CN107563005A - A kind of tension and compression different-stiffness Instantaneous method for optimally controlling - Google Patents
A kind of tension and compression different-stiffness Instantaneous method for optimally controlling Download PDFInfo
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Abstract
The invention discloses a kind of tension and compression different-stiffness Instantaneous method for optimally controlling, there are following steps：Using finite element method, the mass matrix, damping matrix and tensible rigidity matrix of structural system are obtained；Parametric variable is introduced, traditional Bilinear Constitutive Relation of tension and compression different-stiffness is converted into containing the unified constitutive relation of ginseng；The parameter unified equation of tension and compression different-stiffness Structural Dynamics is established, this kind of nonlinear structural dynamics problem is converted into the linear complementary problem of standard, solves the linear complementary problem, introduced parametric variable can be obtained；It is theoretical based on instantaneous self correlation, device design is controlled, to obtain Optimal Control Force.It is of the invention to be avoided compared with traditional tension and compression different-stiffness structural dynamical model method in each time step, the renewal of judgement and stiffness matrix to tension and compression different stress, improve the efficiency of calculating and the stability of control.
Description
Technical field
The present invention relates to active control in structural vibration technical field, especially relates to a kind of tension and compression different-stiffness structure and shakes
Dynamic instantaneous self correlation method.
Background technology
In the Practical Project such as building, machinery, space flight field, vibration problem is very common Mechanics Phenomenon.Structure is being shaken
During dynamic, change drastically may occur for its shape and bearing capacity, and service behaviour and service life to structure can produce sternly
The influence of weight.Therefore, the numerical algorithm of active control structure nuisance vibration is studied, had been to be concerned by more and more people.Many institutes
Known, classical elasticity theoretical description is the elastic behavior for stretching material identical with modulus of elasticity in comperssion.However, a large amount of examinations
Test and research shows, many engineering materials are under absolute value identical tension or action of compressive stress, it may occur that absolute value is different
Stretching strain and compressive strain, i.e. material has the nonlinear characteristic of tension and compression different-stiffness.Such as：Concrete, fiberglass, graphite,
Powdered metallurgical material and composite etc..In recent years, as the Typical Representative with tension and compression different-stiffness structure, rope, film knot
Structure is increasingly applied to the fields such as building, space flight.In engineering design, when this kind of materials in tension and compression property differs larger
When, if still being calculated using classical elastic theory and analysis of finite element method, it is clear that be irrational.In addition, all class formations exist
During use, inevitably due to by earthquake, wind swash and impact etc. external load function and vibrate, for mitigate
The vibration carries out correct mechanics point to this kind of nonlinear material structure to the adverse effect caused by structure function and safety
Analysis, and certain control measure are taken, developing effective control method will be very necessary.
Early in 1864, Saint-Venant just discussed bullet of some materials under tension and compression different conditions
Property property is different.Nineteen forty-one, famous mechanics scientist Timoshenko are proposed first when studying the bending stress of pure bending beam
The concept of bimodular material.Hereafter, this problem causes the close attention of domestic and international related scholar.In the 1980s,
The former Russian scholar Ambartsumyan analysis and summary test data of a large amount of new materials, by the sheet of machine with draw tool material
Structure relation generalization is bilinear model.1989, in algorithm aspect, domestic scholars Zhang Yunzhen etc. constructed bimodular problem and asked
The finite element scheme of solution, and it is iterated solution.Later, Yang Haitian etc. proposed initial stress method and solves bimodular elastic problem,
And discuss the dynamic analysis of one-dimensional problem.2004, the bright grades of Ye Zhi were summarized to the present Research of bimodular problem,
And point out that the solution difficult point of the problem is the conversion of tension and compression state and how to construct rational Iteration.It is 2011, high-strength
Et al. be based on parameter variable's variational principle, give power parameter variable's variational principle and the finite element side of machine with draw tool material
Method.For this kind of strong nonlinearity Structural Dynamics, the subject matter faced is：Iterative process needs to perform lengthy and tedious each time
Unit tension and compression condition adjudgement and structure Bulk stiffness matrix assembling computing；Strong nonlinearity brings the unstability and difficulty of iteration
Convergence.
Nonlinear structural vibration active control is a hot issue of structural vibration control.Solves nonlinear organization at present
The method of Active Vibration Control mainly includes multinomial control, NONLINEAR OPTIMAL CONTROL, Acceleration Control, dynamical linearization side
Method, PREDICTIVE CONTROL, fuzzy control and robust control etc..So, in traditional tension and compression different-stiffness material constitutive relation and existing
Nonlinear control method on the basis of, to above-mentioned nonlinear structure system design a stable, efficient Active Control Method
To be one has the potential research work extensively using value.
