CN110018695B - Active control method for flutter pneumatic wing plate of super-large-span suspension bridge - Google Patents

Active control method for flutter pneumatic wing plate of super-large-span suspension bridge Download PDF

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CN110018695B
CN110018695B CN201810271930.9A CN201810271930A CN110018695B CN 110018695 B CN110018695 B CN 110018695B CN 201810271930 A CN201810271930 A CN 201810271930A CN 110018695 B CN110018695 B CN 110018695B
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flutter
pneumatic
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wing plate
main beam
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CN110018695A (en
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李珂
葛耀君
赵林
陈翰林
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Tongji University
Chongqing University
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Abstract

The active control method for the flutter pneumatic wing plate of the ultra-large span suspension bridge comprises the following steps: (1) identifying dynamic characteristic parameters of the large-span suspension bridge to obtain physical parameters describing the dynamic performance of the structure and physical characteristics describing air flow, (2) designing a closed-loop control rate, eliminating high-order derivative expression of a wing plate corner, and enhancing the stability of a control equation; carrying out dimension reduction processing on the original system; the goal of state feedback control is realized; selecting a matched state observer, and (3) verifying and applying an active wing plate to carry out flutter control, and correcting the posture of the pneumatic wing plate. Through repeated observation and control, the pneumatic wing plate continuously changes the posture to vibrate, and the pneumatic self-excitation force generated by vibration is transmitted to the main beam through the support, so that the vibration of the main beam is restrained, and the flutter critical wind speed of the large-span suspension bridge is improved.

Description

Active control method for flutter pneumatic wing plate of super-large-span suspension bridge
Technical Field
The invention belongs to the field of suspension bridge flutter control, and particularly relates to an active control method for a flutter pneumatic wing plate of a super-large-span suspension bridge.
Background
The rapid development of world economy, the continuous research and development of new materials, and the continuous progress of design concepts, construction technologies and computing power enable people to have the ability to build bridges with larger spans and connect the ways which cannot be connected in the past. The suspension bridge is widely applied to crossing the cutting as a bridge form with the strongest crossing capability. Because of its excellent spanning ability, this bridge type is in a vigorous development stage all over the world, and its span increases very rapidly. However, the ultra-large span suspension bridge has low self rigidity and is often located in a high wind speed region on the sea, and how to avoid the bridge from being damaged due to flutter in the life cycle becomes a key point and a difficult point of design. The flutter control is mainly carried out by changing the structural strength, the auxiliary mechanical energy consumption and the pneumatic adjustment mode, and researches show that the pneumatic measure has the most attractive force in the three flutter control methods, on one hand, the pneumatic measure does not need to modify the bridge structure by a broadsword axe, only exists in the mode of an auxiliary component, and has good economy; on the other hand, the action process of aerodynamic force on the main beam is fundamentally changed by the aerodynamic measures, the source of flutter energy is fundamentally reduced, and the main beam has good effectiveness.
The pneumatic control measures are mainly divided into three types, namely a fixed pneumatic device, a movable pneumatic device and an active pneumatic device. The fixed pneumatic device is characterized in that the appearance arrangement of the bridge is properly changed or some flow guide devices, also called passive pneumatic devices, are added on the premise of not changing the main structure and the use performance of the bridge, and the devices need to be subjected to wind tunnel test selection to find out an effective appearance scheme, can improve the flutter critical wind speed in a certain range, are widely applied to bridge flutter control, and have the advantages of unconditional stability, mature technology and limited damping effect and complicated selection process; the movable pneumatic device enables the pneumatic device to make corresponding posture adjustment along with the vibration of the stiffening beam by arranging a smart mechanical structure, and the device can further improve the flutter critical wind speed on the basis of fixing the pneumatic device, has the defects of complex design of a control device, higher implementation difficulty and difficulty in realizing vibration suppression in an optimal mode, and in addition, the integrity of the stiffening beam can be damaged by some devices; the active pneumatic measure can be controlled and designed on the basis of a mathematical physical model established in advance, the optimal control of the system is completed under the selected performance target, the complicated comparison and selection process of the passive pneumatic measure is avoided, and the control device is simpler than a movable pneumatic device. Compared with mechanical active control measures, the measures have the advantages of low energy consumption, mainly have the disadvantages of conditional stability and energy requirement, and related researches are yet to be further carried out.
Disclosure of Invention
The invention aims to provide an active control method for flutter pneumatic wing plates of a super-large-span suspension bridge, which forms a flutter active control theoretical framework of the large-span suspension bridge based on the pneumatic wing plates, is used as a universal template to guide the design of flutter control of the large-span suspension bridge, and completes analysis of a control mode.
In order to achieve the above purpose, the invention adopts the technical scheme that:
the active control method for the flutter pneumatic wing plate of the ultra-large span suspension bridge comprises the following steps:
the method comprises the following steps: identifying dynamic characteristic parameters of large-span suspension bridge
1) On the basis of finite element analysis, according to a calculation method specified in road bridge wind resistance design specifications (JTG/T D60-01-2004), identifying a torsional mode and a vertical bending mode related to flutter through mode analysis, extracting an equivalent mass m and an equivalent mass inertia moment I of the model according to a formula (1), and finally obtaining physical parameters describing the structural dynamic performance;
Figure GDA0002720422980000021
in the formula (1), phiαAnd phihThe method comprises the following steps that the lowest-order torsional mode and the corresponding lowest-order vertical bending mode of a bridge are subjected to mode quality normalization, and a denominator part respectively integrates modal vertical displacement h or torsional displacement alpha within the length range of a main beam according to a calculation object;
2) providing aerodynamic force information of an aerodynamic wing plate and a main beam to describe the relationship between the aerodynamic force acting on a main beam-wing plate system and the system running state, obtaining self-excitation force parameters to obtain the physical characteristics of the described airflow, wherein the self-excitation force parameters are obtained by the Scanlan frequency domain flutter derivative description and then converted into time domain parameters,
Figure GDA0002720422980000022
in formula (2): fseThe self-excited aerodynamic force acting on the section with unit length consists of a self-excited aerodynamic lift force L and a self-excited aerodynamic lift moment T; ρ is the air density; u is the average wind speed; b is the section width; k is a dimensionless discounted frequency derived from the circle frequency omega; h is the vertical displacement of the section, and alpha is the torsion angle of the section;
