CN108256251A - A kind of symmetrical cable-rod structure of height for considering datum mark mechanical balance looks for shape method - Google Patents
A kind of symmetrical cable-rod structure of height for considering datum mark mechanical balance looks for shape method Download PDFInfo
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- CN108256251A CN108256251A CN201810081326.XA CN201810081326A CN108256251A CN 108256251 A CN108256251 A CN 108256251A CN 201810081326 A CN201810081326 A CN 201810081326A CN 108256251 A CN108256251 A CN 108256251A
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Abstract
The invention discloses a kind of symmetrical cable-rod structures of the height for considering datum mark mechanical balance to look for shape method, for the arbitrary symmetrical cable-rod structure of height, determine the geometrical topology of structure, certain several datum mark is selected, to the Equilibrium Analysis of datum mark progress X, Y, Z-direction, according to geometrically symmetric relationship, acquire each group force in member density relationship, then selected reference point coordinates, the coordinate of remaining all node is acquired by coordinate conversion matrix, finally acquires rod length and initial prestress distribution.The present invention takes full advantage of the symmetric properties of structure and the topological relation of structure, calculates more succinctly, efficiently, suitable those skilled in the art promote the use.
Description
Technical field
The invention belongs to the fields such as space structure, the morphological analysis of symmetrical tension integral structure, design and development, are related to one
Kind considers that the symmetrical cable-rod structure of the height of datum mark mechanical balance looks for shape method.
Background technology
Rope bar tension integral structure is a kind of structure of self-balancing, during design, it is most important that determine structure
Geometry topology and initial internal force state, i.e. form finding analysis.Traditional tension integral structure looks for shape method to have force density
Method, Dynamic Relaxation and FInite Element.
For high symmetrical cable-rod structure, its force density form finding analysis mistake can significantly be simplified by making full use of the high symmetry of structure
Journey.Force density looks for shape method to look for shape method as a kind of important tension integral structure, combines on this basis to construction geometry
The symmetric coordinates system of structure is established in the analysis of topological form, by sparse, zonal distribution the correlation matrix under cartesian coordinate system
(such as stiffness matrix, balancing matrix, force density matrix) is converted into the simple matrix under symmetric coordinates system, and it is high right to significantly improve
Claim the computational efficiency of rope bar tension integral structure form finding analysis.
Invention content
Technical problem：The present invention provides one kind for high symmetrical prestressed cable-strut structures, can easily and fast, accurately count
The symmetrical cable-rod structure of height of the considerations of calculating rod piece force density relationship and corresponding node coordinate datum mark mechanical balance looks for shape side
Method.
Technical solution：The present invention for high symmetrical prestressed cable-strut structures, with force density method establish around datum mark X,
Y, the equilibrium equation of Z-direction, and the force density matrix Q around the datum mark is formed, each point is obtained according to high symmetrical structure
The transition matrix of coordinate, it is zero to make QR ranks of matrix, and analysis acquires each group rod piece force density relationship and under cartesian coordinate system
Node coordinate.
The symmetrical cable-rod structure of height of the considerations of of the invention datum mark mechanical balance looks for shape method, includes the following steps：
The geometrical topology of the clear and definite structure of step 1 is grouped structure drag-line, striker member according to symmetry, place
It it is one group in the drag-line of par, the compression bar in par is one group, and node is grouped according to symmetry, is in
The node of par is one group；
Step 2 establishes coordinate system using symmetrical configuration central point as origin, and each selected node is distinguished in each group node
As datum mark, the benchmark point coordinates is (x1, y1, z1), with force density method establish datum mark X, Y, Z-direction dynamic balance
Equation, and form the force density matrix Q of each datum mark；
Step 3 finds out remaining node in same group node in addition to datum mark relative to benchmark according to geometrically symmetric relationship
The transformational relation of point, show that remaining node is relative to the coordinate conversion matrix of the grouping datum mark in each grouping according to transformational relation
R；
Step 4 solves the product of the force density matrix Q of each datum mark and the coordinate conversion matrix R relative to each datum mark
QR, and enable gained Matrix QR order be zero, acquire the force density relationship of each group component；
The coordinate conversion matrix R relative to datum mark that step 5 is obtained by the step 3, finds out in same group node
Except datum mark remaining each node i relative to datum mark coordinate conversion matrix Ri, acquire cartesian coordinate system and similarly hereinafter organize section
The coordinate of remaining each node in point except datum mark is (xi, yi, zi)=Ri*(x1, y1, z1)；
Step 6 acquires each rod length and initial prestress distribution.
