CC BY
0Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 1, pp. 59-89. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd230102
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 74H10
Nonlinear Dynamics of a Microscale Rate Integrating Gyroscope with a Disk Resonator under Parametric
Excitation
D. A. Indeitsev, E. V. Zavorotneva, A. V. Lukin, I. A. Popov, V. S. Igumnova
This article presents an analytical study of the dynamics of a micromechanical integrating gyroscope with a disk resonator. A discrete dynamic model of the resonator is obtained, taking into account the axial anisotropy of its mass and stiffness properties, as well as the action of the electrical control system of oscillations. An analysis of the spectral problem of disk vibrations in the plane is carried out. The nonlinear dynamics of the resonator in the regimes of free and parametrically excited vibrations are investigated. In the mode of parametric oscillations, qualitative dependencies of the gyroscopic drift on the operating voltage, angular velocity and parameters of defects are obtained.
Keywords: MEMS, MRIG, nonlinear dynamics, BAW, parametric excitation
Received July 19, 2022 Accepted December 08, 2022
This research is funded by the Russian https://rscf.ru/en/project/21-71-10009/.
Science Foundation grant no. 21-71-10009,
Dmitry A. Indeitsev Dmitry.indeitsev@gmail.com
Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Bolshoj prosp. 61, Saint Petersburg, 199178 Russia Peter the Great St. Petersburg Polytechnic University, ul. Polytechnicheskaya 29, Saint Petersburg, 195251 Russia
Ekaterina V. Zavorotneva zavorotneva.ev@gmail.com Alexei V. Lukin lukinalexei@yandex.ru Ivan A. Popov popov_ia@spbstu.ru Vasilisa S. Igumnova igumnova_vs@spbstu. ru
Peter the Great St. Petersburg Polytechnic University, ul. Polytechnicheskaya 29, Saint Petersburg, 195251 Russia
1. Introduction
One of the most important tasks of measuring instrumentation at the present stage is the creation of small-sized inertial navigation systems that provide determination of orientation parameters and coordinates of aircraft, surface and underwater vehicles, ground vehicles, robots, etc. The key area of development here is the creation of microelectromechanical inertial information sensors: MEMS accelerometers and MEMS gyroscopes of the navigation accuracy class [1]. Along with the high requirements for the accuracy characteristics of sensors, the parameters of temperature stability and the stability of systems to external influences (vibration, shock, acoustic, noise) are of decisive importance.
Over the last decade, microsystems have been intensively developed in the global industry to create Coriolis vibrational gyroscopes (CVG) based on bulk acoustic waves (BAW) — gyroscopes with a micromachined disk resonator [2-7]. The advantages of such systems over more common tuning fork architectures (single-mass and multi-mass) include:
1) extremely high resistance to external vibration, shock and acoustic influences due to the high operating frequencies of the resonator (range 1-10 MHz). Vibration stability can also be provided by multi-mass architectures if they are dynamically balanced [8];
2) a high Q-factor of resonant properties caused by a number of factors: a) the smallness of elastic energy dissipation through the anchor due to the symmetry and dynamic balance of the resonator design; b) the smallness of thermoelastic damping on volumetric working forms of vibrations compared with bending (beam) forms of vibrations of tuning fork gyroscopes; c) the smallness of the influence of gas damping on the Q factor due to the high rigidity of working forms;
3) reduced requirements for vacuuming the gyroscope housing due to the above;
4) small dimensions of the device;
5) the ability to operate the gyroscope in the rate integrating mode (free precession mode of the vibration mode of a dynamically symmetric resonator) — the implementation of a microme-chanical rate-integrating gyroscope (MRIG).
Compared to MEMS gyroscopes with the ring resonator, the microgyroscope on the BAW is characterized by orders of magnitude higher frequencies and considerably large values of the modal mass, resulting in a proportionate increase in the conversion ratio of the sensor.
A number of developments of micromechanical gyroscopes with a flat axisymmetric resonator, structurally consisting of a series of concentric rings connected by radial bonds [9-19], are closely adjacent to the class of MEMS gyroscopes at BAW. This direction can be considered as a development of the concept of gyroscopes with ring resonators, caused by the desire to increase the effective mass of the working forms of vibrations.
The study of the nonlinear dynamics of micromechanical vibration gyroscopes and the proposal of constructive and algorithmic solutions to improve the operational characteristics of sensors is an extensive area of scientific research and engineering development. A significant number of papers are devoted to mathematical modeling of the dynamics of Coriolis MEMS gyroscopes with a ring resonator [20-27]. The development of algorithms for generation and control of oscillations of such systems is an active reseach area [20, 28-35]. The papers [36-38] address the issues of parametric excitation and maintenance of MEMS resonator oscillations taking into account nonlinear factors.
There are well-known studies aimed at developing analytical methods for effective modeling of the dynamics of MEMS gyroscopes on the BAW. An important preliminary stage in the preparation of mathematical models of such systems is the solution of the problem of finding
the natural frequencies and forms of bending vibrations of a hollow disk resonator in its plane. There is an extensive literature [39-42, 42-47] on this problem. The influence of the thermoelastic damping on the wave properties of continuous media and the spectral properties of resonators is studied in [48-50]. The works [51-57] take into account geometric, physical-mechanical and electrostatic nonlinearities and analyze their influence on the spectral and resonance characteristics of disk resonators. The current state of research on the nonlinear dynamics of MEMS gyroscopes based on BAW is reflected in [58-62]. Among the works devoted to analytical modeling of the dynamics of micromechanical rate-integrating sensors, one should, first of all. mention [63].
The present work is devoted to the development of a mathematical model and a qualitative study of the nonlinear dynamics of a MRIG with a micromachined disk resonator in the free precession mode under parametric electrostatic excitation. The contribution of the article lies in the construction of an asymptotic solution for a nonlinear system of equations of a solidstate microgyroscope, taking into account the rotation of the base and parametric excitation of oscillations. Separate numerical experiments are computationally time-consuming and cannot capture the general behavior of the system depending on the changing parameter. This problem can be solved by numerical methods of bifurcation theory. Finding the slow evolution of nonlinear oscillations of the system by the averaging method with the representation of the solution in the form of amplitude and phase does not allow one to obtain the steady behavior of the system. Therefore, it was proposed to look for a solution in terms of slowly varying amplitudes, which made it possible to reveal the limit cycle regime, which can later be studied analytically using the methods of the theory of continuation with respect to a parameter. As far as the authors of this paper know, attempts have been made in the literature on this topic to conduct qualitative parametric analyses of such kinds of free-precession modes in the presence of external excitations and structural imperfections, but there were computational difficulties that were not solved. An interesting effect of dependence of the precession coefficient (Bryan factor) on the ratio of the geometrical parameters of the disk is revealed. There is a nonmonotonicity of this dependence and the presence of a local extremum, which indicates a weak sensitivity to external disturbances at a certain ratio of geometric parameters.
In the present work, in Section 2, a mathematical formulation of the problem of vibrations of a linear-elastic isotropic disk with a central hole in its plane is carried out in a nonlinear geometric setting taking into account the mass-stiffness imperfections of the structure, the action of Coriolis inertia forces, the action of an electrostatic system of parametric excitation and maintenance of vibrations. In Section 3, the spectral problem of small free vibrations of a disk on a fixed base is investigated. The effect of gyroscopic frequency splitting in the presence of rotation is considered. In Section 4, a detailed analysis of the RIG nonlinear dynamics is presented. The dynamic effects arising from the geometric nonlinearity of the resonator are investigated. The zone of primary parametric resonance is qualitatively investigated, and the starting voltages are estimated. The nonlinear dynamics of a disk resonator on fixed and rotating platforms is investigated. Equations in slow amplitude variables are obtained using the averaging method. Using numerical methods of bifurcation theory, the dependences of the precession frequency on the angular velocity, the amplitude of the electrostatic forces, and the frequency detuning of the excitation system are found. The influence of mass and stiffness imperfections of the resonator on the nature of the free precession of the working mode of oscillations has been studied.
2. Mathematical model
Resonators of this type are usually made of single-crystal silicon (SCS), which is an aniso-tropic material. It is well known [64] that a disk resonator made of SCS with crystal orienta-
tions (111) is isotropic in the plane relative to vibration modes with the third axial symmetry index (k = 3). Figure 1 shows the two considered orthogonal disk modes with multiple frequency, found using the finite-element method.
Fig. 1. Operating (k = 3) modes of the disk resonator
Table 1 shows the main geometrical and physical-mechanical characteristics of the sensitive element (resonator), adopted for further numerical calculations of the dynamics of MRIG. In order to simplify the calculations and take into account the above-mentioned absence of splitting of the working vibration mode for a single-crystal silicon disk, the anisotropy of the elastic properties of the material can be neglected and taken to be isotropic and linearly elastic. However, in some cases discussed below, the nonlinear distribution of the elastic modulus and density will be taken into account. It will be assumed that these imperfections are due to manufacturing tolerances.
