The paper reports an analytical model of a 4-DOF gyroaccelerometer consisting of 2-DOF drive and 2-DOF sense oscillators configured 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 inphase and quadrature components of the system's output signal. These two components can be utilized in the detection of angular motion and linear acceleration. The in-phase signal can be used for angular rate detection and the quadrature 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.
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 [
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
mathematical modeling plays a key role in device design as well for optical elements [
The scheme for the discrimination of the angular rate and acceleration is presented below. This scheme for angular rate amplitude, Ω , and associated frequency, , along with linear acceleration is applicable in case where angular rate is time dependent. The absolute transformation of (4) yields the solution that comprises both temporally damped and un-damped terms. The temporal decay terms, however, are not trivial 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 contributions of decay terms become insignificant. Therefore the settled solution, of (4), is written as: ̅3( ) = 1 cos{( + ) + ( ) + 2 ( + ) + ( + )} + 2 cos{( − ) + ( ) + 2 ( − ) + ( − )} +ℛ 2 ( ) ( )sin (
+ 2 ( ) + ( )) , where ℛ , external acceleration and, 1,2 = −Ωo 22 ( )( ± 12 ) 2 ( ± ) ( ± ), −2( ) = [( 12 − 2)( 22 − 2)− 2 24 − 4 1 2 2]2 +4[ 1 ( 22 − 2) + 2 ( 12 − 2) ]2, ( ) = −tan−1
2 { 1 ( 22 − 2)+ 2 ( 12 − 2)} ( 12 − 2)( 22 − 2)− 2 24 − 4 1 2 2 , −2( )= [( 22 − 2)( 32 − 2)− 2 34 − 4 2 3 2]2 +4[ 2 ( 32 − 2) + 3 ( 22 − 2) ]2, ( ) = −tan−1 2 { 2 ( 32 − 2)+ 3 ( 22 − 2)} , ( 22 − 2)( 32 − 2)− 2 34 − 4 2 3 2 22 ( ) = [( 22 + 32 − 2)2 + 4 22 2],
2 2 2 ( )= tan−1 22 + 32 − 2 , 12 = ( 1 + 2 )/ 1; 22 = 2 / , 22 = ( 2 + 3 )/ 2; 32 = 3 / 3, 2 22 = 2 / 1; 2 32 = 3 / 2; 2 = / 1, 2 = 3/ 2; = / 1; 1 = 1 /2 1, 2 = 2 /2 ; 2 = 2 /2 2; 3 = 3 /2 3.
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
components defined as ̅ = ̅3( )cos( )and ̅ = ̅3( )sin( )respectively. In order to
arrive at the low-pass-filtered solution after demodulation, we primarily deal with the
in-phase component, ̅ . The quadrature component, ̅ , can be tackled accordingly.
With the aid of trigonometric identities and settled solution (5), the in-phase
component ̅ is rearranged as,
̅ = {cos(2 + ( )+ )+ cos( ( )+ )}cos( + )
− {sin(2 + ( )+ )+ sin( ( )+ )}sin( + )
1
2 2 ( ) ( ){sin( 2 ( )+ ( ))
+ ℛ
+sin(2 + 2 ( )+ ( ))}. (6)
The modified parameters included in (6), as per [
1 Δ = 2[ 2 ( + )− 2 ( − )+ ( + )− ( − )].
Further, the output signals terms cos( + )and sin( + )in (6) led by phase shift, , are related to the angular rate. The phase shift, , in these function is where = as, where = 2 2 + , distorted by frequency, . Hence, it is essential to employ low-pass-filtering in order to eliminate the terms having doubled frequency. Thus, the filtered solution after trigonometric manipulation is given by, ̅ = cos( + ) + 2 2 ( ) ( ) sin ( 2 ( ) + ( )),
ℛ 2 = cos2 ( + ( )) + 2sin2 ( + ( )), = tan−1 [
tan ( + ( ))].
Similarly, the low-pass-filtered quadrature component after demodulation is written ̅ = cos( + ) + 2 2 ( ) ( ) cos ( 2 ( ) + ( )),
ℛ 2 = sin2 ( + ( )) + 2cos2 ( + ( )), (7) (7a) (7b) (8) (8a) (8b) = −tan−1 [
cot ( + ( ))].
It is worth mentioning here that the first terms pertaining to gyro action of the of
respective in-phase and quadrature signals (7) and (8) are distorted both in amplitudes,
and , and in corresponding phases, and , by . At the same time, the
linear acceleration terms in these signals are unaffected by . 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 [
Considering the design equations, the spring constants and structural frequencies by adjusting mass values, 1, 2, 3, frame mass, and subsequently mass ratios, 2 and 2, 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.
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
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, and , calculated with the help of (7a) and (8a) respectively for driving frequencies, = = 1 and = 1.005 1 . From this figure it is observed that a slight deviation of driving frequency, , from 1 within the range of drive and sense bandwidth does not have the considerable effect on the components and . For both values of drive frequency, , the in-phase amplitude is the major component and increases with in both the cases. At the same time the quadrature component, , is nearly zero for initial values of , and increases thereafter with . The results shown in Fig. 2b correspond to phase components, and calculated by using respective expressions (7b) and (8b) under similar conditions as of Fig. 2a. The phase component for both drive frequencies = 1 and = 1.005 1 , is nearly zero for entire calculated range of and related curves overlap with each other. While , for = 1 , is zero at = 0 and abruptly goes gown to -90o as increases and persists with this value for the entire calculated range of . On the other hand for = 1.005 1 , the component rises steeply from -180o to -90o as increases and then persists with latter value for rest of the calculated range of . From this it is inferred that both in-phase and quadrature signal components are at 90o of phase difference.
Figure 2c reveals that for = 0.95 1 , both the components and , have the same nature of variation in their magnitudes as shown in Fig. 2a. However, for = 1.05 1 , the magnitudes of and are almost the same as those for = 0.95 1 , within the range, < 0.04 , but beyond this range of both and rise steeply and peak at about = 0.06 . Thereafter, both amplitude components 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 = 0.95 1 , is zero and invariant with the variation of . At the same time, is about zero at = 0 and then increases with and reaches to 90o which persists for the entire calculated range of . The components and for = 1.05 1 , show different characteristics. The phase component originates at -180o and as increases phase value reaches to 90o at about = .06 . Then it starts decreasing up to -270o, whereas is zero in the beginning and then it fluctuates from zero to negative values by varying . 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, ̅ 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 ( ̅ ) 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
The simultaneous detection scheme of time varying angular rate and linear acceleration 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 oscillators, 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 insignificant. Therefore, in-phase signal can be used for acceleration detection and quadrature one for angular rate extractions. MATLAB®/Simulink model of the gyroaccelerometer system was developed in order to investigate the feasibility of such a detection scheme and simulation results have shown excellent correspondence with analytical results.