The content of the invention
The present invention proposes a kind of tension and compression different-stiffness Instantaneous method for optimally controlling.This method is become based on parameter
Divide principle and instantaneous self correlation theoretical, to solve the problems, such as tension and compression different-stiffness active control in structural vibration, it is therefore intended that avoid
The renewal of judgement and stiffness matrix in numerical procedure to tension and compression different stress, improve control stability and efficiently
Property.
The technological means that the present invention uses is as follows：
A kind of tension and compression different-stiffness Instantaneous method for optimally controlling, has following steps：
S1, using finite element method, obtain the mass matrix M, damping matrix C and tensible rigidity matrix K of structural system(+)；
S2, parametric variable is introduced, traditional Bilinear Constitutive Relation of tension and compression different-stiffness is converted into containing this unified structure of ginseng
Relation：
For the material of tension and compression different-stiffness, it is assumed that it is stretched and compression stiffness is respectively k(+)And k(-), and k(+)≠k(-),
If k(-)=0, then it is class cord material；
Using traditional bilinear model, the constitutive equation of tension and compression different-stiffness material is：
Δ F=k (t) Δ u,
Wherein, Δ u is element deformation amount, and Δ F is the corresponding internal force knots modification of unit, and
By introducing parametric variable λ >=0, construction is as follows containing the unified constitutive relation of ginseng：
Δ F=k(+)(Δu-sλ)
Wherein, s=sign (k(-)-k(+)), symbol sign definition is
S3, the parameter unified equation for establishing tension and compression different-stiffness Structural Dynamics, by this kind of nonlinear structural dynamics
Problem is converted into the linear complementary problem of standard, solves the linear complementary problem, can obtain introduced parametric variable：
Consider n free degree tension and compression different-stiffness Structural Dynamics, its kinetic equation is：
And haveWherein, M, C and K (t) are followed successively by the mass matrix of structural system, damping matrix and just
Degree matrix, q (t),WithIt is followed successively by the motion vector, velocity vector and vector acceleration of n × 1, D1Outer for n × r swashs
Position instruction matrix is encouraged, f (t) is the external excitation vector of r × 1,Do not refer to the summation of simple algebraically, and refer to each moment according to
Each unit tension and compression state is prejudged according to equation, then obtains the element stiffness matrix k inscribed when thise(t), finally by finite element
Method is assembled to obtain system global stiffness matrix；
On the basis of the constitutive relation of unification containing ginseng that step S2 is constructed, according to parameter variable's variational principle and limited configurations
Metatheory, derive the parameter unified equation of following multivariant tension and compression different-stiffness Structural Dynamics：
Parametric variable column vector λ in above formula can be obtained by solving following Linear Complementary Equations,
Wherein, K(+)For system integrally stretching stiffness matrix, A, B and F are constant coefficient matrix, and λ and υ represent parameter respectively
Variable column vector sum slack variable column vector；
It is S4, theoretical based on instantaneous self correlation, device design is controlled, to obtain Optimal Control Force：
Establish the controlled dynamic forces equation containing ginseng：
Wherein, D2For n × m controling power location matrix, for location control power, and u (t) be m × 1 controling power to
Amount；
Using Newmark methods in time domain it is discrete, solve it is above-mentioned containing ginseng controlled dynamic forces equation, obtain controlled dynamic forces sound
Answer state vkRecursive expression；
Further, by vkFollowing output equation is substituted into, obtains output state amount yk,
Wherein,For p × 3n output control matrix, p is output variable number；
The instantaneous self correlation performance indications are taken to be：
In above formula, Q is p × p positive semidefinite weight matrix, and R is m × m positive definite weight matrix；
It is theoretical according to instantaneous self correlation, make controlled system performance indications minimalization at any time, thereforeBy
This can obtain instantaneous self correlation power ukFor：
In the step S2, parameter lambda can be obtained by solving following linear complementary problem,
In the step S3, tensible rigidity matrix displacement K(+), constant coefficient matrix A, B and F, parametric variable column vector λ and pine
Relaxation variable column vector υ is represented by respectively：
Above formula NeFor number of unit, ΦjThe converting vector of overall coordinate is gone to from local coordinate for jth unit, i.e.,
Δqj=Φj·qj；
For space truss structure, it is assumed that a length of l of jth bar bar, node coordinate are respectively (x1,y1,z1)、(x2,y2,
z2), then：
Φj=[- α ,-β ,-γ, α, beta, gamma],
Wherein,
For truss structural, then for：
Φj=[- α ,-β, α, β].
In the step S4, controlled dynamic forces responsive state vkRecursive expression it is specific as follows：
Wherein,
In the step S4, instantaneous self correlation power u of the required current time containing parameterkIn the kth moment parameter
Variable column vector λkFollowing discrete rear solve can be made by the Linear Complementary Equations in step S3 to obtain：
The beneficial good effect of the present invention：
1. the present invention is in preceding processing modeling process, assembling global stiffness matrix, total with traditional structure finite element modeling
Firm assemble method is similar, it is only necessary to which the overall tensible rigidity matrix of package system, the process is well known, is easy to grasp
Make.
