A method for accurately identifying the resonant frequency of a silicon resonant accelerometer

Abstract

The present invention discloses a method for accurately identifying the resonant frequency of a silicon resonant accelerometer, and the steps are as follows: Step 1, constructing an amplitude-frequency characteristic curve model of a silicon micro-resonant accelerometer resonator according to the modal characteristics of the silicon resonant accelerometer; Step 2, finding the amplitude maximum point of the amplitude-frequency curve based on the golden section algorithm; Step 3, implementing the golden section algorithm based on the principle of direct digital frequency synthesis; Step 4, verifying the accuracy and feasibility of the method through numerical simulation. Aiming at the problem that the accuracy of identifying the resonant frequency is not high and the human influence factor is large in the traditional frequency sweeping scheme that identifies the resonant frequency based on the amplitude of the output of the silicon resonant accelerometer meeting the amplitude threshold set manually, the present invention improves the accuracy of resonant frequency identification based on the amplitude-frequency characteristic model of the silicon resonant accelerometer resonator, based on the golden section algorithm and the principle of digital frequency synthesis, and ensures that the silicon resonant accelerometer is in a normal resonant working state.

Description

Technical Field

[0001] The invention belongs to the technical field of inertial navigation and discloses a method for accurately identifying the resonant frequency of a silicon resonant accelerometer. Background Art

[0002] Silicon resonant accelerometer uses the resonance principle to characterize acceleration by the change of resonant frequency. It has the advantages of high sensitivity, high reliability, small size, low price, easy integration, and semi-digital output. It has broad application prospects in both the civilian and military fields. Since the resonant frequency is an important parameter in the closed-loop drive and acceleration characterization of the silicon resonant accelerometer, and in actual engineering applications, the resonant frequencies of the silicon resonant accelerometer heads vary due to manufacturing errors, and online frequency scanning is required to meet the requirement that all products work at the resonant frequency state. Therefore, accurate resonant frequency identification is of great significance to improving the dynamic performance and engineering application of silicon resonant accelerometers.

[0003] The traditional method of resonant frequency identification that can be applied in engineering mainly uses unidirectional frequency sweeping to detect whether the amplitude corresponding to each frequency point reaches the manually set amplitude threshold to determine whether the frequency point is a resonant frequency. The implementation principle of the traditional method is simple, but because this method is a unidirectional frequency sweep, the setting of the amplitude threshold is often determined by human experience, and the frequency sweep resolution is low, so there are human errors and system errors, and the accuracy of the identified resonant frequency is low, which in turn causes the silicon resonant accelerometer to fail to work properly. The low accuracy of resonant frequency identification has an important impact on the accuracy of the output frequency value of the silicon resonant accelerometer to represent the acceleration. However, there is still a lack of methods for accurately identifying the resonant frequency of the silicon resonant accelerometer, which affects the accuracy of the output results of the silicon resonant accelerometer and seriously limits the engineering application of the silicon resonant accelerometer in the field of high-precision navigation. Summary of the invention

[0004] In view of the above problems, the present invention discloses a method for accurately identifying the resonant frequency of a silicon resonant accelerometer. According to the mathematical model of the amplitude-frequency characteristic curve of the silicon resonant accelerometer, the relationship between the resonant frequency and the amplitude is analyzed, and based on the golden section algorithm and the principle of direct digital frequency synthesis, a method for accurately identifying the resonant frequency of a silicon resonant accelerometer is proposed, and the accuracy and feasibility of the method are verified by numerical simulation.

[0005] The present invention discloses a method for accurately identifying the resonant frequency of a silicon resonant accelerometer. The method for accurately identifying the resonant frequency of a silicon resonant accelerometer comprises the following steps:

[0006] Step 1, constructing an amplitude-frequency characteristic curve model of a silicon micro-resonance accelerometer resonator according to the modal characteristics of the silicon resonant accelerometer;

[0007] Step 2, finding the amplitude maximum point of the amplitude-frequency curve based on the golden section algorithm;

[0008] Step 3: Implement the golden section algorithm based on the principle of direct digital frequency synthesis.

