General modeling method for assembling unit multi-body dynamics with non-linear deformation taken into account

Abstract

The invention discloses an assembly unit multi-body dynamics general modeling method with nonlinear deformation included. The method comprises the following steps: (1) pre-processor modeling; and establishing a tetrahedron flexible body model of the assembly unit by adopting an absolute node coordinate method. The shape function of the model is described in a nonlinear mode, and the nonlinear characteristic of deformation is reflected; (2) on the basis of the flexible body information in the step (1), continuously establishing a connection relationship with other tetrahedrons for assembly, determining the dynamic continuity constraint of a connection interface, and completing a hexahedron assembly structure model from one tetrahedron to five tetrahedrons; and (3) model information of the pre-processor serves as an initial state to be input into the main processor, and position, speed and acceleration analysis is conducted on the model. Therefore, the numerical simulation of the antenna assembly process can be realized, the deformation nonlinearity of the assembly unit and the dynamic continuity at the connection interface can be accurately processed, and the dynamic characteristics can be accurately predicted.

Description

Technical Field

[0001] The invention relates to the technical field of large-scale space antennas, in particular to a general multi-body dynamics modeling method for assembly units taking nonlinear deformation into account. Background Art

[0002] Large space antennas are key core equipment in aerospace engineering. Limited by the rocket's upward envelope, space 100-meter / kilometer-level antennas need to be transported upward in a form with a very high storage ratio, and deployed in orbit in a controllable manner after entering orbit. At the same time, for larger antenna structures, it is necessary to further introduce on-orbit assembly technology to achieve the largest application aperture of space antennas. Space antennas with extremely large apertures have become the commanding heights in the field of satellite-borne antenna research at home and abroad.

[0003] The on-orbit implementation forms of very large aperture antennas in space are mainly divided into three categories: independent deployment of antennas in space, assembly of large antennas with human participation based on the space station platform, and unmanned on-orbit construction of large antennas. The first two methods are limited by the antenna caliber and the size of the space station platform, and cannot realize the on-orbit construction of very large aperture antennas in space; the third type of unmanned on-orbit construction of large antennas can launch deployable antenna modules through carrier rockets, and multiple modules are assembled on the spacecraft platform, which is the best way to realize the antenna with a very large aperture. The dynamics technology of assembling very large aperture antennas in space has the following technical difficulties: the dynamics of the assembly process of very large aperture antennas in space will face complex dynamics problems such as configuration growth, parameter changes, orbit-attitude-structure coupling, contact collision, etc. This type of research is extremely difficult and belongs to the dynamics problem of high-dimensional nonlinear multi-flexible mechanical system. Since the ground experiment of the dynamics of very large aperture antennas in space is difficult and cannot completely offset the influence of gravity, the dynamic numerical simulation of its deployment process is the key technology to ensure the successful deployment of the structure in orbit. Summary of the invention

[0004] The present invention provides a general modeling method for multi-body dynamics of assembly units taking into account nonlinear deformation, with the aim of accurately simulating the nonlinear deformation dynamic characteristics of the module.

[0005] In a first aspect, a general modeling method for multi-body dynamics of assembly units taking into account nonlinear deformation is provided, comprising:

[0006] Step 1: Preprocessor modeling: Use the absolute node coordinate method to establish the tetrahedron model of the assembly unit, and obtain the coordinate information and shape function of the tetrahedron model in the volume coordinate system. The shape function of the tetrahedron model is nonlinear.

[0007] Step 2: Based on the tetrahedron model in step 1, establish the connection relationship between the tetrahedron model and other tetrahedron models for assembly, and determine the dynamic continuity constraint of the connection interface; the above model information is input into the main processor as the initial state for dynamic analysis;

[0008] Step 3: Enter the main processor and perform position analysis, velocity analysis and acceleration analysis on the assembly unit respectively.