2. traditional Bilinear Constitutive Relation of tension and compression different-stiffness material is converted into containing the unified constitutive relation of ginseng by the present invention,
Parameter variable's variational principle is then based on, tension and compression different-stiffness Structural Dynamics parameter unified equation is established, by this quasi-nonlinear
The linear complementary problem that Structural Dynamics are converted into standard solves.With traditional tension and compression different-stiffness structural dynamical model side
Method is compared, and is avoided in each time step, the renewal of judgement and stiffness matrix to tension and compression different stress, improves calculating
Efficiency and control stability.
3. the present invention unifies column advantage by the parameter for the tension and compression different-stiffness Structural Dynamics established, based on wink
When the theory of optimal control, to the controller designed by this kind of tension and compression different-stiffness structural system, with other existing control method phases
Than design process is more succinct, control effect is notable, can be applied to solve this kind of new expansion knot with tension and compression different-stiffness
The engineering problem of structure vibration control.
Brief description of the drawings
In order to illustrate more clearly about the embodiment of the present invention or technical scheme of the prior art, below will be to embodiment or existing
There is the required accompanying drawing used in technology description to do simply to introduce, it should be apparent that, drawings in the following description are this hairs
Some bright embodiments, for those of ordinary skill in the art, on the premise of not paying creative work, can be with root
Other accompanying drawings are obtained according to these accompanying drawings.
Fig. 1 is a kind of tension and compression different-stiffness Instantaneous method for optimally controlling in embodiment of the invention
Flow chart.
Fig. 2 is the individual layer plane girder for having in the embodiment of the present invention rope unit.
Fig. 3 be the present invention embodiment in parametric variable change with time figure.
Fig. 4 is the horizontal displacement response without individual layer plane girder in the case of control in embodiment of the invention.
Fig. 5 is the vertical displacement response without individual layer plane girder in the case of control in embodiment of the invention.
Fig. 6 is the double layer planar truss for having in the embodiment of the present invention rope unit.
Fig. 7 is that have uncontrolled dynamic respond correlation curve in the embodiment of the present invention.
Fig. 8 is controling power time-history curves in embodiment of the invention.
Embodiment
To make the purpose, technical scheme and advantage of the embodiment of the present invention clearer, below in conjunction with the embodiment of the present invention
In accompanying drawing, the technical scheme in the embodiment of the present invention is clearly and completely described, it is clear that described embodiment is
Part of the embodiment of the present invention, rather than whole embodiments.Based on the embodiment in the present invention, those of ordinary skill in the art
The all other embodiment obtained under the premise of creative work is not made, belongs to the scope of protection of the invention.
As shown in figure 1, a kind of tension and compression different-stiffness Instantaneous method for optimally controlling, has following steps：
S1, using finite element method, obtain the mass matrix M, damping matrix C and tensible rigidity matrix K of structural system(+)；
S2, parametric variable is introduced, traditional Bilinear Constitutive Relation of tension and compression different-stiffness is converted into containing this unified structure of ginseng
Relation：
For the material of tension and compression different-stiffness, it is assumed that it is stretched and compression stiffness is respectively k(+)And k(-), and k(+)≠k(-)；
Using traditional bilinear model, the constitutive equation of tension and compression different-stiffness material is：
Δ F=k (t) Δ u,
Wherein, Δ u is element deformation amount, and Δ F is the corresponding internal force knots modification of unit, and
By introducing parametric variable λ >=0, construction is as follows containing the unified constitutive relation of ginseng：
Δ F=k(+)(Δu-sλ)
Wherein, s=sign (k(-)-k(+)), symbol sign definition is
S3, the parameter unified equation for establishing tension and compression different-stiffness Structural Dynamics, by this kind of nonlinear structural dynamics
Problem is converted into the linear complementary problem of standard, solves the linear complementary problem, can obtain introduced parametric variable：
Consider n free degree tension and compression different-stiffness Structural Dynamics, its kinetic equation is：
And haveWherein, M, C and K (t) are followed successively by the mass matrix of structural system, damping matrix and just
Degree matrix, q (t),WithIt is followed successively by the motion vector, velocity vector and vector acceleration of n × 1, D1Outer for n × r swashs
Position instruction matrix is encouraged, f (t) is the external excitation vector of r × 1,Do not refer to the summation of simple algebraically, and refer to each moment according to
Each unit tension and compression state is prejudged according to equation, then obtains the element stiffness matrix k inscribed when thise(t), finally by finite element
Method is assembled to obtain system global stiffness matrix；
On the basis of the constitutive relation of unification containing ginseng that step S2 is constructed, according to parameter variable's variational principle and limited configurations
Metatheory, derive the parameter unified equation of following multivariant tension and compression different-stiffness Structural Dynamics：
