CN106326501A - Natural frequency and vibration mode calculation method for building construction dynamic analysis - Google Patents
Natural frequency and vibration mode calculation method for building construction dynamic analysis Download PDFInfo
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Abstract
The invention relates to a natural frequency and vibration mode calculation method for building construction dynamic analysis, and the method comprises the steps of establishing a building construction system natural frequency equation with multi freedoms, and converting the natural frequency equation into a characteristic equation; solving the natural frequency of the building construction system from the natural frequency equation; adopting Schmidt orthogonalization QR decomposition algorithm with "origin displacement" first to solve all characteristic values of the natural frequency equation (lambda), then calculating the natural frequency (omega); calculating a vibration mode of the building construction system based on the characteristic values solved from the natural frequency equation; and solving a characteristic vector through the method of nonhomogeneous linear equations by 1 solution. And the standardized characteristic vector is the vibration mode to be solved. <{EN2}>The calculating speed is fast; the calculating process is stable; no characteristic value is missed; the algorithm can be applied in constructions like bridges and can be used to solve vibration frequency and corresponding vibration mode in case of damping.
Description
Technical field
The present invention relates to civil construction project field, be related specifically to building and skyscraper under system with several degrees of freedom and carry out the natural frequency of vibration and the computational methods of the vibration shape during dynamic structural analysis.
Background technology
The natural frequency of vibration and the vibration shape are the intrinsic properties of building system itself, and they are design and the wind resistance of research structure system, the basis of antidetonation.Under multiple degrees of freedom, the natural frequency of vibration and vibration shape more than one.The natural frequency of vibration is relevant with the stiffness coefficient of itself and Mass Distribution thereof, and unrelated with external loads.And the vibration shape can be understood as the particular form of structural system vibration, can uniquely be determined the relative amplitude of structural system by standardized method.Frequency of vibration and the vibration shape are the basic design parameters that building carries out dynamic structural analysis, and they are the starting points of other dynamic analyses, such as transient dynamic analysis, spectrum response analysis and analysis of spectrum.Transient dynamic analysis is also known as time history analysis, is for determining a kind of method bearing the dynamic response arbitrarily changing over load structure.Spectrum response analysis and analysis of spectrum are being built especially in high-rise building design, are commonly used to calculate equivalent earthquake load.
The vibration problems of building is calculated by system with several degrees of freedom, such as the lateral vibration of multi storied factory building, non-contour framed bent, the vibration etc. of skyscraper, solve frequently with applying quite varied " stiffness method " or " flexibility method " in building (including skyscraper), the former is to solve by setting up the equilibrium equation of power, and the latter is by setting up displacement coordination equation solution.
The system with several degrees of freedom natural frequency of vibration and the vibration shape are solved by prior art, typically use approximate or the method for iteration, although minimum several eigenvalues and the natural frequency of vibration always can be produced, but this point does not obtain mathematical Strict Proof, sometimes there is also not convergence problem.So, in addition it is also necessary to eigenvalue and the natural frequency of vibration of omission is checked whether by sequential test method.Meanwhile, after having had eigenvalue and the natural frequency of vibration, being still and solved by iterative method, it is generally required to repeatedly solve system of linear equations, cause calculating process the most loaded down with trivial details, result of calculation is unstable.
Summary of the invention
It is an object of the invention to overcome above-mentioned the deficiencies in the prior art, it is provided that a kind of new building particularly skyscraper carries out the natural frequency of vibration and the computational methods of the vibration shape vector during dynamic structural analysis.It is fast that the computational methods of the present invention want to accomplish to calculate speed, and result of calculation is stable, does not haves eigenvalue and omits.
In order to reach foregoing invention purpose, the technical scheme that patent of the present invention provides is as follows:
A kind of fabric structure dynamic analysis natural frequency of vibration and the computational methods of the vibration shape, it is characterised in that these computational methods comprise the following steps that
The first step, for selected building, multivariant building structure ties up to undamped when principal oscillation, the natural frequency of vibration equation of architectural structure system is set up with flexibility method, and it is translated into characteristic equation, assuming that architectural structure system only has translation freedoms, its stiffness matrix [K] is expressed as [K]=k [K0], k is constant coefficient, and mass matrix [M] is expressed as [M]=m [M0], m is constant coefficient；Flexibility matrix [δ] is the inverse matrix of stiffness matrix [K], [δ]=[K]-1=δ[δ0], δ=1/k；
Then natural frequency of vibration equation is:
(1)
I.e.
