CN101887492A - Method of correcting singular displacement component to eliminate singularity of stiffness matrix in wrinkle calculation - Google Patents

Abstract

A method of correcting singular displacement component to eliminate the singularity of stiffness matrix in wrinkle calculation relates to the technical field of membrane structural mechanics and structural wrinkle analysis and solves the problem that the existing wrinkling calculation method can not effectively eliminate the singularity of stiffness matrix, accurately introduce wrinkle mode and timely eliminate the influence of wrinkle mode on post-wrinkle characteristic. The method of the invention comprises the following steps: firstly introducing corrected parameter to obtain real displacement solution, determining singular displacement component; secondly introducing wrinkle mode equality, relationship, corresponding to the wrinkle mode of wrinkle critical point, after the elimination of load and singular displacement solution existence condition equality to wrinkle critical point, obtaining nonsingular displacement solution, and correcting singular displacement component to eliminate the singularity of stiffness matrix. The method realizes that the singular displacement component is corrected in wrinkle calculation to eliminate the singularity of stiffness matrix, and is used in the wrinkle calculation of membrane structure.

Description

The method of stiffness matrix singularity when eliminating fold calculating by revising unusual displacement component

Technical field

The invention belongs to membrane structure mechanics and buckling structure analysis technical field.

Background technology

Fold is the distinctive deformation phenomenon of fexible film structure, its inside configuration regional area generation flexing and the distortion of generation the inside and then formation corrugated fold when the fexible film structure is compressed stress.The generation of fold can greatly influence the stress performance of structure, as change the bang path of load in the structure, influence the rigidity of structure and mass distribution and then the structural vibrations characteristic is exerted an influence, change the surface configuration of structure and then influence the thermodynamic behaviour of structure, particularly for the exigent film antenna reflecting surface of shape surface accuracy, the existence of fold is a kind of typical functions failure mode, so the research of fold problem all receives much concern all the time.

The fold computing method mainly contain two big classes at present: one, membrane process, this method is based on the tension field theory, ignore film material bendind rigidity fully, think that any small compressive stress all will form fold, each point is in along the plane stress state of the unidirectional tension of fold direction in the plication region; Two, shell method, this method are considered the small bendind rigidity that the film material has based on Post-buckling Theory, the fold analysis is converted into shell post-buckling problem studies.

Constitutive relation or the distortion tensor of described membrane process when revising fold carries out fold and calculates, its key problem is to determine the actual stress state and the constitutive relation of plication region, this method only can obtain plication region and direction, can not forecast the details and the rugosity unstability overall process of film of single fold.

Described shell method is utilized nonlinear buckling analysis to carry out fold and is calculated, and its key problem is the elimination of stiffness matrix singularity and the accurate assessment of imperfection sensitivity, and this method can be forecast fold wavelength and amplitude, and can obtain fold expansion evolutionary process.The existing firm gust of strange strategy that disappears mainly is to adopt prestress control, adjustment stable factor and restricted boundary condition to carry out, yet these strategies all can't obtain consistent convergence effect, and all fundamentally do not provide the overall process of stiffness matrix singularity elimination; In addition, all be to adopt the strategy of former rank mode linear superposition to obtain initial imperfection in existing the analysis, and it is introduced directly into brings out fold in the analytical model and produce, obtain its post-buckling characteristic, but the defective of introducing in analyzing can not be removed, and this produces a very large impact the rugosity characteristic in back.

It is to eliminate the stiffness matrix singularity that fold calculates key problem, in just battle array disappears strange process, need emphasis to consider the accurate introducing of rugosity mode and elimination in time, existing fold computing method all can not provide the effective ways that the stiffness matrix singularity is eliminated, and all ignored the influence of initial imperfection to the rugosity characteristic in back, therefore, existing fold calculates and can not accurately obtain fold deformation and evolution properties.

Summary of the invention

The present invention is in order to solve existing fold computing method and can not effectively to eliminate the stiffness matrix singularity and can't accurately introduce and in time eliminating the influence of rugosity mode to the rugosity characteristic in back, proposes a kind of when revising unusual displacement component and eliminate fold calculating the method for stiffness matrix singularity.

