A virtual body-based flexible multi-body system dynamics sliding joint modeling method

Abstract

The application discloses a virtual body-based flexible multi-body system dynamics sliding hinge modeling method, and belongs to the technical field of flexible multi-body system dynamics simulation. First, a kinematics model of a floating coordinate system of a flexible body is established, and a sliding point is defined by fixing interpolation parameters; then, a local geometric frame of the sliding point is constructed according to current coordinates of the sliding point and neighbor nodes of the sliding point, and a virtual rigid body is introduced at the sliding point, so that a reference point of the virtual rigid body is coincided with the sliding point, and a local frame of the virtual rigid body keeps a constant relative rotation relationship with the local geometric frame of the sliding point; relative motion between a slider rigid body and the virtual rigid body is expressed as a standard rigid body sliding hinge constraint, and is assembled into a differential-algebraic equation set of a flexible multi-body system for solving. The application can describe sliding motion of a flexible structure under large deformation and a spatial bending path, and is suitable for simulation of rigid-flexible coupling systems such as a flexible guide rail, a cable driving mechanism, a pantograph-catenary system and a pulley mechanism.

Description

Technical Field

[0001] This invention belongs to the field of dynamic simulation technology of flexible multibody systems, and relates to a joint constraint modeling method in rigid-flexible coupled systems. In particular, it relates to a sliding hinge modeling method for flexible multibody systems based on virtual bodies, which is applicable to dynamic modeling and simulation of deformable guide rails, spatial flexible tracks, cable-driven mechanisms, pantograph-contact network systems, pulley mechanisms, and other mechanical systems containing rigid-flexible coupled sliding motion. Background Technology

[0002] Flexible multibody systems are widely found in robotics, vehicle engineering, aerospace structures, cable-driven mechanisms, pantograph-contact network systems, and flexible guide rail mechanisms. In these systems, relative sliding motion frequently occurs between rigid components and flexible structures, such as the movement of a slider along a flexible guide rail, the movement of a pulley along a flexible cable, and sliding contact between the pantograph and the flexible contact network. This type of sliding motion involves both the constrained motion of the rigid components and the deformation of the flexible structure, making it a typical problem in rigid-flexible coupling dynamics modeling.

[0003] Unlike prismatic pairs in rigid body systems, the sliding hinge constraint points in flexible multibody systems move with the deformation of the flexible structure. Their position, local tangential direction, and normal direction all change with the instantaneous configuration of the structure. Therefore, this type of problem typically manifests as a moving boundary problem. Existing modeling methods for sliding hinges in flexible multibody systems usually require the introduction of arc length parameters, intermediate coordinates, variable-length elements, or element switching strategies to track the motion position of the sliding points on the flexible structure. While these methods can describe the sliding motion on flexible structures to some extent, they increase the number of generalized coordinates and the scale of constraint equations, making the model implementation more complex and reducing their applicability in general multibody dynamics solution frameworks.

[0004] In the prior art, Chinese invention patent (application number: 201710260407.1) proposes a rigid-flexible coupling dynamics modeling method based on the Simulink platform. This method achieves parametric modeling and simulation of rigid-flexible coupling systems by dividing the system into rigid and flexible subsystems, performing modal analysis on the flexible components, and constructing subsystem models. While this method focuses on platform-based modeling of rigid-flexible coupling systems and can improve the automation level of modeling, it does not specifically address the issue of slip points on the surface of the flexible body moving with the deformation of the flexible body, nor does it provide a constraint construction method for the local orientation of the slip points as the instantaneous configuration of the flexible body updates.

[0005] Chinese invention patent (application number: 202310486356.X) proposes a method, device, and electronic device for analyzing system state. By establishing a local coordinate system, virtual power equations, and constraint equations, it transforms connection types such as rotational hinges, sliding hinges, and ball hinges into corresponding constraint equations to obtain system dynamic response data. This method can describe the dynamic response of systems with various connection types, but its sliding hinge constraints are mainly based on a defined local coordinate system and component motion variables, focusing on system state analysis and dynamic response solving. For flexible bodies discretized using solid finite element methods and where nodes do not have explicit rotational degrees of freedom, this method does not provide a method for constructing a local geometric frame for the sliding point from the current spatial position of the nodes on the flexible body surface, nor can it directly solve the problem of real-time updating of the sliding direction when the sliding point moves along the deformable flexible structure.

[0006] In summary, while existing technologies can describe the dynamic behavior of rigid-flexible coupled systems to some extent, a dedicated modeling method for sliding hinges on flexible body surfaces is still lacking. Especially for flexible bodies discretized using solid finite element methods and whose nodes lack explicit rotational degrees of freedom, existing methods struggle to achieve continuous updates of the slip point with local deformation of the flexible body, real-time updates of the slip direction, and compatibility with standard rigid body sliding hinge constraints without introducing additional arc-length coordinates, intermediate coordinates, or complex element switching strategies. Therefore, there is an urgent need to propose a modeling method for flexible multibody sliding hinges based on virtual bodies to address the problems of complex slip point tracking, difficulty in updating local directions, numerous additional variables, and insufficient compatibility with general multibody dynamics solvers in existing methods. Summary of the Invention

[0007] This invention addresses the problems of existing flexible multibody sliding hinge modeling methods, such as numerous extraneous variables, complex constraint construction, difficulties in handling moving boundaries, and insufficient compatibility with general multibody dynamics solvers. It proposes a sliding hinge modeling method for flexible multibody systems based on virtual bodies. This invention describes the current sliding point on the flexible body surface using fixed interpolation parameters and introduces a virtual rigid body at the current sliding point, updated with the local geometric frame. This transforms the moving sliding constraints on the flexible body surface into sliding hinge position-level constraints between the sliding rigid body and the virtual rigid body. This invention avoids introducing extra arc-length coordinates, intermediate coordinates, or complex element switching strategies, and can describe the sliding motion of flexible structures under large deformations and spatial bending paths.

[0008] To achieve the above objectives, the technical solution of the present invention is as follows: A method for modeling the sliding hinge dynamics of a flexible multibody system based on virtual bodies includes the following steps.

[0009] Step 1: Establish a kinematic model of the flexible body in a floating coordinate system to obtain the current spatial position of each node of the flexible body in the global coordinate system. Specifically: Step 1.1: Define the global coordinate system, the flexible body reference coordinate system, and the flexible body generalized coordinates; Define the global coordinate system as Define the flexible body reference coordinate system as The flexible body reference coordinate system moves with the overall rigid body motion of the flexible body and is used to describe the overall translation, overall rotation, and elastic deformation of the flexible body relative to the flexible body reference coordinate system.

[0010] The overall motion of the flexible body is decomposed into the translation of the origin of the flexible body's reference coordinate system in the global coordinate system, the rotation of the flexible body's reference coordinate system relative to the global coordinate system, and the elastic deformation of the flexible body relative to the flexible body's reference coordinate system, thus obtaining the generalized coordinate system of the flexible body. : (1) in, For the generalized coordinate vector of the flexible body; This is the translational coordinate vector of the origin of the flexible body's reference coordinate system in the global coordinate system. This is the rotation coordinate vector of the flexible body's reference coordinate system relative to the global coordinate system; This is the modal coordinate vector of the flexible body, used to describe the elastic deformation of the flexible body relative to the reference coordinate system of the flexible body; the superscript T indicates the transpose of the matrix or vector.

[0011] Step 1.2: Based on the generalized coordinates of the flexible body obtained in Step 1.1, establish the expression for the position of the flexible body nodes; The spatial position of a flexible body node is determined by the position of the origin of the flexible body's reference coordinate system, the orientation of the flexible body's reference coordinate system, the node's position in its undeformed configuration, and the node's elastic deformation. (The last part, "Flexible body...," appears to be an incomplete sentence or fragment.) The position vector of each node in the global coordinate system Represented as: (2) in, For flexible bodies The position vector of each node in the global coordinate system; Number the nodes of the flexible body; This is the position vector of the origin of the flexible body's reference coordinate system in the global coordinate system; This is the attitude transformation matrix from the flexible body's reference coordinate system to the global coordinate system; For the first The position vector of each node in the undeformed configuration of the flexible body; For the first The mode matrix at each node; is the modal coordinate vector of the flexible body.

[0012] Step 1.3: Based on the flexible body node position expression established in Step 1.2, obtain the flexible body modal information and calculate the current spatial position of each node of the flexible body in the global coordinate system; The elastic deformation of the flexible body is described by modal coordinates, and the modal information is obtained through finite element modal analysis. For the flexible body discretized by solid finite element, the node coordinates, element topology and modal matrix are extracted from the finite element model, and combined with the modal coordinates of the flexible body, the extracted modal information is substituted into equation (2) to calculate the position of each node under the current configuration of the flexible body, and the current spatial position of each node of the flexible body in the global coordinate system is obtained.

[0013] Step 2: Based on the current spatial positions of each node of the flexible body in the global coordinate system obtained in Step 1, define a sliding path on the surface of the flexible body, select the interpolation region of the sliding path, establish the interpolation expression for the sliding point position, and determine the bilinear interpolation shape function. This yields the interpolation expression for the sliding path and sliding point position on the flexible body surface, jointly described by the quadrilateral interpolation region, interpolation parameters, and the bilinear interpolation shape function. Specifically: Step 2.1: Based on the current spatial position of each node of the flexible body in the global coordinate system obtained in Step 1, define the sliding path on the surface of the flexible body and select the sliding path interpolation region to obtain the quadrilateral interpolation region used to describe the position of the sliding point. In this invention, the flexible body surface refers to the outer surface in the finite element model of the flexible body that allows the slider or sliding member to slide relative to each other; the sliding path refers to the path that is pre-defined on the flexible body surface to define the direction of movement of the slider or sliding member; the sliding path point is a discrete point used to describe the geometry of the sliding path; the current sliding point refers to the position point of the slider or sliding member on the sliding path at a certain moment.

[0014] Several surface nodes are selected on the surface of the flexible body to form the sliding path interpolation region. For the flexible body discretized using solid finite element method, four surface nodes are selected to form a quadrilateral interpolation region, and the position of the sliding point on the surface of the flexible body is described by the quadrilateral interpolation region.

[0015] The slip point is not directly bound to a discrete node, but is defined by a quadrilateral interpolation region and interpolation parameters. This results in a quadrilateral interpolation region describing the slip point's position, avoiding positional discontinuities when the slip point jumps from one node or element to another.

[0016] Step 2.2: Based on the quadrilateral interpolation region obtained in Step 2.1, establish the interpolation expression for the position of the sliding point to obtain the position expression of the sliding point in the global coordinate system; For a quadrilateral interpolation region consisting of four nodes, the slip point position is represented as: (3) in, This represents the position vector of the sliding point in the global coordinate system. This represents the current spatial position of the first node within the interpolation region. This represents the current spatial position of the second node within the interpolation region. This represents the current spatial location of the third node within the interpolation region. This represents the current spatial position of the fourth node within the interpolation region; For the first interpolation parameter, This is the second interpolation parameter; the two together constitute the interpolation parameters. Let be the bilinear interpolation form function corresponding to the first node. The bilinear interpolation form function corresponding to the second node. The bilinear interpolation form function corresponding to the third node. This is the bilinear interpolation form function corresponding to the fourth node.

