CN111046463B - Truss structure deformation decomposition method based on orthogonal theory - Google Patents
Truss structure deformation decomposition method based on orthogonal theory Download PDFInfo
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Abstract
The invention belongs to the technical field of mechanical analysis, and discloses a truss structure deformation decomposition method based on an orthogonal theory, which comprises the following steps of: firstly, obtaining a complete orthogonal mechanical basis matrix under a local coordinate system by adopting an orthogonal theory decomposition method; then projecting coordinate displacement vectors of each rod unit node of the truss structure under the local coordinate system onto the complete orthogonal mechanical basis matrix, and obtaining projection coefficient vectors of the rod units under the local coordinate system through linear transformation; then converting the basic displacement projection coefficient vector under the local coordinate system into a basic displacement projection coefficient vector under a global coordinate system through space coordinate transformation; and judging the basic displacement information and the deformation information of the rod unit according to the basic displacement and deformation projection coefficient vectors in the local and global coordinate systems. The method can intuitively and quickly quantitatively identify the basic displacement and deformation information in the comprehensive deformation of the truss structure.
Description
Technical Field
The invention belongs to the technical field of mechanical analysis, and relates to a truss structure deformation decomposition method based on an orthogonal theory.
Background
The truss structure is widely applied to various engineering structures such as truss bridges, stadiums and high-rise building transfer floors due to the advantages of saving materials, realizing large span and the like. Under the ideal state, each rod unit of the truss is only stressed by axial force, the stress distribution of the cross section is uniform, and each part of each rod unit of the truss can simultaneously reach the stress tolerance value, so that the truss structure is an ideal structural form. However, in recent years, the safety accidents of the truss structure still occur endlessly, because the displacement deformation of the truss structure under the load working condition is still a comprehensive deformation, the existing analysis method cannot directly identify the basic displacement or deformation information in the comprehensive deformation, and further cannot accurately analyze the deformation performance of the truss structure. How to effectively identify the basic displacement or deformation information of the rod unit has important significance on the performance analysis and the optimized design of the truss structure.
The rod unit in the truss structure is only provided with two nodes, and the node comprehensive displacement of the truss structure unit can only be obtained by the existing analysis methods such as finite element analysis, test monitoring and the like, so that further analysis cannot be realized. The method provided by the invention constructs a complete orthogonal mechanical base based on the rod unit based on an orthogonal theory, and further provides a deformation decomposition method of the truss structure, so that corresponding basic displacement or deformation information can be further identified and separated on the basis of the comprehensive displacement of the unit node, and the deformation performance of the truss structure is quantitatively analyzed. At present, a truss structure deformation decomposition method based on an orthogonal theory is not found.
Disclosure of Invention
The invention aims to provide a method for decomposing the deformation of a truss structure based on an orthogonal theory, which can intuitively and quickly quantitatively identify basic displacement and deformation information in the comprehensive deformation of the truss structure, so that the deformation performance of the truss structure can be more accurately judged.
In order to achieve the purpose, the invention adopts the following technical scheme:
a truss structure deformation decomposition method based on an orthogonal theory comprises the following steps:
step 1: establishing a local coordinate system of the three-dimensional two-node rod unit, and obtaining basic displacement and deformation of the rod unit under the local coordinate system by adopting an orthogonal decomposition theory so as to construct a complete orthogonal mechanical basis matrix of the three-dimensional rod unit;
step 2: establishing a truss model in a global coordinate system, dividing a truss structure by adopting two node rod units to obtain node coordinate values of the rod units under the global coordinate system and node coordinate values of the rod units under the corresponding load working conditions under the global coordinate system, and obtaining node coordinate displacement vectors of the rod units under the corresponding load working conditions under a local coordinate system through space coordinate transformation;
and 3, step 3: projecting the node coordinate displacement vector of the lower rod unit of the local coordinate system under the corresponding load working condition onto a complete orthogonal mechanical basis matrix, and obtaining a projection coefficient vector of the lower rod unit of the local coordinate system under the corresponding load working condition through linear transformation;
and 4, step 4: obtaining a projection coefficient vector of basic displacement and a projection coefficient vector of deformation of the rod unit under the corresponding load working condition in a local coordinate system through rigid-flexible separation; converting the projection coefficient vector of the basic displacement of the rod unit under the corresponding load working condition in a local coordinate system into the projection coefficient vector of the basic displacement of the rod unit under the corresponding load working condition in a global coordinate system through space coordinate transformation;
and 5: and judging the basic displacement information and the deformation information of the rod unit according to the projection coefficient size contained in the projection coefficient vector of the rod unit under the local coordinate system and the global coordinate system.
Further, the local coordinate system takes the axial direction of the rod unit as an X 'axis and the middle point of the rod unit as an origin, the local coordinate system is marked as O' -X 'Y' Z ', and the basic displacement and deformation of the rod unit under the local coordinate system comprise X' axis rigid body linear displacement, Y 'axis rigid body displacement, Z' axis rigid body linear displacement, X 'O' Y 'plane rigid body rotational displacement, X' O 'Z' plane rigid body rotational displacement and axial tension and compression deformation.
