Digital-analog coupled 2-beam vibration accelerometer with linear output over the entire range

Abstract

A vibration beam accelerometer (10) can be mathematically modeled as a system having two beams (11, 11') operating in differential push-pull mode, each of the two beams vibrates at corresponding first and second vibration frequencies (f+, f-) where the respective amplitudes (A+, A-) are dominant outside a predetermined lock-in zone where the respective dominant frequencies (,) of the first and second beams are the same. Acceleration is determined within a predetermined range including the lock-in zone as a function of the measured vibration amplitudes of the two beams. Outside the predetermined range, acceleration is derived as a function of the difference between the measured vibration frequencies of the two beams.

Description

Technology Field

[0001] The present invention relates to a vibration beam accelerometer.

[0002] Conventional technology

[0003] References to the prior art considered relevant as background art to the present invention are listed below, the contents of which are incorporated herein by reference. The recognition of references herein should not be inferred to imply that they have any relation to the patentability of the invention disclosed herein. Each reference is identified by a number in brackets, and thus prior art will be referred to by numbers in brackets throughout the specification.

[0004] [1] O. Le Traon, D. Janiaud, M. Pernice, S. Masson, S. Muller and J.-Y. Tridera, “A new quartz monolithic differential vibrating beam accelerometer,” IEEE / ION Position, Location, And Navigation Symposium , Coronado, CA, USA, 2006, pp. 6-15.

[0005] [2] O. Lefort, S. Jaud, R. Quer and A. Milesi, “Inertial grade silicon vibrating beam accelerometer,” IEEE Inertial Sensors and Systems , Karlsruhe, Germany, 2012, pp. 1-19.

[0006] [3] T. Loret et al ., "Navigation grade accelerometer with quartz vibrating beam," DGON Inertial Sensors and Systems (ISS) , Karlsruhe, Germany, 2014, pp. 1-14.

[0007] [4] O. Lefort, I. Thomas and S. Jaud, "To the production of a robust and highly accurate MEMS vibrating accelerometer," IEEE Inertial Sensors and Systems , Karlsruhe, Germany, 2017, pp. 1-19.

[0008] [5] O. Le Traon et al ., "The NG DIVA: A navigation grade differential inertial vibrating beam accelerometer," IEEE / ION Position, Location and Navigation Symposium (PLANS) , Monterey, CA, USA, 2018, pp. 24-30.

[0009] [6] W. C. Albert, "Force sensing using quartz crystal flexure resonators," 38th Annual Symposium on Frequency Control , Philadelphia, PA, USA, 1984, pp. 233-239.

[0010] [7] K. A. Gibbons, "A micromechanical silicon oscillating accelerometer," Master of Science thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology (MIT), USA, 1997, pp. 1-102.

[0011] [8] N. St. Michel, "Force multiplier in a microelectromechanical silicon oscillating accelerometer," Master of Science thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology (MIT), USA, 2000, pp.1-90.

[0012] [9] K. S. Meredith, "Effects of mechanical coupling on oscillator frequency in a micromechanical accelerometer," Bachelor and Master of Science thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology (MIT), USA, 2001, pp.1-76.

[0013]

[10] O. Le Traon, D. Janiaud, B. Lecorre, M. Pernice, S. Muller and J.-Y. Tridera, "Monolithic differential vibrating beam accelerometer within an isolating system between the two resonators," SENSORS, IEEE , Irvine, CA, USA, 2005, pp. 648-651.

[0014]

[11] J. L. Gruver, "A novel mathematical derivation of the lock-in effect for Coriolis vibrating gyroscopes," DGON Inertial Sensors and Systems (ISS), Braunschweig, Germany, 2022, pp. 1-15.

[0015]

[12] J. R. Wilkinson, "Ring Lasers," Prog. Quant. Electr., 1987, vol. 11, pp. 1-103.

[0016]

[13] Y. Arazi and J. L. Gruver, "A new operational mode for axisymmetric Coriolis vibrating gyroscopes: Force Angle Tracking (FAT)," DGON Inertial Sensors and Systems (ISS) , Braunschweig, Germany, 2023, pp. 1-13.

[0017]

[14] Zhen Zhou, Jie Shi, Bao-yin Yao and Li-Shuang Feng, "Theoretical investigation of the mechanical coupling in a differential vibrating beam accelerometer," Micro-syst Technol, Springer-Verlag, Berlin Heidelberg, 2015, vol. 21, pp. 1459-1464.

[0018]

[15] H.M. Zhang, W.Z. Yuan, B.Y. Li, Y.C. Hao, M. Kraft, and H.L. Chang, "A novel resonant accelerometer based on mode localization of weakly coupled resonators," 2015 Transducers - 18th International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS), Anchorage, AK, USA, 2015, pp. 1073-1076.

[0019]

[16] H. Zhang, B. Li, W. Yuan, M. Kraft and H. Chang, "An Acceleration Sensing Method Based on the Mode Localization of Weakly Coupled Resonators," in Journal of Microelectromechanical Systems , vol. 25, no. 2, pp. 286-296, April 2016.

[0020]

[17] H. Zhang et al ., "A Mode-Localized Mems Accelerometer in the Modal Overlap Regime Employing Parametric Pump," 21 st International Conference on Solid-State Sensors, Actuators and Microsystems (Transducers) , Orlando, FL, USA, 2021, pp. 108-111.

[0021]

[18] Omer HaLevy and Slava Krylov, “Modeling of Frequency Locking in a Differential Vibrational Beam Accelerometer,” ENOC2020, July 5-10, 2020, Lyon, France. Background Technology

[0022] Vibrating beam accelerometers are based on the fact that the resonant frequency of a vibrating beam attached to an inertial mass changes when subjected to external acceleration. A beam subjected to a pulling force vibrates at a higher frequency, while a beam subjected to a pushing force vibrates at a lower frequency. This action is similar to that of a violin string, which increases its pitch as tension increases and decreases its pitch as tension (or relative compression) decreases. Typically, two vibrating beams are used in a differential arrangement to reduce common parasitic sensitivities, such as temperature, pressure, or degradation. The two beams operate in a push-pull mode, where one beam is compressed while being pushed, and the other is tensioned while being pulled. During operation, the two beams continuously switch roles, and as described in the literature, there exists a measured vibration frequency for each beam, which may appear as vibrating at two frequencies: one slightly higher corresponding to the pull mode and the other slightly lower corresponding to the push mode.

[0023] These beams are generally based on piezoelectric materials, such as quartz, which generate an electric charge in response to mechanical stress. When the beams are compressed or stretched, they generate an electric voltage across their surfaces. Conversely, an electric field applied across the beams will induce oscillations. To this end, an AC voltage is typically applied to each pair of electrodes deposited on the opposing surfaces of each beam to induce steady-state oscillations of the beams, the amplitudes of which are measured as analog signals across the respective electrodes deposited on the opposing surfaces of the two beams. The two beams oscillate at respective frequencies that can be used to measure acceleration, as is well known in the literature.

[0024] However, the use of two beams coupled by an inertial mass can lead to the existence of "blind spots" where the sensor fails to produce a useful output when external acceleration is present. These blind spots are widely known to occur due to a "lock-in" phenomenon in which the resonant frequencies of both beams are "locked" (i.e., both beams vibrate at the same frequency), causing the output acceleration (proportional to the frequency difference) to be zero even when external acceleration is present. This effect limits the range of acceleration at which the sensor can be used.

