CN102508978B - Method for judging movability of symmetric rod system structure based on group theory - Google Patents

Abstract

The invention discloses a method for judging the movability of a symmetric rod system structure based on a group theory, belonging to the field of building structure designs and space structural designs. The method is particularly suitable for the initial design of a deployable structure. According to the method, a degree of freedom formula of the symmetric rod system structure is simplified to various types of symmetric linear expressions based on inherent irreduable representations and characteristic standard values of a symmetric group; and the symmetric property of a mechanism displacement mode and a self-stress mode is obtained, so that the movability of the symmetric structure can be directly judged. Compared with the conventional movability judgment method in which the symmetric property of the structure is not used fully and the analysis calculation for large structures is complicated, the method has the advantages that: the movability of most symmetric rod systems can be judged by only carrying out simple vector calculation and simplification, and a structure designer can popularize and apply the method conveniently.

Description

A kind of criteria for mobility method of the symmetrical member structure based on the group theory

Technical field

The present invention is a kind of method that is applied to Architectural Structure Design and space structure design, particularly relates to a kind of criteria for mobility method of the symmetrical member structure based on the group theory.

Background technology

In the last thirty years, deployable structure is applied to the fields such as building, machinery, aerospace engineering gradually, and this class formation has movable degree of freedom, thereby can present the plurality of stable configuration.The criteria for mobility of structure, can the transmitting single order rigidity from stress mode and make construction geometry stable of structure, be an important step of research and design deployable structure.The geometrical stability decision method that Calladine and Pellegrino propose, can effectively be applied to the conventional member structure lower from the stress modulus; Subsequently, Luo Yaozhi and Lu Jinyu have considered the stable problem of structure under the load action, further perfect this decision criteria.But, along with structure is complicated gradually, the increasing from the stress modulus of structure, the solution efficiency of this decision criteria significantly descends, and even can't judge.Although there is the scholar to adopt optimization methods such as considering rod member type, joints characteristics, improve the efficiency of construction geometry determination of stability, all ignored the intrinsic symmetric properties of structure.Symmetrical member structure has a plurality of rotations, mirror image symmetry operation, adopts the symmetry analysis method based on trooping theoretical can take full advantage of structural symmetry, avoids the complex calculation of conventional decision method.

In addition, conventional structural stability decision method lays particular emphasis on the structure of checking geometrical stability more at present, in tension integral structure and other tension force systems, has significant application value.In the design analysis process of deployable structure, probe into structure while being whether how much movable (unstable) with conventional decision method, still lack enough cogencyes.

Summary of the invention

Technical matters:

The purpose of this invention is to provide a kind of criteria for mobility method based on the theoretical symmetrical member structure of trooping, emphasis solves the geometrical stability problem of moving indeterminate structure in building structure and space structure, is particularly useful for the primary design analysis of symmetrical deployable structure.While in view of conventional criteria for mobility method, probing into the member structure higher from stress mode, the computation process complexity, and be difficult to verify the movable performance of structure, key technical problem of the present invention is mobility how to determine efficiently and accurately structure.

Technical scheme:

For above problem, the present invention is based on the intrinsic irreducible representation of symmetric group and feature scale value, the linear expression that is all kinds of symmetries by the degree of freedom formula yojan of symmetrical member structure, and learn that Mechanism displacement model reaches the symmetric properties from stress mode, thereby the mobility of symmetrical structure is directly judged.Technical scheme is as follows:

A kind of criteria for mobility method of the symmetrical member structure based on the group theory,

Step 1 is determined the affiliated symmetric group of symmetrical member structure to be determined, obtains the Mechanism displacement model of symmetrical member structure to be determined and, from stress mode, described symmetric group comprises: symmetry operation, and irreducible representation and character,

Step 2 is calculated the Degree of Structure Freedom under each symmetry operation, then the Degree of Structure Freedom under each symmetry operation is formed to the degree of freedom vector, degree of freedom vector Г _m-sfor:

Г _m-s＝M-S＝JT-K-η(T+R)，

Wherein vector M and vectorial S are respectively mechanism displacement moduluses and from the symmetric formulation of stress modulus, vector T and vectorial R are respectively the symmetric formulations of rigid body translation displacement modulus and Rigid Body in Rotation With displacement modulus, vector J and vectorial K are respectively the symmetric formulations of node and the unit of symmetrical member structure to be determined, and the character based on affiliated symmetric group, by the degree of freedom vector Г tried to achieve _m-sthe linear combination that yojan is all kinds of irreducible representations,

