CN111504551B - Strain moment instrument bandwidth expansion method based on least square complex exponential method - Google Patents

Strain moment instrument bandwidth expansion method based on least square complex exponential method Download PDF

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CN111504551B
CN111504551B CN202010161776.7A CN202010161776A CN111504551B CN 111504551 B CN111504551 B CN 111504551B CN 202010161776 A CN202010161776 A CN 202010161776A CN 111504551 B CN111504551 B CN 111504551B
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torquer
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pass filter
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CN111504551A (en
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段发阶
刘昌文
傅骁
谢琛
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Tianjin University
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    • G01—MEASURING; TESTING
    • G01L—MEASURING FORCE, STRESS, TORQUE, WORK, MECHANICAL POWER, MECHANICAL EFFICIENCY, OR FLUID PRESSURE
    • G01L25/00—Testing or calibrating of apparatus for measuring force, torque, work, mechanical power, or mechanical efficiency
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Abstract

The invention discloses a method for expanding the bandwidth of a strain type torque meter based on a least square complex exponential method, which comprises the following steps of firstly, identifying modal parameters by using the least square complex exponential method; and step two, designing a digital correction unit according to the identified modal parameters, wherein the digital correction unit comprises a comb filter and a low-pass filter, and correcting the torquer according to the comb filter and the low-pass filter. The method can identify modal parameters of the torquer according to a least square complex exponential method, then design a digital correction unit according to the modal parameters such as damping natural frequency, damping ratio and the like, accurately correct frequency bands near natural frequency points, ensure that amplitude-frequency response in a passband tends to a constant value, and further expand frequency bands.

Description

Strain moment instrument bandwidth expansion method based on least square complex exponential method
Technical Field
The invention belongs to the technical field of instruments and meters, and particularly relates to a strain type torquer bandwidth expansion method based on a least square complex exponential method.
Background
The strain-type torque meter measurement principle is strain effect. When a moment is loaded, the strain gauge generates torsional deformation along with the measuring shaft, and the resistance value of the strain gauge is correspondingly changed; the torsion angle of the measuring shaft can be obtained by measuring the resistance value of the strain gauge; and calculating the measured torque according to the torsion angle and the torsion elastic modulus of the measuring shaft. When the measuring shaft and the strain gauge with smaller torsional elastic modulus are selected, the torquer has higher sensitivity and can be used for measuring tiny torque.
An important link for detecting the performance of the motor is to measure the magnitude of the ripple torque when the motor rotates, and the greater the ripple torque is, the more unstable the motor rotates. In high-speed motors such as a gyroscope motor and the like, even if the torque has small fluctuation, the performance of the motor can still be greatly influenced; therefore, in the detection of the ripple torque of the high-speed motor, a high-sensitivity strain-type torque meter is often used.
The measuring shaft of the high-sensitivity strain-type torque meter needs to use a material with a small torsional elastic modulus, so that when a small torque is applied, the measuring shaft has a large torsional angle, and the resistance value of the strain gauge is obviously changed. But the smaller torsional elastic modulus makes the natural frequency of the measuring shaft lower, the amplitude-frequency response of the frequency band near the natural frequency is far larger than that of zero frequency, when measuring torque, the measurement of the wave dynamic torque in the frequency band is not accurate, and low-frequency oscillation can be generated, so that the output signal of the torque meter also comprises an oscillation signal caused by the excitation of the measuring shaft by external torque besides the wave dynamic torque signal to be measured.
In order to accurately measure the ripple moment, a low-pass filter is used for filtering an oscillation signal, and the design principle of the low-pass filter is to attenuate the oscillation signal to a value that the measurement of the ripple moment is not influenced, limit the bandwidth of a system and ensure that a pass band is flat, namely the measurement result of the ripple moment in the pass band meets the precision requirement. In the traditional method, before designing a filter, frequency spectrum estimation is carried out on a measurement signal to obtain the frequency of an oscillation signal, and then filter parameters such as cut-off frequency and the like are selected according to experience. The cutoff frequency selected by the empirical estimation method is small, and although the oscillation signals are filtered to enable the pass band to be flat, the bandwidth is greatly reduced, and the measurement performance of the torquer is reduced.