[0025] The applicant’s concurrently pending Israeli Patent IL 315720, filed on September 18, 2024, discloses that the primary cause of lock-in is that conventional approaches for measuring the vibration frequencies of beams measure only the dominant frequency, i.e., the frequency with the higher amplitude, and consequently ignore the less dominant frequency with the lower amplitude. Throughout this disclosure, the applicant will refer to these frequencies as dominant and semi-dominant frequencies. In contrast, Israeli Patent IL 315720 detects both the dominant and semi-dominant frequencies, which, when appropriately considered, can determine a lock-in boundary where the amplitudes of the dominant and semi-dominant frequencies intersect. The frequencies within the lock-in boundary can then be selected to avoid the effect of lock-in. Lock-in is avoided by also measuring the quasi-dominant frequencies in the lock-in zone and calculating the acceleration in the lock-in zone as a function of the difference between the dominant frequency of one beam and the quasi-dominant frequency of another beam. It was found that below a certain acceleration threshold, the relationship between the frequency difference and acceleration is nonlinear.

[0026] Specifically, for each of the beams, there exist corresponding first and second frequencies (f-, f+) with a dominant amplitude. The lock-in zone is defined by respective intersection points for each beam, and the respective amplitudes (A-, A+) at both the first and second frequencies (f-, f+) are identical, defining an intersection zone within the respective dominant amplitudes of the first and second frequencies such that the frequency with a higher amplitude outside the intersection zone has a lower amplitude inside the intersection zone, and vice versa.

[0027] An alternative approach is taken from the applicant’s concurrently pending Israeli patent IL 318446, filed on January 15, 2025, which avoids the lock-in problem by calculating acceleration as a function of vibration amplitude measured directly for each beam as an analog signal, rather than as a function of vibration frequency. It is presented that acceleration can be derived as a linear function of amplitude over the entire range of acceleration. Given that the accelerometer must respond over the entire range, an ideal linear function entails measuring both the dominant and sub-dominant amplitudes for each beam. This requires measuring the vibration amplitude (length dimension) at both dominant and sub-dominant frequencies. However, a very important point is that within the lock-in zone, the alternative function can be based on measurements of only the dominant or only the sub-dominant amplitudes. The alternative function also provides excellent linearity within the lock-in zone, although it is not as significantly accurate as the ideal function.

[0028] The combined teachings of Israeli Patent IL 315720 and Israeli Patent IL 318446 can be summarized as follows. Israeli Patent IL 315720 discloses an accelerometer having linear characteristics outside the lock-in zone, wherein acceleration is a linear function of the difference between the measured vibration frequencies of two beams. However, within the lock-in zone where the dominant frequencies of the two beams are the same, additional fine-tuning is required. On the other hand, Israeli Patent IL 318446 discloses an accelerometer having linear characteristics within the lock-in zone, wherein acceleration is a linear function of the measured vibration amplitudes of two beams.

[0029] For clarity, reference is made to the applicant’s concurrently pending Israeli patents IL 315720 and IL 318446, but it should be noted that neither said patent nor the gist thereof was disclosed prior to the effective filing date of this application. Accordingly, the foregoing discussion regarding lock-in and dominant and quasi-dominant frequencies provides important background art for the present disclosure, but is not prior art.

[0030] According to a broad aspect of the present invention, a method for determining acceleration using a vibrating beam accelerometer is provided, wherein, outside the lock-in zone, acceleration is derived as a function of the difference between the measured vibration frequencies of two beams, and inside the lock-in zone, acceleration is derived as a function of the measured vibration amplitudes of two beams. In some embodiments, the measured frequencies and amplitudes for both cases correspond to the dominant frequencies and amplitudes within each zone, namely, the frequency outside the lock-in zone and the amplitude inside the lock-in zone. Consequently, if the boundary of the lock-in zone is known and thus it can be determined for any measured acceleration whether it is inside or outside the lock-in zone, an appropriate transformation can be applied based on the direct measurement of the frequency difference or the amplitude. This avoids the requirements for supplementary processing required in Israeli Patent IL 315720 and Israeli Patent IL 318446 to measure the semi-dominant frequencies and amplitudes. Brief explanation of the drawing

[0031] To understand the present invention and to see how to implement the invention in practice, embodiments will now be described only as non-limiting examples with reference to the accompanying drawings. Figure 1 schematically illustrates a simple differential vibration beam accelerometer. Figure 2 is an equivalent spring mass model of the device shown in Figure 1. Figure 3 is a graphical representation of the amplitude-acceleration characteristics of the device derived using the model of Figure 2. FIGS. 4 and 5 graph the amplitude-acceleration characteristics of the device shown in FIG. 1 in both inside and outside the lock-in zone as functions of the ratio of vibration amplitudes to different combinations of dominant frequency amplitudes. FIG. 6 graphs amplitude-acceleration curves for a device derived using the model of FIG. 2 according to three different embodiments of the present invention. Figures 7, 8, and 9 graph the deviations from linearity for each of the three curves of Figure 6. FIGS. 10 and FIGS. 11 illustrate the dominant and sub-dominant frequencies of beam 1 and beam 2, respectively, both of which can be measured according to different embodiments of the present invention. Figure 12 is a graphical representation of the frequency-acceleration characteristics of the device derived using the model of Figure 2. Figure 13 graphs the relationship between the vibration amplitudes of both the beams inside and outside the lock-in zone. FIG. 14 is a graphic representation illustrating the corrected frequency difference (Δf(t)) for acceleration using an algorithm according to an embodiment of the present invention. Figures 15 and 16 are flowcharts illustrating an alternative method for determining the boundaries of a lock-in zone during accelerometer calibration. Specific details for implementing the invention

[0032] A proper understanding of the present invention requires a detailed analysis of the vibrating beam accelerometer, many of which will be familiar to those skilled in the art. For completeness, the applicant presents a model of the vibrating beam accelerometer and derives a mathematical formula defining the boundaries of the lock-in zone.

[0033] FIG. 1 schematically illustrates a simple vibrating beam accelerometer that can be mathematically interpreted using the equivalent spring-mass model of FIG. 2. For exemplary purposes and simplification, in the following description, the model illustrated in FIG. 2 considers the case of identical beams, and it is understood that the invention defined by the appended claims encompasses any actual embodiment of a vibrating beam accelerometer that can be reduced to the same or similar mathematical model, including cases where the beams are not identical. Accordingly, FIG. 1 illustrates an accelerometer (10) comprising a pair of collinear or parallel beams (11, 11') in which one end is respectively fixed to a support member (12, 12') and the opposite ends are commonly coupled to an inertial mass body (13). The beams (11, 11') may be formed of a piezoelectric material, such as quartz, which generates an electric charge in response to mechanical stress. When the beams are compressed or stretched, the beams generate an electric voltage across their surfaces. Conversely, an electric field applied across the beams will induce vibration. To this end, an AC voltage (14) is applied in common across each pair of electrodes (15, 15') deposited on the opposing surfaces of each beam, thereby inducing steady-state oscillation of both beams, the amplitude of which is measured as an analog signal across each electrode (16, 16') deposited on the opposing surfaces of the two beams.

[0034] When the structure is subjected to external acceleration, a force component parallel to the longitudinal axis of the beam moves the inertial mass back and forth axially, pushing one of the beams away and pulling the other, causing a slight difference in their respective resonant frequencies. Specifically, when using conventional methods for measuring frequency, the shorter compressed beam will vibrate at a lower frequency, which will be denoted as f- compared to the longer extended beam, which has a higher resonant frequency (f+). In an actual embodiment, the beams are used in a bending mode as sensing elements, and the vibration is maintained by an oscillator circuit schematically indicated by the AC voltage (14). As long as the beams are subjected to axial vibration, the beams will be continuously extended and shortened in reverse phase, and their resonant frequencies will likewise decrease and increase in reverse phase.

[0035] The applied acceleration induces tensile or compressive stress in each beam to modify the resonance frequency, and thus the measured value of the resonance frequency represents the applied acceleration.

