Gyroscopic Motion

Gyroscopic motion is the motion of a spinning rigid body whose spin axis is also rotating or precessing in 3D space. Spinning objects resist changes in its axis of rotation due to their angular momentum.

---

Velocity and Acceleration of a 3D Rigid Body

For a point at $\vec{r}$ from a fixed point $O$ on a body with angular velocity $\omega$:

Constant Spin Rate

If $|\vec{\omega }|$ is constant, we can calculate the total angular velocity of the object using this simplified equation:

$\dot{\vec{\omega }}=\vec{\Omega }\times \vec{\omega }$

Definition bq

$\vec{\Omega }$ is the angular velocity of the spin axis itself.

---

Effect of a Small Moment on Spin Axis

Precession refers to the motion of a spinning body’s spin axis when an external moment is applied to it. Precession rate (${\Omega }_p$) is the speed at which the spin axis rotates or wobbles.
${\Omega }_p=\frac{M}{I\omega }$

As seen in the equation above, the faster an object is spinning ($\omega$) or the larger its mass moment of inertia ($I$) about the spin axis, the slower the precession rate (${\Omega }_p$) for a given external moment ($M$).
${\Omega }_p\propto \frac{1}{I\omega }$

Quote bq

We assume for all following equations that spin rate greatly exceeds precession rate (i.e. the gyroscopic effect).

Vector Form and Direction:

The direction of precession can be determined using the cross product relationship, where $\mathbf{H}=I\omega$.
$\mathbf{M}=\mathbf{\Omega }\times \mathbf{H}$

The spin axis always turns towards the direction of the applied moment vector ($\mathbf{M}$).

---

Steady Precession of a Symmetric Top

A symmetric top, with mass $m$ and spin rate $\omega$, tilts at an angle $\theta$ from the vertical. The centre of mass $G$ is a distance $r$ from the pivot $O$.

Gravity pulls down on $G$, creating a torque (moment $M_O$) :
$M_O=mgr\sin \theta$

The precession rate $p$ is about the vertical, thus the component of $p$ normal to the spin axis is $p\sin \theta$. Then $M_O=I\omega p\sin \theta$ gives:
$mgr\sin \theta =I\omega p\sin \theta ⟹p=\frac{mgr}{I\omega }$

Comment bq

This is known as steady precession where $\theta$ is constant and $p$ is independent of $\theta$.

Precession effect on a spinning top

---

Nutation

If the moment or the starting conditions don’t match steady precession then $p\ne \frac{mgr}{I\omega }$ and $\theta (t)$ oscillates. This is known as unsteady precession which is called nutation.

---

await dv.view("Scripts/topic-backlinks");