Table 1. The main parameters of the resonator
Parameters
a 420 (yU,m) outer radius
b 210 (¿tni) inner radius
d 270 (nm) capacitive gap
h 40 (¿tni) disk height
E0 170 (GPa) Young's modulus mean
V 0.266 Poisson's ratio
Po 2329 (kg/m3) density mean
f 4.5313 (MHz) natural frequency
Q 68 230 Q-factor
The mechanical model of the system is shown in Fig. 2. Neglecting centrifugal forces, we write the kinetic energy of a rotating disk as follows:
2n a
Eu =
^ J J p[(u2 + v2) + 2Q(uv + ùv) + 2Qvr] r dr dip. 0 b
(2.1)
Fig. 2. Schematic representation of an MRIG with a disk resonator
The potential energy of elastic deformations has the form
2n a
Wr£r + + dr dLP'
F h
(2.2)
0 b
where r and p are the radial and angular coordinates, respectively, u(t, r, p), v(t, r, p) are the components of the displacement vector of the material points of the resonator, h is the height of the disk, a and b are the outer and inner radii of the disk, and p is the density.
The components of the stress tensor in the expression (2.2) are related to the components of the strain tensor by virtue of Hooke's law by the following relations:
E
ar ~ l-v2
E
~ l-v2
T ri
(er + vev), (ev + v£r ),
(2.3)
where er, ep
Yrv are the components of the strain tensor in a cylindrical coordinate system, E is Young's modulus, v is Poisson's ratio, and ¡i is the shear modulus.
In this work, the effect of geometric nonlinearity on the resonator dynamics is taken into account. Many works are devoted to the study of nonlinear theory in plates and shells [65, 66]. However, a general resonator model is considered which can be applied to both a disk and a ring. The following nonlinear relations for deformations and displacements are used [67]:
du 1
tr = ~ h ~< --
dr
d v
-ip
dv 1 du
dr r d(p
1 dv u 1
= -TT- + - + Ö
r dp r 2 u du r2 dp
v r
uv
dr r
1 du\' r dp J
1 du dv . . .. r2 dp dp r dr r dr
u dv r2 dp ' v du
u dv
du dv dr dr
(2.4)
2
In the manufacture of the sensor, inevitable technological imperfections occur. The simplest way to investigate density unevenness within a structure is to represent density as a periodic function of the angular coordinate p. In the work [68] it is shown that, due to the symmetry of the working mode of oscillations, the greatest contribution is made by the zero and 2k harmonics:
P = Po[1 + cos(2kp + Pp)]. (2.5)
Similarly, the inhomogeneity of stiffness properties is introduced:
E = E0[1 + ÔE cos(2kp + pE)]. (2.6)
To excite and maintain the desired vibration mode, the electrode structure proposed in the work [69] can be applied. The structure under study is shown in Fig. 3, it consists of 24 electrodes with a round surface, and due to the symmetry of the operating mode of oscillations, it contains 4 independent groups of electrodes located symmetrically relative to the resonator planes. Each electrode is positioned such that its surface is "parallel" to the cylindrical surface of the disk. DC and AC voltage is applied to each group of electrodes:
V(t) = W; + AWtM (t),
(2.7)
where Wi and AWi are the amplitudes of DC and AC voltage, respectively, and M(t) = | + + | sin(Ai) is a meander function. The square of the voltage is equal to
Vl(t)'2 = Wf + WtAWt + \aw'2 + -^AWf + (—AW2 + -WtAWt) sin(At) - cos(2Xt)AW2.
4 n2 \n n ) n2
(2.8)
2Aw
V(ip,t)
te V2 V3 V4 V,
12
AWjM(i)
W,
<P
Fig. 3. Electrode structure for excitation and maintenance of oscillations
Consider an infinitesimal element of disk surface area. Together with a fixed electrode, a small disk surface area dN = ah dp can be described by the model of a plane-parallel capacitor:
dC\ = e°
■ d,N = /°ha „dtp,
d — u(a) d — u(a)
(2.9)
where dCe is the electric capacity of an elementary site, e0 œ 8.854 x 10"12 Fm-1 is the electrical constant, d is the capacitive gap between the disk and the electrodes, and u(a) is the displacement on the outer radius of the disk. The expression for electrical energy will take the form
C.
2n
V2(p, t)
^ u(a) it2(a) it3(a) u4(a)
d
d2
d3
d4
dp, (2.10)
where V2(p, t) is the function of the square of the potential difference, the angular distribution of which is shown in Fig. 3. Since the function of the squared potential difference V2 (p, t) is periodic, it can be expanded into a Fourier series in terms of the angular coordinate:
= Y + cos iLp + bisin ^ •
¿=0
The expression for electrical energy takes the form
€0ha
2n
% 2
+ y^ [a, cos ip + b, sin ip]
i=0
1 +
u(a) u2 (a) u3(a) u4(a)
d
+
d2
+
d3
+
d4
(2.11)
dp. (2.12)
The Lagrangian of the system is
L = Ek - Ep + Ee. (2.13)
The classical approach to the study and construction of a discrete model of MRIG is as follows [70]. The resonator displacement functions are
u = u(r) [C(t) cos kip + S(t) sin kip], v = v(r)[C(t) sin kip — S(t) cos ktp],
(2.14)
where k is the symmetry index and U(r), v(r) are eigenfunctions depending on the radial coordinate corresponding to the operating frequency of the oscillations of an ideal resonator. The question of finding these eigenfunctions and the corresponding natural frequencies will be discussed in Section 3. These eigenfunctions are dimensional functions of displacements, while the funct.ions C(t) and S(t) are dimensionless functions of time t. For convenience, the dimensionless modal coordinates over the capacitive gap are introduced:
C (t) =
cm d,
S (t) =
m d.
(2.15)
Using the Ritz method (by writing Lagrange equations of the second kind for a system with a Lagrangian obtained by substituting expressions (2.14) into (2.13) and integrating over spatial variables), it is possible to derive a system of differential equations with respect to unknown time functions C and S. Linear viscous damping with parameter R is taken into account (terms of the form RC, RS):
d_dL_ dL_ dtdÖ ~ dC d_dL_ _ dL_ dt dS dS
+ RC = 0, + RS = 0.
(2.16)
0
0
Let us introduce dimensionless time
T = ut,
where u is the natural frequency of the resonator on a fixed base. Consider the main parametric resonance
A o
— ~ 2 + <Ti,
u
(2.17)
(2.18)
where a1 is the deviation of the excitation frequency from the doubled natural frequency of the resonator.
The final dimensionless system of nonlinear ODEs will look like
[A + 1]C + A2S + A3C + A4S+
+ [A7 + A8 + A6cos((4 + 2a1)T) + A5sin((2 + a1)T )]C 3+ + [An + A12 + A10cos((4 + 2a1)T) + A9sin((2 + a1)T )]C 2S+
+ [A15 + A14cos((4 + 2a1)T) + A13sin((2 + a1 )t )]CS2+
+
(l + a2x) 2 + A18 + A17Cos((4 + 2a1)T) + A16 sin((2 + a^T)
C+
+ [A21 + A22 + A2ocos((4 + 2a1)T) + A19sin((2 + a1 )t )]S3+ + [A25 + A26 + A24cos((4 + 2a1)T) + A23sin((2 + a1 )t )]S = 0, [1 - A1]S? + A2C - A4C + A3S+
+ [A21 + A22 + A20cos((4 + 2a1)t) + A19 sin((2 + a1)T )]C 3+ + [A15 + A14cos((4 + 2a1)T) + A13sin((2 + a1 )t )]C 2S+ + [A11 + A12 + A10cos((4 + 2a1)T) + A9sin((2 + a1)T )]CS2+ + [A25 + A26 + A24cos((4 + 2a1)T) + A23sin((2 + a1 )t )]C+ + [D23 - A7 + D21 cos((4 + 2a1 )t) + D2osin((2 + a1 )t)]S3+ + [(1 + a22)2 - A18 + D25cos((4 + 2a1 )t) + sin((2 + a1)T)
(2.19)
S = 0,
where A^, Di are coefficients depending on the eigenforms of the stationary resonator and material parameters. In this work, they are not given explicitly since the expressions are cumbersome; they will be partially presented below.