Parametric variable column vector λ in above formula can be obtained by solving following Linear Complementary Equations,
Wherein, K(+)For system integrally stretching stiffness matrix, A, B and F are constant coefficient matrix, and λ and υ represent parameter respectively
Variable column vector sum slack variable column vector；
It is S4, theoretical based on instantaneous self correlation, device design is controlled, to obtain Optimal Control Force：
Establish the controlled dynamic forces equation containing ginseng：
Wherein, D2For n × m controling power location matrix, for location control power, and u (t) be m × 1 controling power to
Amount；
Using Newmark methods in time domain it is discrete, solve it is above-mentioned containing ginseng controlled dynamic forces equation, obtain controlled dynamic forces sound
Answer state vkRecursive expression；
Further, by vkFollowing output equation is substituted into, obtains output state amount yk,
Wherein,For p × 3n output control matrix, p is output variable number；
The instantaneous self correlation performance indications are taken to be：
In above formula, Q is p × p positive semidefinite weight matrix, and R is m × m positive definite weight matrix；
It is theoretical according to instantaneous self correlation, make controlled system performance indications minimalization at any time, thereforeBy
This can obtain instantaneous self correlation power ukFor：
In the step S2, parameter lambda can be obtained by solving following linear complementary problem,
In the step S3, tensible rigidity matrix displacement K(+), constant coefficient matrix A, B and F parametric variable column vector λ and pine
Relaxation variable column vector υ is represented by respectively：
Above formula NeFor number of unit, ΦjThe converting vector of overall coordinate is gone to from local coordinate for jth unit, i.e.,
Δqj=Φj·qj；
For space truss structure, it is assumed that a length of l of jth bar bar, node coordinate are respectively (x1,y1,z1)、(x2,y2,
z2), then：
Φj=[- α ,-β ,-γ, α, beta, gamma],
Wherein,
For truss structural, then for：
Φj=[- α ,-β, α, β].
In the step S4, controlled dynamic forces responsive state vkRecursive expression it is specific as follows：
Wherein,
In the step S4, instantaneous self correlation power u of the required current time containing parameterkIn the kth moment parameter
Variable column vector λkFollowing discrete rear solve can be made by the Linear Complementary Equations in step S3 to obtain：
Simulation example：Using the inventive method, be directed to respectively with tension and compression different-stiffness structural system without control dynamics
Analysis and two problems of instantaneous self correlation, deploy numerical simulation.
For first problem：Such as Fig. 2, by taking the individual layer truss structural with rope unit as an example, carry out without control feelings
Dynamic response analysis under condition.Assuming that the tension and compression rigidity of truss structure bar unit (solid line) is 1e5 (N/m2), the left and right sides
Oblique rope is followed successively by 2e5 (N/m with center vertical rope unit (dotted line) tension rigidity2) and 1e5 (N/m2).System is by first initiating
Amount acts on, wherein：No. 1 knee level momentum is 1, and vertical momentum is 0；No. 2 knee level momentum are 0, and vertical momentum is 1；No. 3
Knee level momentum is -1, and vertical momentum is 0.Step delta t=0.002s is taken, carries out the dynamics simulation that the time is 300s.
What Fig. 3 was provided is the change of rope unit relevant parameter variable, and they are continuous between more than zero and equal to zero
There is tensioning and the frequent switching of relaxation two states in vibration processes in jump, this three rope of explanation, and this reacts well
The dynamic characteristics of this kind of Material Nonlinear Structure Analysis system.Such as Fig. 4 and Fig. 5, what is provided successively is the horizontal position of each free nodes of Fig. 2
Move response and vertical displacement response.As shown in Figure 4, in the horizontal direction, the displacement of No. 1 node and No. 3 nodes is in symmetric periodic
Change, and the displacement of No. 2 nodes is zero.As shown in Figure 5, in the vertical direction, the displacement of No. 1 node and No. 3 nodes are to overlap
, and the displacement of No. 2 nodes is between (- 0.02m, 0.025m), and be in mechanical periodicity.Above simulation result embodies well
The symmetry and periodicity that are matched with expected results.
For Second Problem：It is more by the above-mentioned individual layer truss structural popularization with rope unit such as Fig. 6
Layer, and assume to dispose actuator in the horizontal direction of node 1 and node 6, apply active controlling force, investigate structure in El
The controlled response of the lower horizontal direction of Centro seismic stimulations effect.Wherein, the material parameter of structural system is constant, and assumes initial
State is static.It is all Δ t=0.02s, simulation time 30s to take the simulation step length of algorithm and earthquake sampling period.
When Fig. 7 and Fig. 8 provide controlled successively, the horizontal displacement response and corresponding controling power time-histories at No. 1 node are bent
Line.As shown in Figure 7, controlled front and rear, the peak value of No. 1 modal displacement response is down to 0.076m by 0.418m, reduces 81.8%；Position
Move response root mean square and 0.022m is then down to by 0.182m, reduce 87.9%, control effect is satisfactory.And as shown in Figure 8,1
Corresponding controling power is in (- 149N, 150N) range at number node.Above simulation result shows：Drawing proposed by the invention
Pressure different-stiffness Instantaneous method for optimally controlling can significantly suppress the vibration of this kind of nonlinear organization, and calculate and protect
Hold long-time stable.
