A calculation method for HINGE connector based on implicit flexible structure solver

Abstract

The present application relates to a calculation method for a HINGE connector based on an implicit flexible structure solver, which decomposes the large spatial rotation of the HINGE connector into a rotation axis and a rotation angle, and defines the rotation variable formed by the rotation axis and the rotation angle as a rotation pseudo-vector; constructs a rotation matrix based on the rotation pseudo-vector, performs first-order variation and equivalent substitution on the rotation formula constructed based on the rotation matrix, and obtains a linearized matrix of the rotation matrix; constructs the matrix forms of the kinematic constraint equations of a JOIN connector and a REVOUTE connector based on the rotation matrix represented by quaternions, the rotation pseudo-vector, and the linearized matrix of the rotation matrix; combines the matrix forms of the kinematic constraint equations of the two connectors to obtain the matrix form of the kinematic constraint equations of the HINGE connector, adopts the Newton-Raphson and Lagrange multiplier methods to solve the matrix form, and obtains the displacement increment and constraint force of the desired node in the HINGE connector.

Description

Technical Field

[0001] The present application relates to the technical field of high-performance numerical simulation analysis, and in particular to a HINGE connector calculation method based on an implicit flexible structure solver. Background Art

[0002] At present, when numerical simulation explores multi-rigid body, multi-flexible body and rigid-flexible coupling problems, they are usually modeled as multi-body systems with one or more components exerting elastic effects, and then a multi-body solver is used to calculate the results. This method has the advantage of high computational efficiency, such as commercial software Adams, but a major limitation of this method is that it cannot fully consider material nonlinearity, large deformation of flexible bodies and plastic strain. Secondly, explicit solution is prone to numerical instability when dealing with rigid problems, and a smaller time step is required to ensure stability, which significantly increases the calculation time. In addition, when the penalty method deals with motion constraints, the computational complexity increases with the increase in the number of penalty factors, and the choice of penalty factors directly affects the convergence of the solution. Finally, although the Euler angle and the Cardan angle are easy to understand when representing the rotation of objects in three-dimensional space, the universal lock phenomenon may occur under specific motion sequences and angle combinations, which limits their application.

[0003] Therefore, how to improve the modeling method of multi-body systems and consider the problem of material nonlinearity, how to solve the problem of time step dependence, and how to solve the influence of penalty factors on the results in order to accurately simulate and predict multi-body kinematic problems are key scientific issues that need to be solved urgently. Summary of the invention

[0004] Based on this, it is necessary to provide a HINGE connector calculation method based on an implicit flexible structure solver, which includes:

[0005] S1: According to the Euler principle, the large spatial rotation of the hinge connector is decomposed into a rotation axis and a rotation angle, and the rotation variable formed by the rotation axis and the rotation angle is defined as a rotation pseudo vector;

[0006] S2: constructing a rotation matrix based on the rotation pseudo vector, performing first-order variation and equivalent substitution on the rotation formula constructed based on the rotation matrix, and obtaining a linearized matrix of the rotation matrix;

[0007] S3: using quaternion form to represent the rotation matrix, the rotation pseudo vector, and the linearization matrix of the rotation matrix;

[0008] S4: The HINGE connector is composed of a JOIN connector and a REVOUTE connector. Based on the rotation matrix represented by quaternion, the rotation pseudo vector, and the linearization matrix of the rotation matrix, the matrix forms of the kinematic constraint equations of the JOIN connector and the REVOUTE connector are respectively constructed;

[0009] S5: combining the matrix forms of the kinematic constraint equations of the JOIN connector and the REVOUTE connector to obtain the matrix form of the kinematic constraint equation of the HINGE connector;

[0010] S6: The Newton-Raphson and Lagrange multiplier methods are used to solve the matrix form of the kinematic constraint equations of the HINGE connector, and the displacement increments and constraint forces of the nodes in the HINGE connector are obtained.

[0011] Beneficial effects: This method is based on finite element modeling of multi-body kinematics, adopts an implicit flexible structure solver, uses quaternions to represent large compound rotations in space, re-derives the kinematic constraint equations of the HINGE connector, and establishes solutions to key problems such as multi-rigid bodies, multi-flexible bodies, and rigid-flexible coupled motion, thus achieving accurate prediction of mechanical and kinematic responses. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0013] Figure 1 This is a flowchart of a HINGE connector calculation method based on an implicit flexible structure solver in an embodiment of the present application.