In [69] it was shown that compensation of mass and stiffness imperfections of the resonator can be performed by electronic balancing (application to different groups of electrodes of certain voltage values). The problems of developing a methodology for performing calibration tests of MRIG and algorithms for electronic balancing of the resonator constitute an extensive area of research [71, 72] and are not investigated in this work. Next, the main mode of parametric excitation of oscillations adopted by V2 = V22 = V32 = V42 = V2 will be considered. For an ideal disk, taking into account the basic excitation mode, the system (2.19) takes the form
C + A4S + A.aC + [(1 + <r2)2 + a17cos((4 + 2<71)t) + a16sin((2 + ajr)] C+ + \A8 + A6cos((4 + 2<71)t) + A5sin((2 + (t1)t)]c(c2 + s2) = 0, 5 - A4C + + [(1 + (i2)2 + a17cos((4 + 2<71)t) + A16sin((2 + ajr)] S+ + \A8 + A6cos((4 + 2<71)t) + A5sin((2 + cr1)r)]S'(C2 + 52) = 0,
(2.20)
where Ai = Ait a2 = <r2 = a2 when V2 = V.2 = V32 = V42 = V2. We omit the notation Ai and simply write A{. Below are the values of the coefficients of the system (2.20)
E(U12 + Unu) Ruj (1 - v2) Qp0oj (l — v2)
W V ' 3 2Ehn(U12 + U\\v) ' 4 E(U12 + Unu) '
14 (1 - v2) u(a)4AWe0a(2W + AW) _ 14 (l - v2) u(a)4AW2e0q
— r TT. j3_/rr TTT ~ ' —
5Ed3 n(Ul2 + Ull v) ' 6 5Edf n2(Ul2 + Unv) '
A8 = A8e + A89 ,
7«(a)4 (1 - v2) e0a (8AW2 + 4W2tt2 + ATU2tt2 + AWAWir2) _ 2d2(U4 + vU3)
8e = 20Ed?iT2(UV2 + ' = ^12 + ^11 :
28 (1 - z/2) «(a)2A^t0a(2^ + AW) _ 28 (l - */2) «(a)2A^2t0a
^16 = 1 r rrt_/rr , r, \ ) ^17 =
^2 =
15Ed3 n(Ul2 + Ullv ) ' 17 15 Ed3 n2(Ul2 + Ullv)
7u(a)2e0a, (8A^2 + AW'fir2 + A^vr2 + 4W1AW1vr2) 30T6d3poir2 '
a
1 4 3 3 dv 2 2 2 du 2 dv 2 du dv 2
Uo = /--T 18« + 96u3v + 12« —r + 74u2v2 + V2u2v—r + 140«2v—-r - 12u2-—-r2-
3 32r3 dr dr dr dr dr
b
2 dv 2 2 3 2 du 2 dv du dv 2 dv 2 2 du dv 2 3
-52u2 — v - V2uv6 + 32u,v2—r + 24«v2—r - 28«v-—-r2 - 12uv— r2 - Au-—- r3+ dr dr dr dr dr dr dr dr
_2 du2 2 t^du 2 dv 3 du2 dv 2 4\ ,
+2v2— r - Av----r6 + 2---- r4 dr,
dr dr dr dr dr
a
/1 ( ~4 ~3 dv ~2~2 ~2~ du dv
-T 261« + 204«> + 12« —r + 236« v + 12«2v—-r + 32u2v—r-
32r3 dr dr dr
b
2 du dv 2 2 dv 2 2 2 du du dv 2 du dv 2 3 4 3 dv
-12m2 ——r2 + 2«2—- r2 + 32uv2—r - 28u,v-—-r2 - 4«—— r3 + 3v4 - 12u3—r+ dr dr dr dr dr dr dr dr dr
_Q _Q _Q _ O _Q _Q
_2 du 2 _2 dv 2 ,_du dv 3 _dv 3 du dv 4 dv 4 . , +2tr — r2 + 18v — r2 - 4u— —r3 - 12u— r3 + 2— — r4 + 3— r4 ) d,r, dr dr dr dr dr dr dr dr
a2 1 2 dv du 2 dv du dv 2 2
if u = — / — 9« + — 6u,—r — A—ur + v — 2 v—r — 12—vr + — r ) dr, ll 4r dr dr dr dr dr
b
a
1 / _ _ „_dv ____ __dv „du2 2 dv2
^12 = / — + - 6«—r + 19v - 2v—r + 2— r2 + — r2 ) dr,
l2 4r dr dr dr dr
b
a
T6 = J ^r(u2 +v2)dr. (2.21)
b
3. Spectral properties of the system
3.1. Analysis of the spectral problem for an ideal resonator on a fixed base
Let us consider the oscillations of an ideal isotropic resonator without taking into account geometric nonlinearity, the influence of electric forces, mass and stiffness imperfections, and also the angular velocity. In this formulation, a dynamic plane problem of elasticity theory in polar coordinates is obtained. The equations of the resonator oscillation take the form
d2u E /1 du d2u v d2v 1 dv u\ /1 dv 1 d2u 1 dv
/'-T7T = l--7 -T- + T-T + -T-TT- —7— — • /' -T-TT- +
dt2 1 — v2 V r dr dr2 r dr dp r2 dp r2 J \r dr dp r2 dp2 r2 dp д2v E (v д2u 1 d2v 1 du\ ( 1 dv д2v v д2u 1 du v
p CU9. ~ 1 ..2 „ Я„ Я,„ + Я, „2 Я,- + ^ I , Я„ + Я„2 + „
8t2 1 — v2 y r дr cJp r2 дp2 r2 cjp J y r cjr дН r cjr дp r2 дp r2
(3.1)
A simplified formulation is considered in which the rigidity of the elastic suspension of the resonator is neglected and the boundary conditions are applied, expressing stress-free edges of the disk:
ar (a) = ar (b) = 0, rW rV 7 (3.2)
Trv(a) = Trv(b) =
Let us reduce the system (3.1) to dimensionless form by introducing the following dimensionless variables:
t _ r _ и _ v
T = r = -, и, = -, v = -. (3.3)
! pb2(i—u2) b b b
E
The system takes the form
cßu f 1 дй д2й v d2v 1 dv й\ /1 d2v 1 д2й 1 dv
-— + — + -7^----—--) +
(3.4)
c)t2 y r dr dr2 r dr dp r2 dp r2 J y r dr dp r2 dp2 r2 dp
d2v /v d2u 1 d2v 1 9«\ /1 dv d2v 1 d2u 1 du, v
dr2 y r dr dp r2 dp2 r2 dp J y r dr dr2 r dr dp r2 dp r2
wherea = ^,re [l, f].
The boundary conditions of (3.2) are
du, v ( _ dv dr r y dp
dv 1 ( _ du,
dr r y dp
(3.5)
0.
The solution to the problem (3.4)-(3.5) is expressed as follows:
и, = ît,(r) exp(-iwf) cos (hp), v = v(r) exp(¿wr) sin(fcp),
(3.6)
where u is some natural frequency of the disk and n is the symmetry index of the oscillations. Substituting (3.6) into (3.4) and (3.5), the equations for the unknown functions u(r), v(r) are
obtained:
2~ d2u u 1 du 1 v .1 dv / v ,1 dv j2 u
u U + —T + 3J + = - k=9 + KV=— - a K^J - k-—+k
(jf* 'p j* (j'p f* 'p C/'P \ f* ¥ C/T ¥
d2v dr2
1 dv f dr
v
u ¿2
1 du f dr
2~ I d v 1 dv v , u 1 du \ 1 du 2 v , ^ v^ ^
uj v + a ( + =— — zö — ^=9 — k-— — Kv-— — S j — KU-— = 0,
1 du f dr
1 du f dr
(3.7)
du v, „
7Z + ~{U + kv) dr r
dv 1, „
— - -(v + to) or r
0,
(3.8)
7= l,f
A huge number of works are devoted to derivation of analytical solutions to spectral problems for disks and rings [39-47]. However, within the framework of this study, it is sufficient to obtain a numerical verified solution of this problem. Equations (3.7) together with the boundary conditions (3.8) represent a boundary eigenvalue problem (BVP). To solve such problems, one can use numerical methods for integrating equations in Matlab [73], namely, the bvp4c method. The solver is a finite-difference scheme implementing the three-step Lobatto formula (collocation formula). The collocation method uses a grid of points to divide the integration interval into smaller segments. The solution of a global system of algebraic equations arising from boundary conditions and collocation conditions imposed on all the intervals obtained allows us to find a numerical solution. Then the error of the numerical solution is estimated at each interval. If the solution does not meet the convergence criteria, the grid adapts and the process repeats again. To solve a two-point boundary value problem, it is necessary to provide the points of the initial grid, as well as the initial approximation of the solution at the grid points.
vin6
600 S 500 r
0.3 0.4 0.5 0.6 0.7 0.8 0.9
b
0.3 0.4 0.5 0.6 0.7 0.8 0.9 b
(a)
(b)
Fig. 4. The first natural frequency of the resonator f = on the ratio of the inner and outer radius of the disk
2?r /
with the symmetry index k = 3 depending
The problem was also solved by the finite-element method. The ratio of outer to inner radii of the disk varied within a certain range of values. It can be seen from the graph in Fig. 4 that, with a ratio of the inner and outer disk radii close to unity, the solution is in good agreement with the formula [70] known in the literature and obtained for a thin ring:
w.
ring
k (k2 ~ 1) 2vr (k2 + 1)
Vk2 + u
I EI pSR4'
(3.9)
r=l,f
Figure 5 presents the eigenfunctions u, v corresponding to the waveform k = 3.
—a=660 (¡tin) 6=410 (iim)
—a=420 (pm) 6=240 (/im)
a=240 (/un) - 6=120 (/tm)
33 iSlib
1.6 0.2
1.5 0
1.4 -0.2
1.3 iblib SIS -0.4
1.2 -0.6
1.1 -0.8
1 -1
Fig. 5. The eigenfunctions of the resonator vibrations for k = 3: the solid lines correspond to the variation of the outer radius at b = 210 pm, and the dotted lines correspond to the variation of the inner radius at a = 420 pm
Figure 6 shows the working waveform k obtained using finite-element modeling.
= 3 for various resonator sizes. The results are
6= 120 /¿m a = 420
6 = 210 fim a = 600
6 = 210 nm a = 420
6 = 400 /im a = 420
6 = 210 /um a = 320 iim^^
» 3999,32 71)95,61 10181,8 13286,2 18384,5 19480,8 22577,1 25873,4 3
« J
V/
1550,51 8586,61 15522,7 22508,8 29484^ 36481 43487,1 50453,2 57438,3 04425,4
6 = 210 /¿m a = 240
21887,1 345084 47029,8 58561,3 72072,7
Fig. 6. Operating forms of oscillation when changing the dimensions of the resonator
It can be seen from the analysis of the results obtained that a change in the dimensions of the resonator significantly affects both the eigenfrequencies and the eigenfunctions. However, while maintaining the proportions of the outer and inner radii of the disk, the modes of oscillations do not change significantly. From the above results, it becomes clear that the solution by the BVP method is accurate, which means that it can be used to build a discretized dynamic MRIG model.
