CN111046499A - Method and system for determining installation position of rate gyro - Google Patents
Method and system for determining installation position of rate gyro Download PDFInfo
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- CN111046499A CN111046499A CN201911379486.3A CN201911379486A CN111046499A CN 111046499 A CN111046499 A CN 111046499A CN 201911379486 A CN201911379486 A CN 201911379486A CN 111046499 A CN111046499 A CN 111046499A
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Abstract
The application discloses a method and a system for determining a rate gyro installation position, wherein the method for determining the rate gyro installation position specifically comprises the following steps: according to the rocket body structure of the rocket and the environmental constraints in the rocket body, providing an interval set allowing installation of the rate gyro; calculating average proportional coupling factors of all levels according to the overall level of the rocket body structure; calculating a quantitative evaluation index of the installation position of the rate gyro according to the average proportional coupling factors of all levels; determining the minimum value of the quantitative evaluation index of the installation position of the rate gyro in the interval set allowing the rate gyro to be installed; and outputting the mounting position corresponding to the minimum value of the quantitative evaluation index as the optimal mounting position. The method and the device can reduce the requirement for high attenuation of the notch filter caused by insufficient elastic suppression control margin, and improve the stability margin, dynamic characteristic and control precision of flight.
Description
Technical Field
The application relates to the field of rockets, in particular to a method and a system for determining the installation position of a rate gyro.
Background
In order to avoid the coupling effect between the rigid body attitude control system and the structural elastic vibration caused by the additional rotational angular velocity introduced by the structural elastic vibration at the installation position of an Inertial Measurement Unit (IMU) in the rigid body attitude control of the launch vehicle, one or more rate gyros are often adopted as an independent angular velocity measuring device and installed at the positions with smaller elastic rotational amplitude or capable of being mutually offset, so as to improve the quality of the attitude control system. If the rate gyro is still difficult to meet the control margin requirement, a correction network, usually a digital filter, is required, and the frequency characteristic of the filter plays a role in notch attenuation at the elastic vibration frequency and is matched with the rigid body frequency range to compensate the phase lag. Compared with a liquid carrier rocket, the solid carrier rocket has the advantages that the burning speed of the solid engine is relatively high, the specific impulse is relatively low, in order to overcome the dead weight of the engine after burning as soon as possible and improve the carrying capacity, the solid carrier rocket is designed by adopting more stages, so that the length and the slenderness of the structure are large; on the other hand, although the first order frequency of a solid launch vehicle is relatively high, its frequency changes rapidly with time; furthermore, to minimize launch costs, it is desirable to suppress multi-level elastic vibrations using as few rate gyros as possible; thus, the elastic restraint control of solid launch vehicles has its inherent features and requires the development of a targeted design.
The traditional rate gyroscope arrangement mode roughly gives an installation position according to structural layout constraint, gives a feasibility conclusion only through comprehensive verification, lacks a quantitative optimization design process, causes difficulty in designing a notch filter, influences the margin of a control system, and causes sacrifice on system response bandwidth and noise sensitivity; or separate rate gyros may be used at each stage, but this increases product cost.
Therefore, how to accurately and quickly give the installation position of the rate gyro without increasing the product cost is a problem which needs to be solved urgently by people in the field.
Disclosure of Invention
The application aims to provide a method and a system for determining the installation position of a rate gyro, so that the installation position of the rate gyro is optimized, the requirement of high attenuation of a notch filter caused by insufficient elastic suppression control margin is reduced, and the stability margin, the dynamic characteristic and the control precision of flight are improved.
In order to achieve the above object, the present application provides a method for determining a rate gyro mounting position, which specifically includes the following steps: according to the rocket body structure of the rocket and the environmental constraints in the rocket body, providing an interval set allowing installation of the rate gyro; calculating average proportional coupling factors of all levels according to the overall level of the rocket body structure; calculating a quantitative evaluation index of the installation position of the rate gyro according to the average proportional coupling factors of all levels; determining the minimum value of the quantitative evaluation index of the installation position of the rate gyro in the interval set allowing the rate gyro to be installed; and outputting the mounting position corresponding to the minimum value of the quantitative evaluation index as the optimal mounting position.
