Thermo-mechanical coupling multiscale dynamic solution method for flexible multi-body system with clearance hinge

Abstract

A method for solving the thermal-mechanical coupling multi-scale dynamics of flexible multi-body systems with clearance hinges, the technical field of space deployment mechanism dynamics modeling and numerical simulation. The existing solving method is difficult to accurately capture the change of the dynamics response of the space deployment mechanism under the thermal environment. According to the constructed temperature related hinge gap model, the temperature related material parameter model, the normal contact collision model, the partial slip tangential slip friction model, the tangential total friction force, the adhesion zone displacement and slip velocity model, the thermal viscous damping and tangential friction force model, the static friction growth model and the friction heat feedback transient heat conduction model, the thermal-mechanical coupling mechanical dynamics equation is constructed, which is decomposed into slow field equation and fast field equation, and the slow field equation and the fast field equation are solved through BDF2 and Newton-Raphson method and double parameter adaptive multi-scale implicit integration method, and the motion response of the hinge is obtained. The present application is used for obtaining the motion response of the hinge.

Description

Technical Field

[0001] This invention relates to the field of dynamic modeling and numerical simulation technology of space deployment mechanisms, specifically a thermo-mechanical coupled multi-scale dynamic solution method for a temperature-dependent gap hinge flexible multibody system. Background Technology

[0002] With the continuous development of space science exploration, satellite communication, Earth observation, and deep space missions, large-scale space deployment mechanisms are increasingly widely used in spacecraft. Flexible deployment structures such as ring truss antennas, space solar arrays, deployable support trusses, and space robotic arms typically feature large scale, light mass, low damping, and multiple hinge connections. Their on-orbit deployment accuracy and operational stability directly affect spacecraft mission performance. In the on-orbit environment, space structures are subjected to alternating effects of solar irradiation, Earth shadow blocking, and the cold background of deep space, causing temperature to fluctuate periodically over a wide range. This leads to repeated thermal expansion and contraction of flexible components, pins, and bushings. Due to the different thermal expansion characteristics of various materials, the radial clearance of the hinges dynamically changes with temperature, inducing complex dynamic phenomena such as contact collisions, frictional stick-slip, high-frequency flexible vibrations, and thermal conduction coupling. Therefore, accurately establishing a thermo-mechanical coupled dynamic model of a temperature-dependent clearance hinge flexible multibody system is of great significance for predicting the deployment process, analyzing stability, and assessing the structural reliability of space deployment mechanisms.

[0003] Current research on the dynamics of space deployment mechanisms typically relies on multibody system dynamics, the finite element method, or the absolute nodal coordinate method to establish flexible structural models, combined with contact collision models and friction models to describe the nonlinear dynamic behavior of hinges with gaps. The absolute nodal coordinate method is particularly effective at describing the large deformation motion of flexible beams and rods, making it suitable for modeling large flexible deployment structures. Hertz contact models, hysteresis-damped models, and Coulomb friction models are commonly used to describe normal collisions and tangential friction caused by hinge gaps. However, most existing methods treat hinge gaps as constant or only consider mechanical contact dynamics while neglecting the effects of temperature on material parameters, contact stiffness, frictional characteristics, and gap dimensions. Furthermore, traditional friction models often struggle to accurately characterize phenomena such as partial slip, stick-slip transition, static friction increasing with adhesion time, and thermal-viscosity dissipation during the slip phase, leading to insufficient accuracy in predicting contact friction responses under on-orbit thermal cycling conditions.

[0004] On the other hand, the thermo-mechanical coupling process in space deployment mechanisms exhibits significant bidirectional feedback characteristics: the temperature field alters structural material parameters, thermal deformation, and hinge gaps, thereby affecting contact, collision, and friction behavior; conversely, hinge contact, friction, and collision processes generate frictional heat and heat dissipation, which are fed back to the local temperature field. Existing thermo-mechanical coupling analysis methods sometimes employ separate solution strategies for the thermal and structural fields, which easily leads to information lag between fields and coupling errors; while other studies have established thermo-structural coupling equations, they have not integrated temperature-dependent gap evolution, frictional heat feedback, and non-smooth contact switching processes into a unified closed-loop dynamics framework. Therefore, in flexible multibody systems with temperature-dependent gap hinges, a unified thermo-mechanical coupling modeling method that can simultaneously describe material parameter changes, gap thermal drift, partial sliding friction, frictional heat feedback, and non-smooth switching behavior is still lacking.

[0005] In numerical solutions, the Newmark-β method, HHT-α method, and generalized α method are commonly used implicit integration methods in the dynamics of flexible multibody systems. The Newmark-β method is simple in form, but suffers from insufficient stability under conditions of strong contact collisions and high-frequency nonlinear vibrations. The HHT-α method can introduce numerical dissipation to suppress high-frequency oscillations, but its stability control and dissipation control are usually coupled to a single parameter, making it difficult to simultaneously achieve low-frequency motion accuracy and high-frequency vibration suppression. While the generalized α method can improve high-frequency dissipation performance, it is prone to error accumulation and decreased convergence efficiency in long-term on-orbit thermal cycling simulations, strongly nonlinear contact separation switching, and temperature-dependent time-varying stiffness problems. Furthermore, temperature evolution in space deployment mechanisms is typically a slow process, while contact collisions and stick-slip vibrations are fast processes, exhibiting a significant timescale difference. Using a uniform small-step integration would result in excessively high computational costs; using excessively large step sizes would make it difficult to capture transient impacts and non-smooth switching events.

