Mathematical Modeling


 ANALYTICAL MODELING OF DISCRIMINATION
  SCHEME FOR DETECTION OF ANGULAR RATE
 AND ACCELERATION FOR A 4-DOF MEMS GYRO-
            ACCELEROMETER

                             Payal Verma, S.A. Fomchenkov

                    Samara National Research University, Samara, Russia



       Abstract. The paper reports an analytical model of a 4-DOF gyro-
       accelerometer consisting of 2-DOF drive and 2-DOF sense oscillators config-
       ured orthogonally. A detection scheme for time varying angular rate and linear
       acceleration, by combining the structural-model of gyro-accelerometer with the
       processes of synchronous demodulation and filtration, which leads to the in-
       phase and quadrature components of the system’s output signal. These two
       components can be utilized in the detection of angular motion and linear accel-
       eration. The in-phase signal can be used for angular rate detection and the quad-
       rature signal can be utilized for linear acceleration. Finally, the results of the
       model have been validated by comparing with MATLAB®/Simulink data which
       shows excellent matching with each other.


       Keywords: MEMS, 4-DOF, Gyro-accelerometer, Synchronous demodulation


       Citation: Verma Payal, Fomchenkov SA. Analytical modeling of discrimina-
       tion scheme for detection of angular rate and acceleration for a 4-DOF MEMS
       gyro-accelerometer. CEUR Workshop Proceedings, 2016; 1638: 700-708. DOI:
       10.18287/1613-0073-2016-1638-700-708


1      Introduction

It is a well-known fact that all vibratory gyroscopes operate on the basis of transfer of
energy from one mode to the other. The device may have either single DOF [1] or
multi-DOF oscillators [2-5] which act as two orthogonally configured subsystems, a
self-tuned oscillator forming the drive mode and a micro-g accelerometer, forming the
sense mode. In the event of an angular rate, the transfer of energy from one mode to
the other is detected and processed using suitable circuitry to produce the desired
output. It is quite obvious that all vibratory gyroscopes can also sense linear accelera-
tion in addition to angular rate sensing at their events of occurrence. Considering the
strategy of simultaneous detection of linear acceleration and angular rate at their
events, a controller circuit has been reported for a 2-DOF conventional gyroscope [6].



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Some multi-DOF systems have also been proposed and realized which can sense line-
ar acceleration along with angular rate [7, 8], while offering other advantages such as
increased robustness and immunity to fabrication imperfections.
For the development of superior performance inertial sensors, the characteristics of
the device have to be thoroughly understood and the design optimized which can
achieved by taking proper care in the design and modeling stages. Hence mathemati-
cal modeling plays a key role in device design as well for optical elements [9-13].
Various mathematical models have been reported separately for accelerometer and
gyroscope devices. Some of the mathematical models for gyroscopes have reported
acceleration effect as an error, however confirming its presence. Recently, mathemat-
ical models of multi-DOF structures for simultaneous detection of acceleration effect
and angular rate have also been reported, few of them are, 2-DOF gyro-accelerometer
[14], a 2-DOF drive and 1-DOF sense gyro-accelerometer [7, 15, 16] and a 1-DOF
drive and 2-DOF sense gyro-accelerometer [8].




    Fig. 1. Schematic illustration of mass-spring-damper for a 4-DOF gyro-accelerometer


2      4-DOF Model
The 4-DOF gyro-accelerometer system exploits the dynamic amplification in the
decoupled 2-DOF drive and sense oscillators so as to attain large amplitude of oscilla-
tion without resonance [17]. As shown in Fig. 1, this system is composed of three
proof masses that are interconnected by flexures springs. The mass 𝑚1 is designed
such that it is constrained in the sense direction whilst it is excited in the drive direc-
tion alone. The configuration of the masses 𝑚2 and 𝑚3 within the decoupling frame
mass, 𝑚𝑓 is such that these two masses are mechanically decoupled from the drive
oscillations. Therefore, the first mass 𝑚1 and the combination of mass 𝑚2 , mass 𝑚3
and the decoupling frame mass 𝑚𝑓 i.e. (𝑚2 + 𝑚3 + 𝑚𝑓 ) form the 2-DOF oscillator in
the drive direction. The masses 𝑚2 and 𝑚3 oscillate independantly in the sense direc-
tion thus forming the 2-DOF sense direction oscillator. The larger mass 𝑚2 represents
the primary mass, which generates the Coriolis force that excites the sense oscillator.


