The actual sensing vibration state is in line with the sensing mode. As shown in
Table 1, in the harmonic analysis, the amplitude ratio and phase difference of the displacement between node A and node B are both approximate to the sensing mode. This shows that although the vibration frequency is different from the sensing modal frequency by 200 Hz, when the drive tines vibrate in the Z direction oppositely affected by Coriolis force, the sense tines will also vibrate in the Z direction correspondingly, and the vibration state is almost consistent with the sensing mode. According to Formula (1), at this time, the Coriolis vibration state of the gyroscope is almost completely determined by the sensing mode. Therefore, it is credible to evaluate the transmission of Coriolis vibration by use of the sensing mode, which can greatly reduce the calculation required.
The gyroscope generally works in resonance, and it is known from
Table 1 that the displacement of the drive tines affected by Coriolis force has little connection with the shape of the base. Hence, the key to improving the mechanical sensitivity of the gyroscope is in improving the transmission efficiency of the Coriolis signal from the drive tines to the sense tines as much as possible. The transmission efficiency is denoted as
ŋ and is defined by the formula given below.
where
ZB is the displacement of node
B in the sensing mode, and
ZA is the displacement of node
A.
3.1. Structural Analysis
The mechanical sensitivity of the gyroscope is denoted as
and can be calculated according to Formula (4), where
is sense tine amplitude, Ω is angular velocity,
η is transmission efficiency of the Coriolis vibration,
F is driving force, and the driving direction quality factor and resonant frequency are represented by
and
, respectively. In addition,
is mass of the drive tine, and
and
represent the quality factor and resonant frequency of driving direction, respectively.
According to the calculation formula of mechanical sensitivity, the gyroscope microstructure can be optimized from the following aspects:
- (1)
The transmission efficiency of Coriolis vibration ŋ should be as high as possible;
- (2)
The working modal (driving mode and sensing mode) order and frequency should be as low as possible, which can achieve greater response amplitude in the structural vibration. When the excitation frequency is close to the
r-th natural frequency
ωr, the system response {
q(
t)} is approximately expressed by the following formula.
where {
u(r)} is the
r-th modal vector,
N0r is excitation amplitude, and
ξr is damping ratio.
It is known from Formula (5) that the lower the working modal frequency is, the greater the response amplitude is, and therefore, the higher is the sensitivity of the gyroscope;
- (3)
The driving mode and the sensing mode should be adjacent to reduce the effects of interfering modes.
In addition to the three structures shown in
Figure 4, six typical structures are analyzed, which are shown in
Figure 7. The mode frequency and Coriolis vibration transmission efficiency results of all models are shown in
Table 2.
As seen from
Figure 8, slot structure types have different effects on the transmission efficiency. Based on type D, slotting on the sides of the base will reduce the transmission efficiency, such as in type C. On the contrary, slotting in the middle of the base will improve the transmission efficiency, such as in types E, F and G. According to
Table 1, opening arc slots between the tines can reduce the working modal order and frequency, such as in type I. Therefore, combining the two advantageous features—slotting in the middle of the base and opening arc slots between the tines—can greatly improve the mechanical sensitivity of the gyroscope. The proposed type A not only has the above characteristics, but additionally, its base protrudes slightly outward, so that the sense tine is also symmetrical in a local area, thereby obtaining better vibration performance.
After modal analysis of different models, it was found that type A, with high symmetry, had the best vibration performance. Its transmission efficiency was the highest, and the working mode order and frequency were also the lowest. In addition, the driving and sensing modes were adjacent to each other. In summation, type A perfectly meets the three preset performance requirements.
3.2. Size Optimization with Taguchi Method
The Taguchi method is a reliable design method that guides the optimal design of products in complex environments. In this section, the Taguchi method is used to optimize the relevant feature sizes of the gyroscope. The feature size codes of the gyroscope are shown in
Figure 9. The height of tines
h2, drive-sense interval
d1, drive–drive interval
d2, width of the support beam
w1, and height of the groove
h3, are the signal factors. The noise factor is the mesh size of the finite element model. Parameters and their levels are given in
Table 3 and
Table 4.
The best experimental result is evaluated by the calculation of signal to noise ratio (SNR) in the Taguchi method. The larger the SNR is, the smaller the quality loss and the better the product quality. To obtain the optimum result, three basic categories which are smaller-the-better, larger-the-better and nominal-the-best characteristics are calculated by the formulas given below [
30].
- (a)
Smaller-the-better characteristic:
- (b)
Larger-the-better characteristic:
- (c)
Nominal-the-best characteristic:
where
yi is measured values, and
n and
m are number of experiments and nominal value respectively.