Finally it should be noted that：Various embodiments above is merely illustrative of the technical solution of the present invention, rather than its limitations；To the greatest extent
The present invention is described in detail with reference to foregoing embodiments for pipe, it will be understood by those within the art that：Its according to
The technical scheme described in foregoing embodiments can so be modified, either which part or all technical characteristic are entered
Row equivalent substitution；And these modifications or replacement, the essence of appropriate technical solution is departed from various embodiments of the present invention technology
The scope of scheme.
Claims (5)
1. a kind of tension and compression different-stiffness Instantaneous method for optimally controlling, it is characterised in that there are following steps：
S1, using finite element method, obtain the mass matrix M, damping matrix C and tensible rigidity matrix K of structural system(+)；
S2, parametric variable is introduced, traditional Bilinear Constitutive Relation of tension and compression different-stiffness is converted into containing the unified constitutive relation of ginseng：
For the material of tension and compression different-stiffness, it is assumed that it is stretched and compression stiffness is respectively k(+)And k(-), and k(+)≠k(-)；
Using traditional bilinear model, the constitutive equation of tension and compression different-stiffness material is：
Δ F=k (t) Δ u,
Wherein, Δ u is element deformation amount, and Δ F is the corresponding internal force knots modification of unit, andPass through
Parametric variable λ >=0 is introduced, construction is as follows containing the unified constitutive relation of ginseng：
Δ F=k(+)(Δu-sλ)
Wherein, s=sign (k(-)-k(+)), symbol sign definition is
S3, the parameter unified equation for establishing tension and compression different-stiffness Structural Dynamics, by this kind of nonlinear structural dynamics problem
The linear complementary problem of standard is converted into, solves the linear complementary problem, introduced parametric variable can be obtained：
Consider n free degree tension and compression different-stiffness Structural Dynamics, its kinetic equation is：
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And haveWherein, M, C and K (t) are followed successively by the mass matrix, damping matrix and rigidity square of structural system
Battle array, q (t),WithIt is followed successively by the motion vector, velocity vector and vector acceleration of n × 1, D1For n × r external excitation position
Oriental matrix is put, f (t) is the external excitation vector of r × 1,Do not refer to simple algebraically summation, and refer in each moment foundation side
Journey prejudges each unit tension and compression state, then obtains the element stiffness matrix k inscribed when thise(t), finally by finite element method
Assembled to obtain system global stiffness matrix；
On the basis of the constitutive relation of unification containing ginseng that step S2 is constructed, managed according to parameter variable's variational principle and structure finite element
By the parameter unified equation of the following multivariant tension and compression different-stiffness Structural Dynamics of derivation：
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Parametric variable column vector λ in above formula can be obtained by solving following Linear Complementary Equations,
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Wherein, K(+)For system integrally stretching stiffness matrix, A, B and F are constant coefficient matrix, and λ and υ represent parametric variable respectively
Column vector and slack variable column vector；
It is S4, theoretical based on instantaneous self correlation, device design is controlled, to obtain Optimal Control Force：
Establish the controlled dynamic forces equation containing ginseng：
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Wherein, D2For n × m controling power location matrix, for location control power, and u (t) is the control force vector of m × 1；
Using Newmark methods in time domain it is discrete, solve it is above-mentioned containing ginseng controlled dynamic forces equation, obtain controlled dynamic forces respond shape