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The smaller mass 𝑚3 is a secondary mass and is designed to serve as a dynamic vi-
bration absorber of mass 𝑚2 in the sense direction. In the event of an angular motion,
the sense masses 𝑚2 and 𝑚3 are influenced by Coriolis force and begin to oscillate in
the sense direction. Each of both 2-DOF drive and sense oscillators has two resonance
peaks and flat zone in between peaks. The most important requirement of the overall
4-DOF gyro-accelerometer system is that the flat amplitude regions of both the 2-
DOF oscillators must overlap precisely and the operating frequencies of the system
must be located in their flat amplitude zones, thereby leading to the maximum robust-
ness of the performance against the fluctuations of system parameters. The equations
of motion can be represented by Newton’s second law of motion, [2, 17]:
𝑚1 𝑥̈ 1 + 𝑐1𝑥 𝑥̇ 1 + (𝑘1𝑥 + 𝑘2𝑥 )𝑥1 = 𝑘2𝑥 𝑥2 + 𝐹𝑑 (𝑡) ,                            (1)
𝑀𝑝 𝑥̈ 2 + 𝑐2𝑥 𝑥̇ 2 + 𝑘2𝑥 𝑥2 = 𝑘2𝑥 𝑥1 ,                                             (2)
 𝑚2 𝑦̈ 2 + 𝑐2𝑦 𝑦̇ 2 + (𝑘2𝑦 + 𝑘3𝑦 )𝑦2 = 𝑘3𝑦 𝑦3 − 2𝑚2 Ω𝑧 𝑥̇ 2
−𝑚2 Ω̇𝑧 𝑥2 + 𝑚2 (𝑎𝑥 sin 𝜃 − 𝑎𝑦 cos 𝜃),                                             (3)
𝑚3 𝑦̈ 3 + 𝑐3𝑦 𝑦̇ 3 + 𝑘3𝑦 𝑦3 = 𝑘3𝑦 𝑦2 − 2𝑚3 Ω𝑧 𝑥̇ 2
−𝑚3 Ω̇𝑧 𝑥2 + 𝑚3 (𝑎𝑥 sin 𝜃 − 𝑎𝑦 cos 𝜃),                                             (4)

respectively where, 𝐹𝑑 (𝑡) = 𝐹𝑜 sin 𝜔𝑑 𝑡 is the sinusoidal driving force that excites the
driven mass 𝑚1 at the drive frequency 𝜔𝑑 , Ω𝑧 = Ω𝑜 cos(𝛼𝑡), where Ω𝑜 is the ampli-
tude of angular velocity, 𝛼 is the frequency of angular rate. 𝑐1𝑥 , 𝑐2𝑥 , 𝑐2𝑦 and 𝑐3𝑦 are
the damping coefficients corresponding to the respective stiffness co-efficients 𝑘1𝑥 ,
𝑘2𝑥 , 𝑘2𝑦 and 𝑘3𝑦 as shown in Fig. 1. The term Ω̇𝑜 𝑥2 is the Euler’s acceleration. The
notations 𝑎𝑥 and 𝑎𝑦 are external accelerations along their respective axes and these
have been included for the analysis of gyro-accelerometer system. The terms
2𝑚2 Ω𝑜 𝑥̇ 2 and 2𝑚3 Ω𝑜 𝑥̇ 2 are the Coriolis forces that excite the respective masses, 𝑚2
and 𝑚3 in sense direction. The corresponding displacements of the masses, 𝑚2 and
𝑚3 , are also affected by external linear acceleration.