[0006] Therefore, existing solution methods struggle to accurately capture the dynamic response changes of space deployment mechanisms under thermal conditions. A novel thermo-mechanical coupled dynamic modeling and multi-scale highly stable numerical solution method for temperature-dependent gap hinge flexible multibody systems is urgently needed. This method should be able to describe temperature-dependent material parameters, temperature-dependent hinge gaps, normal contact collisions, partial sliding friction, thermally activated static friction, thermoviscous damping, and frictional thermal feedback within a unified framework. Simultaneously, it should be able to perform multi-timescale decomposition for slow thermal diffusion processes and fast contact-viscosity processes, and achieve accurate maintenance of low-frequency deployment motion, effective dissipation of high-frequency contact vibrations, and stable handling of contact-separation non-smooth events through an implicit integral scheme with adaptive dissipation and accuracy coordination capabilities. Summary of the Invention

[0007] The purpose of this invention is to address the problem that existing solution methods are unable to accurately capture the changes in the dynamic response of space deployment mechanisms under thermal conditions, and to propose a thermo-mechanical coupled multi-scale dynamic solution method for flexible multibody systems with gap hinges.

[0008] A method for solving the thermo-mechanical coupled multi-scale dynamics of a flexible multibody system with gapped hinges, the method comprising the following:

[0009] Step 1, Initial State Assessment: Construct a temperature-dependent hinge gap model, calculate the radial gap, and determine whether the hinge is in a separated or contacted state based on the gap. If it is in a separated state, there is no contact force, proceed to Step 3; if it is in a contacted state, proceed to Step 2.

[0010] Step 2, State Determination: Use the Filippov type non-smooth criterion function to determine whether the contact state is a collision, slip-adhesion, or edge-grabbing motion type;

[0011] Calculate the corresponding force using the appropriate force model based on the type of motion;

[0012] Step 3: Model building: Based on the forces obtained in Step 2 and the temperature-related material parameter model, establish the thermo-mechanical coupled mechanical dynamic equations;

[0013] Step 4: Decompose the thermo-mechanical coupled mechanical dynamics equations into slow-field equations and fast-field equations according to the time scale. Solve the slow-field equations and fast-field equations respectively using the BDF2 method, the Newton-Raphson method and the two-parameter adaptive multi-scale implicit integration method. Superimpose the obtained motion response components to obtain the total motion response of the hinge.

[0014] The beneficial effects of this invention are:

[0015] This invention aims to address the challenges in modeling and solving flexible multibody systems with gap hinges and temperature-dependent gaps in thermal environments. These challenges include difficulties in characterizing temperature-dependent gaps, significant nonlinearities in contact collisions and frictional stick-slip, complex thermo-mechanical bidirectional coupling relationships, difficulty in stably capturing non-smooth state transitions, and low efficiency of multi-timescale numerical integration. These issues lead to the inaccurate capture of the dynamic response changes of space deployment mechanisms under thermal conditions. The invention proposes a multi-scale dynamic solution method for thermo-mechanical coupling of temperature-dependent gap hinge flexible multibody systems. First, a unified thermo-mechanical model of the flexible structure is established based on the absolute nodal coordinate method, incorporating temperature-dependent material parameters and a gap hinge model. Then, a bidirectional coupled dynamic equation for the contact-friction-temperature field is constructed by combining temperature-dependent normal contact, partial slip friction, thermally activated static friction, thermoviscous damping, and frictional thermal feedback. Based on this, Filippov's non-smoothness theory and the G-function are used to determine states such as contact, separation, slip, and edge rubbing. Finally, through fast and slow timescale decomposition and an adaptive multi-scale implicit integration method, high-precision and high-stability predictions of the thermo-mechanical coupling response, contact impact, frictional stick-slip, and stability boundaries of the space deployment mechanism are achieved.

[0016] This invention constructs a temperature-dependent gap hinge model to describe the dynamic changes in gap caused by the difference in thermal expansion between the pin and the bushing, avoiding the problem that constant gap models cannot reflect on-orbit thermal drift. The established thermo-mechanical bidirectional coupling model unifies the description of temperature field, structural deformation, contact collision, friction stick-slip, and frictional heat generation, improving the realism of dynamic response prediction under complex thermal environments. The established partial slip thermal friction model distinguishes between the adhesion and slip regions and considers thermally activated static friction and thermal viscous damping, improving the prediction accuracy of stick-slip transition and frictional dissipation. The established Filippov non-smooth switching discrimination can uniformly judge states such as separation, contact, slip, and edge rubbing, reducing numerical oscillations at contact boundaries and improving simulation stability. The established dual-timescale implicit integration strategy solves the slowly varying thermal field and the rapidly varying contact vibration separately, ensuring the accuracy of low-frequency unfolding motion while suppressing high-frequency vibrations, thus improving computational efficiency.

[0017] By establishing temperature-dependent material parameter models and temperature-dependent gap hinge models, the problem of difficulty in uniformly representing changes in material properties, thermal drift of hinge gaps, and changes in contact stiffness of space deployment mechanisms under on-orbit thermal cycling conditions is solved. This enables the dynamic model of flexible multibody systems with gap hinges to realistically reflect the influence of temperature on structural deformation, contact collisions, and friction behavior, avoiding the problem of inaccurate prediction of on-orbit deployment response by traditional ambient temperature and ambient gap models.

[0018] Furthermore, to address the complex phenomena such as collisions, partial slippage, stick-slip transitions, and frictional heat feedback during the contact process of gap hinges, this method unifies and couples the normal contact collision model, the partial slippage tangential friction model, the thermally activated static friction model, the thermoviscous damping model, and the frictional heat generation model, achieving a synergistic description of contact force, frictional force, frictional dissipation, and local temperature rise. Through this mechanism, the accuracy of predicting the dynamic response of space deployment mechanisms under strongly nonlinear contact conditions can be effectively improved, and a more reliable analytical basis can be provided for hinge gap design, friction pair material selection, and thermal control design.

