Digital twin system for monitoring thermal interaction operation state of bearings of high-speed precision machine tool

Abstract

In order to deal with the challenge of the heat-force interaction of the bearing of the high-speed precision machine tool, the invention discloses a digital twin system for monitoring the running state of the heat-force interaction of the bearing of the high-speed precision machine tool, and a bearing heat-force interaction state monitoring model is embedded in a virtual model. In the operation process of a main shaft bearing system in the material world, a data acquisition system is used for acquiring real-time data, a heat-induced preload is calculated through a bearing heating model, and a state-space equation set is used for describing a heat-force interaction behavior of a bearing part; in consideration of the error between the material world and the virtual world, a state observer is designed to accurately predict the dynamic change of the pre-tightening force of the bearing, and the state observer is integrated into a digital twin system, so that the monitoring of the heat-force interaction state of the bearing of the high-speed precision machine tool is realized.

Description

Technical Field

[0001] The present invention belongs to the technical field of digital twins, and specifically relates to a digital twin system for monitoring the operating status of thermal interaction of high-speed precision machine tool bearings. Background Art

[0002] High-speed spindles are notorious for sudden failures, as thermal bearing failures become increasingly detrimental to the spindle under high-speed conditions. Specifically, the increase in bearing temperature stems from the change in contact load caused by the increased heat generation at high speeds. The interaction between thermal and contact loads reduces the viscosity of the lubricant, resulting in lubrication failure, which leads to a continuous increase in bearing temperature and preload. If a thermal equilibrium state is not achieved, the bearing is susceptible to thermal failure. To prevent sudden failure of high-speed bearings, it is essential to continuously monitor bearing temperature and preload during operation.

[0003] The complex thermomechanical interaction between bearing components, stress states and heat generation leads to complex thermomechanical behavior. Heat generation causes thermal displacement of bearing components, which reduces the problem of bearing heat generation to a thermomechanical interaction problem. Analyzing this interaction lays the foundation for monitoring bearing temperature and preload, however, in the prior art, heat transfer between bearing components, TCR and the viscosity-temperature effect of the lubricant are often ignored.

[0004] Frictional heat is generated within the internal friction interface of the bearing and is transferred to the bearing components and the lubricating medium. Under the influence of thermal loads, the bearing components undergo thermal deformation, and the inconsistent deformation occurring in the contact area leads to changes in the contact pressure. The thermal displacement of the outer ring / bearing housing reduces the contact pressure, while the thermal displacement of the inner ring and rolling elements increases the contact pressure. The overall change in contact pressure is determined by these two factors. Since the contact pressure plays a positive feedback role in the bearing system, the increase in contact pressure will intensify the frictional heat. The additional heat generated will in turn cause new thermal deformation in the bearing components. This cycle continues, resulting in a continuous increase in the temperature of the inner ring, outer ring and rolling elements. When the temperature reaches the temperature resistance limit of the material and the bearing system still does not meet the thermal stability standards, it may lead to bearing thermal failure.

[0005] In the prior art, the monitoring of the operating status of bearings has been extensively studied. Although some significant results have been achieved, the thermal preload of the bearings has not been incorporated as feedback into the bearing heating process. Therefore, the model does not conform to the true thermo-mechanical interaction model. Although digital twin technology can be used for bearing operating status monitoring, the specific method of monitoring the operating status of bearing thermo-mechanical interaction through digital twin systems has not been explored. Summary of the invention

[0006] In view of this, the purpose of the present invention is to provide a digital twin system for monitoring the operating state of thermal interaction of bearings of high-speed precision machine tools. By introducing contact loads between bearing components, the monitoring of the operating state of thermal interaction of bearings is achieved.

[0007] In order to achieve the above object, the present invention provides the following technical solutions:

[0008] A digital twin system for monitoring the operation status of thermal-mechanical interaction of a bearing, comprising a physical world, a virtual world, a twin database and a twin server;

[0009] The material world includes a physical entity, a control system, a cooling system and a data acquisition system of a spindle bearing system; the spindle bearing system includes a spindle and bearings located at both ends of the spindle; the data acquisition system includes a device for collecting real-time data during the operation of the spindle bearing system;

[0010] The virtual world includes a virtual model of the spindle bearing system and a bearing thermal interaction state monitoring model embedded in the virtual model, wherein the bearing thermal interaction state monitoring model includes a bearing heating model for describing the heating of bearing components and a state space equation group for describing the thermal interaction behavior of bearing components;

[0011] The virtual model is used to capture and describe the operating state and behavior of the spindle bearing system, which is expressed as:

[0012] VE=(G v ,P v ,B v ,R v )

[0013] Among them: Geometric model G v It is a three-dimensional representation, including the physical geometry, component dimensions and assembly relationships of the spindle bearing system; the thermal interaction model P v Used to describe multi-dimensional physical properties and practical constraints; behavioral model B v It is used to illustrate the real-time response at various time and space scales, and to describe the time-varying behavior of key boundary conditions in thermal modeling, taking into account external environmental factors and internal mechanisms; knowledge-based model R v Capturing the operating rules, experience and continuous learning of virtual and actual spindle bearing systems, so that the digital twin system can make judgments, assessments and predictions based on evolving knowledge and experience;

[0014] The twin database stores the twin data DD of the material world and the virtual world data:

[0015] DD=(D p ,Dv ,D s ,D k ,D f )

[0016] Where: D p Represents data related to the physical entity of the spindle bearing system, including basic information and operation data of the spindle bearing system, wherein the basic information of the spindle bearing system includes the model, specification, function and performance of the spindle bearing system, and the operation data of the spindle bearing system includes operation status data and environmental status variables; D v Relevant data representing the virtual model of the spindle bearing system, including G v Related data, physical model P v Related data, behavior model B v Related data and rule model B v Related data; D s Includes data related to bearing preload monitoring services and bearing operating condition monitoring services; D k Including data related to functional services; D f It represents the field knowledge related to the modeling principle of thermal interaction, integrates the real-time sensor data, virtual spindle bearing system simulation data, historical data and rule data, and is the most important factor in processing D p , D v , D s and D k The fusion of data provides accurate and comprehensive information for digital twins;

[0017] A state observer is built in the twin server, and the state observer is used for steady-state simulation and prediction of dynamic changes of bearing preload;

[0018] In the material world, the data acquisition system includes a device for collecting real-time data during the operation of the physical entity of the spindle bearing system, and transmitting the real-time data to the virtual model, the twin database and the twin server;

[0019] In the virtual world, the bearing heating model obtains the heating output of the bearing components according to the real-time data; the state space equation group obtains the thermal interaction state between the bearing components according to the real-time data;

[0020] The state observer performs a steady-state simulation on the spindle bearing system according to the twin data DD and predicts the predicted bearing preload.