3      Detection Scheme
The scheme for the discrimination of the angular rate and acceleration is presented
below. This scheme for angular rate amplitude, Ω𝑜 , and associated frequency, 𝛼 ,
along with linear acceleration is applicable in case where angular rate is time depend-
ent. The absolute transformation of (4) yields the solution that comprises both tempo-
rally damped and un-damped terms. The temporal decay terms, however, are not triv-
ial as these are vital for deciding the turn-on time and settling time of the system.
Since the output signal is processed after the device output is settled down, the contri-
butions of decay terms become insignificant. Therefore the settled solution, of (4), is
written as:
𝑦̅3 (𝑡) = 𝐴1 cos{(𝜔 + 𝛼)𝑡 + 𝜙𝑐𝑥 (𝜔) + 𝜙2𝑦 (𝜔 + 𝛼) + 𝜙𝑐𝑦 (𝜔 + 𝛼)}
+𝐴2 cos{(𝜔 − 𝛼)𝑡 + 𝜙𝑐𝑥 (𝜔) + 𝜙2𝑦 (𝜔 − 𝛼) + 𝜙𝑐𝑦 (𝜔 − 𝛼)}
+ℛ𝑒𝑥 𝒜2𝑦 (𝜔)𝒜𝑐𝑦 (𝜔) sin (𝜔𝑡 + 𝜙2𝑦 (𝜔) + 𝜙𝑐𝑦 (𝜔)) ,                                   (5)
where ℛ𝑒𝑥 , external acceleration and,