[0009] Furthermore, in step 1, the following steps are also included:

[0010] Step 11, obtaining the kinematic equation of the silicon resonant accelerometer by performing modal analysis on the resonator;

[0011] Step 12, based on solving the kinematic equation of the silicon resonant accelerometer resonator, obtain the amplitude characteristic curve and the phase-frequency characteristic curve of the silicon resonant accelerometer resonator.

[0012] Furthermore, in step 11, the kinematic equation of the silicon resonant accelerometer resonator is:

[0013]

[0014] Among them, F s represents the driving force, F represents the amplitude of the driving force, f represents the frequency of the driving force, t represents the time, x represents the resonator displacement, represents the resonator movement speed, represents the acceleration of the resonator movement, M is the equivalent mass of the silicon resonant accelerometer resonator, c is the damping of the resonator, and k is the stiffness of the resonator.

[0015] Furthermore, in step 12, the kinematic equations of the silicon resonant accelerometer resonator are solved to obtain:

[0016] x(ω)=A(ω)sin(2πft+φ(ω))

[0017]

[0018] Among them, x(ω) represents the frequency response of the resonator displacement, A(ω) represents the amplitude-frequency characteristic of the resonator displacement, φ(ω) represents the phase-frequency characteristic of the resonator displacement, ω represents the angular frequency, A0 represents the gain of the amplitude-frequency characteristic of the resonator displacement, f0 represents the natural frequency of the resonator, and Q represents the quality factor of the resonator.

[0019] Furthermore, in step 2, based on the golden section algorithm of gradually narrowing the interval, the resonant frequency f is accurately identified by multiple iterations to determine whether the frequency sweep interval is less than the threshold ε. r , the calculation formula of the frequency sweep interval of multiple iterations of the golden section method is:

[0020] L i =λ i (ba)

[0021] In the formula, i is the number of iterations, L i is the length of the frequency sweep interval calculated after the i-th iteration, a and b are the left and right end points of the initially set frequency sweep interval, and λ is the golden ratio.

[0022] Furthermore, in step 3, the principle of direct digital frequency synthesis is to control the frequency of the output signal by the frequency control word. The relationship between the output signal frequency and the frequency control word is:

[0023]

[0024] Where f′ is the output signal frequency of direct digital frequency synthesis, D is the frequency control word, n is the number of bits of the frequency control word accumulator, and f clk is the frequency of the system clock; the two end points a and b of the frequency sweep interval correspond to different frequency control words, so the frequency value can be continuously modified through direct digital frequency synthesis to realize the golden section algorithm.

[0025] Furthermore, the method for accurately identifying the resonant frequency of a silicon resonant accelerometer comprises the following steps:

[0026] Step 4, verifying the accuracy and feasibility of the patented method through numerical simulation;

[0027] According to the amplitude-frequency formula of the resonator, the relationship between the theoretical resonant frequency and the natural frequency can be derived as follows:

[0028]

[0029] Among them, f r represents the resonant frequency;

[0030] The error relative to the theoretical resonant frequency is obtained through the theoretical resonant frequency value, verifying the accuracy and feasibility of the resonant frequency of the silicon resonant accelerometer.

[0031] The beneficial effects achieved by the present invention are:

[0032] The present invention provides a method for accurately identifying the resonant frequency of a silicon resonant accelerometer. According to the amplitude-frequency curve of the silicon resonant accelerometer and based on the golden section algorithm, multiple iterative frequency sweeps are performed, and the frequency sweep interval approaches the resonant frequency point from both ends at an exponential function speed, thereby realizing accurate identification of the resonant frequency.

[0033] The present invention provides a method for accurately identifying the resonant frequency of a silicon resonant accelerometer, which has the advantage of online identification of the resonant frequency. The method is very suitable for a frequency sweeping scheme of an engineered silicon resonant accelerometer.