[0017] Step 2.3: Based on the position expression of the sliding point in the global coordinate system obtained in Step 2.2, determine the bilinear interpolation shape function, and obtain the flexible body surface sliding path and sliding point position interpolation expression jointly described by the quadrilateral interpolation region, interpolation parameters and bilinear interpolation shape function; When the interpolation parameter takes values ​​in the range of , When the bilinear interpolation shape function takes the following form: (4) in, , , and These are the bilinear interpolation shape functions corresponding to the first, second, third, and fourth nodes within the quadrilateral interpolation region, respectively.

[0018] Step 3: Based on the interpolation expressions for the sliding path and sliding point position of the flexible body surface obtained in Step 2, solve for the fixed interpolation parameters of the sliding point under the initial configuration. Keep these fixed interpolation parameters unchanged during the dynamic simulation to obtain the fixed interpolation parameters of the sliding point and the calculation method for updating the sliding point position based on these fixed interpolation parameters. Specifically: Step 3.1: Based on the slip point position interpolation expression obtained in Step 2, given the target slip point position under the initial configuration, obtain the initial interpolation data used to solve the fixed interpolation parameters; In the initial configuration, the target position of the sliding point is given according to the preset sliding path on the surface of the flexible body. .set up Let be the target position vector of the slip point under the initial configuration. , , and These are the position vectors of the first, second, third, and fourth nodes within the quadrilateral interpolation region, respectively, under the initial configuration. The superscript 0 indicates the initial configuration. Let be the position vector of the first node in the initial configuration. Let be the position vector of the second node in the initial configuration. Let be the position vector of the third node in the initial configuration. This is the position vector of the fourth node in the initial configuration.

[0019] Thus, initial interpolation data is obtained for solving the fixed interpolation parameters, the initial interpolation data including the target position of the sliding point. The initial position vectors of the four nodes within the quadrilateral interpolation region. , , , And the bilinear interpolation shape function determined in step 2.

[0020] Step 3.2: Based on the initial interpolation data obtained in Step 3.1, solve the fixed interpolation parameters of the slip point using the least squares problem to obtain the first fixed interpolation parameters and the second fixed interpolation parameters; Based on the target position of the glide point in the initial configuration Using the initial position vectors of the four nodes within the quadrilateral interpolation region, construct the following least squares problem: (5) in, To obtain the fixed interpolation parameters, The first fixed interpolation parameter is... This is the second fixed interpolation parameter; The first interpolation parameter is... This is the second interpolation parameter; The target position vector of the slip point under the initial configuration; Number the nodes within the quadrilateral interpolation region. =1, 2, 3, 4 correspond to the first node, the second node, the third node, and the fourth node, respectively; For the first The position vectors of each node under the initial configuration; For the first The bilinear interpolation shape function corresponding to each node; This represents the Euclidean norm.

[0021] The first fixed interpolation parameter is obtained by solving equation (5). Second fixed interpolation parameter .

[0022] Step 3.3: Based on the first fixed interpolation parameter and the second fixed interpolation parameter obtained in Step 3.2, keep the fixed interpolation parameter unchanged and update the position of the sliding point to obtain the fixed interpolation parameter of the sliding point and the calculation method of updating the position of the sliding point based on the fixed interpolation parameter; Throughout the entire dynamic simulation, the first fixed interpolation parameter is maintained. Second fixed interpolation parameter The positions of the first, second, third, and fourth nodes within the quadrilateral interpolation region remain unchanged as the flexible body deforms. At this time, the first interpolation parameter in equation (3) is updated accordingly. Second interpolation parameter Take them as the first fixed interpolation parameters respectively. Second fixed interpolation parameter Substituting the current spatial positions of the four nodes within the quadrilateral interpolation region into equation (3), the position of the current sliding point in the global coordinate system can be obtained.

[0023] The current spatial positions of the four nodes within the quadrilateral interpolation region are determined by formula (2) in step 1; the first fixed interpolation parameter Second fixed interpolation parameter The slip point remains unchanged during the simulation. Therefore, the slip point can be continuously updated with the local deformation of the flexible body, avoiding the position discontinuity problem caused by directly binding the slip point to discrete nodes or switching elements. This yields fixed interpolation parameters for the slip point and a calculation method for updating the slip point position based on these fixed interpolation parameters.

[0024] Step 4: Based on the fixed interpolation parameters of the slip point obtained in Step 3 and the slip path on the flexible body surface obtained in Step 2, calculate the current position of the current slip point, adjacent path points, and auxiliary points in the global coordinate system, and construct the first basis vector, second basis vector, and third basis vector at the slip point to obtain the local geometric frame at the slip point that is updated synchronously with the local configuration of the flexible body. Specifically: Step 4.1: Based on the fixed interpolation parameters of the sliding point obtained in Step 3, calculate the current position of the current sliding point and the adjacent path points in the global coordinate system; To describe the characteristic of the slip direction changing with the deformation of the flexible body, a local geometric frame is constructed at the current slip point. This local geometric frame consists of a first basis vector, a second basis vector, and a third basis vector. The first basis vector is determined along the local tangential direction of the slip path, the second basis vector is obtained by orthogonalizing the directions of auxiliary points, and the third basis vector is obtained by the cross product of the first and second basis vectors. All of these basis vectors are represented in the global coordinate system.

[0025] Wherein, the position vector of the current sliding point in the global coordinate system is denoted as It is calculated by the interpolation formula shown in equation (3) based on the fixed interpolation parameters and the current node coordinates of the quadrilateral interpolation region: (6) The position vectors of adjacent path points before and after the current slip point in the global coordinate system are denoted as follows: and It also has corresponding interpolation parameters. and Substituting into the interpolation formula shown in equation (3), we obtain: (7) (8) in, This represents the position vector of the current sliding point in the global coordinate system. This is the position vector of the previous adjacent path point in the global coordinate system; This is the position vector of the next adjacent path point in the global coordinate system; For the quadrilateral interpolation region containing the current slip point, the first... The current spatial location of each node; For the quadrilateral interpolation region containing the previous adjacent path point, the first... The current spatial location of each node; For the quadrilateral interpolation region containing the next adjacent path point, the first The current spatial location of each node; Number the nodes within the quadrilateral interpolation region. =1, 2, 3, 4 correspond to the first node, the second node, the third node, and the fourth node, respectively.

[0026] Step 4.2: Based on the preceding and following adjacent path points obtained in Step 4.1, construct local tangential basis vectors; Based on the adjacent path points before and after the current slip point and Define the local tangential vector of the slip path. : (9) The first basis vector of the tangent of the slip path for: (10) in, This is the local tangential vector of the slip path; This is the position vector of the next adjacent path point before the current sliding point in the global coordinate system; This is the position vector of the next adjacent path point after the current sliding point in the global coordinate system; is the first basis vector of the local geometric frame, used to represent the local tangential direction of the slip path near the current slip point; This represents the Euclidean norm.

[0027] Step 4.3: Based on the current glide point and the first basis vector, determine the auxiliary point and construct the second basis vector.

[0028] To determine the second basis vector of the local geometric frame, auxiliary points are selected. The position vector of the auxiliary point in the global coordinate system is denoted as . The auxiliary point can be selected as a surface node near the current sliding point that is not collinear with the local tangential direction of the sliding path, or it can be selected from a quadrilateral interpolation region near the current sliding point using interpolation parameters. The calculation yielded: (11) in, This is the position vector of the auxiliary point in the global coordinate system; The first interpolation parameter corresponding to the auxiliary point; The second interpolation parameter corresponding to the auxiliary point; For the quadrilateral interpolation region containing auxiliary points, the first The current spatial location of each node.

[0029] Defined by the current slip point Point to auxiliary point auxiliary direction vector : (12) To ensure the orthogonality of the local geometric frame, auxiliary direction vectors are removed. First basis The component in the direction, obtained with orthogonal vectors : (13) Then the second basis vector of the local geometric frame for: (14) Step 4.4: Construct a local geometric frame based on the first and second basis vectors; Third basis From the first basis Second basis Cross product yields: (15) This yields the local geometric frame at the slip point. : (16) because , , and All values ​​are calculated from the position of the current node of the flexible body in the global coordinate system. Therefore, when the flexible body deforms, the local geometric frame... It can update synchronously with the local configuration of the flexible body. For flexible bodies discretized using solid finite element methods and whose nodes do not have explicit rotational degrees of freedom, the local geometric frame... It can be used to characterize the local attitude and slip direction at the slip point. Thus, a local geometric frame at the slip point is obtained, which is updated synchronously with the local configuration of the flexible body. .

[0030] Step 5: Based on the local geometric frame at the slip point obtained in Step 4 A virtual rigid body is introduced at the slip point, and a reference point and a local frame for the virtual rigid body are determined to obtain a virtual rigid body used to transmit the local position and attitude of the slip point. Specifically: Step 5.1: Set the virtual rigid body reference point; A virtual rigid body is introduced at the slip point. The virtual rigid body is a massless rigid body or a mass that is negligible. It is not used to represent actual mass and inertia, but to convey the local position and attitude information of the flexible body at the slip path.

[0031] Make the reference point of the virtual rigid body coincide with the slip point: (17) in, This is the position vector of the virtual rigid body reference point in the global coordinate system; This is the position vector of the current sliding point in the global coordinate system.

[0032] Step 5.2: Set up a virtual rigid body local frame; Let the local geometric frame at the slip point be... The virtual rigid body local frame is This ensures a constant relative rotational relationship between the virtual rigid body local frame and the local geometric frame at the slip point: (18) in, This is the constant relative rotation matrix determined under the initial configuration. Specifically, under the initial configuration, it is based on the local geometric frame at the slip point. and the preset virtual rigid body initial local frame Determine the constant relative rotation matrix: (19) During the dynamic simulation process The virtual rigid body reference point remains unchanged. As a result, the virtual rigid body reference point is updated synchronously with the position of the slip point, and the virtual rigid body local frame is updated synchronously with the local geometric frame at the slip point, thus obtaining a virtual rigid body used to transmit the local position and attitude of the slip point.

[0033] Step 6: Based on the virtual rigid body obtained in Step 5, construct the sliding hinge position-level constraint between the slider rigid body and the virtual rigid body, obtaining the complete sliding hinge position-level constraint vector. Specifically: Step 6.1: Define the sliding hinge connection point; The slider rigid body is denoted as a rigid body. The virtual rigid body is denoted as a rigid body. In a rigid body With rigid body Establish position-level constraints at the sliding hinge between the rigid bodies to enable them to... Relative to rigid body Only the relative translational degrees of freedom along the local slip direction are retained.