The present invention is achieved by following proposal: the method for stiffness matrix singularity when eliminating fold calculating by revising unusual displacement component, and the process of described method is:

Step 1: in rugosity space of feature vectors, introduce corrected parameter correction singular stiffness matrix, obtain nonsingular stiffness matrix form;

Step 2: obtain real displacement in conjunction with nonsingular stiffness matrix form and separate;

Step 3: judge the singularity that real displacement is separated, and obtain unusual displacement component;

Step 4: introduce rugosity mode equation in rugosity critical point place, relational equation and the unusual displacement solution existence condition equation of the load of rugosity critical point place is rugosity mode correspondence after eliminating, and utilize described three unusual displacement components of equation correction, obtain nonsingular displacement solution, the singularity of stiffness matrix when calculating by revising unusual displacement solution and then eliminating gauffer.

In rugosity critical point place, the stiffness matrix singularity that fold calculates mainly shows as:

K _Tcrφ _cr＝0

Wherein, K _TcrBe critical tangent stiffness matrix, φ _CrBe critical rugosity mode.

Stiffness matrix singularity elimination problem was converted into the problem of revising unusual displacement component when this method was calculated fold by introducing a corrected parameter.Its core concept is to eliminate the singularity of the rugosity eigenvector in first rank in the space of feature vectors.

When this method is calculated in order to realize fold the stiffness matrix singularity is eliminated, in its rugosity space of feature vectors, at first introduced a corrected parameter singular stiffness matrix is revised, obtain the stiffness matrix form of nonsingular form greater than zero; Introduce rugosity mode afterwards, obtain real displacement and separate in conjunction with revising the nonsingular stiffness matrix form obtain, so far the stiffness matrix singularity problem is converted into the singularity problem of the displacement component that real displacement separates.Utilize rugosity mode equation in rugosity critical point place, relational equation and the unusual displacement solution existence condition equation correction unusual displacement component of the load of rugosity critical point place is rugosity mode correspondence after eliminating, finally obtain nonsingular displacement solution, and then realize the purpose of the stiffness matrix singularity elimination that fold calculates.Wherein, its influence to the rugosity characteristic in back has been avoided in the introducing of the relational equation after the load of the rugosity mode correspondence of rugosity critical point place is eliminated.Introduce disturbance fold is calculated the rugosity stage of back that proceeds to, realize the whole process analysis of fold.

The present invention is applicable to the fold computing method of membrane structure, mainly be applicable to the unusual displacement component of correction of stiffness matrix singularity when eliminating fold calculates, it has realized that fundamentally stiffness matrix in the fold computing method effectively eliminates the problem of singularity, and avoided of the influence of rugosity mode to the rugosity characteristic in back, the overall process that the stiffness matrix singularity was eliminated when reflection fold that can be more deep calculated, obtain nonsingular accurately displacement solution, and can more fully reflect the rugosity characteristic in back after introducing the displacement disturbance, for follow-up forecast of fold configuration and control provide information more accurately.

Description of drawings

Fig. 1 be embodiment one described when revising unusual displacement component and eliminate fold and calculate the process flow diagram of the method for stiffness matrix singularity; Fig. 2 is the fold Distribution calculation result who uses the square film diagonal stretch that method of the present invention obtains; Fig. 3 is the outer The deformation calculation result of fold face who uses the square film diagonal stretch that method of the present invention obtains.

Embodiment

Embodiment one, specify present embodiment below in conjunction with Fig. 1, Fig. 2 and Fig. 3.The method of stiffness matrix singularity when eliminating fold calculating by revising unusual displacement component, the process of described method is:

Step 1: in rugosity space of feature vectors, introduce corrected parameter correction singular stiffness matrix, obtain nonsingular stiffness matrix form;

Step 2: obtain real displacement in conjunction with nonsingular stiffness matrix form and separate;

Step 3: judge the singularity that real displacement is separated, and obtain unusual displacement component;

Step 4: introduce rugosity mode equation in rugosity critical point place, relational equation and the unusual displacement solution existence condition equation of the load of rugosity critical point place is rugosity mode correspondence after eliminating, and utilize described three unusual displacement components of equation correction, obtain nonsingular displacement solution, the singularity of stiffness matrix when calculating by revising unusual displacement solution and then eliminating gauffer.

In rugosity critical point place, the stiffness matrix singularity that fold calculates mainly shows as:

K _Tcrφ _cr＝0

Wherein, K _TcrBe critical tangent stiffness matrix, φ _CrBe critical rugosity mode.

Stiffness matrix singularity elimination problem was converted into the problem of revising unusual displacement component when present embodiment was calculated fold by introducing a corrected parameter.Its core concept is to eliminate the singularity of the rugosity eigenvector in first rank in the space of feature vectors.