[0019] This invention introduces Filippov's nonsmooth dynamics theory and the G-function discrimination mechanism to uniformly judge states such as separation, contact, collision, adhesion, slippage, and edge rubbing. This reduces numerical oscillations during contact-separation and adhesion-slippage switching processes, improving the stability and reliability of nonsmooth dynamics simulations. Simultaneously, addressing the significant timescale difference between thermal diffusion and contact / collision processes, a slow-field averaging-fast-field perturbation decomposition and a two-parameter adaptive multi-scale implicit integration strategy are employed. This effectively suppresses high-frequency contact vibrations while maintaining the accuracy of low-frequency deployment motion, reducing the computational cost of long-term on-orbit thermal cycling simulations. This provides crucial support for the dynamic prediction, stability assessment, and on-orbit reliability analysis of large space deployment mechanisms. It improves the stability, accuracy, and computational efficiency of dynamic simulations of space deployment mechanisms under on-orbit thermal cycling conditions. Attached Figure Description

[0020] Figure 1 A framework for kinematic and thermo-structural modeling;

[0021] Figure 2 Flowchart for contact event sub-step correction and thermal friction state determination;

[0022] Figure 3 A flowchart of the thermo-mechanical coupling multi-scale implicit integral;

[0023] Figure 4 Flowchart for iterative solution of the slow-field average equation and fast-field feedback;

[0024] Figure 5 This is a flowchart of bifurcation tracing based on arc length extension. Detailed Implementation

[0025] 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.

[0026] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other. The present invention will be further described below with reference to the accompanying drawings and specific embodiments, but this is not intended to limit the scope of the invention.

[0027] Example:

[0028] The thermo-mechanical coupled mechanical dynamic equations constructed in this embodiment can accurately determine the hinge motion state in real time. The solution is divided into slow field and fast field. The slow field is used to describe thermal diffusion and quasi-static structural response, while the fast field is used to describe contact collision, stick-slip vibration and transient thermal disturbance. The decomposition solution is fast, the results are more stable and the accuracy is higher.

[0029] Combination Figure 1 This embodiment describes a thermo-mechanical coupled multi-scale dynamics solution method for a flexible multibody system with a gap hinge. The method includes the following:

[0030] Step 1, Initial State Assessment: Construct a temperature-dependent hinge gap model, calculate the radial gap, and determine whether the hinge is in a separated or contacted state based on the gap. If it is in a separated state, there is no contact force, proceed to Step 3; if it is in a contacted state, proceed to Step 2.

[0031] Step 2, State Determination: Use the Filippov type non-smooth criterion function to determine whether the contact state is a collision, slip-adhesion, or edge-grabbing motion type;

[0032] Calculate the corresponding force using the appropriate force model based on the type of motion;

[0033] Step 3: Model building: Based on the forces obtained in Step 2 and the temperature-related material parameter model, establish the thermo-mechanical coupled mechanical dynamic equations;

[0034] Step 4: Decompose the thermo-mechanical coupled mechanical dynamics equations into slow-field equations and fast-field equations according to the time scale. Solve the slow-field equations and fast-field equations respectively using the BDF2 method, the Newton-Raphson method and the two-parameter adaptive multi-scale implicit integration method. Superimpose the obtained motion response components to obtain the total motion response of the hinge.

[0035] Specifically, this embodiment takes the on-orbit deployment process of a large space ring truss antenna as an example, and details the specific implementation steps of this embodiment:

[0036] Step 1 also includes constructing the absolute node coordinates of the hinged flexible multibody system, which are based on the unified modeling of flexible structures and temperature fields in ANCF:

[0037] Figure 1This figure illustrates the overall construction process from geometry to a thermo-mechanical coupled dynamic model, providing a framework for kinematic and thermal configuration modeling. First, a kinematic description of the flexible component is established based on the Absolute Nodal Coordinates (ANCF) method, interpolating the position and temperature fields using the same shape functions. Then, temperature-dependent material parameters (such as elastic modulus and coefficient of thermal expansion) and a temperature-dependent gap hinge model are introduced to calculate the instantaneous gap as a function of temperature. Building upon this, a contact-friction model incorporating normal contact collision, partial sliding friction, thermally activated static friction, and thermoviscous damping is constructed, and frictional heat feedback and transient heat conduction equations are established, ultimately forming a closed-loop thermo-mechanical bidirectional coupled dynamic equation.

[0038] like Figure 1 As shown, this embodiment first establishes a kinematic and thermal configuration modeling framework. The hinged flexible multibody system is discretized into multiple elements, each with multiple preset nodes. The absolute node coordinate method is used to describe the position field of any material point in the flexible deployment structure, and the temperature field is interpolated using the same shape function.

[0039]

[0040]

[0041] In the formula, This represents the position vector of the material point in the global coordinate system. For local material coordinates of the element; For time; The shape function matrix of the ANCF beam element; This is the absolute node coordinate vector of the element, containing the node position and gradient degrees of freedom; The material point temperature; It is the shape function for interpolating the temperature field; Let be the element node temperature vector. By letting... This ensures that the structural field and the temperature field are coupled under the same discrete grid.

[0042] A flexible unfolding structure model is established by using the absolute node coordinate method, which facilitates the description of all models within a unified framework.

[0043] II. Temperature-dependent material parameter model in step 3:

[0044] The thermophysical and mechanical parameters of structural and hinge materials are expressed as temperature functions:

[0045]

[0046] In the formula, Density; It is the elastic modulus; Poisson's ratio; The coefficient of thermal expansion; Thermal conductivity; Specific heat capacity; The current temperature; For reference temperature; These are density, elastic modulus, Poisson's ratio, coefficient of thermal expansion, thermal conductivity, and specific heat capacity at the reference temperature, respectively. This corresponds to the temperature sensitivity coefficient. This step is used to update the system's mass matrix, stiffness matrix, damping matrix, and thermal load in real time.

[0047] III. Temperature-dependent hinge gap model and preliminary judgment of contact state:

[0048] (I) Temperature-dependent hinge gap model constructed in step 1:

[0049] The radial clearance model is constructed based on reference values ​​for the inner radius or hole radius of the bushing, reference values ​​for the outer radius of the pin, the temperature-dependent thermal expansion coefficient of the bushing material, the temperature-dependent thermal expansion coefficient of the pin material, the current temperature, the reference temperature, and correction coefficients that take into account nonlinear geometric thermal deformation under extreme temperatures.