3.2. Frequency splitting analysis 3.2.1. Gyroscopic splitting
This section discusses the spectral problem for a disk resonator on a movable (rotating) base. The effect of gyroscopic splitting will be studied on the basis of a discrete model. All nonlinear terms in Eqs. (2.20) are discarded, take R = 0, and hence A3 = 0. Thus, for an ideal disk, the equations of motion take the form
C + C = -AS, S + S = A4C.
(3.10)
Dimensional gyroscopically separated natural frequencies of the operating mode of oscillations are determined by the expression
2 2 wl,2 = W
1 + Iii ± + 4
M 2
2
(3.11)
where lo = naf urai frequency of the fixed disk and Ad = — " ^Tl
V (1 -v2>T6 Po 4 E(U12+U\
The relative frequency splitting is determined by the expression
v)
Wi — Wn
w
(3.12)
Figure 7 shows the dependence of the deviation of the frequency of the rotating disk from the frequency of the stationary resonator in the presence of rotation. The physical parameters of the resonator are taken from Table 1. The expression (3.11) can be compared with the well-known formula obtained in [70] for a ring:
wr =
k(k2 - 1) / EI
(k2 + 1) I/ pSR4
(k2 + 1) ± 2Q
k
k2 + 1'
(3.13)
e
1.2
(a) (b)
Fig. 7. Relative frequency splitting as a function of disk thickness, the dashed lines correspond to the ring model
iXlO~6
-a=240 (fim) a=300 (/mi) a=420 (/im) — a=630 (/im) =210 (/mi
, xlO
-6
— 6=380 (/jm) 6=300 (/im) 6=210 (/im) 6=120 (/im)
|a=420(/mi|
4 6 Q (rad/s)
(a)
4 6 (rad/s)
(b)
Fig. 8. Relative frequency splitting depending on the angular velocity, dashed lines correspond to the ring model
It can be seen from Figs. 7-8 that for f
1 the dependencies coincide, however, at f < 0.7,
there is a noticeable difference.
Interestingly, as the outer radius increases, the relative frequency splitting first falls and then slightly increases (see Figs. 7a, 8a). This is due to the fact that the natural frequency of a stationary disk also behaves nonmonotonically, and has an extremum (Fig. 4a).
From the results presented above, it can be concluded that the model obtained is in good agreement with the ring model for a disk of small thickness. However, for a sufficiently large thickness of the resonator, a noticeable discrepancy is seen, which requires using a solution to the problem of free vibrations of a hollow disk constructed above.
3.2.2. Influence of mass and stiffness imperfections
In an imperfect gyroscope, the presence of mass and stiffness imperfections inevitably leads to a change in the natural frequency of the system. Consider the influence of imperfections on resonator oscillations. The natural frequencies of a nonideal disk (see system (2.19)) looks like
^1,2 =
\
(A^ + A2A25 - l) ± \J(A1Alg + A2A25 - l)2 - (-A? - A2 + 1) (-A?8 - Al5 + l)
(-A? - A| + 1)
(3.14)
Figures 9-11 show the dependence of relative frequencies of a nonideal resonator on the parameters of mass and stiffness imperfections.
It can be concluded that the presence of mass and stiffness imperfections clearly affects the splitting of the operating vibration frequency. The frequency difference caused by mass defects is significantly greater than that caused by stiffness imperfections. However, for certain combinations of defects, splitting is absent. It is obvious that mass defects compensate for the frequency splitting caused by rigidity defects and vice versa.
3.2.3. Influence of the electrostatic excitation
Consider Eqs. (2.20). The coefficient at cubic nonlinearity Ag is the sum of two terms. The first stands for electrical forces, and the second is responsible for the geometric nonlinearity of
rnl 3
<31
1.003 1.002 1.001 1
0.999 0.998 0.997 0.996
0
2TT
3
7T
Vp
47T 3
5tt 3
(a) (b)
Fig. 9. Dependences of the natural frequency of the resonator on the parameter: (a) y, (b) yE
<313 1
7T 2w
3 3
7T ^P
47T Ö7T
3 3
Fig. 10. Dependences of the natural frequency of the resonator on the parameter yp
1.25 1.2 1.15 1.1 13 1.05 1
0.95 0.9 0.85,
<3
SE = 0. ■¿B = 0.5 —i„ = 0.
¥>„ = 0 Ve = 0
0 0.2 0.4 0.6 0.8
(a)
1.02 1.015 1.01 1.005
3 1
0.995 0.99 0.985 0.98,
X—
-<5„ = 0 ---- --
¿„ = 0.01 ------—
—¿,, = 0.03 ¥>„ = 0
— = 0.05 V>e = °
0 0.2 0.4 0.6 0.8 1
je
(b)
Fig. 11. Dependences of the natural frequency of the resonator on the parameter: (a) Sp, (b) SE
the resonator:
a8 — a8. + a8
Aa — -
7u(a)4 (1 - v2) e0a (8AW2 + 4W2n2 + AW2n2 + 4WAWn2)
20E1 d3n2(U12 + vUll )
(3.15)
A—
2d2 (U4 + vU3)
U12 + vU-
11
where U3, U4, U12, U13 are coefficients depending on the resonator's eigenforms represented in (2.21). Figure 12 shows the dependence of the coefficient A8 on the ratio of the inner and outer radii of the disk.
n n1 —a=420 (H U-Ui 6=210 (/¿m)
0.008 5 4
« 0.006 ^ 3
0.004 2
xlO"
0.002 0
0.2 0.3 0.4 0.5 0.6 /
0.3 0.4 0.5 0.6 0.7 0.8 0.9
b
Fig. 12. Dependence on the ratio of the inner and outer radii of the disk
e
g
e
g
As can be seen from the figure, when the outer radius is varied with decreasing disk thickness, the relative coefficient of cubic nonlinearity becomes larger and, conversely, decreases by several orders of magnitude with increasing thickness. It is worth mentioning one interesting phenomenon: when the inner radius changes, the coefficient first similarly decreases, but then starts growing again.
Figure 13 shows the dependence of the coefficient at the cubic nonlinearity A8 on the constant and variable components of the electric voltage. The graphs are constructed for a resonator with physical parameters in accordance with Table 1.
As can be seen from the figure, when a certain voltage value is reached, the value A8 becomes zero. This is due to the fact that the electrical part of the system balances the geometric one
A8 — A8 and, therefore, the optimal tuning of voltage removes the cubic term in Eq. (2.20) and potentially increases the amplitudes of excited oscillations.
Figure 14 shows the space of parameters under which the relation A8 (AW, W) — 0 is satisfied.
It is worth noting that, as ^ decreases, the voltage first increases and then decreases. Moreover, the voltage drops if the change in the outer radius occurs much earlier than the change in the inner radius.
00
2 3 W (V)
(a)
2 3 AW (V)
(b)
Fig. 13. Dependence of A8 on (a) the DC voltage, (b) the AC voltage
V 0 V(2v)
(b)
Fig. 14. Parameter space for A8 (AW, W) = 0 when changing (a) outer and (b) inner disk radii
4. Nonlinear dynamics of the MRIG
This section is devoted to a qualitative study of the dynamics of the resonator in the regime of parametric excitation, taking into account the geometric and electrostatic nonlinearity of the system. Free oscillations of a moving resonator are considered, an expression for the precession rate of the wave pattern of oscillations is found, the zones of parametric resonance are determined, and the frequency response for a resonator on a fixed base is derived in the region of the main parametric resonance. The dependences of the precession frequency on the angular velocity, the amplitude of the electrostatic force and the frequency detuning of the excitation system are obtained. The influence of mass and stiffness imperfections on the precession frequency of the resonator in the operating mode is investigated.
4.1. Free nonlinear oscillations of a rotating resonator
The equations of free oscillations of the resonator can be obtained by eliminating in (2.20) all terms associated with the electrical part of the system:
C + C + A3C + A4S + A8 C {C2 + S2) = 0,
g
S + S - a4C + A3S + a8 s (C2 + S2) = 0.
2 , ^ ~
The system (4.1) is investigated by the Krylov-Bogolyubov averaging method [74]. For this purpose, the change of variables according to the formulas is performed:
C = a(r) cos ( ( 1 + + m(T) sin + \G\ ) T
5 = b(r) COS ( ( 1 + i<7! J T j + n(r) Sin if 1 + i<7! ) T ) ,
(4.2)
where a(r), m(r), b(r), n(r) are slowly changing variables. The procedure for averaging the equations of motion over fast time leads to the following equations for slow variables:
3
à = ^-Ag a2m + abn — ^A3a + b2,m — ^A4b + m3 + rim2,
1
8"°g
3
3
m = — a3 — -A ab2 — -A am2 — -A an2 — \ar brim — -A-vm — -A,??,, 8 g 8 g 8 g 8 g 4 g 2 3 2
b = -Ao a2n + ^-Ao abm + -A*a + -A b2n - -A-,b + -A m2n + - A, n3
O 8 g A 8g O 4 O 8 g O 3 O 8g O 8g
3
3
(4.3)
3
n = — ^Agji2b — ^Ag^anm — -¿A8nb3 — ^Agbni2 — ^Agbn2 + ^A4ni — ^A3n.