[0021] Further, the bearing heating model is expressed as:

[0022] H total =1.047×10 ―4 ×[n·(Mv +M gy )+n s ·M s +n·f 1 ·(F β +0.9F pt ·cotα)·d m ]

[0023] Where: M v 、M s and M gy represent the viscous friction torque, rotational friction torque and gyroscopic torque respectively; n and n s Respectively represent the bearing speed and rotation speed; n and n s Respectively represent bearing speed and rotation speed; F β represents the equivalent external load; f 1 Indicates load factor; F pt represents the thermally induced preload; α represents the contact angle; d m Indicates the bearing pitch diameter.

[0024] Further, it is assumed that the thermal deformation of the inner ring and the outer ring is respectively 3 and ε 2 It means that the center of the rolling element is from C b Move to C bn , and the center of curvature of the inner circle is from C i Transition to C in ; The center of curvature of the outer ring is from C o Move to C on ; The contact angle is from α 0 changes to α; the sphere radius changes from r b Change to r b ′ , the change of the interference fit in the contact direction is given by Indicates that:

[0025] Total interference fit ε t is the rolling element thermal deformation ε 1 and initial interference fit ε 10 The superposition of is expressed as:

[0026]

[0027] Where: 3 Indicates the deformation of the inner ring; ε 2 Indicates that the outer ring is deformed;

[0028] The thermal deformation of the inner ring and rolling element is a linear function of temperature. The deformation of the rolling element is expressed as:

[0029] ε 1 =α d ·rb ·T b

[0030] Where: α d 、r b and T b They represent the thermal expansion coefficient, ball radius and rolling element temperature respectively;

[0031] The inner ring deformation is expressed as:

[0032] ε 3 =α d ·r i ·T i

[0033] Where: r i and T i represent the inner circle radius and temperature respectively;

[0034] Outer ring / bearing housing deformation ε 2 Related to the thickness of the bearing housing, expressed as:

[0035]

[0036] Where: T(r) represents the outer ring / bearing housing temperature; v represents the Poisson’s ratio; T o and T h Respectively represent the outer ring and bearing housing temperature; r o and r h Respectively represent the outer ring and bearing housing diameters;

[0037] By interference fit t The contact pressure change caused by the change is determined as:

[0038]

[0039] Where: k t represents the normal contact stiffness; β represents the exponent;

[0040] The change in thermally induced preload is expressed as:

[0041] F pt =F t ·sinα·N

[0042] Where: N represents the number of steel balls.

[0043] Further, the state space equation is expressed as:

[0044]

[0045] Where: C oh represents heat capacity; T oh represents the temperature distribution function; T orepresents the temperature of the control volume Ω; T b Indicates rolling element temperature; T i Indicates the inner ring temperature; R ib Indicates the TCR of the inner ring / ball interface; R bo Indicates the TCR of the outer ring / ball interface; R is Indicates the TCR of the inner ring / shaft interface; R oh Indicates the TCR of the outer ring / bearing housing interface; h ∞ is the convection coefficient between the bearing housing and the air; h c is the convection coefficient between the rolling element and the lubricant; r h Indicates the shell diameter; r o Indicates the outer circle radius; h c represents the natural convection coefficient; h ∞ Represents the convection coefficient; A o Indicates the surface area of ​​the outer ring; A h Indicates the surface area of ​​the bearing box; A b is the surface area of ​​the sphere; A i is the surface area of ​​the inner ring of the bearing; η o Indicates the heat absorption coefficient of the outer ring; η o Indicates the heat absorption coefficient of the ball; η i Indicates the heat absorption coefficient of the inner ring; Q f represents the total heat generation; n represents the bearing speed; F β Represents the equivalent external load; F pt represents the change of thermal induced preload; N represents the number of steel balls; b represents steel balls; ρ is the material density; m b Indicates the mass of the steel ball; c b Represents the specific heat capacity of the steel ball; m i Indicates the quality of the inner ring; c i Indicates the specific heat capacity of the inner ring.

[0046] Furthermore, the state observer adopts a stateless observer;

[0047] Speed ​​n and temperature T of the bearing outer ring o Respectively represented as u 1 and u 2 , and as the system input; the state variables are selected as x 1 、x 2 and x 3 Indicates T oh (r,t),T b and T i ; Heat-induced preload F pt is the output, and the state variable vector is X = [x 1 ,x 2 ,x 3 ] indicates that the initial preload function is Fp0 , then: f(X,u)=[f 1 (x,u),f 2 (x,u),f 3 (x,u)], the state space model structure is:

[0048]

[0049] Where: u represents the system input vector, z represents the system output, z=F pt =p(x,F p0 );

[0050] The state space model is:

[0051]

[0052] Among them: A, B and C represent coefficient matrices;

[0053] The coefficient matrix A is of 3×3 dimensions, the matrix B is of 3×1 dimensions, and the matrix C also has 3×1 dimensions; these coefficient matrices in the state-space model are expressed as:

[0054]

[0055] Furthermore, the state observer adopts a full-dimensional observer;

[0056] Taking into account the uncertainty of heat generation calculation, Δq(u,z) is incorporated into the bearing thermal model:

[0057]

[0058] z=p(x,F p0 )

[0059] Where: Δq(u,z) represents the bearing heat value adjusted for temperature error compensation;

[0060] Introduce the difference between observed and experimental values:

[0061]

[0062] The observer dynamic error is:

[0063]

[0064] Among them: δq(e) represents a higher-order infinitesimal; Defined as:

[0065]

[0066] Where: x = [x 1,x 2 ,…,x n ]; q(x) = [q 1 (x),q 2 (x),…,q n (x)].

[0067] Furthermore, the state observer adopts a proportional integral observer;

[0068] The bearing heating process and the additional heat-induced preload generation process are both linear processes. Considering the nonlinear factors such as lubrication, bearing component sliding and wear during the heating process, an integral term is introduced to improve the steady-state tracking accuracy of the observer:

[0069]

[0070] Where: D 1 represents the coefficient matrix; K p and K I represents the matrix of the corresponding dimension, and K p and K I denote the proportional gain and integral gain of the proportional observer and the integral observer respectively;

[0071] get:

[0072]

[0073] in: f∈R m , represents the introduced integral action;

[0074] The state estimation error is defined as but:

[0075]

[0076] in:

[0077] The beneficial effects of the present invention are:

[0078] In order to meet the challenges of thermal-mechanical interaction of bearings, the present invention proposes a digital twin system for monitoring the operating status of thermal interaction of bearings of high-speed precision machine tools. By embedding a bearing thermal interaction state monitoring model in a virtual model, during the operation of the spindle bearing system in the material world, real-time data is collected by a data acquisition system, the thermally induced preload is calculated by the bearing heating model, and the thermal interaction behavior of bearing components is described by a set of state-space equations. Taking into account the error between the material world and the virtual world, a state observer is designed to accurately predict the dynamic change of the bearing preload, and the state observer is integrated into the digital twin system, thereby realizing the monitoring of the thermal interaction state of the bearing. BRIEF DESCRIPTION OF THE DRAWINGS

[0079] In order to make the purpose, technical solution and beneficial effects of the present invention clearer, the present invention provides the following drawings for illustration:

