envelope can
therefore coexist with a changing mean tilt and a changing precession rate. It is
incorrect to infer stability solely from decreasing amplitude over a short record:
strong dissipation can conceal an unstable conservative mode, and a torque from the
support can force a bounded response at one operating point. Separate free-decay
tests from driven tests, and report the source of damping. In a driven test, record
the applied torque spectrum because a periodic support disturbance can excite a
nutation frequency even when the unforced motion would decay.

Instability is identified by the response to a reproducible small perturbation. A
stable state returns to a bounded nutation band or settles toward a nearby
steady-precession state. An unstable state shows increasing tilt excursion, a drift
to a different branch, loss of contact at the pivot, or a transition into large-angle
motion. Low spin can remove the separation between rapid spin and slower transverse
motion that underlies the fast-top approximation. Large tilt, significant rotor
asymmetry, a moving support, and transverse angular momentum of the same order as
spin angular momentum require the full Euler-angle or body-component equations.

A practical stability protocol begins with a balanced rotor and a measured inertia
tensor or an uncertainty range for it. Set a spin rate, tilt, and precession state;
apply the same small angular displacement or calibrated torque pulse on each trial;
then record spin rate, tilt, azimuth, and support force at a sampling rate well above
the expected nutation frequency. Fit the mean precession and the nutation envelope
over separate windows. Repeat across spin rates and perturbation amplitudes. The
reported boundary must state the support geometry, damping condition, sensor frame,
and observation duration. A result from a short, nearly steady video interval does
not establish long-time stability, particularly when spin decay moves the system
through the tested operating range. Repeatability establishes the measured boundary.

**Precession under changing torque and support motion.**

A torque impulse changes angular momentum by its time integral,
$\Delta\vec L=\int\vec\tau\,\d t$. A short lateral push on the support,
an actuator pulse, or a brief cable contact can therefore redirect the rotor axis
even when its duration is far shorter than a precession period. The post-impulse
motion starts from the new angular momentum and the original orientation at the end
of the pulse. It generally contains both a changed mean precession and nutation.
Replacing a time-varying torque by its average is valid for the angular-momentum
change over the interval, but it does not reproduce the orientation history during
the pulse. That distinction matters when a pointing system must remain inside a
small angular tolerance while a control torque is applied.

Resolve applied torque in two complementary ways. In a laboratory frame, vertical
and horizontal components identify the force directions and their lever arms. In a
frame based on the instantaneous angular momentum, the parallel component changes
the spin-angular-momentum magnitude, while the perpendicular component changes its
direction. A horizontal torque can have both parallel and perpendicular parts when
the rotor axis is tilted. A vertical torque can also redirect the axis if the spin
axis is not vertical. Naming a torque only by its laboratory direction is therefore
insufficient for predicting the immediate gyroscopic response. The vector product
with the actual lever arm and the current angular-momentum direction fixes the
response.

$$
% caption: A torque impulse gives a finite change $\Delta\vec L=\int\vec\tau\,\d t$. Its component along the initial $\vec L$ changes spin magnitude; the sideways component turns the axis, and the following free motion can include a shifted precession and nutation.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.45,0.45)--(0.45,3.15) node[above] {vertical};
\draw[->,black] (0.45,0.45)--(5.65,0.45) node[right] {side direction};
\draw[->,acc,very thick] (0.45,0.45)--(2.7,2.15) node[above left] {initial L};
\draw[->,acc,very thick] (0.45,0.45)--(3.6,2.45) node[right] {new L};
\draw[->,black,thick] (2.7,2.15)--(3.6,2.45) node[midway,above] {impulse};
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\draw[black,dashed] (3.6,2.45)--(3.6,0.45);
\end{tikzpicture}
$$

Support acceleration modifies the torque that a fixed-pivot model attributes to
gravity. For a support whose frame translates with acceleration $\vec a_p$ and
does not rotate appreciably during the interval, the centre of mass experiences the
effective load $M(\vec g-\vec a_p)$ in that frame. The pivot torque is then
$\vec r\times M(\vec g-\vec a_p)$, together with bearing and actuator
torques. A horizontal support acceleration creates a horizontal effective load and
can change the precession direction or tilt response. A vertical acceleration
changes the apparent load magnitude; rapid vertical motion can reduce pivot contact
force or produce intermittent contact. This support-frame description requires the
frame acceleration to be measured or estimated. If the support also rotates, the
frame introduces additional inertial terms and the simple translating-frame result
is no longer sufficient.