[0034] Assume a rigid body The position vector of the sliding hinge connection point on the upper surface in the global coordinate system is: ,rigid body The position vector of the sliding hinge connection point on the upper surface in the global coordinate system is: .

[0035] Step 6.2: Define the connection vector between connection points; Define the connection vector between two connection points. for: (20) Step 6.3: Construct translational constraints for the sliding hinge; set up and Virtual rigid body In a local frame, two normal unit vectors orthogonal to the slip direction. The first normal unit vector is orthogonal to the slip direction. The second normal unit vector is orthogonal to the sliding direction; to make the slider a rigid body Only along virtual rigid bodies The defined slip direction undergoes relative translation, which should cause the connecting vector to... respectively with and Orthogonal, we obtain the first and second translational constraints of the sliding hinge: (twenty one) Step 6.4: Construct the rotational constraint of the sliding hinge; set up For the slider rigid body The direction of the sliding axis in the diagram. To ensure the rigidity of the slider... The direction of the slip axis defined in the virtual rigid body The local slip direction defined in the middle is consistent, so that respectively with , Orthogonal constraints result in two rotational constraints: the first rotational constraint and the second rotational constraint. (twenty two) To constrain the rigid body of the slider Relative rotation about a direction other than the sliding axis introduces a rigid body for the slider. A normal direction orthogonal to the slip axis direction and make it with a virtual rigid body normal direction in Satisfying the orthogonality relation yields the third rotation constraint: (twenty three) Step 6.5: Obtain the complete sliding hinge position-level constraint vector; From equation (21) to equation (23), the rigid body of the slider is obtained. With virtual rigid bodies The complete sliding hinge position-level constraint vector between them: (twenty four) in, This is the position-level constraint vector for the sliding hinge, comprising five constraints in total: two translational constraints and three rotational constraints. The two translational constraints are used to restrict the rigid body. Relative to rigid body Relative translation in two normal directions; three rotational constraints are used to confine the rigid body. Relative to rigid body The relative rotation of the rigid body Relative to rigid body Only the relative translational degrees of freedom along the local slip direction are retained.

[0036] Through steps 6.1 to 6.5, the sliding constraints on the flexible body are transformed into sliding hinge position-level constraints between the slider rigid body and the virtual rigid body, resulting in a complete sliding hinge position-level constraint vector.

[0037] Step 7: Based on the sliding hinge position-level constraint vector obtained in Step 6, establish the sliding hinge velocity-level constraint equations and assemble the sliding hinge constraint Jacobian matrix to obtain the sliding hinge velocity-level constraint equations and the sliding hinge constraint Jacobian matrix. Specifically: Step 7.1: Define the position-level constraint equations for the sliding hinge: The sliding hinge position-level constraint vectors obtained in step 6 are uniformly denoted as the sliding hinge position-level constraint equations: (25) in, The vector of the position-level constraint equations for the sliding hinge; For the system's generalized coordinate vector; For time.

[0038] Step 7.2: Establish the velocity level constraint equations for the sliding hinge; Taking the time derivative of the sliding hinge position-level constraint equation shown in equation (25), we obtain the sliding hinge velocity-level constraint equation: (26) in, For the position-level constraint equations of the sliding hinge, apply to the generalized coordinates of the system The Jacobian matrix; The system's generalized velocity vector; This refers to the velocity term corresponding to the explicit time term in the position-level constraint equation of the sliding hinge.

[0039] Step 7.3: Assemble the Jacobian matrix of the sliding hinge constraint; The five constraints in equation (24) are applied to the system's generalized coordinates respectively. Find the partial derivatives and arrange them row by row according to the constraint order to obtain the Jacobian matrix of the sliding hinge constraint. The Jacobian matrix describes the influence of local geometric frame updates of the slider rigid body, virtual rigid body, and flexible body on the sliding hinge constraints, and is assembled into the system dynamics equations along with other constraint Jacobian matrices in subsequent steps. Thus, the velocity-level constraint equations of the sliding hinge and the Jacobian matrix of the sliding hinge constraints are obtained.

[0040] Step 8: Based on the connection relationship between the flexible body and the virtual rigid body obtained in Step 5, the sliding hinge position-level constraints obtained in Step 6, and the sliding hinge constraint Jacobian matrix obtained in Step 7, assemble and solve the dynamic equations of the flexible multibody system to obtain the slider motion response, flexible body deformation response, local sliding direction change, and constraint reaction force. Specifically: Step 8.1: Define the overall system constraint equations; The virtual rigid body reference point coincidence relationship shown in equation (17) and the virtual rigid body local frame update relationship shown in equation (18) are collectively denoted as the connection constraint between the flexible body and the virtual rigid body. It is represented as: (27) in, The constraint is the connection between the flexible body and the virtual rigid body. The first constraint in equation (27) is used to make the virtual rigid body reference point coincide with the current slip point, thereby ensuring that the virtual rigid body reference point is updated synchronously with the slip point position; the second constraint is used to make the virtual rigid body local frame Local geometric frame at the slip point A constant relative rotational relationship is maintained between them, thereby ensuring that the virtual rigid body's attitude is updated synchronously with the local geometric frame at the sliding point.

[0041] Connection constraints between flexible bodies and virtual rigid bodies The sliding hinge position level constraint shown in equation (24) After unified assembly, the overall system constraint equations are obtained: (28) in, The system's total constraint equation vector; For the connection constraint between the flexible body and the virtual rigid body; This is a positional constraint for the sliding hinge between the slider rigid body and the virtual rigid body.

[0042] Step 8.2: Establish the system dynamic equations; Based on the system's generalized coordinates, mass matrix, generalized forces, and the overall system constraint equations obtained in step 8.1, the dynamic equations of the flexible multibody system are established: (29) in, The mass matrix of the flexible multibody system; The generalized acceleration vector of the system; For the system's overall constraint equations with respect to generalized coordinates The Jacobian matrix; These are Lagrange multiplier vectors; For generalized active power; It is a generalized inertial coupling force; It refers to generalized elastic force.

[0043] Step 8.3: Establish the acceleration level constraint equations; Taking the total system constraint equation shown in equation (28) twice in time, we obtain the acceleration level constraint equation: (30) in, This is the right-hand side vector of the acceleration level obtained by differentiating the system's overall constraint equation with respect to time.

[0044] Step 8.4: Establish the augmented dynamic equations; Combining equations (29) and (30), we obtain the augmented dynamic equation: (31) Step 8.5: Time integration and nonlinear iterative solution; Within each time step, the system state is predicted using the time integration method, and equation (31) is solved using a nonlinear iterative method to obtain the system's generalized coordinates, generalized velocity, generalized acceleration, and Lagrange multipliers. After the solution is completed, the slider motion response, flexible body deformation response, local slip direction change, and constraint reaction force are output.

[0045] Through steps 8.1 to 8.5, the connection constraints between the flexible body and the virtual rigid body, as well as the sliding hinge position constraints between the slider rigid body and the virtual rigid body, are uniformly incorporated into the dynamic equations of the flexible multibody system, thereby obtaining the slider motion response, the flexible body deformation response, the local sliding direction change, and the constraint reaction force.

[0046] The beneficial effects of this invention are as follows: First, the present invention describes the current sliding point on the surface of the flexible body by using fixed interpolation parameters and quadrilateral interpolation regions, so that the current sliding point can be continuously updated with the local deformation of the flexible body without introducing additional arc length coordinates, intermediate coordinates or complex unit switching strategies, thereby reducing the number of generalized coordinates and the complexity of constraint equations in the system, and improving the ease of implementation of flexible multibody sliding hinge modeling.

[0047] Second, the present invention introduces a virtual rigid body at the current sliding point, and makes the reference point of the virtual rigid body coincide with the current sliding point. The local frame of the virtual rigid body is updated synchronously with the local geometric frame at the sliding point, thereby enabling the position and attitude information of the sliding point on the surface of the flexible body to be transmitted to the virtual rigid body, providing a basis for constructing the position-level constraint of the sliding hinge between the slider rigid body and the virtual rigid body.

[0048] Third, this invention transforms the sliding constraints on the flexible body into position-level constraints of the sliding hinge between the slider rigid body and the virtual rigid body, enabling these constraints to be assembled together with the connection constraints between the flexible body and the virtual rigid body into the dynamic equations of the flexible multibody system, thereby improving the compatibility of the modeling method with the general multibody dynamics joint constraint forms and solution framework.

[0049] Fourth, the present invention constructs a local geometric frame at the sliding point by using the current sliding point, adjacent path points before and after, and auxiliary points, so that the local sliding direction can be updated synchronously with the local configuration of the flexible body. This can describe the sliding motion of the flexible structure under large deformation and spatial bending path, and avoid the geometric inconsistency problem caused by using a fixed global sliding direction.

[0050] Fifth, this invention is applicable to flexible bodies discretized using solid finite element methods and whose nodes do not have explicit rotational degrees of freedom. It can construct a local geometric frame at the sliding point by the current spatial position of the nodes on the surface of the flexible body. It is suitable for dynamic simulation of rigid-flexible coupling sliding problems such as flexible guide rails, spatial curved tracks, cable-driven mechanisms, pantograph-contact network systems, and pulley mechanisms.

[0051] In summary, this invention enables the continuous updating of the current sliding point as the flexible body undergoes local deformation. Furthermore, by using a local geometric frame and a virtual rigid body, the problem of modeling the sliding hinge on the surface of the flexible body is transformed into a position-level constraint problem of the sliding hinge between the slider rigid body and the virtual rigid body. This improves the versatility, stability, and engineering applicability of sliding hinge modeling in flexible multibody systems. Attached Figure Description

[0052] Figure 1 This is a flowchart of the flexible multibody sliding hinge modeling method based on virtual bodies according to the present invention; Figure 2 This is a schematic diagram illustrating the kinematics of a flexible body using a floating coordinate system; where... Use the global coordinate system; For flexible body reference coordinate system; The origin of the global coordinate system; The origin of the flexible body reference coordinate system; This represents the position vector of the origin of the flexible body's reference coordinate system in the global coordinate system. Represents a node The undeformed position vector in the flexible body reference coordinate system; Represents a node The elastic deformation vector caused by the modal coordinates at the location; Represents a node Current position vector in the flexible body reference coordinate system; Represents a node The position vector in the global coordinate system.

[0053] Figure 3 This diagram illustrates the sliding path and quadrilateral interpolation region on the surface of a flexible body. The flexible body represents a flexible structure that allows relative sliding of a slider or sliding component; the sliding path represents a pre-defined sliding motion path on the surface of the flexible body; path points represent discrete points used to describe the geometry of the sliding path; node 1, node 2, node 3, and node 4 represent the four surface nodes constituting the quadrilateral interpolation region; and the quadrilateral interpolation region represents the four-node interpolation region used to calculate the current sliding point position. This represents the position vector of the current sliding point in the global coordinate system.