[0050] Specifically, considering the difference in thermal expansion between the pin and bushing materials, a temperature-dependent radial clearance model is established:

[0051]

[0052] In the formula, The radial clearance at the current temperature; The nominal gap is at the reference temperature; This is a reference value for the inner radius of the bushing or the radius of the hole; This is a reference value for the outer radius of the pin. The temperature-dependent thermal expansion coefficient of the bushing material; The temperature-dependent thermal expansion coefficient of the pin material; The current temperature; For reference temperature; This is a correction factor to account for nonlinear geometric thermal deformation at extreme temperatures.

[0053] (II) Gap Function and State Determination:

[0054] In step 1, the hinge is determined to be in a disengaged or engaged state based on this gap:

[0055] Establish the clearance function based on the relative position of the pin and the bushing:

[0056] ,

[0057] In the formula, It is the gap function; These are system spatial location variables or generalized coordinate-related variables; For time; This is the relative position vector between the contact surfaces of the pin and the bushing; It is the relative radial distance; Temperature at the current moment The corresponding radial clearance;

[0058] when When the hinge is in the disengaged state; when At this time, the hinge enters a contact or collision state.

[0059] Specifically, the gap function can be used to determine whether the hinge is in a separated or contact state. For the contact state, it can be further divided into through-collision, sliding adhesion, or edge-grabbing motion types.

[0060] IV. Filippov Non-Smooth Switching System and Motion Type Judgment:

[0061] Construct a Filippov-type non-smooth switching function:

[0062] To describe non-smooth states such as separation, contact, collision, adhesion, and slippage, the mechanical and temperature states are unified into an extended state vector:

[0063] ,

[0064] Establish the switching manifold for the temperature drift corresponding to the j-th gap hinge:

[0065] ,

[0066] in, Represents the extended state vector of the system; Represents the coordinate vector of an ANCF node; Represents the node velocity vector; Represents the node temperature vector; Indicates the transpose of a vector or matrix; express 3D real state space; This indicates the dimension of the extended state vector; Indicates the first The switching manifold corresponding to each gap hinge; Indicates the clearance hinge number; Indicates the first The switching function for each hinge; Indicates the first The hinge at the current temperature The radial clearance below; Represented by node coordinates The decision Radial penetration of each hinge; This indicates the current temperature.

[0067] To further distinguish between contact-related collision, slip-adhesion, or edge-grabbing motion types:

[0068] The motion type of the system at the gap boundary is determined based on the switching manifold normal vector and the G function, and a Filippov slip vector field is constructed in the slip state:

[0069]

[0070]

[0071]

[0072]

[0073]

[0074] in, Indicates the first The unit normal vector of the switching manifold; Indicates the switching function For the extended state vector The gradient; This represents the Euclidean norm of the gradient; Indicates the first The zeroth-order G function corresponding to each motion region; Indicates the separated region; Indicates the contact area; Indicates the first The system vector field of each region; Indicates the first [condition] caused by temperature change. The speed of movement at the boundary of the hinge gap; Represents the dot product of vectors; Indicates the first The first-order G function of each region; This represents the total derivative of the system's vector field with respect to time. This represents the total derivative of the unit normal vector of the switching manifold with respect to time; Represents the Filippov slip vector field; Represents the vector field of the separated region; Represents the vector field of the contact region; The weights of the convex combination of the slip vector field satisfy the following conditions: .

[0075] The judgment rule is: when At that time, the system experienced a collision; when When the system enters a slip or stick-slip state; when Furthermore, when higher-order criteria are met, the system undergoes edge-grabbing motion.

[0076] V. Calculation of contact force and friction force:

[0077] (a) Normal contact collision model (crossing collision or edge-grabbing state):

[0078] Combination Figure 1 Furthermore, under three different motion types—collision, slippage and adhesion, and edge friction—the corresponding forces and parameters are determined separately:

[0079] If the collision or edge-grabbing situation occurs, the normal contact force is calculated based on the constructed normal contact collision model.

[0080] If the object is in a slip-adhesion state, the tangential stress is calculated based on the constructed partial slip tangential friction model. The tangential friction force, tangential displacement, and slip velocity are calculated based on the constructed total tangential friction force, adhesion zone displacement and slip velocity model, and thermoviscous damping and tangential friction force model.

[0081] Among them, the normal contact collision model:

[0082] When the gap function satisfies The Hertz contact pressure distribution is used to describe the normal pressure in the contact area, and the improved Hertz model with hysteresis damping is further used to calculate the normal contact collision force.

[0083]

[0084]

[0085]

[0086] in, Indicates the radius position within the contact area Normal contact pressure at the point; Represents the radial coordinates within the contact area; Indicates time; This indicates the maximum normal pressure at the contact center; Indicates the instantaneous contact radius; Indicates the normal contact force; Represents pi; Indicates temperature-dependent normal contact stiffness; Indicates the current temperature of the contact area; Indicates the normal penetration depth, and ; Represents the gap function; Indicates the normal penetration velocity; Indicates the initial collision velocity or reference penetration velocity; Indicates the collision recovery coefficient; Represents the Heaviside step function, when hour ,when hour .

[0087] (II) Partial slip tangential friction model (slip-adhesion state):

[0088] Among them, the partial slip tangential slip friction model:

[0089] To address the difficulty of distinguishing between complete adhesion, partial slip, and complete slip in traditional friction models, the Cattaneo–Mindlin–Jäger partial slip theory is introduced to establish a tangential contact stress model.