[0080] Figure 1 The principle of estimating bearing preload using digital twin system is presented;

[0081] Figure 2 It is a framework diagram of an embodiment of a digital twin system for monitoring the thermal interaction operating state of a high-speed precision machine tool bearing according to the present invention;

[0082] Figure 3 For the experimental device; Figure 4 It is composed of twin data; Figure 5 It is the interface of the digital twin system; Figure 6 It is the thermal-mechanical interaction process of the bearing;

[0083] Figure 7 Comparative analysis of heat generation with and without viscosity-temperature effect; Figure 8 is the viscous friction torque; Fig. 9 is the outer ring temperature; Fig.10 is the total heat generated by the bearing;

[0084] Fig.11 is the position of the center of curvature of the rolling element and the groove; Fig.12 It is the heat transfer and heat dissipation mechanism; Fig.13 It is a two-dimensional temperature node system;

[0085] Fig.14 is the bearing thermal model; Fig.15 The speed and outer ring temperature rise; Fig.16 Temperature rise of bearing components; Fig.17 Dynamic changes in bearing preload;

[0086] Fig.18 The effect of rotation speed on preload; Fig.19 is the preload and total interference fit ε t ; Fig. 20 The temperature and deformation of the bearing components;

[0087] Fig.21 are the thermal deformation and component state values; Fig. 22 is the bearing preload; Fig.23 Calculate temperatures for bearings; Fig.24 Temperature rise of bearing components; Fig.25 for preload and bearing assembly deformation; Fig.26 The temperature rise of the bearing components with PI observer; Fig. 27 Preload for PI observer;

[0088] Fig.28 is the heating nonlinear term; Fig.29 is the test temperature;

[0089] Fig.30 Observed Toh and preload at 1000r / min; Fig.31 Observed Toh and preload at 5000r / min.

[0090]

[0091] DETAILED DESCRIPTION

[0092] The present invention is further described below in conjunction with the accompanying drawings and specific embodiments so that those skilled in the art can better understand the present invention and implement it, but the embodiments are not intended to limit the present invention.

[0093] 1. Digital twin system for monitoring the operating status of bearing thermal interaction

[0094] The positive feedback effect of contact pressure is a potential cause of bearing thermal failure, so it is necessary to analyze its dynamic changes in engineering applications and monitor it in real time. However, it is challenging to directly measure contact pressure. In practical applications, monitoring the temperature of the outer ring of the bearing is often used to evaluate the stability of the bearing system. When the temperature exceeds its specified maximum value, the system stops operating. This embodiment aims to address these challenges. Considering that the preload of the bearing can directly reflect the change of contact pressure, this embodiment proposes for the first time to monitor the operating status of the thermal interaction of the bearing through a digital twin system, such as Figure 1 As shown. The dynamic change of the bearing preload caused by thermal effect is predicted by the test values ​​of spindle speed and bearing outer ring / bearing housing temperature. Real-time reflection of the operating status of the bearing through preload prediction helps to prevent failures caused by thermal effects in the bearing. By analyzing heat transfer and heat dissipation, a set of thermal balance equations for the bearing components is formulated.

[0095] Specifically, Figure 2As shown, the digital twin system for monitoring the thermal interaction operation state of high-speed precision machine tool bearings in this embodiment includes a material world, a virtual world, a twin database and a twin server. The material world includes the physical entity, control system, cooling system and data acquisition system of the spindle bearing system; the spindle bearing system includes a spindle and bearings located at both ends of the spindle; the data acquisition system includes a system for collecting real-time data during the operation of the spindle bearing system.

[0096] The virtual world includes a virtual model of the spindle bearing system and a bearing thermal interaction state monitoring model embedded in the virtual model. The bearing thermal interaction state monitoring model includes a bearing heating model for describing the heating of bearing components and a state space equation group for describing the thermal interaction behavior of bearing components.

[0097] The virtual model is used to capture and describe the operating state and behavior of the spindle bearing system, which is expressed as:

[0098] VE=(G v ,P v ,B v ,R v )

[0099] Among them: Geometric model G v It is a three-dimensional representation, including the physical geometry, component dimensions and assembly relationships of the spindle bearing system; the thermal interaction model P v Used to describe multi-dimensional physical properties and practical constraints; behavioral model B v It is used to illustrate the real-time response at various time and space scales, and to describe the time-varying behavior of key boundary conditions in thermal modeling, taking into account external environmental factors and internal mechanisms; knowledge-based model R v Used to capture the operating rules, experience and continuous learning of virtual and actual spindle bearing systems, enabling the digital twin system to make judgments, assessments and predictions based on evolving knowledge and experience.

[0100] The twin database stores the twin data DD of the material world and the virtual world data:

[0101] DD=(D p ,D v ,D s ,D k ,D f )

[0102] Where: D pRepresents data related to the physical entity of the spindle bearing system, including basic information and operation data of the spindle bearing system, wherein the basic information of the spindle bearing system includes the model, specification, function and performance of the spindle bearing system, and the operation data of the spindle bearing system includes operation status data and environmental status variables; D v Relevant data representing the virtual model of the spindle bearing system, including G v Related data, physical model P v Related data, behavior model B v Related data and rule model B v Related data; D s Includes data related to bearing preload monitoring services and bearing operating condition monitoring services; D k Including data related to functional services; D f It represents the field knowledge related to the modeling principle of thermal interaction, integrates the real-time sensor data, virtual spindle bearing system simulation data, historical data and rule data, and is the most important factor in processing D p , D v , D s and D k The fusion of data provides accurate and comprehensive information for digital twins.

[0103] A state observer is built in the twin server, and the state observer is used for steady-state simulation and prediction of dynamic changes in bearing preload.

[0104] In the material world, the data acquisition system includes a system for collecting real-time data during the operation of the physical entity of the spindle bearing system, and transmitting the real-time data to the virtual model, the twin database and the twin server. In the virtual world, the bearing heating model obtains the heating output of the bearing components according to the real-time data; the state space equation group obtains the thermal interaction state between the bearing components according to the real-time data. The state observer performs a steady-state simulation of the spindle bearing system according to the twin data DD and predicts the predicted bearing preload.

[0105] 1.1. Material World

[0106] The spindle bearing system is the physical entity that forms the basis of the digital twin model, e.g. Figure 3As shown. The entity includes the spindle bearing system itself, the control system, the cooling system and the data acquisition system. The data acquisition system includes temperature and displacement sensors, etc. The spindle bearing system includes the front bearing and the rear bearing. The front and rear bearings are B7008 and B7007. The lubrication method is oil mist lubrication. The compressed air flow rate is 10L / s and the pressure is 0.32Pa. The spindle bearing system runs at a rated speed of 24000r / min, and the rated power of the motor is 12.4kW. The cooling system has excellent performance, with a flow rate of 16L / min and an inlet temperature of 18℃. The room temperature is maintained at a constant 23℃.