[0014] Figure 2 This is a theoretical schematic diagram of the HINGE connector in an embodiment of the present application.

[0015] Figure 3 It is a structural schematic diagram of a simple pendulum calculation example in an embodiment of the present application.

[0016] Figure 4 Schematic diagram of node division and boundary conditions for a single pendulum calculation example in an embodiment of the present application.

[0017] Figure 5 Schematic diagram of the calculation process followed by the single pendulum calculation example in the embodiment of the present application. DETAILED DESCRIPTION

[0018] In order to make the above-mentioned purposes, features and advantages of the present application more obvious and easy to understand, the specific implementation methods of the present application are described in detail below in conjunction with the accompanying drawings. In the following description, many specific details are set forth to facilitate a full understanding of the present application. However, the present application can be implemented in many other ways different from those described herein, and those skilled in the art can make similar improvements without violating the connotation of the present application, so the present application is not limited by the specific embodiments disclosed below.

[0019] In addition, the terms "first" and "second" are used for descriptive purposes only and should not be understood as indicating or implying relative importance or implicitly indicating the number of the indicated technical features. Therefore, the features defined as "first" and "second" may explicitly or implicitly include at least one of the features. In the description of this application, the meaning of "plurality" is at least two, such as two, three, etc., unless otherwise clearly and specifically defined.

[0020] like Figure 1 As shown, this embodiment provides a HINGE connector calculation method based on an implicit flexible structure solver, the method comprising:

[0021] In three-dimensional space, large rotations no longer have the additivity of vectors. Infinitely small rotations can be represented by a vector, which satisfies the commutativity and additivity of vectors. However, for any continuous large rotation, the result depends not only on the size of the rotation angle, but also on the order of rotation. Therefore, large rotations in space cannot be regarded as simple superposition of vectors.

[0022] Euler's principle: The motion of a rigid body relative to a fixed point can be considered as a rotation around a certain axis of the point. Under the definition, a large rotation in space has two characteristics: size and direction, namely the rotation axis and the rotation angle.

[0023] S1: According to the Euler principle, the large spatial rotation of the hinge connector is decomposed into a rotation axis and a rotation angle, and the rotation variable formed by the rotation axis and the rotation angle is defined as a rotation pseudo vector.

[0024] In this embodiment, the expression of the rotation pseudo vector is:

[0025] ;

[0026] in, represents the rotation pseudo vector; Represents the rotation pseudo vector The first component of Represents the rotation pseudo vector The second component of Represents the rotation pseudo vector The third component of The first unit basis vector representing the rotation axis; The second unit basis vector representing the rotation axis; The third unit basis vector representing the axis of rotation.

[0027] S2: construct a rotation matrix based on the rotation pseudo-vector, perform first-order variation and equivalent substitution on the rotation formula constructed based on the rotation matrix, and obtain a linearized matrix of the rotation matrix.

[0028] Specifically, constructing a rotation matrix based on the rotation pseudo vector includes:

[0029] The rotation pseudo vector is skew-symmetrically expanded to obtain the spin tensor; the calculation formula is:

[0030] ;

[0031] in, S represents the spin tensor; represents oblique symmetric expansion; represents the rotation pseudo vector; Represents the rotation pseudo vector The first component of Represents the rotation pseudo vector The second component of Represents the rotation pseudo vector The third component of

[0032] According to the linear algebra definition, the exponential expansion of a skew-symmetric matrix (a tensor of your choice) is an orthogonal rotation matrix, which can represent finite rotations:

[0033] ;

[0034] Then, based on the properties of the spin tensor, a closed expression of the finite rotation is calculated to obtain the rotation matrix;

[0035] The properties of the optional tensor are:

[0036] ;

[0037] in, Indicates the module length; I represents the identity matrix; represents tensor product;

[0038] The expression of the rotation matrix is:

[0039] ;

[0040] in, C represents the rotation matrix; exp represents the exponential expansion; Represents the rotation pseudo vector of the skew-symmetric matrix.

[0041] In geometric nonlinear analysis, since the rotation pseudo vectors cannot be added, the rotation matrix is ​​iterated in multiple steps to derive the iterative relationship of the rotation matrix. The iterative formula is:

[0042] ;

[0043] in, Indicates s The rotation matrix of the iteration step; Indicates s The iteration increment of the iteration step; Indicates s -1 The rotation matrix of the iteration step; the update of the rotation matrix is ​​obtained from this iteration relationship.