(2)
This natural frequency of vibration equation is to solve for the natural frequency of vibrationAnd the vibration shapeEquation, order
,, above-mentioned natural frequency of vibration equation translates into algebra characterisitic value problem:
(3)
Wherein A is n rank real matrixes,For the vibration shape；
Second step, solves All Eigenvalues from natural frequency of vibration equation, use the Schmidt orthogonalization QR decomposition algorithm of band " origin displacement " to solve, for nonsingular n rank real matrix A, orthogonal matrix Q and the product of upper triangular matrix R can be resolved into by Schmidt orthogonalization QR decomposition method:
(4)
The Schmidt orthogonalization QR decomposition method specific algorithm of " band origin displacement " is as follows:
1. QR decomposes: willCarry out QR decomposition, i.e.
2. RQ structure:
WhereinFor unit matrix, hereBy following value:
(5)
WhereinIt is the number being not zero, nonsingular matrix A matrix sequence after QR algorithm, essential convergence is in upper triangular matrix or block upper triangular matrix, and diagonal blocks is 1 rank sub-block and 2 rank sub-blocks, and 1 rank sub-block is exactly the factual investigation of A, and 2 rank sub-blocks contain a pair Conjugate complex roots or the true weight root of A；Calculate All EigenvaluesAfter, can directly calculate the natural frequency of vibration of architectural structure system:
3rd step, the eigenvalue solved according to natural frequency of vibration equationCalculate the vibration shape of architectural structure system: the method solving Linear Equations by 1 time, direct solution characteristic vector, be exactly the required vibration shape after characteristic vector standardization, this vibration shapeRelative amplitude for particle.
In the computational methods of the fabric structure dynamic analysis natural frequency of vibration of the present invention and the vibration shape, the reality that will have calculatedAnd characteristic vectorSubstitution formula (3), result of calculation generally there are error term:
(6)
WhereinFor error, for guaranteeing the stability of equation solution,Can by the difference of parity column value as follows (whereinIt is a number the least):
(7)
Utilize formula (6) to solve Linear Equations direct solution and draw characteristic vector, and the same manner solves all vibration shapes of architectural structure system in building.
In the computational methods of the fabric structure dynamic analysis natural frequency of vibration of the present invention and the vibration shape, in described stiffness matrix and mass matrix, k is constant coefficient, m is also constant coefficient, and it is determined by the horizontal layout of architectural structure system, Mass Distribution, material actual performance, construction quality and foundation in building.
Based on technique scheme, patent of the present invention compared with prior art has a following technological merit:
1. compared with the iterative method of prior art (such as Rayleigh-Ritz theory, inverse power method etc.), when solving the natural frequency of vibration, the second step of the inventive method has algorithm stability, do not have the eigenvalue of omission, but computational accuracy is to rely on a kind of algorithm of specific computer system, its computational accuracy depends on the figure place of computer and the number of significant figures of compiling system.
2. when solving the vibration shape, the computational methods of the present invention are direct method, in addition to algorithmic stability, calculate speed the fastest.Iterative method frequently with inverse power method, it is assumed that iteration is s time, then at least need to solve 2s sublinear equation group；And the method using the 3rd step, it is only necessary to solving 1 sublinear equation group, the time used is the 1/2s of inverse power method, and when the node of model is the biggest, the computational methods of the present invention can save the substantial amounts of calculating time.
Accompanying drawing explanation
Fig. 1 is the schematic flow sheet of the computational methods of the fabric structure dynamic analysis natural frequency of vibration of the present invention and the vibration shape.
Fig. 2 be the fabric structure dynamic analysis natural frequency of vibration of the present invention and the vibration shape computational methods in the targeted building stiffness matrix of embodiment 1 and mass matrix schematic diagram.
Fig. 3 be patent of the present invention embodiment 1 in building ground floor floor produce unit displacement (displacement of other each particles remains zero) time structure stiffness coefficient schematic diagram.
Fig. 4 be patent of the present invention embodiment 1 in building second layer floor produce unit displacement (displacement of other each particles remains zero) time structure stiffness coefficient schematic diagram.
Fig. 5 be patent of the present invention embodiment 1 in building third layer floor produce unit displacement (displacement of other each particles remains zero) time structure stiffness coefficient schematic diagram.
Fig. 6 is fabric structure the first principal mode schematic diagram that the embodiment 1 of patent of the present invention calculates.