Further, step 1 specifically includes: establishing a local coordinate system of the three-dimensional two-node rod unit, and obtaining a basic vector P of basic displacement and deformation of the rod unit under the local coordinate system based on 6 basic displacements and deformations of the rod unit by adopting an orthogonal theory decomposition method 1 ～P 6 The method comprises the following steps:
P 1 base vector for X' axial rigid body linear displacement:
P 1 ＝(0.7071,0,0,0.7071,0,0) T (1)；
P 2 base vectors for the Y' axial rigid body linear displacement:
P 2 ＝(0,0.7071,0,0,0.7071,0) T (2)；
P 3 base vector for Z' axial rigid body linear displacement:
P 3 ＝(0,0,0.7071,0,0,0.7071) T (3)；
P 4 is the base direction of rigid body rotational displacement in the X ' O ' Y ' planeQuantity:
P 4 ＝(0,0.7071,0,0,-0.7071,0) T (4)；
P 5 is the base vector of rigid body rotation in the X ' O ' Z ' plane:
P 5 ＝(0,0,0.7071,0,0,-0.7071) T (5)；
P 6 axial tension and compression deformation:
P 6 ＝(-0.7071,0,0,0.7071,0,0) T (6)；
the complete orthogonal mechanical basis matrix is P,
P＝[P 1 P 2 P 3 P 4 P 5 P 6 ] (7)。
further, the global coordinate system is denoted as O-XYZ, the node coordinate value of the rod unit in the global coordinate system is d, and the node coordinate value d of the rod unit in the local coordinate system is obtained by d through space coordinate transformation 1 ，
d 1 ＝(x 1 y 1 z 1 x 2 y 2 z 2 ) (8)；
Under the working condition of corresponding load, the node coordinate value of the rod unit in the global coordinate system is d 2 ，d 2 Obtaining node coordinate values d of the lower rod unit of the local coordinate system under corresponding load working conditions through space coordinate transformation 3 ，
d 3 ＝(x 1 ' y 1 ' z 1 ' x 2 ' y 2 ' z 2 ') (9)；
The node coordinate displacement vector d of the lower rod unit of the local coordinate system under the corresponding load working condition can be obtained by subtracting the above formula (8) from the above formula (9) e ，
d e ＝(x 1 '-x 1 ,y 1 '-y 1 ,z 1 '-z 1 ,x 2 '-x 2 ,y 2 '-y 2 ,z 2 '-z 2 ) (10)。
Further, the step 3 specifically includes: projecting the node coordinate displacement vector of the lower rod unit of the local coordinate system under the corresponding load working condition onto a complete orthogonal mechanical basis matrix to obtain:
d e ＝p·P T (11)，
wherein P is a complete orthogonal mechanical basis matrix, P T Is a transposed matrix of P, (P) T ) -1 Is P T The inverse matrix of (d);
the projection coefficient vector p of the rod unit under the corresponding load working condition under the local coordinate system can be obtained by carrying out linear transformation on the formula (11),
p＝d e (P T ) -1 ＝d e P (12)，
p is the projection coefficient vector of the basic displacement and deformation in the local coordinate system,
p＝(p 1 ,p 2 ,p 3 ,p 4 ,p 5 ,p 6 ) (13)，
wherein p is 1 Is the projection coefficient, p, of the rod unit on the X' axial rigid body linear displacement base vector under the local coordinate system 2 Is the projection coefficient, p, of the rod unit on the base vector of the linear displacement of the Y' axial rigid body under the local coordinate system 3 Is the projection coefficient, p, of the rod unit on the Z' -axial rigid body linear displacement base vector under a local coordinate system 4 Is the projection coefficient of the rod unit on the rigid body rotation vector of the X ' O ' Y ' plane under the local coordinate system, p 5 Is the projection coefficient, p, of the rod unit in the local coordinate system on the rigid body rotation vector in the X ' O ' Z ' plane 6 The projection coefficient of the lower rod unit of the local coordinate system on the axial tension-compression deformation base vector is obtained;
the projection coefficient vector of the basic displacement of the rod unit under the corresponding load working condition in the local coordinate system is P d I.e. by
p d ＝(p 1 ,p 2 ,p 3 ,p 4 ,p 5 ) (14)，
Projection coefficient vector P under local coordinate system can be transformed by space coordinate d Transforming the projection coefficient vector P into a global coordinate system w ：
p w ＝p d ·T (15)，
Wherein T is a coordinate transformation matrix between a local coordinate system and a global coordinate system, P w Is a projection coefficient vector of the basic displacement of the rod unit under the corresponding load working condition in the overall coordinate system,
p w ＝(p w1 ,p w2 ,p w3 ,p w4 ,p w5 ,p w6 ) (16)，
wherein p is w1 Is a projection coefficient, p, of the rod unit on the X-axis rigid body linear displacement under a global coordinate system w2 Is a projection coefficient, p, of the rod unit on the displacement of the rigid body line in the Y axis direction under the global coordinate system w3 Is a projection coefficient, p, of the rod unit on the Z-axial rigid body line displacement under a global coordinate system w4 Is a projection coefficient, p, of the rod unit on the rotation of the XOY plane rigid body under the global coordinate system w5 Is a projection coefficient, p, of the rod unit on the XOZ plane rigid body rotation under the global coordinate system w6 The projection coefficient of the rod unit on the rigid rotation of the YOZ plane under the global coordinate system is shown.
