Explicit simulation method for new-mode flexible body-cone complementary contact dynamics of non-ideal hinge

Abstract

The invention provides an explicit simulation method for new modal flexible body-cone complementary contact dynamics of a non-ideal hinge, and belongs to the technical field of non-ideal hinge contact mechanics, the constraint condition of the non-ideal hinge existing in actual engineering is converted into a gap-containing contact collision equation with real physical significance through modeling and analysis methods; the method comprises the following steps: carrying out dynamic modeling on a flexible part by adopting a new modal flexible body method based on a force method, introducing a cone complementation theory, converting a nonlinear constraint condition generated in a flexible body contact collision process into a flexible body cone complementation constraint form, and carrying out high-performance explicit simulation of a non-ideal hinge by utilizing a GPU and CPU heterogeneous parallel computing architecture. According to the method, the time step length in calculation can be effectively increased while the calculation precision is kept, and the problem that the calculation efficiency is low when the non-ideal hinge contact problem is processed based on an explicit solver is solved.

Description

Technical Field

[0001] This invention belongs to the field of non-ideal hinge contact mechanics technology, specifically, it relates to an explicit simulation method for the complementary contact dynamics of a novel modal flexible cone of a non-ideal hinge. Background Technology

[0002] Complex mechanical systems are typically composed of multiple components connected by hinges. When modeling multibody dynamics, hinges are usually assumed to be ideal constraints, establishing constraints on the displacements or rotations of the connected components and solving them together with the dynamic equations. However, in practical engineering, due to design and assembly errors, machining accuracy, and wear, hinges inevitably have gaps. This leads to non-smooth motion of the connected components, resulting in violent impacts and collisions between them. The resulting reaction forces of the kinematic pairs can be several times or even ten times greater than normal. Simultaneously, friction exacerbates wear on the kinematic pairs, further increasing the hinge gaps and affecting the lifespan of the mechanism. In such cases, continuing to use the ideal hinge assumption for dynamic modeling results in low modeling accuracy and fails to accurately reflect the precise dynamic characteristics of the system.

[0003] The most direct approach to dynamic modeling of non-ideal hinges is to add nonlinear springs or dampers to the constraint equations of ideal hinges to simulate the contact nonlinearity caused by the gap. However, determining the nonlinear form and coefficients is a challenge of this equivalent modeling method. For specific hinge connections, equivalent models need to be obtained through experimental identification. Since the parameters of the equivalent model are highly dependent on the gap size, experimental identification lacks universality and is costly. To more refined dynamic modeling of non-ideal hinges and describe the influence of gaps on connection nonlinearity, establishing contact models of hinges with gaps has gradually become a trend. Currently known methods include establishing contact collision models of sliding hinges with gaps based on contact friction between rigid bodies and conducting experimental verification; or establishing contact collision dynamic models of non-ideal rotating hinges based on the relative motion of the journal and bearing for spatial rotating hinges. These models are not easily extended to large systems containing complex hinges. Non-smooth dynamics methods can also be used to study the influence of parameters such as gap size, impact coefficient, friction coefficient, and Young's modulus on the dynamic characteristics of the system; however, traditional implicit computation methods cannot fully utilize the parallel advantages of modern high-performance computers. Another common approach is to consider the inertia and material properties of the joint components and establish a contact and collision model for the three-dimensional rotary joint based on the finite element method. However, this method has a large degree of freedom, which limits the computational efficiency.

[0004] Since hinge performance affects the stability of the entire mechanical system, establishing a high-precision dynamic model and achieving efficient dynamic simulation for non-ideal factors such as hinge clearance and lubrication is a crucial problem to be solved in practical applications. With increasingly abundant computing resources, high-performance parallel computing and cloud service platforms have become important development trends in numerical computation methods, and parallel computing of multibody system dynamics is the future direction of real-time simulation. Solving algorithms for multibody system dynamics are mainly divided into two categories: implicit and explicit integration methods. The computational complexity of implicit integration methods increases quadratically or cubically with the dimension of the equation system. Newton iterations during the solution process limit its parallelism and make computation time unpredictable, which is not conducive to real-time simulation. Explicit integration methods, on the other hand, do not require Newton iterations, and their computational complexity increases linearly with the dimension of the equation system, making them very suitable for large-scale CPU and GPU parallel computing to achieve high-performance simulation. However, due to the strong rigidity of contact problems, establishing a non-ideal hinge contact collision model applicable to explicit methods is a key issue.

