Hinge constraint internal force solving method based on Newton-Euler equation

Abstract

The invention discloses a hinge constraint internal force solving method based on a Newton-Euler equation, relates to the technical field of structural dynamics numerical simulation analysis in engineering, and aims to solve the problems of insufficient constraint precision and unstable constraint calculation in finite element analysis in the prior art. The method comprises the following steps: simplifying a hinge structure, and constructing an equivalent constraint equation; solving a rigid body kinetic equation; solving the constraint equation, and calculating the constraint internal force. According to the method, the calculation stability is remarkably enhanced while the calculation precision is effectively improved. According to the hinge force calculation method provided by the invention, introduction of manual parameters is avoided theoretically, so that a calculation result is not influenced by parameter selection.

Description

Technical Field

[0001] This invention relates to the field of numerical simulation analysis technology of structural dynamics in engineering, and in particular to a method for solving the internal forces of hinged constraints based on the Newton-Euler equations. Background Technology

[0002] In explicit dynamic analysis of mechanical structure impacts and vehicle collisions, existing technologies for handling hinged structures mainly fall into two categories: refined modeling methods based on physical shape and simplified modeling methods based on constraint equations. Refined modeling methods based on physical shape mesh the hinge based on its actual geometry and define contact relationships between interconnected components to transfer forces and motion. This method offers high computational accuracy but significantly increases computational cost and reduces computational stability. Simplified modeling methods based on constraint equations, in many dynamic conditions such as whole-vehicle collision analysis, pay less attention to the details of local deformation of the hinge structure itself. To improve computational efficiency and the stability of local positions, simplified equivalent constraint models are often used in modeling. The behavior of this constraint model can be represented by constraint equations, and the solution method is crucial. Currently, commonly used constraint equation solution methods mainly include the Lagrange multiplier method and the penalty function method.

[0003] The Lagrange multiplier method introduces Lagrange multipliers to solve the constraint equations and system equations related to the articulated element simultaneously, obtaining motion parameters and internal forces. However, in explicit finite element analysis, this method undermines the inherent advantage of explicit algorithms—"no need to solve simultaneous equations"—significantly increasing computational complexity and requiring more storage space. Therefore, this method is generally unsuitable for explicit dynamic analysis. The penalty function method implements constraints by introducing penalty terms into the deviations of the constraint equations. When constraints are not strictly satisfied, penalty forces are applied to the relevant nodes to force the constraints to approach satisfaction. This method does not require solving simultaneous equations, has simple computational logic, and is easily parallelized. Therefore, mainstream explicit dynamic analysis software, such as LS-DYNA, ABAQUS, and Radioss, commonly uses this method when handling articulated constraints.

[0004] Currently, the penalty function method is the most widely used approach for analyzing the dynamic response of articulated structures in explicit dynamics programs. However, this method has significant technical drawbacks when dealing with articulated constraints: firstly, the penalty function method relies on a penalty factor for constraint correction, leading to insufficient constraint accuracy of the articulated elements; secondly, the penalty factor is highly sensitive during explicit integration, easily causing solution instability or even program interruption. Selecting a reasonable penalty factor is a very difficult task, as different working conditions often require different values. These problems significantly limit the application of simplified articulated models in high-precision and large-scale vehicle collision simulations. Therefore, insufficient constraint accuracy and unstable constraint calculations in finite element analysis have become urgent scientific problems that need to be solved. Summary of the Invention

[0005] The purpose of this invention is to provide a method for solving the internal forces of articulated constraints based on the Newton-Euler equations, so as to solve the problems of insufficient constraint accuracy and unstable constraint calculation in the existing technology in finite element analysis.

[0006] This invention provides a method for solving the internal forces of hinged constraints based on the Newton-Euler equations, including: Step 1: Simplify the hinge structure and construct the equivalent constraint equations. The hinge structure includes a ball joint and a revolute joint. The constraint characteristic of the ball joint is that there is a fixed spatial co-point relationship between the two mechanical components. A specific node N1 on component A and its corresponding node N2 on component B always coincide in spatial position. The equivalent constraint equations of the ball joint are represented by the coordinates of the two nodes: (1); Where x1 is the coordinate of node N1 and x2 is the coordinate of node N2; The equivalent constraint equations of the revolute joint are expressed based on the position vectors of two sets of four constraint nodes. Compared with the ball joint, an additional set of coincident constraint nodes N3 and N4 are added. The first set of constraint equations is the same as equation (1), and the second set of constraint equations is: (2); Where x3 is the coordinate of node N3, and x4 is the coordinate of node N4; Step two: Solve the rigid body dynamics equations. Under external forces, the dynamic response of a single rigid body follows Newton's and Euler's equations, corresponding to the translational and rotational degrees of freedom, respectively. After calculating the translational and rotational accelerations based on their mathematical expressions, the translational and rotational displacement increments are updated using a central difference algorithm within an explicit time integration framework. The rotational displacement increments of the rigid body are then utilized. To calculate the transformation matrix Based on the transformation matrix and time step Inertia matrix at time step Update time step Inertial tensor of a rigid body at time t And update the coordinates of the subordinate nodes on the rigid body; Step 3: Solve the constraint equations and calculate the constraint internal forces.