3
1
3
Similar equations with different values of the coefficients were encountered in [75] in the study of ring resonators. It is convenient to trace the oscillation pattern of the resonator using the toroidal coordinates r, k, 0, £, called the orbital elements [70]:
C = r cos({ + t) cos(0) — k sin({ + t) sin(0), S = r cos({ + t) sin(0) + k sin({ + t) cos(0), C = —r sin({ + t) cos(0) — k cos({ + t) sin(0), S = —r sin({ + t) sin(0) + k cos({ + t) cos(0).
(4.4)
Fig. 15. Common elliptical orbit and resonator precession in phase space
The total oscillation energy E and the angular momentum K are expressed in terms of k, r as follows:
E = i (c2 + S2 + C2 + S2) = i (r2 + k2), K = CS-CS = rk
(4.5)
The variables k and r describe the main dynamic characteristics of the free oscillations of the resonator and are the semiaxes of the ellipse. The value d is the angle of inclination of the semiaxis of the ellipse to the axis C and £ determines the position of the point on the trajectory.
Substituting (4.5) into (4.2) and performing the necessary transformations, the system can be written in terms of the new slow variables. Averaging the resulting system over the explicitly incoming time, the equations can be written as
r = 0 ^ r = r0, k = 0 ^ k = k0,
Ö = \A4- jÄ8kr, ï = (*2 + r2).
(4.6)
In dimensional form, the expression for d takes the form
n
• ^ 1 „ , [2rU(r)V(r) dr 9 = rjQ- -ujA8 k0r0, rj =-^--\—•
4 9 Jr(uu(r)2 + V(r)2J dr
(4.7)
As expected, analysis of the expression (4.7) shows that the rate of precession of the wave pattern of oscillations is proportional to the angular velocity of the base with a scale factor. Figure 16 shows the dependence of the precession coefficient n on the ratio of the inner and outer radii of the disk.
0.3 0.4 0.5 0.6 0.7 0.8 0.9 b
a
(a)
xlO6 5 0.6
4 0.59
3 ^ K 2^ 0.58 ^0.57 0.56
1 0.55
0 0.54
0.3 0.4 0.5 0.6 0.7 0.8 0.9
b a
(b)
Fig. 16. Dependence of the precession coefficient on the ratio of the inner and outer radii of the disk
As expected, it follows from formula (4.7) that the precession coefficient, or Bryan's coefficient, is a purely geometric parameter that depends only on the shape of the resonator. Figure 16 shows that the coefficient changes nonmonotonically when the size of the disk changes and has an extremum at some point £ œ 0.5. The detected presence of an extremum in this dependence indirectly indicates the low sensitivity of the MRIG to perturbations of its geometric and, probably, mass-inertial characteristics in the vicinity of the specified disk size ratio. The subject of further research may be the study of the possibilities of using this effect in order to develop sensitive elements of MRIGs with minimal parasitic deviation from the operating precession mode in the presence of technologically inevitable imperfections.
The precession frequency of the standing wave is also proportional to the angular momentum K = k0r0. It follows that the nonlinear elastic properties of the resonator material and the nonzero values of the quadrature K lead to a drift of the gyroscope. According to [70, 75], this error is present in all gyroscopes that implement the idea of the Foucault pendulum. In order to eliminate this drift, one of the semiaxes of the ellipse is kept constant and the other is kept equal to zero.
4.2. Analysis of MRIG dynamics on a fixed base
Let us proceed by studying parametrically excited oscillations of the resonator using a nonlinear model of the system. When Q = 0 (A4 = 0) there will be no standing wave precession. Since the disk has two orthogonal eigenforms corresponding to one eigenfrequency, only one of the forms will be excited in this mode. Let us choose the initial conditions C0 = 0, C0 = 0, S0 = 0, S0 = 0. Thus, S = 0, and the system of equations (2.20) degenerates into one equation:
C + A3C + [(1 + a2)2 + A17 cos((4 + 2^)t) + A16 sin((2 + al)r)] C+
+ [A8 + A6 cos((4 + 2^ )t ) + A5 sin((2 + a1)r )]C3 = 0. (4.8)
The small parameter e is introduced as follows:
C + e2A3C + [(1 + ea2)2 + e2Ai7 cos((4 + 2^)t) + e2Ai6 sin((2 + al)r)] C+
+ [A8 + A6 cos((4 + 2a1 )t) + A5 sin((2 + a1)r)]C3 = 0. (4.9)
Let us solve Eq. (4.9) using the method of multiple scales [76] by presenting the solution as a series in the small parameter e:
C(t, e) = eC0(T0, T\) + e2C^, T1) + e3C:i(T0, T\) + ..., (4.10)
where T0 = t , T1 = eT.
The following relationships hold:
d ^ ^ - = D0 + eD1 + ....
d2
— = D2 + 2sD0D1 + ...,
(4.11)
where Dn = jM^.
To estimate the starting voltages in the main resonance zone, it suffices to obtain a firstorder expansion for the linear terms, in this case the calculations are performed up to terms of order e2. Equating to zero the coefficients at successive powers of e, a system of differential equations can be obtained. By isolating the secular terms in the resulting system, one can obtain an expression for the transition curves:
1 . 1
-2^3 + 4 V-+ (4(72 " ^i)2 = (4-12)
The results of calculating the transition curves for a disk with parameters from Table 1 are
shown in Fig. 17.
The areas under the transition curves correspond to damped oscillations, and the areas
inside the curves correspond to the oscillation buildup zones.
—A W= =100
A W= =5 00
A W= =10 (V)
—A W= =15 (V)
-AW= =20 (V)
9.045
9.05 9.055 A (Hz)
9.06 xlO6
20
15
>
10
<1
5
0
—W= 100
W= 5 00
W= 10 (V)
—w= 15 (V)
-w= 20 (V)
9.04 9.045 9.05 9.055 9.06 A (Hz) xlO6
Fig. 17. Transition curves in the region of the main parametric resonance. Variation of the variable voltage component
The solution in the first approximation for a nonlinear system will be
C = a cos(r + 3) + O(e), where a, (3 are from the solution of the system:
.1 3 1 1 3 1 a =-A5a cos(0) — -A3a + — A6a, sm(2ip) +-A16acos(ip), 8 2 16 4
3 1 1 1
= '2a2 + -A8a2 - -A5a2 sin(0) + -A6a2 cos(2tp) - -A16 sin(0) -
^ = 2( - a1T1.
(4.13)
(4.14)
For the system (4.14) it is possible to construct the frequency response analytically. Equating the right-hand sides to zero, the expression for the steady-state amplitude and phase can be found. Figure 18 shows the dependence of the steady-state oscillation amplitude of the resonator on the frequency detuning of the electrode excitation system at fixed values of the variable and constant components of the voltage across the electrodes. The dotted lines indicate unstable branches.
1 r s
N
0.8 0.6
0.2 0
AW = >001
—W = 20 (V)
W = 10 (V)
W = 5 00
—W = 1 (V)
1rs
N
0.8 0.6
0 4 \w=20 001 \
—AW = 20 (V) n9 AW = 15 (V) AW = 10 (V) AW = 5 (V)
9.02 9.03 9.04 9.05 9.06 A (V) xlO6
9.025 9.035 9.045 9.055 A (Hz) xlO6
Fig. 18. Amplitude-frequency characteristics for a parametrically excited resonator
4.3. Nonlinear oscillations of a rotating resonator under parametric excitation
Moving forward to dynamics of the rate-integrating gyroscope on the rotating base, the complete nonlinear system of equations (2.20) taking into account the action of the Coriolis
inertia forces is investigated. Similarly to the previous sections, an asymptotic approximation can be built using the multiple scale method by searching for modal coordinates in the form
C = a cos(t + Bi) + 0(e),
1 (4.15)
S = a2 cos(T + B2) + 0(e),
where a1, a2, B1, B2 are slowly varying amplitudes and phases, respectively, and in the case of an ideal disk are found from the equations:
a'i = -j— \2A5af cos(ip^ - SA^ - 8AAa,2 cos f^- - ^ ) + A6af sin(2-01) + 4A16a,1 cos{ip1)+ 16 \ \ 2 2 J
+A6a1a2 sin0 + 02) + 2A5a1a2 cos(^1) + 2A8a1 a2 sin0 — 02) j, a/2 = ^ (2A54 cos(02) - 8A3a2 + 8A^ cos (y " y) + sin(202) + 4A16a2 cos(02)+ +A6a2a2 sin(^1 + 02) + 2A5af a2 cos(02) — 2A8a\a2 sin(^1 — 02) j,
ip[ = (1 Qala2 - 8a1a1 + 2^4«! + 6A8af - 4Aba\ sin^) + 8A4a2 sin (- ) + 1 8a1 1 2 1 1 4 1 8 1 5 1 1 4 2 2 2
+A6a3 cos(201) + 4A8a1a2 — 4A16a1 sin(^1) + A6a1a2 cos0 + 02) — 2A5a1a2 sin(^1) — —2A5a1a2 sin(^) + 2A8 a1 a2 cos(01 — 02 )),
02 = (l6a2<72 - 80,2(7! + 2Aia2 + 6A8a| - 4A5a% sin(02) + 8A^ sin (y " y) +
+A6a3 cos(202) + 4A8a^a2 — 4A16a2 sin(02) + A6a2a2 cos0 + 02) — 2A5afa2 sin0) — —2A5a^a2 sin(02) + 2A8a\a2 cos0 — 02)
01 = 2B1 — a^,
02 = 2B2 — (4.16)
In the operating mode of oscillations, due to the presence of the angular velocity, the standing wave excited in the resonator begins to precess, which means that the amplitude of the steady-state oscillations should periodically change with time, while the phase of the oscillations will take a constant value. Such a mode for a system in slow variables should represent a limit cycle. However, due to some peculiarities of the numerical integration of the obtained system (resulting from the fact that in free-precession mode slow-time amplitudes in (4.16) cross zero value once per period of precession), at certain times there is a sharp jump in the amplitude and phase of Fig. 19, which complicates further analysis of the resulting system.