Further, in the method for the present invention, in step 1 definition of the drag-line and compression bar of par be：Structure is a certain
Under symmetry operation, a certain drag-line may switch to another drag-line position by the symmetry operation, then this two drag-lines are in
Par, a certain compression bar may switch to another compression bar position by the symmetry operation, then this two compression bars are in same
Wait status.
Further, in the method for the present invention, the node definition of par is in step 1：Structure is in a certain symmetry operation
Under, certain node may switch to the position of another node by the symmetry operation, then the two nodes are in par.
Further, in the method for the present invention, the datum mark of each group node is selected in step 2 as follows：At same group
An arbitrarily selected node chooses a datum mark per group node as datum mark in node.
Further, in the method for the present invention, the force density relationship of each group component is acquired in step 4 as follows：It enables
All elements are equal to 0 in the corresponding Matrix QR of each datum mark, obtain QR (j, k)=0, wherein QR (j, k) representing matrix QR jth
The element of row kth row, j=1,2 ..., m, m are the line number of Matrix QR, and k=1,2 ..., n, n is the columns of Matrix QR, and simultaneous owns
The corresponding QR (j, k) of datum mark=0 forms equation group, and solves equation group, obtains the force density relationship of each group rod piece.
Further, in the method for the present invention, step 5 interior joint i relative to datum mark coordinate conversion matrix RiFor step 3
In the obtained 3i rows of coordinate conversion matrix R to 3i+2 rows.
The present invention is on the basis of existing force density of analyzing and researching looks for the method for shape, with reference to the particularity of high symmetrical structure
The converting expressing mode of matter and each node under cartesian coordinate system, it is proposed that it is a kind of new the considerations of datum mark mechanical balance
The symmetrical cable-rod structure of height look for shape method, enormously simplify the form finding analysis to high symmetrical prestressed cable bar.
The method of the present invention selectes a certain datum mark of structure, study its X, Y, Z-direction power balance, and establish remaining section
The transition matrix relative to datum mark is put, analysis acquires the force density relationship of different type rod piece and the coordinate of each node.
Advantageous effect：Compared with prior art, the present invention it has the following advantages：
The advantage of the invention is that structural symmetry and geometric topo-relationship are taken full advantage of, from the mechanics of single feature point
Balance is set out, and force density fundamental equation is simplified, and directly analyzes each nodal force density relationship matrix and coordinate conversion matrix
Be multiplied gained matrix.The force density matrix that existing method acquires for n tie up square formation, n be structure in free node number, calculation amount compared with
Greatly.When existing force density method is used for tensegrity structures, since rod piece force density value may be negative, it is impossible to ensure force density matrix
Nonsingularity, will appear when solving force density fundamental equation without solution, the situation of infinite multiresolution, can cause can not ask solution or
Shape need to be looked for again by adding obligatory point.So computational efficiency is relatively low.The method of the invention is first by each node force density
Equilibrium relation writes out force density matrix；Then according to geometrically symmetric relationship, find out remaining node and closed relative to the conversion of datum mark
System, obtains corresponding coordinate conversion matrix.Then the expression matrix of the mechanical balance of single feature point is obtained, obtains a three-dimensional
With the relevant matrix of force density.The matrix is analyzed, it is zero to enable its order, can acquire each group force in member density relationship.Most successor takes
Benchmark point coordinates acquires remaining node coordinate under cartesian coordinate system by coordinate conversion matrix.Any structure is only needed
A three-dimensional matrice is analyzed, greatly reduces calculation amount, significantly improves the computational efficiency of high symmetrical cable-rod structure form finding analysis,
Convenient for this field, engineers and technicians promote the use of.
Description of the drawings
Fig. 1 is to consider that the symmetrical cable-rod structure of height of datum mark mechanical balance looks for the flow of shape method to illustrate.
Fig. 2 is a D4Symmetrical tension integral structure.
Fig. 3 is D4Symmetrical tension integral structure symmetry operation signal.