Figure GDA0002720422980000023
the frequency domain flutter derivative of the Scanlan is a function of the reduction frequency K; the method comprises the following steps that (1) the pneumatic derivatives of the pneumatic wing plates and the main beam are obtained by the following method a and method b respectively, and the method for converting the Scanlan frequency domain flutter derivatives into time domain parameters is shown in step c;
a. for the aerodynamic wing plate, the aerodynamic force characteristic is simplified by an ideal flat plate, and the frequency domain flutter derivative of the scanlan is obtained by deduction according to Theodorsen theory;
Figure GDA0002720422980000031
in formula (3): k is 0.5K and is the dimensionless reduction frequency; j. the design is a squareiRepresenting a first class Bessel function of order i; y isiRepresenting a Bessel function of the second class of the order i;
b. for the main beam, the aerodynamic characteristics are obtained through wind tunnel test or CFD calculation analysis, and the Scanlan frequency domain flutter self-excitation force parameter obtaining mode is as follows:
the wind tunnel test or CFD calculation adopts a forced vibration method, the basic principle is that a main beam vibrates along the vertical bending and torsional freedom degrees at a certain folding and reducing frequency K to obtain aerodynamic force acting on the main beam, and then the aerodynamic force is identified as a Scanlan frequency domain flutter derivative according to the following method:
make the girder vibrate with the single-frequency in vertical and torsion direction respectively, specific form is as follows:
h＝Ahsin(ωt) (4)
α＝Aαsin(ωt) (5)
when only vertical vibration is carried out, the flutter derivative under the reduction frequency K is obtained according to self-spectrum identification of the lift force L time course and the lift moment T time course
Figure GDA0002720422980000032
And
Figure GDA0002720422980000034
Figure GDA0002720422980000035
Figure GDA0002720422980000036
Figure GDA0002720422980000037
when only torsional vibration is carried out, the flutter derivative under the reduction frequency K is obtained according to the self-spectrum identification of the lift force L time course and the lift moment T time course
Figure GDA0002720422980000038
And
Figure GDA0002720422980000039
Figure GDA00027204229800000310
Figure GDA00027204229800000311
Figure GDA00027204229800000312
Figure GDA0002720422980000041
in equations (6) to (13), the phase angle θ is derived from the cross-spectrum of the aerodynamic force and the displacement:
θL＝tan-1(Im(Sxy(L，h)|ω)/Re(Sxy(L，h)|ω)) (14)
θT＝tan-1(Im(Sxy(T，α)|ω)/Re(Sxy(T，α)|ω)) (15)
c. the method for converting the flutter derivative into the time domain flutter self-excitation force parameter comprises the following steps:
approximate time domain expression for flutter self-excitation:
Figure GDA0002720422980000042
in the formula (16), m pneumatic state variables phi are introducedk(k 1-m) describe the lag states of the flutter self-excitation force, each pneumatic state only affects one pneumatic self-excitation force component, increasing the number of pneumatic state variables can improve the precision of the approximate time domain expression, wherein phi iskAnd k is 1 to m, and further satisfies the following relationship:
Figure GDA0002720422980000043
λkthe attenuation rate of the lag term is used, so that the conversion process of the Scanlan time-frequency mixed expression mode to the pure time-domain approximate expression mode is completed;
key parameter A1，A2，A3，Ak+3K is 1 to m and λkObtaining k 1-m, and indirectly fitting based on Scanlan pneumatic derivatives;
to obtain a time-domain self-excitation parameter A1，A2，A3，Ak+3K is 1 to m and λkAnd k is 1-m, fitting needs to be carried out based on the Scanlan frequency domain flutter derivative, and a frequency response function Q in the Laplace domain is constructed according to the Roger rational function description, as shown in formula (18):
Figure GDA0002720422980000044
in the formula (18), the superscript ^ represents the approximate estimation; a. the1，A2，A3Respectively representing self-excitation pneumatic rigidity, pneumatic damping and pneumatic quality; a. thek+3K is 1 to m, λ is a memory effect considering the hysteresis of self-excited aerodynamic force to the structurekK is 1 to m, and is the attenuation rate of the hysteresis effect
Figure GDA0002720422980000045
When fitting is carried out, the flutter critical state is considered as simple harmonic vibration, the real number part in the pull variable is omitted, p is equal to Ki, and the reduced wind speed v is equal to 2 pi/K, so that:
Figure GDA0002720422980000051
in the formula (19), the
Figure GDA0002720422980000052
The real part and the imaginary part are expressed separately, and an objective function is constructed:
Figure GDA0002720422980000053
for such least squares problems, an internal confidence domain method is used to solve the time-domain pneumatic parameter fitting problem, here let { A }iParameter A in the ith iteration for the objective function1，A2，A3，…Ak+3…, it needs to search for { A }iSo that J ({ A })i) The minimum value of the objective function is reached, and then the current iteration step parameter { A } is definediNeighborhood Ω ofiSo that:
Ωi＝{{A}i∈R|||{A}-{A}i||≤Δi} (21)
in the formula (21), ΔiFor confidence radius, assuming that the objective function J is continuously differentiable in the real number domain R at second order, the problem translates into an appropriate quadratic model approximation q ({ A }) for finding the objective function J ({ A }) in the neighborhood, let s ═ A } - { A }iCalculating the minimum value-taking point s of the quadratic modeliSo that | | si||≤ΔiThe fitting problem of the time-domain pneumatic parameters can be converted into:
Figure GDA0002720422980000054
in the formula (22), the reaction mixture is,
Figure GDA0002720422980000055
according to a quadratic model q(i)(s) adjusting the confidence radius by fitting to the objective function J ({ A }), and setting a consistency parameter
Figure GDA0002720422980000056
Through continuous iteration and calculation of the consistency parameter after each iteration, the adjustment values of the confidence domain and the estimation value can be determined, and finally acceptable approximation errors are achieved;
the method realizes the purpose of converting the Scanlan frequency domain flutter derivative Hi *,Ai *I is 1-4 directional time domain parameter A1，A2，A3And Ak+3K is a transformation from 1 to m;
step two: design of closed loop control rate
This part of the content is to control the flutter stability of the girder-strake system:
1) firstly, considering the convenience of engineering application and experimental research, the relative rotation angle of a pneumatic wing plate is used as system input, the vibration response of a main beam is system output, and the high-order derivative expression of the rotation angle of the wing plate is eliminated by reconstructing the state variable of the system, so that the stability of a control equation is enhanced;
reselecting the system state variable as x ═ x1，x2，…x3，k…x4，k…x5，k…}TWherein:
Figure GDA0002720422980000061
for the main beam-wing system, the definition system is described by the reconstructed system variable x (t) and the output of the system to the environment is represented by y (t),
Figure GDA0002720422980000062
in formula (25), the subscript c represents a continuous time description, matrix Ac，Bc，Cc，DcIs a coefficient matrix independent of the system state x and the environmental input u, when the system is a linear time-invariant system, where AcThe relation among all state variables in the system is represented, and a system change mechanism is reflected and is called as a system matrix; b iscThe state variables are expressed how each input variable controls the state variable and are called control matrixes; ccRepresenting how the output variable reacts to the state variable, called the observation matrix; dcRepresenting the direct effect of the input on the output, called the direct transfer matrix;
the coefficient matrix is as follows:
Figure GDA0002720422980000063
Figure GDA0002720422980000071
