The MEMS vibrating ring gyroscope is an inertial sensor that delivers precise and reliable angular rate measurements in dynamic motion environments. Utilizing Coriolis force and resonant frequency principles, this miniaturized device offers excellent sensitivity and stability, ideal for applications such as navigation, stabilization, and motion tracking. Dual-axis gyroscopes, which measure angular motion in two orthogonal axes, tend to be larger and consume more power than the MEMS vibrating ring gyroscope. The MEMS vibrating ring gyroscope outshines dual-axis and tuning fork gyroscopes in several key aspects, including size, power consumption, sensitivity, and reliability. Its advanced design and unique operating principles make it a highly attractive sensor for various applications, including navigation, stabilization, and motion tracking [
17,
18,
19].
2.2.2. Various Energies Effect
To understand the motion equations of the vibrating ring structure, we will evaluate the effects of various energies on the ring resonator and the main mode of vibration of the ring structure, considered as the coordinates of 2
θ elliptical vibrations, as shown in
Figure 6. We will use the Lagrange equation to determine the motion equation for the vibrating ring gyroscope. The given scheme of the energy equation is shown below. Here
represents the Lagrange equation, which is
). Where
is the kinetic energy,
is the Rayleigh damping,
is the strain energy and
is the strain energy of the semicircular support beams.
To find the kinetic energy of the vibrating ring, we need to consider the displacements of the ring and the central anchor. The absolute displacements of the ring are denoted by
and
and for the central anchor, denoted as
and
. To determine the absolute displacements denoted as
and
of the vibrating ring system, we need to combine the displacements terms of the vibrating ring and centrally placed anchor. The absolute velocities
and
of the ring are given in equations 21 and 22.
Hence, the kinetic energy of the vibrating ring is written in equation (23)
Here
is the density,
is the radius, and
is the cross-sectional area of the vibrating ring gyroscope. By putting equations 21 and 22 in equation 23 and further solving it, we find out the kinetic energy of the ring as equation 24 where
represents the mass of the vibrating ring.
Equation 24 provides a detailed dynamic movement of the ring through the kinetic energy equation. The equation considered both the rigid body displacement of the ring and the elliptical mode of the ring structure, which pertain to the displacements with inplane of the motion of the ring. Moreover, the equation considers the influence of base excitation, which refers to external forces applied on the anchors that the ring is connected to. One important feature is the decoupling of the generalized coordinates and that represent the motion of the ring. The coordinates, represented as in the equation, are not mutually dependent in calculating kinetic energy. The phenomenon of uncoupling suggests that the various components of motion, such as inplane vibrations and base excitation, contribute to the overall kinetic energy of the vibrating ring gyroscope system.
In relation to the concept of the elastic strain energy, it is supposed that the vibrating ring possesses elastic properties and corresponds to Hook's law. The law states that stress is directly proportional to strain, with constant proportionality being Young's Modulus "E" for the material. Hence the elastic strain energy "
" is.
Where
is the radius of the ring and
is the normal strain of the ring. The normal strain of the ring can be calculated as.
The term "
" refers to the distance of a point of the ring from the central axis. As mentioned, the inextensionality of the ring, which makes
at
. Therefore, equation 27 becomes simpler and shows the extensional strain of the ring.
We have the elastic strain energy equation by putting Equations 26 and 27 into equation 25.
The elastic strain energy equation shows the elliptical modes of the ring and also shows that the generalized ring coordinates are not connected in the strain energy equation.
The flexible beams are the integral parts of any MEMS vibrating gyroscope system. The semicircular beams are the supporting structure to the vibrating ring in the vibrating ring gyroscope system. In
Figure 4, the eight semicircular beams are connected to support the ring structure. As we can see, the semicircular beams are relatively small to the ring structure, so we can neglect the mass of the semicircular beams. In this scenario, the kinetic energy of the semicircular beam could be considered zero.
When the force is applied to the vibrating structure, the semicircular beams undergo radial and tangential direction. We will consider semicircular beam radial and tangential stiffness separately, as described in
Figure 7 [
25].
The strain energy equation can determine the stiffness constant for a semicircular beam when subjected to an applied force or an external rotation, which results in tangential and radial displacements, respectively. Therefore further, we will determine tangential and radial stiffness constant.
The tangential strain energy equation for tangential displacement is presented as equation 30. Where
is Young's modulus,
is the moment of inertia,
is the bending moment experienced, and
is the differential width of the semicircular beam.
The above equation is further solved [
25] and presented below.
Where
is the net applied force,
is the radius of the semicircular support beam, and
is the imaginary moment experienced by the semicircular support beam when subjected to the force. Further solving the tangential strain energy equation determines the tangential stiffness constant.
The radial strain energy equation for radial displacement is presented as equation 33.
The above equation further solved and determined the radial stiffness constant which is shown in equation 34, where
is the length of the beam.
The complete stiffness constant equation for the semicircular support beam can be written as.
There are eight semicircular support beams are attached to the vibrating ring. Therefore, the total strain energy
for semicircular beams could be written as equation 36. Where
ith represents the semicircular support beam position attached to the ring.
By putting equations 18 and 20 into equation 36, we can further determine the total strain energy
for eight semicircular beams attached to the vibrating ring.
The damping energy for MEMS inertial sensors in the vacuum is the thermoelastic damping [
3,
26]. The energy dissipation factor is considered as Rayleigh damping function. The Rayleigh damping function for the MEMS vibrating ring gyroscope system [
27] can be expressed as.
The given expressions and are elliptical mode shape which is generally represented by 2.
To derive the equations of motion for the vibrating internal ring gyroscope, we will be using Lagrange's equation, and which is given as equation 20.
The vibrating internal ring gyroscope resonator mechanical model is shown in
Figure 4. The derived energy equations will be put into equation 20 to find the motion equation of a vibrating ring gyroscope.
To simplify the equations of motion, we consider
,
, and
.
The natural frequency of the gyroscope
and the Quality factor of the gyroscope
. Therefore, the refined equations of motion for the vibrating ring gyroscope are presented below.