Further, the step 5 specifically includes:
comparing the absolute value of each projection coefficient according to the projection coefficient vector of the basic displacement of the rod unit under the local coordinate system, judging the main displacement of the rod unit when the absolute value is the largest, and identifying the basic displacement information of the rod unit under the local coordinate system by analogy; comparing the absolute value of each projection coefficient according to the projection coefficient vector of the basic displacement of the rod unit in the global coordinate system, judging the main displacement of the rod unit when the absolute value is the maximum, and identifying the basic displacement information of the rod unit in the global coordinate system by analogy; and identifying deformation information of the rod unit according to the projection coefficient vector of the deformation of the rod unit under the local coordinate system.
Further, the projection coefficient p of the rod unit on the axial tension-compression deformation base vector under the local coordinate system 6 Positive, indicating that the rod unit is axially pulled; projection coefficient p of lower rod unit in local coordinate system on axial tension-compression deformation base vector 6 Negative indicates that the rod unit is axially compressed.
Compared with the prior art, the invention has the beneficial effects that:
according to the invention, the comprehensive deformation of the truss structure can be quantitatively identified and analyzed by an orthogonal decomposition method, the comprehensive deformation of the truss structure is subjected to deformation decomposition by utilizing the constructed complete orthogonal mechanical basis vector, and the basic displacement and deformation information of the truss structure are identified, so that the basic displacement and deformation information in the comprehensive deformation of the truss structure are quantitatively identified, the deformation performance of the truss structure is more accurately judged, and a theoretical basis is provided for the optimal design and accident analysis of the truss structure.
Drawings
Fig. 1 is a schematic flow chart of a truss structure deformation decomposition method based on an orthogonal theory according to the invention.
FIG. 2 is a schematic view of the overall and local coordinate system of the three-dimensional two-node pole unit of the present invention.
FIG. 3 is a schematic diagram of the spatial deformation of the three-dimensional two-node rod unit in the local coordinate system according to the present invention.
FIG. 4 is a schematic diagram of the X' axial rigid body displacement in local coordinates of the three-dimensional two-node rod unit according to the present invention.
FIG. 5 is a schematic diagram of the Y' axial rigid body displacement in local coordinates of the three-dimensional two-node rod unit according to the present invention.
FIG. 6 is a schematic view of the Z' axial rigid body displacement in local coordinates of the three-dimensional two-node rod unit according to the present invention.
FIG. 7 is a schematic diagram of rigid rotation displacement of an X ' O ' Y ' plane in a local coordinate system of a three-dimensional two-node rod unit according to the present invention.
FIG. 8 is a schematic diagram of rigid body rotation displacement of the X ' O ' Z ' plane in the local coordinate system of the three-dimensional two-node rod unit according to the present invention.
FIG. 9 is a schematic view of the axial tension-compression deformation of the three-dimensional two-node rod unit according to the present invention in a local coordinate system.
Fig. 10 is a schematic view of a truss structure according to an embodiment of the invention.
FIG. 11 is a graph comparing the X-direction translation coefficient and the node displacement of the finite element software according to the first embodiment of the present invention.
Fig. 12 is a schematic diagram of a gantry structure according to a second embodiment of the present invention.
Detailed Description
The following examples are intended to illustrate the invention, but are not intended to limit the scope of the invention. Unless otherwise specified, the technical means used in the examples are conventional means well known to those skilled in the art.
Fig. 1 shows a schematic flow chart of the method for decomposing the deformation of the truss structure based on the orthogonal theory. As shown in fig. 2 to 9, any three-dimensional two-node rod unit is provided, the local coordinate system and the global coordinate system of which are shown in fig. 2, the spatial deformation of which in the local coordinate system is shown in fig. 3, and the basic displacement and deformation of which in the local coordinate system is shown in fig. 4 to 9.