[0054] Figure 4 This is a schematic diagram of the local geometric frame construction at the slip point. This represents the position vector of the current sliding point in the global coordinate system; This represents the position vector of the next adjacent path point before the current slip point in the global coordinate system; This represents the position vector of the next adjacent path point after the current slip point in the global coordinate system; This represents the position vector of the auxiliary point in the global coordinate system; Denotes the first basis vector of the local geometric frame; The second basis vector of the local geometric frame; The third basis vector of the local geometric frame; This represents the local geometric frame at the slip point.

[0055] Figure 5 This is a schematic diagram of the virtual rigid body construction at the slip point. This represents the position vector of the current sliding point in the global coordinate system; and These represent the position vectors of the adjacent path points before and after the current slip point in the global coordinate system; This represents the position vector of the auxiliary point in the global coordinate system; , and Let represent the first, second, and third basis vectors of the local geometric frame at the slip point, respectively; This represents the local geometric frame at the slip point; Represents a local frame of a virtual rigid body; This represents the position vector of the virtual rigid body reference point in the global coordinate system. , and This represents the direction vector within the local frame of the virtual rigid body; Represents the local geometric frame at the slip point With virtual rigid body local frame The constant relative rotation matrix between them.

[0056] Figure 6 This is a schematic diagram illustrating the positional constraint of the sliding hinge between the slider rigid body and the virtual rigid body. The slider rigid body... A rigid body component constrained to move along the local slip direction; a virtual rigid body. This represents a virtual rigid body introduced at the slip point; Use the global coordinate system; , , Represents a rigid body slider Local coordinate axes; , , Represents a virtual rigid body Local coordinate axes; Represents a rigid body slider The position vector of the reference point in the global coordinate system; Represents a virtual rigid body The position vector of the reference point in the global coordinate system; Represents a rigid body slider The position vector of the upper sliding hinge connection point in the global coordinate system; Represents a virtual rigid body The position vector of the upper sliding hinge connection point in the global coordinate system; Represents a rigid body slider Reference point to connector The local position vector; Represents a virtual rigid body Reference point to connector The local position vector; Indicates connection point and The connection vector between them; Represents a rigid body slider The direction of the sliding axis; Represents a virtual rigid body Defined local slip direction; and Represents a virtual rigid body Two normal unit vectors orthogonal to the local slip direction in the local frame; and Represents a rigid body slider The normal direction vector in the local frame that is related to the direction of the slip axis.

[0057] Figure 7 This is a schematic diagram of an embodiment of the present invention applied to the sliding simulation of a large deformation flexible beam. In the diagram, 1 is a fixed virtual rigid body used to fix one end of the slender flexible beam to the ground; 2 is a slider rigid body used to represent a slider component moving along the sliding path on the surface of the slender flexible beam; 3 is a sliding virtual rigid body used to transmit the position and local geometric frame information of the sliding point on the surface of the slender flexible beam; 4 is a slender flexible beam used to represent a flexible body undergoing large deformation; 5 is a load-bearing virtual rigid body used to apply an external load to the end of the slender flexible beam; 6 is a fixed hinge; and 7 is a sliding hinge. The direction of the velocity of the rigid body sliding along the sliding path; This refers to the external load applied to the end of a slender, flexible beam. This is the global coordinate system.

[0058] Figure 8 This is a representative configuration diagram of the slider moving along a flexible beam with large deformation. The blue grid structure represents the flexible beam, and the green blocks represent the rigid body of the slider. This represents the horizontal position coordinates in the global coordinate system, in meters (m). Represents the vertical position coordinates in the global coordinate system, in meters (m). The simulation time is expressed in seconds. Figure 8 (a) in the diagram represents the initial configuration at t=0.00s; Figure 8 (b) in the diagram represents the configuration at t = 0.72 s; Figure 8 (c) in the figure represents the configuration at t=0.84s; Figure 8 In this context, (d) represents the configuration at t = 0.90 s; Figure 8 (e) in the figure represents the configuration at t = 0.94 s; Figure 8 In this context, (f) represents the configuration at t = 0.97 s; Figure 8 In this context, (g) represents the configuration at t = 0.98 s; Figure 8 (h) in the figure represents the configuration at t=1.00s. Detailed Implementation

[0059] The technical solution of the present invention will be further described below with reference to specific embodiments.

[0060] Example 1: Dynamic simulation of slider sliding along a large deformation flexible beam; This embodiment illustrates the application of the present invention in the simulation of sliding motion of flexible structures under large deformation. For example... Figure 1 As shown, the overall modeling process of this invention includes establishing a kinematic model of a floating coordinate system for a flexible body, defining the sliding path on the surface of the flexible body, determining the fixed interpolation parameters at the sliding point, constructing a local geometric frame at the sliding point, introducing a virtual rigid body, constructing position-level constraints of the sliding hinge between the slider rigid body and the virtual rigid body, establishing velocity-level constraint equations and constraint Jacobian matrices for the sliding hinge, and assembling and solving the dynamic equations of the flexible multibody system.

[0061] like Figure 7 As shown, the simulation objects in this embodiment include a slender flexible beam 4 and a slider rigid body 2. One end of the slender flexible beam 4 is fixedly connected to the ground through a fixed virtual rigid body 1 and a fixed hinge 6, while the other end is subjected to an external load through a load-bearing virtual rigid body 5. This causes the slender flexible beam 4 to undergo significant bending deformation. The relative motion between the fixed virtual rigid body 1 and the ground is constrained by the fixed hinge 6, achieving fixed support at one end of the slender flexible beam 4. The slider rigid body 2 is located on the surface of the slender flexible beam 4 and slides along a preset path in the direction of velocity. Motion. A sliding virtual rigid body 3 is positioned at a sliding point on the surface of the slender flexible beam 4 to transmit the position of the sliding point and local geometric frame information; the slider rigid body 2 is connected to the sliding virtual rigid body 3 via a sliding hinge 7, allowing the slider rigid body 2 to move along the local sliding direction at the sliding point. This embodiment illustrates that when the slender flexible beam 4 undergoes large deformation, the present invention can update the local sliding direction at the sliding point in real time and maintain the sliding constraint relationship between the slider rigid body 2 and the slender flexible beam 4.

[0062] In this embodiment, the geometric parameters, material parameters, and modal truncation settings of the flexible beam and the slider rigid body are shown in Table 1.

[0063] Table 1: Geometric and material parameters of flexible beam structures and slider rigid bodies

[0064] Step 1: Establish a floating coordinate system kinematic model of the slender flexible beam 4 to obtain the current spatial position of each node of the slender flexible beam 4 in the global coordinate system. Specifically: Step 1.1: As Figure 2 As shown, a global coordinate system is defined. Flexible body reference coordinate system and flexible body generalized coordinates; The overall motion of the slender flexible beam 4 is decomposed into the translation of the origin of the flexible body reference coordinate system in the global coordinate system, the rotation of the flexible body reference coordinate system relative to the global coordinate system, and the elastic deformation of the slender flexible beam 4 relative to the flexible body reference coordinate system, thus obtaining the generalized coordinates of the slender flexible beam 4. : (1) in, For the generalized coordinate vector of the flexible body; This is the translational coordinate vector of the origin of the flexible body's reference coordinate system in the global coordinate system. This is the rotation coordinate vector of the flexible body's reference coordinate system relative to the global coordinate system; This is the modal coordinate vector of the flexible body, used to describe the elastic deformation of the flexible body relative to the reference coordinate system of the flexible body; the superscript T indicates the transpose of the matrix or vector.

[0065] Step 1.2: Based on the generalized coordinates of the flexible body obtained in Step 1.1, establish the nodal position expressions for the slender flexible beam 4; The spatial positions of the four nodes of the slender flexible beam are determined by the position of the origin of the flexible body reference coordinate system, the attitude of the flexible body reference coordinate system, the position of the nodes in the undeformed configuration, and the elastic deformation of the nodes. The position vector of each node in the global coordinate system Represented as: (2) in, For the 4th slender flexible beam The position vector of each node in the global coordinate system; Number the nodes of the flexible body; This is the translational coordinate vector of the origin of the flexible body's reference coordinate system in the global coordinate system. This is the attitude transformation matrix from the flexible body's reference coordinate system to the global coordinate system; For the first The position vector of each node in the undeformed configuration of the flexible body; For the first The mode matrix at each node; is the modal coordinate vector of the flexible body.

[0066] like Figure 2 As shown, This represents the position vector of the origin of the flexible body's reference coordinate system in the global coordinate system. Represents a node Undeformed position vector in the flexible body reference coordinate system Represents a node The elastic deformation vector caused by modal coordinates at that point. Represents a node The position vector in the global coordinate system.

[0067] Step 1.3: Based on the node position expression established in Step 1.2, obtain the flexible body modal information and calculate the current spatial position of each node of the slender flexible beam 4 in the global coordinate system; In this embodiment, the slender flexible beam 4 is modeled using a floating coordinate system method. Its elastic deformation is described by the modal information obtained from finite element modal analysis, and the 7th to 30th elastic modes are retained. The finite element model of the slender flexible beam 4 is established according to the geometric and material parameters in Table 1. The node coordinates, element topology, and modal matrix are extracted from the finite element model. The flexible body modal coordinates are then substituted into equation (2) to calculate the current spatial position of each node in the global coordinate system under the current configuration of the slender flexible beam 4.

[0068] Steps 1.1 to 1.3 are used to obtain the current spatial position of each node of the slender flexible beam 4 in the global coordinate system, which provides a basis for defining the sliding path, calculating the position of the sliding point and constructing the local geometric frame.

[0069] Step 2: Based on the current spatial positions of each node of the slender flexible beam 4 in the global coordinate system obtained in Step 1, define a sliding path on the surface of the slender flexible beam 4, select the sliding path interpolation region, establish the interpolation expression for the sliding point position, and determine the bilinear interpolation shape function. This yields the sliding path and the interpolation expression for the sliding point position on the flexible body surface, jointly described by the quadrilateral interpolation region, interpolation parameters, and the bilinear interpolation shape function. Specifically: Step 2.1: Based on the current spatial position of each node of the slender flexible beam 4 in the global coordinate system obtained in Step 1, define the sliding path on the surface of the slender flexible beam 4 and select the sliding path interpolation region to obtain the quadrilateral interpolation region used to describe the position of the sliding point.

[0070] like Figure 3 As shown, a sliding path is preset on the surface of the slender flexible beam 4. This sliding path is the path along which the slider rigid body 2 moves along the surface of the slender flexible beam 4. The sliding path is described by multiple path points, and the current sliding point is the position point of the slider rigid body 2 on the sliding path at a certain moment. Nodes 1, 2, 3, and 4 are selected near the current sliding point to form a quadrilateral interpolation region, which is used to describe the position of the current sliding point on the surface of the slender flexible beam 4.