[0036] The equation of motion describing the dynamics of the model in Fig. 2 is as follows.

[0037]

[0038] In the above formula is the externally applied acceleration.

[0039]

[0040] From mathematical formula (2), and It is evident that is a function of Γ. However, in all the following mathematical expressions, to prevent them from becoming difficult to handle, the dependency on Γ and Omitting the [unclear], simply and It will be referred to as.

[0041] Each and Divided as follows.

[0042]

[0043] Subtracting the first and third mathematical expressions from the second mathematical expression yields the following.

[0044]

[0045] New variable as follows and Defines.

[0046]

[0047] Substituting into mathematical formula (4) yields the following:

[0048]

[0049] New variable as follows and Defines.

[0050]

[0051] Substituting into mathematical formula (5) yields the following:

[0052]

[0053] Mathematical formula (7) can be written more conveniently in matrix form.

[0054]

[0055] It is defined as follows.

[0056]

[0057] Mathematical formula (8) can be rewritten as follows.

[0058]

[0059] Mathematical formula (9) has a solution of the following form.

[0060]

[0061] and

[0062]

[0063] In the above formula,

[0064]

[0065] The above mathematical equations form the basis for deriving acceleration based on both the amplitude and frequency of the vibrations of the two beams. However, since it is well known that outside the lock-in zone, acceleration is an almost linear function of the difference between the measured frequencies, the description of the present invention here will be limited to amplitude measurements within the lock-in zone. In this context, it will be recognized that it is necessary to know the boundaries of the lock-in zone before determining whether acceleration is calculated as a function of the frequency difference or the amplitude. This issue will be addressed at an appropriate time.

[0066] Calculation of acceleration from vibration amplitude

[0067] The vibration amplitudes of each beam can be calculated from mathematical equation (10) as follows.

[0068]

[0069] For clarity, regarding Γ in mathematical formula (13) and and and It is explained again that the dependency is omitted for simplification. Each beam is and It vibrates at two frequencies given by, and the respective vibration amplitude is , and , To our knowledge, existing technology makes no mention of the effect of the amplitudes of non-dominant frequencies on the determination of the lock-in zone. Likewise, there is no discussion of the possible influence of non-dominant amplitudes on the amplitude-acceleration characteristics inside or outside the lock-in zone.

[0070] Therefore, the literature [Zhang et al.

[15] ,

[16] and

[17] ] describes the ratio of the dominant vibration frequency amplitudes for the two beams within the lock-in zone This shows that it is sensitive to acceleration but nonlinear. Similar results are found in the literature [HaLevy et al.

[18] ], where Figure 2c plots the ratio between the measured dominant amplitudes of the two beams and illustrates that there appears to be a near-linear relationship in the lock-in zone. However, the apparent linearity can be misleading due to the highly compressed scale. Additionally, it can be seen that outside the lock-in zone, there is almost no sensitivity.

[0071] FIG. 3 is a graph showing the amplitude-acceleration characteristics of the device derived using mathematical equation (13). A specific acceleration ( and It can be seen that (indicated by) exists and defines opposite boundaries of the lock-in zone. Acceleration In the case of, the dominant amplitude is and And; In the case of, the dominant amplitude is and And; finally In the case of, the dominant amplitude is and It can be seen that... To avoid the need to designate the frequency with the dominant amplitude, this frequency will be referred to as the "dominant frequency," and it is understood that the dominant frequency can be f- or f+. For clarity, in Fig. 2, the positive acceleration is It refers to acceleration in the direction of increase, and negative acceleration is It should be noted that this indicates deceleration in one direction or acceleration in the opposite direction. The opposing boundaries of the lock-in zone are and It can be seen that this occurs at the point where the respective amplitudes of the dominant and sub-dominant frequencies for each beam become equal. Numbers 1 and 2 refer to Beam 1 and Beam 2, respectively, labeled as reference numbers 11 and 11' in Fig. 1. To distinguish the four frequencies (two for each of the two beams), the amplitude-acceleration curves are displayed in different colors. In jurisdictions where the use of color drawings is permitted, the four curves appear as follows.

[0072] [Table 1]

[0073]

[0074] In addition, since the curves are labeled according to the amplitudes shown in the table, those appearing in grayscale can be identified. From the drawing, it can be seen that the vibration amplitudes of both beams change in a continuous manner.

[0075] It can be seen that the intersection determines the negative and positive accelerations at the ends of the lock-in zone. Furthermore, it can be seen that the previously dominant frequency becomes semi-dominant at the intersections, and vice versa. Specifically, for positive acceleration outside the lock-in zone, the dominant frequency for Beam 1 is (Diagrammed in red), and this also remains the dominant frequency within the lock-in zone. In the case of negative acceleration outside the lock-in zone, the only frequency measured for Beam 1 is (Illustrated in blue). What this actually means is that if the acceleration applied to Beam 1 is higher than -2g, the only measured frequency is While it will be, if the acceleration is less than -2g, the only measured frequency is It will be. Likewise, if Beam 2 receives a positive acceleration greater than +2g, i.e., outside the lock-in zone, the only measured frequency is While it will be, if the acceleration is less than +2g, the only measured frequency is It will be. Inside the lock-in zone, the amplitudes for beams 1 and 2 are, respectively (indicated in red) and (Illustrated in green), and these are the dominant frequencies and It corresponds to vibration amplitudes, and the importance of this will soon become clear by referring to the mathematical equation (14) below.

[0076] FIGS. 4 and FIGS. 5 are each at an expanded scale and The amplitude ratios for the dominant frequencies within the lock-in zone, denoted by , are illustrated. These figures demonstrate that while the amplitude ratios are indeed sensitive to acceleration within the lock-in zone as depicted in the prior art, the relationship is certainly not linear. Additionally, the figures show the amplitude acceleration characteristics outside the lock-in zone based on the ratios between the respective dominant amplitudes, indicated by dashed lines. Thus, Figure 4 relates to negative acceleration. and for positive acceleration These are plots, and they are the ratios of the dominant amplitudes for Beam 1 to Beam 2 in each range. Therefore, as can be seen from Fig. 3, in the case of negative acceleration outside the lock-in zone, the dominant amplitude for Beam 1 is (Blue) and the dominant amplitude for beam 2 is (Green), on the other hand, for positive acceleration outside the lock-in zone, the dominant amplitude for Beam 1 is (Red) and the dominant amplitude for beam 2 is It is (bright blue). Similarly, Fig. 5 is for negative acceleration and for positive acceleration The plots represent the ratios of dominant amplitudes for beam 2 to beam 1 in each range. It can be seen that the amplitude ratios based on dominant amplitudes measured outside the lock-in zone are sensitive to acceleration to at least some extent, but the relationships are also not linear.

[0077] Before proceeding further, it may be useful at this point to summarize the results, as this establishes the essential components of the present invention. In this invention, it is shown that there are two vibration frequencies for each beam. The one with the larger amplitude is called the dominant frequency, and the other is called the sub-dominant frequency. Over the entire range of the accelerometer, the two frequencies can swap roles at one of the boundaries of the lock-in zone, so that the dominant frequency becomes the sub-dominant frequency and vice versa. The manner in which this occurs is graphed in FIG. 3. By definition, the vibration frequency whose amplitude is directly measured is always the dominant frequency, and in this invention, that amplitude is called the dominant amplitude. In this invention, it is also acknowledged that within the lock-in zone, the ratio of the respective dominant amplitudes for the two beams is sensitive to acceleration, but the relationship is not linear.