In Fig. 2, fine line represents drag-line component, and heavy line represents striker member.The connection of first level drag-line 1 first
Node A and second node B, the second jump stay 2 connection second node B and third node C, third jump stay 3 connect third
Node C and fourth node D, the 4th jump stay 4 connection first node A and fourth node D, the 5th jump stay 5 connection the 5th
Node E and the 6th node F, the 6th jump stay 6 connection the 6th node F and the 7th node G, the 7th jump stay 7 connection the 7th
Node G and the 8th node H, the 8th jump stay 8 connection the 5th node E and the 8th node H, the first vertical connection of drag-line 9 first
Node A and the 5th node E, the second vertical drag-line 10 connect second node B and the 6th node F, the vertical connection of drag-line 11 of third the
Three node C and the 7th node G, the 4th vertical drag-line 12 connect fourth node D and the 8th node H, the first compression bar 13 connection first
Node A and the 8th node H, the second compression bar 14 connection second node B and the 5th node E, third compression bar 15 connect third node C and
6th node F, the 4th compression bar 16 connection fourth node D and the 7th node G.
In Fig. 3, fine line represents drag-line component, and heavy line represents striker member, and dotted line represents double rotary shaft.The
One double rotary shaft a passes through construction geometry central point and the midpoint of second node B and the 7th node G lines, the second double rotation
Axis b passes through construction geometry central point and the vertical drag-line midpoint of third, and the double rotary shaft c of third is by construction geometry central point and the
The midpoint of three node C and the 8th node H lines, the 4th double rotary shaft d pass through construction geometry central point and the 4th vertical drag-line
Midpoint.
Specific embodiment
With reference to embodiment and Figure of description, the present invention is further illustrated.
There are 8 nodes in structure, rod piece according to symmetry, is divided into three groups, jump stay, vertical drag-line by 16 rod pieces
And compression bar, q is used respectivelyc1、qc2、qsRepresent the force density of jump stay, vertical drag-line and compression bar, all nodes are in comparably
Position, all nodes belong to same group node.
Coordinate system is established using symmetrical configuration center as origin, point on the basis of first node A is taken, with force density method to first
Node A carries out the mechanical balance analysis of X, Y, Z-direction：
(2qc1+qc2+qs)×x1-qc1×x2-qc1×x4-qc2×x5-qs×x8=0；
(2qc1+qc2+qs)×y1-qc1×y2-qc1×y4-qc2×y5-qs×y8=0；
(2qc1+qc2+qs)×z1-qc1×z2-qc1×z4-qc2×z5-qs×z8=0.
Above three equation is expressed as with matrix form：
Wherein
Q=[Q1, Q2, Q3, Q4, Q5, Q6, Q7, Q8]；
Q3=Q6=Q7=0.
Structure as shown in Figure 2 has eight symmetry operations：Identical transformation (R1), it is rotated by 90 ° (R about the z axis2)、180°(R3)、
270°(R4), rotate 180 ° of (R around the first double rotary shaft a5), around the second double rotary shaft b rotate 180 ° of (R6), around the three or two
Respin shaft c rotates 180 ° of (R7), around the 4th double rotary shaft d rotate 180 ° of (R8), then each node coordinate can convert square by coordinate
Battle array is obtained with benchmark point coordinates, i.e. (xi, yi, zi)=Ri(x1, y1, z1), wherein i=1,2,3,4,5,6,7,8；
Wherein
It enables
By transformation matrix R,Abbreviation isQR matrixes are studied, because with any one in group node
Point can serve as datum mark, and equation to be made to set up the coordinate of any one node in structure, that is, need to meet gained
QR ranks of matrix are 0.All elements in QR matrixes is enabled to be equal to 0, the parsing that can be calculated between each unit force density is closed
System：
The coordinate for taking datum mark is (x1, y1, z1), remaining node under cartesian coordinate system can be obtained by coordinate conversion matrix
Coordinate, acquire the length of each rod piece, the product of force density and length is the structural initial pre stress of the high symmetry cable-rod structure
Distribution.
Therefore, a kind of symmetrical cable-rod structure of height for considering datum mark mechanical balance looks for shape method, selected reference point, with power
Densimetry establishes the equilibrium equation around datum mark X, Y, Z-direction, finds out transformational relation of remaining node relative to datum mark,
To dimension, smaller force density matrix is analyzed, directly acquire force density relationship and the node coordinate of each rod piece of structure to get
The initial prestress distribution of high symmetrical cable-rod structure.