[0071] Step 2.2: Based on the quadrilateral interpolation region obtained in Step 2.1, establish the interpolation expression for the position of the sliding point to obtain the position expression of the sliding point in the global coordinate system; For a quadrilateral interpolation region consisting of four nodes, the slip point position is represented as: (3) in, This represents the position vector of the sliding point in the global coordinate system. This represents the current spatial position of the first node within the interpolation region. This represents the current spatial position of the second node within the interpolation region. This represents the current spatial location of the third node within the interpolation region. This represents the current spatial position of the fourth node within the interpolation region; For the first interpolation parameter, This is the second interpolation parameter; the two together constitute the interpolation parameters. Let be the bilinear interpolation form function corresponding to the first node. The bilinear interpolation form function corresponding to the second node. The bilinear interpolation form function corresponding to the third node. This is the bilinear interpolation form function corresponding to the fourth node.

[0072] , , and correspond Figure 3 The current spatial positions of nodes 1, 2, 3 and 4 are determined. The current sliding point position of the slider rigid body 2 is described by the quadrilateral interpolation region on the surface of the slender flexible beam 4 using equation (3).

[0073] Step 2.3: Based on the position expression of the sliding point in the global coordinate system obtained in Step 2.2, determine the bilinear interpolation shape function, and obtain the flexible body surface sliding path and sliding point position interpolation expression jointly described by the quadrilateral interpolation region, interpolation parameters and bilinear interpolation shape function; When the interpolation parameter takes values ​​in the range of , When the bilinear interpolation shape function takes the following form: (4) in, , , and These are the bilinear interpolation shape functions corresponding to the first, second, third, and fourth nodes within the quadrilateral interpolation region, respectively.

[0074] In this embodiment, when the rigid slider 2 moves along the sliding path on the surface of the slender flexible beam 4, the current sliding point can be determined by equations (3) and (4). When the sliding path on the surface of the slender flexible beam 4 crosses multiple quadrilateral interpolation regions, the corresponding quadrilateral interpolation regions are selected sequentially along the sliding path, and multiple path points on the sliding path are defined by interpolation using equations (3) and (4) respectively. Thus, the interpolation expression for the sliding path and sliding point position of the flexible body surface, which is jointly described by the quadrilateral interpolation region, interpolation parameters, and bilinear interpolation shape function, is obtained.

[0075] Step 3: Based on the interpolation expressions for the sliding path and sliding point position of the flexible body surface obtained in Step 2, solve for the fixed interpolation parameters of the sliding point under the initial configuration. Keep these fixed interpolation parameters unchanged during the dynamic simulation to obtain the fixed interpolation parameters of the sliding point and the calculation method for updating the sliding point position based on these fixed interpolation parameters. Specifically: Step 3.1: Based on the slip point position interpolation expression obtained in Step 2, given the target slip point position under the initial configuration, obtain the initial interpolation data used to solve the fixed interpolation parameters; In the initial configuration, the target position of the sliding point is given according to the pre-set sliding path on the surface of the slender flexible beam 4. .set up Let be the target position vector of the slip point under the initial configuration. , , and These are the position vectors of the first, second, third, and fourth nodes within the quadrilateral interpolation region, respectively, under the initial configuration. The superscript 0 indicates the initial configuration. Let be the position vector of the first node in the initial configuration. Let be the position vector of the second node in the initial configuration. Let be the position vector of the third node in the initial configuration. This is the position vector of the fourth node in the initial configuration.

[0076] This yields initial interpolation data for solving the fixed interpolation parameters, the initial interpolation data including the target position of the sliding point. The initial position vectors of the four nodes within the quadrilateral interpolation region. , , , And the bilinear interpolation shape function determined in step 2.

[0077] Step 3.2: Based on the initial interpolation data obtained in Step 3.1, solve the fixed interpolation parameters of the slip point using the least squares problem to obtain the first fixed interpolation parameters and the second fixed interpolation parameters; Based on the target position of the glide point in the initial configuration Using the initial position vectors of the four nodes within the quadrilateral interpolation region, construct the following least squares problem: (5) in, To obtain the fixed interpolation parameters, The first fixed interpolation parameter is... This is the second fixed interpolation parameter; The first interpolation parameter is... This is the second interpolation parameter; The target position vector of the slip point under the initial configuration; Number the nodes within the quadrilateral interpolation region. =1, 2, 3, 4 correspond to the first node, the second node, the third node, and the fourth node, respectively; For the first The position vectors of each node under the initial configuration; For the first The bilinear interpolation shape function corresponding to each node; This represents the Euclidean norm.

[0078] The first fixed interpolation parameter is obtained by solving equation (5). Second fixed interpolation parameter .

[0079] Step 3.3: Based on the first fixed interpolation parameter and the second fixed interpolation parameter, keep the fixed interpolation parameter unchanged and update the position of the sliding point to obtain the fixed interpolation parameter of the sliding point and the calculation method of updating the position of the sliding point based on the fixed interpolation parameter; Throughout the entire dynamic simulation, the first fixed interpolation parameter is maintained. Second fixed interpolation parameter The position remains unchanged. When the slender flexible beam 4 deforms, the current spatial positions of the first, second, third, and fourth nodes within the quadrilateral interpolation region are updated as the flexible body moves. At this time, the first interpolation parameter in equation (3) is changed. Second interpolation parameter Take them as the first fixed interpolation parameters respectively. Second fixed interpolation parameter Substituting the current spatial positions of the four nodes within the quadrilateral interpolation region into equation (3), the position of the current sliding point in the global coordinate system can be obtained.

[0080] The current spatial positions of the four nodes within the quadrilateral interpolation region are determined by the current spatial positions of each node of the slender flexible beam 4 in the global coordinate system obtained in step 1, i.e., by formula (2); the first fixed interpolation parameter Second fixed interpolation parameter The slip point remains unchanged during the simulation. Therefore, the slip point can be continuously updated with the local deformation of the flexible body, avoiding the position discontinuity problem caused by directly binding the slip point to discrete nodes or switching elements. This yields fixed interpolation parameters for the slip point and a calculation method for updating the slip point position based on these fixed interpolation parameters.

[0081] Step 4: Based on the fixed interpolation parameters of the slip point and the slip path on the flexible body surface, calculate the current position of the current slip point, adjacent path points, and auxiliary points in the global coordinate system, and construct the first basis vector, second basis vector, and third basis vector at the slip point to obtain the local geometric frame at the slip point that is updated synchronously with the local configuration of the flexible body. Specifically: like Figure 4As shown, in this embodiment, the construction of the local geometric frame at the slip point is based on the current slip point, the previous adjacent path point, the next adjacent path point, and the auxiliary point. First, the local tangential direction of the slip path near the current slip point is determined based on the previous and next adjacent path points, and normalized into a first basis vector. Then, based on the auxiliary direction pointing from the current slip point to the auxiliary point, the projection component of the auxiliary direction in the direction of the first basis vector is removed, and the orthogonalized direction is normalized into a second basis vector. Finally, the first basis vector and the second basis vector are cross-multiplied to obtain a third basis vector. The first basis vector, the second basis vector, and the third basis vector together constitute the local geometric frame at the slip point. Since the positions of the current slip point, adjacent path points, and auxiliary points are all obtained by interpolating the coordinates of the current nodes on the surface of the slender flexible beam, this local geometric frame can be updated synchronously with the bending deformation of the slender flexible beam and the change of the slip path direction, providing a local slip direction for the subsequent attitude update of the virtual rigid body. Step 4.1: Based on the fixed interpolation parameters of the sliding point obtained in Step 3, calculate the current position of the current sliding point and the adjacent path points in the global coordinate system; The position vector of the current sliding point in the global coordinate system is denoted as It is calculated by the interpolation formula shown in equation (3) based on the fixed interpolation parameters and the current node coordinates of the quadrilateral interpolation region: (6) The position vectors of adjacent path points before and after the current slip point in the global coordinate system are denoted as follows: and It also has corresponding interpolation parameters. and Substituting into the interpolation formula shown in equation (3), we obtain: (7) (8) in, This represents the position vector of the current sliding point in the global coordinate system. This is the position vector of the previous adjacent path point in the global coordinate system; This is the position vector of the next adjacent path point in the global coordinate system; For the quadrilateral interpolation region containing the current slip point, the first... The current spatial location of each node; For the quadrilateral interpolation region containing the previous adjacent path point, the first... The current spatial location of each node; For the quadrilateral interpolation region containing the next adjacent path point, the first The current spatial location of each node; Number the nodes within the quadrilateral interpolation region. =1, 2, 3, 4 correspond to the first node, the second node, the third node, and the fourth node, respectively.

[0082] Step 4.2: Based on the preceding and following adjacent path points obtained in Step 4.1, construct local tangential basis vectors; Based on the adjacent path points before and after the current slip point and Define the local tangential vector of the slip path. : (9) The first basis vector of the tangent of the slip path for: (10) in, This is the local tangential vector of the slip path; This is the position vector of the next adjacent path point before the current sliding point in the global coordinate system; This is the position vector of the next adjacent path point after the current sliding point in the global coordinate system; is the first basis vector of the local geometric frame, used to represent the local tangential direction of the slip path near the current slip point; This represents the Euclidean norm.

[0083] Step 4.3: Based on the current glide point and the first basis vector, determine the auxiliary point and construct the second basis vector.

[0084] To determine the second basis vector of the local geometric frame, auxiliary points are selected. The position vector of the auxiliary point in the global coordinate system is denoted as . The auxiliary point can be selected as a surface node near the current sliding point that is not collinear with the local tangential direction of the sliding path, or it can be selected from a quadrilateral interpolation region near the current sliding point using interpolation parameters. The calculation yielded: (11) in, This is the position vector of the auxiliary point in the global coordinate system; The first interpolation parameter corresponding to the auxiliary point; The second interpolation parameter corresponding to the auxiliary point; For the quadrilateral interpolation region containing auxiliary points, the first The current spatial location of each node.

[0085] Defined by the current slip point Point to auxiliary point auxiliary direction vector : (12) To ensure the orthogonality of the local geometric frame, auxiliary direction vectors are removed. First basis The component in the direction, obtained with orthogonal vectors : (13) Then the second basis vector of the local geometric frame for: (14) Step 4.4: Construct a local geometric frame based on the first and second basis vectors; Third basis From the first basis Second basis Cross product yields: (15) This yields the local geometric frame at the slip point. : (16) because , , and All values ​​are calculated from the position of the current node of the slender flexible beam 4 in the global coordinate system. Therefore, when the slender flexible beam 4 deforms, the local geometric frame... It can update synchronously with the local configuration of the flexible body. For flexible bodies discretized using solid finite element methods and whose nodes do not have explicit rotational degrees of freedom, the local geometric frame... This characterizes the local attitude and slip direction at the slip point. Thus, a local geometric frame at the slip point is obtained, updated synchronously with the local configuration of the flexible body. .