Disclosure of Invention
The invention aims to overcome the defects in the prior art, and provides a bandwidth extension method of a strain-type torquer based on a least square complex exponential method, which can be used for identifying modal parameters of the torquer according to the least square complex exponential method, designing a digital correction unit according to the modal parameters such as damping natural frequency, damping ratio and the like, accurately correcting a frequency band near a natural frequency point, and ensuring that the amplitude-frequency response in a passband tends to a constant value, thereby extending the frequency band. The invention also provides a typical case for expanding the bandwidth of the strain-type torquer, modal parameter identification is carried out on the torquer, a correction unit is designed according to the identification result, a result of comparison with the traditional low-pass filter is provided, and the feasibility, the applicability and the superiority of the method provided by the invention are verified.
The purpose of the invention is realized by the following technical scheme:
a strain-type torquer bandwidth extension method based on a least square complex exponential method comprises the following steps
Step one, modal parameter identification is carried out by using a least square complex exponential method;
and step two, designing a digital correction unit according to the identified modal parameters, wherein the digital correction unit comprises a comb filter and a low-pass filter, and correcting the torquer according to the comb filter and the low-pass filter.
Further, the step one specifically comprises the following steps:
(101) the loading measurement system applies impact torque to the torquer by using the impact hammer before the measurement system performs measurement, and collects torque signals output by the torquer;
(102) selecting a data segment h representing free response of a torquer in the collected signalsn(n＝0,1,2…)；
(103) H is to benDividing into m +1 time sequence samples, each sample having 2n sampling data (m +1)>2n) that is
hl＝[hl hl+1 … hl+2n-1]T(l＝0,1,,m) (1)
Wherein h islRepresented in bold, representing a vector, is a free response data segment hnA 2n continuous data segment.
Can be derived from a system of regression equations:
Ha＝-h2n+ε (2)
wherein
Figure BDA0002406059370000021
a＝[a0 a1 … a2n-1]T (4)
h2n＝[h2n h2n+1 … h2n+m]T (5)
ε＝[ε0 ε1 … εm]T (6)
Wherein H is (m +1) multiplied by 2n order Hankel matrix, a is autoregressive coefficient matrix, H2nIs a (m +1) -order real array, and epsilon is an error vector
ε＝h2n+Ha (7)；
(104) Constructing an objective function:
e＝εTε＝(h2n+Ha)T(h2n+Ha) (8)
order to
Figure BDA0002406059370000031
Least squares solution of a
a＝-(HTH)-1HTh2n (10)；
(105) Taking the least squares solution of a as the coefficient of the Prony equation:
Figure BDA0002406059370000032
solving the equation to obtain 2n conjugated complex roots ziWherein z isi(i-1, 2 … n) and zi(i ═ n +1, n +2 … 2n) are conjugated to each other;
(106) taking the first n roots to respectively calculate modal parameters of n-order modes:
Figure BDA0002406059370000033
Figure BDA0002406059370000034
Figure BDA0002406059370000035
Figure BDA0002406059370000036
wherein ω issTo sample the circular frequency, Rezi、ImziReal and imaginary parts, ω, of the complex root, respectivelydi、σi、ωni、ξiRespectively, the damping natural frequency, the attenuation coefficient, the undamped natural frequency and the damping ratio of the ith-order mode.
Furthermore, in the second step, the strain-type torque meter is equivalent to a small-damping oscillation link (0< xi < 1), and the transfer function and the amplitude-frequency response of the oscillation link are respectively as follows:
Figure BDA0002406059370000037
Figure BDA0002406059370000038
in ω ∈ (0, ω)d) The time-amplitude-frequency response is monotonously increased, and is within the range of omega e (omega)dInfinity) the amplitude-frequency response decreases monotonically, so only the first-order mode needs to be identified in the correction; setting the cut-off frequency of the pass band as the damping natural frequency omegadFor the pass band ω e (0, ω)d) Correcting to ensure that the amplitude-frequency response in the pass band tends to a constant value; according to the mode identification step of the first part, n is 1 to obtain each parameter omega of the first-order moded,σ,ωnXi, then carrying out the following steps to design a digital correction unit;
(201) determining wave power moment measurement decibel error delta dB according to the measurement technical index, namely:
Figure BDA0002406059370000041
in the formula Am,ArAmplitude and true values are measured for the ripple torque, respectively.