State vkRecursive expression；
Further, by vkFollowing output equation is substituted into, obtains output state amount yk,
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Wherein,For p × 3n output control matrix, p is output variable number；
The instantaneous self correlation performance indications are taken to be：
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</msub>
<mo>+</mo>
<msubsup>
<mi>u</mi>
<mi>k</mi>
<mi>T</mi>
</msubsup>
<msub>
<mi>Ru</mi>
<mi>k</mi>
</msub>
<mo>,</mo>
</mrow>
In above formula, Q is p × p positive semidefinite weight matrix, and R is m × m positive definite weight matrix；
It is theoretical according to instantaneous self correlation, make controlled system performance indications minimalization at any time, thereforeThus may be used
Obtain instantaneous self correlation power ukFor：
<mrow>
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>u</mi>
<mi>k</mi>
</msub>
<mo>=</mo>
<mo>-</mo>
<msup>
<mrow>
<mo>&lsqb;</mo>
<msup>
<mrow>
<mo>(</mo>
<mover>
<mi>C</mi>
<mo>^</mo>
</mover>
<msub>
<mi>wD</mi>
<mn>2</mn>
</msub>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
<mi>Q</mi>
<mover>
<mi>C</mi>
<mo>^</mo>
</mover>
<msub>
<mi>wD</mi>
<mn>2</mn>
</msub>
<mo>+</mo>
<mi>R</mi>
<mo>&rsqb;</mo>
</mrow>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>&CenterDot;</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mover>
<mi>C</mi>
<mo>^</mo>
</mover>
<msub>
<mi>wD</mi>
<mn>2</mn>
</msub>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
<mi>Q</mi>
<mover>
<mi>C</mi>
<mo>^</mo>
</mover>
<mo>{</mo>
<msub>
<mi>hv</mi>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>w</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>F&lambda;</mi>
<mi>k</mi>
</msub>
<mo>+</mo>
<msub>
<mi>D</mi>
<mn>1</mn>
</msub>
<msub>
<mi>f</mi>
<mi>k</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>}</mo>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>.</mo>
</mrow>
A kind of 2. tension and compression different-stiffness Instantaneous method for optimally controlling according to claim 1, it is characterised in that：
In the step S2, parameter lambda can be obtained by solving following linear complementary problem,
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>s</mi>
<mrow>
<mo>(</mo>
<msup>
<mi>k</mi>
<mrow>
<mo>(</mo>
<mo>+</mo>
<mo>)</mo>
</mrow>
</msup>
<mo>-</mo>
<msup>
<mi>k</mi>
<mrow>
<mo>(</mo>
<mo>-</mo>
<mo>)</mo>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mi>&Delta;</mi>
<mi>u</mi>
<mo>-</mo>
<msup>
<mi>k</mi>
<mrow>
<mo>(</mo>
<mo>+</mo>
<mo>)</mo>
</mrow>
</msup>
<mi>&lambda;</mi>
<mo>+</mo>
<mi>&upsi;</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>&lambda;</mi>
<mo>&GreaterEqual;</mo>
<mn>0</mn>
<mo>,</mo>
<mi>&upsi;</mi>
<mo>&GreaterEqual;</mo>
<mn>0</mn>
<mo>,</mo>
<mi>&lambda;</mi>
<mi>&upsi;</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>.</mo>
</mrow>
A kind of 3. tension and compression different-stiffness Instantaneous method for optimally controlling according to claim 1, it is characterised in that
In the step S3, tensible rigidity matrix displacement K(+), constant coefficient matrix A, B and F, parametric variable column vector λ and slack variable arrange
Vectorial υ is represented by respectively：
<mrow>
<msup>
<mi>K</mi>
<mrow>
<mo>(</mo>
<mo>+</mo>
<mo>)</mo>
</mrow>
</msup>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<msub>
<mi>N</mi>
<mi>e</mi>
</msub>
</munderover>
<msubsup>
<mi>k</mi>
<mi>j</mi>
<mrow>
<mo>(</mo>
<mo>+</mo>
<mo>)</mo>
</mrow>
</msubsup>
<msubsup>
<mi>&Phi;</mi>
<mi>j</mi>
<mi>T</mi>
</msubsup>
<msub>
<mi>&Phi;</mi>
<mi>j</mi>
</msub>
<mo>,</mo>
<mi>F</mi>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<msub>
<mi>N</mi>
<mi>e</mi>
</msub>
</munderover>
<msub>
<mi>s</mi>
<mi>j</mi>
</msub>
<msubsup>
<mi>k</mi>
<mi>j</mi>
<mrow>
<mo>(</mo>
<mo>+</mo>
<mo>)</mo>
</mrow>
</msubsup>
<msubsup>
<mi>&Phi;</mi>
<mi>j</mi>
<mi>T</mi>
</msubsup>
<mo>,</mo>
</mrow>
<mrow>
<mi>A</mi>
<mo>=</mo>
<mi>d</mi>
<mi>i</mi>
<mi>a</mi>
<mi>g</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mi>k</mi>
<mi>j</mi>
<mrow>
<mo>(</mo>
<mo>+</mo>
<mo>)</mo>
</mrow>
</msubsup>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mi>B</mi>
<mo>=</mo>
<mo>-</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<msub>