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               2
                                 1
𝐴1,2 = −Ωo 𝑓𝑜 𝜔2𝑥 𝒜𝑐𝑥 (𝜔) (𝜔 ± 𝛼) 𝒜2𝑦 (𝜔 ± 𝛼)𝒜𝑐𝑦 (𝜔 ± 𝛼),
                                 2
  −2 (𝜔)       2           2                4
𝒜𝑐𝑥      = [(𝜔1𝑥  − 𝜔2 )(𝜔2𝑥  − 𝜔2 ) − 𝜇𝑥2 𝜔2𝑥 − 4𝜆1𝑥 𝜆2𝑥 𝜔2 ]2
          2                    2
+4[𝜆1𝑥 (𝜔2𝑥 − 𝜔2 )𝜔 + 𝜆2𝑥 (𝜔1𝑥   − 𝜔2 )𝜔]2 ,
                                   2                 2
                       2𝜔{𝜆1𝑥 (𝜔2𝑥   − 𝜔2 ) + 𝜆2𝑥 (𝜔1𝑥 − 𝜔2 )}
𝜙𝑐𝑥 (𝜔) = − tan−1 2                2               4                 ,
                   (𝜔1𝑥 − 𝜔 2 )(𝜔2𝑥 − 𝜔 2 ) − 𝜇𝑥2 𝜔2𝑥 − 4𝜆1𝑥 𝜆2𝑥 𝜔 2
  −2 (𝜔)       2           2                 4                 2
𝒜𝑐𝑦      = [(𝜔2𝑦  − 𝜔2 )(𝜔3𝑦  − 𝜔2 ) − 𝜇𝑦2 𝜔3𝑦 − 4𝜆2𝑦 𝜆3𝑦 𝜔2 ]
         2                  2                2
+4[𝜆2𝑦 (𝜔3𝑦 − 𝜔2 )𝜔 + 𝜆3𝑦 (𝜔2𝑦 − 𝜔2 )𝜔] ,
                                 2                 2
                        2𝜔{𝜆2𝑦 (𝜔3𝑦 − 𝜔2 ) + 𝜆3𝑦 (𝜔2𝑦 − 𝜔2 )}
𝜙𝑐𝑦 (𝜔) = − tan−1     2           2                 4
                                                                         ,
                    (𝜔2𝑦 − 𝜔 2 )(𝜔3𝑦 − 𝜔 2 ) − 𝜇𝑦2 𝜔3𝑦 − 4𝜆2𝑦 𝜆3𝑦 𝜔 2
 2 (𝜔)      2     2              2
𝒜2𝑦    = [(𝜔2𝑦 + 𝜔3𝑦 − 𝜔2 ) + 4𝜆22𝑦 𝜔2 ],
                        2𝜆2𝑦 𝜔
𝜙2𝑦 (𝜔) = tan−1     2       2        ,
                 𝜔2𝑦 + 𝜔3𝑦      − 𝜔2
  2                          2
𝜔1𝑥  = (𝑘1𝑥 + 𝑘2𝑥 )/𝑚1 ; 𝜔2𝑥     = 𝑘2𝑥 /𝑀𝑝 ,
  2                           2
𝜔2𝑦 = (𝑘2𝑦 + 𝑘3𝑦 )/𝑚2 ; 𝜔3𝑦 = 𝑘3𝑦 /𝑚3 ,
     2                   2
𝜇𝑥2 𝜔2𝑥 = 𝑘2𝑥 /𝑚1 ; 𝜇𝑦2 𝜔3𝑦  = 𝑘3𝑦 /𝑚2 ; 𝜇𝑥2 = 𝑀𝑝 /𝑚1 ,
𝜇𝑦2 = 𝑚3 /𝑚2 ; 𝑓𝑜 = 𝐹𝑜 /𝑚1 ; 𝜆1𝑥 = 𝑐1𝑥 /2𝑚1 ,
𝜆2𝑥 = 𝑐2𝑥 /2𝑀𝑝 ; 𝜆2𝑦 = 𝑐2𝑦 /2𝑚2 ; 𝜆3𝑦 = 𝑐3𝑦 /2𝑚3 .
As is evident from (5), the output signal is modulated by a sinusoidal function. The
synchronous demodulation of this signal yields the in-phase and quadrature compo-
nents defined as 𝑦̅𝑝 = 𝑦̅3 (𝑡)cos(𝜔𝑡) and 𝑦̅𝑞 = 𝑦̅3 (𝑡)sin(𝜔𝑡) respectively. In order to
arrive at the low-pass-filtered solution after demodulation, we primarily deal with the
in-phase component, 𝑦̅𝑝 . The quadrature component, 𝑦̅𝑞 , can be tackled accordingly.
With the aid of trigonometric identities and settled solution (5), the in-phase compo-
nent 𝑦̅𝑝 is rearranged as,
𝑦̅𝑝 = 𝐴{cos(2𝜔𝑡 + 𝜙𝑐𝑥 (𝜔) + 𝜙) + cos(𝜙𝑐𝑥 (𝜔) + 𝜙)} cos(𝛼𝑡 + 𝛥𝜙)
−𝛿𝐴{sin(2𝜔𝑡 + 𝜙𝑐𝑥 (𝜔) + 𝜙) + sin(𝜙𝑐𝑥 (𝜔) + 𝜙)} sin(𝛼𝑡 + 𝛥𝜙)
   1
+ ℛ𝑒𝑥 𝒜2𝑦 (𝜔)𝒜𝑐𝑦 (𝜔) {sin (𝜙2𝑦 (𝜔) + 𝜙𝑐𝑦 (𝜔))
   2
+ sin (2𝜔𝑡 + 𝜙2𝑦 (𝜔) + 𝜙𝑐𝑦 (𝜔))}.                                                    (6)
The modified parameters included in (6), as per [6, 7], are defined as,
      1                   1
𝐴̅ = (𝐴1 + 𝐴2 ); 𝛿𝐴 = (𝐴1 − 𝐴2 ),
      2                   2
      1
 ̅
𝜙 = [𝜙2𝑦 (𝜔 + 𝛼) + 𝜙2𝑦 (𝜔 − 𝛼) + 𝜙𝑐𝑦 (𝜔 + 𝛼) + 𝜙𝑐𝑦 (𝜔 − 𝛼)],
      2
        1
Δ𝜙 = [𝜙2𝑦 (𝜔 + 𝛼) − 𝜙2𝑦 (𝜔 − 𝛼) + 𝜙𝑐𝑦 (𝜔 + 𝛼) − 𝜙𝑐𝑦 (𝜔 − 𝛼)].
        2
Further, the output signals terms cos(𝛼𝑡 + 𝛥𝜙) and sin(𝛼𝑡 + 𝛥𝜙) in (6) led by phase
shift, 𝛥𝜙, are related to the angular rate. The phase shift, 𝛥𝜙, in these function is