[0078] In one embodiment, the present invention discloses determining acceleration based on measurements of only dominant frequencies and amplitudes, while achieving greater linearity over the entire range. Accordingly, according to some embodiments, within the lock-in zone, acceleration is calculated as a function of the measured dominant amplitudes of vibrations for two beams, while outside the lock-in zone, acceleration is calculated as a function of the difference between the measured dominant frequencies of vibrations for two beams. For clarity, in this context, the term 'measurement' means that the dominant frequencies and amplitudes are measured directly across the output electrodes (16, 16') (shown in FIG. 1). In contrast, sub-dominant frequencies and amplitudes can be derived only by further processing of the signals across the output electrodes and are therefore measured only indirectly. The ability to obtain linear characteristics over the entire range of the accelerometer based solely on the direct measurement of dominant frequencies and amplitudes represents a substantial advantage over known approaches, including those taught in the applicant’s concurrently pending Israeli patents IL 315720 and IL 318446, which require additional processing for determining sub-dominant frequencies or amplitudes to achieve linearity over the entire range. This requirement is eliminated according to one aspect of the present invention, which allows acceleration to be determined, appropriately, as a linear function of amplitude or differential frequency over the entire range based solely on the measurement of dominant amplitude and frequency.

[0079] This will also involve all possibilities for a full disclosure, as it is still feasible to calculate acceleration within the lock-in zone as a linear function of quasi-dominant amplitudes. To this end, the present invention relates to amplitude-acceleration characteristics for dominant and quasi-dominant amplitudes and Defines as follows.

[0080]

[0081] Using these mathematical equations, the effective amplitude that varies as a function of acceleration Constitutes.

[0082]

[0083] Value from mathematical formula (14) and By substituting into mathematical formula (15), mathematical formula (15) can be rewritten as follows.

[0084]

[0085] The terms of mathematical formula (16) are rearranged as follows.

[0086]

[0087] Mathematical formula (17) can be physically expressed as follows.

[0088]

[0089] Equation (17) is an acceleration-dependent amplitude This defines the difference between the respective amplitude ratios of f- to f+ and f+ to f- for Beam 2 to Beam 1 minus the difference between the same respective amplitude ratios for Beam 1 to Beam 2. In both cases, the amplitudes of the frequencies f- and f+ will be dominant or semi-dominant, as shown in FIG. 3, depending on whether the associated acceleration is inside or outside the lock-in zone. Of particular importance is that, as can be seen in FIG. 3, within the lock-in zone, the dominant amplitudes are always for Beam 1 and Beam 2. and Corresponds to each.

[0090] Using mathematical equations 13, 14, and 15, it can be seen that the "amplitude ratio - acceleration" relationship has an accurate mathematical form in terms of the physical parameters of the system.

[0091]

[0092] port It is referred to as the effective mass of the beam. It is represented as. From mathematical equation (19), the nonlinear terms are It can be seen that it depends as a power of. Also, in mathematical equation (2), the spring constant and Is It will be recalled that it is equal to. λ is the ratio of the inertial mass to the Euler buckling load and is set low by design. Spring constant Since cannot be negative, must be less than 1. To simplify mathematical formula (19), It can be assumed that, and therefore The terms in can be ignored, and mathematical formula (19) can be approximated as follows.

[0093]

[0094] A less accurate "amplitude-acceleration" relationship for this can be obtained from mathematical equation (14).

[0095]

[0096] Therefore, acceleration is an approximate linear function of the measured vibration amplitudes of the two beams for both dominant and sub-dominant amplitudes.

[0097] Generally, output acceleration can be recorded as follows.

[0098]

[0099] In the above formula is a scale factor, and The terms consider the nonlinearity of the scale factor. In fact is taken as a measure of scale factor nonlinearity. These terms are calculated by fitting the accelerometer output using a polynomial function.

[0100] What is revealed from the analysis above is that within the lock-in zone, Equation (21a) can be used to calculate acceleration as a linear function of the measured amplitudes of the vibrations of the two beams, because they will always correspond to the dominant amplitudes within the lock-in zone. Alternatively, Equation (21b) can be used based on semi-dominant amplitudes, although additional signal processing is required. Likewise, Equation (20) can be used based on combined measurements of dominant and semi-dominant amplitudes.

[0101] Examples

[0102] Acceleration is generally expressed in units of the local value of the gravitational constant g, which is 10ms for the simulation -2 It was approximated as. For example, an inertial mass , mass In beam, spring constant , and Consider a specific embodiment. The initial conditions are and is the effective mass of the beam It can be calculated as follows.

[0103]

[0104] By substituting this value into mathematical formulas (20) and (21), the theoretical scale factor (SF) can be calculated as follows.

[0105]

[0106] Using mathematical formulas (14) and (15), as shown in FIG. 6 , , and Calculate the new "amplitude ratio" acceleration relationship corresponding to it. To distinguish the three graphs, the amplitude-acceleration curves are displayed using different colors and line styles. In jurisdictions where colored drawings are permitted, the three curves appear as follows.

[0107] [Table 2]

[0108]

[0109] In addition, since the curves are labeled according to the amplitudes shown in the table, it is possible to identify the curves that appear in grayscale. From this figure, it can be seen that the acceleration varies substantially linearly as a function of the measured vibration amplitude over the entire range, and within the lock-in zone, only the measured dominant amplitude is required.

[0110] Although the amplitude-acceleration relationship shown in Fig. 6 appears linear, it is only approximately linear. In practice, to estimate the nonlinearity of the scale factor, the actual accelerations are plotted as a function of the measured amplitudes, and then a polynomial fit of the resulting curves of the following form is estimated using least squares fitting or any other suitable technique.

[0111]

[0112] In the above formula refers to the accelerometer bias, and is a scale factor, and and This characterizes the nonlinearity of the amplitude-ratio acceleration relationship. From this polynomial fitting, the measured polynomial coefficients are estimated as follows.

[0113]

[0114] From mathematical equation (26), several interesting facts can be observed. First, the scale factor obtained from the polynomial fitting is in close agreement with the theoretical factor calculated in mathematical equation (24). Table 3 summarizes the results.

[0115] [Table 3]

[0116]

[0117] This ratio is the bias value (ideally should be 0).

[0118] [Table 4]

[0119]

[0120] It can be seen that the bias for the embodiment defined by mathematical formula (19) is nearly zero. However, the bias defined by mathematical formula (21) is not negligible but has a small value. Finally, the nonlinear coefficients of order 2 and order 3 are calculated as follows.

[0121] [Table 5]

[0122]

[0123] Again, non-linear coefficients is SF 0 It is very small or even negligible. However, The term is SF + and SF - It is a few digits larger than . The same applies to protests.

[0124] Figures 7, 8, and 9 graph the deviation from linearity for each of the three curves in Figure 6. Each curve plots the deviation of the amplitude-acceleration characteristic from the respective best linear fit for each curve. and The maximum deviation of acceleration from the ideal linear value is approximately 0.007g, i.e., approximately 0.07m / s² 2 On the other hand, The deviation in acceleration in the lock-in zone for is shown to be nearly zero. This implies that there is a trade-off for the possibility of slight inaccuracy in the case of calculated acceleration in the lock-in zone based solely on the measured dominant vibration amplitudes. On the other hand, the present invention and It avoids the need to determine the amplitudes of the semi-dominant amplitudes required for. As can be seen in Fig. 9, while requiring additional processing to determine the semi-dominant amplitude, Because it experiences the same deviation as, It is determined that there is little incentive to calculate the acceleration within the lock-in zone based solely on the dominant amplitudes. Nevertheless, it is still possible to calculate the acceleration within the lock-in zone from the semi-dominant amplitudes. More importantly, if slight inaccuracies in calculating the acceleration based solely on the measured dominant amplitudes are unacceptable, both the dominant amplitude and the semi-dominant amplitude can be determined, and then, according to Equation (20). Acceleration can be calculated based on [this]. As shown in mathematical formulas (16) and (17). Is and Since it is a function of, this is a measurement of both dominant and sub-dominant amplitudes, and It requires the calculation of. Then, the corresponding acceleration is determined from the corresponding amplitude-acceleration shown in FIG. 6. This is, of course, applicable over the entire range of the accelerometer, even if the applicability to the present invention is mainly limited to within the lock-in zone. In any case, only when slight inaccuracies in Equation (14) are unacceptable It is repeatedly explained that it is required to calculate acceleration based on [it].