Therefore, it is convenient to investigate the original system by the Krylov - Bogolyubov averaging method [74]
C = a(r) cos + + m(T) sin + T
S = b(r) cos + r) + n{r) sin + ±<7^ r) ,
where a(T), m(T), b(T), u(t) are slowly varying amplitude variables. Substituting (4.17) into (2.20) and averaging over the fast variable, a system of differential equations for slowly varying
40 60 80 100
t (sec)
10
5
<D fi c§ U
* -5 -10
—M*) <h®
20
40 60
t (sec)
80 100
Fig. 19. Integration results of the system in slow variables (W = 10 (V), AW = 4 (V), a1 Q = !0 (rad/sec))
-6.5e - 4,
amplitudes can be obtained:
a = fi(a, m, b, n), m = f2(a, m, b, n), b = f3(a, m, b, n), n = f4(a, m, b, n).
(4.18)
The explicit form of the functions /1, f2, f3, f4 is not given in this work since the expressions are cumbersome. The operating mode of the resonator for the system (4.18) is the limit cycle whose period is the period of the standing wave precession. Figure 20 shows the results of numerical integration of the resulting system in slow variables for a disk with parameters from Table 1.
0.5
1
t (sec)
1.5
Fig. 20. Integration results of the system in slow variables (W = 10 (V), AW = 4 (V), ai = —6.5 x 10 4, Q = 10 (rad/s))
Using the numerical continuation methods of the Matcont software package [77], the dependence of the standing wave precession frequency on the angular velocity is obtained and shown
in Fig. 21. The graph also shows the frequency of precession of free oscillations of the resonator in the absence of quadrature error caused by geometric nonlinearity (see expression (4.7)).
fi (rad/sec)
Fig. 21. Dependence of the precession frequency on the angular velocity, (W = 10 (V), AW = 4 (V), ax = -6.5 x 10"4)
It can be seen from the graph that the precession frequency of the wave pattern of free oscillations is quite close to the precession frequency of parametrically excited oscillations, but the dependences do not completely coincide. Similarly, the obtained dependence of the precession frequency on the value of the alternating voltage is shown in Fig. 22a. It can be seen from the figure that, as the constant voltage acting on the resonator increases, the precession frequency of the wave pattern decreases, and with decreasing voltage, it tends, as expected, to the precession frequency of free oscillations of the resonator.
The dependence of the precession frequency on the DC voltage is shown in Fig. 22b. As can be seen from the figure, with an increase in the voltage acting on the resonator, the frequency of the precession of the wave pattern decreases, and with decreasing voltage, it tends to the precession frequency of free oscillations of the resonator.
3.5 4 4.5 AW (V)
(a)
9
W (V)
(b)
Fig. 22. Dependence of the precession frequency on the magnitude of the (a) variable component and (b) constant component of the electric field (Q = 10 (rad/sec), ai = -6.5 x 10~4)
Let us see how the presence of mass and stiffness imperfections affects the resonator dynamics. Figure 23 shows the deviation of the precession frequency of the resonator in the presence of imperfections from the precession of an ideal disk from the parameters 5p and 5E. The results were obtained by numerical continuation of the found limit cycle (precession mode) in slow-time-variables phase space with active (continued) parameters being values of imperfections.
—a =10 (rad/s)
a =20 (rad/s)
-si =5 (rad/s)
(a)
(b)
Fig. 23. Dependence of the precession frequency on the parameter (a) SE and (b) Sp (W = 10 (V), AW = 4 (V), a1 = -6.5 x 10"4)
Figures 24, 25 show the envelopes of modal coordinates for the dimensionless period of the limit cycle for various parameters of imperfections of mass and stiffness.
<5n = 0
Sw = le —06
öp = 1.5e—06
Fig. 24. Evolution of the limit cycle with changing rigidity imperfections (W = 10 (V), AW = 4 (V), a1 = -6.5 x 10~4, Q = 10 (rad/sec))
It should be noted that, despite the obvious change in the amplitude parameters, the trajectory of the precession of the standing wave on the plane C, S, which is an ellipse, remains unchanged in the presence of defects in mass and rigidity.
0 0.5 1 0 0.5 1 0 0.5 1
C1
A'l
t t t
Fig. 25. Evolution of the limit cycle with changing mass imperfections (W = 10 (V), AW = 4 (V), a1 = -6.5 x 10~4, i = 10 (rad/sec))
The above results indicate that the proposed approach to modeling the nonlinear dynamics of MRIG allows, in contrast to the currently known methods of analysis, to carry out a qualitative (parametric) study of the nature of the precession of an excited elastic wave to determine the period of this precession and its shape depending on the parameters of electrostatic excitation system, mass and stiffness imperfections of the resonator, its geometry and other factors.
5. Conclusion
The nonlinear dynamics of the microscale rate integrating gyroscope with disk resonator was investigated in the region of the main parametric resonance. Differential equations of parametric oscillations of the MRIG were obtained considering the geometric and electrical nonlinearities of the system in the presence of mass and stiffness imperfections of the resonator. The starting voltages were estimated in accordance with the specific parameters of the possible design of the MRIG. Resonance curves were constructed using asymptotic methods, and the stability of the found stationary solutions was investigated. The influence of mass and stiffness defects of the disk on the spectral properties, as well as on the nature of free precession in the operating mode of resonator oscillations, was shown. An effect of dependence of the Bryan factor on the ratio of the geometrical parameters of the disk was revealed. A fairly general method for the qualitative study of the free precession regime in systems of the Foucault pendulum type was proposed, based on the application of numerical continuation methods. The proposed dynamic model of MRIG can be used to develop calibration and dynamic balancing methodology in the presence of material and geometric imperfections, as well as algorithms for controlling the oscillations of the sensitive element.
Conflict of interest
The authors declare that they have no conflicts of interest.
References
[1] Peshekhonov, V. G., The Outlook for Gyroscopy, Gyroscopy Navig., 2020, vol. 11, no. 3, pp. 193-197; see also: Giroskop. Navig., 2020, vol. 28, no. 2 (109), pp. 3-10.
[2] Pourkamali, S., Zhili Hao, Ayazi, F., VHF Single Crystal Silicon Capacitive Elliptic Bulk-Mode Disk Resonators: Part 2. Implementation and Characterization, J. Microelectromech. Syst., 2004, vol. 13, no. 6, pp. 1054-1062.
[3] Johari, H. and Ayazi, F., High-Frequency Capacitive Disk Gyroscopes in (100) and (111) Silicon, in
2007 IEEE 20th Internat. Conf. on Micro Electro Mechanical Systems (MEMS, Hyogo, Japan, Jan 2008), pp. 47-50.
[4] Johari, H., Shah, J., and Ayazi, F., High Frequency XYZ-Axis Single-Disk Silicon Gyroscope, in
2008 IEEE 21st Internat. Conf. on Micro Electro Mechanical Systems (MEMS, Tucson, Ariz., Jan 2008), pp. 856-859.
[5] Mirjalili, R., Wen, H., Serrano, D.E., and Ayazi, F., Substrate-Decoupled Silicon Disk Resonators Having Degenerate Gyroscopic Modes with Q in Excess of 1-Million, in Proc. of the 18th Internat. Conf. on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS, Anchorage',, Alaska, Jun 2015), pp. 15-18.
[6] Serrano, D.E., Zaman, M.F., Rahafrooz, A., Hrudey, P., Lipka, R., Younkin, D., Nagpal, S., Jafri, I., and Ayazi, F., Substrate-Decoupled, Bulk-Acoustic Wave Gyroscopes: Design and Evaluation of Next-Generation Environmentally Robust Devices, Microsyst. Nanoeng., 2016, vol. 2, Art. 16015, 10 pp.
[7] Serrano, D.E., Lipka, R., Younkin, D., Hrudey, P., Tovera, J., Rahafrooz, A., Zaman, M.F., Nagpal, S., Jafri, I., and Ayazi, F., Environmentally-Robust High-Performance Tri-Axial Bulk Acoustic Wave Gyroscopes, in 2016 IEEE/ION Position, Location and Navigation Symposium (PLANS, Savannah, Ga., Apr 2016), pp. 5-8.
[8] Acar, C. and Shkel, A., MEMS Vibratory Gyroscopes: Structural Approaches to Improve Robustness, New York: Springer, 2008.
[9] Challoner, A.D., Ge, H.H., and Liu, J.Y., Boeing Disc Resonator Gyroscope, in 2014 IEEE/ION Position, Location and Navigation Symposium (PLANS, Monterey, Calif., May 2014), pp. 504-514.
[10] Su, T.-H., Nitzan, S.H., Taheri-Tehrani, P., Kline, M.H., Boser, B.E., and Horsley, D.A., Silicon MEMS Disk Resonator Gyroscope with an Integrated CMOS Analog Front-End, IEEE Sens. J., 2014, vol. 14, no. 10, pp. 3426-3432.