For any three-dimensional two-node rod unit, a local coordinate system O ' -X ' Y ' Z ' is constructed with the axial direction of the rod unit as the X ' axis and the midpoint of the rod unit as the origin (see fig. 2). For any three-dimensional two-node rod unit, the basic displacement and deformation of the three-dimensional two-node rod unit under the local coordinate system comprise X ' axis rigid body linear displacement, Y ' axis rigid body displacement, Z ' axis rigid body linear displacement, X ' O ' Y ' plane rigid body rotary displacement, X ' O ' Z ' plane rigid body rotary displacement and axial tension and compression deformation (see figures 4-9). Base vector P of the above 6 basic displacements and deformations 1 ～P 6 The method comprises the following steps:
P 1 base vector for X' axial rigid body linear displacement:
P 1 ＝(0.7071,0,0,0.7071,0,0) T (1)；
P 2 base vectors for the Y' axial rigid body linear displacement:
P 2 ＝(0,0.7071,0,0,0.7071,0) T (2)；
P 3 base vector for Z' axial rigid body linear displacement:
P 3 ＝(0,0,0.7071,0,0,0.7071) T (3)；
P 4 is the base vector of rigid body rotation displacement in the X ' O ' Y ' plane:
P 4 ＝(0,0.7071,0,0,-0.7071,0) T (4)；
P 5 is the base vector of rigid body rotation in the X ' O ' Z ' plane:
P 5 ＝(0,0,0.7071,0,0,-0.7071) T (5)；
P 6 and (3) axial tension and compression deformation:
P 6 ＝(-0.7071,0,0,0.7071,0,0) T (6)；
so as to obtain a complete orthogonal mechanics basis matrix P,
P＝[P 1 P 2 P 3 P 4 P 5 P 6 ] (7)。
constructing a truss structure model of the three-dimensional rod unit, establishing an overall coordinate system O-XYZ of the truss structure, dividing the truss structure by adopting two-node rod units to obtain a node coordinate value d of the rod unit under the overall coordinate system and a node coordinate value d of the rod unit under the corresponding load working condition under the overall coordinate system 2 ，
d obtaining the node coordinate value d of the lower rod unit of the local coordinate system through space coordinate transformation 1 ，
d 1 ＝(x 1 y 1 z 1 x 2 y 2 z 2 ) (8)；
d 2 Obtaining node coordinate values d of the lower rod unit of the local coordinate system under corresponding load working conditions through space coordinate transformation 3 ，
d 3 ＝(x 1 ' y 1 ' z 1 ' x 2 ' y 2 ' z 2 ') (9)；
Subtracting the formula (8) from the formula (9), so as to obtain a node coordinate displacement vector d of the rod unit under the corresponding load working condition in the local coordinate system e ，
d e ＝(x 1 '-x 1 ,y 1 '-y 1 ,z 1 '-z 1 ,x 2 '-x 2 ,y 2 '-y 2 ,z 2 '-z 2 ) (10)；
Node coordinate displacement vector d of lower rod unit of local coordinate system under corresponding load working condition e On the complete orthogonal mechanics basis matrix P, we can obtain:
d e ＝p·P T (11)，
wherein P is a complete orthogonal mechanical basis matrix, P T Is a transposed matrix of P, (P) T ) -1 Is P T The inverse matrix of (d);
the projection coefficient vector p of the lower rod unit of the local coordinate system under the corresponding load working condition can be obtained by carrying out linear transformation on the formula (11),
p＝d e (P T ) -1 ＝d e P (12)，
p is the vector of the basic displacement and deformation projection coefficients in the local coordinate system,
p＝(p 1 ,p 2 ,p 3 ,p 4 ,p 5 ,p 6 ) (13)，
wherein p is 1 Is the projection coefficient, p, of the rod unit on the X' axial rigid body linear displacement base vector under the local coordinate system 2 Is the projection coefficient, p, of the rod unit on the linear displacement base vector of the Y' axial rigid body under the local coordinate system 3 Is the projection coefficient, p, of the rod unit on the linear displacement base vector of the Z' axial rigid body under the local coordinate system 4 Is the projection coefficient of the rod unit on the rigid body rotation vector of the X ' O ' Y ' plane under the local coordinate system, p 5 Is the projection coefficient, p, of the rod unit in the local coordinate system on the rigid body rotation vector in the X ' O ' Z ' plane 6 The projection coefficient of the rod unit on the axial tension-compression deformation base vector under the local coordinate system is shown.
The projection coefficient vector of the basic displacement of the rod unit under the corresponding load working condition in the local coordinate system is P d I.e. by
p d ＝(p 1 ,p 2 ,p 3 ,p 4 ,p 5 ) (14)，
Projection coefficient vector P under local coordinate system can be obtained through space coordinate transformation d Transforming the projection coefficient vector P into a global coordinate system w ：
p w ＝p d ·T (15)，
Wherein T is a coordinate transformation matrix between a local coordinate system and a global coordinate system, P w Is a projection coefficient vector of the basic displacement of the rod unit under the corresponding load working condition in the overall coordinate system,
p w ＝(p w1 ,p w2 ,p w3 ,p w4 ,p w5 ,p w6 ) (16)，
wherein p is w1 For the rod unit under the global coordinate systemProjection coefficient, p, on X-axis rigid body linear displacement w2 Is a projection coefficient, p, of the rod unit on the displacement of the rigid body line in the Y axis direction under the global coordinate system w3 Is a projection coefficient, p, of the rod unit on the Z-axial rigid body line displacement under a global coordinate system w4 Is a projection coefficient, p, of the rod unit on the rotation of the XOY plane rigid body under the global coordinate system w5 Is a projection coefficient, p, of the rod unit on the rotation of the XOZ plane rigid body under a global coordinate system w6 The projection coefficient of the rod unit on the rigid rotation of the YOZ plane under the global coordinate system is shown.
Order to
To P d And (3) carrying out normalization processing on the projection coefficient of the basic displacement of the lower rod unit in the middle local coordinate system to obtain:
p d '＝(p 1 ',p 2 ',p 3 ',p 4 ',p 5 ') (17)，
wherein p is i ' (i =1,2,3,4,5) represents the proportion of the ith basic displacement of the rod unit in the total basic displacement under the local coordinate system.
Order to
To P w And (3) carrying out normalization processing on the projection coefficient of the basic displacement of the rod unit under the middle overall coordinate to obtain:
p w '＝(p w1 ',p w2 ',p w3 ',p w4 ',p w5 ',p w6 ') (18)，
wherein p is wi ' (i =1,2,3,4,5,6) represents the proportion of the ith basic displacement of the rod unit in the total basic displacement under the overall coordinate system.