[0005] Traditional flexible body models based on modal order reduction typically rely on implicit numerical methods for solution. However, when dealing with highly nonlinear problems such as contact and collision, the computational step size decreases drastically due to stability constraints, and iterative convergence difficulties or even computational failures are common. In contrast, novel modal flexible body representation methods based on force methods can improve numerical stability by appropriately reducing system stiffness, thus making them suitable for explicit integration strategies. Building upon this foundation, further development of explicit time integration methods suitable for this type of model, combined with the massively parallel computing capabilities of modern GPUs, and the development of corresponding explicit parallel acceleration techniques, can significantly improve simulation efficiency and scale, ultimately potentially enabling high-performance explicit simulation of complex multibody system dynamics.

[0006] In summary, traditional methods for solving non-ideal hinge contact collision problems require a small step size to ensure numerical stability, which leads to problems such as low computational accuracy, large degree of freedom, and inapplicability to explicit calculations (simulations). Summary of the Invention

[0007] To address the aforementioned issues, this invention proposes an explicit simulation method for the complementary contact dynamics of non-ideal hinges in a novel modal flexible body. This method transforms the non-ideal hinge into a contact collision problem involving multiple rigid or flexible bodies with gaps. The novel modal flexible body method is used to model the dynamics of the flexible body. By locally parameterizing the surface regions where contact may occur, a precise description of the hinge contact region geometry is provided. Furthermore, a novel computational method for flexible contact dynamics based on cone optimization is developed.

[0008] This invention is achieved through the following technical solution: an explicit simulation method for the complementary contact dynamics of a novel modal flexible cone with a non-ideal hinge, the method specifically including the following steps: Step 1: Transform the constraints of non-ideal hinges in actual engineering into contact collision equations with gaps that have real physical meaning through modeling and analysis methods. Step 2: Perform dynamic modeling of the flexible component using the novel modal flexible body method based on the combined force approach. Step 3: Introduce the theory of conical complementarity to transform the nonlinear constraints generated during actual contact and collision into a rigid-flexible coupled conical complementary constraint form: Step 4: Utilize the heterogeneous parallel computing architecture of GPU and CPU to perform high-performance explicit simulation of non-ideal hinges.

[0009] Further, in step 1, the following is included: Step 1.1: Analyze the non-ideal hinge to determine its key parameters and structural characteristics, including material properties, hinge clearance, coefficient of friction, possible contact area, and contact force frequency.

[0010] Step 1.2: Based on the manufacturing tolerances, wear, or assembly errors in actual mechanical systems, establish complex non-smooth dynamic behaviors of contact-collision-separation, and transform the traditional constraint equations into real contact and collision equations with gaps between multiple rigid or flexible bodies.

[0011] Furthermore, in step 2, Under the assumption of small deformation, the deformation of flexible bodies is divided into overall deformation caused by distributed forces and local deformation caused by concentrated forces; the overall deformation is expressed by a low-order mode matrix, and the local deformation is expressed by a high-frequency compliance matrix. The flexible body is divided into several elements and the generalized node vector of each element is determined. The relationship between the material point on the flexible body and the generalized node vector of its element is established through shape functions, and the displacement of the material point is calculated.

[0012] Furthermore, in step 2, The high-frequency compliance matrix is ​​calculated in advance based on the stiffness matrix, low-frequency frequency and modes of the flexible body, and only reflects the deformation effects related to high frequencies. For high-frequency components caused by concentrated forces (moments), the calculation is based on quasi-static values. The influence of distributed forces on mid-to-high-frequency components is ignored relative to the high-frequency mode effects caused by concentrated forces.