(202) Omega identified according to step onenXi, a three-parameter comb filter is designed for correction, and the transfer function is as follows:
Figure BDA0002406059370000042
wherein ω is0Is a natural frequency, taken as ωn；K1Taking the depth coefficient of the stop band as 2 xi; k2For the coefficient of the stop band width, K can be obtained according to the following formula2：
Figure BDA0002406059370000043
Namely:
Figure BDA0002406059370000044
(203) transfer function H obtained in step 22(s) into pole-zero form
Figure BDA0002406059370000045
Wherein z isi,piIs the zero-pole of the continuous time system, and K is the gain of the continuous time system;
then using zero pole matching method to transfer function H2(s) discretization, i.e. H is obtained from the following equation2(z)
Figure BDA0002406059370000046
Wherein z'i,p'iZero pole of discrete time system, K' discrete time system gain, TsIs the sampling time;
arranging to obtain a rational polynomial form:
Figure BDA0002406059370000047
(204) designing a low-pass filter to filter noise outside a pass band, and obtaining the cut-off frequency omega of the pass band of the low-pass filter according to the following formulalp：
20log(H1(jωlp))+20log(H2(jωlp))＝-δ (25)
Stop band cut-off frequency omega of low-pass filterlsThe following can be taken:
ωls＝ωlp+6π
the low pass filter stopband amplitude-frequency response may be taken as Als＝-80dB；
According to the parameters, the equal ripple low-pass filter is designed by using Matlab filter design software to obtain an FIR low-pass filter Hlp(z)；
(205) H obtained from the steps (203) and (204) by using signal processing software such as Matlab2(z),Hlp(z) measurement of wave momentsi(n) (k is 0,1,2, …) and a correction signal x is obtained by filteringo(n) (n is 0,1,2, …), the torquer is calibrated to widen the bandwidth while maintaining the original sensitivity.
Compared with the prior art, the technical scheme of the invention has the following beneficial effects:
(1) the defect that the dynamic measurement result of the strain type torquer is inaccurate is overcome, and an amplitude-frequency response correction method is provided to further improve the accuracy of dynamic measurement;
(2) the defect of narrow bandwidth of the strain type torque meter is overcome, and an amplitude-frequency response correction method is provided to further expand the bandwidth of the strain type torque meter.
Drawings
FIG. 1 shows an amplitude-frequency response curve before correction of a torquer;
FIG. 2 shows a phase frequency response curve before correction of a torquer;
FIG. 3 shows a comb filter amplitude-frequency response curve;
FIG. 4 shows a comb filter phase frequency response curve;
FIG. 5 shows the amplitude-frequency response curve of the moment meter after correction by the comb filter;
FIG. 6 shows the phase-frequency response curve of a torquer corrected by a comb filter;
FIG. 7 shows a low pass filter amplitude frequency response curve;
FIG. 8 shows a low pass filter phase frequency response curve;
FIG. 9 shows a corrected amplitude-frequency response curve of a comb filter and a low-pass filter;
FIG. 10 shows a comb-notch, low-pass filter corrected phase-frequency response curve;
fig. 11 shows the bandwidth extension effect.
Detailed Description
The invention is described in further detail below with reference to the figures and specific examples. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
The invention provides a strain type torquer bandwidth extension method based on a least square complex exponential method, which comprises the following parts:
a first part, performing modal parameter identification by using a least square complex exponential method;
step 101: the loading measurement system applies impact torque to the torquer by using the impact hammer before the system performs measurement, and collects torque signals output by the torquer;
step 102: selecting a data segment h representing free response of a torquer in the collected signalsn(n＝0,1,2…)；
Step 103: handle hnDividing into m +1 time sequence samples, each sample having 2n sampling data (m +1)>2n) that is
hl＝[hl hl+1 … hl+2n-1]T(l＝0,1,,m) (1)
Wherein h islRepresented in bold, representing a vector, is a free response data segment hnA 2n continuous data segment.