Rigid body rotational displacement error analysis
The rigid body rotation is nonlinear displacement, and an error is generated during linear decomposition, namely, the rigid body rotation displacement vector of the rod unit has a projection coefficient not only on a rigid body rotation base vector, but also on other displacement and deformation vectors, and has an error with a theoretical condition, so that the numerical value of the projection coefficient of the rigid body rotation of the rod unit on other deformation and displacement base vectors needs to be analyzed, and whether the rigid body rotation displacement affects the calculation precision is judged.
The side length of the three-dimensional two-node rod unit is set to be 2a, the axial direction of the rod unit is along the X direction, the center of the rod unit is located at the origin of coordinates, at the moment, the local coordinate system is overlapped with the whole coordinate system, and deformation decomposition can be directly carried out under the whole coordinate system. When the three-dimensional two-node rod unit rotates around the rod center clockwise theta, namely, rotates on the XOY plane theta. Therefore, the coordinate displacement vectors of the two nodes of the three-dimensional rod unit are as follows:
(a-acosθ,asinθ,0,acosθ-a,-asinθ,0) (19)。
and projecting node displacement vectors obtained by rotating the three-dimensional two-node rod unit on an XOY plane onto a constructed complete orthogonal mechanical matrix, and finding that the rigid body rotation node displacement vectors of the rod unit only have projection coefficients on an XOY plane rigid body rotation displacement basis vector and an axial tension-compression deformation basis vector, and the projection coefficients on other basic displacement vectors are 0. Therefore, the 6 constraint equations obtained by the node displacement vector projection of the rigid body rotation of the unit can be simplified into 2 independent constraint equations as follows:
wherein: p is a radical of 4 、p 6 The projection coefficients of the displacement vector of the rigid body rotation node of the rod unit on the XOY plane rigid body rotation displacement base vector and the axial tension-compression deformation base vector are respectively.
Solving equation (20) above yields:
taylor series expansion is performed on sin theta and cos theta terms in the formula (21), and high-order small quantity is ignored, so that the following can be obtained:
comparison of p by mathematical means 4 、p 6 The magnitude of (a) can be given by:
from the above equation (23), in the case of small deformation, the three-dimensional rod unit XOY plane rigid body rotation node displacement vector projects the coefficient p on the tension-compression deformation base vector 6 Relative to XOY plane rigid body rotation displacement base vector projection coefficient p 4 The displacement vector of the rigid body rotation node of the XOY plane is a high-order infinitesimal quantity of theta, so that the projection coefficient of the displacement vector of the rigid body rotation node of the XOY plane on a tension-compression deformation base can be ignored, and the error is in an allowable range.
Example one embodiment of the present invention will be described in detail below with reference to fig. 10.
As shown in fig. 10, a bottom-hinged vertical truss is taken as an example, wherein the length and width of the bottom surface of the truss are both 0.2m, the height of the truss is 0.6m, the length and width directions of the bottom surface are the X axis and the Y axis, and the height direction is the Z axis. The modulus of elasticity of the truss structure is 3030Mpa, the Poisson ratio is 0.2, and the radius of the rod unit is 10mm. And respectively applying a load 500N in the X direction to the two nodes at the top of the truss. Three vertical rod units of the truss are selected and marked as No. 1, no. 2 and No. 3 rod units respectively, and the three rods are deformed and decomposed respectively.
And (3) carrying out stress calculation on the truss structure to obtain the displacement of each node of the truss structure, and carrying out deformation decomposition on the vertical No. 1, no. 2 and No. 3 rod units of the truss structure respectively.
Firstly, a local coordinate system which takes the axial direction of a truss structure No. 1 rod unit as an axis and the midpoint of the rod unit as an origin is established, and a node coordinate displacement vector d of the No. 1 rod unit under the local coordinate system can be obtained through truss stress analysis and space coordinate transformation e ：
d e ＝(0,0,0,2.0787,-7.1805,-3.1498)。
Through calculation, the projection coefficients and the occupied ratio of each basic displacement of the rod unit No. 1 in the local coordinate system are shown in Table 1. The proportion of the linear displacement of the rigid body on the Y 'axis and the rigid body rotation on the X' O 'Y' plane in the total basic displacement is the same, and both are 31.58%, and the absolute value of the projection coefficient is the largest, so that the basic displacement of the No. 1 rod unit in the local coordinate system is mainly the linear displacement of the rigid body on the Y 'axis and the rigid body rotation on the X' O 'Y' plane.
Table 1 truss structure No. 1 pole unit local coordinate system deformation decomposition result
The projection coefficients and the occupied ratios of the No. 1 rod unit in each basic displacement of the whole coordinate system can be obtained through space coordinate transformation, as shown in table 2, the proportion of the X-axis rigid body linear displacement of the No. 1 rod unit in the whole coordinate system and the proportion of the rigid body rotation of the XOZ plane in the total basic displacement are the same, both are 31.58%, and the absolute value of the projection coefficient is the maximum, so that the basic displacement of the No. 1 rod unit in the whole coordinate system mainly comprises the X-axis rigid body linear displacement and the rigid body rotation of the XOZ plane. The axial tension-compression deformation projection coefficient of the truss No. 1 rod unit is 1.4699, so that the truss No. 1 rod is in tension deformation.