[0013] Furthermore, in step 3, Local deformation was calculated using the high-frequency compliance matrix and contact-friction force. The contact-friction force adopts the Coulomb contact model and must satisfy the non-intrusion condition that the contacting bodies do not embed into each other, as well as the maximum dissipation principle that optimizes the frictional force dissipation power. Each contact point consists of two markers. First, a right-handed frame matrix containing a unit normal vector and two unit tangent vectors is constructed. The contact-friction force vector is set as the three-dimensional component under this frame. By calculating the virtual work done by the contact force and the relative velocity of the contact point, and then combining the Karush-Kuhn-Tucker condition, the contact-friction force is solved.

[0014] Furthermore, in step 4, the GPU is used to process tasks with high parallelism, while the CPU is used to process serial tasks.

[0015] An explicit simulation system for the complementary contact dynamics of a novel modal flexible cone with a non-ideal hinge; The system includes a movable robotic arm, a base, a connecting rod, and an actuator. The base is connected to the guide rail via a sliding hinge, the connecting rod is connected to the base via a ball joint, and the actuator is connected to the connecting rod via a rotary hinge. The ball joint and rotary hinge are non-ideal hinges with contact areas of different radii and a preset gap between each contact area.

[0016] In the absolute coordinate system, the base is subjected to an external force that increases linearly with time along the x-axis, the connecting rod is subjected to an external force along the x-axis and an external torque along the y-axis, and the actuator is subjected to an external torque along the x-axis; in addition, to assist the ball joint in contacting, an additional external force is applied along the z-axis in the connecting rod body coordinate system. Under the aforementioned load, the system drives the actuator to move to the designated position to complete the contact.

[0017] Furthermore, each component adopts a new modal flexible body modeling method. When the overall system is modeled using contact, it behaves as an ODE system, and numerical simulation is performed using explicit contact dynamics based on the CCP complementary scheme. The simulation results of contact under low friction conditions are compared with the simulation results of the ideal constraint (DAE) model; then the differences between the contact simulation results and the ideal constraint are examined under different values ​​of friction coefficient μ.

[0018] Compared with the prior art, the present invention has the following beneficial effects: This invention transforms a non-ideal hinge into a contact collision problem involving multiple rigid or flexible bodies with gaps. A novel modal flexible body method is employed to model the dynamics of the flexible body. By locally parameterizing the surface regions where contact may occur, a precise geometric description of the hinge contact area is provided, leading to the development of a novel computational scheme for flexible contact dynamics based on cone optimization. This method is applicable not only to explicit methods but also ensures computational accuracy and stability. Specifically, for low-frequency deformations of the flexible body, its structural dynamic response is directly calculated; while for the high-frequency components of local deformations caused by contact, only the quasi-static response needs to be calculated. This simplification effectively improves the time step size while maintaining computational accuracy. Finally, high-performance explicit simulation of non-ideal hinges is performed using heterogeneous parallel processing of GPU and CPU.

[0019] This invention can effectively improve the time step in the calculation while maintaining the calculation accuracy, and overcome the problem of low calculation efficiency faced by explicit solvers when dealing with non-ideal hinge contact problems.

[0020] This invention combines a novel modal flexible body method, a cone complementarity method, and an explicit integration method. It is applicable not only to the explicit simulation of non-ideal hinges with gap friction but also to contact and collision problems in gears and bearings. This invention provides technical support for the dynamic simulation analysis and life prediction analysis of high-end equipment systems such as robot joints, spacecraft hinges, and high-speed mechanisms. Attached Figure Description

[0021] Figure 1 This is a flowchart illustrating the thought process of the explicit simulation method for the novel modal flexible cone complementary contact dynamics of the non-ideal hinge of the present invention. Figure 2 This is a schematic diagram of the movable robotic arm of the present invention; Figure 3 This illustrates the relationship between actuator displacement (μ = 0.001) and time in the embodiment. Figure 4 The actuator speed in the embodiment is μ = 0.001. Figure 5 The contact force / constraint force (μ = 0.001) experienced by the actuator in the x and z directions in the embodiment. Figure 6 The contact force in the corresponding region of the rotating hinge in the positive x direction under different friction coefficients μ in the embodiments; Figure 7 The frictional force in the corresponding region of the positive x-direction of the rotating hinge under different friction coefficients μ in the embodiments; Figure 8 The contact force of the actuator in the x-direction in the embodiment; Figure 9 The contact force of the actuator in the z-direction in the embodiment; Figure 10 The x-direction displacement of the actuator in the embodiment; Figure 11 The displacement of the actuator in the z-direction in the embodiment is shown. Detailed Implementation