Fig. 7 is fabric structure the second principal mode schematic diagram that the embodiment 1 of patent of the present invention calculates.
Fig. 8 is fabric structure the 3rd principal mode schematic diagram that the embodiment 1 of patent of the present invention calculates.
Detailed description of the invention
We combine accompanying drawing and specific embodiment and are further elaborated the computational methods of the patent fabric structure dynamic analysis natural frequency of vibration of the present invention and the vibration shape below; in the hope of being more fully apparent from understand its Computing Principle and step, but the protection domain of patent of the present invention can not be limited with this.
The present invention is the brand-new natural frequency of vibration of a kind of fabric structure dynamic analysis and the computational methods of the vibration shape.As it is shown in figure 1, these computational methods comprise the following steps that
The first step, sets up multivariant architectural structure system natural frequency of vibration equation, and is translated into characteristic equation problem.
Second step, solves the natural frequency of vibration of architectural structure system from natural frequency of vibration equation, first uses the Schmidt orthogonalization QR decomposition algorithm of band " origin displacement " to solve the All Eigenvalues of natural frequency of vibration equation, then calculate the natural frequency of vibration。
3rd step, the eigenvalue solved according to natural frequency of vibration equationCalculate the vibration shape of architectural structure system, by solving the method direct solution characteristic vector of Linear Equations for 1 time, be exactly the required vibration shape after characteristic vector standardization。
In the first step of above-mentioned computational methods, set up architectural structure system natural frequency of vibration equation by " flexibility method ", and be translated into characteristic equation problem.The vibration problems (such as the lateral vibration of multi storied factory building, non-contour framed bent, the vibration etc. of skyscraper) of building is calculated by system with several degrees of freedom, often (including skyscraper) under construction applies quite varied employing " stiffness method " or " flexibility method " to solve, the former is to solve by setting up the equilibrium equation of power, and the latter passes through displacement coordination equation solution.The inventive method uses " flexibility method ", and the natural frequency of vibration equation of " flexibility method " can be derived indirectly by the equation of " stiffness method ", afterwards solving of the natural frequency of vibration is converted into solving of characteristic equation problem.
When multivariant building structure ties up to undamped-free vibration, it is assumed that architectural structure system only has translation freedoms, its stiffness matrix [K] and mass matrix [M] are just represented by: [K]=k [K0]、[M]=m[M0], k and m is respectively constant coefficient.It is to say, all can accurately determine in situations such as assuming the horizontal layout of architectural structure system, Mass Distribution, material actual performance, construction quality, foundation, the most once building determines, k and m is constant constant.[K0] and [M0] it is respectively the relative rigidity between particle and relative mass, and flexibility matrix [δ] is inverse matrix ([δ]=[K] of stiffness matrix-1=δ[δ0], δ=1/k).Element k in [K]ijFor the stiffness coefficient of structure, it is the power making a j apply needed for an i when producing unit displacement (displacement of other each points remains zero).The stiffness coefficient of structure, can be solved by the stiffness equations met between the power suffered by structure and the displacement of structure according to D'Alembert's principle.Stiffness matrix used by " flexibility method " that the present invention uses, is consistent with seeking the stiffness matrix used by the architectural structure system Static Calculation such as internal force, displacement.
After having had [K], [M] and [δ], it is possible to " flexibility method " natural frequency of vibration equation setting up architectural structure system (is to solve for the natural frequency of vibrationAnd the vibration shapeEquation):
(1)
I.e.
(2)
Order,, then natural frequency of vibration equation has translated into and has solved algebra characterisitic value problem:
(3)
In above-mentioned formula, A is n rank real matrixes,For the vibration shape.
In the second step of above-mentioned computational methods, from the natural frequency of vibration equation of formula (2), solve All Eigenvalues, it is that the Schmidt orthogonalization QR decomposition algorithm using band " origin displacement " solves.For nonsingular n rank real matrix A, orthogonal matrix Q and the product of upper triangular matrix R can be resolved into by Schmidt orthogonalization method:
(4)
The Schmidt orthogonalization QR decomposition method specific algorithm of " band origin displacement " is as follows:
1. QR decomposes: willCarry out QR decomposition, i.e.
2. RQ structure:
WhereinFor unit matrix, hereBy following value:
(5)
WhereinIt it is the number being not zero；
Nonsingular matrix A matrix sequence after QR algorithm, essential convergence is in upper triangular matrix or block upper triangular matrix, and diagonal blocks is 1 rank sub-block and 2 rank sub-blocks, and 1 rank sub-block is exactly the factual investigation of A, and 2 rank sub-blocks contain a pair Conjugate complex roots or the true weight root of A.So-called matrixNonsingular, it is simply that to refer to that the determinant of A is not zero.