TABLE 2 deformation decomposition results of truss structure No. 1 rod unit in global coordinate system
Similarly, by performing stress calculation on the truss structure, a node coordinate displacement vector d under the local coordinate system of the No. 2 rod unit of the truss structure can be obtained e ：
d e ＝(2.0787,-7.1805,-3.1498,3.0876,-18.5386,-6.3279)。
The projection coefficients and the occupied ratio of the 2 # rod unit in each basic displacement of the local coordinate system are obtained through calculation and are shown in table 3. The proportion of the displacement of the Y ' axis rigid body line of the No. 2 rod unit in the local coordinate system to the total rigid body displacement is 46.85 percent, the displacement of the Y ' axis rigid body line of the No. 2 rod unit is far greater than the proportion of the displacement of other basic rigid bodies to the total rigid body displacement, and the absolute value of a projection coefficient is the maximum, so that the basic displacement of the No. 2 rod unit in the local coordinate system is mainly the displacement of the Y ' axis rigid body line.
Table 3 deformation decomposition results under local coordinate system of truss structure No. 2 rod unit
The projection coefficients and the occupied ratios of the 2 # rod unit in each basic displacement of the whole coordinate system can be obtained through space coordinate transformation, as shown in table 4, the X-axis rigid body linear displacement of the 2 # rod unit in the whole coordinate system accounts for 46.85% of the maximum total rigid body displacement, and the absolute value of the projection coefficient is the maximum, so that the basic displacement of the 2 # rod unit in the whole coordinate system mainly accounts for the X-axis rigid body linear displacement. The projection coefficient of tension and compression deformation of the truss No. 2 rod unit is 0.7134, so that the truss No. 2 rod unit is in tension deformation.
Table 4 deformation decomposition result of truss structure No. 2 rod unit in whole coordinate system
Similarly, by performing stress calculation on the truss structure, a node coordinate displacement vector d under the local coordinate system of the No. 3 rod unit of the truss structure can be obtained e Comprises the following steps:
d e ＝(3.0876,-18.5386,-6.3279,3.0876,-31.794,-9.7074)。
the projection coefficients and the occupied ratio of the No. 3 rod unit in each basic displacement of the local coordinate system obtained by calculation are shown in Table 5. The proportion of the Y 'axis rigid linear displacement of the No. 3 rod unit in the local coordinate system to the total rigid displacement is 56.44 percent, and the absolute value of the projection coefficient is maximum, so that the basic displacement of the No. 3 rod unit in the local coordinate system is mainly the Y' axis rigid linear displacement.
TABLE 5 deformation decomposition result of truss structure No. 3 rod unit under local coordinate system
The projection coefficients and the occupied ratios of the basic displacements of the rod unit No. 3 in the global coordinate system can be obtained through space coordinate transformation, as shown in Table 6, the proportion of the linear displacement of the rigid body of the rod unit No. 3 in the global coordinate system is 56.44% of the total rigid body displacement, and the absolute value of the projection coefficient is the maximum, so that the basic displacement of the rod unit No. 3 in the global coordinate system is mainly the linear displacement of the rigid body of the X axis. The tension-compression deformation projection coefficient of the truss No. 3 rod unit is 0, so that the truss structure No. 3 rod unit has no deformation.
TABLE 6 deformation decomposition result of truss structure No. 3 rod unit under integral coordinate system
In order to verify the correctness of the method, the translational displacement of the No. 1, no. 2 and No. 3 rod units in the X direction is extracted, which respectively comprises the following steps: 5.0073, 18.1847 and 35.5890.
The displacements of the nodes at the two ends of the No. 1,2 and 3 rod units in the X direction are extracted from finite element software, and compared with the translation of the three rod units in the X direction calculated by the method, as shown in FIG. 11.
As can be seen from fig. 11, the X-direction translation coefficient of the deformation decomposition of the three vertical rods of the truss is consistent with the displacement trend of the finite element calculated node, which can verify the correctness of the method, but the finite element software can only calculate the comprehensive displacement of the node of each rod unit of the truss structure and cannot accurately identify the basic displacement and deformation of the rod unit. Therefore, the method can accurately identify the basic displacement and the deformation of the truss structure rod unit, and compare the projection coefficients of the node displacement vector on the basic displacement and the deformation, thereby determining the main deformation and the main displacement of the truss structure rod unit.
Example two
Next, the No. 4 diagonal rods in the gantry structure column in fig. 12 are deformed and decomposed.
As shown in FIG. 12, the portal frame has a height of 6m, a width of 22.5 m, a beam main truss diameter of 9.58cm, a column middle main truss diameter of 11.2cm, diagonal rods diameter of 7.98cm, and a rod member elastic modulus of 7206000N/mm, density 7.8X 10 -6 . The bottom of the portal frame is hinged.
Through modal analysis of the portal frame structure, a node coordinate displacement vector d of the portal frame structure under a No. 4 rod unit local coordinate system under a first-order mode can be obtained e Comprises the following steps:
d e ＝(-0.0136,0.0034,0.0586,-0.009,0.0048,0.01)。
the projection coefficients and the occupied ratio of the 4 th rod unit in each basic displacement of the local coordinate system obtained by calculation are shown in table 7. The proportion of Z '-axis rigid body linear displacement of the No. 4 rod unit in the total rigid body displacement under the local coordinate system is 45.88 percent, and the absolute value of the projection coefficient is the maximum, so that the basic displacement of the No. 4 rod unit under the local coordinate system is mainly the Z' -axis rigid body linear displacement.
