Frequency response

In order to develop a stabilization system for MEMS gyroscope, it is important to understand the

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gyroscope is likely to have a direct impact on the controller design and can help identify

potential stability issues—especially when considering wider-bandwidth solutions for next

generation designs. This information is also useful for predicting the gyroscopes‘ response to

vibration (See e.g., Looney, July 2012).

Taking Laplace transformation of system equations (2.3) and (2.4) presented in Chapter 2, the

equations in the s domain can be expressed as

𝑠2_𝑄 1(𝑠) +𝜔_𝑄𝑥 𝑥𝑠𝑄1(𝑠) − 2𝛺𝑠𝑄2(𝑠) + 𝜔𝑥 2_{− 𝛺}2 _𝑄 1(𝑠) = 𝐹1(𝑠) (3.5) 𝑠2_𝑄 2(𝑠) + 2𝛺𝑠𝑄1(𝑠) +𝜔_𝑄𝑦 𝑦 𝑠𝑄2(𝑠) + 𝜔𝑦 2_{− 𝛺}2 _𝑄 2(𝑠) = 0 (3.6)

where 𝐹₁ 𝑠 , 𝑄₁(𝑠) and 𝑄₂(𝑠) represent, respectively, the Laplace transform of 𝑓₁ 𝑡 , 𝑞₁(𝑡) and

𝑞2 𝑡 .

From Equations (3.5) and (3.6), the amplitude ratio of the displacement in the sensing direction

to the displacement in the driving direction (i.e., 𝑄2/𝑄1 ) is evaluated considering frequency mismatch. Similarly, the forced frequency response magnitude 𝑄2/𝐹1 is also evaluated. For this purpose, the magnitudes of

𝑄₂(𝑠) 𝑄₁(𝑠)

= −

2𝛺𝑠 𝑠2+𝜔 𝑦 𝑄𝑦𝑠+(𝜔𝑦2−𝛺2) , and (3.7) 𝑄₂(𝑠) 𝐹₁(𝑠)

= −

2𝛺𝑠 𝐴𝑠4_+𝐵𝑠3_+𝐶𝑠2_{+𝐷𝑠+ 𝜔} 𝑥 2_−𝛺2 _(𝜔 𝑦 2_−𝛺2₎ , (3.8)

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are used, where

𝐴 = 1,

(3.9a)

𝐵 =

𝜔𝑥 𝑄𝑥

+

𝜔𝑦 𝑄𝑦 , (3.9b)

𝐶 =

𝜔𝑥 𝑄𝑥 𝜔𝑦 𝑄𝑦

+ 𝜔

𝑥 2

_{− 𝛺}

2

_{+ 𝜔}

𝑦2

− 𝛺

2

+ 4𝛺

2, (3.9c)

𝐷 =

𝜔𝑥 𝑄𝑥

𝜔

𝑦 2

_{− 𝛺}

2

₊

𝜔𝑦 𝑄𝑦

(𝜔

𝑥 2

_{− 𝛺}

2

_{) ,}

_(3.9d)

The parameters given in Table 2-1 are used for the numerical calculations of the amplitude ratio

as well as forced frequency responses. From Equation (3.6), the amplitude ratio of the

displacement in the sensing direction to the displacement in the driving direction (i.e., 𝑄2/𝑄1 ) is evaluated and depicted for quality factors 1 × 108 and 1000 in Figures 3-11 and Figure 3-12, respectively. The figures show that the amplitude ratio has the maximum value near the non-

rotating mass-spring natural frequency ω_y and that the magnitude of the amplitude ratio increases with an increase in the input angular rate.

Similarly, the frequency response magnitude 𝑄2/𝐹1 , evaluated from Equation (3.7), is illustrated for quality factor 1 × 108 and 1000, respectively, are in Figures 3-13 and 3-14. In addition, the magnitude of frequency response is shown to increase as the input angular rate

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Figure 3-11. Variation of amplitude ratio for different input angular rates (frequency mismatch with 0.01%, Quality factor, 𝑄_𝑥 = 𝑄_𝑦 = 1 × 108)

Figure 3-12. Variation of amplitude ratio for different input angular rates (frequency mismatch with 0.01% mismatch, Quality factor, 𝑄𝑥 = 𝑄𝑦 = 1000)

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Figure 3-13. Variation of frequency response for different input angular rates (frequency mismatch with 0.01% mean, Quality factor, 𝑄_𝑥 = 𝑄_𝑦 = 1 × 108)

Figure 3-14. Variation of frequency response for different input angular rates (frequency mismatch with 0.01% mean, Quality factor, 𝑄_𝑥 = 𝑄_𝑦 = 1000)

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In order to analyze the uncertainty quantification in the frequency domain with frequency

mismatch, 50 random samples are employed in the numerical simulation and the resulting

responses are demonstrated in Figure 3-15 where the input angular rate and quality factor are

fixed at 2π rad/sec and 1000, respectively.