The above, wherein the calculating of the average proportional coupling factor of each stage, further comprises the following sub-steps: determining first order elastic vibration frequency f1(t); determining function omega of rigid body attitude control loop cut-off frequency changing with timec(t); function omega for controlling circuit cut-off frequency to change along with time according to rigid body attitudec(t) first order elastic vibration frequency f1(t) determining the average proportional coupling factor for each stage.
As above, wherein the first order elastic vibration frequency f1And (t) performing mathematical modeling calculation on the structural form and mass distribution of the carrier rocket by adopting a finite element method.
As above, wherein the first order elastic vibration frequency f1Cut-off frequency omega of rigid body attitude control loopc(t) the ratio is a proportional coupling factor, defining a normalized integral of the proportional coupling factor to time of flight as the average proportional coupling factorSub wkAverage proportional coupling factor omega of each stagekThe concrete expression is as follows:
wherein f is1(t) is a function of the time variation of the first-order elastic vibration frequency, ωc(t) is a function of the rigid body attitude control loop cut-off frequency with time; integral operation represents continuous solution;
and
respectively representing the starting time and the stopping time of the dynamic flight of the series.
As above, the quantitative evaluation index j (x) is represented by:
wherein, wkWhich represents the average proportional coupling factor and is,
and
respectively representing the starting time and the stopping time of the series dynamic flight,
represents the weight of the vibration of this order, δ represents the control command, ωc(t) is a function of the change of the cut-off frequency of the rigid attitude control loop with time, qiRepresenting the generalized coordinates of vibration of each order, i represents the current order, W'i(t, x) represents the variation of the rotation angle caused by vibration with the length of the structure, i.e., the mode slope. Wherein t represents a certain moment of flight, x represents a rate gyro installation position, k is the number of stages, and N is the total number of stages of the rocket.
As above, wherein the weight of the present order vibration is determined
The method comprises the following substeps: calculating a control command to each order of vibration generalized coordinate qiTransfer function of
Determining the amplitude of the change of the transfer function of the control command to the generalized coordinates with the flight time according to the transfer function
And determining the weight of the order vibration according to the amplitude of the change of the transfer function along with the flight time.
As above, wherein the cutoff frequency w of each orderc(t) control commands to amplitude of transfer function variation with time of flight
The time history of (A) is expressed as
I.e. the weight of the present order vibration.
A system for determining the installation position of a rate gyro specifically comprises: a mounting position design processor and a position output unit; wherein the mounting position processor is used for the design method of the mounting position of the rate gyro; and the position output unit is used for receiving and outputting the optimal installation position.
As above, wherein the mounting position processor further comprises the following sub-modules: the device comprises an installation set determining module, an average proportional coupling factor determining module, a quantitative evaluation index determining module and an installation position determining unit; the installation set determining module is used for giving an interval set allowing the rate gyro to be installed according to the rocket body structure and the environmental constraint in the rocket body; the average proportional coupling factor determining module is used for calculating average proportional coupling factors of all levels according to the total number of stages of the rocket; the quantitative evaluation index determining module is used for calculating the quantitative evaluation index of the installation position of the rate gyro according to the average proportional coupling factors of all levels; and the mounting position determining unit is used for determining the minimum value of the quantitative evaluation index of the mounting position of the rate gyro according to the interval set of the mounting rate gyro.
The above, wherein the average proportional coupling factor determining module further comprises the following sub-modules: the device comprises a first-order elastic vibration frequency determining module, a loop cut-off frequency function determining module and a coupling factor calculating module; wherein the first order elastic vibration frequency determining module is used for determining the first order elastic vibration frequency f1(t); the loop cut-off frequency function determining module is connected with the first-order elastic vibration frequency determining module and is used for determining the function w of the change of the cut-off frequency of the rigid body attitude control loop along with the timec(t); the coupling factor calculation module is respectively connected with the first-order elastic vibration frequency determination module and the loop cut-off frequency function determination module and is used for controlling a function w of the loop cut-off frequency changing along with time according to the posture of the rigid bodyc(t) first order elastic vibration frequency f1(t) determining the average proportional coupling factor w for each stagek。
The beneficial effect of this application is: the method and the device can reduce the requirement for high attenuation of the notch filter caused by insufficient elastic suppression control margin, and improve the stability margin, dynamic characteristic and control precision of flight. The use requirements of two stages can be effectively considered, the single-rate gyroscope of each channel is used to adapt to multi-stage flight as much as possible, and the product cost is reduced. And a quantitative evaluation index of the installation position of the rate gyro is provided, and an optimal installation position is sought according to elastic theory calculation or test data, so that a refined design is realized.