[0009] In combination with the first aspect, in certain implementations of the first aspect, the coordinate information e of the tetrahedron model in the volume coordinate system is k satisfy:

[0010]

[0011] The superscript k represents the number of nodes in the tetrahedron model, and r k is the position coordinate of node k in the tetrahedron model in the Cartesian coordinate system, ξ,η,ζ,χ are the four volume coordinate directions corresponding to the four nodes one by one,

[0012] In combination with the first aspect, in some implementations of the first aspect, the shape function S of the tetrahedron model is:

[0013] S=[s 1 I 2 I 3 I 4 I 5 I 6 I 7 I 8 I 9 I 10 I 11 I 12 I 13 I 14 I 15 I 16 I]

[0014] I is the 3×3 identity matrix; s 1 to 16 are the 16 shape function parameters of shape function S, satisfying:

[0015]

[0016] Among them, ξ 0 ,η 0 ,ζ 0 ,x 0 are the volume coordinates of the tetrahedral model.

[0017] In combination with the first aspect, in some implementations of the first aspect, assembling a tetrahedron model with other tetrahedron models includes:

[0018] The coordinate information of the tetrahedron model in the volume coordinate system is kConvert to Cartesian coordinate system;

[0019] Assemble and connect multiple tetrahedral models in the Cartesian coordinate system. During dynamic modeling, ensure the position continuity constraints and gradient continuity constraints of the common connection points on the connection interface, and apply gradient continuity constraints to the common nodes: in The superscript number k represents the node number, and the subscript α represents the node position coordinate r k The volume coordinate direction for the first differentiation, subscript β represents the node position coordinate r k The volume coordinate direction in which the second differentiation is performed.

[0020] In combination with the first aspect, in some implementations of the first aspect, the coordinate information e of the tetrahedron model in the volume coordinate system is k Converted to Cartesian coordinate system, it satisfies:

[0021] e k =T k p k

[0022]

[0023] Among them, p k Represents the coordinate information of the tetrahedron model in the Cartesian coordinate system. The superscript k represents the number of nodes of the tetrahedron model. k is the position coordinate of node k in the tetrahedron model in the Cartesian coordinate system, X 1 ,X 2 ,X 3 are the three directions of the Cartesian coordinate system, T 1 , T 2 , T 3 , T 4 They are all 12×12 square matrices, containing the parametric conversion relationship between the volume coordinate information of the four nodes of the tetrahedron model and the Cartesian coordinate information.

[0024] In combination with the first aspect, in certain implementations of the first aspect, the position analysis includes: calculating the Jacobian matrix of the constraint based on the mechanical constraint side information; after determining the Jacobian matrix of the constraint equation, determining the types of independent coordinates and non-independent coordinates by using the Gaussian elimination method; based on the numerical determination of the independent position coordinates, using the Newton-Raphson algorithm to solve the constraint equation and determine the numerical size of the non-independent position coordinates.

[0025] In combination with the first aspect, in certain implementations of the first aspect, the speed analysis includes: solving the time derivative of the mechanical sub-constraint equation according to the system's mechanical sub-constraint equation; determining the numerical size of the non-independent speed coordinate based on the numerical determination of the independent speed coordinate; the above completes the speed analysis.

[0026] In combination with the first aspect, in certain implementations of the first aspect, the acceleration analysis includes: after completing the position analysis and velocity analysis, the generalized position coordinates and the generalized velocity coordinates are determined, and further organized into system state space vectors for calculating the acceleration vector and the Lagrange multiplier vector; the above completes the acceleration analysis of the flexible body model.

[0027] In conjunction with the first aspect, in some implementations of the first aspect, the main processor simulation process includes:

[0028] In the nth iteration, the constraint Jacobian matrix C is calculated based on the mechanical constraint side information n,q ; In determining the Jacobian matrix C of the constraint equation n,q Then, the independent position coordinates q are determined by Gaussian elimination method. n,i and the dependent position coordinate q n,d Type; independent position coordinates q n,i The value is determined based on the result of the n-1th iteration, according to formula C n,qi δq n,i =-C n,qd δq n,d , determine the dependent coordinates q n,d The numerical value of C n,qi ,C n,qd is the Jacobian matrix C n,q For independent position coordinates q n,i and the dependent position coordinate q n,d The partial derivative of δ is the variation symbol; the position coordinates are calculated based on the result of the n-1th iteration. Updates;