[0044] Furthermore, the first-order variation and equivalent replacement of the rotation formula constructed based on the rotation matrix includes:

[0045] The rotation formula constructed based on the rotation matrix is ​​expressed as:

[0046] ;

[0047] in, C represents the rotation matrix; A represents any vector in space; a Represents vector A A vector after rotation in space; Represents vector dot product;

[0048] The rotation formula is subjected to first-order variation, and the calculation formula is:

[0049] ;

[0050] in, represents variation;

[0051] Vector a The variation of is replaced by the variation of the rotation field variable, and the calculation formula is:

[0052] ;

[0053] in, represents the rotation field variable; represents vector cross product; Represents the skew-symmetric matrixed rotation field variable.

[0054] Based on the rotation formula, the first-order variation of the rotation formula and the vector a The variational substitution of , we get the variational of the rotation matrix, and the calculation formula is:

[0055] ;

[0056] The variation of the rotation field is represented by the linearization of the rotation pseudo vector;

[0057] ;

[0058] in, The linearization matrix representing the rotation matrix; represents the rotation pseudo vector;

[0059] Based on the variational calculation formula of the rotation matrix, the variational linearized representation of the rotation field is inverted to obtain the linearized matrix of the rotation matrix; the linearized matrix expression of the rotation matrix is:

[0060] ;

[0061] in, Indicates the module length; I represents the identity matrix; Represents the rotation pseudo vector of the skew-symmetric matrix.

[0062] From the above, it can be seen that large spatial rotations need to be represented by rotation pseudo-vectors and their corresponding rotation matrices, while continuous rotations in three-dimensional space cannot be obtained by simply adding rotation pseudo-vectors, but need to be updated by multiplying rotation matrices. If the rotation matrix is ​​directly used to store and update the rotation degrees of freedom, it will consume huge storage space and computing time.

[0063] There are usually three ways to express large rotations in three-dimensional space: Euler angles, Cardan angles, and quaternions. In the above rotation expressions, Euler angles are limited by the rotation order and cannot determine the only rotation; Cardan angles will have a gimbal lock phenomenon under certain specific rotation orders and angle combinations. Therefore, quaternions are used in this embodiment to describe the rotation state of the object.

[0064] S3: The rotation matrix, the rotation pseudo vector, and the linearization matrix of the rotation matrix are represented in quaternion form.

[0065] In this embodiment, the expression of the rotation matrix represented by the quaternion form is:

[0066] ;

[0067] ;

[0068] ;

[0069] in, Represents a rotation matrix expressed in quaternion form; Represents the real part of the quaternion; I represents the identity matrix; q Represents a quaternion; Represents the quaternion t real numbers, t ∈[1,2,3]; i represents the first imaginary unit; j represents the second imaginary unit; k represents the third imaginary unit; Represents the first real number of the quaternion; Represents the second real number of the quaternion; Represents the third real number of the quaternion; represents the rotation pseudo vector; The first pseudo vector representing the rotation t Quantity; Indicates the module length; Represents a skew-symmetric matrixed quaternion.

[0070] The expression of the rotation pseudo vector and its modulus expressed in quaternion form is:

[0071] ;

[0072] in, The first pseudo vector representing the rotation in quaternion form t Quantity; Indicates the magnitude of the rotation pseudo vector expressed in quaternion form; Represents the inverse tangent function.

[0073] The rotation pseudo vector and its modulus in the linearization matrix expression of the rotation matrix are replaced with the rotation pseudo vector and its modulus expressed in the form of quaternion, so as to obtain the linearization matrix of the rotation matrix expressed in the form of quaternion.

[0074] In this embodiment, quaternions are similar to rotation matrices and also do not satisfy the commutative rate of multiplication. They also need to follow the iterative relationship of the rotation matrix, which is expressed as:

[0075] ;

[0076] in, Indicates s The quaternion of the iteration step; Represents the iterative increment of a quaternion; Indicates s -1 The quaternion for the iteration step.

[0077] From the above analysis, it can be seen that in the process of numerical solution, large spatial rotations can be stored and updated using quaternions. When rotation pseudo-vectors, rotation matrices, and linearization matrices of rotation matrices are needed, they can all be expressed by quaternions. This can greatly reduce memory consumption during the calculation process and improve calculation efficiency.

[0078] S4: The HINGE connector is composed of a JOIN connector and a REVOUTE connector. Based on the rotation matrix represented by quaternions, the rotation pseudo vector, and the linearization matrix of the rotation matrix, the matrix forms of the kinematic constraint equations of the JOIN connector and the REVOUTE connector are constructed respectively.