Cc＝[E … 0 …] (28)
Figure GDA0002720422980000072
wherein the quality matrix
Figure GDA0002720422980000073
Damping matrix
Figure GDA0002720422980000074
And a stiffness matrix
Figure GDA0002720422980000075
The damping ratio xi of vertical bending mode and torsion mode is composed of the mass m of the main beam in unit length, the mass inertia moment I in unit length and the damping ratio xi of the vertical bending mode and the torsion modehAnd xiαAnd the circular frequencies ω of the vertical bending mode and the torsional modehAnd ωαDetermining; other parameters are defined as follows:
Figure GDA0002720422980000076
Figure GDA0002720422980000077
Figure GDA0002720422980000078
Figure GDA0002720422980000079
Figure GDA00027204229800000710
Figure GDA00027204229800000711
Figure GDA00027204229800000712
Figure GDA00027204229800000713
Figure GDA00027204229800000714
Figure GDA00027204229800000715
Figure GDA00027204229800000716
Figure GDA0002720422980000081
Figure GDA0002720422980000082
Figure GDA0002720422980000083
Figure GDA0002720422980000084
Figure GDA0002720422980000085
wherein, the superscript d represents the girder, and the superscript w represents the girderThe common characters of the pneumatic wing plates are shown, and the superscripts l and t respectively represent the characteristic matrixes of the pneumatic wing plates at the windward side and the pneumatic wing plates at the leeward side,
Figure GDA0002720422980000086
e is a unit diagonal matrix, and the meaning of other parameters is the same as the meaning of the parameters corresponding to the time domain pneumatic parameters solved in the step one, so that the state space expression of the flutter control model of the girder-wing plate system is completed;
2) based on the result of structural modal analysis, selecting a flutter-related mode, and performing dimension reduction processing on the original system by using a Schur decomposition method to eliminate the dimension which is not observable in the original system:
to the original system matrix AcCarrying out Schur decomposition
Ac＝VSVT (46)
In the formula (46), the diagonal elements of the matrix S are the eigenvalues of the original system, and the matrix V is transformed to make the eigenvalue λ in the matrix Ss,iAnd i is 1 to n, which are arranged in descending order according to the size of the real part:
Figure GDA0002720422980000087
get
Figure GDA0002720422980000088
And order
Figure GDA0002720422980000089
It is possible to obtain:
Figure GDA00027204229800000810
from the control point of view, will be with
Figure GDA00027204229800000811
The related terms are omitted without affecting the low-order dynamic response and stability of the system, and the following results are obtained:
Figure GDA00027204229800000812
because unstable modes in the original system are all included in the dimension reduction system
Figure GDA0002720422980000098
The input and output of the system are not changed, and the output controllability of the system is reserved;
3) introducing an optimal control theory, setting a weighting matrix for vibration amplitudes of the main beam and the wing plate, designing to obtain a relation between a relative corner of the wing plate and a system state, and realizing a state feedback control target:
the flutter stability problem of the main beam-wing plate system is solved, the running condition of the current system is obtained by utilizing the virtual system estimation, the relative rotation angle u of the wing plate is further determined through a given expression, and at the moment, a feedback gain matrix K is givencAnd enabling the relative rotation angle u of the wing plate to satisfy the following relation:
Figure GDA0002720422980000091
the operation of the virtual system and the actual system is required to be in accordance with the formula (51):
Figure GDA0002720422980000092
feedback gain matrix KcComprises the following steps:
Figure GDA0002720422980000093
where P is the solution of the Riccati algebraic equation:
Figure GDA0002720422980000094
4) according to the characteristic value of the controlled system, a matched state observer is selected to achieve the aim of estimating the state of the system by using the vibration of the main beam, and the specific process is as follows:
the virtual system uses the input variable u and output vector y which can be directly measured in the original system as input signals to make the self state of the virtual system as an input signal
Figure GDA0002720422980000099
Approaching to the actual system state x, and satisfying the following relation:
Figure GDA0002720422980000095
given that the change in state of the virtual system conforms to equation (54):
Figure GDA0002720422980000096
at this time, the error between the state of the virtual system and the state of the actual system satisfies equation (55):
Figure GDA0002720422980000097
the solution is as follows:
Figure GDA0002720422980000101
at this time, due to the system matrix AcAnd an observation matrix CcDepending on the actual system, only the appropriate K needs to be foundoLet A bec-KoCcAll the characteristic values have negative real parts, namely the target of the attenuation of the virtual system state to the real system can be met, at the moment, although the initial state of the virtual system is unknown, the error between the virtual system and the actual system is exponentially attenuated, and the method is based on the current situationIn the generation control theory, because the system state of the dimensionality reduction system is completely observable, a proper matrix K can be foundoThus, the design of feedback control parameters and a state observer is completed;
step three: validating and applying active strakes for flutter control
1) First, the proper sampling and control time interval is selected in combination with the dither frequency and the plant capability
Assuming that the sampling time interval and the control time interval are equal, performing equidistant control and sampling with a constant T as a period, and when the sampling consumed time is ignored, obtaining that the equation (58) is satisfied between the discrete system state and the continuous system state:
Figure GDA0002720422980000102
at the moment, the sampling time T needs to satisfy Shannon sampling theorem, in order to recover the analog signal without distortion, the sampling frequency is not less than 2 times of the highest frequency in the frequency spectrum of the analog signal, when the wind speed is close to the flutter critical wind speed, other modes except the defibrillation mode correspond to higher damping ratios, and at the moment, the response peak value of the suspension bridge is concentrated on the flutter circular frequency omegacrNearby, considering that the high frequency component of aerodynamic forces has a limited effect on flutter stability, the minimum requirement for sampling frequency is given:
T≤π/ωcr (59)
combining the time interval with the key parameters of the state observer obtained in the second step to obtain observation parameters of the discrete system;
acquiring discrete data of the vibration of the main beam through a sensor, and using the acquired data for a state observer to carry out state estimation to obtain an observation parameter B of a discrete systemeAnd Ee，
In addition, the relative rotation angle of the aerodynamic wing plates is controlled by a zero-order retainer, namely the rotation angle of the previous control node is kept unchanged before the next control node:
u(t)＝u(iT)，iT≤t＜(i+1)T (60)
at this time, equation (55) can be converted into a discrete expression:
Figure GDA0002720422980000103
wherein:
Figure GDA0002720422980000104
Figure GDA0002720422980000111
Figure GDA0002720422980000112
so far, the conversion of the total continuous time differential equation of the observation equation to the discrete time algebraic equation is completed;
2) correcting the attitude of the pneumatic wing plate based on the feedback gain matrix obtained by design
Through repeated observation and control, the pneumatic wing plate continuously changes the posture to vibrate, and the pneumatic self-excitation force caused by vibration is transmitted to the main beam through the support, so that the vibration of the main beam is restrained, and the flutter critical wind speed of the large-span suspension bridge is improved.