There is All EigenvaluesAfter, it is possible to directly calculate the natural frequency of vibration of architectural structure system:
In the 3rd step of above-mentioned computational methods, the eigenvalue solved according to natural frequency of vibration equation calculates the vibration shape of architectural structure system: the method solving Linear Equations by 1 time, direct solution characteristic vector, be exactly the required vibration shape after characteristic vector standardization, and the vibration shape here is exactly the relative amplitude of particle；
If the reality that will have tried to achieveAnd characteristic vectorSubstitution formula (3), result of calculation generally there are error term:
(6)
WhereinFor error.
In the computer program design carried out based on computational methods of the present invention, for guaranteeing the stability of equation solution,Can by the difference of parity column value as follows (whereinIt is a number the least):
(7)
Formula (6) can use the method direct solution solving Linear Equations to draw, in like manner can try to achieve other All Eigenvalues characteristic of correspondence vectors.
Embodiment
1
The present embodiment is that the computational methods utilizing the present invention are to ask the natural frequency of vibration and the principal mode of multiple degrees of freedom rigid frame shown in Fig. 2.Assuming that the deformation of crossbeam is omitted to disregard, the storey stiffness coefficient of first, second, and third layer is respectivelyWith.The quality of rigid frame all concentrates on floor, and the quality at first, second, and third floor plate is respectively 2m, m and m.The present embodiment is to use " flexibility method " to calculate.
The process utilizing the computational methods of the present invention to solve is as follows:
The first step:The stiffness coefficient of rigid frame is as shown in Fig. 3, Fig. 4 and Fig. 5, (a), (b) and (c) that wherein Fig. 3, Fig. 4 are corresponding with Fig. 5 represents the power applied needed for each floor plate when ground floor, the second layer and third layer floor produce unit displacement (displacement of other each floor plates remains zero), the i.e. stiffness coefficient of structure respectively.The calculating process of stiffness coefficient is omitted, and stiffness matrix and mass matrix are respectively as follows:
Element k in [K]ijFor the stiffness coefficient of structure, it is the power making a j apply needed for an i when producing unit displacement (displacement of other each points remains zero).And the flexibility matrix of rigid frame is the inverse matrix of stiffness matrix:
Wherein
Then have:
Required natural frequency of vibration equation is:
I.e.
Order
Natural frequency of vibration equation translates into algebra characterisitic value problem:
Second step:The natural frequency of vibration is solved from natural frequency of vibration equation
The All Eigenvalues that the Schmidt orthogonalization QR decomposition algorithm using band " origin displacement " solves natural frequency of vibration equation is as follows:
After solving All Eigenvalues from natural frequency of vibration equation, it is possible to directly calculate the natural frequency of vibration according to following formula:
The natural frequency of vibration is:
3rd step:Eigenvalue calculation principal mode according to natural frequency of vibration equation
, solve character pair vector, here.The most useless by formula (7), equation is carried out pretreatment.
1. row Linear Equations
Row Linear Equations is as follows:
Linear Equations to be solved is as follows:
2. the vibration shape is obtained after solving characteristic vector standardization
Schmidt orthogonalization QR decomposition method is used to solve Linear Equations, the most unitization vector solved and norm thereof are as follows:
Have after unitization:
If pressing the standardized method of structural mechanics teaching material, as long as by the 3rd element normalization, the result of calculation of the first principal mode is consistent with structural mechanics teaching material, as shown in Figure 6:
In like manner, can try to achieve、Corresponding second and the 3rd principal mode.