[0086] Step 5: Based on the local geometric frame at the slip point A virtual rigid body is introduced at the slip point, and a reference point and a local frame for the virtual rigid body are determined to obtain a virtual rigid body used to transmit the local position and attitude of the slip point. Specifically: like Figure 5As shown, a virtual rigid body is introduced at the current slip point. The reference point of the virtual rigid body coincides with the current slip point, allowing the virtual rigid body reference point to update synchronously with the position change of the current slip point in the global coordinate system. Simultaneously, a constant relative rotation relationship is established between the local frame of the virtual rigid body and the local geometric frame constructed in step 4 at the slip point, enabling the slip direction and the two normal directions orthogonal to the slip direction in the local frame of the virtual rigid body to update synchronously with the local geometric frame. This virtual rigid body is not used to represent actual mass and inertia, but rather to transfer the position, local tangential direction, and normal direction of the current slip point to the subsequent slip hinge constraint construction process, thus providing an intermediate connection carrier for converting the flexible body surface slip constraint into a standard rigid body slip hinge position-level constraint.

[0087] Step 5.1: Set the virtual rigid body reference point; A virtual rigid body is introduced at the slip point. The virtual rigid body is a massless rigid body or a mass that is negligible. It is not used to represent actual mass and inertia, but to convey the local position and attitude information of the flexible body at the slip path.

[0088] Make the reference point of the virtual rigid body coincide with the slip point: (17) in, This is the position vector of the virtual rigid body reference point in the global coordinate system; This is the position vector of the current sliding point in the global coordinate system.

[0089] Step 5.2: Set up a virtual rigid body local frame; Let the local geometric frame at the slip point be... The virtual rigid body local frame is This ensures a constant relative rotational relationship between the virtual rigid body local frame and the local geometric frame at the slip point: (18) in, This is the constant relative rotation matrix determined under the initial configuration. Under the initial configuration, it is based on the local geometric frame at the slip point. and the preset virtual rigid body initial local frame Determine the constant relative rotation matrix: (19) During the dynamic simulation process The virtual rigid body reference point is updated synchronously with the position of the slip point, and the virtual rigid body local frame is updated synchronously with the local geometric frame at the slip point, resulting in a virtual rigid body used to transmit the local position and attitude of the slip point.

[0090] Step 6: Based on the virtual rigid body obtained in Step 5, construct the sliding hinge position-level constraint between the slider rigid body 2 and the sliding virtual rigid body 3, obtaining the complete sliding hinge position-level constraint vector. Specifically: Step 6.1: Define the sliding hinge connection point; like Figure 6 As shown, the slider rigid body 2 is considered as a slider rigid body. Treat the sliding virtual rigid body 3 as a virtual rigid body In the rigid body of the slider With virtual rigid bodies Establish positional constraints at the sliding hinge level between them to make the slider rigid body Relative to virtual rigid body Only the relative translational degrees of freedom along the local slip direction are retained.

[0091] Assume a rigid body The position vector of the sliding hinge connection point on the upper surface in the global coordinate system is: ,rigid body The position vector of the sliding hinge connection point on the upper surface in the global coordinate system is: .

[0092] Step 6.2: Define the connection vector between connection points; Define the connection vector between two connection points. for: (20) Step 6.3: Construct translational constraints for the sliding hinge; set up and Virtual rigid body In a local frame, two normal unit vectors orthogonal to the slip direction. The first normal unit vector is orthogonal to the slip direction. The second normal unit vector is orthogonal to the sliding direction; to make the slider a rigid body Only along virtual rigid bodies The defined slip direction undergoes relative translation, which should cause the connecting vector to... respectively with and Orthogonal, we obtain the first and second translational constraints of the sliding hinge: (twenty one) Step 6.4: Construct the rotational constraint of the sliding hinge; set up For the slider rigid body The direction of the sliding axis in the diagram. To ensure the rigidity of the slider... The direction of the slip axis defined in the virtual rigid body The local slip direction defined in the middle is consistent, so that respectively with , Orthogonal constraints result in two rotational constraints: the first rotational constraint and the second rotational constraint. (twenty two) To constrain the rigid body of the slider Relative rotation about a direction other than the sliding axis introduces a rigid body for the slider. A normal direction orthogonal to the slip axis direction To make it consistent with the virtual rigid body normal direction in Satisfying the orthogonality relation yields the third rotation constraint: (twenty three) Step 6.5: Obtain the complete sliding hinge position-level constraint vector; From equation (21) to equation (23), the rigid body of the slider is obtained. With virtual rigid bodies The complete sliding hinge position-level constraint vector between them: (twenty four) in, This is the position-level constraint vector for the sliding hinge, which includes two translational constraints and three rotational constraints.

[0093] Through steps 6.1 to 6.5, the sliding constraints on the flexible body are transformed into sliding hinge position-level constraints between the slider rigid body and the virtual rigid body, resulting in a complete sliding hinge position-level constraint vector.

[0094] Step 7: Based on the sliding hinge position-level constraint vector obtained in Step 6, establish the sliding hinge velocity-level constraint equations and assemble the sliding hinge constraint Jacobian matrix to obtain the sliding hinge velocity-level constraint equations and the sliding hinge constraint Jacobian matrix. Specifically: Step 7.1: Define the position-level constraint equations for the sliding hinge: The sliding hinge position-level constraint vectors obtained in step 6 are uniformly denoted as the sliding hinge position-level constraint equations: (25) in, The vector of the position-level constraint equations for the sliding hinge; For the system's generalized coordinate vector; For time.

[0095] Step 7.2: Establish the velocity level constraint equations for the sliding hinge; Taking the time derivative of the sliding hinge position-level constraint equation shown in equation (25), we obtain the sliding hinge velocity-level constraint equation: (26) in, For the position-level constraint equations of the sliding hinge, apply to the generalized coordinates of the system The Jacobian matrix; The system's generalized velocity vector; This refers to the velocity term corresponding to the explicit time term in the position-level constraint equation of the sliding hinge.

[0096] Step 7.3: Assemble the Jacobian matrix of the sliding hinge constraint; The five constraints in equation (24) are applied to the system's generalized coordinates respectively. Find the partial derivatives and arrange them row by row according to the constraint order to obtain the Jacobian matrix of the sliding hinge constraint. The Jacobian matrix describes the rigid body of the slider. Virtual rigid body The influence of the local geometric frame update of the slender flexible beam 4 on the sliding hinge constraint is also considered, and in subsequent steps, it is assembled into the system dynamics equations along with other constraint Jacobian matrices. Thus, the sliding hinge velocity level constraint equations and the sliding hinge constraint Jacobian matrix are obtained.

[0097] Step 8: Assemble and solve the dynamic equations of the flexible multibody system to obtain the slider motion response, flexible body deformation response, local slip direction changes, and constraint reactions. Specifically: Step 8.1: Define the overall system constraint equations; The virtual rigid body reference point coincidence relationship shown in equation (17) and the virtual rigid body local frame update relationship shown in equation (18) are collectively denoted as the connection constraint between the flexible body and the virtual rigid body. It is represented as: (27) in, The constraint is the connection between the flexible body and the virtual rigid body. The first constraint in equation (27) is used to make the virtual rigid body reference point coincide with the current slip point, thereby ensuring that the virtual rigid body reference point is updated synchronously with the slip point position; the second constraint is used to make the virtual rigid body local frame Local geometric frame at the slip point A constant relative rotational relationship is maintained between them, thereby ensuring that the virtual rigid body's attitude is updated synchronously with the local geometric frame at the sliding point.

[0098] Connection constraints between flexible bodies and virtual rigid bodies The sliding hinge position level constraint shown in equation (24) After unified assembly, the overall system constraint equations are obtained: (28) in, The system's total constraint equation vector; For the connection constraint between the flexible body and the virtual rigid body; This is a positional constraint for the sliding hinge between the slider rigid body and the virtual rigid body.

[0099] Step 8.2: Establish the system dynamic equations; Based on the system's generalized coordinates, mass matrix, generalized forces, and the overall system constraint equations obtained in step 8.1, the dynamic equations of the flexible multibody system are established: (29) in, The mass matrix of the flexible multibody system; The generalized acceleration vector of the system; For the system's overall constraint equations with respect to generalized coordinates The Jacobian matrix; These are Lagrange multiplier vectors; For generalized active power; It is a generalized inertial coupling force; It refers to generalized elastic force.

[0100] Step 8.3: Establish the acceleration level constraint equations; Taking the total system constraint equation shown in equation (28) twice in time, we obtain the acceleration level constraint equation: (30) in, This is the right-hand side vector of the acceleration level obtained by differentiating the system's overall constraint equation with respect to time.

[0101] Step 8.4: Establish the augmented dynamic equations; Combining equations (29) and (30), we obtain the augmented dynamic equation: (31) Step 8.5: Time integration and nonlinear iterative solution; Within each time step, the system state is predicted using the time integration method, and equation (31) is solved using a nonlinear iterative method to obtain the system's generalized coordinates, generalized velocity, generalized acceleration, and Lagrange multipliers. After the solution is completed, the slider motion response, flexible body deformation response, local slip direction change, and constraint reaction force are output. Through steps 8.1 to 8.5, the connection constraints between the flexible body and the virtual rigid body, as well as the sliding hinge position-level constraints between the slider rigid body and the virtual rigid body, are uniformly incorporated into the dynamic equations of the flexible multibody system, resulting in the slider motion response, flexible body deformation response, local slip direction change, and constraint reaction force.

[0102] Step 8.6: Analysis of simulation results.

[0103] After the solution is completed, the result is as follows: Figure 8 The shown diagram illustrates a representative configuration of the rigid slider 2 moving along the slender, flexible beam 4 during large deformation. Figure 8 In the diagram, the blue mesh structure represents the slender flexible beam 4, and the green block represents the rigid slider body 2; X represents the horizontal position coordinate in the global coordinate system, in meters (m); Z represents the vertical position coordinate in the global coordinate system, in meters (m); and t represents the simulation time, in seconds (s). Specifically: Figure 8 (a) in the figure represents the initial configuration at t=0.00s, at which point the slender flexible beam 4 is basically in the initial straight beam state, and the slider rigid body 2 is located near the left side of the slender flexible beam 4. Figure 8 (b) in the figure represents the configuration at t=0.72s, at which point the slender flexible beam 4 begins to bend under the action of the external load F at the end, and the slider rigid body 2 slides to the right along the surface of the slender flexible beam 4. Figure 8 (c) represents the configuration at t=0.84s. At this time, the bending degree of the slender flexible beam 4 further increases, and the slider rigid body 2 is still located near the sliding path on the surface of the flexible beam, without any detachment or jump phenomenon. Figure 8 In the diagram, (d) represents the configuration at t=0.90s. At this time, the bending deformation at the ends of the slender flexible beam 4 is more obvious, and the local tangential direction of the slip path near the current slip point changes relative to the initial state. Figure 8 (e) represents the configuration at t=0.94s, where the rigid body 2 of the slider is located slightly to the right of the middle of the slender flexible beam 4. The local geometric frame at the current sliding point is updated synchronously with the current configuration of the slender flexible beam 4. Figure 8 In the figure (f), the configuration is at t=0.97s. At this time, the slender flexible beam 4 has a large bending deformation, and the slider rigid body 2 continues to move along the local sliding direction, indicating that the position level constraint of the sliding hinge can still maintain the constraint relationship under large deformation. Figure 8 In the diagram, (g) represents the configuration at t=0.98s. At this time, the rigid body 2 of the slider continues to move towards the right end of the slender flexible beam 4. The position of the slider changes continuously with time, and there is no problem of position discontinuity caused by node switching or element switching. Figure 8 In the figure, (h) represents the configuration at t=1.00s. At this time, the rigid body 2 of the slider moves to the vicinity of the right end of the slender flexible beam 4, and the system can still maintain the sliding constraint relationship between the rigid body 2 of the slider and the slender flexible beam 4.