Drawings
In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings needed to be used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments described in the present application, and other drawings can be obtained by those skilled in the art according to the drawings.
FIG. 1 is a flow chart of a method for determining a rate gyro mounting location provided in accordance with an embodiment of the present application;
FIG. 2 is a schematic illustration of a range of installation positions of a rate gyro according to an embodiment of the present application;
FIG. 3 is an internal block diagram of a system for determining a rate gyro mounting location provided in accordance with an embodiment of the present application;
fig. 4 is a block diagram of internal sub-modules of a system for determining a rate gyro mounting position according to an embodiment of the present application.
Detailed Description
The technical solutions in the embodiments of the present application are clearly and completely described below with reference to the drawings in the embodiments of the present application, and it is obvious that the described embodiments are some, but not all, embodiments of the present application. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present application.
The application relates to a method and a system for determining a rate gyro installation position. According to the application, the installation position of the rate gyro is optimized, the requirement of high attenuation of the notch filter caused by insufficient elastic suppression control margin is reduced, and the stability margin, the dynamic characteristic and the control precision of the flight are improved. The single-rate gyroscope in each channel is used to adapt to multi-level flight as much as possible, and the product cost is reduced. Quantitative evaluation indexes of the installation positions of the rate gyros are given, the optimal installation positions are sought according to elastic theory calculation or test data, and refined design is achieved.
The application provides a method for determining a rate gyro installation position, as shown in fig. 1, specifically comprising the following steps:
step S110: and obtaining an interval set allowing the rate gyro to be installed according to the rocket body structure and the environmental constraint in the rocket body.
Specifically, according to the structural layout and environmental conditions of the carrier rocket, the installation positions of the instruments and equipment are in gaps formed by the shrinkage of cylindrical sections at two ends of the engine towards the spray pipe or the seal head, and the positions at two sides of the interstage separation surface. The instrument device generally refers to various electrical or mechanical devices required for flight, and the rate gyro provided by the embodiment is a specific instrument device.
Further, the degree of tolerance of the instrumentation to environmental conditions such as vibration, noise and interstage separation shock from engine operation is also a consideration. Taking FIG. 2 as an example, the installation position of the rate gyro is L1And L2Interval, so the set of intervals allowed to be installed is L ═ L { (L)1，L2}。
Step S120: and calculating the average proportional coupling factor of each stage according to the overall stage number of the rocket body structure.
Wherein the total number of stages of the rocket is N, the current number of stages is k, and the starting and stopping moments of the dynamic flight of the stage are respectively
And
the total elastic vibration order is n and the current order is i.
Before calculating the average proportional coupling factors of the stages, the step S120 further includes the following sub-steps:
step Q1: determining first order elastic vibration frequency f1(t)。
Wherein the first order elastic vibration frequency is defined as f1(t), where t represents a certain moment of flight, f1(t) the structural form and the mass distribution of the launch vehicle can be calculated by mathematical modeling with FEM (Finite Element Method), and the algorithm can refer to the prior art and is not described herein.
Step Q2: determining function omega of rigid body attitude control loop cut-off frequency changing with timec(t)。
Specifically, the control system may be obtained by performing frequency domain synthesis on design parameters of the control system and a dynamic model of the launch vehicle, the method is a reference method in the prior art, and the specific process is not described herein again.
Step Q3: function omega for controlling circuit cut-off frequency to change along with time according to rigid body attitudec(t) first order elastic vibration frequency f1(t) determining the average proportional coupling factor omega for each stagek。
Wherein the first order elastic vibration frequency f is set1Cut-off frequency omega of rigid body attitude control loopc(t) the ratio is the proportional coupling factor, defining the normalized integral of the proportional coupling factor over time of flight as the average proportional coupling factor wkThus, each stage average proportional coupling factor wkThe concrete expression is as follows:
wherein f is1(t) is a function of the time variation of the first-order elastic vibration frequency, ωc(t) is a function of the change of the cut-off frequency of the rigid body attitude control loop along with time (called the cut-off frequency for short); integral operation represents continuous solution;
and
respectively representing the starting time and the stopping time of the dynamic flight of the series.