[0029] According to the mechanical side constraint equation of the system, solve the time derivative C of the mechanical side constraint equation n,t ; Independent velocity coordinates The value is determined based on the result of the n-1th iteration, according to the formula Determining dependent velocity coordinates The value of the velocity coordinate is based on the result of the n-1th iteration. Updates;

[0030] According to the formula Determine the acceleration vector and the Lagrange multiplier vector λ, where M is the mass matrix, Q vis the secondary velocity vector, which includes the gyroscopic force term and the Corry force term of the system, Q c is the system constraint force vector; Q g is the generalized force vector; Q e is the elastic force vector; λ is the Lagrange multiplier vector;

[0031] According to the calculated velocity vector and the acceleration vector Arranged into the first-order derivative form of the system state space And by using the numerical integration algorithm to perform time integration, the updated state equation is obtained According to the updated state equation y n+1 Get the next independent position coordinates q n+1,i and independent velocity coordinates

[0032] Let n=n+1 to perform the n+1th iteration simulation until the end of the simulation time.

[0033] In a second aspect, an electronic device is provided, wherein the electronic device is used to execute the method described in any one of the implementations of the first aspect.

[0034] Compared with the prior art, the solution provided by the present invention includes at least the following beneficial technical effects:

[0035] Compared with the problem that the traditional floating coordinate method cannot accurately describe the nonlinear deformation of the structure, the present invention uses the absolute node coordinate method as the flexible body modeling theory, and the established nonlinear constitutive equation describes the deformation nonlinearity, which can more accurately reflect the nonlinear deformation characteristics of the assembly unit. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 The present invention is a flow chart of the general modeling and design method for multi-body dynamics of assembly units taking into account nonlinear deformation.

[0037] Figure 2 It is a schematic diagram of the absolute node coordinate assembly unit tetrahedron of the present invention.

[0038] Figure 3 Schematic diagram of the traditional assembly unit tetrahedron model.

[0039] Figure 4 It is a schematic diagram of a hexahedral assembly structure assembled from assembly units of the present invention.

[0040] Figure 5 This is the interface continuity constraint effect of the present invention. (a) Before applying the constraint; (b) After applying the constraint.

[0041] Figure 6The 5 tetrahedral models of the present invention are assembled into a model. (a) 1 hexahedral unit; (b) 5 tetrahedral models are assembled into 1 hexahedral unit. DETAILED DESCRIPTION

[0042] The present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0043] Reference Figure 1 The specific steps of the general modeling method of multi-body dynamics of assembly units with nonlinear deformation of the present invention are as follows.

[0044] Step 1: Establish a tetrahedral flexible model of the assembly unit in the preprocessor. Establish a nonlinear shape function S to reflect the nonlinear characteristics of deformation.

[0045] 1) For tetrahedron modeling, such as Figure 2 As shown, the absolute node coordinate tetrahedron model is used for simulation. The unit has 4 nodes, each node has 12 coordinate information, and a total of 48 coordinate information is as follows:

[0046]

[0047] The superscript k = 1, 2, 3, 4 represents the number of nodes in the tetrahedron model, r k is the position coordinate of node k in the tetrahedron model in the Cartesian coordinate system, They are r k The gradient of ξ,η,ζ,χ in the volume coordinate direction to describe the deformation along the volume coordinate direction.

[0048] The shape function S of the tetrahedron model proposed in this patent is:

[0049]

[0050] Where: I is the 3×3 unit matrix. 1 to 16 are the 16 shape function parameters of the shape function S. The expressions of these 16 shape function parameters are:

[0051]

[0052] Among them, ξ 0 ,η 0 ,ζ 0 ,x 0 are the volume coordinates of the tetrahedral model.