In the first step, for the segment model test, according to a calculation method specified in the bridge wind resistance specification, the equivalent mass and the equivalent mass moment of inertia of the model are further extracted.
The invention has the beneficial effects that:
(1) the measurable performance and the controllable performance of output of the state of the girder-wing plate system are analyzed, the necessity of system dimension reduction is pointed out, and after the system dimension reduction is carried out by introducing a Schur decomposition method, the dynamic stability of the system can be ensured to be similar to that of the original system, and a foundation is laid for the subsequent flutter control.
(2) The state estimation of the dimensionality reduction system is realized based on a pole allocation method, the current state of the system can be conveniently estimated by using the output of the system, and the exponential decay rate of the state estimator is recommended to be set to be about three times of that of the system after control due to the consideration of sensitivity and effectiveness.
(3) Based on linear quadratic performance indexes, the optimal control of a dimensionality reduction system is realized, a key principle is explained, wherein the physical meanings of main beam control weight and wing plate control weight are particularly pointed out, and the parameter sensitivity and the corresponding control effect are verified in detail on a wind tunnel test plate.
(4) For practical consideration, a time discrete method of system observation and system control is set forth, and a discretization equation is given.
(5) By integrating all the results, a flutter active control theoretical framework of the large-span suspension bridge based on the pneumatic wing plate is formed, and the framework can be used as a universal template to guide the flutter control design of the large-span suspension bridge; and then, simulating the theoretical framework by adopting a numerical simulation mode, and completing the analysis of the control mode.
Drawings
FIG. 1 is the ideal plate flutter derivative at high deflection wind speeds, (a) H _ i ^ (i ^ 1-4), (b) A _ i ^ (i ^ 1-4);
FIG. 2 shows the fitting result of pneumatic lag term with different orders of main beam (a)
Figure GDA0002720422980000121
(b)
Figure GDA0002720422980000122
FIG. 3 shows the fitting results of pneumatic lag terms of different orders with pneumatic wings, (a)
Figure GDA0002720422980000123
(b)
Figure GDA0002720422980000124
FIG. 4 is a graph of girder vibration under uniform flow conditions at different control parameters.
Detailed Description
Examples
The active control method for the flutter pneumatic wing plate of the ultra-large span suspension bridge comprises the following steps:
the method comprises the following steps: identifying dynamic characteristic parameters of large-span suspension bridge
(1) Structural dynamic characteristic acquisition
In the aspect of bridge dynamic characteristics, two typical large-span suspension bridges are selected as reference. One is that the established Danish Storebaelt East sea-crossing bridge spanning 1624m is widely analyzed internationally and has a large amount of comparable data; and secondly, the 5000m main span ultra-large span suspension bridge designed by the paradox university Navy and the Kumazu is complete in design and detailed in data and can represent the limit of the span design of the suspension bridge. On the basis of finite element analysis, according to a calculation method specified in road bridge wind resistance design specification (JTG/T D60-01-2004), a torsional mode and a vertical bending mode related to flutter are identified through mode analysis, and an equivalent mass m and an equivalent mass moment of inertia I of the model, a mode frequency and the like are extracted according to formula (1) and are shown in table 1.
Figure GDA0002720422980000125
In the formula (1), phiαAnd phihThe method comprises the following steps that the lowest-order torsional mode and the corresponding lowest-order vertical bending mode of a bridge are subjected to mode quality normalization, and a denominator part respectively integrates modal vertical displacement h or torsional displacement alpha within the length range of a main beam according to a calculation object;
TABLE 1 bridge segment model Power parameters
Figure GDA0002720422980000126
Figure GDA0002720422980000131
(2) Obtaining aerodynamic characteristics
Providing aerodynamic force information of an aerodynamic wing plate and a main beam to describe the relationship between the aerodynamic force acting on a main beam-wing plate system and the system running state, obtaining self-excitation force parameters to obtain the physical characteristics of the described airflow, wherein the self-excitation force parameters are obtained by the Scanlan frequency domain flutter derivative description and then converted into time domain parameters,
Figure GDA0002720422980000132
in formula (2): fseThe self-excited aerodynamic force acting on the section with unit length consists of a self-excited aerodynamic lift force L and a self-excited aerodynamic lift moment T; ρ is the air density; u is the average wind speed; b is the section width; k is a dimensionless discounted frequency derived from the circle frequency omega; h is the vertical displacement of the section, and alpha is the torsion angle of the section;
Figure GDA0002720422980000133
the frequency domain flutter derivative of the Scanlan is a function of the reduction frequency K; the method comprises the following steps that (1) the pneumatic derivatives of the pneumatic wing plates and the main beam are obtained by the following method a and method b respectively, and the method for converting the Scanlan frequency domain flutter derivatives into time domain parameters is shown in step c;
a. for the pneumatic wing plate, the factors such as landform, reappearance period and the like are considered, the flutter critical wind speed of most of large-span suspension bridges is generally required to be more than 80m/s, the vertical bending and torsion fundamental frequency of the structure is integrated, and the flutter frequency is generally 0.1 Hz-0.2 Hz. The width of the pneumatic wing plate is far smaller than the width of the cross section of the bridge, the corresponding reduction wind speed is usually more than 100, and based on the formula (3), the wing plate Scanlan frequency domain flutter self-excitation force parameter can be obtained by using the ideal flat plate expression of Theodorsen, as shown in figure 1.
b. For the main beam, the main beam of the Storebaelt East bridge vibrates in a single frequency in the vertical direction and the torsional direction respectively. The self-excitation force nonlinear effect and the identification precision of the flutter derivative of the section of the blunt body are comprehensively considered, the vertical amplitude and the torsional amplitude are respectively set to be 0.3 times of the height of the main beam and 5 degrees, and the parameters of the flutter self-excitation force in the Scanlan frequency domain are obtained based on the formulas (4) to (15) and are shown in a table 2:
TABLE 2 Storebaelt East bridge flutter derivative identification results
Figure GDA0002720422980000134
TABLE 2 Storebaelt East bridge flutter derivative identification results
Figure GDA0002720422980000141
h＝Ahsin(ωt) (4)
α＝Aαsin(ωt) (5)
When only vertical vibration is carried out, the flutter derivative under the reduction frequency K is obtained according to self-spectrum identification of the lift force L time course and the lift moment T time course
Figure GDA0002720422980000142
And
Figure GDA0002720422980000143
Figure GDA0002720422980000144
Figure GDA0002720422980000145
Figure GDA0002720422980000146
Figure GDA0002720422980000147
when only torsional vibration is carried out, the flutter derivative under the reduction frequency K is obtained according to the self-spectrum identification of the lift force L time course and the lift moment T time course
Figure GDA0002720422980000148
And
Figure GDA0002720422980000149
Figure GDA00027204229800001410
Figure GDA00027204229800001411
Figure GDA00027204229800001412
Figure GDA00027204229800001413
in equations (6) to (13), the phase angle θ is derived from the cross-spectrum of the aerodynamic force and the displacement:
θL＝tan-1(Im(Sxy(L，h)|ω)/Re(Sxy(L，h)|ω)) (14)
θT＝tan-1(Im(Sxy(T，α)|ω)/Re(Sxy(T，α)|ω)) (15)
the calculated flutter critical wind speed 73.6m/s of the Storebaelt East bridge is well consistent with the Larsen result 73m/s, and the reliability of aerodynamic force identification under the condition of a dynamic grid is demonstrated.