Preferably, ω iscAnd (t) performing mathematical modeling calculation by using finite element method FEM according to the structural form and mass distribution of the carrier rocket.
Step S130: and calculating the quantitative evaluation index of the installation position of the rate gyro according to the average proportional coupling factors of all levels.
Specifically, the quantitative evaluation index j (x) is represented by:
wherein, wkWhich represents the average proportional coupling factor and is,
and
respectively representing the start and stop of dynamic flight of the seriesThe end-of-time is the time,
represents the weight of the vibration of this order, δ represents the control command, ωc(t) is a function of the change of the cut-off frequency of the rigid attitude control loop with time, qiRepresenting the generalized coordinates of vibration of each order, i represents the current order, W'i(t, x) represents the variation of the rotation angle caused by vibration with the length of the structure, i.e., the mode slope. Wherein t represents a certain moment of flight, x represents a rate gyro installation position, k is the number of stages, and N is the total number of stages of the rocket.
In the formula one, the average proportional coupling factor of each stage represents the proximity of each stage of elastic vibration and rigid body frequency. The larger the proportional coupling factor and the smaller the elastic influence, the smaller the weighting factor should be, typically the reciprocal 1/w of the average proportional coupling factorkAs a weighting factor for equation two.
Preferably, wherein the weight of the order vibration
The determination of (b) comprises the sub-steps of:
step D1: calculating a control command to each order of vibration generalized coordinate qiThe transfer function of (2).
Specifically, the control command includes a thrust vector-controlled nozzle yaw angle, a yaw angle of a gas rudder or an air rudder, and the like.
Wherein the transfer function is defined as
Representing the complex frequency domain variation in the laplacian transform.
Preferably, the transfer function is obtained by frequency domain synthesis of the design parameters of the control system with a dynamic model of the launch vehicle.
Step D2: and determining the amplitude of the change of the transfer function of the control command to the generalized coordinate along with the flight time according to the transfer function.
Wherein the amplitude of the transfer function is defined as
Representing the complex frequency domain variation in the laplace transform, and t represents a certain time of flight.
Determining a transfer function
The amplitude method comprises the following steps: s is replaced by j omega, wherein j is an imaginary unit, omega is a circular frequency, and a transfer function
Amplitude at a certain circular frequency omega
A modulus of complex number g (jw) ═ a (ω) + jb (ω)
Step D3: and determining the weight of the order vibration according to the amplitude of the change of the transfer function along with the flight time.
Wherein the cutoff frequency ω of each orderc(t) control commands to
The time history of (A) is expressed as
I.e. the weight of the order vibration.
Preferably, the weighting of the order vibrations is still obtained by frequency domain synthesis of the design parameters of the control system with the dynamic model of the launch vehicle.
Step S140: and determining the minimum value of the quantitative evaluation index of the installation position of the rate gyro according to the interval set for installing the rate gyro.
Wherein the minimum value of the quantitative evaluation index is represented as:
where L represents an allowable installation interval that satisfies structural and environmental constraints, and x e L represents that the rate gyro installation position is located in the allowable installation interval.
The meaning of the formula three is: on the premise that x ∈ L, x is determined so that J (x) takes a minimum value, and the mounting position x is redefined as xgsWhere gs is an abbreviation for rate gyro.
Preferably, x is foundgsThe process of (2) can adopt an optimized search algorithm or a traversal mode and the like.
Step S150: and outputting the optimal mounting position corresponding to the minimum value of the quantitative evaluation index.
Wherein x in step S140gsNamely the optimal position for installing the rate gyro, and outputting the position.
The present application provides a system for determining a rate gyro mounting position, as shown in fig. 3, specifically including: a position design processor 301 and a position output unit 302 are installed.
Wherein the mounting position processor 301 is configured to determine a mounting position where the rate gyro is optimal and transmit the position to the position output unit 302.
The position output unit 302 is connected to the mounting position processor 301, and is used for receiving and outputting the optimal mounting position.
Specifically, as shown in fig. 4, the mounting position processor 301 further includes the following sub-modules: an installation set determination module 401, an average proportional coupling factor determination module 402, a quantitative evaluation index determination module 403, and an installation position determination unit 404.