[0007] Furthermore, in step two, the mathematical expressions for the translational and rotational degrees of freedom of the rigid body are: (3) (4) in, Represents the mass of a rigid body. It is the acceleration of the center of mass of the rigid body. It is a combined external force; It is the inertia tensor of a rigid body. It is the angular velocity of the rigid body. It is the angular acceleration of a rigid body. It is the resultant external torque.

[0008] Furthermore, the mass of a rigid body Set of subordinate nodes of a rigid body Middle node quality The sum of: (5) Inertia tensor of a rigid body Given by the following formula: (6) in, For nodes The location.

[0009] Furthermore, in step two, within the explicit time integration framework, the central difference algorithm is used to update the translational and rotational displacement increments of the rigid body. The update formula for the translational coordinates of the rigid body's center of mass is as follows: (7) in, Let n be the position of the center of mass of the rigid body at time step n. It is the position of the rigid body's center of mass at time step n+1. This represents the increment of translational displacement.

[0010] Furthermore, in step two, based on the Hughes-Winget algorithm, the rotational displacement increment of the rigid body is utilized. To calculate the transformation matrix : (8) (9) in, It is the identity matrix. It is the intermediate matrix. It is any vector.

[0011] Furthermore, in step two, based on the transformation matrix... and time step Inertia matrix at time step Update time step based on equation (10) Inertial tensor of a rigid body at time t And update the coordinates of the subordinate nodes on the rigid body using equation (11): (10) (11) in, and Let represent the coordinates of node i at time step n and time step n+1, respectively. This represents the increment of the translational displacement of a rigid body. This represents the position vector of node i relative to the center of mass of the rigid body.

[0012] Furthermore, in step three, the constraint equations of the ball joint are equivalently expressed by the following equations: (12) in, This represents the node displacement at time step n+1. This represents the nodal displacement at step n. express relatively The position vector; (13) The dynamic equations of the rigid body have been given in equations (3) and (4), and are expressed in the following form: (14) (15) in, It is a combination of external forces. Let 'a' represent the mass of the rigid body, and 'a' be the translational acceleration of the rigid body. It is the angular acceleration of a rigid body. Let represent the inertia tensor matrix of the rigid body in the global coordinate system, which is updated according to equation (10) at each time step. This represents the equivalent external torque, while The resultant torques acting on a rigid body are represented by the following relationships: (16).

[0013] Furthermore, in step three, the method also includes: In the explicit time integral, the central difference algorithm is used to discretize in time, and equations (14) and (15) are discretized as follows: (17) (18) in, It is the displacement of the rigid body's center of mass. It is the rotational displacement of a rigid body. It is the velocity of a rigid body. It is the rigid body angular velocity. This represents the sum of external forces excluding the internal forces at the hinges. It is a rigid body torque caused by the internal forces of the hinge. It is the rigid body torque caused by forces other than the hinge forces. It is the internal force of the hinge. This is the time increment between step n and step (n+1). Indicates the first Step and the first The time increment between steps; and These are the translational displacements of the rigid body at time steps n and n+1, respectively. and These are the rotational displacements of the rigid body at time steps n and n+1, respectively. It is the translational velocity of the rigid body at time step n+1 / 2. It is the rotational velocity of the rigid body at time step n+1 / 2; For any point on a rigid body, its displacement With the displacement of the center of mass The relationship between them is: (19) in, This represents the position vector of the constraint node relative to the center of mass of the rigid body. , This represents the rotational displacement of a rigid body.