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distorted by frequency, 𝛼. Hence, it is essential to employ low-pass-filtering in order
to eliminate the terms having doubled frequency. Thus, the filtered solution after trig-
onometric manipulation is given by,
                           1
𝑦̅𝑙𝑝 = 𝐴𝑝 cos(𝛼𝑡 + 𝜓𝑝 ) + ℛ𝑒𝑥 𝒜2𝑦 (𝜔)𝒜𝑐𝑦 (𝜔) sin (𝜙2𝑦 (𝜔) + 𝜙𝑐𝑦 (𝜔)),                (7)
                           2
where
        2
𝐴2𝑝 = 𝐴 cos2 (𝜙 + 𝜙𝑐𝑥 (𝜔)) + 𝛿𝐴2 sin2 (𝜙 + 𝜙𝑐𝑥 (𝜔)),                          (7a)
𝜓𝑝 = 𝛥𝜙 + 𝛥𝜑𝑝 ,                                                               (7b)
               𝛿𝐴
𝛥𝜑𝑝 = tan−1 [ tan (𝜙 + 𝜙𝑐𝑥 (𝜔))].
                𝐴
Similarly, the low-pass-filtered quadrature component after demodulation is written
as,
                           1
𝑦̅𝑙𝑞 = 𝐴𝑞 cos(𝛼𝑡 + 𝜓𝑞 ) + ℛ𝑒𝑥 𝒜2𝑦 (𝜔)𝒜𝑐𝑦 (𝜔) cos (𝜙2𝑦 (𝜔) + 𝜙𝑐𝑦 (𝜔)),           (8)
                           2
where
        2
𝐴2𝑞 = 𝐴 sin2 (𝜙 + 𝜙𝑐𝑥 (𝜔)) + 𝛿𝐴2 cos2 (𝜙 + 𝜙𝑐𝑥 (𝜔)),                               (8a)
𝜓𝑞 = 𝛥𝜙 + 𝛥𝜑𝑞 ,                                                                    (8b)
                  𝛿𝐴
𝛥𝜑𝑞 = −tan−1 [ cot (𝜙 + 𝜙𝑐𝑥 (𝜔))].
                   𝐴
It is worth mentioning here that the first terms pertaining to gyro action of the of re-
spective in-phase and quadrature signals (7) and (8) are distorted both in amplitudes,
𝐴𝑝 and 𝐴𝑞 , and in corresponding phases, 𝜓𝑝 and 𝜓𝑞 , by 𝛼. At the same time, the line-
ar acceleration terms in these signals are unaffected by 𝛼. From equations (7) and (8)
it is also inferred that at the optimum operating condition of the present device, the
system must be driven at zero-phase frequency, as already emphasized [17], of the 2-
DOF drive oscillator and the corresponding structural frequencies, 𝜔1𝑥 and 𝜔2𝑥 must
be equal and these must match with driving frequency that results into 𝜙𝑐𝑥 (𝜔) = 0.
Likewise, the anti-resonance frequency of passive mass (𝑚2 ) amplitude must also
match with drive frequency and sense mass related structural frequencies, 𝜔2𝑦 and
𝜔3𝑦 √(1 + 𝜇2 ), must be equal [17]. That leads to the phase, 𝜙2𝑦 (𝜔) + 𝜙𝑐𝑦 (𝜔) = 0.
As a consequence of this the acceleration term in in-phase signal (7) vanishes and
only the gyro action related term exists. On the other hand in quadrature signal (8),
the acceleration term exist and gyro related term is almost insignificant. Therefore,
the in-phase signal (7) can be used for angular rate detection and the quadrature one
(8) can be utilized for linear acceleration extraction under such optimized operating
condition of the gyro-accelerometer system.