[0107] The basic parameters of the B7008 front bearing are shown in Table 1. The basic rated static load value of the bearing is 7400N, and the inner diameter (D i ) is 40mm, the outer diameter (D o ) is 68mm, material density (ρ) is 7800kg / m3, elastic modulus (E) is 2.06×105MPa, bearing width (B) is 15mm, thermal conductivity (K) is 0.8W / (m℃), number of steel balls (N) is 18, thermal expansion coefficient (δ) is 11×10-6m / ℃. According to SKF maintenance and lubrication product specifications, the lubricant selected for the bearing is No. 32 turbine oil, the base oil viscosity at 40℃ is 32cSt, the oil viscosity at 100℃ (v 0 ) is 7.5 cSt, and the viscosity index (f v ) is 0.0242.

[0108] Table 1 Basic bearing parameters of B7008

[0109]

[0110] A spring-compressed Pt100 resistance temperature sensor with a spring preload structure is used to measure the temperature of the bearing. The end face of the temperature sensor is in close contact with the curved surface of the bearing outer ring, ensuring fast and accurate temperature measurement. The temperature sensor has an accuracy grade of A, a measurement error of (0.15±0.0002t)℃, a measurement range of -50-200℃, and a size of φ6×18.3mm. The output current of the Pt100 temperature sensor is collected using a Yokogawa MAX100 acquisition instrument with a maximum resolution of 0.1℃. For displacement measurement, a KD2300-5SUM Kaman eddy current sensor is used. The sensor has a range of 0-0.5mm, a probe diameter of 2mm, and a head length of 4.6mm. The displacement sensitivity is 1000mV / mm and the linearity is 1.0% FSO.

[0111] 1.2 Virtual World

[0112] The Virtual Entity (VE) of the Virtual Spindle Bearing System is a comprehensive, multi-dimensional, high-fidelity model designed to accurately reflect the physical spindle bearing system. This virtual model effectively captures and describes the operating status and behavior.

[0113] VE=(G v ,P v ,B v ,R v )

[0114] Geometry Model G v It is a three-dimensional representation, including the physical geometry, component dimensions and assembly relationships of the spindle bearing system; using common three-dimensional (3D) modeling tools such as Solidworks and CATIA, the geometric model G v The geometric visualization is realized in digital space. Thermomechanical interaction model P v Usually built using MATLAB or Python, describing multi-dimensional physical properties and practical constraints. Behavioral Model B v The real-time response at various time and space scales is illustrated, taking into account external environmental factors (such as CNC command changes and manual operations) and internal mechanisms (such as heat transfer patterns); simulations are performed using MATLAB or Python to describe the time-varying behavior of key boundary conditions in thermal modeling. Knowledge-based Models R v The operating rules, experience and continuous learning of the virtual and actual spindle bearing systems are captured, enabling the system to make judgments, assessments and predictions based on evolving knowledge and experience.

[0115] The twin data DD of the spindle bearing system acts as a driving force, influencing its function and response.

[0116] DD=( D p,D v ,D s ,D k ,D f )

[0117] Where: D p Represents data related to the actual spindle bearing system, including basic information such as the spindle bearing system model, specifications, functions and performance, as well as operational data such as operating status data and environmental status variables. v Indicates VE related data, including G v Related data, physical model P v Related data, behavior model B v Related data and rule model B v Related data. s Includes data related to bearing preload monitoring services and bearing operating condition monitoring services. kThis includes data related to functional services, such as model algorithms and update records, and data related to business services, such as thermal interaction modeling results. f Represents field knowledge related to the principles of thermomechanical interaction modeling, which is the processing of D p , D v , D s and D k Data fusion data. It integrates real-time sensor data, virtual spindle bearing system simulation data, historical data and rule data to provide accurate and comprehensive information for digital twins.

[0118] The spindle bearing system twin data of this embodiment integrates all the physical information and data of the spindle bearing system digital twin model as the driving force behind the spindle bearing digital twin operation. Due to the multi-source and complexity of these data, a mainstream database management system is used to establish a dual database for the spindle bearing system. This helps to organize and uniformly manage the data of each model in the digital twin structure, such as Figure 4 shown.

[0119] According to the requirements of bearing thermal interaction modeling, the digital twin system software is carefully designed, such as Figure 4 As shown. The Python program is used to calculate the internal heat generation, TCR, and convection coefficient. In addition, MATLAB is called to simulate the thermal interaction behavior, and the simulation results are displayed on the interface of the digital twin system software. The software interface integrates a variety of functions, such as model selection, spindle parameter configuration for preload and external load, environmental settings for adjusting ambient temperature, material property assignment for each component of the spindle bearing system, contact interface configuration for setting contact load and interference fit, clearance fit value, bearing heat generation model for calculating heat generation, bearing thermal model for heat transfer modeling, and state observer for operating status monitoring.

[0120] The heat flux of the front and rear bearings is 41070W / m 2 and 32010W / m 2 Forced convection coefficient #1 is the convection coefficient of the bearing assembly / compressed air, and its value is 343.55W / (m 2 ℃). Forced convection coefficient #2 is the convection coefficient of the shaft core end / air, and its value is 30.40W / (m 2 ℃). Forced convection coefficient #3 is the convection coefficient of the bearing box / air, and its value is 9.7W / (m 2 ℃), such as Figure 5 shown.

[0121] 1.3 Model Interaction and Mapping

[0122] Figure 5The framework of the digital twin system for bearing thermal interaction condition monitoring is presented. The physical spindle bearing system runs in real time and collects data such as bearing housing temperature, stator temperature, ambient temperature, and speed. Subsequently, these data are transmitted to the virtual spindle bearing system, the bearing thermal interaction condition monitoring service platform, and the dual database through the OLE Unified Architecture for Process Control (OPC-UA) protocol. The virtual system uses the data from the physical system to update the thermomechanical interaction condition monitoring model in real time. The behavioral model produces outputs of bearing heat flux, TCR, and convection coefficient based on real-time operating conditions, while the physical model updates the boundary conditions. This helps to perform bearing thermal interaction modeling and realize operational condition monitoring. The results of modeling and simulation, including the simulated temperature rise of bearing components, the dynamic change of bearing preload, and the nonlinear terms of bearing deformation and heating, are written to the dual database through the Microsoft Open Database Connectivity (ODBC) interface. Finally, these results are transmitted to the spindle bearing system service platform through the Transmission Control Protocol / Internet Protocol (TCP / IP) interface and displayed on the digital twin system software.

[0123] 2. Bearing thermal interaction state monitoring model

[0124] Figure 6 The thermomechanical interaction process under given load and speed conditions is described. Frictional heat is generated and subsequently transferred from the contact interface to the bearing components, and the thermal load causes significant thermal deformations in the bearing components and the shaft core. These uncoordinated deformations lead to additional preloads, which in turn affect the generation of frictional heat. It is worth noting that the process of bearing heating and uncoordinated deformation is nonlinear, while the process of heat transfer and thermal deformation is linear.