[0090]

[0091] in, Indicates the contact surface at Tangential stress in the direction; Indicates the axial-circumferential tangential direction; Represents the radial coordinates within the contact area; Represents the circumferential angular coordinates within the contact area; Indicates time; Indicates the temperature-dependent coefficient of friction; Indicates the contact radius as Normal pressure distribution at time; Indicates the instantaneous contact radius; Indicates the radius of the adhesion region as Normal pressure distribution at time; Denotes the radius of the adhesion region in the partial slip model, satisfying ; Represents the Heaviside step function; This represents the gap function.

[0092] (III) Model of total tangential friction, adhesive zone displacement and slip velocity:

[0093] After obtaining the tangential stress distribution, the total tangential friction force is obtained by integrating the contact area, and the tangential displacement and local slip velocity in the adhesive region are calculated.

[0094]

[0095]

[0096]

[0097]

[0098] in, This represents the total tangential frictional force within the contact area; Indicates the temperature-dependent coefficient of friction; Indicates the contact radius as The corresponding normal contact force at that time; Indicates the radius of the adhesion region as The corresponding normal contact force at that time; Indicates the instantaneous contact radius; Indicates the radius of the adhesion region; Indicates the axial length of the contact area; Indicates local tangential contact stress; Represents radial coordinates; Represents the circumferential angular coordinates; Represents a radial integral infinitesimal element; Represents a circumferential integral infinitesimal element; This represents the equivalent tangential displacement in the adhesion region; Indicates the temperature-dependent equivalent elastic modulus; Indicates the temperature-dependent equivalent shear modulus; Indicates the contact radius as The normal approach quantity at that time; Indicates the radius of the adhesion region as The normal approach quantity at that time; This indicates the local sliding velocity at the point of contact; This indicates the relative angular velocity between the pin and the bushing; This represents the tangential displacement velocity in the adhesive region.

[0099] (iv) Thermoviscous damping and tangential friction model:

[0100] During the slip phase, an Arrhenius-type thermoviscous damping term is introduced, so that the tangential friction force simultaneously includes both the Coulomb friction term and the thermoviscous damping term. The first term is the Coulomb friction term, and the second term is the thermoviscous damping term. The combination of the two is the tangential friction force.

[0101]

[0102]

[0103]

[0104] in, This represents the temperature-dependent thermoviscosity coefficient. Indicates the reference viscosity coefficient; Indicates thermal activation energy; Represents the universal gas constant; This indicates absolute temperature, usually measured in Kelvin (K). This represents tangential frictional force; Indicates the macroscopic tangential relative slip velocity; Indicates the normal contact force; Indicates the temperature-dependent coefficient of kinetic friction; The sign function is used to indicate that the direction of frictional force is opposite to the direction of sliding velocity. Indicates the instantaneous contact area; This represents the static friction coefficient under the combined influence of temperature and adhesion time. This represents the absolute value of the tangential frictional force.

[0105] In this model, the second formula is used to calculate the friction force in the slip state, and the third formula is used to determine the static friction constraint in the adhesive state.

[0106] VI. Frictional heat feedback and temperature field update:

[0107] If the object is in a slip-adhesion or cross-collision state, the frictional heat generation power is calculated by integrating the slip velocity and tangential stress over the contact area. Combined with the transient heat conduction model based on frictional heat feedback, a new temperature is obtained. Using this new temperature, initial state assessment, fine state assessment, and model establishment are performed sequentially to update the thermo-mechanical coupled dynamic model. The frictional heat generation power is:

[0108] ,

[0109] ,

[0110] ,

[0111] according to This further generates heat, affecting the temperature of each model. Therefore, during slip adhesion, As a heat source Substitute this into the constructed transient heat conduction model of frictional heat feedback:

[0112] ,

[0113] in, Assign a matrix to the heat source. Represents the temperature-dependent heat capacity matrix; Represents the vector of the rate of change of node temperature; Represents the temperature-dependent heat conduction matrix; Represents the node temperature vector; This represents the vector of nodal heat sources formed by frictional slip and collision dissipation; Represents the orbital radiative heat input vector; Indicates the power of frictional heat generation; Indicates the axial length of the contact area; Indicates circumferential angle The radius of the adhesion zone boundary at the location; Indicates circumferential angle The radius of the contact boundary at that location; Indicates local tangential contact stress; Indicates local slip velocity; Represents radial coordinates; Represents the circumferential angular coordinates; Represents a radial integral infinitesimal element; Represents a circumferential integral infinitesimal element; This represents the frictional heat power allocated to the first contacting body; This represents the frictional heat power allocated to the second contacting body; This represents the thermal permeability of the first contacting body; This indicates the thermal permeability of the second contact body; This represents the total frictional heat generation power.

[0114] Obtain the new temperature and update the temperature-dependent models, such as the temperature-dependent hinge gap model, the temperature-dependent material parameter model, and the constructed thermally activated static friction growth model, with the corresponding model parameters updated. Simultaneously, incorporate the new temperature into the constructed thermally activated static friction growth model to obtain the new adhesion time. Temperature is static friction coefficient at time This is used to replace the static friction coefficient in the thermoviscous damping and tangential friction force model, thereby regenerating the corresponding force and updating the thermo-mechanical coupled dynamic model. The thermo-activated static friction growth model is then implemented.

[0115]

[0116]

[0117]

[0118]

[0119] in, Represents the temperature-dependent heat capacity matrix; Represents the vector of the rate of change of node temperature; Represents the temperature-dependent heat conduction matrix; Represents the node temperature vector; This represents the vector of nodal heat sources formed by frictional slip and collision dissipation; Represents the orbital radiative heat input vector; Indicates the power of frictional heat generation; Indicates the axial length of the contact area; Indicates circumferential angle The radius of the adhesion zone boundary at the location; Indicates circumferential angle The radius of the contact boundary at that location; Indicates local tangential contact stress; Indicates local slip velocity; Represents radial coordinates; Represents the circumferential angular coordinates; Represents a radial integral infinitesimal element; Represents a circumferential integral infinitesimal element; This represents the frictional heat power allocated to the first contacting body; This represents the frictional heat power allocated to the second contacting body; This represents the thermal permeability of the first contacting body; This indicates the thermal permeability of the second contact body; This represents the total frictional heat generation power.