[0104] Depend on Figure 8 (a) to Figure 8As shown in (h), as the slender flexible beam 4 undergoes bending deformation under the external load F, the slider rigid body 2 continuously moves along the sliding path on the surface of the slender flexible beam 4. This result demonstrates that, by fixing interpolation parameters, local geometric frames, and virtual rigid bodies, this invention can update the current sliding point position and local sliding direction in real time when the slender flexible beam 4 undergoes large deformation, and transform the sliding constraint on the flexible body surface into a sliding hinge position-level constraint between the slider rigid body a and the virtual rigid body b, thereby realizing the dynamic simulation of the slider sliding along the large deformation flexible beam.

[0105] The above embodiments are merely illustrative of the implementation methods of the present invention, but should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the protection scope of the present invention.

Claims

1. A method for modeling sliding hinges in the dynamics of flexible multibody systems based on virtual bodies, characterized in that, The method for modeling the dynamic sliding hinge of a flexible multibody system includes the following steps; Step 1: Establish a kinematic model of the flexible body in a floating coordinate system to obtain the current spatial position of each node of the flexible body in the global coordinate system; Step 2: Based on the current spatial position, define the sliding path on the surface of the flexible body, select the interpolation region of the sliding path, establish the interpolation expression of the sliding point position and determine the bilinear interpolation shape function, and obtain the sliding path and the interpolation expression of the sliding point position of the flexible body surface described by the quadrilateral interpolation region, interpolation parameters and bilinear interpolation shape function. Step 3: Based on the interpolation expression of the sliding path and sliding point position on the flexible body surface, solve the fixed interpolation parameters of the sliding point under the initial configuration, and keep the fixed interpolation parameters unchanged during the dynamic simulation to obtain the fixed interpolation parameters of the sliding point and the calculation method of updating the sliding point position based on the fixed interpolation parameters; Step 4: Based on the fixed interpolation parameters of the slip point obtained in Step 3 and the slip path on the flexible body surface obtained in Step 2, calculate the current position of the current slip point, adjacent path points, and auxiliary points in the global coordinate system, and construct the first basis vector, second basis vector, and third basis vector at the slip point to obtain the local geometric frame at the slip point that is updated synchronously with the local configuration of the flexible body. ; Step 5: Based on the local geometric frame at the slip point, introduce a virtual rigid body at the slip point, and determine the virtual rigid body reference point and the virtual rigid body local frame to obtain the virtual rigid body used to transmit the local position and attitude of the slip point; Step 6: Based on the virtual rigid body, construct the sliding hinge position level constraint between the slider rigid body and the virtual rigid body to obtain the complete sliding hinge position level constraint vector; Step 7: Based on the position-level constraint vector of the sliding hinge, establish the velocity-level constraint equation of the sliding hinge and assemble the constraint Jacobian matrix of the sliding hinge to obtain the velocity-level constraint equation of the sliding hinge and the constraint Jacobian matrix of the sliding hinge. Step 8: Assemble and solve the dynamic equations of the flexible multibody system to obtain the slider motion response, flexible body deformation response, local sliding direction change, and constraint reaction force.

2. The method for modeling sliding hinges in the dynamics of a flexible multibody system based on a virtual body according to claim 1, characterized in that, Specifically, step 1 is as follows: Step 1.1: Define the global coordinate system Flexible body reference coordinate system and flexible body generalized coordinates; The flexible body reference coordinate system moves with the overall rigid body motion of the flexible body. The overall motion of the flexible body is decomposed into the translation of the origin of the flexible body reference coordinate system in the global coordinate system, the rotation of the flexible body reference coordinate system relative to the global coordinate system, and the elastic deformation of the flexible body relative to the flexible body reference coordinate system, thus obtaining the generalized coordinate system of the flexible body. : （1） in, For the generalized coordinate vector of the flexible body; This is the translational coordinate vector of the origin of the flexible body's reference coordinate system in the global coordinate system. This is the rotation coordinate vector of the flexible body's reference coordinate system relative to the global coordinate system; This is the modal coordinate vector of the flexible body, used to describe the elastic deformation of the flexible body relative to the reference coordinate system of the flexible body; the superscript T indicates transpose; Step 1.2: Based on the generalized coordinates of the flexible body obtained in Step 1.1, establish the expression for the position of the flexible body nodes; The spatial position of a flexible body node is determined by the position of the origin of the flexible body reference coordinate system, the attitude of the flexible body reference coordinate system, the position of the node in the undeformed configuration, and the elastic deformation of the node; the flexible body's first... The position vector of each node in the global coordinate system Represented as: （2） in, For flexible bodies The position vector of each node in the global coordinate system; Number the nodes of the flexible body; This is the position vector of the origin of the flexible body's reference coordinate system in the global coordinate system; This is the attitude transformation matrix from the flexible body's reference coordinate system to the global coordinate system; For the first The position vector of each node in the undeformed configuration of the flexible body; For the first The mode matrix at each node; The modal coordinate vector of the flexible body; Step 1.3: Based on the flexible body node position expression established in Step 1.2, obtain the flexible body modal information and calculate the current spatial position of each node of the flexible body in the global coordinate system; The elastic deformation of the flexible body is described by modal coordinates, and the modal information is obtained through finite element modal analysis. For the flexible body discretized by solid finite element, the node coordinates, element topology and modal matrix are extracted from the finite element model. Combined with the modal coordinates of the flexible body, the extracted modal information is substituted into formula (2) to calculate the position of each node under the current configuration of the flexible body, and the current spatial position of each node of the flexible body in the global coordinate system is obtained.

3. The method for modeling sliding hinges in the dynamics of a flexible multibody system based on a virtual body according to claim 2, characterized in that, Step 2 specifically includes: Step 2.1: Based on the current spatial position of each node of the flexible body in the global coordinate system obtained in Step 1, define the sliding path on the surface of the flexible body and select the sliding path interpolation region to obtain the quadrilateral interpolation region used to describe the position of the sliding point. The flexible body surface refers to the outer surface in the finite element model of the flexible body that allows relative sliding of the slider or sliding component; the sliding path refers to the path pre-defined on the flexible body surface to define the direction of movement of the slider or sliding component; the current sliding point refers to the position point of the slider or sliding component on the sliding path at a certain moment; Several surface nodes are selected on the surface of the flexible body to form the sliding path interpolation region; for the flexible body discretized by solid finite element method, four surface nodes are selected to form a quadrilateral interpolation region, and the position of the sliding point on the surface of the flexible body is described by the quadrilateral interpolation region. Step 2.2: Based on the quadrilateral interpolation region obtained in Step 2.1, establish the interpolation expression for the position of the sliding point to obtain the position expression of the sliding point in the global coordinate system; For a quadrilateral interpolation region consisting of four nodes, the slip point position is represented as: （3） in, This represents the position vector of the sliding point in the global coordinate system. This represents the current spatial position of the first node within the interpolation region. This represents the current spatial position of the second node within the interpolation region. This represents the current spatial location of the third node within the interpolation region. This represents the current spatial position of the fourth node within the interpolation region; For the first interpolation parameter, This is the second interpolation parameter; the two together constitute the interpolation parameters. Let be the bilinear interpolation form function corresponding to the first node. The bilinear interpolation form function corresponding to the second node. The bilinear interpolation form function corresponding to the third node. This is the bilinear interpolation shape function corresponding to the fourth node; Step 2.3: Based on the position expression of the sliding point in the global coordinate system obtained in Step 2.2, determine the bilinear interpolation shape function, and obtain the flexible body surface sliding path and sliding point position interpolation expression jointly described by the quadrilateral interpolation region, interpolation parameters and bilinear interpolation shape function; When the interpolation parameter takes values ​​in the range of , When the bilinear interpolation shape function takes the following form: （4） in, , , and These are the bilinear interpolation shape functions corresponding to the first, second, third, and fourth nodes within the quadrilateral interpolation region, respectively.

4. The method for modeling sliding hinges in the dynamics of a flexible multibody system based on a virtual body according to claim 3, characterized in that, Step 3 specifically includes: Step 3.1: Based on the slip point position interpolation expression obtained in Step 2, given the target slip point position under the initial configuration, obtain the initial interpolation data used to solve the fixed interpolation parameters; In the initial configuration, the target position of the sliding point is given according to the preset sliding path on the surface of the flexible body. ;set up Let be the target position vector of the slip point under the initial configuration. , , and These are the position vectors of the first, second, third, and fourth nodes within the quadrilateral interpolation region under the initial configuration, respectively; where the superscript 0 indicates the initial configuration; Let be the position vector of the first node in the initial configuration. Let be the position vector of the second node in the initial configuration. Let be the position vector of the third node in the initial configuration. This is the position vector of the fourth node under the initial configuration; This yields the initial interpolation data used to solve for the fixed interpolation parameters. The initial interpolation data includes the target position of the sliding point. The initial position vectors of the four nodes within the quadrilateral interpolation region. , , , And the bilinear interpolation shape function determined in step 2; Step 3.2: Based on the initial interpolation data obtained in Step 3.1, solve the fixed interpolation parameters of the slip point using the least squares problem to obtain the first fixed interpolation parameters and the second fixed interpolation parameters; Based on the target position of the glide point in the initial configuration Using the initial position vectors of the four nodes within the quadrilateral interpolation region, construct a least-squares problem: （5） in, The first fixed interpolation parameter is... This is the second fixed interpolation parameter; The first interpolation parameter is... This is the second interpolation parameter; The target position vector of the slip point under the initial configuration; Number the nodes within the quadrilateral interpolation region. =1, 2, 3, 4 correspond to the first node, the second node, the third node, and the fourth node, respectively; For the first The position vectors of each node under the initial configuration; For the first The bilinear interpolation shape function corresponding to each node; Represents the Euclidean norm; Step 3.3: Based on the first fixed interpolation parameter and the second fixed interpolation parameter obtained in Step 3.2, keep the fixed interpolation parameter unchanged and update the position of the sliding point to obtain the fixed interpolation parameter of the sliding point and the calculation method of updating the position of the sliding point based on the fixed interpolation parameter; Throughout the entire dynamic simulation, the first fixed interpolation parameter is maintained. Second fixed interpolation parameter The first interpolation parameter remains unchanged; when the flexible body deforms, the current spatial positions of the first, second, third, and fourth nodes within the quadrilateral interpolation region are updated as the flexible body moves; at this time, the first interpolation parameter in equation (3) is changed. Second interpolation parameter Take them as the first fixed interpolation parameters respectively. Second fixed interpolation parameter Substituting the current spatial positions of the four nodes within the quadrilateral interpolation region into equation (3), we can obtain the position of the current sliding point in the global coordinate system.