c. Conversion of flutter derivative into time domain flutter self-excitation force parameter
Approximate time domain expression for flutter self-excitation:
Figure GDA0002720422980000151
in the formula (16), m pneumatic state variables phi are introducedk(k1-m), each pneumatic state only affects one pneumatic self-excitation force component, increasing the number of pneumatic state variables can improve the precision of the approximate time domain expression, wherein phikAnd k is 1 to m, and further satisfies the following relationship:
Figure GDA0002720422980000152
λkthe attenuation rate of the lag term is used, so that the conversion process of the Scanlan time-frequency mixed expression mode to the pure time-domain approximate expression mode is completed;
key parameter A1，A2，A3，Ak+3K is 1 to m and λkObtaining k 1-m, and indirectly fitting based on Scanlan pneumatic derivatives;
to obtain a time-domain self-excitation parameter A1，A2，A3，Ak+3K is 1 to m and λkAnd k is 1-m, fitting needs to be carried out based on the Scanlan frequency domain flutter derivative, and a frequency response function Q in the Laplace domain is constructed according to the Roger rational function description, as shown in formula (18):
Figure GDA0002720422980000153
in the formula (18), the superscript ^ represents the approximate estimation; a. the1，A2，A3Respectively representing self-excitation pneumatic rigidity, pneumatic damping and pneumatic quality; a. thek+3K is 1 to m, λ is a memory effect considering the hysteresis of self-excited aerodynamic force to the structurekK is 1 to m, and is the attenuation rate of the hysteresis effect
Figure GDA0002720422980000154
When fitting is carried out, the flutter critical state is considered as simple harmonic vibration, the real number part in the pull variable is omitted, p is equal to Ki, and the reduced wind speed v is equal to 2 pi/K, so that:
Figure GDA0002720422980000155
in the formula (19), the
Figure GDA0002720422980000156
The real part and the imaginary part are expressed separately, and an objective function is constructed:
Figure GDA0002720422980000161
for such least squares problems, an internal confidence domain method is used to solve the time-domain pneumatic parameter fitting problem, here let { A }iParameter A in the ith iteration for the objective function1，A2，A3，…Ak+3…, it needs to search for { A }iSo that J ({ A })i) The minimum value of the objective function is reached, and then the current iteration step parameter { A } is definediNeighborhood Ω ofiSo that:
Ωi＝{{A}i∈R|||{A}-{A}i||≤Δi} (21)
in the formula (21), ΔiFor confidence radius, assuming that the objective function J is continuously differentiable in the real number domain R at second order, the problem translates into an appropriate quadratic model approximation q ({ A }) for finding the objective function J ({ A }) in the neighborhood, let s ═ A } - { A }iCalculating the minimum value-taking point s of the quadratic modeliSo that | | si||≤ΔiThe fitting problem of the time-domain pneumatic parameters can be converted into:
Figure GDA0002720422980000162
in the formula (22), the reaction mixture is,
Figure GDA0002720422980000163
according to a quadratic model q(i)(s) adjusting the confidence radius by fitting to the objective function J ({ A }), and setting a consistency parameter
Figure GDA0002720422980000164
According to the time-domain self-excitation parameter method of the formula (16) -formula (23), the time-domain parameter A can be obtained based on the Scanlan frequency-domain flutter self-excitation parameter of the main beam and the wing plate1，A2，A3And Ak+3,k＝1～m。
And the folding and reducing wind speeds corresponding to the main beam and the pneumatic wing plate in the flutter control stage are different. Therefore, the order m of different aerodynamic lag terms and the same attenuation coefficient lambda are respectively adoptedkThe fit effect of the flutter derivative for the low and high discounted wind speed intervals is compared at 0.5.
FIG. 2 shows a fitting result of the reduced wind speed of the main beam within a range of 0-20. The discrete data points are theoretical solutions of the flutter derivative of the ideal flat plate, and the color lines are flutter derivative curves obtained by inverse calculation of the fitted time-domain parameters. The red line represents the case where the pneumatic lag term correction is not taken into account, i.e. m is 0. At this time, there is only A representing the self-excited aerodynamic stiffness1Characterization of the aerodynamic damping2And A characterizing the aerodynamic mass3Participate in the time domain description. And the green and black lines respectively correspond to the fitting results of the first-order pneumatic hysteresis term and the second-order pneumatic hysteresis term. It can be seen that the influence of the aerodynamic lag term on the discrete flutter derivative is more obvious under the condition of low breaking wind speed. When the pneumatic lag term correction is not considered, the time domain fitting result obviously deviates from the steady state condition described by the flutter derivative; and when the order of the pneumatic correction term is gradually increased, the fitting precision is improved. When m is 2, the fitting result is already very close to the original flutter derivative.
FIG. 3 shows the fitting result of the wind reduction speed of the aerodynamic wing plate within the range of 100-200. What is different from the fitting result of the low refraction and reduction wind speed is that the fitting precision of the flutter derivative is not greatly influenced by adjusting the order of the pneumatic lag term. When the order of the pneumatic correction term is increased, the fitting precision cannot be obviously improved. Likewise, when m is 2, the fitting result is already very close to the original flutter derivative. Therefore, in the research, the time-domain self-excitation forces of the main beam and the pneumatic wing plate are uniformly expressed by adopting the condition that m is 2, and the requirements can be better met.
Step two: design of closed loop control rate
According to the flutter active control theory framework, key parameters influencing the final main beam-wing plate control relationship can be divided into two types. One category belongs to objective parameters, and the parameters comprise the dynamic characteristics of a bridge structure and the self-excitation pneumatic characteristics of a main beam and a pneumatic wing plate. These parameters are determined by the actual conditions of the study subject, are unique and are not subject to change by the subjective knowledge of the designer. Another type of parameter belongs to subjective parameters, and the parameters comprise design weight values for controlling wind speed U and controlling the amplitude of the relative vibration of the main beam and the wing plate, namely a feedback gain matrix K is calculated in the formula (53)cAnd (4) weighting matrixes Q and R for the main beam and the pneumatic wing plate. In both the theoretical analysis and the CFD numerical simulation sections, it has been demonstrated that it is appropriate to design the control wind speed to be selected as the flutter critical wind speed of the original profile. Such selection can effectively improve the critical wind speed that shimmys, and can not influence the stability of girder before shimmys. Therefore, this section will focus on the impact of the constraint weighting matrix on the main beam-wing system stability performance and control process.