The orthogonal experiment table was created according to
Table 3 and
Table 4, and the corresponding finite element model was established to analyze according to different combinations of each factor level. The observed experiment results include the following items:
- (1)
The transmission efficiency of Coriolis vibration which meets the larger-the-better characteristic;
- (2)
Frequency difference between the driving mode and sensing mode. When the frequency difference between the two is large, the working bandwidth of the gyroscope meets Formula (9) [
31]:
where ΔF is the frequency difference between the driving mode and sensing mode.
Given that the applications of tactical and inertial gyroscope require bandwidths of around 100 Hz [
9], the frequency difference Δ
F should be maintained at about 200 Hz. For the convenience of calculation, this paper defines the difference between Δ
F and 200 Hz as a new variable Δ
f, which meets the smaller-the-better characteristic and follows the Formula (10) given below:
- (3)
The driving modal frequency which meets the smaller-the-better characteristic;
- (4)
The drive coupling coefficient which meets the smaller-the-better characteristic. Similar to
Section 3.2, the drive coupling coefficient is denoted as
λ and defined by the Formula (11) given below.
where
XB is the displacement of node
B in the driving mode, and
XA is the displacement of node
A.
Since it is preferable for the undesired coupling motion between the drive tines and the sense tines to be separated, it is better to keep the sense tines stationary when the drive tines are vibrating in the opposite direction, and, therefore, the drive coupling coefficient needs to be as low as possible.
According to the different quality characteristics of each result index, the corresponding SNR calculation formula is used for data processing. The relationship between the SNR of different result indexes and the signal factors is shown in
Figure 10. Moreover, variance analysis is performed to examine the influence degree of each factor on different outcome indicators, as shown in
Figure 11.
The influence degree of every signal factor on each outcome indicator is different. As can be seen from
Figure 11, for the indicator (a), the contribution rate of each signal factor is ranked as
, where the effect of factor B and factor C is huge. For indicator (b), the contribution rate of each signal factor is ranked as
, where the effect of factor A is powerful. For the indicator (c), the contribution rate of each signal factor is ranked as
, where the effect of factor A is significant. For the indicator (d), the contribution rate of each signal factor is ranked as
, where the influence degree of each factor is little different.
In the Taguchi method, the factor that has a significant impact is prioritized, and the factor with insignificant effect is comprehensively considered according to actual needs. According to
Figure 10a, the levels of factor B and factor C are selected as B1 and C5, respectively. According to
Figure 10b,c, the level of factor A is selected as A5. The levels of factor D and factor E are selected as D1 and E4, taking
Figure 10a–d into account. Therefore, when considering only the above four performance indicators, the optimal parameter according to the Taguchi method is A5B1C5D1E4. However, the performance indicators that need to be considered in practical application also include miniaturization, impact resistance, processing difficulty, etc. Therefore, it is necessary to further optimize the results obtained by the Taguchi method in accordance with the gyroscope application requirements.
It was found that the modal frequency will be too concentrated when the height of tines
h2 is too high. In order to isolate the influence of the interference mode, the level of factor A was optimized from A5 to A4. Combined with the specific structure of the gyroscope, the difference between factor B and factor C cannot be too large, otherwise the two sides of the drive tine will not be sufficiently symmetrical, which will affect the gyroscope mode shape. As known from
Figure 10a, within the appropriate range, a smaller value of factor B and bigger value of factor C will result in better vibration performance. Considering miniaturization and the index, the difference between factor B and factor C cannot be too large, therefore, the level of factor B was selected as B1, and the value of factor C was selected as 0.72 mm after further calculation, which was near C2 and was denoted as C2′. Considering the processing difficulty, the level of the factor D was optimized from D1 to D3. Considering the miniaturization, the level of the factor E was optimized from E4 to E2. Therefore, combined with practical application requirements, the further optimization result based on the Taguchi method is A4B1C2′D3E2.
The feature sizes of the gyroscope optimized by the Taguchi method are shown in
Figure 12, where the gyroscopic thickness is 0.5 mm.
The feature sizes of the gyroscope before and after Taguchi optimization are shown in
Table 5, where the relevant size codes have been shown in
Figure 9. The related vibration performance is shown in
Table 6.
As can be seen from
Table 6, compared with the initial structure, the transmission efficiency
ŋ of the optimized gyroscope has improved by about 18%, and the working modal frequency has been reduced by about 2.7 kHz. According to the previous analysis, the improvement of these two indicators will further improve the mechanical sensitivity of the gyroscope.