When present embodiment is calculated in order to realize fold the stiffness matrix singularity is eliminated, in its rugosity space of feature vectors, at first introduced a corrected parameter singular stiffness matrix is revised, obtain the stiffness matrix form of nonsingular form greater than zero; Introduce rugosity mode afterwards, obtain real displacement and separate in conjunction with revising the nonsingular stiffness matrix form obtain, so far the stiffness matrix singularity problem is converted into the singularity problem of the displacement component that real displacement separates.Utilize rugosity mode equation in rugosity critical point place, relational equation and the unusual displacement solution existence condition equation correction unusual displacement component of the load of rugosity critical point place is rugosity mode correspondence after eliminating, finally obtain nonsingular displacement solution, and then realize the purpose of the stiffness matrix singularity elimination that fold calculates.Wherein, its influence to the rugosity characteristic in back has been avoided in the introducing of the relational equation after the load of the rugosity mode correspondence of rugosity critical point place is eliminated.Introduce disturbance fold is calculated the rugosity stage of back that proceeds to, realize the whole process analysis of fold.

When Fig. 2 and Fig. 3 calculate for adopting the described method of present embodiment to eliminate fold after the stiffness matrix singularity, analysis result figure to fold, the fold Distribution calculation result of Fig. 2 side of being film diagonal stretch, the outer The deformation calculation result of the fold face of Fig. 3 side of being film diagonal stretch.

Present embodiment is applicable to the fold computing method of membrane structure, mainly be applicable to the unusual displacement component of correction of stiffness matrix singularity when eliminating fold calculates, it has realized that fundamentally stiffness matrix in the fold computing method effectively eliminates the problem of singularity, and avoided of the influence of rugosity mode to the rugosity characteristic in back, the overall process that the stiffness matrix singularity was eliminated when reflection fold that can be more deep calculated, obtain nonsingular accurately displacement solution, and can more fully reflect the rugosity characteristic in back after introducing the displacement disturbance, for follow-up forecast of fold configuration and control provide information more accurately.

Embodiment two, present embodiment be to embodiment one described when revising unusual displacement component and eliminate fold and calculate the further specifying of the step 1 in the method for stiffness matrix singularity, introducing corrected parameter correction singular stiffness matrix described in the step 1, the process of obtaining nonsingular stiffness matrix form is specially:

The column vector form of the first rank eigenvector is in the space of feature vectors:

φ＝(1，0，0，…，0) ^T

To be the 1st be 1 to the first rank eigenvector φ in the described space of feature vectors, and all the other are 0 column vector,

Introduce corrected parameter β＞0, and obtain revised nonsingular stiffness matrix in conjunction with the first rank eigenvector φ in the space of feature vectors

Form is:

$K_T^{*}=K_T+K_{\mathrm{\φ}}=K_T+\mathrm{\β\φ}{\mathrm{\φ}}^T$

Wherein, K _TBe singular stiffness matrix, K _φBe the correction matrix of introducing, φ ^TTransposition for the first rank eigenvector φ in the space of feature vectors.

Embodiment three, present embodiment be to embodiment one or two described when revising unusual displacement component and eliminate fold and calculate the further specifying of the step 2 in the method for stiffness matrix singularity, obtaining the process that real displacement separates in conjunction with nonsingular stiffness matrix form and be specially described in the step 2:

Supposing that real displacement is separated is u, with itself and revised nonsingular stiffness matrix

Multiply each other, obtain:

$K_T^{*}u=K_{T}u+\mathrm{\β}\left({\mathrm{\φ}}^{T}u\right)\mathrm{\φ}=q+\mathrm{\β}\left({\mathrm{\φ}}^{T}u\right)\mathrm{\φ}$

Wherein, q is the load of corresponding tangent stiffness matrix,

At this moment

No unusual, obtain real displacement and separate u:

$u=u_q+\frac{{\mathrm{\β\φ}}^T{u}_q}{1-{\mathrm{\β\φ}}^T{u}_{\mathrm{\φ}}}{u}_{\mathrm{\φ}}$

Wherein, $u_{\mathrm{\φ}}={\left(K_T^{*}\right)}^{-1}\mathrm{\φ},$ $u_q={\left(K_T^{*}\right)}^{-1}q.$

Embodiment four, present embodiment be to embodiment one, two or three described when revising unusual displacement component and eliminate fold and calculate the further specifying of the step 3 in the method for stiffness matrix singularity, the singularity that judgement real displacement described in the step 3 is separated, and the process of obtaining unusual displacement component is specially:

When true displacement solution:

Second displacement component in 1-β φ ^Tu _φ=0 o'clock, presence bit was transfered from one place to another under escort unusual, and then unusual displacement component is