[0034] The present invention provides a method for accurately identifying the resonant frequency of a silicon resonant accelerometer, which can be extended and applied to other resonant sensor frequency scanning schemes and has universality and versatility. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 A flowchart of a specific implementation plan;

[0036] Figure 2 This is the flow chart of the golden section algorithm;

[0037] Figure 3 Bode plot of the silicon resonant accelerometer resonator;

[0038] Figure 4 This is the simulation result diagram of the change curve of the frequency sweep interval length with the number of iterations;

[0039] Figure 5 It is the simulation diagram of the change curve of the two end points a and b;

[0040] Figure 6 This is a graph showing the change in the resonant frequency calculated by this method versus the number of iterations. DETAILED DESCRIPTION

[0041] The present invention will be further described below in conjunction with specific embodiments, and the advantages and features of the present invention will become clearer as the description proceeds. However, these embodiments are exemplary only and do not constitute any limitation to the scope of the present invention. It should be understood by those skilled in the art that the details and forms of the technical solution of the present invention may be modified or replaced without departing from the spirit and scope of the present invention, but these modifications and replacements all fall within the scope of protection of the present invention.

[0042] like Figure 1 As shown, this embodiment provides a method for accurately identifying the resonant frequency of a silicon resonant accelerometer, the method comprising the following steps:

[0043] Step 1, constructing an amplitude-frequency characteristic curve model of a silicon micro-resonance accelerometer resonator according to the modal characteristics of the silicon resonant accelerometer;

[0044] The specific steps also include:

[0045] Step 11, obtaining the kinematic equation of the silicon resonant accelerometer by performing modal analysis on the resonator;

[0046] Through the modal analysis of the silicon resonant accelerometer resonator, it can be found that the resonant motion of the silicon resonant accelerometer resonator can be regarded as a low-damped second-order resonant system. Its kinematic equation can be expressed as:

[0047]

[0048] Among them, Fs represents the driving force, F represents the amplitude of the driving force, f represents the frequency of the driving force, t represents the time, x represents the displacement of the resonator, represents the resonator movement speed, represents the acceleration of the resonator movement, M is the equivalent mass of the silicon resonant accelerometer resonator, c is the damping of the resonator, and k is the stiffness of the resonator.

[0049] In the formula, the equivalent mass of the silicon resonant accelerometer resonator is taken as M = 10 -14 kg, the equivalent stiffness of the resonator k = 2.9856 × 10 -4 N / m, damping c=10 -11 N·s / m.

[0050] Step 12, solving the kinematic equation of the silicon resonant accelerometer resonator to obtain an amplitude characteristic curve and a phase-frequency characteristic curve of the silicon resonant accelerometer resonator;

[0051] From the above formula (1), it can be concluded that the resonator is a second-order resonant system. The input driving force is a sinusoidal change, so the output of the resonator can be regarded as the sinusoidal response of the second-order resonant system. According to the sinusoidal response principle of the second-order system, the above formula is solved as follows:

[0052] x(ω)=A(ω)sin(2πft+φ(ω)) (2)

[0053]

[0054] Among them, x(ω) represents the frequency response of the resonator displacement, A(ω) represents the amplitude-frequency characteristic of the resonator displacement, φ(ω) represents the phase-frequency characteristic of the resonator displacement, ω represents the angular frequency, A0 represents the gain of the amplitude-frequency characteristic of the resonator displacement, f0 represents the natural frequency of the resonator, and Q represents the quality factor of the resonator.

[0055] Equation (2) is the solution of the resonator's sinusoidal response, and Equation (3) is the amplitude-frequency formula of the resonator, where is the natural frequency of the resonator, its value is 27500Hz, The quality factor of the driving mode is 172.788. Since the resonator is equivalent to a low damping system, its resonant frequency is close to the natural frequency of the resonator. The various parameters of the second-order system of the resonator are known, and finally the Bode diagram consisting of the amplitude characteristic curve and the phase-frequency characteristic curve of the silicon resonant accelerometer resonator can be obtained as shown in Figure 3 shown.

[0056] Step 2: Find the amplitude maximum point of the amplitude-frequency curve based on the golden section algorithm.

[0057] According to formula (3), the amplitude characteristic curve of the resonator has a maximum value only at the resonant frequency. As long as the maximum value of the resonator amplitude is found, the resonant frequency is found. Therefore, identifying the resonant frequency can be transformed into the problem of finding the maximum value of the A(ω) amplitude. The golden section algorithm is an algorithm for searching for the maximum value of a one-dimensional function. The golden section algorithm based on gradually narrowing the interval only requires a small amount of calculation and can accurately identify the resonant frequency through multiple iterations. The calculation formula for the search interval of multiple iterations of the golden section method is:

[0058] L i =λ i (ba) (4)

[0059] In the formula, i is the number of iterations, L i is the length of the frequency sweep interval calculated after the i-th iteration, a and b are the left and right end points of the initially set frequency sweep interval, respectively. In the simulation, the left end point a=20400 Hz and the right end point b=27600 Hz are taken, and λ is the golden ratio, whose value is 0.618.