[0125] 2.1 Bearing heating model

[0126] 2.1.1. Initial heating of bearings

[0127] The heating of the bearing is caused by friction and the resulting relative sliding at the ball / groove interface. According to reference

[25] , the total heat generated in the bearing, Q f for:

[0128] Q f =Q l +Q v +Q s +Q gy =1.047×10 ―4 ×[n·(M v +M l +M gy )+n s ·M s ]

[0129] Where: Q l, Q v , Q s and Q gy are the heat generated by external load, lubricant viscous friction, rotational motion and gyroscopic motion respectively; M v 、M l 、M s and M gy They represent viscous friction torque, load friction torque, rotational friction torque and gyroscopic torque respectively; n and n s Represent the bearing speed and rotation speed respectively.

[0130] Figure 7 It shows that considering the viscosity-temperature effect of the lubricant significantly reduces the overall friction heat generation Q of the lubricant f and viscosity-induced heat generation Q v For example, at a speed of 16000 r / min, without considering the viscosity-temperature effect of the lubricant, the total friction heat Q f is 589.8W. After taking this effect into account, the total friction heat Q f The heat generated by the viscosity of the lubricant is Q v It dropped from 393.9W to 194.3W. Obviously, considering this effect, the total heat Q f and the portion of Q caused by lubricant viscosity v This shows that there is a direct relationship between the lubricant temperature and its viscosity, which affects the heat generated by friction. As the speed increases, the total heat generation Q without considering this effect decreases compared to when including this effect. f This trend indicates that at higher speeds, the heat generated by friction becomes more sensitive to changes in lubricant properties. When this effect is not considered, the friction heat generated by the lubricant viscosity generates Q v also increases significantly with increasing speed. However, when this effect is taken into account, Q v is still low and increases only slightly with speed, indicating that the reduced viscosity of the lubricant at higher temperatures helps mitigate the increase in frictional heat. The difference between the two models with and without the viscosity-temperature effect suggests that the traditional model (without this effect) may not be suitable for high-speed applications where heat generation affects the thermomechanical interaction behavior. f and Q v The trend line for is significantly flatter, implying a more stable heat generation behavior. This can be interpreted as a potential agreement with the actual operating characteristics, where the temperature increase at high speeds does not translate linearly into increased frictional heat due to lubricant thinning. For design and operation considerations, Figure 7 It is shown that lubricant selection should take into account viscosity-temperature characteristics to ensure that bearings operate within safe thermal limits, especially in high-speed environments. Figure 7 The criticality of incorporating this effect into the analysis of frictional heat generation in bearings, especially at higher rotational speeds, is emphasized. This consideration is essential for accurate predictions of the thermomechanical interaction behavior and reliable bearing performance.

[0131] Figure 8 It is shown that without considering this effect, the viscous friction torque M v It shows an increasing trend with the increase of speed. However, when the viscosity-temperature effect of the lubricant is considered, the viscous friction torque M v The main reason for this behavior is the decrease in lubricant viscosity caused by the increase in temperature. The decrease in viscosity leads to a corresponding decrease in viscous friction torque, which offsets the increase in viscous torque caused by the higher rotational speed. Figure 8 The viscous friction torque M is shown v There are two different trends of M without viscosity-temperature effect. v The magenta line shows that as the speed increases, M v In contrast, the red line considering the viscosity-temperature effect maintains a relatively flat profile. In the presence of the viscosity-temperature effect, the relatively constant M v shows that as the operating temperature increases with speed, the viscosity of the lubricant decreases. The decrease in viscosity offsets the increase in friction torque that would normally result from an increase in speed, resulting in a more stable M v . Figure 8 A clear physical explanation is provided that classical models that do not account for the effect of temperature on lubricant viscosity may significantly overestimate torque, especially at higher speeds. This overestimation may lead to inaccurate predictions of the power required to overcome friction. v The stability under viscosity-temperature effects suggests that bearings can be designed to operate more efficiently at higher speeds if proper consideration is given to lubricant properties. This could have implications for energy savings and extended bearing life. v The flat trend with the viscosity-temperature effect also means that the model may be more accurate over a certain speed range and require less adjustment or correction based on empirical data, thus enhancing the model's predictive ability. The reduction in friction torque due to the viscosity-temperature effect may indicate that the bearing may operate more efficiently at high speeds than previously expected because the reduction in lubricant viscosity results in lower energy losses. In summary, Figure 8 The data in highlight the importance of accounting for viscosity-temperature effects in models predicting viscous friction torque in bearings. Such considerations are essential for accurate and efficient bearing design, especially in applications involving a wide range of operating speeds.

[0132] Fig. 9It is shown that if this effect is not taken into account, there is a significant difference between the calculated and experimental temperature. The main reason for this growing difference is that the classical calculation of the viscous friction torque is proportional to the 2 / 3 power of the rotational speed n. However, when the influence of the lubricant effect is taken into account, the difference between the theoretical calculation and the experimental data is significantly reduced, highlighting the effectiveness of incorporating this effect into the analysis. Fig. 9 A clear trend is shown, where the theoretical values ​​without this effect (black squares) consistently overestimate the temperature over the entire speed range. This overestimation becomes more pronounced as the speed increases. The steep slope of the line indicates that at higher speeds the model diverges increasingly from reality. The theoretically calculated values ​​with the viscosity-temperature effect (red circles) are closer to the experimental data (blue triangles). The slope of this line is less steep than that without the effect, suggesting that a more accurate model can better explain the changes in lubricant properties at different temperatures. The model that considers the viscosity-temperature effect not only agrees more closely with the experimental data at lower speeds, but also maintains this agreement as speed increases. This is in stark contrast to the model without the effect, where the disagreement with the experimental data widens with increasing speed, indicating that the latter model fails to capture the key dynamics that affect bearing temperature at higher operating speeds. At the highest speed of 16,000 rpm, the error of the model with the viscosity-temperature effect is significantly reduced compared to the model without the viscosity-temperature effect. The experimental data show that the actual temperature does not rise as sharply with speed as predicted by the model without the viscosity-temperature effect. When this effect is ignored, the error ratio for temperature is in the range of approximately 36% to 46%, which is unsatisfactorily high for predictive modeling. In contrast, the error ratio with this effect is much lower, around 5% to 8%, indicating a more reliable model for practical applications. Fig. 9 The effectiveness of incorporating viscosity-temperature effects into the model was confirmed. The reduction in error ratio and the closer agreement with experimental data highlight the necessity of considering temperature-induced viscosity changes in the design and analysis of bearing systems, especially in applications involving a wide range of operating speeds. The analysis clearly shows that accounting for viscosity-temperature effects significantly improves the model's predictive accuracy, making it an important factor in bearing thermal analysis.