[0079] like Figure 2 As shown, the HINGE connector is composed of a JOIN connector and a REVOUTE connector, wherein the JOIN connector is used to make the node a , b The relative position of the nodes remains fixed, and there is no restriction on rotation; the REVOUTE connector is used to constrain the nodes a , b The two local rotations can only rotate freely around the rotation axis, and the rotation direction of the first local rotation is opposite to that of the second local rotation.

[0080] Specifically, the process of constructing the matrix form of the kinematic constraint equation of the JOIN connector includes:

[0081] The kinematic constraint equation of the JOIN connector is expressed as:

[0082] ;

[0083] in, Represents the kinematic constraint equations of the JOIN connector; express b The displacement vector of the node; express a The displacement vector of the node; express a The node's rotation matrix in quaternion form; Indicates the rotation axis t unit basis vectors, t ∈[1,2,3]; express a The rotation pseudo vector of the node in quaternion form; T means transpose; Represents vector dot product;

[0084] The kinematic constraint equation of the JOIN connector is subjected to first-order variation and considered aThe linearization matrix of the rotation matrix represented by the quaternion form of the node obtains the first constraint equation of the first-order variation. The calculation formula of the derivation process includes:

[0085] ;

[0086] ;

[0087] ;

[0088] ;

[0089] ;

[0090] ;

[0091] in, represents the first constraint equation of the first-order variation; represents variation; express a The linearization matrix of the node's quaternion representation of the rotation matrix; represents vector cross product;

[0092] Rewrite the first constraint equation of the first-order variation into a matrix format to obtain the matrix form of the kinematic constraint equation of the JOIN connector, which is expressed as follows:

[0093] .

[0094] Furthermore, the process of constructing the matrix form of the kinematic constraint equations of the REVOUTE connector includes:

[0095] The kinematic constraint equations of the REVOUTE connector are expressed as:

[0096] ;

[0097] in, Represents the constraint equation for the first local rotation; Represents the constraint equation for the second local rotation; express a The node's rotation matrix in quaternion form; express b The node's rotation matrix in quaternion form; The first unit basis vector representing the rotation axis; The second unit basis vector representing the rotation axis; The third unit basis vector representing the rotation axis; T represents transpose; Represents vector dot product;

[0098] Perform first-order variation on the kinematic constraint equation of the REVOUTE connector to obtain the second constraint equation of the first-order variation. The calculation formula is:

[0099] ;

[0100] in, The constraint equations representing the first local rotation of the variation; The constraint equations representing the second local rotation of the variation; represents variation;

[0101] consider a , b The linearization matrix of the rotation matrix represented by the quaternion form of the node rewrites the constraint equation of the first local rotation of the variation into the linearization of the first rotation constraint equation. The calculation formula of the derivation process includes:

[0102] ];

[0103] ];

[0104] ;

[0105] ;

[0106] in, express a The linearization matrix of the rotation matrix represented by the quaternion form of the point; express b The linearization matrix of the rotation matrix represented by the quaternion form of the point; express a The node's rotation pseudo vector expressed in quaternion form; express b The node's rotation pseudo vector expressed in quaternion form; represents vector cross product;

[0107] Based on the linearized representation of the first rotation constraint equation and according to the rotation symmetry, the linearized representation of the second rotation constraint equation is obtained, and the calculation formula is:

[0108] ;

[0109] The linearized representation of the first rotation constraint equation and the linearized representation of the second rotation constraint equation are combined and rewritten into a matrix format to obtain the matrix form of the kinematic constraint equation of the REVOUTE connector, which is expressed as follows:

[0110] .

[0111] S5: Combining the matrix forms of the kinematic constraint equations of the JOIN connector and the REVOUTE connector to obtain the matrix form of the kinematic constraint equation of the HINGE connector.

[0112] Specifically, the matrix form of the kinematic constraint equation of the HINGE connector is expressed as:

[0113] ;

[0114] in, express a The displacement vector of the node; express b The displacement vector of the node; express a The node's rotation matrix in quaternion form; express b The node's rotation matrix in quaternion form; The first unit basis vector representing the rotation axis; The second unit basis vector representing the rotation axis; The third unit basis vector representing the rotation axis; T represents transpose; express a The node's rotation pseudo vector expressed in quaternion form; express b The node's rotation pseudo vector expressed in quaternion form; express a The linearization matrix of the node's quaternion representation of the rotation matrix; express b The linearization matrix of the rotation matrix represented by the quaternion form of the point; represents vector cross product; Represents variation.