[0060] The golden section algorithm flow chart is as follows Figure 2 As shown, by inputting the initial endpoint frequencies a and b, a1=b-λ(ba) and a2=a+λ(ba) are calculated respectively, and the sizes of A(2πa1) and A(2πa2) are determined; if A(2πa1)>A(2πa2), the endpoint frequency b=a2, and if A(2πa1)≤A(2πa2), the endpoint frequency a=a1 is determined, and it is determined whether |ba| is less than the threshold ε. If it is less than the threshold ε, the resonant frequency f r =(a+b) / 2, if it is greater than the threshold ε, the iterative endpoint frequencies a and b are calculated again.

[0061] From the above formula (4), the sweep interval of the golden section algorithm decreases exponentially. The simulation results show that the sweep interval length changes as follows: Figure 4 As shown, the interval length of the simulation results is also compressed at an exponential speed in accordance with the theoretical formula. According to the principle of the golden section algorithm, the maximum amplitude of the resonator is in the middle of the sweep interval. Therefore, the more iterations, the shorter the sweep interval, the more accurate the search for the maximum amplitude of the resonator, and the higher the accuracy of the found resonant frequency.

[0062] Step 3: Implement the golden section algorithm based on the principle of direct digital frequency synthesis.

[0063] The principle of direct digital frequency synthesis is to control the frequency of the output signal through the frequency control word. The relationship between the output signal frequency and the frequency control word is:

[0064]

[0065] Where f′ is the output signal frequency of direct digital frequency synthesis, D is the frequency control word, n is the number of bits of the frequency control word accumulator, which is 40; f clk is the frequency of the system clock, and its value is 100MHz.

[0066] The two end points a and b of the frequency sweep interval correspond to different frequency control words, so the frequency value can be continuously modified through direct digital frequency synthesis to realize the golden section algorithm.

[0067] Since the golden section algorithm needs to continuously modify the frequency values ​​at both ends of the sweep frequency interval, and the frequency control word is linearly related to the frequency of the output signal, the direct frequency synthesis method can directly and linearly control the frequency change, so this frequency synthesis method facilitates the golden section algorithm to approximate the resonant frequency. In addition, the direct frequency synthesis algorithm also improves the accuracy of identifying the resonant frequency.

[0068] Step 4: Verify the accuracy and feasibility of the patented method through numerical simulation.

[0069] Using Matlab software to perform numerical simulation, we get the curves of the end points a and b as follows: Figure 5 As shown, from Figure 4 It can be seen that the two end points gradually move toward the resonant frequency f r Approximation, or from Figure 6 The curve of the resonant frequency calculated by the patent method changing with the number of iterations shows that the resonant frequency calculated by this method is very close to the theoretical resonant frequency after about 15 iterations. In addition, according to formula (3), the relationship between the theoretical resonant frequency and the natural frequency can be deduced as follows:

[0070]

[0071] Among them, f r Indicates the resonant frequency.

[0072] The theoretical resonant frequency is calculated by formula (6) to be 27499.53946 Hz, and then the errors of the traditional method and the present method relative to the theoretical resonant frequency are obtained, as shown in Table 1. Since the silicon resonant accelerometer characterizes acceleration by measuring the resonant frequency, in high-precision or even strategic-level inertial navigation systems, the accuracy of identifying the resonant frequency of the resonant accelerometer requires an error within 1 Hz or even more stringent, while the traditional method in Table 1 has an error of about 22 Hz, and the method proposed in this patent has an error of 0.23 Hz. The accuracy of this method is more than 100 times higher than that of the traditional method, meeting the engineering requirements of the silicon resonant accelerometer with higher precision, proving that this method improves the accuracy and feasibility of identifying the resonant frequency of the silicon resonant accelerometer.