[0014] Furthermore, in step three, the method also includes: based on the principles of rigid body kinematics, the displacement increment between time step n and time step n+1 is derived from the translational and rotational displacements of the center of mass. (20) in, The position vector of the constraint node relative to the center of mass of the rigid body is represented by equation (20), which is approximately expressed as: (twenty one) Hinged internal forces This refers to the constraint forces between rigid bodies A and B at hinged constraint points 1 and 2. This represents the constraint force exerted by rigid body B on rigid body A at node 1 on rigid body A. This represents the constraint force exerted by rigid body A on rigid body B at node 2. and For interaction forces, satisfying: (twenty two) Resultant force acting on a rigid body Constraints from other articulated units They are mutually coupled.

[0015] Furthermore, in step three, the method also includes: Approximately expressed as: (twenty three) in, For the present The net force acting on the rigid body at all times, i=1 represents the net force acting on rigid body A, and i=2 represents the net force acting on rigid body B. express The internal forces constrained by the ball joint at the current moment of study; Substituting equations (17), (18), and (21)-(23) into equation (12), we get: (twenty four) in, It is a coefficient matrix. Indicates the time step and The increment of the hinge constraint force between them; It is a three-dimensional vector, generated by the resultant force acting on the rigid body. and torque and the velocity at the constraint nodes and displacement The calculation yields the following:

[0016] It is the inverse matrix of the inertia tensor of rigid body A. It is the inverse matrix of the rigid body B inertia tensor.

[0017] This invention offers the following advantages: A method for solving the internal forces of hinged constraints based on the Newton-Euler equations, using a simplified hinged structure and combining an explicit time integration algorithm with the rigid body Newton-Euler equations, directly solves the internal forces between multiple rigid body constraint nodes. This eliminates the need for artificial parameters, enabling transient dynamic behavior analysis of the hinged structure under an explicit scheme. Compared to traditional penalty function methods and Lagrange multiplier methods, this invention fundamentally avoids the uncertainties and computational errors caused by the selection of penalty factors in traditional penalty function methods, ensuring that the constraint equations are accurately satisfied. Compared to the Lagrange multiplier method, this algorithm directly constructs the constraint equations, bypassing the solution process of the augmented system equations, and significantly improves computational efficiency through an explicit solution scheme, while also being naturally compatible with parallel computing architectures. Since no artificial parameters are introduced, computational stability is no longer affected by parameter values, resulting in more robust performance compared to traditional penalty function methods. Attached Figure Description

[0018] To more clearly illustrate the technical solution of the present invention, the drawings used in the embodiments will be briefly introduced below. Obviously, those skilled in the art can obtain other drawings based on these drawings without creative effort.

[0019] Figure 1 Here are schematic diagrams of a ball joint structure, where (a) is a three-dimensional schematic diagram of the ball joint structure and (b) is a cross-sectional view of the ball joint structure. Figure 2 The diagram shows the structure of the revolute joint, where (a) is a three-dimensional schematic diagram of the revolute joint structure and (b) is a schematic diagram showing the positions of nodes N1, N2, N3, and N4. Detailed Implementation

[0020] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below in conjunction with specific embodiments and corresponding drawings. Obviously, the described embodiments are only a part of the embodiments of this invention, and not all of them. Based on the embodiments of this invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this invention. The technical solutions provided by various embodiments of this invention will be described in detail below with reference to the accompanying drawings.

[0021] Please see Figure 1 and Figure 2 An embodiment of the present invention provides a method for solving the internal forces of hinged constraints based on the Newton-Euler equations, comprising the following steps: Step 1: Simplify the hinged structure and construct equivalent constraint equations.

[0022] In mechanical structures, ball joints and revolute joints are the two most common types of hinge structures. Both structures can be simplified to construct equivalent constraint equations. The basic structure of a ball joint is as follows: Figure 1 As shown. The constraint characteristic of a ball joint is that there is a fixed spatial co-point relationship between the two mechanical parts, that is, a specific node N1 on part A must always coincide with the corresponding node N2 on part B in spatial position. Under this constraint, the mechanical structure can rotate about the constraint point, but its translational degrees of freedom are completely constrained. Its equivalent constraint equation can be expressed by the coordinates of the two nodes: (1) Where x1 is the coordinate of node N1 and x2 is the coordinate of node N2.

[0023] like Figure 2The revolute joint constrains three translational degrees of freedom and two rotational degrees of freedom between the components, retaining only one rotational degree of freedom around the central axis. Essentially, it is equivalent to a combination of two ball joints. Therefore, the equivalent constraint equations of the revolute joint can be expressed based on the position vectors of two sets of constraint nodes, totaling four. Compared to the ball joint, it adds a set of overlapping constraint nodes N3 and N4. The first constraint equation is the same as equation (1), and the second set of constraint equations is... (2) Where x3 is the coordinate of node N3 and x4 is the coordinate of node N4.