The installation set determining module 401 is configured to obtain an interval set allowing installation of the rate gyro according to a rocket body structure and an environmental constraint in the rocket body.
The average proportional coupling factor determining module 402 is connected to the installation set determining module 401, and is configured to calculate average proportional coupling factors of each stage according to the total number of stages of the rocket.
Further, the average proportional coupling factor determining module 402 further includes the following sub-modules: the device comprises a first-order elastic vibration frequency determining module, a loop cut-off frequency function determining module and a coupling factor calculating module.
Wherein the first order elastic vibration frequency determining module is used for determining the first order elastic vibration frequency f1(t)。
The loop cut-off frequency function determining module is connected with the first-order elastic vibration frequency determining module and is used for determining a function omega of the change of the cut-off frequency of the rigid body attitude control loop along with timec(t)。
The coupling factor calculation module is respectively connected with the first-order elastic vibration frequency determination module and the loop cut-off frequency function determination module and is used for controlling a function omega of the loop cut-off frequency changing along with time according to the posture of the rigid bodyc(t) first order elastic vibration frequency f1(t) determining the average proportional coupling factor omega for each stagek。
The quantitative evaluation index determining module 403 is connected to the average proportional coupling factor determining module 402, and is configured to calculate a quantitative evaluation index of the installation position of the rate gyro according to each level of average proportional coupling factors.
Further, the quantitative evaluation index determining module 403 specifically includes the following sub-modules: the device comprises a transfer function determining module, a transfer function amplitude determining module and a weight determining module of order vibration.
The transfer function determining module is used for determining vibration generalized coordinates q of each order of the control commandiThe transfer function of (2).
And the transfer function amplitude determining module is connected with the transfer function determining module and is used for determining the amplitude of the change of the transfer function from the control command to the generalized coordinate along with the flight time according to the transfer function.
And the weight determining module of the order vibration is connected with the transfer function amplitude determining module and is used for determining the weight of the order vibration according to the amplitude of the transfer function changing along with the flight time.
The mounting position determination unit 404 is connected to the quantitative evaluation index determination module 403, and is configured to determine a minimum value of the quantitative evaluation index of the rate gyro mounting position according to the interval set in which the rate gyro is mounted.
The beneficial effect of this application is: the method and the device can reduce the requirement for high attenuation of the notch filter caused by insufficient elastic suppression control margin, and improve the stability margin, dynamic characteristic and control precision of flight. The use requirements of two stages can be effectively considered, the single-rate gyroscope of each channel is used to adapt to multi-stage flight as much as possible, and the product cost is reduced. And a quantitative evaluation index of the installation position of the rate gyro is provided, and an optimal installation position is sought according to elastic theory calculation or test data, so that a refined design is realized.
Although the present application has been described with reference to examples, which are intended to be illustrative only and not to be limiting of the application, changes, additions and/or deletions may be made to the embodiments without departing from the scope of the application.
The above description is only for the specific embodiments of the present application, but the scope of the present application is not limited thereto, and any person skilled in the art can easily conceive of the changes or substitutions within the technical scope of the present application, and shall be covered by the scope of the present application. Therefore, the protection scope of the present application shall be subject to the protection scope of the claims.
Claims (10)
1. A method for determining the installation position of a rate gyro is characterized by comprising the following steps:
according to the rocket body structure of the rocket and the environmental constraints in the rocket body, providing an interval set allowing installation of the rate gyro;
calculating average proportional coupling factors of all levels according to the overall level of the rocket body structure;
calculating a quantitative evaluation index of the installation position of the rate gyro according to the average proportional coupling factors of all levels;
determining the minimum value of the quantitative evaluation index of the installation position of the rate gyro in the interval set allowing the rate gyro to be installed;
and outputting the mounting position corresponding to the minimum value of the quantitative evaluation index as the optimal mounting position.
2. The method of determining a rate gyro mounting location of claim 1 wherein calculating the average proportional coupling factors for each stage further comprises the substeps of:
determining first order elastic vibration frequency f1(t)；
Determining function omega of rigid body attitude control loop cut-off frequency changing with timec(t)；
Function omega for controlling circuit cut-off frequency to change along with time according to rigid body attitudec(t) first order elastic vibration frequency f1(t) determining the average proportional coupling factor for each stage.