Embodiment five, present embodiment be to embodiment one, two, three or four described when revising unusual displacement component and eliminate fold and calculate the further specifying of the step 4 in the method for stiffness matrix singularity, described in the step 4 introduce rugosity mode equation in rugosity critical point place, relational equation and the unusual displacement solution existence condition equation of the load of rugosity critical point place is rugosity mode correspondence after eliminating be respectively:

Rugosity mode equation:

u _φ＝φ；

Relational equation after the load of mode correspondence that rugosity critical point place is rugosity is eliminated:

$u^T{u}_{\mathrm{\φ}}=u_q^T{u}_{\mathrm{\φ}}=0;$

Unusual displacement solution existence condition equation:

${\mathrm{\φ}}^T{u}_q={\left(K_T^{*}{u}_{\mathrm{\φ}}\right)}^T{u}_q=u_{\mathrm{\φ}}^{T}q=0.$

The influence of rugosity mode to the rugosity characteristic in back can accurately be introduced and in time eliminated to the singularity that relational equation after the load of mode correspondence that present embodiment is utilized rugosity mode equation that rugosity critical point place introduces, rugosity critical point place is rugosity is eliminated and unusual displacement solution existence condition equation are eliminated displacement component.

The described rugosity mode equation of present embodiment shows: rugosity mode is the first rank mode;

Relational equation after the load of the rugosity mode correspondence of described rugosity critical point place of present embodiment is eliminated shows: rugosity critical point place is the load corresponding with rugosity mode not, and promptly the rugosity mode of Yin Ruing must be eliminated after fold produces immediately;

The described unusual displacement solution existence condition equation of present embodiment shows: unusual displacement existence of solution condition.

Embodiment six, present embodiment be to embodiment one, two, three, four or five described when revising unusual displacement component and eliminate fold and calculate the further specifying of the step 4 in the method for stiffness matrix singularity, described in the step 4 utilize rugosity mode equation, relational equation and the unusual displacement solution existence condition equation correction unusual displacement component of the load of rugosity critical point place is rugosity mode correspondence after eliminating, the process that obtains nonsingular displacement solution is specially:

Make that unusual displacement component parameter is C, then

Wherein, C is a constant,

It is updated to real displacement separates

In, can get u-u _q=Cu _φ,

Put in order after passing through transposition and introducing rugosity mode and obtain:

$C=\frac{u^T{u}_{\mathrm{\φ}}-u_q^T{u}_{\mathrm{\φ}}}{u_{\mathrm{\φ}}^T{u}_{\mathrm{\φ}}},$

Utilize described rugosity mode equation u _φ=φ ₁, rugosity critical point place is rugosity mode correspondence the relational equation of load after eliminating

And unusual displacement solution existence condition equation

Can obtain:

$C=\frac{\mathrm{\β}{\mathrm{\φ}}^T{u}_q}{1-\mathrm{\β}{\mathrm{\φ}}^T{u}_{\mathrm{\φ}}}=\frac{-u_q^T{u}_{\mathrm{\φ}}}{u_{\mathrm{\φ}}^T{u}_{\mathrm{\φ}}},$

And then, obtain nonsingular displacement solution u ^*For:

The unusual problem of stiffness matrix when present embodiment solves the calculating of elimination fold by revising unusual displacement component.

Claims (6)

1. the method for stiffness matrix singularity when eliminating fold calculating by the unusual displacement component of correction, it is characterized in that: the process of described method is:

Step 1: in rugosity space of feature vectors, introduce corrected parameter correction singular stiffness matrix, obtain nonsingular stiffness matrix form;

Step 2: obtain real displacement in conjunction with nonsingular stiffness matrix form and separate;

Step 3: judge the singularity that real displacement is separated, and obtain unusual displacement component;

Step 4: introduce rugosity mode equation in rugosity critical point place, relational equation and the unusual displacement solution existence condition equation of the load of rugosity critical point place is rugosity mode correspondence after eliminating, and utilize described three unusual displacement components of equation correction, obtain nonsingular displacement solution, the singularity of stiffness matrix when calculating by revising unusual displacement solution and then eliminating gauffer.