[0125] The above description with reference to FIGS. 6 through 9 relates to the calculation of amplitude-acceleration characteristics over the entire range of the accelerometer, that is, both outside and inside the lock-in range. FIG. 6 shows three different linear amplitude-acceleration characteristics, among which, based on Equation (21) and The two corresponding to are Valid for, Other corresponding characteristics do not make such assumptions. However, the present invention is limited in all embodiments to a hybrid accelerometer that calculates acceleration within a defined range including a lock-in zone based on amplitude measurements and calculates acceleration outside the defined range based on differential frequency measurements.

[0126] In summary, within the lock-in zone, in one embodiment, vibration amplitude associated with the dominant frequency A substantially linear characteristic based on direct measurement is obtained. This means that within the lock-in zone, vibration amplitudes as measured across the electrodes (16 and 16') of FIG. 1 are sufficient to derive acceleration using Equation (14). In this embodiment, quasi-dominant frequencies (illustrated in blue) and Vibration amplitudes for (shown in light blue) and There is no need to determine. Outside the lock-in zone, acceleration is calculated as a function of the difference in the vibration frequencies of the two beams in a known way.

[0127] It is well known that lock-in occurs when the frequencies of the amplitude-dominant vibrations are the same for two beams. Conventional differential two-beam accelerometers determine acceleration as a function of the frequency difference, and since only the dominant frequencies are measured and the frequency difference is zero, lock-in occurs when the measured frequencies of the two beams are the same. The applicant’s concurrently pending Israeli patent IL 315720 shows that for each beam, there exist corresponding first and second frequencies (f-, f+) with respective amplitudes. The lock-in zone is bounded by respective intersection points for each beam, and the respective amplitudes (A-, A+) at both the first and second frequencies (f-, f+) are the same, defining an intersection zone within the respective dominant amplitudes of the first and second frequencies such that the higher frequency outside the intersection zone has a lower amplitude inside the intersection zone, and vice versa.

[0128] To determine when to measure the amplitude of the quasi-dominant frequency, the boundaries of the lock-in zone must be known. This is determined at the factory and can be communicated to the end user by the manufacturer. However, for the sake of completeness, before detailing how to measure accurate amplitudes depending on whether a given acceleration is inside or outside the lock-in zone, I will now explain how to derive the lock-in zone.

[0129] Derivation of the lock-in zone

[0130] From mathematical equation (12), both beams have natural frequencies and - Here It can be seen that it vibrates at -. This means that a single beam attached to an inertial mass vibrates at a single frequency, but two beams attached to the same mass vibrate at two natural frequencies. and It means that it vibrates through a combination of.

[0131] In actual applications, beams have an initial zero velocity, i.e., It comes to vibrate as. That is, the horizontal axis, i.e., and By pulling the beams along the path, the beams are deviated from equilibrium. Equations (10) and (13) can be simplified as follows.

[0132]

[0133] It is defined as follows.

[0134]

[0135] In the above formula and It is as follows.

[0136]

[0137] The amplitudes are the values ​​of external acceleration It will vary depending on. The lock-in boundary is defined by the following conditions.

[0138]

[0139] Find the solution to condition (30) for and obtain the following.

[0140]

[0141] In a special case, that is, when both beams receive the same initial lateral displacement,

[0142]

[0143] Equation (32) follows the range of the lock-in zone measured by the literature [Le Traon, 2006 ([1])]. It can be seen that if the masses of the two beams are not equal, the lock-in range will not be centered at zero frequency, but will be centered at a positive or negative acceleration position depending on the size of the two beams. For small differences between the masses of the beams, Equation (32) remains a very good approximation. It will be understood that Equation (32) specifies negative and positive acceleration at the boundary of the lock-in zone. In practice, since the amplitude is measured to determine acceleration, it is not necessary to know the frequency at the boundary of the lock-in zone, but to know the vibration amplitude.

[0144] With this in mind, this institution now [describes] the amplitude-acceleration characteristics for quasi-dominant and dominant frequencies and The effective amplitude defined in mathematical formulas (14) and (17) showing It returns to. To calculate this, the vibration amplitude of frequency f+ must be measured. However, as can be seen in Fig. 3, f+ will be the dominant or sub-dominant frequency depending on whether we are inside or outside the lock-in zone. Of course, measuring the vibration amplitude of frequency f- is necessary and, measurement of the respective amplitudes of both f+ and f- is required The same applies to . However, within the lock-in zone, directly measured amplitudes These are always the amplitudes of the dominant frequency.

[0145] Therefore, if it is known that the measured acceleration is within the lock-in zone, the dominant amplitudes can be measured directly. In other words, within the lock-in zone, the measured vibration amplitude is always Since it is the same, acceleration can be calculated directly from mathematical equation (21) as follows.

[0146]

[0147] Alternatively, as mentioned above, After determining the quasi-dominant amplitudes corresponding to, the acceleration can be derived from mathematical equation (21) as follows.

[0148]

[0149] likewise, After determining the quasi-dominant amplitudes corresponding to, the acceleration can be derived from mathematical equation (17) as follows.

[0150]

[0151] For acceleration outside the lock-in zone, the vibration frequency is measured in the conventional manner rather than the vibration amplitude of the two beams, and the acceleration is calculated as a function of the frequency difference between the two beams. To do this, we will return to Equation (12) regarding frequency-acceleration characteristics, and for simplification, for Γ and The omission of dependency is repeated. Differential frequency is given by the following.

[0152]

[0153] It should be recalled that at any given moment, the beams will vibrate at slightly different frequencies (f-, f+). Specifically, one beam will vibrate at a slightly higher frequency f+ corresponding to the full mode, and the other will vibrate at a slightly lower frequency f- corresponding to the push mode. These two frequencies are indicated in Equation (12), and therefore Therefore, the measured frequency difference between the two beams is provided by mathematical formula (33).

[0154] and In the case of, The mathematical formula (33) can be ignored and can be approximated as follows.

[0155]

[0156] We use a partial Taylor expansion for, where is, and If the terms are ignored, it is as follows.

[0157]

[0158] This results in the following outcome.

[0159]

[0160] In this case.

[0161]

[0162] In this case.

[0163]

[0164] In the above formula is a scale factor.

[0165] Examples

[0166] For example, an inertial mass , mass In beam, spring constant , Consider a specific embodiment. The initial conditions are and is. Output signal using mathematical formula (10) Calculate.

[0167] To measure the vibration frequency of each beam, a frequency counter based on the signal zero-crossing at a predetermined time interval is used. The frequency counter provides a single output frequency, which always corresponds to the dominant frequency. If the measured acceleration is outside the lock-in zone, the frequency difference Δf is proportional to the applied external acceleration. The vibration amplitudes of each beam can be calculated from Equation (10) as follows.

[0168]

[0169] Again, for clarity, in mathematical formula (39), for Γ and and and It is emphasized again that the dependency is omitted for the sake of simplification.

[0170] Figure 10 illustrates the frequency-acceleration characteristics for beam 1, showing both the dominant frequency and the quasi-dominant frequency. The dominant frequencies are, respectively and is plotted by solid red and blue lines. For completeness, the respective quasi-dominant frequencies are also plotted by dashed red and blue lines. Likewise, Fig. 11 plots the frequency-acceleration characteristics for beam 2, showing both the dominant and quasi-dominant frequencies. The dominant frequencies are, respectively and This is plotted by solid green and light blue lines. Likewise, the respective quasi-dominant frequencies are plotted by dashed green and light blue lines. As expected for identical beams, it can be seen that the frequency-acceleration characteristics for both beams are identical.