4       Results and discussion

Considering the design equations, the spring constants and structural frequencies by
adjusting mass values, 𝑚1 , 𝑚2 , 𝑚3 , frame mass, 𝑚𝑓 and subsequently mass ratios, 𝜇𝑥2
and 𝜇𝑦2 , have been decided optimally. The values of these and other parameters are
listed in Table 1. The following figures have been calculated by using these values
unless it is specified.


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                        Table 1. Parameter values used for calculations

Parameters                            Values
Active mass (m1)                      201.9 x 10-9 kg
Passive mass (m2)                     57.24 x 10-9 kg
Sense mass (m3)                       5.6 x 10-9 kg
Frame mass (mf)                       10.5 x 10-9 kg
Spring constant (k1x; k2x)            153.5 N/m; 87.59 N/m
Spring constant (k2y; k3y)            62.26 N/m; 6.1 N/m
Frequencies (ω1x = ω2x = ω2y)         5.5 kHz
Frequency (ω3y)                       5.25 kHz
Fo; Angular rate (Ωo)                 2.171x10-5 N; 200 rad
Figure 2a-d are Bode plots of the demodulated and low-pass-filtered in-phase and
quadrature components of amplitudes and corresponding emerged phases of Coriolis
and Euler’s signal of gyro-accelerometer for different values of driving frequency.
Figure 2a illustrates the results of respective in-phase and quadrature components, 𝐴𝑝
and 𝐴𝑞 , calculated with the help of (7a) and (8a) respectively for driving frequencies,
𝜔 = 𝜔𝑑 = 𝜔1𝑥 and 𝜔𝑑 = 1.005 𝜔1𝑥 . From this figure it is observed that a slight
deviation of driving frequency, 𝜔𝑑 , from 𝜔1𝑥 within the range of drive and sense
bandwidth does not have the considerable effect on the components 𝐴𝑝 and 𝐴𝑞 . For
both values of drive frequency, 𝜔𝑑 , the in-phase amplitude 𝐴𝑝 is the major component
and increases with 𝛼 in both the cases. At the same time the quadrature component,
𝐴𝑞 , is nearly zero for initial values of 𝛼, and increases thereafter with 𝛼.
The results shown in Fig. 2b correspond to phase components, 𝜓𝑝 and 𝜓𝑞 calculated
by using respective expressions (7b) and (8b) under similar conditions as of Fig. 2a.
The phase component 𝜓𝑝 for both drive frequencies 𝜔𝑑 = 𝜔1𝑥 and 𝜔𝑑 = 1.005 𝜔1𝑥 ,
is nearly zero for entire calculated range of 𝛼 and related curves overlap with each
other. While 𝜓𝑞 , for 𝜔𝑑 = 𝜔1𝑥 , is zero at 𝛼 = 0 and abruptly goes gown to -90o as 𝛼
increases and persists with this value for the entire calculated range of 𝛼. On the other
hand for 𝜔𝑑 = 1.005 𝜔1𝑥 , the component 𝜓𝑞 rises steeply from -180o to -90o as 𝛼
increases and then persists with latter value for rest of the calculated range of 𝛼. From
this it is inferred that both in-phase and quadrature signal components are at 90o of
phase difference.
Figure 2c reveals that for 𝜔𝑑 = 0.95 𝜔1𝑥 , both the components 𝐴𝑝 and 𝐴𝑞 , have the
same nature of variation in their magnitudes as shown in Fig. 2a. However, for
𝜔𝑑 = 1.05 𝜔1𝑥 , the magnitudes of 𝐴𝑝 and 𝐴𝑞 are almost the same as those for
𝜔𝑑 = 0.95 𝜔1𝑥 , within the range, 𝛼 < 0.04 𝜔𝑑 , but beyond this range of 𝛼 both 𝐴𝑝
and 𝐴𝑞 rise steeply and peak at about 𝛼 = 0.06 𝜔𝑑 . Thereafter, both amplitude com-
ponents decrease sharply and interchange their magnitudes, thus, showing balancing
natures with each other.
The significant changes have occurred in the phase characteristics for such a deviation
in the drive frequency as shown in the Fig. 2d. For 𝜔𝑑 = 0.95 𝜔1𝑥 , 𝜓𝑝 is zero and
invariant with the variation of 𝛼. At the same time, 𝜓𝑞 is about zero at 𝛼 = 0 and then
increases with 𝛼 and reaches to 90o which persists for the entire calculated range of 𝛼.