2. according to claim 1 when revising unusual displacement component and eliminate fold and calculate the method for stiffness matrix singularity, it is characterized in that: the introducing corrected parameter correction singular stiffness matrix described in the step 1, the process of obtaining nonsingular stiffness matrix form is specially:

The column vector form of the first rank eigenvector is in the space of feature vectors:

φ＝(1，0，0，…，0) ^T

To be the 1st be 1 to the first rank eigenvector φ in the described space of feature vectors, and all the other are 0 column vector,

Introduce corrected parameter β＞0, and obtain revised nonsingular stiffness matrix in conjunction with the first rank eigenvector φ in the space of feature vectors

Form is:

$K_T^{*}=K_T+K_{\mathrm{\φ}}=K_T+{\mathrm{\β\φ\φ}}^T$

Wherein, K _TBe singular stiffness matrix, K _φBe the correction matrix of introducing, φ ^TTransposition for the first rank eigenvector φ in the space of feature vectors.

3. according to claim 1 when revising unusual displacement component and eliminate fold and calculate the method for stiffness matrix singularity, it is characterized in that: obtaining the process that real displacement separates in conjunction with nonsingular stiffness matrix form and be specially described in the step 2:

Supposing that real displacement is separated is u, with itself and revised nonsingular stiffness matrix Multiply each other, obtain:

$K_T^{*}u=K_{T}u+\mathrm{\β}\left({\mathrm{\φ}}^{T}u\right)\mathrm{\φ}=q+\mathrm{\β}\left({\mathrm{\φ}}^{T}u\right)\mathrm{\φ}$

Wherein, q is the load of corresponding tangent stiffness matrix,

At this moment

No unusual, obtain real displacement and separate u:

$u=u_q+\frac{{\mathrm{\β\φ}}^T{u}_q}{1-{\mathrm{\β\φ}}^T{u}_{\mathrm{\φ}}}{u}_{\mathrm{\φ}}$

Wherein, $u_{\mathrm{\φ}}={\left(K_T^{*}\right)}^{-1}\mathrm{\φ},$ $u_q={\left(K_T^{*}\right)}^{-1}q.$

4. according to claim 1 when revising unusual displacement component and eliminate fold and calculate the method for stiffness matrix singularity, it is characterized in that: the singularity that the judgement real displacement described in the step 3 is separated, and the process of obtaining unusual displacement component is specially:

When true displacement solution:

Second displacement component in 1-β φ ^Tu _φ=0 o'clock, presence bit was transfered from one place to another under escort unusual, and then unusual displacement component is

5. according to claim 1 when revising unusual displacement component and eliminate fold and calculate the method for stiffness matrix singularity, it is characterized in that: described in the step 4 introduce rugosity mode equation in rugosity critical point place, relational equation and the unusual displacement solution existence condition equation of the load of rugosity critical point place is rugosity mode correspondence after eliminating be respectively:

Rugosity mode equation:

u _φ＝φ；

Relational equation after the load of mode correspondence that rugosity critical point place is rugosity is eliminated:

$u^T{u}_{\mathrm{\φ}}=u_q^T{u}_{\mathrm{\φ}}=0;$

Unusual displacement solution existence condition equation:

${\mathrm{\φ}}^T{u}_q={\left(K_T^{*}{u}_{\mathrm{\φ}}\right)}^T{u}_q=u_{\mathrm{\φ}}^{T}q=0.$

6. according to claim 5 when revising unusual displacement component and eliminate fold and calculate the method for stiffness matrix singularity, it is characterized in that: described in the step 4 utilize rugosity mode equation, relational equation and the unusual displacement solution existence condition equation correction unusual displacement component of the load of rugosity critical point place is rugosity mode correspondence after eliminating, the process that obtains nonsingular displacement solution is specially:

Make that unusual displacement component parameter is C, then Wherein, C is a constant,

It is updated to real displacement separates

In, can get u-u _q=Cu _φ,

Put in order after passing through transposition and introducing rugosity mode and obtain:

$C=\frac{u^T{u}_{\mathrm{\φ}}-u_q^T{u}_{\mathrm{\φ}}}{u_{\mathrm{\φ}}^T{u}_{\mathrm{\φ}}},$

Utilize described rugosity mode equation u _φ=φ ₁, rugosity critical point place is rugosity mode correspondence the relational equation of load after eliminating

And unusual displacement solution existence condition equation

Can obtain:

$C=\frac{\mathrm{\β}{\mathrm{\φ}}^T{u}_q}{1-\mathrm{\β}{\mathrm{\φ}}^T{u}_{\mathrm{\φ}}}=\frac{-u_q^T{u}_{\mathrm{\φ}}}{u_{\mathrm{\φ}}^T{u}_{\mathrm{\φ}}},$

And then, obtain nonsingular displacement solution u ^*For:

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