Transient data must be compared with a model over matched time windows. Record the
support acceleration, applied torque, wheel or rotor speed, tilt, azimuth, and any
contact-force signal with a common clock. Integrating the measured external torque
gives a predicted change in angular momentum; the orientation record tests how that
change was distributed between precession, nutation, and spin. Finite sensor
bandwidth smooths short torque pulses, while finite actuator bandwidth delays the
commanded correction. An encoder attached to a moving support measures a relative
angle unless its reference motion is removed. Camera tracking has similar limits:
exposure time averages rapid motion and perspective changes the apparent tilt.

Torque reconstruction also requires the force application point and the coordinate
origin used for each moment arm. A force measured at the support can contain a
structural reaction unrelated to the rotor torque. Align force, acceleration, and
orientation records before integrating; a fixed timing offset converts a sharp
control pulse into an erroneous angular-momentum estimate.

Tracking fails first when the response contains frequencies outside the sensing or
control bandwidth, when the applied torque saturates an actuator, or when the rotor
leaves the range in which the fast-top approximation separates spin from transverse
motion. Large impulses can make the symmetry axis and angular momentum appreciably
misaligned. A model that updates only $\Omega=Mgr/(I_s\omega_s)$ after each event
then omits the transient angular-velocity components. Use the full torque history
and rigid-body equations whenever the measured axis turn occurs on the same time
scale as the spin, nutation, or support motion.

**Impulsive torques, vibration, and rotor response.**

An impulsive torque is short compared with the rotor response time, but it need not
be small. Its first effect is the angular-momentum increment
$\Delta\vec L=\int\vec\tau\,\d t$. The subsequent axis motion depends on
which modes that increment excites. A nearly rigid, fast rotor commonly shows a
change in mean precession together with nutation. A rotor carried by a compliant
shaft, gimbal, or test stand can also excite bending, bearing, and support modes.
Those modes add small rotations or translations to the measured axis trace. Their
presence does not alter angular-momentum conservation, but it changes the mapping
from a torque pulse to the observed sensor signal.

The pulse duration sets part of the mode selection. A broad torque pulse has little
spectral content at high rate and may leave flexible modes weakly excited. A sharp
pulse has components over a wider rate range and can excite a support resonance even
when its total angular impulse is modest. The same impulse applied through a force
at different distances from the pivot produces different torque. Contact compliance
also spreads an apparent impact over a finite interval. Force and acceleration data
must therefore be retained at their actual sampling rate before assigning an
impulse from a peak reading.

Vibration is diagnosed from phase and rate content as well as amplitude. A static
mass imbalance usually produces a force component near the spin rate. Shaft
misalignment, bearing defects, and nonlinear contact can add harmonics or sidebands.
A support mode appears at a rate set mainly by the support stiffness and effective
mass; its peak can remain nearly fixed while spin rate changes. Nutation is different:
its rate changes with rotor spin, geometry, and torque state. Sweeping the spin rate
while recording a fixed-frame accelerometer, a body-rate sensor, and the rotor phase
separates these contributions. The sensor axes and mounting stiffness
must be recorded because a flexible mount can create a peak absent from the rotor.

Rotor response measurements need a defined disturbance and an uncertainty model. A
hammer or actuator test requires the contact location, force direction, pulse
duration, and time base shared by the force and motion channels. Repeat the test at
several spin rates and at rest. A peak present only while spinning may arise from
imbalance, gyroscopic coupling, or a spin-shifted structural mode. A peak present
at rest is associated with the support or sensor assembly unless the rotor is driven.
Compare phase as well as amplitude; a resonance often shows a rapid phase change
across its peak.

Rigid-top formulas have clear limits in this setting. They omit shaft bending,
distributed rotor mass, bearing clearance, housing flexure, and finite sensor
bandwidth. They describe the low-rate change in total angular momentum when those
effects are small relative to the required pointing or torque accuracy. Near a
flexible resonance, a single-axis precession model can fit the average motion while
missing the force that controls fatigue, contact loss, or tracking error. Use a
multibody or flexible-rotor model when the vibration amplitude, rate content, or
support deformation is comparable with the resolved gyroscopic response.

## Spin transfer and rotor response

Motor torque changes the rotor angular momentum along the motor shaft. For a rotor
with fixed symmetry axis and nearly constant $I_s$,
$\tau_{\rm motor}=I_s\,\d\omega_s/\d t$ after bearing drag and other shaft torques are
included with their signs. The motor applies an equal and opposite torque to its
stator. If the stator is bolted to a housing, the housing receives the reaction
torque during spin-up. A rotor-only free-body diagram contains motor torque as an
external interaction. A rotor-plus-housing diagram contains that motor interaction
internally; the support torque, air drag, cable forces, and gravity moments then
control the angular momentum of the combined assembly.