Can be derived from a system of regression equations:
Ha＝-h2n+ε (2)
wherein
Figure BDA0002406059370000061
a＝[a0 a1 … a2n-1]T (4)
h2n＝[h2n h2n+1 … h2n+m]T (5)
ε＝[ε0 ε1 … εm]T (6)
Wherein H is (m +1) multiplied by 2n order Hankel matrix, a is autoregressive coefficient matrix, H2nIs a (m +1) order real array, and epsilon is an error vector.
ε＝h2n+Ha (7)
Step 104: constructing an objective function:
e＝εTε＝(h2n+Ha)T(h2n+Ha) (8)
order to
Figure BDA0002406059370000062
Least squares solution of a
a＝-(HTH)-1HTh2n (10)
Step 105: taking the least squares solution of a as the coefficient of the Prony equation:
Figure BDA0002406059370000071
solving the equation to obtain 2n conjugated complex roots ziWherein z isi(i-1, 2 … n) and zi(i ═ n +1, n +2 … 2n) are conjugated to each other.
Step 106: taking the first n roots to respectively calculate modal parameters of n-order modes:
Figure BDA0002406059370000072
Figure BDA0002406059370000073
Figure BDA0002406059370000074
Figure BDA0002406059370000075
wherein ω issTo sample the circular frequency, Rezi、ImziReal and imaginary parts, ω, of the complex root, respectivelydi,σi,ωni,ξiDamping natural frequency, attenuation coefficient, undamped natural frequency and damping ratio of the ith order mode.
And a second part, designing a digital correction unit according to the identified modal parameters, wherein the digital correction unit comprises a comb filter and a low-pass filter, and correcting the torquer.
Because the strain type torquer can be equivalent to a small damping oscillation link (0< xi < 1), the transfer function and amplitude-frequency response are respectively:
Figure BDA0002406059370000076
Figure BDA0002406059370000077
as shown in fig. 1, at ω ∈ (0, ω)d) The time-amplitude-frequency response is monotonously increased, and is within the range of omega e (omega)dAnd infinity) the amplitude-frequency response decreases monotonically, so only the first-order mode needs to be identified in the correction. Since the correction of the monotone interval is easy to realize, the cut-off frequency of the pass band is set as the damping natural frequency omegadFor the pass band ω e (0, ω)d) And correcting to ensure that the amplitude-frequency response in the passband tends to a constant value. According to the mode identification step of the first part, n is 1 to obtain each parameter omega of the first-order moded,σ,ωnξ, the modal parameter identification results are shown in Table 1. Then, the following steps are carried out to design the digital correction unit.
Table 1 results of modal parameter identification using least square complex exponential method
Serial number ωn ωd σξ
1 180.93 180.93 0.35150.0019
2 180.92 180.92 0.35220.0019
3 180.89 180.89 0.34440.0019
4 180.65 180.65 0.33120.0018
5 180.70 180.70 0.33890.0019
6 180.62 180.62 0.35460.0020
7 180.54 180.53 0.37480.0021
8 180.57 180.57 0.40050.0022
9 180.57 180.57 0.34220.0019
10 180.57 180.57 0.34850.0019
Mean value 180.70 180.70 0.35390.0020
Step 201: determining wave power moment measurement decibel error delta dB according to the measurement technical index, namely:
Figure BDA0002406059370000081
in the formula Am,ArAmplitude and true values are measured for the ripple torque, respectively.
Step 202: omega identified from the first partnXi, a three-parameter comb filter is designed for correction, and the transfer function is as follows:
Figure BDA0002406059370000082
wherein ω is0Is a natural frequency, taken as ωn；K1Taking the depth coefficient of the stop band as 2 xi; k2For the coefficient of the stop band width, K can be obtained according to the following formula2：
Figure BDA0002406059370000083
Namely:
Figure BDA0002406059370000084
step 203: transfer function H obtained in step 2022(s) into pole-zero form
Figure BDA0002406059370000085
Wherein z isi,piIs the zero pole of the continuous time system and K is the continuous time system gain.