[0073] Table 1 Error comparison between the traditional method and the resonance frequency identification method proposed in this patent

[0074]

[0075] The above are only specific steps of the present invention and do not constitute any limitation to the protection scope of the present invention; any technical solutions formed by equivalent transformation or equivalent replacement shall fall within the protection scope of the present invention; some parts not elaborated in detail in the present invention belong to the known technologies of those skilled in the art.

Claims

1. A method for accurately identifying the resonant frequency of a silicon resonant accelerometer, characterized in that: The method for accurately identifying the resonant frequency of a silicon resonant accelerometer comprises the following steps: Step 1, constructing an amplitude-frequency characteristic curve model of a silicon micro-resonance accelerometer resonator according to the modal characteristics of the silicon resonant accelerometer; Step 2, finding the amplitude maximum point of the amplitude-frequency curve based on the golden section algorithm; Step 3, realizing the golden section algorithm based on the principle of direct digital frequency synthesis; In step 2, based on the golden section algorithm of gradually narrowing the interval, the resonant frequency f is accurately identified by judging whether the frequency sweep interval is less than the threshold ε through multiple iterations. r , the calculation formula of the frequency sweep interval of multiple iterations of the golden section method is: L i =λ i (ba) In the formula, i is the number of iterations, L i is the length of the frequency sweep interval calculated after the i-th iteration, a and b are the left and right end points of the initially set frequency sweep interval, and λ is the golden ratio; In step 3, the principle of direct digital frequency synthesis is to control the frequency of the output signal through the frequency control word. The relationship between the output signal frequency and the frequency control word is: Where f′ is the output signal frequency of direct digital frequency synthesis, D is the frequency control word, n is the number of bits of the frequency control word accumulator, and f clk is the frequency of the system clock; the two end points a and b of the frequency sweep interval correspond to different frequency control words, so the frequency value can be continuously modified through direct digital frequency synthesis to realize the golden section algorithm.

2. The method for accurately identifying the resonant frequency of a silicon resonant accelerometer according to claim 1, characterized in that: In step 1, the following steps are also included: Step 11, obtaining the kinematic equation of the silicon resonant accelerometer by performing modal analysis on the resonator; Step 12, based on solving the kinematic equation of the silicon resonant accelerometer resonator, obtain the amplitude characteristic curve and the phase-frequency characteristic curve of the silicon resonant accelerometer resonator.

3. The method for accurately identifying the resonant frequency of a silicon resonant accelerometer according to claim 2, characterized in that: In step 11, the kinematic equation of the silicon resonant accelerometer resonator is: Among them, F s represents the driving force, F represents the amplitude of the driving force, f represents the frequency of the driving force, t represents the time, x represents the resonator displacement, represents the resonator movement speed, represents the acceleration of the resonator movement, M is the equivalent mass of the silicon resonant accelerometer resonator, c is the damping of the resonator, and k is the stiffness of the resonator.

4. The method for accurately identifying the resonant frequency of a silicon resonant accelerometer according to claim 3, characterized in that: In step 12, the kinematic equations of the silicon resonant accelerometer resonator are solved to obtain: x(ω)=A(ω)sin(2πft+φ(ω)) Among them, x(ω) represents the frequency response of the resonator displacement, A(ω) represents the amplitude-frequency characteristic of the resonator displacement, φ(ω) represents the phase-frequency characteristic of the resonator displacement, ω represents the angular frequency, A0 represents the gain of the amplitude-frequency characteristic of the resonator displacement, f0 represents the natural frequency of the resonator, and Q represents the quality factor of the resonator.

5. The method for accurately identifying the resonant frequency of a silicon resonant accelerometer according to claim 1, characterized in that: The method for accurately identifying the resonant frequency of a silicon resonant accelerometer comprises the following steps: Step 4, verifying the accuracy and feasibility of the method for accurately identifying the resonant frequency of the silicon resonant accelerometer through numerical simulation; According to the amplitude-frequency formula of the resonator, the relationship between the theoretical resonant frequency and the natural frequency can be derived as follows: Among them, f r represents the resonant frequency, f0 represents the natural frequency of the resonator, M is the equivalent mass of the silicon resonant accelerometer resonator, c is the damping of the resonator, and k is the stiffness of the resonator; The error relative to the theoretical resonant frequency is obtained through the theoretical resonant frequency value, verifying the accuracy and feasibility of the resonant frequency of the silicon resonant accelerometer.

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