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Mathematical Modeling                          Verma Payal, Fomchenkov SA. Analytical…


The components 𝜓𝑝 and 𝜓𝑞 for 𝜔𝑑 = 1.05 𝜔1𝑥 , show different characteristics. The
phase component 𝜓𝑞 originates at -180o and as 𝛼 increases phase value reaches to -
90o at about 𝛼 = .06 𝜔𝑑 . Then it starts decreasing up to -270o, whereas 𝜓𝑝 is zero in
the beginning and then it fluctuates from zero to negative values by varying 𝛼.




 Fig. 2. Bode plots of demodulated and then filtered in-phase and quadrature components of
   amplitudes and corresponding phases, a amplitudes b phases for 𝜔 = 𝜔𝑑 = 𝜔1𝑥 and for
𝜔𝑑 = 1.005 𝜔1𝑥 , c amplitudes d phases for 𝜔𝑑 = 1.05 𝜔1𝑥 and for 𝜔𝑑 = 0.95 𝜔1𝑥 , (symbols
                                   are Simulink results)

The transient response of the in-phase component with respect to time which is a
result of linear acceleration is shown in Fig. 3, after demodulation and filtration. The
Simulink results use the same conditions as that for the analytical analysis, which
considers the settled transient response of the device. It can be observed in Fig. 3 that
the device settles down at about 0.24 ms. The time required for settling down depends
on the ambient pressure at which the device is operated. The in-phase displacement
component, 𝑦̅𝑙𝑝 is plotted using equation (7) and it clearly shows an excellent match
between analytical and Simulink results. Figure 3 also shows that when the device is
driven at frequency, zero-phase frequency, acceleration part is zero and gyro action is
dominant, indicated by a sinusoidal variation Hence, angular rate can be measured
through the in-phase output.
Figure 4 shows the transient response of the quadrature output signal (𝑦̅𝑙𝑞 ) calculated
by using (8), which is a result of the combined action of Coriolis and Euler’s forces. It
is evident from the plot that the acceleration action is dominant and gyro action is
zero. Hence, acceleration action can be determined from the quadrature output. This



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Mathematical Modeling                          Verma Payal, Fomchenkov SA. Analytical…


figure also includes Simulink data showing excellent agreement with the analytical
results.




                                   ̅𝒍𝒒 ) displacement variation with time, (ℛ𝑒𝑥 = 10g)
     Fig. 4. Quadrature component (𝒚


Conclusion
The simultaneous detection scheme of time varying angular rate and linear accelera-
tion utilizes the synchronous demodulation that yields in-phase and quadrature output
signals of the systems. In case of matched zero phase frequencies of both the oscilla-
tors, the associated acceleration term in in-phase component becomes ineffective and
the device deliver only angular rate related signals, whereas the quadrature signal is
dominant by acceleration action and that related to angular rate becomes almost in-
significant. Therefore, in-phase signal can be used for acceleration detection and
quadrature one for angular rate extractions. MATLAB®/Simulink model of the gyro-
accelerometer system was developed in order to investigate the feasibility of such a
detection scheme and simulation results have shown excellent correspondence with
analytical results.


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