The core of the dynamics of a fixed-point rotating rigid body is the angular momentum theorem and the Euler dynamic equations derived from it. These are important tools for analyzing the dynamics of gyroscopes.
1. The definition and formula of angular momentum
For a rigid body rotating about an axis, as shown in the following Figure,
the sum of the momentum of each particle within the rigid body and its distance from the axis, that is, the sum of the momentum of each particle within the rigid body about the axis, is called the angular momentum or moment of momentum of the rigid body about that axis. This definition can be expressed by a formula as:\[ \boldsymbol{ H_l = \sum r_i m_i v_i } \]
Among them, \( \boldsymbol{ H_l } \) represents the angular momentum of the rigid body about axis \( \boldsymbol{ l } \); \( \boldsymbol{ r_i} \) is the distance from this particle to axis \( \boldsymbol{ l } \); \( \boldsymbol{ m_i } \) is the mass of any particle within the rigid body; \( \boldsymbol{ v_i } \) is the velocity of this particle.
Let the angular velocity of the rigid body rotating around axis \( \boldsymbol{ l } \) be \( \boldsymbol{ \omega_i } \). Then the velocity \( \boldsymbol{ v_i} \) of any particle within the rigid body can be expressed as \[ \boldsymbol{ v_i = r_i \omega_l } \]
Substituting it into the equation \( \boldsymbol{ H_l = \sum r_i m_i v_i } \) to obtain \[ \boldsymbol{ H_l = \sum m_i r_i^2 \omega_l } \]
Since the angular velocity \( \boldsymbol{ \omega_i } \) of rotation for all the particles within the rigid body is the same, we obtain \[ \boldsymbol{ H_l = \omega_l \sum m_i r_i^2 } \]
Among them, \( \boldsymbol{ \sum m_i r_i^2 } \) represents the moment of inertia \( \boldsymbol{ J_l } \) of a rigid body about axis \( \boldsymbol{ l } \). Therefore, the formula for the angular momentum of a rigid body rotating about axis \( \boldsymbol{ l } \) is \[ \boldsymbol{ H_l = J_l \omega_l } \]
2. The angular momentum of the gyroscope rotor
Angular momentum is a very important characteristic parameter of a gyroscope. The greater the angular momentum of the rotor, the less likely the spatial orientation of the spin axis will change, and the more obvious the gyroscope’s characteristics will be.
The rotor of the gyroscope rotates at high speed around its spin axis, while also undergoing transport motion around the frame axis. The angular momentum of the rotor should include the angular momentum generated by both the self-rotation motion and the transport motion.
2.1 Spin Angular Momentum
First, we study the angular momentum generated by the spin motion of the rotor. This angular momentum is called the spin angular momentum. Let the moment of inertia of the rotor about the spin axis be \( \boldsymbol{ J_z } \), and its spin angular velocity around the spin axis be \( \boldsymbol{\Omega} \). Then, the magnitude of the spin angular momentum of the rotor is \[ \boldsymbol{H_z = J_z \Omega} \]
The direction is along the axis of rotation and is consistent with the direction of the spin angular velocity \( \boldsymbol{\Omega} \) of the rotor.
2.2 Transport Motion Angular Momentum
The angular momentum generated by the transport motion of the rotor is called the transport motion angular momentum of the rotor. Let the moment of inertia of the rotor about the equatorial axis be \( \boldsymbol{ J_e } \), and its angular velocity about the equatorial axis be \( \boldsymbol{\omega_e} \). Then the transport motion angular momentum of the rotor is \[ \boldsymbol{H_e = J_e \omega_e} \]
The direction is along the equatorial axis (frame axis) and is consistent with the rotational angular velocity \( \boldsymbol{ \omega_e } \) of the rotor around the equatorial axis.
Refer to the relationship shown in the figure:
When considering both the spin angular momentum and the transport angular momentum simultaneously, the rotor angular momentum is \[ \boldsymbol{ H = \sqrt{(J_z \Omega)^2 + (J_e \omega_e)^2} } \]
At this point, there is an angle\[ \boldsymbol{ \varepsilon = \arctan \frac{J_e \omega_e}{J_z \Omega} } \] between the rotor angular momentum and the spin axis.
In actual gyroscopes, the ratio of the equatorial rotational inertia of the rotor to the rotational inertia of the axis is generally \( \boldsymbol{ \frac{J_e}{J_z} \approx 0.6 } \), indicating that the two have the same order of magnitude. However, when the gyroscope is in a normal operating state, the rotor’s rotational speed is generally 22,000 to 24,000 r/min, while the rotational angular velocity of the rotor around the frame axis (transport angular velocity) is generally within a few degrees per minute. Therefore, it can be seen that the rotational angular velocity of the rotor around the frame axis is only one millionth of the rotational angular velocity of the rotor around its spin axis, so the rotor’s spin angular momentum \(\boldsymbol{J_z \Omega}\) is much greater than the transport angular momentum \(\boldsymbol{J_e \omega_e}\).
Therefore, the magnitude of the rotor angular momentum is very close to that of the self-rotation angular momentum, and the direction is also very close to that of the self-rotation angular momentum. We can ignore the influence of non-self-rotation angular momentum and consider that the magnitude and direction of the rotor angular momentum are the same as those of the self-rotation angular momentum.
From this, we obtain a basic concept: The magnitude of the angular momentum of the gyroscope rotor is equal to the product of the rotor’s moment of inertia \(\boldsymbol{J_z\) about its axis of rotation and the rotor’s spin angular velocity, that is \[ \boldsymbol{ H = J_z \Omega } \]
Its direction is along the spin axis and is in the same direction as the spin angular velocity of the rotor.
To obtain a larger rotor angular momentum within the limited instrument volume, the rotor should be designed as having the largest possible moment of inertia and a higher rotational speed.