Spin-up changes the magnitude of the dominant spin angular momentum and can change
the precession response even if the applied motor torque is nearly parallel to the
spin axis. For a supported top, a larger $I_s\omega_s$ reduces the slow-branch
precession rate when the gravitational torque and tilt remain approximately fixed.
During a rapid ramp, those conditions are not automatically satisfied: the housing
can twist, the axis can nutate, and the motor torque can have a transverse component
from shaft misalignment. The current value of $\omega_s$ must be paired with the
precession record from the same interval. Applying a steady-precession formula to a
spin ramp can confuse a transient axis motion with a changed equilibrium rate.

Spin-down has several distinct paths. With the motor unpowered, bearing drag and
air drag transfer angular momentum to the housing and surrounding air. With active
electrical braking, the motor torque reverses and transfers rotor angular momentum
to the stator; an electrical load or energy-storage circuit receives the associated
energy. Regenerative braking does not remove the mechanical reaction torque from the
housing. A free rotor-housing assembly can rotate in response unless an external
support torque constrains it. The energy destination and the angular-momentum
destination must both be stated; they need not be the same component.

Measure spin rate with an encoder or optical marker and establish the sign convention
before calculating $\d\omega_s/\d t$. A motor-current record gives an independent
torque estimate only after the torque constant, current offset, and controller
limits have been calibrated. Bearing loss can be determined from a coast-down test
at the same temperature and support configuration. The angular-momentum cross-check
compares $I_s[\omega_s(t_b)-\omega_s(t_a)]$ with the time integral of the net shaft
torque over the identical interval. For the combined rotor-housing assembly, add the
measured support impulse and any external gravitational moment before comparing the
total angular-momentum change.

During a programmed ramp, record the command as well as the measured current. A
current controller can limit torque without maintaining the requested acceleration.
Temperature changes winding resistance and bearing loss, so two ramps with the same
command can have different slopes. Reversing the commanded current tests the sign
convention: rotor deceleration, housing reaction, and the integrated shaft torque
must reverse together under an unchanged coordinate convention.

Several measurement errors have characteristic signatures. Encoder quantization
amplifies numerical differentiation at low speed. A current sensor can report motor
current during saturation without establishing the torque delivered to the rotor.
An assumed constant inertia fails when a test rig couples an added wheel, slips at a
clutch, or changes fuel distribution. Agreement between energy loss and motor
electrical power does not by itself verify the angular-momentum balance. Use matched
time windows, measured moment arms, and a stated system boundary for every reported
spin-transfer result.

## Experiments and design limits

Torque calibration precedes a comparison with the rigid-top prediction. In a
gravity-driven-top test, measure the total mass $M$, pivot-to-centre-of-mass
distance $r$, and tilt angle $\theta$, then compute the nominal moment
$\tau_g=Mgr\sin\theta$. The pivot location and centre of mass require separate
measurements; using the visible rim of a rotor for $r$ introduces a systematic
torque error. A suspended calibration mass at a measured lever arm provides an
independent static torque check. A load cell at the support can measure force, but
its signal becomes a torque only after its line of action and moment arm have been
established. Calibrate torque in the same support geometry used for the spinning
test because cable tension, bearing preload, and bracket deflection can change the
effective moment.

Extract the precession rate from azimuth versus time rather than from frame-to-frame
angular differences. Unwrap the recorded azimuth across complete turns, select a
time interval with bounded nutation, and obtain $\Omega_{\rm meas}$ from the slope
of a linear regression. Periodic nutation or a changing rate appears in regression
residuals more clearly than in a single period measurement. Camera timing must be
checked against a clock, and the camera axis must be related to the vertical used in
the torque model. A camera pointed obliquely at the pivot distorts azimuth and tilt;
an encoder mounted on a gimbal reports relative angle unless the gimbal reference is
also measured.

On the slow branch, the nominal prediction is
$\Omega_{\rm pred}=Mgr/(I_s\omega_s)$. With independent small input uncertainties,
the fractional scale estimate is
$\delta\Omega_{\rm pred}/\Omega_{\rm pred}\simeq[(\delta M/M)^2+
(\delta r/r)^2+(\delta I_s/I_s)^2+(\delta\omega_s/\omega_s)^2]^{1/2}$.
This expression omits correlated errors, pivot-friction torque, and model error; it
is a measurement-scale estimate rather than a complete uncertainty statement. The
tilt angle cancels from the leading formula, but its measured variation tests the
steady-precession assumption. Report its range and nutation amplitude with every
rate comparison.