[0120] VII. Equations of thermo-mechanical coupling:

[0121] By integrating temperature-dependent material parameters, thermal loads, external loads, and contact friction forces of the clearance hinges obtained from steps one through seven into the ANCF dynamic equations, we obtain the thermo-mechanical coupled mechanical dynamic equations:

[0122] ,

[0123] ,

[0124] in, Represents the temperature-dependent mass matrix; Represents the nodal acceleration vector; Represents the temperature-dependent structural damping matrix; Represents the node velocity vector; Represents the temperature-dependent elastic stiffness matrix; Represents the absolute node coordinate vector in ANCF; Represented by the node temperature vector The resulting thermal load vector; This indicates external deployment force, track disturbance force, or other external loads; Indicates the first Generalized force of the clearance hinge in each motion region; Indicates the separated region; Indicates the contact area; This represents the gap force in the separated state, and is set to zero. This represents the gap force under contact conditions; This represents the summation of all clearance hinges in the system; The Jacobian matrix representing the normal contact force to the generalized coordinates; The Jacobian matrix representing the tangential frictional force to the generalized coordinates; and These represent the transposes of the corresponding Jacobian matrices; Indicates the normal contact force; This represents tangential frictional force.

[0125] VIII. Decomposition of Fast and Slow Time Scales:

[0126] like Figure 3 As shown, the outer layer is a slow-timescale cycle, and the inner layer is a fast-timescale cycle. Within each slow time step, the temperature field and material parameters are first updated, then the fast-timescale integrator is invoked to perform multiple sub-step integrations of the fast-variable equations, and the average value of the contact friction force is fed back to the slow-variable equations. For example... Figure 4 As shown, the fast field solver integrates over multiple rapidly changing cycles, statistically calculates the average values ​​of the contact force and friction force, as well as the average value of the equivalent heat source, and returns them as correction terms for the slow field equations until the entire coupled system converges. Figure 4 The focus is on solving the slow-field equations and their interaction with the fast-field equations. The flowchart begins with the assembly of the slow-field averaged equations, including the slowly varying stiffness matrix, thermal loads, and external loads. This is followed by an iterative loop. In each iteration, the current slowly varying state is used as a basis to invoke the fast-field solver. The fast-field solver integrates over multiple fast-varying cycles, calculating the average values ​​of contact forces and friction forces, as well as the average value of the equivalent heat source. These average values ​​are returned as correction terms for the slow-field equations. The slow-field equations are then updated with these terms to determine the slowly varying displacement, velocity, and temperature fields, and the residuals are checked for convergence. If convergence is not achieved, the iteration continues; if convergence is achieved, the process moves to the next slow time step. Figure 3 As shown, considering the slow thermal diffusion process and the fast contact collision and stick-slip vibration, the system state is decomposed into a slow-varying principal component and a fast-varying disturbance component. The slow field is used to describe thermal diffusion and quasi-static structural response, while the fast field is used to describe contact collision, stick-slip vibration and transient thermal disturbance.

[0127] The classification of slow and fast fields is based on the characteristic timescales of the physical processes: thermal diffusion, thermal expansion, and overall unfolding responses change slowly and are classified as slow fields; contact collisions, adhesion-slip switching, and high-frequency contact vibrations change rapidly and are classified as fast fields. The timescale of the slow field is approximately 10^2 s, while the timescale of the fast field is 10^-2 s, determined by the speed of the response. The equations for the slow and fast fields derived from the decomposition are as follows:

[0128] (Formula 1)

[0129] (Formula 2)

[0130] (Formula 3)

[0131] (Formula 4)

[0132] , (Formula 5)

[0133] in, This represents the temperature-dependent elastic stiffness matrix under slow time scales. Represents the principal components of slowly varying nodal coordinates; Represents the nonlinear stiffness matrix that is related to geometric nonlinearity and temperature; This represents the average value of the external load over a fast time scale; This represents the average value of the gap force under contact conditions on a fast time scale. Represents the principal component of the velocity at slowly varying nodes; This represents the thermal load caused by the slowly varying node temperature; This represents the heat capacity matrix at a slow time scale; This represents the derivative of the slowly varying nodal temperature with respect to a slow time scale. This represents the heat conduction matrix at a slow time scale; This represents the average value of the contact friction heat source over a fast time scale; This represents the orbital radiative heat input on a slow timescale; Represent any fast function The periodic average value; Indicates the characteristic period of fast vibration; A function that represents how a fast-time variable changes; Represents a fast-time integral infinitesimal; This represents the mass matrix under slow temperature conditions. This represents the second derivative of the rapidly changing nodal coordinates with respect to the fast time scale; This represents the first derivative of the rapidly changing nodal coordinates with respect to a fast time scale. The rapidly changing disturbance force represents the relative average value of the instantaneous gap contact force; Represents the rapidly changing disturbance component of the thermal load; This represents the temperature perturbation component at rapidly changing nodes; This represents the derivative of the rapidly changing nodal temperature with respect to a fast-time variable; This represents the rapidly changing disturbance component of the frictional heat source.