[0053] For the convenience of comparison, the traditional tetrahedron model is provided here. Figure 3 As shown, its shape function has only 12 degrees of freedom and can be written as:

[0054] N=[N1 IN 2 IN 3 IN 4 I] (4)

[0055] Where: I is the 3×3 unit matrix. The expressions of the four shape function parameters are:

[0056]

[0057] Where: V is the volume of the tetrahedron, node k is a vertex of the tetrahedron, a k ,b k ,c k ,d k The coefficient expression is:

[0058]

[0059] Where: m, n, p are the other three vertices of the tetrahedron model (excluding node k).

[0060] By comparing formula (3) and formula (6), it can be seen that the shape function of the absolute node coordinate tetrahedron model of the present invention is nonlinear, which can characterize the nonlinear characteristics of the unit deformation.

[0061] Step 2: Based on the flexible body information in step 1, continue to establish the connection relationship with other tetrahedrons for assembly, and complete the assembly structure model from a tetrahedron to a hexahedron assembled by five tetrahedrons. The above model information is input into the main processor as the initial state for dynamic analysis.

[0062] Define the coordinate information of the tetrahedron model in the Cartesian coordinate system k=1,2,3,4,X 1 ,X 2 ,X 3 are the 3 directions of the Cartesian coordinate system. They are r k In the Cartesian coordinate direction X 1 ,X 2 ,X 3 The gradient of is used to describe the deformation along the Cartesian coordinate direction. The transformation relationship between the Cartesian coordinate information and the volume coordinate information of the tetrahedron model is defined as a linear transformation, satisfying e k =T k p k , combined with the coordinate information of the tetrahedron model in the volume coordinate system in formula (1) Matrix T k The parametric transformation of two types of coordinate systems is defined, and the expression is:

[0063]

[0064] Where: T 1 , T 2 , T 3 , T 4 All are 12×12 square matrices, T 1 It is a 12×12 unit matrix, and T contains the parameterized transformation relationship between the volume coordinates of the four nodes of the tetrahedron and the Cartesian coordinates.

[0065] Figure 4 When assembling and connecting two tetrahedrons, the position continuity constraint and gradient continuity constraint of the common connection points 2 and 4 on the connection interface must be ensured during dynamic modeling (the specific operation method of applying continuity constraints can refer to the existing technology and will not be repeated here). In order to achieve gradient continuity and ensure the unit dynamic continuity after multiple tetrahedrons are assembled, it is necessary to apply gradient continuity constraints to the common nodes, that is, in The superscript number k represents the node number, and the subscript α represents the node position coordinate r k The volume coordinate direction for the first differentiation, subscript β represents the node position coordinate r k The volume coordinate direction in which the second differentiation is performed. That is,

[0066]

[0067]

[0068] Figure 5 The following are the dynamic analysis diagrams of the two tetrahedral assembly units before and after applying the continuity constraint. It can be seen that after applying the continuity constraint, the gradient vectors (arrows) at the interface of the assembly unit completely overlap, ensuring the continuity of the dynamics at the interface. Using the above tetrahedral model and interface dynamics continuity constraints, the tetrahedral model can be used to assemble a hexahedral unit, such as Figure 6 shown.

[0069] Step 3: Enter the main processor for system dynamics analysis. Perform position, velocity and acceleration analysis on each flexible body separately, as described below.

[0070] Step 4: Position analysis. In the nth iteration, the constraint Jacobian matrix C is calculated based on the mechanical constraint side information. q , while checking for constraint violations; in determining the Jacobian matrix C of the constraint equation q Then, the independent position coordinates q are determined by Gaussian elimination method. n,i and the dependent position coordinate q n,d Type; in independent position coordinates q n,i Based on the numerical determination (independent position coordinates q n,iThe value can be determined based on the result of the previous iteration), and the Newton-Raphson algorithm is used to solve the constraint equations and determine the non-independent coordinates q n,d The numerical value of is as shown in formula (8), where C n,qi ,C n,qd is the Jacobian matrix C q For independent position coordinates q n,i and the dependent position coordinate q n,d The above completes the position analysis of the flexible body model. After completing the position analysis, the position coordinates can be calculated based on the results of the previous iteration. Updates.