$M-S=\underset{i=1}{\overset{\mathrm{\μ}}{\mathrm{\Σ}}}{\mathrm{\α}}_i{\mathrm{\Γ}}^{\left(i\right)},$

Г ⁽ⁱ⁾for the i class irreducible representation of symmetric group under structure, total number of types that μ is affiliated symmetrical irreducible representation of a group, the factor alpha of i class irreducible representation _ifor

${\mathrm{\α}}_i=\frac{l_i}{h}\·\underset{\mathrm{\γ}=1}{\overset{h}{\mathrm{\Σ}}}{\mathrm{\χ}}^{\left(i\right)}\left(\mathrm{\γ}\right)\·{\mathrm{\Γ}}_{m-s}\left(\mathrm{\γ}\right)$

Wherein, l _ii class irreducible representation Г for affiliated symmetric group ⁽ⁱ⁾number, the sum that h is different symmetry operations in affiliated symmetric group, χ ⁽ⁱ⁾(γ) be the character of i class irreducible representation under the γ class symmetry operation of affiliated symmetric group, Г _m-s(γ) be the Degree of Structure Freedom under the γ class symmetry operation of affiliated symmetric group,

Step 3 is worked as α _i>0 o'clock,

the Mechanism displacement model of symmetrical member structure to be determined has the corresponding symmetric properties of i class irreducible representation of affiliated symmetric group; Work as α _i<0 o'clock,

symmetrical member structure to be determined there is the corresponding symmetric properties of i class irreducible representation of affiliated symmetric group from stress mode, and then determine the Mechanism displacement model of symmetrical member structure to be determined and from the high order symmetry attribute of stress mode,

If the high order symmetry attribute of step 4 Mechanism displacement model is complete symmetry, high order symmetry attribute from stress mode is non-complete symmetry, the movement tendency that Mechanism displacement model that can't balanced structure from stress mode produces, can't transmit single order rigidity, structure is movable, and decision process finishes;

If Mechanism displacement model and be non-complete symmetry from the high order symmetry attribute of stress mode, enter step 5;

Step 5 is according to the high order symmetry attribute of the Mechanism displacement model of symmetrical member structure to be determined, affiliated symmetric group is carried out to depression of order, obtain the affiliated symmetric group of new symmetrical member structure to be determined, repeating step 2,3 again, obtain the Mechanism displacement model of symmetrical member structure to be determined and from the high order symmetry attribute of stress mode, and enter step 6

Affiliated symmetric group is carried out to depression of order and adopt following method: make symmetrical member structure to be determined, along the mechanism displacement path of described high order symmetry attribute, small deformation occur, the sub-symmetric group that the affiliated symmetric group of the symmetrical member structure to be determined after the generation small deformation is symmetric group under original structure

If the high order symmetry attribute of step 6 Mechanism displacement model is complete symmetry, high order symmetry attribute from stress mode is non-complete symmetry, the movement tendency that Mechanism displacement model that can't balanced structure from stress mode produces, can't transmit single order rigidity, and structure is movable.

Beneficial effect:

The invention has the advantages that the intrinsic symmetric properties that takes full advantage of structure, without the complex calculation by conventional decision method, and only carry out simple vector operation and yojan, can complete to the symmetrical leverage of major part the criteria for mobility of structure.When the mechanism displacement modulus of structure or larger from the stress modulus, or structure is while being high order symmetry, and counting yield of the present invention improves significantly.

In addition, because decision method disclosed by the invention only need be considered unit topology information and the node relative position of structure to be highly suitable for the primary design analysis of deployable structure, and avoided structure to thering is identical symmetric properties and topology to be repeated to judge and calculated.

The accompanying drawing explanation

Fig. 1 is for carrying out the process flow diagram of criteria for mobility based on the theory of trooping.

Fig. 2 is C _4vthe three-dimensional plot of the symmetrical member structure of type.

Fig. 3 is C _4vthe planimetric map of the symmetrical member structure of type.