[0134] IX. Solving Implicit Integrals at Multiple Scales:

[0135] The fast field perturbation equations are solved using the two-parameter adaptive dissipation-accuracy coordinated multi-scale implicit integration method (ADPC-MSIIA), and Equation 4 is rearranged into a standard second-order dynamic form:

[0136] (Formula 6),

[0137] Performing ADPC-MSIIA time discretization on Equation 6 yields:

[0138] (Formula 7)

[0139] (Formula 8)

[0140] , (Formula 9)

[0141] , (Formula 10)

[0142] In the formula, This represents the mass matrix at a fast time scale; Represents the rapidly changing nodal acceleration vector; Represents the damping matrix at fast time scales; Represents the velocity vector of the rapidly changing node; Represents the stiffness matrix at fast time scales; This represents the rapidly changing nodal coordinate perturbation vector; This represents the resultant force of external load, contact force, friction force, and thermal load disturbance on a fast timescale. Represents fast-time variables; Indicates the weighting parameters for the quality term; This represents the weighted parameters for the force, damping, and stiffness terms; This indicates the low-frequency stability and accuracy control parameters; This represents the high-frequency numerical dissipation control parameters; Indicates the first The node coordinate perturbation vector at a fast time step; Indicates the first The node coordinate perturbation vector at a fast time step; Indicates the first A node velocity perturbation vector for a fast time step; Indicates the first A node velocity perturbation vector for a fast time step; Indicates the first The node acceleration perturbation vector at a fast time step; Indicates the first The node acceleration perturbation vector at a fast time step; Indicates the first The nodal accelerator at a fast time step, i.e., the third derivative of displacement with respect to time; Indicates a fast time step; This represents the integral parameter in the displacement update format; This refers to the integral parameter in the speed update format; This represents the third-order precision compensation coefficient.

[0143] Among them, after obtaining the nodal coordinates, velocities, and accelerations of the fast and slow fields respectively, the total nodal coordinate vector, total nodal velocity vector, and total nodal temperature vector obtained by superimposing the two fields are as follows:

[0144] (Formula 11)

[0145] (Formula 12)

[0146] (Formula 13)

[0147] In the formula,

[0148] , , Represents variables with slow time scales; Represents actual physical time; Represents variables on a fast time scale; The small parameter representing the separation of fast and slow time scales typically satisfies ; Represents the coordinate vector of the total node; Represents the principal components of nodal coordinates at a slow time scale; This represents the node coordinate perturbation component at a fast time scale; Represents the total node velocity vector; This represents the principal component of the node velocity at a slow time scale. This represents the nodal velocity perturbation component at a fast time scale; Represents the total node temperature vector; This represents the principal component of nodal temperature at a slow time scale. This represents the nodal temperature perturbation component at a fast timescale. Indicates a small quantity of second order or higher; Represents the first total derivative with respect to real time; This represents the second total derivative with respect to real time; and Let represent the partial derivatives with respect to the slow-time variable and the fast-time variable, respectively.

[0149] 10. Contact switching substep correction and bifurcation tracking:

[0150] Figure 5This paper describes a bifurcation tracking procedure for system stability analysis. First, based on the previous state and a given arc length step size, the Euler prediction solution for the next state is calculated. Then, the prediction solution is corrected using Newton's iteration method to ensure it simultaneously satisfies the dynamic equations and arc length constraints. During the correction process, changes in the function sign are monitored to determine if a limit point or bifurcation point has been reached. When a bifurcation is detected, the tracking branch is automatically switched or more precise step size control is applied. Therefore, Figure 5 The arc-length extension method maintains the direction and step size of the search along the equilibrium path by calculating the state differences and control parameter differences between adjacent steps, and identifies the bifurcation point when the differences change abruptly.

[0151] (a) Contact switching sub-step correction:

[0152] like Figure 2 As shown, the gap function is monitored in real time during the fast field integration process. When a sign change in the gap function is detected within the current time step, the switching time is determined using a binary search method:

[0153] ,

[0154] The original time step is subdivided into two sub-steps, and the state determination and integration calculation are re-performed in each sub-step until convergence is achieved before proceeding to the next time step.

[0155] (ii) Arc-length extension bifurcation tracing:

[0156] like Figure 5 As shown, for system stability and bifurcation analysis, arc length extension constraints are further introduced to track limit points and bifurcation points. The arc length constraint equations are as follows:

[0157] ,

[0158] in, Indicates the first Fast time substep size after sub-sub ... Indicates the original fast time step; Indicates the number of sub-steps; Indicates the first The next binary subdivision multiple; Indicates the first A slow-changing node coordinate vector at a slow time step; Indicates the first A slow-changing node coordinate vector at a slow time step; The Euclidean norm represents the state increment between two adjacent slow time steps; Indicates the first The extension parameters for each extension step; Indicates the first The extension parameters for each extension step; This represents the given arc length step size; It represents the square of the arc length step.

[0159] Among them, the extension parameters Options include average track temperature and initial clearance. External load amplitude or other key parameters that affect system stability.

[0160] Through the above steps, this embodiment realizes the thermo-mechanical coupled multi-scale dynamic solution of the temperature-dependent gap hinge flexible multibody system of the space deployment mechanism under on-orbit thermal environment. It can effectively predict the deployment process, contact impact, friction stick-slip and stability boundary, and provide technical support for the design and analysis of space deployment mechanisms.

[0161] Combination Figure 3 and Figure 4 Furthermore, as illustrated by the above formulas, the process of solving the slow-field and fast-field equations using the two-parameter adaptive multi-scale implicit integration method is as follows:

[0162] 1. Initialize the structural state, temperature field, and temperature-dependent material parameters, temperature-dependent gaps, external loads, etc., and substitute them into the slow-field equations in Equations 1 and 2. Solve the slow-field equations using the Newton-Raphson and BDF2-Newton iterative methods to obtain the principal components of the nodal coordinates in the slow timescale. , representing the principal component of nodal velocity at a slow time scale Principal components of nodal temperature on slow timescales ;

[0163] 2. The motion response components are obtained step by step using the ADPC-MSIIA multiscale implicit integral scheme of formula 7-10. , , ;

[0164] 3. Using formulas 11 to 13, the components of the slow field and the fast field are superimposed to obtain the total node coordinates, total node velocity, and total node temperature.

[0165] In each integral substep of the fast field, the gap function is calculated in real time to determine the contact or separation state. When the function changes, the contact switching substep correction is initiated (the time step is refined by the bisection method to accurately locate the switching moment).