[0071] C n,qi δq n,i =-C n,qd δq n,d (8)

[0072] Step 5: Speed ​​analysis: According to the mechanical side constraint equation of the system, further solve the time derivative C of the mechanical side constraint equation n,t In the independent velocity coordinate Based on the numerical determination (independent velocity coordinates The value can be determined based on the result of the previous iteration) to determine the non-independent velocity coordinates The value of is as shown in formula (9). The above completes the velocity analysis. After completing the velocity analysis, the velocity coordinates can be calculated based on the results of the previous iteration. Updates.

[0073]

[0074] Step 6: Acceleration Analysis: Determine the Acceleration Vector and the Lagrange multiplier vector λ to complete the solution of the model, as shown in formula (10).

[0075]

[0076] Where: M is the mass matrix, Q v is the secondary velocity vector (including the secondary velocity vector of the system gyroscopic force term and Corry force term), Q c is the system constraint force vector; Q g is the generalized force vector; Q e is the elastic force vector; λ is the Lagrange multiplier vector.

[0077] Step 7: Based on the calculated velocity vector and the acceleration vector Arranged into the first-order derivative form of the system state space And by using the numerical integration algorithm to perform time integration, the updated state equation is obtained According to the updated state equation y n+1 Get the next independent position coordinates q n+1,i and independent velocity coordinates

[0078] Step 8: Let n=n+1, and repeat steps 4 to 7 until the end of the simulation time.

[0079] Although the present invention is disclosed as above in the form of a preferred embodiment, it is not intended to limit the present invention. Any person skilled in the art may make possible changes and modifications without departing from the spirit and scope of the present invention. Therefore, the scope of protection of the present invention shall be based on the scope defined by the claims of the present invention.

Claims

1. A general modeling method for multi-body dynamics of assembly units taking into account nonlinear deformations, characterized in that: include: Step 1: Pre-processor modeling; The tetrahedron model of the assembly unit is established by using the absolute node coordinate method, and the coordinate information and shape function of the tetrahedron model in the volume coordinate system are obtained. The shape function of the tetrahedron model is nonlinear. Step 2: Based on the tetrahedron model in step 1, establish the connection relationship between the tetrahedron model and other tetrahedron models for assembly, and determine the dynamic continuity constraint of the connection interface; the above model information is input into the main processor as the initial state for dynamic analysis; Step 3: Enter the main processor and perform position analysis, velocity analysis and acceleration analysis on the assembly unit respectively.

2. The method according to claim 1, characterized in that: Coordinate information of the tetrahedron model in the volume coordinate system k satisfy: The superscript k represents the number of nodes in the tetrahedron model, and r k is the position coordinate of node k in the tetrahedron model in the Cartesian coordinate system, ξ,η,ζ,χ are the four volume coordinate directions corresponding to the four nodes one by one, 3. The method according to claim 1, characterized in that The shape function S of the tetrahedron model is: S=[s1I s2I s3I s4I s5I s6I s7I s8I s9I s 10 I s 11 I s 12 I s 13 I s 14 I s 15 I s 16 I] I is the 3×3 identity matrix; s1 to s 16 are the 16 shape function parameters of shape function S, satisfying: s1=ξ0(ξ0 2 +3ξ0(η0+ζ0+χ0)+2(η0ζ0+χ0(η0+ζ0))), s5=η0(η0 2 +3η0(ζ0+χ0+ξ0)+2(ζ0χ0+ξ0(ζ0+χ0))), s9=ζ0(ζ0 2 +3ζ0(χ0+ξ0+η0)+2(χ0ξ0+η0(χ0+ξ0))), s 13 =χ0(χ0 2 +3x0(ξ0+η0+ζ0)+2(ξ0η0+ζ0(ξ0+η0))), Among them, ξ0, η0, ζ0, χ0 are the volume coordinates of the tetrahedral model.

