CN109299511B - A kind of reanalysis method of the Flexible Truss based on stiffness effect - Google Patents
A kind of reanalysis method of the Flexible Truss based on stiffness effect Download PDFInfo
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Abstract
A kind of reanalysis method the present invention provides Flexible Truss based on stiffness effect, it is primarily based on finite element analysis and obtains flexible flat truss displacement frequency response function, construction displacement frequency response matrix, when the elasticity modulus of a certain unit changes, determine overall situation global stiffness matrix variable quantity, based on matrix correction formula, revised structural response is quickly obtained according to initial displacement frequency response matrix, frequency response dynamic weight analysis is completed and solves.It is therefore not necessary to carry out multiple FEM calculation, changed using the local stiffness of initial frequency response dynamic response signal and clear structure, the dynamic analysis of structure after rigidity perturbation can be completed, simplifies computational efficiency, it is more convenient, there is engineering significance.
Description
Technical field
The present invention relates to a kind of reanalysis methods, and in particular to a kind of reanalysis method based on stiffness effect.
Background technique
Reanalysis method can quickly estimate the quick calculating side of structure after modification as one kind according to initial calculation result
Method has obtained extensive research in recent decades, and achieves a series of achievements with theoretical value and engineering significance, in machine
The field of structural design such as tool, building are widely used.
In Practical Project, it is sometimes necessary to carry out localized design adjustment to Flexible Truss, such as change specific truss knot
The elasticity modulus of structure inevitably results in partial structurtes stiffness matrix and changes, to influence the stiffness matrix point of total
Cloth, the dynamic characteristic of structure also change therewith.The signal for how utilizing initial communication, avoids re-starting finite element meter
It calculates, to fast and effeciently obtain dynamic response after rigidity of structure amendment, it has also become Practical Project problem urgently to be resolved.
Summary of the invention
Goal of the invention: it is based in view of the above-mentioned deficiencies in the prior art, it is an object of the present invention to provide a kind of Flexible Truss
The reanalysis method of stiffness effect.
Technical solution: the reanalysis method the present invention provides a kind of Flexible Truss based on stiffness effect, including with
Lower step:
(1) finite element analysis is carried out based on Matlab software, obtains flexible flat truss displacement frequency response function, construction displacement
Frequency response matrix calculates global stiffness matrix variable quantity, and determine it and take the photograph when the elasticity modulus of a certain unit of structure changes
The relationship of displacement frequency response matrix after dynamic；
(2) it is based on matrix correction formula, is calculated according to step (1) initial displacement frequency response function and obtains revised structure sound
It answers, completes rigidity weight analysis and solve.
Further, step (1) the following steps are included:
(11) freedom degree of structure is N, and the flexible flat truss of acquisition is displaced frequency response matrix are as follows:
Wherein, hpqIt indicates under structure node q function unit pulse, the dynamic respond function of structure node p, p=1,2 ...
N, q=1,2 ... N；
(12) by taking a certain structural unit t of flexible flat truss as an example, corresponding node number is respectively i, j, and unit and x-axis are just
Direction is counterclockwise, and inclination angle is θ, the stiffness matrix of the unit are as follows:
Wherein, E is elasticity modulus, and A is cell cross-section product, and l is element length；
It (13) is θ it is found that transition matrix by inclination angle are as follows:
(14) when elastic modulus E increases Δ E, corresponding element stiffness matrix variable quantity is Δ Kt:
(15) the element stiffness matrix variable quantity under world coordinates is obtained according to formula (3) (4):
Wherein, S, G are respectively
(16) by the property of stiffness matrix assembling it is found that the element stiffness matrix variable quantity under world coordinatesIn member
Element is individually positioned in global stiffness matrix variation delta K (2i-1), (2i), (2j-1), at (2j) row and column crossover location, then
Global stiffness matrix variation delta K may be expressed as:
WhereinIndicate (2i-1) in the matrix, (2i), (2j-1), (2j) ranks, which intersect, punishes
Matrix S, the element in G are not corresponded to；
(17) according to the relationship of displacement frequency response matrix and dynamic stiffness matrix, derivation can be obtained:
Wherein, H*For the later displacement frequency response matrix that perturbs.
Further, step (2) is based on matrix correction formula, the displacement frequency response matrix after obtaining weight analysis:
Wherein, I ∈ RN×NFor unit matrix.
The utility model has the advantages that the method for the present invention, which is primarily based on finite element analysis, obtains flexible flat truss displacement frequency response function, structure
Displacement frequency response matrix is made, when the elasticity modulus of a certain unit changes, overall situation global stiffness matrix variable quantity is determined, is based on square
Battle array correction formula is quickly obtained revised structural response according to initial displacement frequency response matrix, completes frequency response dynamic weight analysis and asks
Solution.It is therefore not necessary to carry out multiple FEM calculation, the local stiffness of initial frequency response dynamic response signal and clear structure is utilized
The dynamic analysis of structure after rigidity perturbation can be completed in variation, simplify computational efficiency, more convenient, have engineering significance.
Detailed description of the invention
Fig. 1 is the system schematic of 9 truss elements composition in embodiment；
Fig. 2 is the displacement frequency response function h of the initial system of system structure33；
Fig. 3 is the displacement frequency response function after system weight analysisWith exact value curve.