[0133] According to Table 2, for the bearing heating model without the stick-temperature effect, the error ratio between the measured and modeled temperatures of the outer ring is much higher, ranging from 36.17% to 46.88%. This indicates a significant deviation from the experimental data, indicating that the model without this effect cannot accurately capture the thermal behavior of the bearing. For the thermal model with this effect, the inclusion of this effect leads to a significant reduction in the error ratio, ranging from 5.00% to 7.81%. The large reduction in the error ratio shows that incorporating the stick-temperature effect can more accurately predict the experimental temperature. As the speed increases, the theoretical values ​​without the stick-temperature effect consistently overestimate the experimental temperature. In contrast, the theoretical values ​​with the stick-temperature effect are closer to the experimental data, following a more parallel trend. The absolute values ​​of the temperatures predicted by the two models show that the model without the stick-temperature effect overestimates the temperature at all speeds. However, the temperatures predicted by the model with this effect are closer to the experimental results. The model with the stick-temperature effect shows a significantly improved performance in predicting thermal behavior. The lower error ratio and closer alignment with the experimental data indicate that the model is more robust and reliable for practical applications. In conclusion, incorporating the viscosity-temperature effect into the theoretical model can predict the bearing temperature more accurately and reliably with lower error rate and closer to the experimental data at various speeds, as demonstrated.

[0134] Table 2 Theoretical calculation values ​​and experimental temperature results

[0135]

[0136]

[0137] Considering the viscosity-temperature effect of the lubricant, the quantification of heat generation becomes apparent, e.g. Fig.10 On the one hand, Fig.10 (a) depicts the increase in total bearing heating as the speed and preload increase. This is due to the proportional increase in preload, which is equivalent to an increase in external load. At the same time, the increase in speed leads to an increase in centrifugal force and gyroscopic torque acting on the ball, which increases the heat generation in the bearing. On the other hand, the heat generated in the bearing is affected by the initial contact angle, which confirms that geometry and size have a significant impact on the thermal model, such as Fig.10 (b). In addition, Fig.10 As shown in (c), the effect of the number of rolling elements on the thermal model can be ignored.

[0138] 2.1.2. Thermally induced preload

[0139] Assumption: The thermal deformation of the inner and outer rings is given by ε 3 and ε 2 It means that the center of the rolling element is from C b Move to C bn , and the center of curvature of the inner circle is from C iTransition to C in ; The center of curvature of the outer ring is from C o Move to C on ; The contact angle is from α 0 changes to α; the sphere radius changes from r b Change to r b ',like Fig.11 As shown, the change of the interference fit in the contact direction is given by If it means:

[0140] Total interference fit ε t is the rolling element thermal deformation ε 1 and initial interference fit ε 10 The superposition of is expressed as:

[0141]

[0142] Where: 3 Indicates the deformation of the inner ring; ε 2 Indicates that the outer ring is deformed.

[0143] The thermal deformation of the inner ring and rolling element is a linear function of temperature. The deformation of the rolling element is expressed as:

[0144] ε 1 =α d ·r b ·T b

[0145] Where: α d 、r b and T b They represent the thermal expansion coefficient, ball radius and rolling element temperature respectively.

[0146] The inner ring deformation is expressed as:

[0147] ε 3 =α d ·r i ·T i

[0148] Where: r i and T i represent the inner circle radius and temperature respectively.

[0149] Outer ring / bearing housing deformation ε 2 Related to the thickness of the bearing housing, expressed as:

[0150]

[0151] Where: T(r) represents the outer ring / bearing housing temperature; v represents the Poisson’s ratio; T o and T h Respectively represent the outer ring and bearing housing temperature; r oand r h Represent the outer ring and bearing housing diameters respectively.

[0152] By interference fit t The contact pressure change caused by the change is determined as:

[0153]

[0154] Where: k t represents the normal contact stiffness; β represents the exponent.

[0155] The Hertz elastic contact load deformation coefficient is Γ and Σ represent the first complete elliptic integral, E ′ represents the equivalent elastic modulus of the two contact bodies, κ represents the ellipse eccentricity parameter, and ∑ρ represents the sum of the curvatures of the contact bodies at the contact points on the orthogonal planes. According to the Hertz elastic contact load deformation coefficient K, the normal contact stiffness k t Determined as k t =2.0029×10 8 N / m. β = 1.5.

[0156] The change in thermally induced preload is expressed as:

[0157] F pt =F t ·sinα·N

[0158] Where: N represents the number of steel balls.

[0159] The initial interference fit is expressed as:

[0160]

[0161] The change in the interference fit will result in a load friction torque M l The change in the friction torque M affects the total friction torque, which in turn generates additional forces and affects the heating of the bearing:

[0162] H total =1.047×10 ―4 ×[n·(M v +M gy )+n s ·M s +n·f 1 ·(F β +0.9F pt ·cotα)·d m ]

[0163] Among them: F β represents the equivalent external load; f 1 Indicates the load factor.

[0164] 2.2 State Space Equations

[0165] 2.2.1 Bearing heat transfer mechanism

[0166] (1) Heat conduction

[0167] Heat Q transferred between rolling element and inner ring ib ,for:

[0168]

[0169] Where: R ib Indicates the TCR of the inner ring / ball interface; T i Indicates the inner ring temperature; T b Indicates the rolling element temperature.

[0170] Heat Q transferred between rolling element and outer ring bo ,for:

[0171]

[0172] Where: R bo Indicates the TCR of the outer ring / ball interface; T o Indicates the outer ring temperature.

[0173] Heat transferred between the main shaft and the inner ring Q is ,for:

[0174]

[0175] Where: R is Indicates the TCR of the inner ring / shaft interface; T s Indicates the temperature of the spindle.

[0176] Heat transferred between the outer ring and the bearing housing Q oh ,for:

[0177]

[0178] Where: R oh Indicates the TCR of the outer ring / bearing housing interface; T h∞ Indicates the bearing housing temperature.

[0179] (2) Convective heat transfer

[0180] Heat dissipated by convection heat transfer between rolling elements and lubricant Q bc ,for:

[0181] Q bc =h c ·A b ·(T b ―T l)

[0182] Where: h c represents the convection coefficient; T l Indicates lubricant temperature; A b Represents the surface area of ​​the sphere.

[0183] The heat dissipated by convection heat transfer between the outer ring and the lubricant Q oc ,for:

[0184] Q oc =h c ·A o ·(T o ―T l )

[0185] Among them: A o Represents the outer surface area.

[0186] Convection heat transfer between the bearing housing and the air bc ,for:

[0187] Q hc =h ∞ ·A h ·(T h ―T ∞ )

[0188] Where: h ∞ represents the convection coefficient; T ∞ Indicates ambient temperature; A h Indicates the surface area of ​​the bearing housing.

[0189] Convection heat transfer between the inner ring and the lubricant ic ,for:

[0190] Q ir =h c ·A i ·(T i ―T l )

[0191] Among them: A i Indicates the surface area of ​​the inner ring of the bearing.

[0192] Heat flows from the bearing into the rolling elements and the inner and outer rings, i.e.:

[0193] Q f =Q bf +Q if +Q of

[0194] Where: Q bf Indicates the heat flowing into the rolling element, Q bf =0.5Q f ;Qif and Q of represent the heat flowing into the inner and outer rings, Q of =Q if =0.25Q f .