[0115] S6: The kinematic constraint equations of the above-mentioned hinge connector are nonlinear equations. In classical mechanics, the Newton-Raphson method is usually used to solve the nonlinear equations. There are three methods for processing constraint equations: constraint elimination method, penalty method and Lagrange-Multiplier method. The constraint elimination method will eliminate the degrees of freedom of the slave nodes, reduce the number of equations, and improve the efficiency of solution. However, when there are coupled constraints, the elimination process will involve overly complex transformations, which increases the difficulty of solution. The penalty method converts the constraint equation into an additional part of the objective function by adding a penalty term to the objective function, thereby converting the constraint problem into an unconstrained problem. However, the calculation results are closely related to the selection of penalty parameters, and appropriate penalty parameters must be selected to obtain correct results. The Lagrange multiplier method merges the constraint equation of the objective function into the objective function by introducing additional multipliers. Although this method will increase the number of equations, it does not involve complex transformations or the selection of penalty parameters, and is easier to program and calculate complex problems. Therefore, this embodiment will use the Newton-Raphson and Lagrange multiplier methods to solve the nonlinear constraint equations and obtain the displacement increment and constraint force of the node in the HINGE connector.

[0116] The specific linearization format is derived as follows. Consider a target optimization problem to minimize the target optimization function:

[0117] ;

[0118] Among them, F( q ) is the target optimization function, is the constraint function.

[0119] Assume that the target constraint exists n degrees of freedom, the number of equations of the constraint function is m , we can define n ×1 size residual vector:

[0120] ;

[0121] in, r ( q ) is the residual vector, Represents partial derivative.

[0122] Define the constraint Jacobian matrix:

[0123] ;

[0124] in, express m × n The size of the constrained Jacobian matrix, ; .

[0125] Define the objective function Hessian matrix:

[0126] ;

[0127] in, for n × n The Hessian matrix of the objective function of size, .

[0128] By introducing the Lagrange multiplier, the objective optimization function considering the constraint function can be rewritten as:

[0129] ;

[0130] in, To introduce a new objective optimization function, is the Lagrange multiplier, and the superscript T indicates transposition. The first-order variation of the above equation is:

[0131] ;

[0132] After simplification, it can be written as:

[0133] ;

[0134] For any and All of them hold true, so we can get the control equation:

[0135] ;

[0136] To solve the nonlinear equation using the Newton-Raphson method, q and The linearized format of can be written as:

[0137] ;

[0138] Among them, the superscript " * " represents the convergence value before this iteration step, and the symbol " " represents the increment. Taking the first-order differential of the control equation, substituting the residual vector, constraint Jacobian matrix and Hessian matrix, and considering the linearized format, we get:

[0139] ;

[0140] in, H is the Hessian matrix, B is the constraint Jacobian matrix, is the Lagrange multiplier, q is the unknown quantity, The residual vector before this iteration step is This is the constraint margin before this iteration. In general, the second-order derivative of the constraint condition is It is a minimum value relative to the Hessian matrix and can be ignored during the calculation process.

[0141] From the above derivation, the Newton-Raphson incremental iteration format of the Lagrange multiplier method is obtained, which can be used to solve nonlinear constraint problems.

[0142] like Figure 3 As shown, this embodiment gives a simple pendulum example to verify the correctness and efficiency of this method.

[0143] Since the simple pendulum is a rigid rod, it is only necessary to divide the nodes and apply boundary conditions to the simple pendulum example: divide it into 3 nodes, where a , b It is the hinge point of the HINGE connector. b , c A rigid rod, c is a particle; a Node 1-5 degrees of freedom constraints, c Node application x A predetermined displacement with a magnitude of 3.0 in the direction, such as Figure 4 shown.

[0144] For fixed constraints, this embodiment uses the method of setting large numbers to process, and other constraints and boundaries are processed using constraint equations, and the following constraint equations can be obtained:

[0145] ;

[0146] in, Represents the HINGE connector constraint. represents a rigid rod constraint, express c Predetermined displacement constraints of nodes; express a The displacement vector of the node; express b The node's displacement vector; express c The displacement vector of the node; express a The node's rotation pseudo vector expressed in quaternion form; express bThe node's rotation pseudo vector expressed in quaternion form; express c The node's rotation pseudo vector expressed in quaternion form; express x The displacement increment in the direction. In the initialized total rigid matrix, the corresponding a The main diagonal elements of the 1-5 degrees of freedom of the node are set to 1E36, indicating fixed constraints; then the total stiffness matrix can be written in the matrix form of the kinematic constraint equation of the hinge connector, and follows Figure 5 The calculation process is shown.