Specific embodiment
Technical solution of the present invention is described in detail below, but protection scope of the present invention is not limited to the implementation
Example.
The present embodiment is verified using truss structural, as shown in Figure 1, by taking 9 truss elements 1~9 as an example, totally 6
Node 1.~6., 12 freedom degree z1~z12.Due to clamped in 1 node, the direction z1 and z2 can be constrained, in No. 6 node letters
Branch, can constrain the direction z12, therefore practical structures have 9 freedom degrees, understand implementation steps for convenience, still carried out with 12 freedom degrees
Analysis, elastic modulus E=70000MPa, cell density ρ=2700kg/m3, cell cross-section product A=1 × 10-4m2, including with
Lower step:
Step 1, finite element analysis is carried out based on Matlab software, obtains displacement frequency response function, construction displacement frequency response matrix；
When the elastic modulus E of structure Unit 3 changes, global stiffness matrix variable quantity is calculated, and determine itself and the displacement after perturbation
The relationship of frequency response matrix.
1.1) freedom degree of structure is 12, obtains flexible flat truss based on FEM calculation and is displaced frequency response matrix are as follows:
Wherein, dpqIt indicates under structure node q function unit pulse, the dynamic respond function of structure node p, p=1,2 ...
12, q=1,2 ... 12；
2. 4. 1.2) the corresponding node serial number of No. 3 units of structure is respectively that unit is with positive direction of the x-axis inclination angle counterclockwise
Zero degree, the stiffness matrix of the unit:
It 1.3) is 315 ° it is found that transition matrix by inclination angle are as follows:
1.4) when E increases Δ E=20000, corresponding element stiffness matrix variable quantity is Δ K3:
1.5) the element stiffness matrix variable quantity under world coordinates is obtained according to formula (3) (4):
Wherein, S, G are respectively as follows:
1.6) by the property of stiffness matrix assembling it is found that the element stiffness matrix variable quantity under world coordinatesIn member
Element is individually positioned in global stiffness matrix variation delta K at the 3rd, 4,7,8 row and column crossover location, then global stiffness matrix variable quantity
Δ K may be expressed as:
WhereinIndicate that the 3rd, 4,7,8 row, column infall in the matrix respectively corresponds matrix S, G
In element, specifically:
1.7) according to the relationship of displacement frequency response matrix and dynamic stiffness matrix, derivation can be obtained:
Wherein, H*For the later displacement frequency response matrix that perturbs.
Step 2, it is based on matrix correction formula, revised structural response is quickly obtained according to initial displacement frequency response matrix,
Rigidity weight analysis is completed to solve:
Wherein, I ∈ R12×12For unit matrix.
In order to verify effectiveness of the invention, with initial displacement function h33For, see Fig. 2, using the above method, is weighed
After analysisCompared with exact value, Fig. 3 is seen.It can be seen that the result and theoretical calculation of weight analysis are preferable, present invention benefit
Flexible truss rigidity weight analysis, nothing are realized based on the structural stiffness matrix transformation after rigidity perturbation with initial response signal
FEM calculation need to be carried out again, and method is more succinct effective.
Claims (1)
1. a kind of reanalysis method of Flexible Truss based on stiffness effect, it is characterised in that: the following steps are included:
(1) finite element analysis is carried out based on Matlab software, obtains flexible flat truss displacement frequency response function, construction displacement frequency response
Matrix, when the elasticity modulus of a certain unit of structure changes, calculate global stiffness matrix variable quantity, and determine its with after perturbation
Displacement frequency response matrix relationship；
(2) it is based on matrix correction formula, is calculated according to step (1) initial displacement frequency response function and obtains revised structural response,
Rigidity weight analysis is completed to solve；
Specifically, step (1) the following steps are included:
(11) freedom degree of structure is N, and the flexible flat truss of acquisition is displaced frequency response matrix are as follows:
Wherein, hpqIndicate p=1,2 ... N, q under structure node q function unit pulse, the dynamic respond function of structure node p
=1,2 ... N；
(12) by taking a certain structural unit t of flexible flat truss as an example, corresponding node numbers respectively i, j, unit and positive direction of the x-axis
Inclination angle counterclockwise is θ, the stiffness matrix of the unit are as follows:
Wherein, E is elasticity modulus, and A is cell cross-section product, and l is element length；
It (13) is θ it is found that transition matrix by inclination angle are as follows:
(14) when elastic modulus E increases Δ E, corresponding element stiffness matrix variable quantity is Δ Kt:
(15) the element stiffness matrix variable quantity under world coordinates is obtained according to formula (3) (4):
Wherein, S, G are respectively
(16) by the property of stiffness matrix assembling it is found that the element stiffness matrix variable quantity under world coordinatesIn element point
It is not placed in global stiffness matrix variation delta K (2i-1), (2i), (2j-1), it is at (2j) row and column crossover location, then total rigid
Degree matrix variation delta K may be expressed as:
WhereinIndicate (2i-1) in the matrix, (2i), (2j-1), (2j) ranks infall is right respectively
Answer matrix S, the element in G；
(17) according to the relationship of displacement frequency response matrix and dynamic stiffness matrix, derivation can be obtained:
Wherein, H*For the later displacement frequency response matrix that perturbs；
Step (2) is based on matrix correction formula, the displacement frequency response matrix after obtaining weight analysis:
Wherein, I ∈ RN×NFor unit matrix.
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