According to the optimal control theory, it is necessary to specify relative constraints on the system input u and the system output y. For the two-dimensional section model test of the bridge, the model is placed in a wind field through a spring suspension system, and the model is generally regarded as only containing a vertical bending degree of freedom and a torsional degree of freedom. At this time, the system output y ═ h, α]TAnd is a two-dimensional vector. Furthermore, for practical reasons, the control variable is the angle of the windward and leeward flanks relative to the main beam. At this time, the system input u ═ βl，βt]TAlso a two-dimensional vector. Therefore, the weighting matrix Q output by the control system and the weighting matrix R input by the control system are both two-dimensional square matrices. The principal element determines the amplitude of each controlled variable in the control process, and the secondary element determines the correlation of each controlled variable in the control process. In the process of controlling the flutter of the bridge, only the vertical bending amplitude and the torsional amplitude of the girder are requiredThe magnitude is controlled without controlling the phase between the vertical bending and torsion of the main beam, so the weighting matrix Q is a diagonal matrix containing only principal elements. Similarly, the relative vibration amplitudes of the windward and leeward strakes need only be controlled for stall flutter and mechanical constraint considerations, and the phase between the two is automatically derived by optimal control theory. Therefore, the weighting matrix R is also a diagonal matrix containing only principal elements. On the other hand, it is noted that the vertical bending displacement and the torsional displacement of the main beam in the segmental model test are similar, and the wing plates on the two sides are reasonably restrained identically, so that Q, R main elements can be made to be identical in size, and different restraint degrees on the main beam and the wing plates are reflected by adjusting the relative sizes of the main beam and the wing plates.
For the reasons, the design control wind speed U is set as the original section flutter critical wind speed UcrAnd 1/10 girder torsion cycles are taken at control time intervals, Q: R is respectively 1:2, 1:1 and 2:1, and the control parameters under different weighting conditions can be obtained by substituting the control time intervals into the calculation process of the theoretical frame. After calculation and arrangement, three groups of parameters are shown in Table 3.
TABLE 3 State observer parameters and feedback control gains under different constraints
Figure GDA0002720422980000181
Step three: validating and applying active strakes for flutter control
The flutter active control effect is tested under the condition of uniform inflow, the control parameters obtained under three different weighting conditions in the table 3 are applied to the main beam-wing plate system under the condition of uniform inflow, the change rule of the vibration conditions of the vertical bending freedom degree and the torsional freedom degree of the main beam along with the wind speed under different wind speeds can be obtained, and the root mean square value of the vibration is shown in a figure 4.
In fig. 4, the dotted line a indicates the theoretical solution U of the critical wind speed for main beam flutter when the aerodynamic wing panel is not actively controlled in the closed loopcr. Line B refers to the vibration condition of the main beam when the wind tunnel test measures that the main beam is not controlled, the flutter critical wind speed of the main beam is between 8m/s and 9m/s, and the main beam is in good accordance with the theoretical analysis solution of 8.8 m/s.
Changing the control weight of the vibration of the main beam and the wing plate to obtain three curves of a line C, a line D and a line E, wherein the line C corresponds to the weakest control weight of the main beam, namely Q: R is 1: 2; line D represents the strongest girder control weight, i.e., Q: R ═ 2: 1.
Referring to the previous explanation, the weight strength is shown as the sensitivity of the wing plates to the vibration of the main beam in the control process, when the control weight of the main beam is high, the weak vibration of the main beam can cause the larger response of the wing plates, and when the control weight of the main beam is low, the response of the wing plates is more obvious when the main beam has the larger vibration.
The test results in the observation chart can show that when the control weight of the main beam is great, the lifting amplitude of the flutter critical wind speed is the largest and reaches 33 percent; when the control weight of the main beam is low, the flutter critical wind speed of the main beam is relatively small in lifting amplitude, and reaches 14%. Such differences represent the pertinence and correctness of the optimal control theory introduced in the theoretical framework, and for the phenomenon that the low-weight and medium-weight conditions correspond to almost the same flutter critical wind speed amplification, the differences are considered to be closely related to the bridge section type adopted in the test.
The section of the flat box girder with the tuyere used in the test belongs to a nearly streamlined section, and the flutter of the section generally belongs to hard flutter, namely, the flutter is expressed as sudden divergence of vibration displacement of the main girder after the wind speed exceeds a flutter critical wind speed point (which is in accordance with a curve in a figure). In this case, when the main beam control weight is low or medium, the wing panel is weak in sensitivity, and thus cannot suppress the main beam vibration having a large amplitude. The test results finally show that the vibration of the main beam can not inhibit the flutter through the wing plate after a certain specific wind speed is exceeded.

Claims (1)

1. The active control method for the flutter pneumatic wing plate of the ultra-large span suspension bridge is characterized by comprising the following steps of:
the method comprises the following steps: identifying dynamic characteristic parameters of large-span suspension bridge
1) On the basis of finite element analysis, according to a calculation method specified in road bridge wind resistance design specifications (JTG/T D60-01-2004), identifying a torsional mode and a vertical bending mode related to flutter through mode analysis, extracting an equivalent mass m and an equivalent mass inertia moment I of the model according to a formula (1), and finally obtaining physical parameters describing the structural dynamic performance;
Figure FDA0002720422970000011
in the formula (1), phiαAnd phihThe method comprises the following steps that the lowest-order torsional mode and the corresponding lowest-order vertical bending mode of a bridge are subjected to mode quality normalization, and a denominator part respectively integrates modal vertical displacement h or torsional displacement alpha within the length range of a main beam according to a calculation object;
2) providing aerodynamic force information of an aerodynamic wing plate and a main beam to describe the relationship between the aerodynamic force acting on a main beam-wing plate system and the system running state, obtaining self-excitation force parameters to obtain the physical characteristics of the described airflow, wherein the self-excitation force parameters are obtained by the Scanlan frequency domain flutter derivative description and then converted into time domain parameters,
Figure FDA0002720422970000012
in formula (2): fseThe self-excited aerodynamic force acting on the section with unit length consists of a self-excited aerodynamic lift force L and a self-excited aerodynamic lift moment T; ρ is the air density; u is the average wind speed; b is the section width; k is a dimensionless discounted frequency derived from the circle frequency omega; h is the vertical displacement of the section, and alpha is the torsion angle of the section;
Figure FDA0002720422970000013