[11] HyuckAhn, Ch., Nitzan, S., Ng, E., Hong, V., Yang, Y., Kimbrell, T., Horsley, D., and Kenny, T., Encapsulated High Frequency (235 kHz), High-Q (100 k) Disk Resonator Gyroscope with Electrostatic Parametric Pump, Appl. Phys. Lett., 2014, vol. 105, no. 24, 243504, 5 pp.
[12] HyuckAhn, Ch., Ng, E., Hong, V., Yang, Y., Lee, B., Flader, I., and Kenny, Th., Mode-Matching of Wineglass Mode Disk Resonator Gyroscope in (100) Single Crystal Silicon, J. Microelectromech. Syst., 2015, vol. 24, no. 2, pp. 343-350.
[13] HyuckAhn, Ch., Shin, D. D., Hong, V., Yang, Y., Ng, E. J., Chen, Y., Flader, I. B., and Kenny, Th., Encapsulated Disk Resonator Gyroscope with Differential Internal Electrodes, in 2016 IEEE 29th Internat. Conf. on Micro Electro Mechanical Systems (MEMS, Shanghai, China, Jan 2016), pp. 962-965.
[14] Li, Q., Xiao, D., Zhou, X., Xu, Y., Zhuo, M., Hou, Z., He, K., Zhang, Y., and Wu, X., 0.04 Degree-Per-Hour MEMS Disk Resonator Gyroscope with High-Quality Factor (510 k) and Long Decaying Time Constant (74.9 s), Microsyst. Nanoeng., 2018, vol. 4, Art. 32, 11 pp.
[15] Zhou, X., Xiao, D., Li, Q., Hou, Zh., He, K., Chen, Zh., Wu, Y., and Wu, X., Decaying Time Constant Enhanced MEMS Disk Resonator for High Precision Gyroscopic Application, IEEE ASME Trans. Mechatron, 2018, vol. 23, no. 1, pp. 452-458.
[16] Li, Q., Xiao, D., Zhou, X., Hou, Zh., Xu, Y., and Wu, X., Quality Factor Improvement in the Disk Resonator Gyroscope by Optimizing the Spoke Length Distribution, J. Microelectromech. Syst., 2018, vol. 27, no. 3, pp. 414-423.
[17] Fan, Q., Lin, Ch., Liu, M., Su, Y., Zhao, W., and Zheng, D., High Performance MEMS Disk Gyroscope with Force-to-Rebalance Operation Mode, in Proc. IEEE Sensors (Montreal, Canada, Oct 2018), 3 pp.
[18] Bo, F., Guo, Sh., Cheng, M., Yu, L., Zhou, M., Hu, W., Chen, Z., and Xu, D., A Novel High-Symmetry Cobweb-Like Disk Resonator Gyroscope, Sensors J., 2019, vol. 19, pp. 10289-10297.
[19] Ren, X., Zhou, X., Tao, Y., Li, Q., Wu, X., and Xiao, D., Radially Pleated Disk Resonator for Gyroscopic Application, J. Microelectromech. Syst., 2021, vol. 30, no. 6, pp. 825-835.
[20] Martynenko, Yu., Merkuriev, I., and Podalkov, V., Dynamics of a Ring Micromechanical Gyroscope in the Forced-Oscillation Mode, Gyroscopy Navig., 2010, vol. 1, no. 1, pp. 43-51; see also: Giroskop. Navig, 2009, no. 3, pp. 10-22.
[21] Taheri-Tehrani, P., Guerrieri, A., Defoort, M., and Horsley, D., Mutual 3 : 1 Subharmonic Synchronization in a Micromacined Silicon Disk Resonator, Appl. Phys. Lett., 2017, vol. 111, no. 18, 183505, 5 pp.
[22] Jie, D., Simulating the Performance of Ring-Based Coriolis Vibrating Gyroscopic Sensors, Microsyst. Technol., 2018, vol. 25, no. 1, pp. 139-149.
[23] Chouvion, B., Mcwilliam, S., and Popov, A., Effect of Nonlinear Electrostatic Forces on the Dynamic Behaviour of a Capacitive Ring-Based Coriolis Vibrating Gyroscope under Severe Shock, Mech. Syst. Signal. Process., 2018, vol. 106, pp. 395-412.
[24] Di Napoli, V., McWilliam, S., and Popov, A.A., Frequency Splitting in MEMS Ring-Based Coriolis Vibrating Gyroscopes Caused by Support Non-Linearity, Proceedings, 2018, vol. 2, no. 13, Art. 755, 4 pp.
[25] Maslov, D. and Merkuryev, I., Increase in the Accuracy of the Parameters Identification for a Vibrating Ring Microgyroscope Operating in the Forced Oscillation Mode with Nonlinearity Taken into Account, Russian J. Nonlinear Dyn., 2018, vol. 14, no. 3, pp. 377-386.
[26] Li, Q., Xiao, D., Zhou, X., Hou, Zh., Zhuo, M., Xu, Y., and Wu, X., Dynamic Modeling of the Multiring Disk Resonator Gyroscope, Micromachines, 2019, vol. 10, no. 3, Art. 181, 20 pp.
[27] Maslov, A.A., Maslova, D., Merkuryev, I., and Podalkov, V. V., Development of Methods for Identification of Nonlinear Mathematical Model Parameters of Solid-State Wave Gyroscope, in Proc. of the 27th Saint Petersburg Internat. Conf. on Integrated Navigation Systems (ICINS, St. Petersburg, Russia, May 2020), 4 pp.
[28] Flader, I.B., HyuckAhn, Ch., Gerrard, D.D., Ng, E. J., Yang, Y., Hong, V.A., Pavone, M., and Kenny, Th. W., Autonomous Calibration of MEMS Disk Resonating Gyroscope for Improved Sensor Performance, in 2016 American Control Conference (ACC, Boston, Mass., Jul 2016), pp. 5803-5810.
[29] Hu, Zh. and Gallacher, B. J., Precision Mode Tuning towards a Low Angle Drift MEMS Rate Integrating Gyroscope, Mechatronics, 2017, vol. 56, pp. 306-317.
[30] Hu, Zh. and Gallacher, B. J., A Mode-Matched Force-Rebalance Control for a MEMS Vibratory Gyroscope, Sens. Actuator A Phys., 2018, vol. 273, pp. 1-11.
[31] Maslov, A.A., Maslov, D. A., Merkuryev, I.V., and Podalkov, V. V., Methods to Eliminate Nonlin-earity of Electrostatic Control Sensors of the Wave Solid-State Gyroscope, in Proc. of the 25th Saint Petersburg Internat. Conf. on Integrated Navigation Systems (ICINS, St. Petersburg, Russia, May 2018), 3 pp.
[32] Li, Ch., Wen, H., Wisher, S., Norouzpour-Shirazi, A., Lei, J., Chen, H., and Ayazi, F., An FPGA-Based Interface System for High-Frequency Bulk-Acoustic-Wave Microgyroscopes with In-Run Automatic Mode-Matching, IEEE Trans. Instrum. Meas., 2020, vol. 69, no. 4, pp. 1783-1793.
[33] Ruan, Zh., Ding, X., Qin, Zh., Jia, J., and Li, H., Automatic Mode-Matching Method for MEMS Disk Resonator Gyroscopes Based on Virtual Coriolis Force, Micromachines, 2020, vol. 11, no. 2, Art. 210, 22 pp.
[34] Li, Ch., Wen, H., Chen, H., and Ayazi, F., A Digital Force-to-Rebalance Scheme for High-Frequency Bulk-Acoustic-Wave Micro-Gyroscopes, Sens. Actuator A Phys, 2020, vol. 313, 112181, 9 pp.
[35] Liu, J., Yang, Y., Guo, Sh., Fan, B., Xuan, L., and Xu, D., A Novel Compensation Method for Eliminating Precession Angular-Rate Bias in MEMS Rate-Integrating Gyroscopes, IEEE Access, 2020, vol. 8, pp. 145611-145619.
[36] Nitzan, S., Zega, V., Li, M., HyuckAhn, Ch., Corigliano, A., Kenny, Th., and Horsley, D., Self-Induced Parametric Amplification Arising from Nonlinear Elastic Coupling in a Micromechaical Resonating Disk Gyroscope, Sci. Rep., 2015, vol. 5, Art. 9036, 6 pp.
[37] Polunin, P.M. and Shaw, S.W., Self-Induced Parametric Amplification in Ring Resonating Gyroscopes, Internat. J. Non-Linear Mech., 2017, vol. 94, pp. 300-308.
[38] Gallacher, B. J., Hu, Zh., Harish, K.M., Bowles, S., and Grigg, H., The Application of Parametric Excitation in MEMS Gyroscopes, Applied Non-Linear Dynamical Systems, J. Awrejcewicz (Ed.), Springer Proc. Math. Stat., vol. 93, Cham: Springer, 2014, pp. 473-492.
[39] Chen, S. S. H. and Liu, T. M., Extensional Vibration of Thin Plates of Various Shapes, J. Acoust. Soc. Am., 1975, vol. 58, no. 4, pp. 828-831.
[40] Ambati, G., Bell, J.F.W., and Sharp, J.C.K., In-Plane Vibrations of Annular Rings, J. Sound Vibration, 1976, vol. 47, no. 3, pp. 415-432.