Embodiment

The invention discloses a kind of criteria for mobility method based on the theoretical symmetrical member structure of trooping.Symmetrical member structure is comprised of a series of rod members that are hinged, and node, unit and boundary constraint exist a plurality of rotations and mirror image symmetry operation about the structure centre point.The concrete steps of decision method are as follows:

1) analyze and prepare

The geometric model information of structure clearly to be determined, comprise coordinate and the numbering thereof of each node of structure, the topological mode of each unit (being the node serial number that each two ends, unit connect), and the boundary condition of structure.And the symmetry operation had according to structure, determine the symmetric group type that structure to be determined is affiliated.Symmetry operation refer to original structure be rotated, mirror image etc. is effectively after operational transformation, structural configuration does not still change.When structure only has n effective Rotational Symmetry operation, under structure, symmetric group is C _nsymmetric group; When structure has n Rotational Symmetry operation and has n mirror image symmetry operation, under structure, symmetric group is C _nvsymmetric group.According to the theoretical reference books of trooping (Point-Group Theory Tables), can consult and obtain the described symmetrical irreducible representation of a group of structure and character.

2) set up the body force balancing matrix of member structure, the Mechanism displacement model of computation structure reaches from stress mode

If total j the free node of structure and k unit, t is the rigid body translation displacement modulus (plane leverage t=2, space bar t=3) under structure coordinate system of living in.By the internal force of each unit and the equilibrium relation between the node external load, by conventional matrix integrated approach, can obtain integrally-built equilibrium equation:

HP＝F (1)

In formula, H is integrally-built dynamic balance matrix, the element force vector that P is structure, and F is node external load vector.The kernel of dynamic balance matrix H be structure from stress mode, and from stress modulus s be:

s＝k-r (2)

In formula, the order that r is the dynamic balance matrix H.According to the principle of virtual work, the displacement coordination matrix equals the transposition of dynamic balance matrix H, and the kernel of the displacement coordination matrix Mechanism displacement model that is structure, and mechanism displacement modulus m is:

m＝jt-r-η(t+r) (3)

In formula, η is Boundary Condition Effect coefficient (η when structure is not subject to any displacement constraint=1, otherwise η=0), and r is the Rigid Body in Rotation With displacement modulus (plane leverage r=1, space bar r=3) under structure coordinate system of living in.

3) calculate the Degree of Structure Freedom under each symmetry operation, and based on character, the linear combination that is irreducible representation by the vectorial yojan of the degree of freedom of trying to achieve

Above formula (3) and formula (2) are subtracted each other, are the freedom calculation formula of structure,

m-s＝jt-k-η(t+r) (4)

Based on the theory of trooping, calculate the Degree of Structure Freedom under each symmetry operation, and the Degree of Structure Freedom under each symmetry operation is formed to the degree of freedom vector, obtain the symmetric formulation Г of the Degree of Structure Freedom _m-sfor:

Г _ms＝M-S＝JT-K-η(T+R) (5)

In formula, vector M and S are respectively mechanism displacement moduluses and from the symmetric formulation of stress modulus; Vector T and R are respectively the symmetric formulations of rigid body translation displacement modulus and Rigid Body in Rotation With displacement modulus; Vector J and K are respectively the symmetric formulations of node and unit, and wherein arbitrary element J (γ), K (γ) are the motionless node of structure under γ class symmetry operation in affiliated symmetric group, the quantity of moving cell not.According to affiliated symmetrical irreducible representation of a group and character, but the linear combination that the Degree of Structure Freedom vector M-the S yojan is all kinds of irreducible representations:

$M-S=\underset{i=1}{\overset{\mathrm{\&mu;}}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_i{\mathrm{\&Gamma;}}^{\left(i\right)}---\left(6\right)$

Г in formula ⁽ⁱ⁾for the i class irreducible representation of symmetric group under structure, total number of types that μ is irreducible representation, factor alpha _ifor

${\mathrm{\&alpha;}}_i=\frac{l_i}{h}\&CenterDot;\underset{\mathrm{\&gamma;}=1}{\overset{h}{\mathrm{\&Sigma;}}}{\mathrm{\&chi;}}^{\left(i\right)}\left(\mathrm{\&gamma;}\right)\&CenterDot;{\mathrm{\&Gamma;}}_{m-s}\left(\mathrm{\&gamma;}\right)---\left(7\right)$

Wherein, l _ibe the number of i class irreducible representation, the sum that h is different symmetry operations in symmetric group, χ ⁽ⁱ⁾(γ) be the character of i class irreducible representation under γ class symmetry operation in trooping, Г _m-s(γ) be the degree of freedom of structure under γ class symmetry operation.