For the second main formation, Linear Equations to be solved is as follows:
The most unitization vector solved and norm thereof are as follows:
Have after unitization:
If pressing the standardized method of document, as long as by the 3rd element normalization, the result of calculation of the second principal mode is consistent with structural mechanics teaching material, as shown in Figure 7:
For the 3rd main formation, Linear Equations to be solved is as follows:
The most unitization vector solved and norm thereof are as follows:
Have after unitization:
If pressing the standardized method of document, as long as by the 3rd element normalization, the result of calculation of the 3rd principal mode is consistent with structural mechanics teaching material, as shown in Figure 8:
Use the computational methods of the present invention, the stability of its algorithm is embodied in second step, the raising calculating speed is embodied in the 3rd step, i.e. solving in the vibration shape, because having only to solve 1 Linear Equations, so its time used is the 1/2s using inverse power method to calculate the time (assuming iteration s time) used.N principal mode of inverse power method calculating according to prior art needs iteration s=1000 time, solve system of linear equations every time and need 0.01 second, so inverse power method calculates the vibration shape needs to run the t=2*1000*0.01=20 second, and use the method for the present invention only to need 0.01 second (note: the time that solves of the system of linear equations of two kinds of methods is identical), it is that inverse power method calculates the time used by the vibration shape。
It is fast that the computational methods of the present invention calculate speed, calculates process stabilization, does not haves the situation that eigenvalue is omitted, but computational accuracy depends on specific computer system, and its precision depends on the figure place of computer and the number of significant figures of compiling system.Meanwhile, the algorithm of the amendment present invention, it is possible not only in the engineerings such as bridge application, it is also possible to solve the frequency of vibration in the case of damping and the corresponding vibration shape.
Claims (3)
1. the fabric structure dynamic analysis natural frequency of vibration and the computational methods of the vibration shape, it is characterised in that these computational methods comprise the following steps that
The first step, for selected building, when multivariant building structure ties up to undamped-free vibration, the natural frequency of vibration equation of architectural structure system is set up with flexibility method, and it is translated into characteristic equation, assuming that architectural structure system only has translation freedoms, its stiffness matrix [K] is expressed as [K]=k [K0], k is constant coefficient, and mass matrix [M] is expressed as [M]=m [M0], m is constant coefficient；Flexibility matrix [δ] is the inverse matrix of stiffness matrix [K], [δ]=[K]-1=δ[δ0], δ=1/k；
Then natural frequency of vibration equation is:
(1)
I.e.
(2)
This natural frequency of vibration equation is to solve for the natural frequency of vibrationAnd the vibration shapeEquation, order
,, above-mentioned natural frequency of vibration equation translates into algebra characterisitic value problem:
(3)
Wherein A is n rank real matrixes,For the vibration shape；
Second step, solves All Eigenvalues from natural frequency of vibration equation, use the Schmidt orthogonalization QR decomposition algorithm of band " origin displacement " to solve, for nonsingular n rank real matrix A, resolve into orthogonal matrix Q and the product of upper triangular matrix R by Schmidt orthogonalization QR decomposition method:
(4)
The Schmidt orthogonalization QR decomposition method specific algorithm of " band origin displacement " is as follows:
1. QR decomposes: willCarry out QR decomposition, i.e.
2. RQ structure:
WhereinFor unit matrix, hereBy following value:
(5)
WhereinIt is the number being not zero, nonsingular matrix A matrix sequence after QR algorithm, essential convergence is in upper triangular matrix or block upper triangular matrix, and diagonal blocks is 1 rank sub-block and 2 rank sub-blocks, and 1 rank sub-block is exactly the factual investigation of A, and 2 rank sub-blocks contain a pair Conjugate complex roots or the true weight root of A；Calculate All EigenvaluesAfter, can directly calculate the natural frequency of vibration of architectural structure system:
；
3rd step, the eigenvalue solved according to natural frequency of vibration equationCalculate the vibration shape of architectural structure system: the method solving Linear Equations by 1 time, direct solution characteristic vector, be exactly the required vibration shape after characteristic vector standardization, this vibration shapeRelative amplitude for particle.
The fabric structure dynamic analysis natural frequency of vibration the most according to claim 1 and the computational methods of the vibration shape, it is characterised in that the reality that will have calculatedAnd characteristic vectorSubstitution formula (3), result of calculation generally there are error term:
(6)
WhereinFor error, for guaranteeing the stability of equation solution,Can by the difference of parity column value as follows (whereinIt is a number the least):
(7)
Utilize formula (6) to solve Linear Equations direct solution and draw characteristic vector, and the same manner solves all vibration shapes of architectural structure system in building.
The fabric structure dynamic analysis natural frequency of vibration the most according to claim 1 and the computational methods of the vibration shape, it is characterized in that, in described stiffness matrix and mass matrix, k is constant coefficient, m is also constant coefficient, and it is determined by the horizontal layout of architectural structure system, Mass Distribution, material actual performance, construction quality and foundation in building.
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