[0147] Figure 5 The calculation flow shown includes:

[0148] Step 1: Calculate the incremental quaternion based on the initial displacement of the node;

[0149] Step 2: Update the current quaternion using the incremental quaternion;

[0150] Step 3: Calculate the rotation pseudo vector and rotation matrix based on the updated quaternion;

[0151] Step 4: Calculate the tangent operator matrix (i.e., the linearization matrix of the rotation matrix) based on the rotation pseudo vector;

[0152] Step 5: Collect the rotation matrix and the tangent operator matrix and calculate the constrained Jacobian matrix;

[0153] Step 6: Calculate the multiplier constraint force based on the constraint Jacobian matrix and assemble the constraint Jacobian matrix into the total rigid matrix;

[0154] Step 7: Calculate the residual of the constraint equation based on the rotation pseudo vector;

[0155] Step 8: Assemble the multiplier constraint force and the residual of the constraint equation into a residual force vector;

[0156] Step 9: Determine whether the residual force vector meets the accuracy requirement. If it does, end; if not, continue.

[0157] Step 10: Calculate the incremental displacement based on the total stiffness matrix and the residual force vector;

[0158] Step 11: Calculate the incremental quaternion according to the incremental displacement;

[0159] Step 12: Repeat steps 2-9 until convergence, and obtain the calculation results as shown in Table 1.

[0160] Table 1 Calculation results

[0161] ;

[0162] For this example, only c The motion of a point can be simply solved theoretically:

[0163] ;

[0164] in, express x The displacement increment in the direction; express y The displacement increment in the direction;

[0165] By comparing the calculation results with the theoretical solution, it can be seen that the calculation results are consistent with the theoretical solution after only one iteration, which proves the correctness and efficiency of the method.

[0166] In this embodiment, the algorithm provided in this embodiment is implemented by using the Mxsim general structure software to calculate actual engineering examples (eg, engine piston rigid body motion model), and correct results are obtained.

[0167] The HINGE connector calculation method based on the implicit flexible structure solver provided in this embodiment has the following beneficial effects:

[0168] 1. The multi-body kinematic model based on finite element mesh modeling can automatically represent the geometric shape, and the large deformation / large rotation effect can be built into the finite element formula. Under the premise of considering the nonlinearity of the material, a series of mechanical responses such as stress and strain of the flexible body can be obtained.

[0169] 2. Solving multi-body kinematic problems based on implicit structural solvers can perform high-precision analysis of complex structures and eliminate the influence of time steps on calculation results.

[0170] 3. Quaternion-based storage and update represent large compound rotations in space, which reduces memory consumption during the calculation process and improves calculation efficiency.

[0171] 4. The constraint equations are processed based on the Lagrange multiplier method, which avoids the influence of penalty factors on the calculation results. At the same time, it is easy to implement and the multiplier output constraint reaction force and moment can be directly extracted.

[0172] The technical features of the above-described embodiments may be arbitrarily combined. To make the description concise, not all possible combinations of the technical features in the above-described embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0173] The above-described embodiments only express several implementation methods of the present application, and the descriptions thereof are relatively specific and detailed, but they cannot be construed as limiting the scope of the patent application. It should be pointed out that, for a person of ordinary skill in the art, several variations and improvements can be made without departing from the concept of the present application, and these all belong to the protection scope of the present application. Therefore, the protection scope of the patent application shall be subject to the attached claims.

Claims

1. A HINGE connector calculation method based on an implicit flexible structure solver, characterized in that: include: S1: According to the Euler principle, the large spatial rotation of the hinge connector is decomposed into a rotation axis and a rotation angle, and the rotation variable formed by the rotation axis and the rotation angle is defined as a rotation pseudo vector; S2: constructing a rotation matrix based on the rotation pseudo vector, performing first-order variation and equivalent substitution on the rotation formula constructed based on the rotation matrix, and obtaining a linearized matrix of the rotation matrix; S3: using quaternion form to represent the rotation matrix, the rotation pseudo vector, and the linearization matrix of the rotation matrix; S4: The HINGE connector is composed of a JOIN connector and a REVOUTE connector. Based on the rotation matrix represented by quaternion, the rotation pseudo vector, and the linearization matrix of the rotation matrix, the matrix forms of the kinematic constraint equations of the JOIN connector and the REVOUTE connector are respectively constructed; S5: combining the matrix forms of the kinematic constraint equations of the JOIN connector and the REVOUTE connector to obtain the matrix form of the kinematic constraint equation of the HINGE connector; S6: The Newton-Raphson and Lagrange multiplier methods are used to solve the matrix form of the kinematic constraint equations of the HINGE connector, and the displacement increments and constraint forces of the nodes in the HINGE connector are obtained.