[0171] Outside the lock-in zone, the dominant and sub-dominant frequencies for each beam are both frequencies, i.e. and It can also be seen that it forms a continuum with respect to . Therefore, referring to FIG. 10 with respect to beam 1, The red solid line curve representing it stops at the negative boundary of the lock-in zone at acceleration -2g, and at acceleration below -2g, the quasi-dominant frequency indicated by the red dashed line at this point continues smoothly. The opposite applies equally to the blue curves representing, and these curves are based on the horizontal axis They form mirror images of the curves. The same applies to Beam 2, where each curve is complementary to the curves of Beam 1. That is, is the dominant frequency and extends for negative acceleration, stopping at the positive boundary of the lock-in zone at an acceleration of +2g. At exactly this point for accelerations greater than +2g, the sub-dominant frequency, illustrated by the green dashed line, continues smoothly. A similar but complementary observation can be made regarding the sub-dominant frequency as well.

[0172] As can be seen from FIGS. 10 and 11, if the boundaries of the lock-in zone are known, these figures can be used to determine the crossover frequencies at the two boundaries of the lock-in zone, that is, the respective upper frequencies at which the dominant frequency crosses the boundaries of the lock-in zone. In the case of a balanced system where the two beams are identical, the frequency-acceleration curve is symmetric with respect to the vertical frequency axis, i.e., zero acceleration. In this case, the dominant crossover frequencies of beam 1 will be the same, and therefore the dominant crossover frequencies of beam 2 will also be the same. For both beams, it can be seen that the crossover frequency is approximately 69,910 Hz. In the case of a non-balanced system, the crossover frequencies at the opposing boundaries of the lock-in zone may differ, but the principle is the same. In both FIGS. 10 and 11, lower frequency characteristics are primarily shown as dashed lines because they represent most of the sub-dominant frequencies and also intersect with the boundaries of the lock-in zone at the same frequencies. For both beams, it can be seen that the sub-dominant crossover frequency is approximately 69,810 Hz. If only the upper frequency-acceleration characteristics are considered, it can be seen that the dominant frequencies within the lock-in zone are dipping, that is, they are lower than the crossover frequencies. This means that if the dominant frequency measured for beam 1 is higher than the crossover frequency, it can be inferred that it is not in the lock-in zone. In this case, it is necessary to measure the sub-dominant amplitude of one or the other beam. One way to do this is to sample the signals for the electrodes (16, 16') for each beam over a given time period and convert them from the time domain to the frequency domain using, for example, the FFT as described above. This allows for the determination of both f+ and f- and their respective amplitudes.

[0173] Conversely, if the dominant frequency measured for beam 1 is lower than the crossover frequency, it is in the lock-in zone, and in this case, it can be inferred that there is no need to measure the sub-dominant amplitudes. The same applies to beam 2, which can be seen in Fig. 11. The dominant frequency can be measured using a zero-cross frequency counter in a conventional manner, or using any other suitable method. The vibration amplitude of the beam vibrating at the dominant frequency is measured directly. To determine the vibration amplitude associated with the sub-dominant frequency, the FFT can be used as previously described. Of course, in all cases, the FFT can be used to determine the respective vibration amplitude associated with the dominant or sub-dominant frequency.

[0174] For the sake of completeness, it may be shown that while the previous analysis concerns a balanced system in which the masses of the two beams are equal, for an accelerometer with beams of different masses, the lock-in range will not be centered at zero frequency, but will be centered at a positive or negative acceleration position depending on the masses of the two beams. This does not affect the way the boundaries of the lock-in zone are determined, and thus allows the same principle of the present invention to be applied to the inside and outside of the lock-in zone.

[0175] FIG. 12 plots the measured output frequencies of two beams as measured by conventional accelerometers on a common axis; these are, of course, the dominant frequencies by definition, since only these are measured using the conventional approach. In jurisdictions where color coding is permitted, the frequency curve for Beam 1 is blue and the frequency curve for Beam 2 is orange. By using the signal zero-cross detection method, the range of acceleration It can be seen that the frequency output is identical for . In terms of color, the blue and orange curves for the dominant frequencies, i.e., the two dominant frequencies and It overlaps in the lock-in zone. It is emphasized once again that 'dominant frequency' refers to the frequency in which the amplitude is dominant. This can be explained by the fact that this method detects only the dominant frequency of each beam. In this document, with reference to Fig. 3, the dominant amplitude in this range and It was proven that the detected frequency would be f+. It can be seen that the frequency measured in the lock-in region is f- or f+ depending on the initial conditions. For example, In this case, the lock-in frequency is f+, and In this case, the lock-in frequency is f. Also, it can be seen that the boundary of the lock-in region is [-2,2]g.

[0176] Figure 13 summarizes the relationship between the vibration amplitudes of both beams inside and outside the lock-in zone. The colors of the amplitude-acceleration curves for the two beams are the same as in Figure 3, and when viewing the grayscale of the curves, they can be interpreted according to [Table 1].

[0177] As can be seen in the diagram, outside the lock-in zone, both dominant amplitudes have different associated frequencies for each beam. In the case of negative acceleration, the dominant amplitude for beam 1 is the corresponding frequency having And, the dominant amplitude for beam 2 is the corresponding frequency having is. If you subtract the frequency of the second beam from the frequency of the first beam, ...becomes. In the case of positive acceleration, outside the lock-in range, the dominant amplitude of Beam 1 is the corresponding frequency. having And, the dominant amplitude for beam 2 is the corresponding frequency having It is. To reiterate, if you subtract the frequency of the second beam from the frequency of the first beam, It becomes.

[0178] Fig. 14 is The frequency difference as a function of external acceleration Γ when calculated according to the above procedure It is illustrated. It can be seen that when the measured frequency difference is zero, the acceleration is zero. However, when the acceleration is zero, the frequency difference is discontinuous and non-linear, but according to the present disclosure, this is not a concern because the acceleration in the lock-in zone within is calculated based on vibration amplitude rather than frequency. Furthermore, it can be seen that most of the frequency-acceleration characteristics are approximately linear. Specifically, when the absolute value of the differential frequency exceeds 200 Hz, which corresponds to an acceleration of approximately ±5g, the acceleration is substantially a linear function of the differential frequency. This avoids the need to store the complete frequency-acceleration characteristics and facilitates the rapid calculation of acceleration for most of the range once the differential frequency is measured. Additionally, according to the present invention, since the linear frequency-acceleration characteristics are used only outside the lock-in zone, the differential frequency can be easily calculated as the difference between the dominant frequencies of the two beams measured directly across the output electrodes (16, 16'). Consequently, the determination of the differential frequency and the resulting acceleration is fast and requires very little processing.

[0179] According to the preceding explanation, we must know the boundaries of the lock-in zone in advance or determine them on-site. For acceleration outside the lock-in zone, the dominant vibration frequency for both beams can be measured, and the frequency-acceleration function can be calculated, for example, as shown in Equation (38). Alternatively, even at the cost of additional processing, the quasi-dominant vibration frequency for both beams can be measured because the frequency-acceleration characteristics, such as those in Fig. 14, are the same for the dominant frequency and the quasi-dominant frequency. This can be empirically known from Figs. 10 and 11, as the two frequency-acceleration characteristics for each beam are identical within the lock-in zone and complementary (i.e., anti-symmetric) outside the lock-in zone. For example, at an acceleration of 3g, the dominant frequency for beam 1 is It can be seen from Fig. 10 that it is (solid line) and is approximately equal to 69,925 Hz. Likewise, from Fig. 11, at the same acceleration, the dominant frequency for beam 2 is It can be seen that it is (solid line) and is approximately equal to 69,800 Hz. Similarly, from Fig. 10, at an acceleration of 3g, the quasi-dominant frequency for beam 1 is It can be seen that it is (dotted line) and is approximately equal to 69,800 Hz, and in Fig. 11, at the same acceleration, the quasi-dominant frequency for beam 2 is (Dashed line) shows that it is approximately equal to 69,925 Hz. Therefore, it can be seen that the absolute difference between the frequencies is the same regardless of whether the dominant frequency or the sub-dominant frequency is measured.