the frequency domain flutter derivative of the Scanlan is a function of the reduction frequency K; the method comprises the following steps that (1) the pneumatic derivatives of the pneumatic wing plates and the main beam are obtained by the following method a and method b respectively, and the method for converting the Scanlan frequency domain flutter derivatives into time domain parameters is shown in step c;
a. for the aerodynamic wing plate, the aerodynamic force characteristic is simplified by an ideal flat plate, and the frequency domain flutter derivative of the scanlan is obtained by deduction according to Theodorsen theory;
Figure FDA0002720422970000021
in formula (3): k is 0.5K and is the dimensionless reduction frequency; j. the design is a squareiRepresenting a first class Bessel function of order i; y isiRepresenting a Bessel function of the second class of the order i;
b. for the main beam, the aerodynamic characteristics are obtained through wind tunnel test or CFD calculation analysis, and the Scanlan frequency domain flutter self-excitation force parameter obtaining mode is as follows:
the wind tunnel test or CFD calculation adopts a forced vibration method, the basic principle is that a main beam vibrates along the vertical bending and torsional freedom degrees at a certain folding and reducing frequency K to obtain aerodynamic force acting on the main beam, and then the aerodynamic force is identified as a Scanlan frequency domain flutter derivative according to the following method:
make the girder vibrate with the single-frequency in vertical and torsion direction respectively, specific form is as follows:
h＝Ahsin(ωt) (4)
α＝Aαsin(ωt) (5)
when only vertical vibration is carried out, the flutter derivative under the reduction frequency K is obtained according to self-spectrum identification of the lift force L time course and the lift moment T time course
Figure FDA0002720422970000022
And
Figure FDA0002720422970000023
Figure FDA0002720422970000024
Figure FDA0002720422970000025
Figure FDA0002720422970000026
Figure FDA0002720422970000027
when only torsional vibration is carried out, the flutter derivative under the reduction frequency K is obtained according to the self-spectrum identification of the lift force L time course and the lift moment T time course
Figure FDA0002720422970000028
And
Figure FDA0002720422970000029
Figure FDA00027204229700000210
Figure FDA00027204229700000211
Figure FDA00027204229700000212
Figure FDA00027204229700000213
in equations (6) to (13), the phase angle θ is derived from the cross-spectrum of the aerodynamic force and the displacement:
θL＝tan-1(Im(Sxy(L，h)|ω)/Re(Sxy(L，h)|ω)) (14)
θT＝tan-1(Im(Sxy(T，α )|ω)/Re(Sxy(T，α)|ω)) (15)
c. the method for converting the flutter derivative into the time domain flutter self-excitation force parameter comprises the following steps:
approximate time domain expression for flutter self-excitation:
Figure FDA0002720422970000031
in the formula (16), m pneumatic state variables phi are introducedk(k 1-m) describe the lag states of the flutter self-excitation force, each pneumatic state only affects one pneumatic self-excitation force component, increasing the number of pneumatic state variables can improve the precision of the approximate time domain expression, wherein phi iskAnd k is 1 to m, and further satisfies the following relationship:
Figure FDA0002720422970000032
λkthe attenuation rate of the lag term is used, so that the conversion process of the Scanlan time-frequency mixed expression mode to the pure time-domain approximate expression mode is completed;
key parameter A1，A2，A3，Ak+3K is 1 to m and λkObtaining k 1-m, and indirectly fitting based on Scanlan pneumatic derivatives;
to obtain a time-domain self-excitation parameter A1，A2，A3，Ak+3K is 1 to m and λkAnd k is 1-m, fitting needs to be carried out based on the Scanlan frequency domain flutter derivative, and a frequency response function Q in the Laplace domain is constructed according to the Roger rational function description, as shown in formula (18):
Figure FDA0002720422970000033
in the formula (18), the top mark Λ represents an approximate estimate; a. the1，A2，A3Respectively representing self-excitation pneumatic rigidity, pneumatic damping and pneumatic quality; a. thek+3K is 1 to m, λ is a memory effect considering the hysteresis of self-excited aerodynamic force to the structurekK is 1 to m, and is the attenuation rate of the hysteresis effect
Figure FDA0002720422970000034
When fitting is carried out, the flutter critical state is considered as simple harmonic vibration, the real number part in the pull variable is omitted, p is equal to Ki, and the reduced wind speed v is equal to 2 pi/K, so that:
Figure FDA0002720422970000035
in the formula (19), the
Figure FDA0002720422970000041
The real part and the imaginary part are expressed separately, and an objective function is constructed:
Figure FDA0002720422970000042
for such least squares problems, an internal confidence domain method is used to solve the time-domain pneumatic parameter fitting problem, here let { A }iParameter A in the ith iteration for the objective function1，A2，A3，...Ak+3.., it needs to search { A }iSo that J ({ A })i) The minimum value of the objective function is reached, and then the current iteration step parameter { A } is definediNeighborhood Ω ofiSo that:
Ωi＝{{A}i∈R|||{A}-{A}i||≤Δi} (21)
in the formula (21), ΔiFor confidence radius, assuming that the objective function J is continuously differentiable in the real number domain R at second order, the problem translates into an appropriate quadratic model approximation q ({ A }) for finding the objective function J ({ A }) in the neighborhood, let s ═ A } - { A }iCalculating the minimum value-taking point s of the quadratic modeliSo that | | si||≤ΔiThereby can beThe fitting problem of the time domain pneumatic parameters is converted into:
Figure FDA0002720422970000043
in the formula (22), the reaction mixture is,
Figure FDA0002720422970000044
according to a quadratic model q(i)(s) adjusting the confidence radius by fitting to the objective function J ({ A }), and setting a consistency parameter
Figure FDA0002720422970000045
Through continuous iteration and calculation of the consistency parameter after each iteration, the adjustment values of the confidence domain and the estimation value can be determined, and finally acceptable approximation errors are achieved;
the method realizes the flutter derivative of the Scanlan frequency domain
Figure FDA0002720422970000046
To the time domain parameter A1，A2，A3And Ak+3K is a transformation from 1 to m;
step two: design of closed loop control rate
This part of the content is to control the flutter stability of the girder-strake system:
1) firstly, considering the convenience of engineering application and experimental research, the relative rotation angle of a pneumatic wing plate is used as system input, the vibration response of a main beam is system output, and the high-order derivative expression of the rotation angle of the wing plate is eliminated by reconstructing the state variable of the system, so that the stability of a control equation is enhanced;
reselecting the system state variable as x ═ x1，x2，...x3，k...x4，k...x5，k...}TWherein:
Figure FDA0002720422970000051
for the main beam-wing system, the definition system is described by the reconstructed system variable x (t) and the output of the system to the environment is represented by y (t),
Figure FDA0002720422970000052
in formula (25), the subscript c represents a continuous time description, matrix Ac，Bc，Cc，DcIs a coefficient matrix independent of the system state x and the environmental input u, when the system is a linear time-invariant system, where AcThe relation among all state variables in the system is represented, and a system change mechanism is reflected and is called as a system matrix; b iscThe state variables are expressed how each input variable controls the state variable and are called control matrixes; ccRepresenting how the output variable reacts to the state variable, called the observation matrix; dcRepresenting the direct effect of the input on the output, called the direct transfer matrix;
the coefficient matrix is as follows:
Figure FDA0002720422970000053
Figure FDA0002720422970000054
Cc＝[E … 0 …] (28)
Figure FDA0002720422970000061
wherein the quality matrix