Then using zero pole matching method to transfer function H2(s) discretization, i.e. H is obtained from the following equation2(z)
Figure BDA0002406059370000091
Wherein z'i,p'iZero pole of discrete time system, K' discrete time system gain, TsIs the sampling time.
The arrangement can obtain a rational polynomial form:
Figure BDA0002406059370000092
the amplitude-frequency response and the phase-frequency response are shown in fig. 3 and 4.
Step 204: designing a low pass filter to filter out noise outside the pass band according to the following equationThe formula can obtain the cut-off frequency omega of the passband of the low-pass filterlp：
20log(H1(jωlp))+20log(H2(jωlp))＝-δ (25)
Stop band cut-off frequency omega of low-pass filterlsThe following can be taken:
ωls＝ωlp+6π
the low pass filter stopband amplitude-frequency response may be taken as Als＝-80dB；
According to the parameters, equal ripple low-pass filter design is carried out by using filter design software such as Matlab and the like, and an FIR low-pass filter H can be obtainedlp(z), the amplitude-frequency response and the phase-frequency response are shown in fig. 7 and 8.
Step 205: using signal processing software such as Matlab, based on H obtained in step 3 and step 42(z),Hlp(z) measurement of wave momentsi(n) (k is 0,1,2, …) and a correction signal x is obtained by filteringo(n) (n is 0,1,2, …), the amplitude-frequency response and the phase-frequency response of the torquer after filtering by the comb filter are shown in fig. 5 and 6, the amplitude-frequency response and the phase-frequency response of the torquer after correcting by the comb filter and the low-pass filter are shown in fig. 9 and 10, and the bandwidth expansion effect of the corrected torquer is shown in fig. 11. After the torquer is corrected, the original sensitivity is kept, and meanwhile, the bandwidth is widened.
Taking δ as an example, the modal parameter identification result based on the least square complex exponential method is shown in table 1; the amplitude-frequency response and the phase-frequency response of the identified torquer are shown in the figures 1 and 2; the amplitude-frequency response and the phase-frequency response of the comb filter designed according to the identified modal parameters are shown in fig. 3 and 4; the amplitude-frequency response and the phase-frequency response of the torquer after being filtered by the comb filter are shown in figures 5 and 6; the designed amplitude-frequency response and phase-frequency response of the low-pass filter are shown in FIGS. 7 and 8; the amplitude-frequency response and the phase-frequency response of the torquer after being corrected by the comb filter and the low-pass filter are shown in figures 9 and 10; the corrected band expansion effect is shown in fig. 11.
The present invention is not limited to the above-described embodiments. The foregoing description of the specific embodiments is intended to describe and illustrate the technical solutions of the present invention, and the specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many changes and modifications to the invention without departing from the spirit and scope of the invention as defined in the appended claims.

Claims (1)

1. A strain type torquer bandwidth extension method based on a least square complex exponential method is characterized by comprising the following steps
Step one, modal parameter identification is carried out by using a least square complex exponential method; specifically, the method comprises the following steps:
(101) the loading measurement system applies impact torque to the torquer by using the impact hammer before the measurement system performs measurement, and collects torque signals output by the torquer;
(102) selecting a data segment h representing free response of a torquer in the collected signalsn(n＝0,1,2…)；
(103) H is to benDividing into m +1 time sequence samples, each sample having 2n sampling data (m +1)>2n) that is
hl＝[hl hl+1 … hl+2n-1]T(l＝0,1,…,m) (1)
Wherein h islRepresented in bold, representing a vector, is a free response data segment hnA 2n continuous data segment;
can be derived from a system of regression equations:
Ha＝-h2n+ε (2)
wherein
Figure FDA0003495017420000011
a＝[a0 a1 … a2n-1]T (4)
h2n＝[h2n h2n+1 … h2n+m]T (5)
ε＝[ε0 ε1 … εm]T (6)
Wherein H is (m +1) multiplied by 2n order Hankel matrix, a is autoregressive coefficient matrix, H2nIs (m +1) order real array, and epsilon is error vector
