1. Calculation Formula
According to kinematic theory, in the case where the transport motion is rotation, the acceleration of a particle within a rigid body, in addition to the relative acceleration and transport acceleration, includes an additional acceleration term. This additional acceleration is called the Coriolis acceleration.
At this time, the absolute acceleration \(\boldsymbol{a}\) of the moving point should be equal to the vector sum of the relative acceleration \(\boldsymbol{a_r}\), transport acceleration \(\boldsymbol{a_e}\), and Coriolis acceleration \(\boldsymbol{a_c}\), that is, the following relationship exists:
\[\boldsymbol{a = a_r + a_e + a_c} \]
When the transport motion of moving point undergoes rotational motion \( \boldsymbol{\omega} \), the transport rotation causes the direction of the relative velocity \(\boldsymbol{v_r}\) to constantly change, and the relative motion also causes the magnitude of the transport velocity to constantly change. Both of these reasons result in an additional rate of change of velocity in the same direction, which is the Coriolis acceleration. Or in simpler terms, the Coriolis acceleration is formed due to the mutual influence of the relative motion and the transport motion. The expression of the Coriolis acceleration is:
\[ \boldsymbol{a_c = 2 \omega \times v_r} \]
That is, in general, the magnitude of the Coriolis acceleration is
\[ \boldsymbol{a_c = 2 \omega v_r \sin(\omega, v_r)} \]
The direction of Coriolis acceleration \(\boldsymbol{a_c}\) is perpendicular to the plane formed by the co-rotational angular velocity \( \boldsymbol{\omega} \) and the relative velocity \(\boldsymbol{v_r}\). The direction of force ac is the right-hand rotation direction of the shortest path towards \(\boldsymbol{v_r}\), which is the direction of \( \boldsymbol{\omega} \). As shown in the figure:
The following figure shows a top-down view of the gyroscope from the \(\boldsymbol{z}\) axis of the gyroscope’s rotation. Assume that the rotor rotates uniformly at an angular velocity \( \boldsymbol{\Omega} \) relative to the inner ring along the positive \(\boldsymbol{z}\) axis direction. The rotor, along with the inner and outer rings, rotates uniformly at an angular velocity \( \boldsymbol{\omega_x} \) relative to the base along the positive outer ring \(\boldsymbol{x}\) axis direction. That is, all the particles of the rotor participate in this relative motion and co-rotation, and the co-rotation is a fixed-axis rotation.
Now let’s analyze the acceleration of each particle of the rotor.
2. Relative Acceleration
The relative motion of each particle of the rotor is uniform rotation around the rotational axis. The magnitude of the relative velocity of each particle does not change, so there is no tangential acceleration. However, the direction of the relative velocity of each particle changes, indicating the existence of centripetal acceleration. Therefore, when the rotor rotates uniformly around the rotational axis, the relative acceleration of each particle of the rotor is all centripetal acceleration. Its direction is perpendicular to the rotational axis \(\boldsymbol{z}\) and points towards that axis, and its magnitude is \( \boldsymbol{a_r = r \Omega^2} \), where \(\boldsymbol{r}\) is the vertical distance from that particle to the rotational axis.
3. Transport Acceleration
The transport motion of each particle of the rotor is uniform rotation around the outer ring axis. The magnitude of the transport velocity of each particle does not change, so there is no tangential acceleration. However, the direction of the transport velocity of particle changes, indicating the existence of centripetal acceleration. Therefore, when the rotor rotates uniformly around the outer ring axis, the transport acceleration of each particle of the rotor is all centripetal acceleration. Its direction is perpendicular to the outer ring axis \(\boldsymbol{x}\) and points towards the rotation axis, and its magnitude is \( \boldsymbol{a_e = L \omega_x^2 = r \omega_x^2 \cos \theta} \), where \(\boldsymbol{L}\) is the vertical distance from the particle to the outer ring axis.
4. Coriolis Acceleration
Each particle of the rotor performs relative motion with respect to the frame, while the frame also undergoes transport motion. Due to the interaction between the relative motion and the transport motion, each particle of the rotor has Coriolis acceleration. In the following figure, the magnitude of the relative velocity of each particle is \( \boldsymbol{v_r = r \Omega} \), and its direction is along the tangent direction. The magnitude of the transport angular velocity of each particle is all \( \boldsymbol{w_x}\), and its direction is all parallel to the outer ring axis \( \boldsymbol{x}\). The magnitude of the Coriolis acceleration of each rotor particle is \( \boldsymbol{a_c = 2 \omega_x \Omega r \sin \theta} \), and its direction is determined according to the above right-hand rule. In the first and fourth quadrants, the direction of the Coriolis acceleration is perpendicular to the rotation plane of the rotor and the vector is upward; in the second and third quadrants, the direction of the Coriolis acceleration is perpendicular to the rotation plane of the rotor and the vector is downward.
From this formula \( \boldsymbol{a_c = 2 \omega_x \Omega r \sin \theta} \), it can be seen that the magnitude of the Coriolis acceleration of each particle of the rotor is related to the position of that particle. It varies sinusoidally with the angle \( \boldsymbol{\theta}\) and proportionally with the radius \( \boldsymbol{r}\). The above figure shows the distribution law of the Coriolis acceleration of each particle of a thin circular sheet on a solid cylindrical rotor. If an arbitrary thin circular sheet is taken on the rotor, the distribution law is the same as this.
The above analysis is conducted under the condition where the rotor performs transport rotation around the outer ring axis. If the rotor performs transport rotation around the inner ring axis, it can also be analyzed using a similar method.