[0024] Step 2: Solve the equations of motion for the rigid body.

[0025] When a single rigid body is subjected to external forces, its dynamic response follows Newton's equations and Euler's equations, corresponding to the translational and rotational degrees of freedom of the rigid body, respectively. Their mathematical expressions are as follows: (3) (4) in, Represents the mass of a rigid body. It is the location of the center of mass of the rigid body. It is the acceleration of the center of mass. It is the inertia tensor of a rigid body. It is the angular velocity of the rigid body. It is the angular acceleration of a rigid body. It is a combination of external forces. It is the net external torque. The mass of the rigid body. Set of subordinate nodes of a rigid body Middle node quality The sum of: (5) Inertia tensor of a rigid body Given by the following formula: (6) in For nodes The location.

[0026] After calculating the translational and rotational accelerations of the rigid body using equations (3) and (4), the translational and rotational displacement increments of the rigid body are updated using the central difference algorithm within the explicit time integration framework. The update formula for the translational coordinates of the rigid body's center of mass is as follows: (7) in, Let n be the position of the center of mass of the rigid body at time step n. That is its position at time step n+1. This represents the increment of translational displacement.

[0027] Based on the Hughes-Winget algorithm, the rotational displacement increment of a rigid body can be utilized. To calculate the transformation matrix : (8) (9) in, It is the identity matrix. It is the intermediate matrix. It is any vector.

[0028] Based on the transformation matrix and time step Inertia matrix at time step Update time step based on equation (10) Inertial tensor of a rigid body at time t And update the coordinates of the subordinate nodes on the rigid body using equation (11): (10) (11) in, and Let represent the coordinates of node i at time step n and time step n+1, respectively. This represents the increment of the translational displacement of a rigid body. This represents the position vector of node i relative to the center of mass of the rigid body.

[0029] Step 3: Solve the constraint equations and calculate the constraint internal forces.

[0030] The constraint equations for the ball joint can be found here (1), and can be equivalently expressed by the following equations: (12) in, This represents the node displacement at time step n+1, while This represents the nodal displacement at step n. express relatively The position vector, that is: (13) The dynamic equations of the rigid body have been given in equations (3) and (4). For ease of calculation, they can be expressed in the following form. (14) (15) In this expression, 'a' is the translational acceleration of the rigid body. It is the angular acceleration of a rigid body. Let represent the inertia tensor matrix of the rigid body in the global coordinate system, which is updated according to equation (10) at each time step. This represents the equivalent external torque, while The resultant torques acting on a rigid body are represented by the following relationships: (16) In explicit time integration, the central difference algorithm is used to discretize the time integrals, and equations (14) and (15) can be discretized as follows: (17) (18) in, It is the displacement of the rigid body's center of mass. It is the rotational displacement of a rigid body. It is the velocity of a rigid body. It is the rigid body angular velocity. This represents the sum of external forces excluding the internal forces at the hinges. It is a rigid body torque caused by the internal forces of the hinge. It is the rigid body torque caused by forces other than the hinge forces. It is the internal force of the hinge. This is the time increment between step n and step (n+1), and Indicates the first Step and the first The time increment between steps. and These are the translational displacements of the rigid body at time steps n and n+1, respectively. and These are the rotational displacements of the rigid body at time steps n and n+1, respectively. It is the translational velocity of the rigid body at time step n+1 / 2. It is the rotational velocity of the rigid body at time step n+1 / 2.

[0031] For any point on a rigid body, its displacement With the displacement of the center of mass The relationship between them is (19) in, This represents the position vector of the point relative to the center of mass of the rigid body, i.e. . This represents the rotational displacement of a rigid body.

[0032] According to the principles of rigid body kinematics, the displacement increment between time step n and time step n+1 can be derived from the translational and rotational displacements of the center of mass: (20) in, This represents the position vector of the constraint node relative to the center of mass of the rigid body. Since in the explicit time integration scheme, the time step... The position vectors before and after a single time step are very small. The change is negligible. Therefore, equation (20) can be approximated as: (twenty one) Hinged internal forces This refers to the constraint forces between rigid bodies A and B at hinged constraint points 1 and 2. This represents the constraint force exerted by rigid body B on rigid body A at node 1 on rigid body A. This represents the constraint force exerted by rigid body A on rigid body B at node 2. and For interaction forces, satisfying: (twenty two) The resultant force acting on a rigid body (excluding the articulated internal forces currently under study). The main sources of this force are the constraint forces from other hinged units and external forces such as gravity and contact forces. However, since multiple hinged structures may exert constraints on the same rigid body, the resultant force... Essentially related to the constraint forces from other articulated units They are mutually coupled.