[0022] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0023] Unless otherwise specified, the experimental methods used in the following examples are conventional methods. Unless otherwise specified, the materials, reagents, methods, and instruments used are all conventional materials, reagents, methods, and instruments in the art, and can be obtained commercially by those skilled in the art.

[0024] like Figure 1 As shown, this invention proposes an explicit simulation method for the complementary contact dynamics of a novel mode of non-ideal hinge flexible cone, comprising the following steps: Step 1: Transform the constraints of the non-ideal hinge into a real contact collision problem with gaps: The constraints of non-ideal hinges in practical engineering are transformed into real-world physical contact and collision problems with gaps through modeling and analysis methods, thus more accurately reflecting the dynamic characteristics and mechanical behavior of the system. Specifically, the process involves: first, a detailed analysis of the non-ideal hinge is conducted to determine its key parameters and structural characteristics, including but not limited to material properties, hinge gaps, friction coefficients, potential contact areas, and contact force frequencies; then, based on manufacturing tolerances, wear, or assembly errors in actual mechanical systems, a complex non-smooth dynamic behavior of contact-collision-separation is established based on these parameters and characteristics. This accurately transforms the traditional "constraint equations" into a real "contact and collision problem with gaps" between multiple rigid or flexible bodies, achieving a leap from algebraic constraints to non-smooth dynamics and providing an accurate foundation for subsequent modeling and calculations.

[0025] Step 2: Perform dynamic modeling of the flexible component based on the new modal flexible body method: In traditional modal flexible bodies, the position (rotation) degree of freedom of the concentrated force (moment) must be treated as the key degree of freedom without simplification to ensure calculation accuracy. To calculate local high-frequency deformation caused by concentrated forces with high precision and to address the problem of inaccurate forces due to local geometric inaccuracies, a new modal flexible body model based on the force method can be used. When the low-order frequencies of the structure are much higher than the maximum excitation frequency, the low-order frequencies can be omitted, and only the high-frequency compliance matrix can be used to describe the approximate rigid body situation, thereby establishing the contact dynamics equations of the rigid-flexible coupled system. This invention is also applicable to collisions between rigid and flexible bodies.

[0026] At a given moment, the number of concentrated forces (torques) acting on a flexible body is finite; we can represent them as unknown vectors.

[0027] These forces are typically given in an absolute coordinate system, and their corresponding unknown vectors in the body coordinate system of the flexible body are denoted as...

[0028] Under the small deformation assumption, the deformation of a modal flexible body is caused by distributed forces and concentrated forces. The overall deformation caused by the former can be expressed using a low-order modal matrix; while the latter often requires consideration of local and concentrated forces. The relationship is linear and can be represented by a high-frequency compliance matrix. Therefore, the displacements of each node on the flexible body in volume coordinates can be written as...

[0029] in, It is a low-frequency mode matrix; and The deformation of high-frequency modal components caused by concentrated forces (moments) is described and can be calculated as follows. Assume... and To solve for the mass matrix and stiffness matrix of a flexible body undergoing linear vibration, For a higher-order mode matrix, such that Let be an invertible square matrix composed of all modal vectors; let the corresponding low-order frequencies be denoted as . The higher-order frequencies are .

[0030] The above only includes the displacement on the generalized node vector after the flexible body mesh is generated. However, the displacement and rotation vector at each material point on the flexible body need to be considered in the calculation.

[0031] By orthogonal relation It can be obtained The flexibility matrix of the flexible body describes the displacement of the nodal vectors caused by a unit action at each node.

[0032] therefore, This is the high-frequency compliance matrix of the flexible body; it can be proven by direct calculation.