[0195] In this embodiment, the control volume Ω represents the bearing components, including the bearing inner ring, the bearing outer ring and the balls. Fig.13 Based on the thermal balance principle, the thermal balance equation of the control volume Ω is derived. The control volume Ω represents the bearing components, including the bearing inner ring, the bearing outer ring and the rolling element. Based on the thermal balance principle, the thermal balance equation of the control volume Ω is obtained:

[0196]

[0197] Where: T 0 represents the temperature of the control volume Ω; T 1 , T 2 , T 3 and T 4 represents the temperature of the contact body; Q represents the internal heat source; R 0―1 , R 0―2 , R 0―3 and R 0―4 Indicates TCR.

[0198] The bearing system is divided into different temperature nodes and a thermal resistance network is established, such as Fig.14 (a) and 14(b). The state space equations are obtained:

[0199]

[0200] Where: C oh represents heat capacity; T h∞ , T s∞ and T ∞ Respectively represent the bearing housing, the far end temperature of the shaft and the ambient temperature; R bo , R oh , R ib , R is , R ib 、h ∞ and h c is an unknown parameter.

[0201] 2.2.2 Contact thermal resistance

[0202] If a L c2 , then TCC is:

[0203]

[0204] Among them: a c2 ​represents the critical plastic contact area; represents the dimensionless actual contact area; a L represents the maximum contact area; D is the fractal dimension; k s represents the equivalent thermal conductivity;

[0205] If a c2 <a<a c1 , then TCC is:

[0206]

[0207] Among them: a c1 represents the critical elastic contact area; a represents the micro-contact area; f(a) represents the template function;

[0208] if a>a c1 , then TCC is:

[0209]

[0210] TCR is:

[0211]

[0212] Where: H c Represents thermal contact resistance.

[0213] 2.2.3 Convection coefficient

[0214] The bearing ring undergoes forced convection, and the cooling area exposed to the airflow is expressed as:

[0215]

[0216] Where: D o Indicates the outer diameter of the bearing; D i Indicates the inner diameter of the bearing;

[0217] The air velocity is calculated as:

[0218]

[0219] Where: V air Indicates the flow rate of compressed air; D m Indicates the median diameter; A ax represents the axial airflow through the area between the inner ring and the outer ring; n represents the bearing speed;

[0220] The convection coefficient is:

[0221]

[0222] Where: c 0 、c 1 and c 2are all constants;

[0223] The convection heat transfer area between the bearing components and the compressed air is:

[0224]

[0225] Where: d i and d o Respectively represent the thickness of the inner and outer rings; r i and r o Respectively represent the radius of the inner circle and the outer circle; r b Indicates the rolling element radius;

[0226] The amount of heat dissipated by convection between the bearing components and the compressed air is:

[0227]

[0228] Where: h c represents the natural convection coefficient; ΔT represents the temperature rise;

[0229] The natural convection coefficient between the bearing outer ring and the air is:

[0230] h c =2.3×(T – T ∞ ) 0.25

[0231] Where: T represents the outer ring temperature;

[0232] The area of ​​the bearing outer ring is:

[0233]

[0234] Where: D h Indicates the bearing housing diameter, W h Indicates the thickness of the bearing housing;

[0235] The heat dissipated by convection heat transfer between the outer ring of the bearing and the air is:

[0236]

[0237] The bearing heat balance equation is:

[0238] H total =Q A +Q B

[0239] The heat balance relationship is:

[0240]

[0241] Where: M v 、M s and Mgy They represent viscous friction torque, load friction torque, rotational friction torque and gyroscopic torque respectively; n and n s Represent the bearing speed and rotation speed respectively.

[0242] 2.2.4. Bearing component state equations

[0243] In the heat balance equations, T b and T i are independent, and T o and T h and the temperature distribution function T oh (r, t); therefore, choose T oh (r,t) as state variables:

[0244]

[0245] The state space equations simplify to:

[0246]

[0247] 2.3 State Observer

[0248] 2.3.1. Stateless Observer

[0249] Speed ​​n and temperature T of the bearing outer ring o Respectively represented as u 1 and u 2 , and as the system input; the state variables are selected as x 1 、x 2 and x 3 Indicates T oh (r,t),T b and T i ; Heat-induced preload F pt is the output, and the state variable vector is X = [x 1 ,x 2 ,x 3 ] indicates that the initial preload function is F p0 , then: f(X,u)=[f 1 (x,u),f 2 (x,u),f 3 (x,u)], the state space model structure is:

[0250]

[0251] Where: u represents the system input vector, z represents the system output, z=F pt =p(x,F p0 ).

[0252] The state space model is:

[0253]

[0254] Where: A, B and C represent coefficient matrices.

[0255] The coefficient matrix A is of 3×3 dimensions, the matrix B is of 3×1 dimensions, and the matrix C also has 3×1 dimensions; these coefficient matrices in the state-space model are expressed as:

[0256]

[0257] in:

[0258] 2.3.2 Full-dimensional Observer

[0259] Taking into account the uncertainty of heat generation calculation, Δq(u,z) is incorporated into the bearing thermal model:

[0260]

[0261] z=p(x,F p0 )

[0262] Where: Δq(u,z) represents the bearing heat value adjusted for temperature error compensation.

[0263] To achieve an accurate bearing heating estimate, the most effective approach is to select the signal that is most sensitive to friction heating. While bearing heating itself constitutes the most direct signal, its direct measurement has proven to be challenging. An alternative approach is to measure the power consumption of the drive motor and use this as a surrogate indicator for the heat dissipation of the bearing. However, real-world scenarios introduce other forms of energy dissipation, making this approach less effective. To address this issue, the temperature difference between the predicted and measured data is determined. Subsequently, the temperature difference is adjusted through a proportional link and input into the heat transfer module. In the event that the predicted temperature exceeds the test data, a feedback loop mitigates the bearing heat generated to the heat transfer module.

[0264] Introduce the difference between observed and experimental values:

[0265]

[0266] The observer dynamic error is:

[0267]

[0268] Among them: δq(e) represents a higher-order infinitesimal; Defined as:

[0269]

[0270] Where: x = [x 1 ,x 2 ,…,x n ]; q(x) = [q 1 (x),q 2 (x),…,q n (x)].

[0271] 2.3.3 Proportional Integral (PI) Observer

[0272] Figure 6 The bearing heating process, heat transfer process, thermal bearing component deformation process and additional thermally induced preload generation process are shown. The bearing heating process and the additional thermally induced preload generation process are both linear processes. Considering the nonlinear factors such as lubrication, sliding of bearing components and wear during the heating process, an integral term is introduced to improve the steady-state tracking accuracy of the observer.

[0273]

[0274] Where: D 1 represents the coefficient matrix; K p and K I represents the matrix of the corresponding dimension, and K p and K I denote the proportional gain and integral gain of the proportional observer and the integral observer respectively;

[0275] get:

[0276]

[0277] in: f∈R m , represents the introduced integral action;

[0278] The state estimation error is defined as but:

[0279]

[0280] in:

[0281] 3. Experimental research and verification

[0282] This embodiment only considers the change of preload force. The calculation conditions are as follows: the initial preload is F p0 =400N. Radial load is F r = 0. The required calculation parameters are listed in Table 3.