[0180] To measure the quasi-dominant amplitude or quasi-dominant frequency, it is necessary to isolate the quasi-dominant frequency component of the output signal. Any frequency measurement method that provides a distinction between the two harmonics, namely f+ and f-, may be used. For example, the Fast Fourier Transform (FFT) can be considered the oldest and best-known in the frequency domain, and it can plot frequency spikes representing the dominant and quasi-dominant frequencies, with the respective amplitudes given by the height of the spikes. Other estimation methods, such as the least squares method and Kalman and extended Kalman filters, are described in the literature. Additionally, signal removal can be used by determining the dominant frequency using a phase-locked loop or a frequency counter, synthesizing an equivalent sinusoidal signal of the same frequency, and then subtracting the synthesized signal from the original to derive the quasi-dominant signal. Once the dominant and quasi-dominant frequencies are known, their respective amplitudes can be calculated using Equation (27) or determined by the Fourier Transform as described above. Any method capable of detecting harmonic components of a given signal without loss of generality may be used.

[0181] Actual implementation example

[0182] The discussion thus far has been largely analytical. In this document, for any accelerometer that can be mathematically modeled as a two-beam accelerometer or, in fact, a system having two beams operating in differential push-pull mode, we have shown that each beam will oscillate at two frequencies, one of which has a higher amplitude and is therefore referred to as the dominant frequency; and the other has a lower amplitude and is referred to as the non-dominant frequency. We have also demonstrated that there are respective points for both beams during negative and positive acceleration, where the amplitudes of both frequencies for each beam are identical. In this document, we call these points intersections, and they define the range or boundary of the lock-in range. To avoid doubt, the amplitudes of the respective frequencies at the intersections may differ for the two beams if the beams are not identical. However, while physical properties of the beams, such as mass or stiffness, may cause differences in the amplitudes of the two frequencies at the intersection of each beam, the fact remains that the intersections for the two beams always define opposite extremes of the lock-in range.

[0183] In an alternative embodiment, as described above, a crossover frequency—that is, a higher frequency at which the dominant frequency crosses the boundary of the lock-in zone—can be determined. Knowing this, for any given acceleration to be measured, the measured frequency can be evaluated based on whether it is greater or smaller than the crossover frequency within any specified tolerance threshold. The threshold sets a predefined range that includes the lock-in zone but may extend beyond the lock-in zone. For example, referring to FIG. 10, it can be seen that when the acceleration is ±3g, the dominant frequency is approximately 69,925 Hz. In the present invention, a range limited by this frequency can be defined for both positive and negative accelerations, so that if the measured dominant frequency is less than 69,925 Hz, the acceleration is calculated based only on the measured amplitude. This can be done according to Equation (17), which is applicable to the entire range of the accelerometer. Alternatively, a two-sided approach may be adopted so that if the measured dominant frequency is smaller than the predetermined cross frequency at the opposite boundary of the lock-in range, the acceleration can be calculated based on the measured dominant amplitude of both beams or the derived semi-dominant amplitude of both beams. This allows using Equation (14). or It can be calculated, and then the acceleration can be determined using Equation (21a) and Equation (21b), respectively. Otherwise, if the dominant frequency is greater than the predetermined crossover frequency at the opposing boundaries of the lock-in range, this means that it is outside the lock-in range. However, since the acceleration is still within a defined range where it is calculated as a function of amplitude rather than frequency, the effective amplitude It should be calculated based on the measured dominant and sub-dominant amplitudes of the two beams using mathematical formula (17), and then the acceleration can be determined using mathematical formula (20).

[0184] Conversely, if the measured dominant frequency exceeds 69,925 Hz, a function of the difference between the measured frequencies of the two beams according to Equation (39). Calculate acceleration as.

[0185] In the simplest and most direct embodiment, the predefined range is bounded by crossover frequencies at the precise boundaries of the lock-in range. In this case, if the measured frequency is smaller than the crossover frequency, it is certainly within the lock-in zone, and the acceleration can be calculated as a function of the respective measured (i.e., dominant) vibration amplitudes of the two beams using the upper equation of Equation (14) most simply. Conversely, if the measured frequency is larger than the crossover frequency, the acceleration is divided by the scale factor. It is calculated as a function of. In either case, if the defined range is set precisely at the boundaries of the lock-in zone, it is possible to avoid measuring both the quasi-dominant frequency or amplitude, thereby saving resources and achieving faster calculations.

[0186] All embodiments of the present invention require prior calibration to determine the boundaries of the lock-in range and the cross frequencies. This may be performed, for example, by the manufacturer, or by the end user before actual use by exposing the accelerometer to a range of acceleration and measuring the corresponding amplitudes so that for each beam, there exist corresponding first and second frequencies (f-, f+) whose amplitudes vary according to the dominant acceleration. Each cross point for each beam is determined at the point where the respective amplitudes (A-, A+) at both the first and second frequencies (f-, f+) are the same. Thus, it will be apparent that prior calibration to determine the boundaries of the lock-in zone serves to facilitate subsequent rapid measurement of acceleration, particularly for these embodiments that require direct measurement of only the dominant frequencies or amplitudes of vibration. In an alternative embodiment, vibration frequencies for both beams may be measured. If the frequencies measured for both beams are the same, it is within the lock-in zone, in which case only the amplitudes measured for both beams need to be considered. Conversely, if the frequencies measured for both beams are different, it is outside the lock-in zone, in which case acceleration can be calculated as a function of the difference between the measured frequencies.

[0187] Figure 12 plots the output frequencies of two beams as measured by a conventional accelerometer on a common axis; these are the dominant frequencies by definition, as only these are measured using the conventional approach. In jurisdictions where color coding is permitted, the frequency curve for Beam 1 is blue and the frequency curve for Beam 2 is orange. By using the signal zero-cross detection method, the range of acceleration It can be seen that the frequency output is identical for . In terms of color, the blue and orange curves for the dominant frequencies, i.e., the two dominant frequencies and It overlaps in the lock-in zone. It is emphasized once again that "dominant frequency" refers to the frequency in which the amplitude is dominant. This can be explained by the fact that this method detects only the dominant frequency of each beam. In this document, with reference to Fig. 3, the dominant amplitude in this range and It was proven that the detected frequency would be f+. It can be seen that the frequency measured in the lock-in region is f- or f+ depending on the initial conditions. For example, In this case, the lock-in frequency is f+, and In this case, the lock-in frequency is f. Also, the boundary of the lock-in region is It can be seen that it is g.

[0188] FIGS. 13 and 14 are flowcharts illustrating an alternative method for determining the boundaries of a lock-in zone during accelerometer calibration based on the characteristics of the lock-in zone described above. In both cases, to simplify the flowchart, only the positive boundaries of the lock-in zone are calibrated, and the same procedure is performed to determine the negative boundaries. Thus, in FIG. 13, the device receives variable acceleration increasing from zero to positive within a predetermined range. The boundaries are established by measuring acceleration where the amplitudes of the dominant frequency and the quasi-dominant frequency are equal.