Figure 3-15. Variation of amplitude ratio for different samples (frequency mismatch with 0.01%, Quality factor, 𝑄_𝑥 = 𝑄_𝑦 = 1000, 𝛺 = 2𝜋 rad/sec)

For the purposes of quantifying the effect of the uncertainty frequency mismatch, magnitudes of

the amplitude ratio peaks are computed considering uncertainties in frequency mismatch. Figure

3-16 illustrates the gradually increasing nonlinear relationship between the standard deviation of

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Figure 3-16. Standard deviation of frequency mismatch vs. standard deviation of magnitude of amplitude ratio 𝑄2/𝑄1 (𝛺 = 2𝜋 rad/sec, Quality factor, 𝑄_𝑥 = 𝑄_𝑦 = 1000)

Least-square method is used to get a parametric relationship between two standard deviations

using the MATLAB command ‗polyfit‘ (degree 2) :

𝜍 _𝑄2/𝑄1 = −0.0030𝜍_{𝑓.𝑚𝑖𝑠𝑚𝑎𝑡𝑐 𝑕}2 _{+ 1.1185 × 10}−6_𝜍

𝑓.𝑚𝑖𝑠𝑚𝑎𝑡𝑐 𝑕 − 1.2234 × 10−12, (3.10)

where 𝜍 _𝑄2/𝑄1 and 𝜍_{𝑓.𝑚𝑖𝑠𝑚𝑎𝑡𝑐 𝑕} , respectively, symbolize the standard deviation of magnitude of amplitude ratio 𝑄2/𝑄1 and standard deviation of frequency mismatch. This process is illustrated in Figure 3-17.

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Figure 3-17. Standard deviation of frequency mismatch vs. standard deviation of magnitude of amplitude ratio 𝑄2/𝑄1 (𝛺 = 2𝜋 rad/sec, Quality factor, 𝑄_𝑥 = 𝑄_𝑦 = 1000)

The standard deviation of frequency corresponding to the peak amplitude ratio is also evaluated

for varying standard deviations of frequency mismatch and depicted in Figure 3-18 to illustrate

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Figure 3-18. Standard deviation of frequency mismatch vs. standard deviation of frequency of peak amplitude ratio 𝑄2/𝑄1 (𝛺 = 2𝜋 rad/sec, Quality factor, 𝑄_𝑥 = 𝑄_𝑦 = 1000)

With the intention of quantifying the effect of the uncertainty of peak frequency, magnitudes of

the frequency response peaks are computed considering uncertainties in frequency mismatch.

Figure 3-19 illustrates the gradually increasing nonlinear relationship between the Standard

deviations of frequency mismatch and magnitude of frequency response 𝑄2/𝐹1 . It is interesting to note that this variation has a similar pattern to that exhibited previously in Figure 3-

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Figure 3-19. Standard deviation of input frequency mismatch vs. standard deviation of magnitude of frequency response 𝑄2/𝐹1 (𝛺 = 2𝜋 rad/sec, Quality factor, 𝑄𝑥 = 𝑄𝑦 = 1000)

Least-square method is used to get a parametric relationship between two standard deviations

using the MATLAB command ‗polyfit‘ (degree 2) :

𝜍 _𝑄2/𝐹1 = −1.2121 × 10−15_𝜍

𝑓.𝑚𝑖𝑠𝑚𝑎𝑡𝑐 𝑕2 + 4.4624 × 10−19𝜍𝑓.𝑚𝑖𝑠𝑚𝑎𝑡𝑐 𝑕 − 4.8745 × 10−25,

(3.11)

where 𝜍 _𝑄2/𝐹1 and 𝜍𝑓.𝑚𝑖𝑠𝑚𝑎𝑡𝑐 𝑕, respectively, denote as standard deviation of magnitude of

frequency response 𝑄2/𝐹1 and standard deviation of frequency mismatch. This process is depicted in Figure 3-20.

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Figure 3-20. Standard deviation of frequency mismatch vs. standard deviation of magnitude of frequency response 𝑄2/𝐹1 (𝛺 = 2𝜋 rad/sec, Quality factor, 𝑄𝑥 = 𝑄𝑦 = 1000)

The standard deviation of frequency that corresponds to the magnitude of peak of frequency

response is also evaluated for varying frequency mismatch standard deviation and depicted in

Figure 3-21, which illustrates that the mismatch uncertainty has negligible influence as exhibited

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Figure 3-21. Standard deviation of frequency mismatch (rad/sec) vs. standard deviation of frequency of frequency response 𝑄2/𝐹1 (non-dimensional) (𝛺 = 2𝜋 rad/sec, Quality factor,

𝑄_𝑥 = 𝑄_𝑦 = 1000)

3.6. Closure

In this chapter, optimal temporal sample paths determined in Chapter 2 have been employed for

uncertainty quantification for mass-spring gyroscope. In order to predict response statistics,

dynamic response simulations have been used for quantifying standard deviation of output

response when parameters such as input angular rate, frequency mismatch and quality factor are

subjected to uncertainty. Uncertainty quantification in the frequency domain has also been

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response magnitudes. Least-square algorithm is used in both time and frequency domain in an

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Chapter 4

In document Uncertainty Quantification for a Class of MEMS-based Vibratory Angular Rate Sensors (Page 68-81)