2. The HINGE connector calculation method based on implicit flexible structure solver according to claim 1 is characterized in that: The expression of the rotation pseudo vector is: ； in, represents the rotation pseudo vector; Represents the rotation pseudo vector The first component of Represents the rotation pseudo vector The second component of Represents the rotation pseudo vector The third component of The first unit basis vector representing the rotation axis; The second unit basis vector representing the rotation axis; The third unit basis vector representing the axis of rotation.

3. The HINGE connector calculation method based on implicit flexible structure solver according to claim 1 is characterized in that: The constructing a rotation matrix based on the rotation pseudo vector comprises: The rotation pseudo vector is skew-symmetrically expanded to obtain the spin tensor; the calculation formula is: ； in, S represents the spin tensor; represents oblique symmetric expansion; represents the rotation pseudo vector; Represents the rotation pseudo vector The first component of Represents the rotation pseudo vector The second component of Represents the rotation pseudo vector The third component of Based on the properties of the spin tensor, the spin tensor is exponentially expanded to obtain the rotation matrix; The properties of the optional tensor are: ； in, Indicates the module length; I represents the identity matrix; represents tensor product; The expression of the rotation matrix is: ； in, C represents the rotation matrix; exp represents the exponential expansion; Represents the rotation pseudo vector of the skew-symmetric matrix.

4. The HINGE connector calculation method based on implicit flexible structure solver according to claim 1 is characterized in that: The first-order variation and equivalent replacement of the rotation formula constructed based on the rotation matrix include: The rotation formula constructed based on the rotation matrix is ​​expressed as: ； in, C represents the rotation matrix; A represents any vector in space; a Represents vector A A vector after rotation in space; Represents vector dot product; The rotation formula is subjected to first-order variation, and the calculation formula is: ； in, represents variation; Vector a The variation of is replaced by the variation of the rotation field variable, and the calculation formula is: ； in, represents the rotation field variable; represents the linearization of the rotation field variables; represents vector cross product; represents the skew-symmetric matrixed rotation field variable; Based on the rotation formula, the first-order variation of the rotation formula and the vector a The variational substitution of , we get the variational of the rotation matrix, and the calculation formula is: ； The variation of the rotation field is represented by the linearization of the rotation pseudo vector; ； in, The linearization matrix representing the rotation matrix; represents the rotation pseudo vector; Based on the variational calculation formula of the rotation matrix, the variational linearized representation of the rotation field is inverted to obtain the linearized matrix of the rotation matrix; the linearized matrix expression of the rotation matrix is: ； in, Indicates the module length; I represents the identity matrix; Represents the rotation pseudo vector of the skew-symmetric matrix.

5. The HINGE connector calculation method based on implicit flexible structure solver according to claim 4 is characterized in that: The rotation matrix is ​​iterated in multiple steps to derive the iterative relationship of the rotation matrix. The iterative formula is: ； in, Indicates s The rotation matrix of the iteration step; Indicates s The iteration increment of the iteration step; Indicates s -1 rotation matrix for the iteration step.

6. The HINGE connector calculation method based on implicit flexible structure solver according to claim 1 is characterized in that: The expression of the rotation matrix expressed in quaternion form is: ； ； ； in, Represents a rotation matrix expressed in quaternion form; Represents the real part of the quaternion; I represents the identity matrix; q Represents a quaternion; Represents the quaternion t real numbers, t ∈[1,2,3]; i represents the first imaginary unit; j represents the second imaginary unit; k represents the third imaginary unit; Represents the first real number of the quaternion; Represents the second real number of the quaternion; Represents the third real number of the quaternion; represents the rotation pseudo vector; The first pseudo vector representing the rotation t Quantity; Indicates the module length; Represents a skew-symmetric matrixed quaternion; The expression of the rotation pseudo vector and its modulus expressed in quaternion form is: ； in, The first pseudo vector representing the rotation in quaternion form t Quantity; Indicates the magnitude of the rotation pseudo vector expressed in quaternion form; represents the inverse tangent function; The rotation pseudo vector and its modulus in the linearization matrix expression of the rotation matrix are replaced with the rotation pseudo vector and its modulus expressed in the form of quaternion, so as to obtain the linearization matrix of the rotation matrix expressed in the form of quaternion.