3. The method of determining a rate gyro mounting position of claim 2 wherein the first order elastic vibration frequency f1And (t) performing mathematical modeling calculation on the structural form and mass distribution of the carrier rocket by adopting a finite element method.
4. The method of determining a rate gyro mounting position as claimed in claim 2, wherein the first order elastic vibration frequency f is set1Cut-off frequency omega of rigid body attitude control loopc(t) is a proportional coupling factor, and a normalized integral of the proportional coupling factor to time of flight is defined as an average proportional coupling factor ωkAverage proportional coupling factor omega of each stagekThe concrete expression is as follows:
wherein f is1(t) is a function of the time variation of the first-order elastic vibration frequency, ωc(t) is a function of the rigid body attitude control loop cut-off frequency changing with time, integral operation represents continuous solution,
and
respectively representing the starting time and the stopping time of the dynamic flight of the series.
5. The method of determining a rate gyro mounting position according to claim 1, wherein the quantitative evaluation index j (x) is expressed as:
wherein, ω iskWhich represents the average proportional coupling factor and is,
and
respectively representing the starting time and the stopping time of the series dynamic flight,
represents the weight of the vibration of this order, δ represents the control command, ωc(t) is a function of the change of the cut-off frequency of the rigid attitude control loop with time, qiRepresenting the generalized coordinates of vibration of each order, i represents the current order, W'i(t, x) represents the change of a rotation angle caused by vibration along with the length of the structure, namely the slope of the mode shape, t represents a certain flying time, x represents the installation position of the rate gyro, k is the stage number, and N is the total stage number of the rocket.
6. The method of determining a rate gyro mounting location of claim 5 wherein the weight of the present order vibration is determined
The method comprises the following substeps:
calculating a control command to each order of vibration generalized coordinate qiTransfer function of
Determining the variation with time of flight of the transfer function of the control command to the generalized coordinates based on the transfer functionAmplitude value
And determining the weight of the order vibration according to the amplitude of the change of the transfer function along with the flight time.
7. The method of determining a rate gyro mounting position of claim 5 wherein each order of cutoff frequency ω isc(t) control commands to amplitude of transfer function variation with time of flight
The time history of (A) is expressed as
I.e. the weight of the present order vibration.
8. A system for determining the installation position of a rate gyro is characterized by comprising the following specific steps: a mounting position design processor and a position output unit;
wherein the mounting location processor is configured to perform the method of designing a rate top mounting location of any of claims 1-7;
and the position output unit is used for receiving and outputting the optimal installation position.
9. The system for determining a rate gyro mounting position of claim 8 wherein the mounting position processor further includes the sub-modules of: the device comprises an installation set determining module, an average proportional coupling factor determining module, a quantitative evaluation index determining module and an installation position determining unit;
the installation set determining module is used for giving an interval set allowing the rate gyro to be installed according to the rocket body structure and the environmental constraint in the rocket body;
the average proportional coupling factor determining module is used for calculating average proportional coupling factors of all levels according to the total number of stages of the rocket;
the quantitative evaluation index determining module is used for calculating the quantitative evaluation index of the installation position of the rate gyro according to the average proportional coupling factors of all levels;
and the mounting position determining unit is used for determining the minimum value of the quantitative evaluation index of the mounting position of the rate gyro according to the interval set of the mounting rate gyro.
10. The system for determining a rate gyro mounting position of claim 8 wherein the average proportional coupling factor determination module further includes the sub-modules of: the device comprises a first-order elastic vibration frequency determining module, a loop cut-off frequency function determining module and a coupling factor calculating module;
wherein the first order elastic vibration frequency determining module is used for determining the first order elastic vibration frequency f1(t)；
The loop cut-off frequency function determining module is connected with the first-order elastic vibration frequency determining module and is used for determining a function omega of the change of the cut-off frequency of the rigid body attitude control loop along with timec(t)；
The coupling factor calculation module is respectively connected with the first-order elastic vibration frequency determination module and the loop cut-off frequency function determination module and is used for controlling a function omega of the loop cut-off frequency changing along with time according to the posture of the rigid bodyc(t) first order elastic vibration frequency f1(t) determining the average proportional coupling factor omega for each stagek。
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