5. The method for modeling the sliding hinge of a flexible multibody system based on a virtual body according to claim 4, characterized in that, Step 4 specifically includes: Step 4.1: Based on the fixed interpolation parameters of the sliding point obtained in Step 3, calculate the current position of the current sliding point and the adjacent path points in the global coordinate system; To describe the characteristic of the slip direction changing with the deformation of the flexible body, a local geometric frame is constructed at the current slip point. The local geometric frame consists of a first basis vector, a second basis vector, and a third basis vector. The first basis vector is determined along the local tangential direction of the slip path, the second basis vector is obtained by orthogonalizing the directions of auxiliary points, and the third basis vector is obtained by cross product of the first and second basis vectors. All of the above basis vectors are represented in the global coordinate system. The position vector of the current sliding point in the global coordinate system is denoted as It is calculated by the interpolation formula shown in equation (3) based on the fixed interpolation parameters and the current node coordinates of the quadrilateral interpolation region: （6） The position vectors of adjacent path points before and after the current slip point in the global coordinate system are denoted as follows: and It also has corresponding interpolation parameters. and Substituting into the interpolation formula shown in equation (3), we obtain: （7） （8） in, This represents the position vector of the current sliding point in the global coordinate system. This is the position vector of the previous adjacent path point in the global coordinate system; This is the position vector of the next adjacent path point in the global coordinate system; For the quadrilateral interpolation region containing the current slip point, the first... The current spatial location of each node; For the quadrilateral interpolation region containing the previous adjacent path point, the first... The current spatial location of each node; For the quadrilateral interpolation region containing the next adjacent path point, the first The current spatial location of each node; Step 4.2: Based on the preceding and following adjacent path points obtained in Step 4.1, construct local tangential basis vectors; Based on the adjacent path points before and after the current slip point and Define the local tangential vector of the slip path. : （9） The first basis vector of the tangent of the slip path for: （10） in, This is the local tangential vector of the slip path; This is the position vector of the next adjacent path point before the current sliding point in the global coordinate system; This is the position vector of the next adjacent path point after the current sliding point in the global coordinate system; is the first basis vector of the local geometric frame, used to represent the local tangential direction of the slip path near the current slip point; Step 4.3: Based on the current glide point and the first basis vector, determine the auxiliary point and construct the second basis vector; To determine the second basis vector of the local geometric frame, an auxiliary point is selected; the position vector of the auxiliary point in the global coordinate system is denoted as . ; （11） in, This is the position vector of the auxiliary point in the global coordinate system; The first interpolation parameter corresponding to the auxiliary point; The second interpolation parameter corresponding to the auxiliary point; For the quadrilateral interpolation region containing auxiliary points, the first The current spatial location of each node; Defined by the current slip point Point to auxiliary point auxiliary direction vector : （12） To ensure the orthogonality of the local geometric frame, auxiliary direction vectors are removed. First basis The component in the direction, obtained with orthogonal vectors : （13） Then the second basis vector of the local geometric frame for: （14） Step 4.4: Construct a local geometric frame based on the first and second basis vectors; Third basis From the first basis Second basis Cross product yields: （15） This yields the local geometric frame at the slip point. : （16） When a flexible body deforms, the local geometric frame is updated synchronously with the local configuration of the flexible body. The local geometric frame is used to characterize the local attitude and sliding direction at the slip point. Thus, the local geometric frame at the slip point is obtained synchronously with the local configuration of the flexible body.

6. The method for modeling sliding hinges in the dynamics of a flexible multibody system based on a virtual body according to claim 5, characterized in that, Step 5 specifically includes: Step 5.1: Set the virtual rigid body reference point; A virtual rigid body is introduced at the slip point to transmit the local position and attitude information of the flexible body at the slip path; Make the reference point of the virtual rigid body coincide with the slip point: （17） in, This is the position vector of the virtual rigid body reference point in the global coordinate system; This represents the position vector of the current sliding point in the global coordinate system. Step 5.2: Set up a virtual rigid body local frame; Let the local geometric frame at the slip point be... The virtual rigid body local frame is This ensures a constant relative rotational relationship between the virtual rigid body local frame and the local geometric frame at the slip point: （18） in, The constant relative rotation matrix is ​​determined under the initial configuration; Specifically, under the initial configuration, based on the local geometric frame at the slip point. and the preset virtual rigid body initial local frame Determine the constant relative rotation matrix: （19） During the dynamic simulation process The virtual rigid body reference point remains unchanged; thus, the virtual rigid body reference point is updated synchronously with the position of the slip point, and the virtual rigid body local frame is updated synchronously with the local geometric frame at the slip point, thereby obtaining a virtual rigid body used to transmit the local position and attitude of the slip point.

7. The method for modeling sliding hinges in the dynamics of a flexible multibody system based on a virtual body according to claim 6, characterized in that, Specifically, step 6 includes: Step 6.1: Define the sliding hinge connection point; The slider rigid body is denoted as a rigid body. The virtual rigid body is denoted as a rigid body. ;exist and Establish position-level constraints between the sliding hinges to make Compared to Only the relative translational degrees of freedom along the local slip direction are retained; assuming The position vector of the sliding hinge connection point on the upper surface in the global coordinate system is: , The position vector of the sliding hinge connection point on the upper surface in the global coordinate system is: ; Step 6.2: Define the connection vector between connection points; Define the connection vector between two connection points. for: （20） Step 6.3: Construct translational constraints for the sliding hinge; set up and Virtual rigid body In a local frame, two normal unit vectors orthogonal to the slip direction. The first normal unit vector is orthogonal to the slip direction. The second normal unit vector is orthogonal to the sliding direction; the slider is a rigid body. Only along virtual rigid bodies The defined slip direction undergoes relative translation, causing the connecting vector to... respectively with and Orthogonal, we obtain the first and second translational constraints of the sliding hinge: （21） Step 6.4: Construct the rotational constraint of the sliding hinge; set up For the slider rigid body The direction of the sliding axis in the middle; to ensure the rigidity of the slider. The direction of the slip axis defined in the virtual rigid body The local slip direction defined in the middle is consistent, so that respectively with , Orthogonal constraints result in two rotational constraints: the first rotational constraint and the second rotational constraint. （22） To constrain the rigid body of the slider Relative rotation about a direction other than the sliding axis introduces a rigid body for the slider. A normal direction orthogonal to the slip axis direction and make it with a virtual rigid body normal direction in Satisfying the orthogonality relation yields the third rotation constraint: （23） Step 6.5: Obtain the complete sliding hinge position-level constraint vector; From equation (21) to equation (23), the rigid body of the slider is obtained. With virtual rigid bodies The complete sliding hinge position-level constraint vector between them: （24） in, This is the position-level constraint vector for the sliding hinge, which includes two translational constraints and three rotational constraints, for a total of five constraints. Through steps 6.1 to 6.5, the sliding constraints on the flexible body are transformed into sliding hinge position-level constraints between the slider rigid body and the virtual rigid body, resulting in a complete sliding hinge position-level constraint vector.

8. The method for modeling sliding hinges in the dynamics of a flexible multibody system based on a virtual body according to claim 7, characterized in that, Specifically, step 7 is as follows: Step 7.1: Define the position-level constraint equations for the sliding hinge: The sliding hinge position-level constraint vectors obtained in step 6 are uniformly denoted as the sliding hinge position-level constraint equations: （25） in, The vector of the position-level constraint equations for the sliding hinge; For the system's generalized coordinate vector; For time; Step 7.2: Establish the velocity level constraint equations for the sliding hinge; Taking the time derivative of the sliding hinge position-level constraint equation shown in equation (25), we obtain the sliding hinge velocity-level constraint equation: （26） in, For the position-level constraint equations of the sliding hinge, apply to the generalized coordinates of the system The Jacobian matrix; The system's generalized velocity vector; The velocity term corresponding to the explicit time term in the position-level constraint equation of the sliding hinge; Step 7.3: Assemble the Jacobian matrix of the sliding hinge constraint; The five constraints in equation (24) are applied to the system's generalized coordinates respectively. Find the partial derivatives and arrange them row by row according to the constraint order to obtain the Jacobian matrix of the sliding hinge constraint. .

9. The method for modeling sliding hinges in the dynamics of a flexible multibody system based on a virtual body according to claim 8, characterized in that, Specifically, step 8 includes: Based on the connection relationship between the flexible body and the virtual rigid body obtained in step 5, the sliding hinge position level constraint obtained in step 6, and the sliding hinge constraint Jacobian matrix obtained in step 7, the dynamic equations of the flexible multibody system are assembled and solved. Step 8.1: Define the overall system constraint equations; The virtual rigid body reference point coincidence relationship shown in equation (17) and the virtual rigid body local frame update relationship shown in equation (18) are collectively denoted as the connection constraint between the flexible body and the virtual rigid body. It is represented as: （27） in, For the connection constraint between the flexible body and the virtual rigid body; Connection constraints between flexible bodies and virtual rigid bodies The sliding hinge position level constraint shown in equation (24) After unified assembly, the overall system constraint equations are obtained: （28） in, The system's total constraint equation vector; For the connection constraint between the flexible body and the virtual rigid body; For the positional constraint of the sliding hinge between the slider rigid body and the virtual rigid body; Step 8.2: Establish the system dynamic equations; Based on the system's generalized coordinates, mass matrix, generalized forces, and the overall system constraint equations obtained in step 8.1, the dynamic equations of the flexible multibody system are established: （29） in, The mass matrix of the flexible multibody system; The generalized acceleration vector of the system; For the system's overall constraint equations with respect to generalized coordinates The Jacobian matrix; These are Lagrange multiplier vectors; For generalized active power; It is a generalized inertial coupling force; It is a generalized elastic force; Step 8.3: Establish the acceleration level constraint equations; Taking the total system constraint equation shown in equation (28) twice in time, we obtain the acceleration level constraint equation: （30） in, This is the vector of the right-hand side of the acceleration level obtained by differentiating the system's overall constraint equations with respect to time. Step 8.4: Establish the augmented dynamic equations; Combining equations (29) and (30), we obtain the augmented dynamic equation: （31） Step 8.5: Time integration and nonlinear iterative solution; Within each time step, the system state is predicted by the time integration method, and equation (31) is solved by the nonlinear iterative method to output the slider motion response, flexible body deformation response, local sliding direction change and constraint reaction force; Through steps 8.1 to 8.5, the connection constraints between the flexible body and the virtual rigid body, as well as the sliding hinge position constraints between the slider rigid body and the virtual rigid body, are uniformly incorporated into the dynamic equations of the flexible multibody system, thereby obtaining the slider motion response, the flexible body deformation response, the local sliding direction change, and the constraint reaction force.

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