7. The HINGE connector calculation method based on implicit flexible structure solver according to claim 1 is characterized in that: The process of constructing the matrix form of the kinematic constraint equations of the JOIN connector includes: The JOIN connector is used to make the node a , b The relative position remains fixed, and there is no restriction on rotation; the kinematic constraint equation of the JOIN connector is expressed as: ； in, Represents the kinematic constraint equations of the JOIN connector; express b The node's displacement vector; express a The displacement vector of the node; express a The node's rotation matrix in quaternion form; The first t unit basis vectors, t ∈[1,2,3]; express a The rotation pseudo vector of the node in quaternion form; T means transpose; Represents vector dot product; The kinematic constraint equation of the JOIN connector is subjected to first-order variation and considered a The linearization matrix of the rotation matrix expressed in the quaternion form of the node obtains the first constraint equation of the first-order variation, which is calculated as follows: ； in, represents the first constraint equation of the first-order variation; represents variation; express a The linearization matrix of the node's quaternion representation of the rotation matrix; represents vector cross product; Rewrite the first constraint equation of the first-order variation into a matrix format to obtain the matrix form of the kinematic constraint equation of the JOIN connector, which is expressed as follows: 。 8. The HINGE connector calculation method based on implicit flexible structure solver according to claim 7 is characterized in that: The process of constructing the matrix form of the kinematic constraint equations for the REVOUTE connector includes: The REVOUTE connector is used to constrain the node a , b The two local rotations of the REVOUTE connector can only rotate freely around the rotation axis; the kinematic constraint equation of the REVOUTE connector is expressed as: ； in, Represents the constraint equation for the first local rotation; Represents the constraint equation for the second local rotation; express a The node's rotation matrix in quaternion form; express b The node's rotation matrix in quaternion form; The first unit basis vector representing the rotation axis; The second unit basis vector representing the rotation axis; The third unit basis vector representing the rotation axis; T represents transpose; Represents vector dot product; Perform first-order variation on the kinematic constraint equation of the REVOUTE connector to obtain the second constraint equation of the first-order variation. The calculation formula is: ； in, The constraint equations representing the first local rotation of the variation; The constraint equations representing the second local rotation of the variation; represents variation; consider a , b The linearization matrix of the rotation matrix represented by the quaternion form of the node rewrites the constraint equation of the first local rotation of the variation into the linearization of the first rotation constraint equation. The calculation formula is: ； in, express a The linearization matrix of the rotation matrix represented by the quaternion form of the point; express b The linearization matrix of the rotation matrix represented by the quaternion form of the point; express a The node's rotation pseudo vector expressed in quaternion form; express b The node's rotation pseudo vector expressed in quaternion form; represents vector cross product; Based on the linearized representation of the first rotation constraint equation and according to the rotation symmetry, the linearized representation of the second rotation constraint equation is obtained, and the calculation formula is: ； The linearized representation of the first rotation constraint equation and the linearized representation of the second rotation constraint equation are combined and rewritten into a matrix format to obtain the matrix form of the kinematic constraint equation of the REVOUTE connector, which is expressed as follows: 。 9. The HINGE connector calculation method based on implicit flexible structure solver according to claim 8 is characterized in that: The matrix form of the kinematic constraint equation of the HINGE connector is expressed as: ； in, express a The displacement vector of the node; express b The displacement vector of the node; express a The node's rotation matrix in quaternion form; express b The node's rotation matrix in quaternion form; The first unit basis vector representing the rotation axis; The second unit basis vector representing the rotation axis; The third unit basis vector representing the rotation axis; T represents transpose; express a The node's rotation pseudo vector expressed in quaternion form; express b The node's rotation pseudo vector expressed in quaternion form; express a The linearization matrix of the node's quaternion representation of the rotation matrix; express b The linearization matrix of the rotation matrix represented by the quaternion form of the point; represents vector cross product; Represents variation.

10. The HINGE connector calculation method based on implicit flexible structure solver according to claim 8, characterized in that: The first partial rotation is opposite to the second partial rotation.

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