4) solve respectively the structural mechanism displacement modes and from the high order symmetry attribute of stress mode

Based on the 2nd), 3) the step calculated results, the mechanism displacement modulus of structure and all non-negative, known from the stress modulus, the symmetric properties that can judge respectively Mechanism displacement model and comprise from stress mode by the Degree of Structure Freedom vector M-S:

(1) work as α _i>0 o'clock,

mechanism displacement model has the corresponding symmetric properties of i class irreducible representation;

(2) work as α _i<0 o'clock, there is the corresponding symmetric properties of i class irreducible representation from stress mode.

Finally, all symmetric properties that have according to Mechanism displacement model, determine its higher order term (corresponding Mechanism displacement model has the symmetry operation of maximum kinds); In like manner, obtain the high order symmetry attribute from stress mode.

5) structure criteria for mobility

Judge the criteria for mobility condition for first: the high order symmetry attribute of Mechanism displacement model is complete symmetry, from the high order symmetry attribute of stress mode, is non-complete symmetry;

The movement tendency that Mechanism displacement model that now can't balanced structure from stress mode produces, can't transmit single order rigidity, and structure is movable, and decision process finishes;

Judge the criteria for mobility condition for second: Mechanism displacement model and be non-complete symmetry from the high order symmetry attribute of stress mode;

Now can not directly judge, need further in the low order symmetric group, to probe into the mobility of structure, enter the 6th) step;

If structure to be determined exists serious asymmetric initial imperfection or Mechanism displacement model and when the high order symmetry attribute of stress mode is the symmetric properties of complete symmetry or structure depression of order has been lowest-order, can't be by the mobility of the direct decision structure of decision method disclosed by the invention, decision process finishes, and needs to adopt the further decision analysis of conventional decision method.

6) symmetric group under structure is carried out to depression of order

When the high order symmetry attribute of the Mechanism displacement model of structure is non-complete symmetry, putative structure is along the mechanism displacement generation infinitesimal deflection with this non-complete symmetry attribute, and structure will produce new configuration.Now, under original structure, the part symmetry operation of symmetric group is no longer set up, and new construction only has the part symmetric properties that original structure has.Therefore, the sub-symmetric group that under the new construction configuration, symmetric group is symmetric group under original structure.High order symmetry attribute according to Mechanism displacement model, carry out depression of order to the symmetric group under original structure, obtains the symmetric group under the new construction configuration.Can consult and obtain according to the subgroup of each symmetric group in the theoretical reference books of trooping (Point-Group Theory Tables) and corresponding depression of order mode.

For above-mentioned steps more clearly is described, drawn symmetrical member structure criteria for mobility techniqueflow chart, as shown in Figure 1.Now with a C _4vsymmetrical member structure is the embodiment that example is set forth this method, as follows:

The symmetrical member structure of this three-dimensional (Fig. 2) has 8 nodes and 12 root units, and wherein node 1-4 is boundary node, is subject to x, y, and z is to displacement constraint, and table 1 has been listed the coordinate (dimensionless number) of each node.

Each node coordinate of table 1

Because structural configuration is being rotated respectively 0.5 π counterclockwise around the z axle, π, (be the Rotational Symmetry operation under 1.5 π and 2 π operation

), and plane mirror image operation σ ₁, σ ₂, σ ₃, σ ₄under remain unchanged, known this structure has four class Rotational Symmetries operations and four class mirror image symmetry operations (n=4), the affiliated symmetric group of structure is C _4vthe type symmetric group, as shown in Figure 3.

The dynamic balance rank of matrix of trying to achieve as calculated structure is 11, and it is m=1 from the stress modulus that the mechanism displacement modulus reaches, s=1.Obtained the symmetric formulation M-S=[0 of the Degree of Structure Freedom by formula (5), 0,0 ,-2,2], yojan is

M-S＝B ₂-B ₁ (8)

And known m=s=1, so the symmetric formulation that the mechanism displacement modulus of structure reaches from the stress modulus is respectively:

M＝B ₂，S＝B ₁ (9)

This Mechanism displacement model M=B ₂for non-complete symmetry, now can not directly judge that whether this structure is movable, putative structure is along having B ₂the mechanism displacement generation infinitesimal deflection of symmetric properties, according to the depression of order mode of symmetric group, under structure, symmetric group is by C _4vdepression of order is C _2v.Further at low order symmetric group C _2vin probe into Mechanism displacement model symmetric properties, repeating step 3) and 4), obtain

M＝A ₁，S＝A ₂ (10)

The highest symmetric properties A due to Mechanism displacement model ₁for complete symmetry, and from the highest symmetric properties A of stress mode ₂for non-complete symmetry, the movement tendency can't balanced structure had from stress mode, so this symmetry member structure is movable.