4. The method according to claim 1, characterized in that: Tetrahedron models assemble with other tetrahedron models, including: The coordinate information of the tetrahedron model in the volume coordinate system is k Convert to Cartesian coordinate system; Assemble and connect multiple tetrahedral models in the Cartesian coordinate system. During dynamic modeling, ensure the position continuity constraints and gradient continuity constraints of the common connection points on the connection interface, and apply gradient continuity constraints to the common nodes: in The superscript number k represents the node number, and the subscript α represents the node position coordinate r k The volume coordinate direction for the first differentiation, subscript β represents the node position coordinate r k The volume coordinate direction in which the second differentiation is performed.

5. The method according to claim 1, characterized in that The coordinate information of the tetrahedron model in the volume coordinate system is k Converted to Cartesian coordinate system, it satisfies: e k =T k p k Among them, p k Represents the coordinate information of the tetrahedron model in the Cartesian coordinate system. The superscript k represents the number of nodes of the tetrahedron model. k is the position coordinate of node k in the tetrahedron model in the Cartesian coordinate system, X1, X2, and X3 are the three directions of the Cartesian coordinate system, T 1 , T 2 , T 3 , T 4 They are all 12×12 square matrices, containing the parametric conversion relationship between the volume coordinate information of the four nodes of the tetrahedron model and the Cartesian coordinate information.

6. The method according to claim 1, characterized in that The position analysis includes: calculating the Jacobian matrix of the constraint based on the mechanical constraint side information; after determining the Jacobian matrix of the constraint equation, determining the types of independent coordinates and non-independent coordinates through the Gaussian elimination method; based on the numerical determination of the independent position coordinates, using the Newton-Raphson algorithm to solve the constraint equation and determine the numerical size of the non-independent position coordinates.

7. The method according to claim 1, characterized in that The speed analysis includes: solving the time derivative of the mechanical sub-constraint equation according to the system's mechanical sub-constraint equation; determining the numerical value of the non-independent speed coordinate based on the numerical determination of the independent speed coordinate; the above completes the speed analysis.

8. The method according to claim 1, characterized in that The acceleration analysis includes: after completing the position analysis and velocity analysis, the generalized position coordinates and generalized velocity coordinates are determined, and further organized into system state space vectors for calculating the acceleration vector and Lagrange multiplier vector; the above completes the acceleration analysis of the flexible body model.

9. The method according to claim 1, characterized in that: The main processor emulation process includes: In the nth iteration, the constraint Jacobian matrix C is calculated based on the mechanical constraint side information n,q ; In determining the Jacobian matrix C of the constraint equation n,q Then, the independent position coordinates q are determined by Gaussian elimination method. n,i and the dependent position coordinate q n,d Type; independent position coordinates q n,i The value is determined based on the result of the n-1th iteration, according to formula C n,qi δq n,i =-C n,qd δq n,d , determine the dependent coordinates q n,d The numerical value of C n,qi ,C n,qd is the Jacobian matrix C n,q For independent position coordinates q n,i and the dependent position coordinate q n,d The partial derivative of δ is the variation symbol; the position coordinates are calculated based on the result of the n-1th iteration. Updates; According to the mechanical side constraint equation of the system, solve the time derivative C of the mechanical side constraint equation n,t ; Independent velocity coordinates The value is determined based on the result of the n-1th iteration, according to the formula Determining dependent velocity coordinates The value of the velocity coordinate is based on the result of the n-1th iteration. Updates; According to the formula Determine the acceleration vector and the Lagrange multiplier vector λ, where M is the mass matrix, Q v is the secondary velocity vector, which includes the gyroscopic force term and the Corry force term of the system, Q c is the system constraint force vector; Q g is the generalized force vector; Q e is the elastic force vector; λ is the Lagrange multiplier vector; According to the calculated velocity vector and the acceleration vector Arranged into the first-order derivative form of the system state space And by using the numerical integration algorithm to perform time integration, the updated state equation is obtained According to the updated state equation y n+1 Get the next independent position coordinates q n+1,i and independent velocity coordinates Let n=n+1 to perform the n+1th iteration simulation until the end of the simulation time.

10. An electronic device, characterized in that: The electronic device is configured to execute the method according to any one of claims 1 to 9.

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