Damping biases the comparison through more than one channel. Spin decay makes
$\Omega_{\rm pred}$ increase as $1/\omega_s$ if the gravitational torque is fixed.
Bearing friction can also add a torque that changes the axis direction, while air
drag can produce a torque whose magnitude changes with spin speed. Pair each fitted
precession slope with the mean spin rate over the same interval, then repeat the
fit in adjacent windows. A systematic drift in the residual
$\Omega_{\rm meas}-\Omega_{\rm pred}$ indicates that the constant-spin or
torque-only model is incomplete.

Replicate runs quantify repeatability separately from calibration uncertainty. Keep
the same mass, lever arm, and inertia estimate across a run set, then treat their
errors as shared rather than independent scatter. The regression slope and intercept
should both be reported. A nonzero intercept or curvature in a plot against
$1/\omega_s$ can identify a residual support torque, spin-dependent loss, or an
unmatched time interval before numerical agreement is claimed.

Support friction requires a direct bound or an independent measurement. Measure a
coast-down with the same pivot load, or reverse the spin direction while retaining
the same geometry. A support torque fixed in the laboratory frame changes differently
under spin reversal from a torque tied to rotor drag. The final comparison should
list calibrated torque, $I_s$, spin interval, tilt range, rate-extraction method,
timing uncertainty, support-friction estimate, and residual. Agreement within the
input uncertainty supports the rigid-top approximation only for that tested range of
spin, tilt, and support conditions.

**Design limits, balancing, and bearing losses.**

Rotor geometry sets both the available spin angular momentum and the mechanical
limits on spin speed. For a fixed mass, placing material farther from the axis
increases $I_s$ and hence $L=I_s\omega_s$. A thin rim has large inertia per unit
mass, but its rim speed $v=R\omega_s$ and centrifugal stress rise with radius and
spin rate. For an ideal thin ring, the hoop-stress scale is proportional to
$\rho R^2\omega_s^2$. A solid disk has a different stress distribution, so its
permissible speed cannot be inferred from the ring expression. Material strength,
fatigue margin, hub geometry, attachment method, and any stress concentration set
the allowed operating range. A design that maximizes inertia alone can reduce the
safe speed margin or introduce a flexible mode inside the intended spin band.

Static imbalance occurs when the mass centre lies a distance $e$ from the rotation
axis. At spin rate $\omega_s$, the associated rotating force scale is
$F_u=m_e e\omega_s^2$, where $m_e$ denotes the unbalanced mass contribution. The
force acts at the spin rate and is transmitted through the bearings to the support.
If its line of action has a lever arm about the pivot, it produces a periodic torque
that can modulate precession and excite nutation. Couple imbalance arises when mass
errors in separated axial planes produce a net rotating moment even when the mass
centre lies on the nominal axis. Correcting only one plane can remove a measured
force while leaving that rotating couple.

$$
% caption: Rotor geometry and balancing. A small off-axis mass creates a spin-rate force transmitted through the bearings to a pivot moment; correction must address both the centre-of-mass offset and any separated-plane rotating couple.
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Balance correction uses measured amplitude and phase at one or more spin rates. In
single-plane balancing of a rigid rotor, a trial mass provides the phase reference
needed to select a correction mass and radius. A long rotor requires two-plane
balancing because the bearing signals contain both force and couple information.
Balancing should be repeated after changing a hub, fastener, sensor target, or rotor
temperature. A correction mass changes $I_s$ slightly, so a high-accuracy
precession experiment should recompute the inertia used in its prediction after
balancing.

Bearing losses change both spin magnitude and torque direction. A resisting shaft
torque gives $I_s\dot\omega_s=-\tau_{\rm loss}$ for an otherwise fixed axis, and
the dissipated power is $P_{\rm loss}=\tau_{\rm loss}\omega_s$. Rolling-contact
loss, lubricant shear, seal drag, and preload need not have the same speed law.
Misalignment can add transverse bearing forces and a pivot moment, so it affects
precession beyond the spin-down rate. A coast-down measurement at several speeds
separates an approximately constant loss torque from a speed-dependent drag term
only if the rotor temperature and support load are controlled.

The slow rigid-top estimate responds to these limits through both $I_s\omega_s$ and
the actual external torque. Spin decay raises the nominal $Mgr/(I_s\omega_s)$ rate,
whereas a bearing moment can shift the direction and magnitude of the net torque.
Imbalance can introduce periodic departures around that mean rate. Report the spin
limit, balance criterion, bearing condition, temperature, and coast-down torque with
any observed precession curve. A precession record without those design parameters
cannot separate gravitational precession from a support-driven axis motion.

Design acceptance should specify a maximum residual vibration at each bearing, a
maximum coast-down torque, and a permitted spin interval below the calculated stress
and resonance limits. Test the rotor after assembly and after any maintenance that
changes preload or correction mass. Passing a balance test at one speed does not
establish acceptable response across a broad spin sweep.