[0033] visible It only includes high-frequency effects and can be pre-calculated from the stiffness matrix and low-frequency and mode characteristics. If compared with a given... The generalized force at the equivalent generalized node vector is denoted as Then it can be written as

[0034] This expression gives the high-frequency displacement caused by unit action at the generalized nodal vector, while The concentrated force (moment) vector in the matrix does not necessarily act directly on the generalized nodal vector; therefore, an interpolation process is needed for transition. A material point on a flexible body must belong to a certain element after mesh generation. Assume the coordinates (volume coordinates) of this material point in the reference configuration of the flexible body are... x The displacement at this point can be written using a shape function as follows:

[0035] in for Some components of this correspond to the generalized node vector on the corresponding unit, and can be expressed as: Therefore, the above formula can be written as

[0036] in ,and , ,

[0037] In the derivation of the equation, the high-frequency response is a quasi-static effect, therefore the concentrated effect The high-frequency components are calculated only for their quasi-static values; the distributed force acts on... The influence of mid-to-high frequency components is negligible compared to the high-frequency mode effect caused by concentrated action.

[0038] Step 3: Transform the flexible body contact collision problem into a flexible body cone complementary explicit dynamics problem: By introducing the theory of conical complementarity, the nonlinear constraints generated by flexible bodies during contact and collision are transformed into a mathematically easier-to-handle form of conical complementarity constraints. This transformation can accurately describe the mechanical properties of flexible bodies during contact and collision, including the magnitude and direction of the contact force and the deformation of the contact area. Simultaneously, combined with explicit dynamics methods, a time integration algorithm is used to solve the transformed conical complementarity constraint problem, thereby obtaining the dynamic response of the flexible body during contact and collision, providing a reliable basis for subsequent simulation analysis and optimization design.

[0039] Local deformation can be calculated from the compliance matrix and contact forces, and local deformation can prevent the embedding of contacting bodies. Therefore, the Lagrangian method can be used to calculate complex contact problems. The flexible contact problem is transformed into a linear conical complementarity problem for efficient solution, thus extending the calculation method of multi-rigid-body contact forces to small-deformation flexible bodies. The linear conical complementarity problem can be described as follows:

[0040] in, It is an unknown quantity; and For constant matrices and vectors; For a certain cone in mathematics; and For their complementary cones, satisfying

[0041] This problem arises when matrix N is positive definite, for any d Its solution is unique. If the contact collision problem can be transformed into an LCP problem and the matrix N is ensured to be symmetric and positive definite, high-frequency oscillations caused by the penalty function can be avoided. This invention will model the flexible contact problem based on this approach.

[0042] Assume the first i Each contact calculation point consists of two markers. and Composition, its normal vector Its gap function

[0043] Meets non-invasive conditions The unit normal vector at the point of contact. The two unit tangent vectors are denoted as follows: and These constitute the right-hand frame matrix of the contact system.

[0044] Contact-friction force vector Taking the three-dimensional components under this frame, the contact-friction force in the absolute coordinate system is: The normal contact force at the point of contact is Then the gap function and the normal contact force satisfy the complementarity condition.

[0045] Assuming the contact-friction force satisfies the Coulomb contact model, the contact force In the i Within each contact cone ,Right now

[0046] On the other hand, the i Each contact pair corresponds to two material points. and Its corresponding absolute coordinates and Generalized coordinates of multibody systems can be used. This means that the velocity and infinitesimal displacement of two points can be expressed as:

[0047] Therefore, the virtual work done by contact-friction force is

[0048] Contact frame coordinate system Below, two matter points and The relative velocity is

[0049] The frictional force near the contact point satisfies the coefficient of friction as follows: According to Coulomb's law, the tangential frictional force satisfies the principle of maximum dissipation, meaning that the power dissipated by friction reaches its maximum value.

[0050] This problem is a typical convex optimization problem, and its solution is unique, which can be solved using the Lagrange multiplier method.

[0051] The Karush-Kuhn-Tucker (KKT) conditions for this convex optimization problem can be written out.