TABLE 7 decomposition result of deformation under local coordinate system of No. 4 bar unit of gantry structure
The projection coefficients and the occupation ratios of the 4 # rod unit in each basic displacement of the global coordinate system can be obtained through space coordinate transformation, for example, as shown in table 8 below, the proportion of the Y-axis rigid body linear displacement of the 4 # rod unit in the global coordinate system to the total rigid body displacement is 40.79% at the maximum, and the absolute value of the projection coefficient is the maximum, so that the basic displacement of the 4 # rod unit in the global coordinate system is mainly the Y-axis rigid body linear displacement. The projection coefficient of tension-compression deformation of the No. 4 rod unit of the portal frame is 0.0032, so that the No. 4 rod unit of the truss structure is in tension deformation.
TABLE 8 deformation decomposition result of No. 4 rod unit of gantry structure in integral coordinate system
The above-mentioned embodiments are merely preferred embodiments of the present invention, which are merely illustrative and not restrictive, and it should be understood that other embodiments may be easily made by those skilled in the art by replacing or changing the technical contents disclosed in the specification, and therefore, all changes and modifications that are made on the principle of the present invention should be included in the scope of the claims of the present invention.
Claims (6)
1. A truss structure deformation decomposition method based on an orthogonal theory is characterized by comprising the following steps:
step 1: establishing a local coordinate system of the three-dimensional two-node rod unit, and obtaining basic displacement and deformation of the rod unit under the local coordinate system by adopting an orthogonal decomposition theory so as to construct a complete orthogonal mechanical basis matrix of the three-dimensional rod unit; the step 1 specifically comprises the following steps: establishing a local coordinate system of the three-dimensional two-node rod unit, and obtaining a basic vector P of basic displacement and deformation of the rod unit under the local coordinate system based on 6 basic displacements and deformations of the rod unit by adopting an orthogonal theory decomposition method 1 ～P 6 The method comprises the following steps:
P 1 base vector for X' axial rigid body linear displacement:
P 1 ＝(0.7071,0,0,0.7071,0,0) T (1)；
P 2 base vectors for the Y' axial rigid body linear displacement:
P 2 ＝(0,0.7071,0,0,0.7071,0) T (2)；
P 3 base vector for Z' axial rigid body linear displacement:
P 3 ＝(0,0,0.7071,0,0,0.7071) T (3)；
P 4 is the base vector of the rigid body rotational displacement in the X ' O ' Y ' plane:
P 4 ＝(0,0.7071,0,0,-0.7071,0) T (4)；
P 5 is the base vector of rigid body rotation in the X ' O ' Z ' plane:
P 5 ＝(0,0,0.7071,0,0,-0.7071) T (5)；
P 6 axial tension and compression deformation:
P 6 ＝(-0.7071,0,0,0.7071,0,0) T (6)；
further, a three-dimensional pole unit complete orthogonal mechanics base matrix is constructed as P,
P＝[P 1 P 2 P 3 P 4 P 5 P 6 ] (7)；
step 2: establishing a truss model in a global coordinate system, dividing a truss structure by adopting two node rod units to obtain node coordinate values of the rod units under the global coordinate system and node coordinate values of the rod units under the corresponding load working conditions under the global coordinate system, and obtaining node coordinate displacement vectors of the rod units under the corresponding load working conditions under a local coordinate system through space coordinate transformation;
and 3, step 3: projecting node coordinate displacement vectors of the lower rod unit of the local coordinate system under the corresponding load working condition onto a complete orthogonal mechanical basis matrix, and obtaining projection coefficient vectors of the lower rod unit of the local coordinate system under the corresponding load working condition through linear transformation;
and 4, step 4: obtaining a projection coefficient vector of basic displacement and a projection coefficient vector of deformation of the rod unit under the corresponding load working condition in a local coordinate system through rigid-flexible separation; converting the projection coefficient vector of the basic displacement of the rod unit under the corresponding load working condition in a local coordinate system into the projection coefficient vector of the basic displacement of the rod unit under the corresponding load working condition in a global coordinate system through space coordinate transformation;
and 5: and determining the basic displacement information and the deformation information of the rod unit according to the projection coefficient size contained in the projection coefficient vector of the rod unit under the local coordinate system and the global coordinate system.
2. The method of decomposing deformation of a truss structure based on orthogonal theory as claimed in claim 1, wherein the local coordinate system takes the axial direction of the rod unit as the X 'axis and the midpoint of the rod unit as the origin, the local coordinate system is marked as O' -X 'Y' Z ', and the basic displacement and deformation of the rod unit under the local coordinate system include X' axis rigid body linear displacement, Y 'axis rigid body displacement, Z' axis rigid body linear displacement, X 'O' Y 'plane rigid body rotational displacement, X' O 'Z' plane rigid body rotational displacement and axial tension and compression deformation.