[0033] Considering that the change in constraint force of each hinge element can be ignored within one time increment step, It can be approximated as: (twenty three) in, For the present Acting on the rigid body at all times The resultant force, i=1 represents the resultant force acting on rigid body A, and i=2 represents the resultant force acting on rigid body B. Then it means The internal forces constrained by the ball joint at the current moment of study.

[0034] Substituting equations (17), (18), and (21)-(23) into equation (12), we get: (twenty four) in, It is a coefficient matrix. Indicates the time step and The increment of the hinge constraint force between them. It is a three-dimensional vector that can be represented by the resultant force acting on a rigid body. and torque and the velocity at the constraint nodes and displacement The calculation yields the following:

[0035] It is the inverse matrix of the inertia tensor of rigid body A. It is the inverse matrix of the rigid body B inertia tensor.

[0036] In summary, this invention addresses the problems of insufficient computational accuracy and poor computational stability in the transient dynamic analysis of automotive articulated structures by proposing a method for calculating constraint forces in articulated structures suitable for explicit time integration schemes. This method effectively improves computational accuracy while significantly enhancing computational stability. The proposed articulated force calculation method theoretically avoids the introduction of artificial parameters, making the calculation results unaffected by parameter selection. Therefore, this method exhibits greater adaptability under different operating conditions and is more suitable for engineering applications.

[0037] The embodiments of the present invention described above do not constitute a limitation on the scope of protection of the present invention.

Claims

1. A method for solving the internal forces of hinged constraints based on the Newton-Euler equations, characterized in that, include: Step 1: Simplify the hinge structure and construct the equivalent constraint equations. The hinge structure includes a ball joint and a revolute joint. The constraint characteristic of the ball joint is that there is a fixed spatial co-point relationship between the two mechanical components. A specific node N1 on component A and its corresponding node N2 on component B always coincide in spatial position. The equivalent constraint equations of the ball joint are represented by the coordinates of the two nodes: (1)； Where x1 is the coordinate of node N1 and x2 is the coordinate of node N2; The equivalent constraint equations of the revolute joint are expressed based on the position vectors of two sets of four constraint nodes. Compared with the ball joint, an additional set of coincident constraint nodes N3 and N4 are added. The first set of constraint equations is the same as equation (1), and the second set of constraint equations is: (2)； Where x3 is the coordinate of node N3, and x4 is the coordinate of node N4; Step two: Solve the rigid body dynamics equations. Under external forces, the dynamic response of a single rigid body follows Newton's and Euler's equations, corresponding to the translational and rotational degrees of freedom, respectively. After calculating the translational and rotational accelerations based on their mathematical expressions, the translational and rotational displacement increments are updated using a central difference algorithm within an explicit time integration framework. The rotational displacement increments of the rigid body are then utilized. To calculate the transformation matrix Based on the transformation matrix and time step Inertia matrix at time step Update time step Inertial tensor of a rigid body at time t And update the coordinates of the subordinate nodes on the rigid body; Step 3: Solve the constraint equations and calculate the constraint internal forces.

2. The method for solving the internal forces of hinged constraints based on the Newton-Euler equations as described in claim 1, characterized in that, In step two, the mathematical expressions for the translational and rotational degrees of freedom of the rigid body are: (3) (4) in, Represents the mass of a rigid body. It is the acceleration of the center of mass of the rigid body. It is a combined external force; It is the inertia tensor of a rigid body. It is the angular velocity of the rigid body. It is the angular acceleration of a rigid body. It is the resultant external torque.

3. The method for solving the internal forces of hinged constraints based on the Newton-Euler equations as described in claim 2, characterized in that, mass of a rigid body Set of subordinate nodes of a rigid body Middle node quality The sum of: (5) Inertia tensor of a rigid body Given by the following formula: (6) in, For nodes The location.

4. The method for solving the internal forces of hinged constraints based on the Newton-Euler equations as described in claim 3, characterized in that, In step two, within the explicit time integration framework, the central difference algorithm is used to update the translational and rotational displacement increments of the rigid body. The update formula for the translational coordinates of the rigid body's center of mass is as follows: (7) in, Let n be the position of the center of mass of the rigid body at time step n. It is the position of the rigid body's center of mass at time step n+1. This represents the increment of translational displacement.