Claims (1)

1. the criteria for mobility method of the symmetrical member structure based on the group theory, is characterized in that,

Step 1 is determined the affiliated symmetric group of symmetrical member structure to be determined, obtains the Mechanism displacement model of symmetrical member structure to be determined and, from stress mode, described symmetric group comprises: symmetry operation, and irreducible representation and character,

Step 2 is calculated the Degree of Structure Freedom under each symmetry operation, then the Degree of Structure Freedom under each symmetry operation is formed to the degree of freedom vector, degree of freedom vector Γ _m-sfor:

Γ _m-s＝M-S＝JT-K-η(T+R)，

Wherein vector M and vectorial S are respectively mechanism displacement moduluses and from the symmetric formulation of stress modulus, vector T and vectorial R are respectively the symmetric formulations of rigid body translation displacement modulus and Rigid Body in Rotation With displacement modulus, vector J and vectorial K are respectively the symmetric formulations of node and the unit of symmetrical member structure to be determined, η is the Boundary Condition Effect coefficient, η=1 when symmetrical member structure to be determined is not subject to any displacement constraint, otherwise η=0, and the character based on affiliated symmetric group, by the degree of freedom vector Γ tried to achieve _m-sthe linear combination that yojan is all kinds of irreducible representations,

$M-S=\underset{i=1}{\overset{\mathrm{\&mu;}}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_i{\mathrm{\&Gamma;}}^{\left(i\right)},$

Γ ⁽ⁱ⁾for the i class irreducible representation of symmetric group under structure, total number of types that μ is affiliated symmetrical irreducible representation of a group, the factor alpha of i class irreducible representation _ifor

${\mathrm{\&alpha;}}_i=\frac{l_i}{h}\&CenterDot;\underset{\mathrm{\&gamma;}=1}{\overset{h}{\mathrm{\&Sigma;}}}{\mathrm{\&chi;}}^{\left(i\right)}\left(\mathrm{\&gamma;}\right)\&CenterDot;{\mathrm{\&Gamma;}}_{m-s}\left(\mathrm{\&gamma;}\right)$

Wherein, l _ii class irreducible representation Γ for affiliated symmetric group ⁽ⁱ⁾number, the sum that h is different symmetry operations in affiliated symmetric group, χ ⁽ⁱ⁾(γ) be the character of i class irreducible representation under the γ class symmetry operation of affiliated symmetric group, Γ _m-s(γ) be the Degree of Structure Freedom under the γ class symmetry operation of affiliated symmetric group,

Step 3 is worked as α _i>0 o'clock, the Mechanism displacement model of symmetrical member structure to be determined has the corresponding symmetric properties of i class irreducible representation of affiliated symmetric group; Work as α _i<0 o'clock,

symmetrical member structure to be determined there is the corresponding symmetric properties of i class irreducible representation of affiliated symmetric group from stress mode, and then determine the Mechanism displacement model of symmetrical member structure to be determined and from the high order symmetry attribute of stress mode,

If the high order symmetry attribute of step 4 Mechanism displacement model is complete symmetry, high order symmetry attribute from stress mode is non-complete symmetry, the movement tendency that Mechanism displacement model that can't balanced structure from stress mode produces, can't transmit single order rigidity, structure is movable, and decision process finishes;

If Mechanism displacement model and be non-complete symmetry from the high order symmetry attribute of stress mode, enter step 5;

Step 5 is according to the high order symmetry attribute of the Mechanism displacement model of symmetrical member structure to be determined, affiliated symmetric group is carried out to depression of order, obtain the affiliated symmetric group of new symmetrical member structure to be determined, repeating step 2,3 again, obtain the Mechanism displacement model of symmetrical member structure to be determined and from the high order symmetry attribute of stress mode, and enter step 6

Affiliated symmetric group is carried out to depression of order and adopt following method: make symmetrical member structure to be determined, along the mechanism displacement path of described high order symmetry attribute, small deformation occur, the sub-symmetric group that the affiliated symmetric group of the symmetrical member structure to be determined after the generation small deformation is symmetric group under original structure

If the high order symmetry attribute of step 6 Mechanism displacement model is complete symmetry, high order symmetry attribute from stress mode is non-complete symmetry, the movement tendency that Mechanism displacement model that can't balanced structure from stress mode produces, can't transmit single order rigidity, and structure is movable.

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