[0166] While the invention has been described herein with reference to specific embodiments, it should be understood that these embodiments are merely examples of the principles and applications of the invention. Therefore, it should be understood that many modifications can be made to the exemplary embodiments, and other arrangements can be designed without departing from the spirit and scope of the invention as defined by the appended claims. It should be understood that different dependent claims and features described herein can be combined in ways different from those described in the original claims. It is also understood that features described in conjunction with individual embodiments can be used in other described embodiments.

Claims

1. A method for solving the thermo-mechanical coupled multi-scale dynamics of a flexible multibody system with gapped hinges, characterized in that, The method includes the following: Step 1, Initial State Assessment: Construct a temperature-dependent hinge gap model, calculate the radial gap, and determine whether the hinge is in a separated or contacted state based on the gap. If it is in a separated state, there is no contact force, proceed to Step 3; if it is in a contacted state, proceed to Step 2. Step 2, State Determination: Use the Filippov type non-smooth criterion function to determine whether the contact state is a collision, slip-adhesion, or edge-grabbing motion type; Calculate the corresponding force using the appropriate force model based on the type of motion; Step 3: Model building: Based on the forces obtained in Step 2 and the temperature-related material parameter model, establish the thermo-mechanical coupled mechanical dynamic equations; Step 4: Decompose the thermo-mechanical coupled mechanical dynamics equations into slow-field equations and fast-field equations according to the time scale. Solve the slow-field equations and fast-field equations respectively using the BDF2 method, the Newton-Raphson method and the two-parameter adaptive multi-scale implicit integration method. Superimpose the obtained motion response components to obtain the total motion response of the hinge.

2. The method for solving the thermo-mechanical coupled multi-scale dynamics of a flexible multibody system with a gapped hinge according to claim 1, characterized in that, The temperature-dependent hinge clearance model is constructed based on reference values ​​for the inner radius or hole radius of the bushing, reference values ​​for the outer radius of the pin, the temperature-dependent thermal expansion coefficient of the bushing material, the temperature-dependent thermal expansion coefficient of the pin material, the current temperature, the reference temperature, and correction coefficients that take into account nonlinear geometric thermal deformation under extreme temperatures.

3. The method for solving the thermo-mechanical coupled multi-scale dynamics of a flexible multibody system with a gapped hinge according to claim 1, characterized in that, The hinge is either in a disengaged or engaged state based on this gap, specifically: Construct the gap function: ， In the formula, It is the gap function; These are system spatial location variables or generalized coordinate-related variables; For time; This is the relative position vector between the contact surfaces of the pin and the bushing; It is the relative radial distance; Temperature at the current moment The corresponding radial clearance; when When the hinge is in the disengaged state; when At this time, the hinge enters a contact or collision state.

4. The method for solving the thermo-mechanical coupled multi-scale dynamics of a flexible multibody system with a gapped hinge according to claim 1, characterized in that, Based on the type of motion, the corresponding force is calculated using the appropriate force model, specifically: If the collision or edge-grabbing situation occurs, the normal contact force is calculated based on the constructed normal contact collision model. If the object is in a slip-adhesion state, the tangential stress is calculated based on the constructed partial slip tangential friction model. The tangential friction force, tangential displacement, and slip velocity are calculated based on the constructed total tangential friction force, adhesion zone displacement and slip velocity model, and thermoviscous damping and tangential friction force model.

5. The thermo-mechanical coupled multi-scale dynamic solution method for a flexible multibody system with a gapped hinge according to claim 4, characterized in that, The partial tangential slip friction model is as follows: ， In the formula, Indicates the contact surface at Tangential stress in the direction; Indicates the axial-circumferential tangential direction; Represents the radial coordinates within the contact area; Represents the circumferential angular coordinates within the contact area; Indicates time; Indicates the temperature-dependent coefficient of friction; Indicates the contact radius as Normal pressure distribution at time; Indicates the instantaneous contact radius; Indicates the radius of the adhesion region as Normal pressure distribution at time; Denotes the radius of the adhesion region in the partial slip model, satisfying ; Represents the Heaviside step function; This represents the gap function.

6. The method for solving the thermo-mechanical coupled multi-scale dynamics of a flexible multibody system with a gapped hinge according to claim 1 or 5, characterized in that, The method further includes: If the object is in a slip-adhesion or cross-collision state, the frictional heat generation power is calculated by integrating the slip velocity and tangential stress in the contact area. Combined with the frictional heat feedback transient heat conduction model, a new temperature is obtained. The new temperature is then used to perform initial state judgment, fine state judgment, and model establishment in sequence, and the thermo-mechanical coupling dynamic model is updated.

7. The method for solving the thermo-mechanical coupled multi-scale dynamics of a flexible multibody system with a gapped hinge according to claim 6, characterized in that, The method further includes: If the object is in a slip-adhesion state, the static friction coefficient is calculated using the constructed thermally activated static friction growth model. This coefficient is then used as the static friction coefficient in the thermoviscous damping and tangential friction model to update the static friction coefficient in the thermoviscous damping and tangential friction model, thereby updating the thermo-mechanical coupling dynamic model.

8. The method for solving the thermo-mechanical coupled multi-scale dynamics of a flexible multibody system with a gapped hinge according to claim 4, characterized in that, The normal contact collision model is constructed based on temperature-dependent normal contact stiffness, normal penetration velocity, collision recovery coefficient, radial coordinates within the contact area, and instantaneous contact radius.

9. The method for solving the thermo-mechanical coupled multi-scale dynamics of a flexible multibody system with a gapped hinge according to claim 4, characterized in that, The total tangential friction force is constructed based on the tangential contact stress and the tangential stress.

10. The method for solving the thermo-mechanical coupled multi-scale dynamics of a flexible multibody system with a gapped hinge according to claim 4, characterized in that, The thermoviscous damping and tangential friction model is constructed based on the temperature-dependent thermoviscous coefficient, normal contact force, and temperature-dependent kinetic friction coefficient.

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