<mi>N</mi>
<mi>e</mi>
</msub>
</munderover>
<msub>
<mi>s</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<msubsup>
<mi>k</mi>
<mi>j</mi>
<mrow>
<mo>(</mo>
<mo>+</mo>
<mo>)</mo>
</mrow>
</msubsup>
<mo>-</mo>
<msubsup>
<mi>k</mi>
<mi>j</mi>
<mrow>
<mo>(</mo>
<mo>-</mo>
<mo>)</mo>
</mrow>
</msubsup>
<mo>)</mo>
</mrow>
<msub>
<mi>&Phi;</mi>
<mi>j</mi>
</msub>
<mo>,</mo>
</mrow>
<mrow>
<mi>&lambda;</mi>
<mo>=</mo>
<msup>
<mrow>
<mo>&lsqb;</mo>
<msub>
<mi>&lambda;</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>&lambda;</mi>
<mn>2</mn>
</msub>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<msub>
<mi>&lambda;</mi>
<msub>
<mi>N</mi>
<mi>e</mi>
</msub>
</msub>
<mo>&rsqb;</mo>
</mrow>
<mi>T</mi>
</msup>
<mo>,</mo>
<mi>&upsi;</mi>
<mo>=</mo>
<msup>
<mrow>
<mo>&lsqb;</mo>
<msub>
<mi>&upsi;</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>&upsi;</mi>
<mn>2</mn>
</msub>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<msub>
<mi>&upsi;</mi>
<msub>
<mi>N</mi>
<mi>e</mi>
</msub>
</msub>
<mo>&rsqb;</mo>
</mrow>
<mi>T</mi>
</msup>
<mo>,</mo>
</mrow>
Above formula NeFor number of unit, ΦjThe converting vector of overall coordinate, i.e. Δ q are gone to from local coordinate for jth unitj=
Φj·qj；
For space truss structure, it is assumed that a length of l of jth bar bar, node coordinate are respectively (x1,y1,z1)、(x2,y2,z2), then：
Φj=[- α ,-β ,-γ, α, beta, gamma],
Wherein,
<mrow>
<mi>&alpha;</mi>
<mo>=</mo>
<mfrac>
<mrow>
<msub>
<mi>x</mi>
<mn>2</mn>
</msub>
<mo>-</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
</mrow>
<mi>l</mi>
</mfrac>
<mo>,</mo>
<mi>&beta;</mi>
<mo>=</mo>
<mfrac>
<mrow>
<msub>
<mi>y</mi>
<mn>2</mn>
</msub>
<mo>-</mo>
<msub>
<mi>y</mi>
<mn>1</mn>
</msub>
</mrow>
<mi>l</mi>
</mfrac>
<mo>,</mo>
<mi>&gamma;</mi>
<mo>=</mo>
<mfrac>
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
<mo>-</mo>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
<mi>l</mi>
</mfrac>
<mo>;</mo>
</mrow>
For truss structural, then for：
Φj=[- α ,-β, α, β].
A kind of 4. tension and compression different-stiffness Instantaneous method for optimally controlling according to claim 1, it is characterised in that
In the step S4, controlled dynamic forces responsive state vkRecursive expression it is specific as follows：
<mrow>
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>v</mi>
<mi>k</mi>
</msub>
<mo>=</mo>
<msub>
<mi>hv</mi>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>w</mi>
<msub>
<mover>
<mi>S</mi>
<mo>^</mo>
</mover>
<mi>k</mi>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<mo>{</mo>
<msub>
<mi>hv</mi>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>w</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>F&lambda;</mi>
<mi>k</mi>
</msub>
<mo>+</mo>
<msub>
<mi>D</mi>
<mn>1</mn>
</msub>
<msub>
<mi>f</mi>
<mi>k</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>}</mo>
<mo>+</mo>
<msub>
<mi>wD</mi>
<mn>2</mn>
</msub>
<msub>
<mi>u</mi>
<mi>k</mi>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>,</mo>
</mrow>
Wherein,
<mrow>
<msub>
<mi>v</mi>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<msub>
<mi>q</mi>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mover>
<mi>q</mi>
<mo>&CenterDot;</mo>
</mover>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mover>
<mi>q</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>,</mo>
<msub>
<mi>v</mi>
<mi>k</mi>
</msub>
<mo>=</mo>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<msub>
<mi>q</mi>
<mi>k</mi>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mover>
<mi>q</mi>
<mo>&CenterDot;</mo>
</mover>
<mi>k</mi>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mover>
<mi>q</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mi>k</mi>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>,</mo>
<mi>h</mi>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msub>
<mi>a</mi>
<mn>11</mn>
</msub>
</mtd>
<mtd>
<msub>
<mi>a</mi>
<mn>12</mn>
</msub>
</mtd>
<mtd>
<msub>
<mi>a</mi>
<mn>13</mn>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>a</mi>
<mn>21</mn>
</msub>
</mtd>
<mtd>
<msub>
<mi>a</mi>
<mn>22</mn>
</msub>
</mtd>
<mtd>
<msub>
<mi>a</mi>
<mn>23</mn>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>a</mi>
<mn>31</mn>
</msub>
</mtd>
<mtd>
<msub>
<mi>a</mi>
<mn>32</mn>
</msub>
</mtd>
<mtd>
<msub>