Figure FDA0002720422970000062
Damping matrix
Figure FDA0002720422970000063
And a stiffness matrix
Figure FDA0002720422970000064
The damping ratio xi of vertical bending mode and torsion mode is composed of the mass m of the main beam in unit length, the mass inertia moment I in unit length and the damping ratio xi of the vertical bending mode and the torsion modehAnd xiαAnd the circular frequencies ω of the vertical bending mode and the torsional modehAnd ωαDetermining; other parameters are defined as follows:
Figure FDA0002720422970000065
Figure FDA0002720422970000066
Figure FDA0002720422970000067
Figure FDA0002720422970000068
Figure FDA0002720422970000069
Figure FDA00027204229700000610
Figure FDA00027204229700000611
Figure FDA00027204229700000612
Figure FDA00027204229700000613
Figure FDA00027204229700000614
Figure FDA00027204229700000615
Figure FDA00027204229700000616
Figure FDA00027204229700000617
Figure FDA00027204229700000618
Figure FDA00027204229700000619
Figure FDA0002720422970000071
wherein, the superscript d represents the main beam, the superscript w represents the commonality of the pneumatic wing plate, the superscripts l and t respectively represent the characteristic matrixes of the pneumatic wing plate at the windward side and the pneumatic wing plate at the leeward side,
Figure FDA0002720422970000072
e is a unit diagonal matrix, and the meaning of other parameters is the same as the meaning of the parameters corresponding to the time domain pneumatic parameters solved in the step one, so that the state space expression of the flutter control model of the girder-wing plate system is completed;
2) based on the result of structural modal analysis, selecting a flutter-related mode, and performing dimension reduction processing on the original system by using a Schur decomposition method to eliminate the dimension which is not observable in the original system:
to the original system matrix AcCarrying out Schur decomposition
Ac＝VSVT (46)
In the formula (46), the diagonal elements of the matrix S are the eigenvalues of the original system, and the matrix V is transformed to make the eigenvalue λ in the matrix Ss，iAnd i is 1 to n, which are arranged in descending order according to the size of the real part:
Figure FDA0002720422970000073
get
Figure FDA0002720422970000074
And order
Figure FDA0002720422970000075
It is possible to obtain:
Figure FDA0002720422970000076
from the control point of view, will be with
Figure FDA0002720422970000077
The related terms are omitted without affecting the low-order dynamic response and stability of the system, and the following results are obtained:
Figure FDA0002720422970000078
because unstable modes in the original system are all included in the dimension reduction system
Figure FDA0002720422970000079
The input and output of the system are not changed, and the output controllability of the system is reserved;
3) introducing an optimal control theory, setting a weighting matrix for vibration amplitudes of the main beam and the wing plate, designing to obtain a relation between a relative corner of the wing plate and a system state, and realizing a state feedback control target:
the flutter stability problem of the main beam-wing plate system is solved, the running condition of the current system is obtained by utilizing the virtual system estimation, the relative rotation angle u of the wing plate is further determined through a given expression, and at the moment, a feedback gain matrix K is givencAnd enabling the relative rotation angle u of the wing plate to satisfy the following relation:
Figure FDA0002720422970000081
the operation of the virtual system and the actual system is required to be in accordance with the formula (51):
Figure FDA0002720422970000082
feedback gain matrix KcComprises the following steps:
Figure FDA0002720422970000083
where P is the solution of the Riccati algebraic equation:
Figure FDA0002720422970000084
4) according to the characteristic value of the controlled system, a matched state observer is selected to achieve the aim of estimating the state of the system by using the vibration of the main beam, and the specific process is as follows:
the virtual system uses the input variable u and output vector y which can be directly measured in the original system as input signals to make the self state of the virtual system as an input signal
Figure FDA0002720422970000085
Approaching to the actual system state x, and satisfying the following relation:
Figure FDA0002720422970000086
given that the change in state of the virtual system conforms to equation (54):
Figure FDA0002720422970000087
at this time, the error between the state of the virtual system and the state of the actual system satisfies equation (55):
Figure FDA0002720422970000088
the solution is as follows:
Figure FDA0002720422970000089
at this time, due to the system matrix AcAnd an observation matrix CcDepending on the actual system, only the appropriate K needs to be foundoLet A bec-KoCcAll the characteristic values have negative real parts, namely, the target that the virtual system state is attenuated to the real system can be met, at the moment, although the initial state of the virtual system isThe state is unknown, but the error between the virtual system and the actual system is exponentially attenuated, and according to the modern control theory, because the system state of the dimensionality reduction system is completely observable, a proper matrix K can be foundoThus, the design of feedback control parameters and a state observer is completed;
step three: validating and applying active strakes for flutter control
1) First, the proper sampling and control time interval is selected in combination with the dither frequency and the plant capability
Assuming that the sampling time interval and the control time interval are equal, performing equidistant control and sampling with a constant T as a period, and when the sampling consumed time is ignored, obtaining that the equation (58) is satisfied between the discrete system state and the continuous system state:
Figure FDA0002720422970000091
at the moment, the sampling time T needs to satisfy Shannon sampling theorem, in order to recover the analog signal without distortion, the sampling frequency is not less than 2 times of the highest frequency in the frequency spectrum of the analog signal, when the wind speed is close to the flutter critical wind speed, other modes except the defibrillation mode correspond to higher damping ratios, and at the moment, the response peak value of the suspension bridge is concentrated on the flutter circular frequency omegacrNearby, considering that the high frequency component of aerodynamic forces has a limited effect on flutter stability, the minimum requirement for sampling frequency is given:
T≤π/ωcr (59)
combining the time interval with the key parameters of the state observer obtained in the second step to obtain observation parameters of the discrete system;
acquiring discrete data of the vibration of the main beam through a sensor, and using the acquired data for a state observer to carry out state estimation to obtain an observation parameter B of a discrete systemeAnd Ee，
In addition, the relative rotation angle of the aerodynamic wing plates is controlled by a zero-order retainer, namely the rotation angle of the previous control node is kept unchanged before the next control node:
u(t)＝u(iT)，iT≤t＜(i+1)T (60)
at this time, equation (55) can be converted into a discrete expression:
Figure FDA0002720422970000092
wherein:
Figure FDA0002720422970000093
Figure FDA0002720422970000094
Figure FDA0002720422970000095
so far, the conversion of the total continuous time differential equation of the observation equation to the discrete time algebraic equation is completed;
2) correcting the attitude of the pneumatic wing plate based on the feedback gain matrix obtained by design
Through repeated observation and control, the pneumatic wing plate continuously changes the posture to vibrate, and the pneumatic self-excitation force caused by vibration is transmitted to the main beam through the support, so that the vibration of the main beam is restrained, and the flutter critical wind speed of the large-span suspension bridge is improved.
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