ε＝h2n+Ha (7)；
(104) Constructing an objective function:
e＝εTε＝(h2n+Ha)T(h2n+Ha) (8)
order to
Figure FDA0003495017420000012
Least squares solution of a
a＝-(HTH)-1HTh2n (10)；
(105) Taking the least squares solution of a as the coefficient of the Prony equation:
Figure FDA0003495017420000021
solving the equation to obtain 2n conjugated complex roots ziWherein z isi(i-1, 2 … n) and zi(i ═ n +1, n +2 … 2n) are conjugated to each other;
(106) taking the first n roots to respectively calculate modal parameters of n-order modes:
Figure FDA0003495017420000022
Figure FDA0003495017420000023
Figure FDA0003495017420000024
Figure FDA0003495017420000025
wherein ω issTo sample the circular frequency, Rezi、ImziReal and imaginary parts, ω, of the complex root, respectivelydi、σi、ωni、ξiDamping natural frequency, attenuation coefficient, undamped natural frequency and damping ratio of the ith-order mode are respectively set;
designing a digital correction unit according to the identified modal parameters, wherein the digital correction unit comprises a comb filter and a low-pass filter, and correcting the torquer according to the comb filter and the low-pass filter; specifically, the method comprises the following steps:
the strain-type torquer is equivalent to a small-damping oscillation link (0< xi < 1), and the transfer function and amplitude-frequency response of the strain-type torquer are respectively as follows:
Figure FDA0003495017420000026
Figure FDA0003495017420000027
in ω ∈ (0, ω)d) The time-amplitude-frequency response is monotonously increased, and is within the range of omega e (omega)dInfinity) the amplitude-frequency response decreases monotonically, so only the first-order mode needs to be identified in the correction; setting the cut-off frequency of the pass band as the damping natural frequency omegadFor the pass band ω e (0, ω)d) Correcting to ensure that the amplitude-frequency response in the pass band tends to a constant value; obtaining each parameter omega of the first-order mode according to the mode parameter identification result of the step oned,σ,ωnXi, then carrying out the following steps to design a digital correction unit;
(201) determining wave power moment measurement decibel error delta dB according to the measurement technical indexes, namely:
Figure FDA0003495017420000028
in the formula Am,ArRespectively measuring amplitude and true value for the wave dynamic moment;
(202) omega identified according to step onenXi, a three-parameter comb filter is designed for correction, and the transfer function is as follows:
Figure FDA0003495017420000031
wherein ω is0Is a natural frequency, taken as ωn；K1Taking the depth coefficient of the stop band as 2 xi; k2For the coefficient of the stop band width, K is obtained according to the following formula2：
Figure FDA0003495017420000032
Namely:
Figure FDA0003495017420000033
(203) transfer function H obtained in step 22(s) into pole-zero form
Figure FDA0003495017420000034
Wherein z isi,piIs the zero-pole of the continuous time system, and K is the gain of the continuous time system;
then using zero pole matching method to transfer function H2(s) discretization, i.e. determining H from the following equation2(z)
Figure FDA0003495017420000035
Wherein z'i,p′iZero pole of discrete time system, K' discrete time system gain, TsIs the sampling time;
arranging to obtain a rational polynomial form:
Figure FDA0003495017420000036
(204) designing a low-pass filter to filter noise outside a pass band, and obtaining a cut-off frequency omega of the pass band of the low-pass filter according to the following formulalp：
20log(H1(jωlp))+20log(H2(jωlp))＝-δ (25)
Stop band cut-off frequency omega of low-pass filterlsTaking the following steps:
ωls＝ωlp+6π
the stopband amplitude-frequency response of the low-pass filter is taken as Als＝-80dB；
According to the parameters, the equal ripple low-pass filter is designed by using Matlab filter design software to obtain an FIR low-pass filter Hlp(z)；
(205) Using Matlab signal processing software, from the H found in step (203) and step (204)2(z),Hlp(z) measurement of wave momentsi(n) (k is 0,1,2, …) and a correction signal x is obtained by filteringo(n) (n is 0,1,2, …), the torquer is calibrated to widen the bandwidth while maintaining the original sensitivity.
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