<mi>a</mi>
<mn>33</mn>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>,</mo>
<mi>w</mi>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msup>
<mover>
<mi>K</mi>
<mo>^</mo>
</mover>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mi>b</mi>
<mn>1</mn>
</msub>
<msup>
<mover>
<mi>K</mi>
<mo>^</mo>
</mover>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mi>b</mi>
<mn>4</mn>
</msub>
<msup>
<mover>
<mi>K</mi>
<mo>^</mo>
</mover>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
<mrow>
<mover>
<mi>K</mi>
<mo>^</mo>
</mover>
<mo>=</mo>
<msub>
<mi>b</mi>
<mn>4</mn>
</msub>
<mi>M</mi>
<mo>+</mo>
<msub>
<mi>b</mi>
<mn>1</mn>
</msub>
<mi>C</mi>
<mo>+</mo>
<msup>
<mi>K</mi>
<mrow>
<mo>(</mo>
<mo>+</mo>
<mo>)</mo>
</mrow>
</msup>
<mo>,</mo>
<msub>
<mover>
<mi>S</mi>
<mo>^</mo>
</mover>
<mi>k</mi>
</msub>
<mo>=</mo>
<msub>
<mi>F&lambda;</mi>
<mi>k</mi>
</msub>
<mo>+</mo>
<msub>
<mi>D</mi>
<mn>1</mn>
</msub>
<msub>
<mi>f</mi>
<mi>k</mi>
</msub>
<mo>+</mo>
<msub>
<mi>D</mi>
<mn>2</mn>
</msub>
<msub>
<mi>u</mi>
<mi>k</mi>
</msub>
</mrow>
A kind of 5. tension and compression different-stiffness Instantaneous method for optimally controlling according to claim 1, it is characterised in that
In the step S4, instantaneous self correlation power u of the required current time containing parameterkIn the kth moment parametric variable arrange to
Measure λkFollowing discrete rear solve can be made by the Linear Complementary Equations in step S3 to obtain：
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<msub>
<mi>&upsi;</mi>
<mi>k</mi>
</msub>
<mo>-</mo>
<mi>A</mi>
<msub>
<mi>&lambda;</mi>
<mi>k</mi>
</msub>
<mo>-</mo>
<mi>B</mi>
<msub>
<mi>q</mi>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>&lambda;</mi>
<mi>k</mi>
</msub>
<mo>&GreaterEqual;</mo>
<mn>0</mn>
<mo>,</mo>
<msub>
<mi>&upsi;</mi>
<mi>k</mi>
</msub>
<mo>&GreaterEqual;</mo>
<mn>0</mn>
<mo>,</mo>
<msubsup>
<mi>&lambda;</mi>
<mi>k</mi>
<mi>T</mi>
</msubsup>
<msub>
<mi>&upsi;</mi>
<mi>k</mi>
</msub>
<mo>=</mo>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>.</mo>
</mrow>
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| CN110704905A (en) * | 2019-09-16 | 2020-01-17 | 东南大学 | Optimal design method for viscous damper for stay cable multistage modal vibration control |
| CN111460586A (en) * | 2020-03-13 | 2020-07-28 | 浙江大胜达包装股份有限公司 | Method for simultaneously identifying gap value and gap contact stiffness of sliding pair gap |
| CN111475980A (en) * | 2020-04-09 | 2020-07-31 | 西北工业大学 | Thin-wall part dynamic parameter acquisition method integrating actuator quality influence |
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| CN109739091A (en) * | 2019-01-16 | 2019-05-10 | 大连理工大学 | A kind of intelligent tensioning overall structure vibration multi-layer distributed model predictive control method based on Substructure Techniques |
| CN110704905A (en) * | 2019-09-16 | 2020-01-17 | 东南大学 | Optimal design method for viscous damper for stay cable multistage modal vibration control |
| CN110704905B (en) * | 2019-09-16 | 2023-04-18 | 东南大学 | Optimal design method for viscous damper for stay cable multistage modal vibration control |
| CN111460586A (en) * | 2020-03-13 | 2020-07-28 | 浙江大胜达包装股份有限公司 | Method for simultaneously identifying gap value and gap contact stiffness of sliding pair gap |
| CN111460586B (en) * | 2020-03-13 | 2023-05-12 | 浙江大胜达包装股份有限公司 | Method for identifying clearance value and clearance contact stiffness of sliding pair clearance simultaneously |
| CN111475980A (en) * | 2020-04-09 | 2020-07-31 | 西北工业大学 | Thin-wall part dynamic parameter acquisition method integrating actuator quality influence |
| CN112129617A (en) * | 2020-08-25 | 2020-12-25 | 江苏大学 | Equivalent soil stiffness online detection method for intelligent agricultural machinery |
| CN112129617B (en) * | 2020-08-25 | 2022-09-16 | 江苏大学 | Equivalent soil stiffness online detection method for intelligent agricultural machinery |
| CN112257145A (en) * | 2020-09-30 | 2021-01-22 | 上海建工集团股份有限公司 | Method for identifying structural damping and rigidity by utilizing dynamic response |
| CN112257145B (en) * | 2020-09-30 | 2023-12-22 | 上海建工集团股份有限公司 | Method for identifying structural damping and rigidity by utilizing dynamic response |
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