3. According to claimThe method for decomposing the deformation of the truss structure based on the orthogonal theory is characterized in that the overall coordinate system is recorded as O-XYZ, the node coordinate value of the rod unit in the overall coordinate system is d, and the node coordinate value d of the rod unit in the local coordinate system is obtained by d through space coordinate transformation 1 ，
d 1 ＝(x 1 y 1 z 1 x 2 y 2 z 2 ) (8)；
The node coordinate value of the rod unit under the corresponding load working condition in the global coordinate system is d 2 ，d 2 Obtaining node coordinate values d of the lower rod unit of the local coordinate system under corresponding load working conditions through space coordinate transformation 3 ，
d 3 ＝(x 1 ' y 1 ' z 1 ' x 2 ' y 2 ' z 2 ') (9)；
The node coordinate displacement vector d of the lower rod unit of the local coordinate system under the corresponding load working condition can be obtained by subtracting the above expression (8) from the above expression (9) e ，
d e ＝(x 1 '-x 1 ,y 1 '-y 1 ,z 1 '-z 1 ,x 2 '-x 2 ,y 2 '-y 2 ,z 2 '-z 2 ) (10)。
4. The method for decomposing the deformation of the truss structure based on the orthogonal theory as claimed in claim 3, wherein the step 3 is specifically as follows: projecting the node coordinate displacement vector of the lower rod unit of the local coordinate system under the corresponding load working condition onto a complete orthogonal mechanical basis matrix to obtain:
d e ＝p·P T (11)，
wherein P is a complete orthogonal mechanical basis matrix, P T Is a transposed matrix of P, (P) T ) -1 Is P T The inverse matrix of (d);
the projection coefficient vector p of the lower rod unit of the local coordinate system under the corresponding load working condition can be obtained by carrying out linear transformation on the formula (11),
p＝d e (P T ) -1 ＝d e P (12)，
p is the projection coefficient vector of the basic displacement and deformation in the local coordinate system,
p＝(p 1 ,p 2 ,p 3 ,p 4 ,p 5 ,p 6 ) (13)，
wherein p is 1 Is the projection coefficient, p, of the rod unit on the X' axial rigid body linear displacement base vector under the local coordinate system 2 Is the projection coefficient, p, of the rod unit on the linear displacement base vector of the Y' axial rigid body under the local coordinate system 3 Is the projection coefficient, p, of the rod unit on the linear displacement base vector of the Z' axial rigid body under the local coordinate system 4 Is the projection coefficient of the rod unit on the rigid body rotation vector of the X ' O ' Y ' plane under the local coordinate system, p 5 Is the projection coefficient, p, of the rod unit on the rigid body rotation vector of the X ' O ' Z ' plane under the local coordinate system 6 The projection coefficient of the lower rod unit of the local coordinate system on the axial tension-compression deformation base vector is obtained;
the projection coefficient vector of the basic displacement of the rod unit under the corresponding load working condition in the local coordinate system is P d I.e. by
p d ＝(p 1 ,p 2 ,p 3 ,p 4 ,p 5 ) (14)，
Projection coefficient vector P under local coordinate system can be obtained through space coordinate transformation d Transforming the projection coefficient vector P into a whole coordinate system w ：
p w ＝p d ·T (15)，
Wherein T is a coordinate transformation matrix between a local coordinate system and a global coordinate system, P w Is a projection coefficient vector of the basic displacement of the rod unit under the corresponding load working condition in the global coordinate system,
p w ＝(p w1 ,p w2 ,p w3 ,p w4 ,p w5 ,p w6 ) (16)，
wherein p is w1 Is a projection coefficient, p, of the rod unit on the rigid body displacement in the X axis direction under a global coordinate system w2 Is a projection coefficient, p, of the rod unit on the rigid body displacement in the Y axis direction under a global coordinate system w3 Is the displacement of the rod unit in the Z-axial rigid body line under the global coordinate systemProjection coefficient of (p) w4 Is a projection coefficient, p, of the rod unit on the rotation of the XOY plane rigid body under the global coordinate system w5 Is a projection coefficient, p, of the rod unit on the rotation of the XOZ plane rigid body under a global coordinate system w6 The projection coefficient of the rod unit on the rotation of the rigid body in the YOZ plane under the integral coordinate system.
5. The method for decomposing the deformation of the truss structure based on the orthogonal theory as claimed in claim 1, wherein the step 5 specifically comprises:
comparing the absolute value of each projection coefficient according to the projection coefficient vector of the basic displacement of the rod unit under the local coordinate system, judging the main displacement of the rod unit when the absolute value is the largest, and identifying the basic displacement information of the rod unit under the local coordinate system by analogy;
comparing the absolute value of each projection coefficient according to the projection coefficient vector of the basic displacement of the rod unit in the global coordinate system, judging the main displacement of the rod unit when the absolute value is the maximum, and identifying the basic displacement information of the rod unit in the global coordinate system by analogy;
and identifying deformation information of the rod unit according to the projection coefficient vector of the deformation of the rod unit under the local coordinate system.
6. The method for decomposing the deformation of the truss structure based on the orthonormal theory as claimed in claim 4, wherein the projection coefficient p of the rod unit on the axial tension-compression deformation basis vector under the local coordinate system is 6 Positive, indicating that the rod unit is axially pulled; projection coefficient p of lower rod unit in local coordinate system on axial tension-compression deformation base vector 6 Negative indicates that the rod unit is axially compressed.
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