[0189] In FIG. 14, the device is exposed to various accelerations exceeding the expected acceleration at the intersection boundary, and the absolute value decreases until the dominant frequencies of each of the two beams become equal for the first time. At this point, we have reached the boundary of the lock-in zone. If we reduce the positive acceleration or increase the negative acceleration (i.e., reduce the negative value), we reach the opposite boundary where the measured dominant frequencies of the two beams differ while remaining within the lock-in zone. We refer to this as another alternative approach that can be adopted to determine the boundary of the lock-in zone. In both cases, the intersection points correspond to the positive and negative accelerations at which the dominant or measured vibration frequencies of the two beams have the highest maximum absolute value.

[0190] Finally, for the sake of completeness, damping was not discussed, but it can be seen that the same principle applies to damped vibrations in accelerometers. In a real system, damping will occur, but the AC voltage (14) applied to each pair of electrodes (15 and 15') across the two beams can be adjusted using feedback control to offset the damping effect.

[0191] conclusion

[0192] The present invention differs from known approaches in that it utilizes measurements of vibration amplitudes within a predefined range including a lock-in zone and vibration frequencies outside the predefined range. When the predefined range is set at the precise boundary of the lock-in zone, the measured amplitudes and frequencies may correspond to dominant frequencies, the measured values ​​are used "as is," and there is no requirement to determine sub-dominant amplitudes or frequencies.

Claims

Claim 1 A method for determining acceleration using a vibration beam accelerometer (10) which can be mathematically modeled as a system having two beams (11, 11') operating in a differential push-pull mode, wherein each of the two beams has a dominant frequency of the first and second beams, respectively ( f+ , f- Outside the same predetermined lock-in zone, each amplitude ( A+ , A- ) dominant corresponding first and second vibration frequencies ( , A method comprising vibrating as ), and comprising: (a) deriving acceleration as a function of the measured vibration amplitudes of the two beams within a predefined range including the lock-in zone; and (b) deriving acceleration as a function of the difference between the measured vibration frequencies of the two beams outside the predefined range. Claim 2 In claim 1, (a) applying acceleration to the accelerometer (10) and each dominant vibration frequency for each beam ( , (b) a step of measuring ) when the dominant frequency exceeds a predetermined crossover frequency at opposing boundaries of a predefined range, i) a step of measuring a corresponding differential frequency (Δf) equal to the difference of each of the measured dominant frequencies for each beam; and ii) determining acceleration (Γ) from the measured differential frequency (Δf) and the pre-corrected frequency-acceleration characteristics; (c) when the dominant frequency at the opposing boundaries of the pre-defined range is smaller than the crossover frequency, i) A function satisfying the following It includes a step of calculating acceleration (Γ) as, and In the above formula is the effective amplitude from which acceleration is derived according to a known approximate linear function, and A method wherein the measured or derived amplitudes of vibrations α and β for each beam, or functions thereof, are, where if the dominant frequency is smaller than a predetermined crossover frequency at the opposing boundaries of the lock-in range, α and β are the respective dominant amplitudes or respective sub-dominant amplitudes of the two beams, and if the dominant frequency is larger than the predetermined crossover frequency at the opposing boundaries of the lock-in range, α and β are functions of both the dominant and sub-dominant amplitudes of the two beams. Claim 3 A method according to paragraph 2, wherein the predefined range is set at the boundary of the lock-in zone, α is the amplitude (A1+) of the dominant vibration frequency of beam 1 inside the lock-in zone, and β is the amplitude (A2+) of the dominant vibration frequency of beam 2 inside the lock-in zone. Claim 4 In paragraph 3, acceleration (Γ) is a solution to the following mathematical formula, and In the above formula A method in which is a function of the vibration amplitudes for both beams at the respective dominant vibration frequencies (f+) for each beam, and k is a constant. Claim 5 In paragraph 4, is a method given by the following mathematical formula. Claim 6 A method according to paragraph 2, wherein the predefined range is set at the boundary of the lock-in zone, α is the amplitude (A1-) of the quasi-dominant vibration frequency of beam 1 inside the lock-in zone, and β is the amplitude (A2-) of the quasi-dominant vibration frequency of beam 2 inside the lock-in zone. Claim 7 In paragraph 6, acceleration (Γ) is a solution to the following mathematical formula, and In the above formula A method in which is a function of the vibration amplitudes for both beams at each quasi-dominant vibration frequency (f-) for each beam, and k is a constant. Claim 8 In Paragraph 7, is a method given by the following mathematical formula. Claim 9 In paragraph 4 or 7, the constant k is given by the following mathematical formula, and In the above equation, M is the mass of the inertial mass, and And, is the mass of each beam, and λ is It is a scale factor satisfying, and is the spring stiffness at acceleration = 0, and is the spring stiffness at acceleration = Γ, method. Claim 10 In claim 1 or 2, acceleration Γ within the aforementioned predefined range is a solution to the following mathematical formula, and In the above formula is the first and second vibration frequencies for each beam ( f- , f+ ) is a function of the vibration amplitudes for the two beams in both, M is the mass of the inertial mass, and And, is the mass of each beam, and λ is It is a scale factor satisfying, and is the spring stiffness at acceleration = 0, and is the spring stiffness at acceleration = Γ, method. Claim 11 In Paragraph 10, is given by the following mathematical formula, and In the above formula and is the first vibration frequency of beams 1 and 2 ( f+ These are the respective amplitudes of ), and is the second vibration frequency of beams 1 and 2 ( f- The method, which consists of the respective amplitudes of ). Claim 12 In Article 10 or Article 11, and the acceleration Γ within the above-mentioned predefined range is an approximate solution to the following mathematical formula, method. Claim 13 In paragraph 12, acceleration within the lock-in zone is approximated as shown in the following mathematical formula, and In the above formula , that is, a function of the dominant amplitude or the quasi-dominant amplitude, respectively, and M is the mass of the inertial mass, and is acceleration, and And, is the mass of each beam, and λ is It is a scale factor satisfying, and is the spring stiffness at acceleration = 0, and is the spring stiffness at acceleration = Γ, method. Claim 14 A method according to any one of claims 2 through 13, wherein the prior-corrected frequency-acceleration characteristic is a linear function. Claim 15 In Clause 14, the above linear function is as follows, and In the above equation, ω=2πf, and Δf = the measured frequency difference, and = acceleration and, = The mass of each beam, and = It is the stiffness of each beam, and λ = a known scale factor, method. Claim 16 In any one of claims 1 to 13, i) corresponding first and second frequencies in which the respective amplitudes for each beam are dominant at different accelerations over the range ( f- , f+ ii) obtaining each amplitude-acceleration characteristic pair by exposing the accelerometer to a range of known accelerations for each of the beams and measuring the corresponding signal amplitudes at each acceleration, so that ) exists, and ii) determining each intersection point for each beam - at each intersection point the first and second frequencies ( f- , f+ ) the respective amplitudes mentioned above in both ( A- , A+ A method further comprising the step of calibrating the vibration beam accelerometer by ) being identical and equal to A1 and A2 for each of the above beams. Claim 17 In any one of claims 1 to 13, i) for each beam, corresponding first and second frequencies in which one is dominant and the other is semi-dominant ( f- , f+ A method further comprising the step of calibrating the vibration beam accelerometer by exposing the accelerometer to a known range of positive and negative accelerations such that ) exists; ii) measuring the respective dominant frequency of each beam for each known acceleration; and iii) determining the respective intersection point where the dominant vibration frequencies of the two beams are the same and the absolute values ​​of the positive and negative accelerations for both beams are maximum. Claim 18 A method according to any one of claims 1 to 13, further comprising: measuring the dominant vibration frequencies of each of the two beams; and establishing that the acceleration is within the lock-in zone when the measured frequencies are the same for both beams.

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