Above-described embodiment is only the preferred embodiment of the present invention, it should be pointed out that：For the ordinary skill of the art
For personnel, without departing from the principle of the present invention, several improvement and equivalent replacement can also be made, these are to the present invention
Claim be improved with the technical solution after equivalent replacement, each fall within protection scope of the present invention.
Claims (6)
1. a kind of symmetrical cable-rod structure of height for considering datum mark mechanical balance looks for shape method, which is characterized in that this method include with
Lower step：
The geometrical topology of the clear and definite structure of step 1 is grouped structure drag-line, striker member according to symmetry, in same
The drag-line for waiting status is one group, and the compression bar in par is one group, node is grouped according to symmetry, in equal
The node of status is one group；
Step 2 establishes coordinate system using symmetrical configuration central point as origin, and each selected node conduct is distinguished in each group node
Datum mark, the benchmark point coordinates are (x1, y1, z1), with force density method establish datum mark X, Y, Z-direction equilibrium equation,
And form the force density matrix Q of each datum mark；
Step 3 finds out remaining node in same group node in addition to datum mark relative to datum mark according to geometrically symmetric relationship
Transformational relation show that remaining node is relative to the coordinate conversion matrix R of the grouping datum mark in each grouping according to transformational relation；
Step 4 solves the product QR of the force density matrix Q of each datum mark and the coordinate conversion matrix R relative to each datum mark, and
The order for enabling gained Matrix QR is zero, acquires the force density relationship of each group component；
The coordinate conversion matrix R relative to datum mark that step 5 is obtained by the step 3, benchmark is found out in same group node
Except point remaining each node i relative to datum mark coordinate conversion matrix Ri, cartesian coordinate system is acquired similarly hereinafter in group node
The coordinate of remaining each node except datum mark is (xi, yi, zi)=Ri*(x1, y1, z1)；
Step 6 acquires each rod length and initial prestress distribution.
2. the height symmetrical cable-rod structure according to claim 1 for considering datum mark mechanical balance looks for shape method, feature exists
In, in the step 1 definition of the drag-line and compression bar of par be：Under a certain symmetry operation, a certain drag-line leads to structure
It crosses the symmetry operation and may switch to another drag-line position, then this two drag-lines are in par, and a certain compression bar passes through
The symmetry operation may switch to another compression bar position, then this two compression bars are in par.
3. the height symmetrical cable-rod structure according to claim 1 for considering datum mark mechanical balance looks for shape method, feature exists
In the node definition of par is in the step 1：For structure under a certain symmetry operation, certain node passes through the symmetry operation
The position of another node is may switch to, then the two nodes are in par.
4. the symmetrical cable-rod structure of height according to datum mark mechanical balance the considerations of described of claim 1,2 or 3 looks for shape method, special
Sign is, selectes the datum mark of each group node in the step 2 as follows：An arbitrarily selected section in same group node
Point chooses a datum mark per group node as datum mark.
5. the symmetrical cable-rod structure of height according to datum mark mechanical balance the considerations of described of claim 1,2 or 3 looks for shape method, special
Sign is, acquires the force density relationship of each group component in the step 4 as follows：Enable the corresponding Matrix QR of each datum mark
Middle all elements are equal to 0, obtain QR (j, k)=0, wherein the element of QR (j, k) representing matrix QR jth rows kth row, j=1,
The line number of 2 ..., m, m for Matrix QR, k=1,2 ..., n, columns of the n for Matrix QR, the corresponding QR (j, k) of all datum marks of simultaneous
=0 composition equation group, and equation group is solved, obtain the force density relationship of each group rod piece.
6. the symmetrical cable-rod structure of height according to datum mark mechanical balance the considerations of described of claim 1,2 or 3 looks for shape method, special
Sign is, the step 5 interior joint i relative to datum mark coordinate conversion matrix RiCoordinate to be obtained in step 3 converts square
The 3i rows of battle array R are to 3i+2 rows.
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Cited By (1)
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| CN115017682A (en) * | 2022-05-10 | 2022-09-06 | 西北工业大学 | Mechanical behavior analysis method for tensioning integral module in space force thermal environment |
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