[41] Farag, N. and Pan, J., Modal Characteristics of In-Plane Vibration of Circular Plates Clamped at the Outer Edge, J. Acoust. Soc. Am., 2003, vol. 113, no. 4, pp. 1935-1946.
[42] Zhili Hao, Pourkamali, S., and Ayazi, F., VHF Single-Crystal Silicon Elliptic Bulk-Mode Capacitive Disk Resonators-Part: 1. Design and Modeling, J. Microelectromech. Syst., 2004, vol. 13, no. 6, pp. 1043-1053.
[43] Bashmal, S. Bhat, R., and Rakheja, S., In-Plane Free Vibration of Circular Disks Using Characteristic Orthogonal Polynomials in Rayleigh-Ritz Method, Can. Acoust., 2007, vol. 35, no. 3, pp. 164-165.
[44] Park, Ch., Frequency Equation for the In-Plane Vibration of a Clamped Circular Plate, J. Sound Vibration, 2008, vol. 313, nos. 1-2, pp. 325-333.
[45] Bashmal, S., Bhat, R., and Rakheja, S., Frequency Equations for the In-Plane Vibration of Circular Annular Disks, Adv. Acoust. Vib., 2010, vol. 2010, Art. 501902, 8 pp.
[46] Shi, X., Li, W., and Shi, D., Free In-Plane Vibrations of Annular Sector Plates with Elastic Boundary Supports, J. Acoust. Soc. Am, 2012, vol. 132, no. 3, Art. 2067.
[47] Wei, X. and Seshia, A. A., Analytical Formulation of Modal Frequency Split in the Elliptical Mode of SCS Micromechanical Disk Resonators, J. Micromech. Microeng., 2014, vol. 24, no. 2, 025011, 7 pp.
[48] Lychev, S., Manzhirov, A., and Joubert, S., Closed Solutions of Boundary-Value Problems of Coupled Thermoelasticity, Mech. Solids, 2010, vol. 45, no. 4, pp. 610-623; see also: Izv. Akad. Nauk. Mekh. Tverd. Tela, 2010, no. 4, pp. 138-154.
[49] Sharma, J., Grover, D., and Kaur, D., Mathematical Modelling and Analysis of Bulk Waves in Rotating Generalized Thermoelastic Media with Voids, Appl. Math. Model., 2011, vol. 35, no. 7, pp. 3396-3407.
[50] Jani, S. M. H. and Kiani, Y., Generalized Thermo-Electro-Elasticity of a Piezoelectric Disk Using Lord-Shulman Theory, J. Therm. Stresses, 2020, vol. 43, no. 4, pp. 473-488.
[51] Luo, A. and Mote, C., Nonlinear Vibration of Rotating Thin Disks, J. Vib. Acoust., 2000, vol. 122, no. 4, pp. 376-383.
[52] Luo, A., An Approximate Theory for Geometrically Nonlinear Thin Plates, Int. J. Solids Struct., 2000, vol. 37, no. 51, pp. 7655-7670.
[53] Lacarbonara, W., Arena, A., and Antman, S. S., Flexural Vibrations of Nonlinearly Elastic Circular Rings, Meccanica, 2015, vol. 50, no. 3, pp. 689-705.
[54] Yang, Y., Ng, E.J., Polunin, P.M., Chen, Y., Flader, I.B., Shaw, S.W., Dykman, M.I., and Kenny, Th. W., Nonlinearity of Degenerately Doped Bulk-Mode Silicon MEMS Resonators, J. Microelectromech. Syst., 2016, vol. 25, no. 5, pp. 859-869.
[55] Chorsi, M., Chorsi, H., and Gedney, S., A Conceptual Study of Microelectromechanical Disk Resonators, IEEE J. Multiscale Multiphysics Comput. Tech., 2017, vol. 2, pp. 29-37.
[56] Xianglin, W., Jiao, Y., Chen, Zh., and Ma, W., Nonlinear Transverse and In-Plane Vibrations of a Thin Rotating Disk, in IFToMM'2018: Proc. of the 10th Internat. Conf. on Rotor Dynamics, K.Cavalca, H.Weber (Eds.), Mech. Mach. Sci., vol. 62, Cham: Springer, 2019, pp. 135-147.
[57] Lu, T., Tsouvalas, A., and Metrikine, A., A High-Order Model for In-Plane Vibrations of Rotating Rings on Elastic Foundation, J. Sound Vibration, 2019, vol. 455, pp. 118-135.
[58] Joubert, S., Shatalov, M., and Manzhirov, A., Bryan's Effect and Isotropic Nonlinear Damping, J. Sound Vibration, 2013, vol. 332, no. 23, pp. 6169-6176.
[59] Joubert, S.V., Shatalov, M.Yu., and Spoelstra, H., On Electronically Restoring an Imperfect Vibratory Gyroscope to an Ideal State, in Mechanics for Materials and Technologies, H. Altenbach, R. Goldstein, E. Murashkin (Eds.), Adv. Struct. Mater., vol. 46, Cham: Springer, 2017, pp. 231-256.
[60] Joubert, S.V., Shatalov, M.Yu., Fay, T.H., and Sedebo, G. T., On Bryan's Law and Anisotropic Nonlinear Damping, in Proc. of the 24th Internat. Conf. on Sound and Vibration (ICSV, London, UK, Jul 2017), 8 pp.
[61] Sedebo, G. T., Joubert, S.V., and Shatalov, M.Yu., An Analytical Theory for a Three-Dimensional Thick-Disc Thin-Plate Vibratory Gyroscope, J. Phys. Conf. Ser., 2018, vol. 991, no. 1, 012072, 8 pp.
[62] Parajuli, M., Sobreviela, G., Pandit, M., Zhang, H., and Seshia, A.A., Sub-Deg-per-Hour Edge-Anchored Bulk Acoustic Wave Micromachined Disk Gyroscope, J. Microelectromech. Syst., 2021, vol. 30, no. 6, pp. 836-842.
[63] Hu, Zh. and Gallacher, B. J., Effects of Nonlinearity on the Angular Drift Error of an Electrostatic MEMS Rate Integrating Gyroscope, IEEE Sens. J., 2019, vol. 19, no. 22, pp. 10271-10280.
[64] Daruwalla, A., Wen, H., Mirjalili, R., and Ayazi, F., Epitaxially-Grown Thick Polysilicon for BAW Disk Resonator Gyroscopes with Very Low Dissipation, in 2018 IEEE Micro Electro Mechanical Systems (MEMS, Belfast, UK, Apr 2018), pp. 1008-1011.
[65] Stein, M., Nonlinear Theory for Plates and Shells including the Effects of Transverse Shearing, AIAA J, 1986, vol. 24, no. 9, pp. 1537-1544.
[66] Lu, T., Tsouvalas, A., and Metrikine, A. V., A High-Order Model for In-Plane Vibrations of Rotating Rings on Elastic Foundation, J. Sound Vibration, 2019, vol. 455, pp. 118-135.
[67] Polunin, P.M. and Shaw, S.W., Self-Induced Parametric Amplification in Ring Resonating Gyroscopes, Internat. J. Non-Linear Mech., 2017, vol. 94, pp. 300-308.
[68] Joubert, S.V., Shatalov, M.Yu., and Coetzee, Ch.E., Using Fourier Series to Analyse Mass Imperfections in Vibrtory Gyroscopes, J. Symb. Comput., 2014, vols. 61-62, pp. 116-127.
[69] Joubert, S.V., Shatalov, M.Yu., and Spoelstra, H., On Electronically Restoring an Imperfect Vibratory Gyroscope to an Ideal State, in Mechanics for Materials and Technologies, H. Altenbach, R. Goldstein, E. Murashkin (Eds.), Adv. Struct. Mater., vol. 46, Cham: Springer, 2017, pp. 231-256.
[70] Klimov, D. M., Zhuravlev, V. F., and Zhbanov, Yu. K., Quartz Hemispherical Resonator (Wave SolidState Gyroscope), Moscow: L.A.Kim, 2017 (Russian).
[71] Joubert, S.V., Shatalov, M.Yu., and Spoelstra, H., Optimisation of the Mass Balancing Process in Vibratory Gyroscopes, in Proc. of the 22nd Internat. Congr. on Sound and Vibration (Florence, Italy, Jul 2015), 8 pp.
[72] Kim, J.-H. and Kim, J.-H., Thermoelastic Damping Effect of the Micro-Ring Resonator with Irregular Mass and Stiffness, J. Sound Vibration, 2016, vol. 369, pp. 168-177.
[73] MATLAB R2019b 9.7.0.1190202, https://www.mathworks.com (2018).
[74] Mitropolsky, Yu. A. and Dao, Nguyen, Applied Asymptotic Methods in Nonlinear Oscillations, Solid Mech. Appl., vol. 55, Dordrecht: Springer, 2013.
[75] Merkuriev, I. and Podalkov, V., Dynamics of Micromechanical and Wave Solid-State Gyroscopes, Moscow: Fizmatlit, 2009 (Russian).
[76] Nayfeh, A. H. and Mook, D.T., Nonlinear Oscillations, New York: Wiley, 1979.
[77] Dhooge, A., Govaerts, W., and Kuznetsov, Yu. A., MATCONT: A Matlab Package for Numerical Bifurcation Analysis of ODEs, ACM Trans. Math. Softw, 2003, vol. 29, no. 2, pp. 141-164.