[0052] Known To transform the contact-friction problem into an LCP problem, a set of equations needs to be constructed from the complementarity relation and the above equation. Contact point geometric vector

[0053] in,

[0054] The generalized coordinates predicted when all contact forces are ignored, and Geometric corrections due to contact forces. All contact forces. The virtual work done at all points of contact can be written as

[0055] Assuming all hinges are replaced by a flexible body contact problem, the system dynamics equations can be written as follows:

[0056] Among them, M and D follow q Slow change, and f Follow q and v Slow change. Let the step size be... h It adopts the simplest explicit discrete scheme.

[0057] here The step size should be small enough to solve explicitly and accurately. However, it is still much larger than the high-frequency vibration period caused by local deformation.

[0058] Step 4: High-performance explicit simulation of non-ideal hinges based on heterogeneous parallelism of GPU and CPU: By leveraging a heterogeneous parallel computing architecture combining GPUs and CPUs, this approach accelerates high-performance explicit simulations of non-ideal hinges, fully utilizing the powerful parallel computing capabilities of GPUs and the serial processing advantages of CPUs. Computational tasks are allocated strategically: parts suitable for parallel processing, such as large-scale matrix operations and iterative calculations, are handled by the GPU, while serial tasks like logic control and task scheduling are handled by the CPU. This heterogeneous parallel computing approach significantly improves simulation efficiency and reduces computation time while maintaining accuracy and stability. This results in more accurate and efficient analysis of the dynamic characteristics of non-ideal hinges, providing strong support for the design and optimization of hinges in complex mechanical systems.

[0059] like Figure 2 As shown, the movable robotic arm consists of three parts: a base, a connecting rod, and an actuator. The base measures 0.4m × 0.4m × 0.2m, the connecting rod is 0.8m long with a radius of 0.05m, and the actuator measures 0.4m × 0.1m × 0.1m. All three components are made of the same material with a density of 7.8 × 10⁻⁶ m³ / s. 3 kg / m 3The Young's modulus is 200 GPa, and Poisson's ratio is 0.29. The base and guide rail are connected by a sliding hinge, the connecting rod and base by a ball joint, and the actuator and connecting rod by a rotary hinge. Considering the hinges to be non-ideal, the blue contact area has a radius of 50 mm, the red contact area has a radius of 60 mm, and the green contact area has a radius of 30 mm. There is a 15 μm gap between each contact area.

[0060] In the absolute coordinate system, a 600tN external force is applied to the base along the x-axis, increasing linearly with time. A 300tN external force is applied to the connecting rod along the x-axis, and a 20N·m external torque is applied along the y-axis. The actuator is subjected to a 5N·m external torque along the x-axis. Additionally, to assist in the ball joint engagement, an extra 100N external force is applied along the z-axis in the connecting rod's coordinate system. Under these loads, the system drives the actuator to the designated position to complete the task.

[0061] Each component is modeled using a novel modal flexible body method. The overall system behaves as an ODE system when contact modeling is employed, and numerical simulations are performed using explicit contact dynamics based on a CCP complementary scheme. First, the contact simulation results under low friction conditions are compared with those of the ideal constraint (DAE) model. Furthermore, the differences in dynamic response based on the contact model under different friction coefficient μ values ​​are examined. Figure 3-5 This corresponds to the comparison between the results when μ = 0.001 and the ideal constraints. Figure 6-11 This corresponds to a comparison of the results for four different μ values.

[0062] The above provides a detailed description of the explicit simulation method for the novel modal flexible cone complementary contact dynamics of non-ideal hinges proposed in this invention, and elucidates the principles and implementation methods of this invention. The above description of the embodiments is only for the purpose of helping to understand the method and core ideas of this invention. At the same time, for those skilled in the art, there will be changes in the specific implementation methods and application scope based on the ideas of this invention. Therefore, the content of this specification should not be construed as a limitation of this invention.

Claims

1. An explicit simulation method for the complementary contact dynamics of a novel modal flexible cone with a non-ideal hinge, characterized in that: The method specifically includes the following steps: Step 1: Transform the constraints of non-ideal hinges in actual engineering into contact collision equations with gaps that have real physical meaning through modeling and analysis methods. Step 2: Perform dynamic modeling of the flexible component using the novel modal flexible body method based on the combined force approach. Step 3: Introduce the theory of conical complementarity to transform the nonlinear constraints generated during actual contact and collision into a rigid-flexible coupled conical complementary constraint form: Step 4: Utilize the heterogeneous parallel computing architecture of GPU and CPU to perform high-performance explicit simulation of non-ideal hinges.