5. The method for solving the internal forces of hinged constraints based on the Newton-Euler equations as described in claim 4, characterized in that, In step two, based on the Hughes-Winget algorithm, the rotational displacement increment of the rigid body is utilized. To calculate the transformation matrix : (8) (9) in, It is the identity matrix. It is the intermediate matrix. It is any vector.

6. The method for solving the internal forces of hinged constraints based on the Newton-Euler equations as described in claim 5, characterized in that, In step two, based on the transformation matrix and time step Inertia matrix at time step Update time step based on equation (10) Inertial tensor of a rigid body at time t And update the coordinates of the subordinate nodes on the rigid body using equation (11): (10) (11) in, and Let represent the coordinates of node i at time step n and time step n+1, respectively. This represents the increment of the translational displacement of a rigid body. This represents the position vector of node i relative to the center of mass of the rigid body.

7. The method for solving the internal forces of hinged constraints based on the Newton-Euler equations as described in claim 6, characterized in that, In step three, the constraint equations of the ball joint are equivalently expressed by the following equations: (12) in, This represents the node displacement at time step n+1. This represents the nodal displacement at step n. express relatively The position vector; (13) The dynamic equations of the rigid body have been given in equations (3) and (4), and are expressed in the following form: (14) (15) in, It is a combination of external forces. Let denot _a_ represent the mass of the rigid body, and _a_ represent the translational acceleration of the rigid body. It is the angular acceleration of a rigid body. Let represent the inertia tensor matrix of the rigid body in the global coordinate system, which is updated according to equation (10) at each time step. This represents the equivalent external torque, while The resultant torques acting on a rigid body are represented by the following relationships: (16)。 8. The method for solving the internal forces of hinged constraints based on the Newton-Euler equations as described in claim 7, characterized in that, In step three, the method further includes: In the explicit time integral, the central difference algorithm is used to discretize in time, and equations (14) and (15) are discretized as follows: (17) (18) in, It is the displacement of the rigid body's center of mass. It is the rotational displacement of a rigid body. It is the velocity of a rigid body. It is the rigid body angular velocity. This represents the sum of external forces excluding the internal forces at the hinges. It is a rigid body torque caused by the internal forces of the hinge. It is the rigid body torque caused by forces other than the hinge forces. It is the internal force of the hinge. This is the time increment between step n and step (n+1). Indicates the first Step and the first The time increment between steps; and These are the translational displacements of the rigid body at time steps n and n+1, respectively. and These are the rotational displacements of the rigid body at time steps n and n+1, respectively. It is the translational velocity of the rigid body at time step n+1 / 2. It is the rotational velocity of the rigid body at time step n+1 / 2; For any point on a rigid body, its displacement With the displacement of the center of mass The relationship between them is: (19) in, This represents the position vector of the constraint node relative to the center of mass of the rigid body. , This represents the rotational displacement of a rigid body.

9. The method for solving the internal forces of hinged constraints based on the Newton-Euler equations as described in claim 8, characterized in that, In step three, the method further includes: based on the principles of rigid body kinematics, the displacement increment between time step n and time step n+1 is derived from the translational and rotational displacements of the center of mass. (20) in, The position vector of the constraint node relative to the center of mass of the rigid body is represented by equation (20), which is approximately expressed as: (21) Hinged internal forces This refers to the constraint forces between rigid bodies A and B at hinged constraint points 1 and 2. This represents the constraint force exerted by rigid body B on rigid body A at node 1 on rigid body A. This represents the constraint force exerted by rigid body A on rigid body B at node 2. and For interaction forces, satisfying: (22) Resultant force acting on a rigid body Constraints from other articulated units They are mutually coupled.

10. The method for solving the internal forces of hinged constraints based on the Newton-Euler equations as described in claim 9, characterized in that, In step three, the method further includes: Approximately expressed as: (23) in, For the present The net force acting on the rigid body at all times, i=1 represents the net force acting on rigid body A, and i=2 represents the net force acting on rigid body B. express The internal forces constrained by the ball joint at the current moment of study; Substituting equations (17), (18), and (21)-(23) into equation (12), we get: (24) in, It is a coefficient matrix. Indicates the time step and The increment of the hinge constraint force between them; It is a three-dimensional vector, generated by the resultant force acting on the rigid body. and torque and the velocity at the constraint nodes and displacement The calculation yields the following: It is the inverse matrix of the inertia tensor of rigid body A. It is the inverse matrix of the rigid body B inertia tensor.

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