2. The method according to claim 1, characterized in that: Step 1 includes: Step 1.1: Analyze the non-ideal hinge to determine its key parameters and structural characteristics, including material properties, hinge clearance, coefficient of friction, possible contact area and contact force frequency; Step 1.2: Based on the manufacturing tolerances, wear, or assembly errors in actual mechanical systems, establish complex non-smooth dynamic behaviors of contact-collision-separation, and transform the traditional constraint equations into real contact and collision equations with gaps between multiple rigid or flexible bodies.

3. The method according to claim 2, characterized in that: In step 2, Under the assumption of small deformation, the deformation of flexible bodies is divided into overall deformation caused by distributed forces and local deformation caused by concentrated forces; the overall deformation is expressed by a low-order mode matrix, and the local deformation is expressed by a high-frequency compliance matrix. The flexible body is divided into several elements and the generalized node vector of each element is determined. The relationship between the material point on the flexible body and the generalized node vector of its element is established through shape functions, and the displacement of the material point is calculated.

4. The method according to claim 3, characterized in that: In step 2, The high-frequency compliance matrix is ​​calculated in advance based on the stiffness matrix, low-frequency frequency and modes of the flexible body, and only reflects the deformation effects related to high frequencies. For high-frequency components caused by concentrated forces (moments), the calculation is based on quasi-static values. The influence of distributed forces on mid-to-high-frequency components is ignored relative to the high-frequency mode effects caused by concentrated forces.

5. The method according to claim 4, characterized in that: In step 3, Local deformation was calculated using the high-frequency compliance matrix and contact-friction force. The contact-friction force adopts the Coulomb contact model and must satisfy the non-intrusion condition that the contacting bodies do not embed into each other, as well as the maximum dissipation principle that optimizes the frictional force dissipation power. Each contact point consists of two markers. First, a right-handed frame matrix containing a unit normal vector and two unit tangent vectors is constructed. The contact-friction force vector is set as the three-dimensional component under this frame. By calculating the virtual work done by the contact force and the relative velocity of the contact point, and then combining the Karush-Kuhn-Tucker condition, the contact-friction force is solved.

6. The method according to claim 5, characterized in that: In step 4, The GPU is used to process tasks with high parallelism, while the CPU is used to process serial tasks.

7. An explicit simulation system for the complementary contact dynamics of a novel modal flexible cone with a non-ideal hinge, characterized in that: The explicit simulation system is implemented based on the explicit simulation method of complementary contact dynamics of novel modal flexible cones with non-ideal hinges as described in any one of claims 1 to 6. The system includes a movable robotic arm, a base, a connecting rod, and an actuator. The base is connected to the guide rail via a sliding hinge, the connecting rod is connected to the base via a ball joint, and the actuator is connected to the connecting rod via a rotary hinge. The ball joint and rotary hinge are non-ideal hinges with contact areas of different radii and a preset gap between each contact area. In the absolute coordinate system, the base is subjected to an external force that increases linearly with time along the x-axis, the connecting rod is subjected to an external force along the x-axis and an external torque along the y-axis, and the actuator is subjected to an external torque along the x-axis; in addition, to assist the ball joint in contacting, an additional external force is applied along the z-axis in the connecting rod body coordinate system. Under the aforementioned load, the system drives the actuator to move to the designated position to complete the contact.

8. The system according to claim 7, characterized in that: Each component adopts a new modal flexible body modeling method. When the overall system is modeled using contact, it behaves as an ODE system. Numerical simulation is performed using explicit contact dynamics based on the CCP complementary scheme. The simulation results of contact under low friction conditions are compared with the simulation results of the ideal constraint (DAE) model; then the differences between the contact simulation results and the ideal constraint are examined under different values ​​of friction coefficient μ.

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