[0283] Table 3 Required calculation parameters: size and material performance parameters

[0284]

[0285] 3.1. Stateless Observer

[0286] Fig.15 The speed is shown to increase gradually from 1000 rpm to 4000 rpm. The temperature is also shown and appears to be correlated with the speed. The spindle maintains a speed of 1000 rpm from the beginning until about 3776 seconds. It then accelerates to 2000 rpm between about 3776s and 7876s. The next increase to 3000 rpm occurs between about 7876s and 13240s. Finally, the speed reaches 4000 rpm between about 13240s and 18692s. The temperature data was recorded using a Max100 data logger, which is commonly used for high-precision temperature measurements. The temperature measured by the Pt100 temperature sensor appears to increase with each increase in speed. The increase is gradual without any sudden peaks or drops, indicating that the temperature responds consistently to the change in speed. As the speed increases, friction also increases, which causes the temperature to increase.

[0287] Simulated temperature Fig.16 As shown. Among the four components, the inner ring temperature Ti reaches the highest value. The sharp increase corresponds to the period of increased operating pressure or energy input. This high temperature may be due to the TCR at the inner ring / shaft interface limiting the heat dissipation from the inner ring to the shaft core. The increase in the ball temperature Tb is less obvious compared to the inner ring temperature Ti, indicating better heat dissipation. The temperature Toh increases gradually, indicating that it is affected by the heat of other components, but to a lesser extent due to the cooling effect of the lubricant. The outer ring temperature To is the lowest temperature among the four components observed here. The relatively flat and low temperature distribution means that the heat of the outer ring is effectively absorbed by the bearing housing. Due to its large heat capacity, the bearing housing acts as a heat sink. The temperature curve shows the presence of a step change, similar to Fig.15 This indicates that these temperature changes are likely the result of a step change in operating speed. The hysteresis in the temperature response after each step change indicates the time it takes for heat to transfer through the system and reach a new equilibrium.

[0288] The dynamic changes of bearing preload are as follows: Fig.17As shown. During the speed conversion process, the preload showed a fluctuating trend. The preload started from approximately 440N and showed fluctuating behavior throughout the duration, with several obvious increases and decreases. These fluctuations correspond to the transition period of the bearing speed change, because the increase in speed will affect the preload due to thermal expansion and other factors. When the speed is 4000r / min, the preload finally stabilizes at approximately 540N. The initial increase in preload and subsequent decrease is caused by the inconsistent response time constants of the bearing components. The observed fluctuations may be caused by the differences in thermal expansion rates between the various bearing components. Since each component has a different response time constant based on its material properties and cooling conditions, the preload will change until thermal equilibrium is reached. The bearing component temperature is regarded as a transient response under the excitation of a constant body heat source. The time constant is given by Given, where: ρ represents the material density, C p ρC represents the specific heat capacity of the material, V represents the volume, h represents the convection coefficient, and A represents the surface area. p V represents specific heat capacity and hA represents cooling conditions. The larger the heat capacity and the better the cooling conditions, the faster the component reaches thermal equilibrium. However, the bearing housing, which generally has a larger heat capacity than the inner ring and the ball, will have a larger time constant and therefore a slower rate of temperature change. This indicates that even though the bearing housing may dissipate heat more effectively, it will take longer to reach thermal equilibrium, resulting in a delay and long-term stabilization of the preload. Together, they determine the rate of response to transient processes. The heat capacity of the bearing housing exceeds that of the inner ring and the ball. Although the bearing housing has better heat dissipation conditions than the inner ring and the ball, the time constant of the bearing housing exceeds that of the inner ring and the ball. Therefore, the thermal equilibrium time of the bearing housing is extended and its temperature change rate is smaller than that of the inner ring and the ball.

[0289] like Fig.18The relationship between dynamic preload and speed is simulated as shown in Figure 2. The additional preload increases with increasing speed, showing an initial upward trend that eventually turns into a downward trajectory as the running time passes. This behavior stems from the fact that the outer ring / bearing housing has a larger heat capacity and more superior cooling conditions compared to the inner ring and ball. Therefore, the temperature change rate of the inner ring and ball is relatively large. Each line represents the change of preload over time. When the speed is applied, the preload immediately peaks. This is probably due to thermal expansion when the temperature rises, which initially increases the preload. After the initial peak, the preload at each speed level stabilizes to a relatively constant value over time. This stability indicates that thermal equilibrium has been achieved within the bearing system, where the rate of heat generation is balanced with the rate of heat dissipation. The stable preload increases with increasing speed. This is consistent with the understanding that increasing speed will generate more heat due to friction, which in turn increases thermal expansion and thus increases preload. The 3000r / min line starts at a preload of more than 2000N and quickly stabilizes. The 6000r / min line starts at around 1000N, while the 9000r / min line stabilizes above 1000N. The 12000r / min line shows the most impressive stability at a preload close to 2500N.

[0290] The observed behavior above reflects the fact that the bearing housing, with its greater heat capacity and superior cooling, does not experience the same rapid temperature changes as the inner race and balls. The inner race and balls, with their lower heat capacity and potentially less efficient cooling, experience more significant temperature changes, which can affect preload. Over time, as the spindle system operates, the bearing housing temperature rises more slowly than the internal components, causing the outer race to gradually increase in size, thereby increasing the preload. However, due to the large heat capacity of the housing, the system eventually reaches a new thermal equilibrium at the elevated temperature and the preload stabilizes.

[0291] When the speed is 3000r / min, the preload and total interference fit are as follows Fig.19 and 20 As shown. Fig.19(a), with a maximum preload of 456.8N. The peak at the beginning of the preload may represent an initial overload or transient response as the bearing system adapts to the operating conditions. It may be interesting to analyze why this peak occurs and whether it corresponds to the moment when the speed reaches 3000r / min. The point at which the preload level decreases indicates that the preload has reached a steady state. This stabilization may mean that the bearing system has reached the operating conditions for which it was designed and that the bearing load has remained constant. Changes in preload over time may also be affected by temperature, as the components of the system expand or contract due to the heat generated by high-speed friction. The final preload stabilization at around 380N indicates that a new equilibrium state has been reached by accounting for the thermal deformation of the bearing components.

[0292] according to Fig.19 (b), the rate of change of thermal displacement of the bearing components is the same at 164.5 seconds, and then the total thermal displacement of the inner ring and the ball is equal to the thermal displacement of the outer ring at about 814.9 seconds. The preload returns to its initial value, i.e. 400N, and the bearing outer ring / bearing housing reaches thermal equilibrium at about 2000 seconds. The preload eventually stabilizes at around 380N. The initial peak may indicate a deformation overshoot, possibly caused by rapid heating at the start-up of the bearing system. After this, the bearing material begins to expand more uniformly, and the inconsistency of the bearing deformation decreases until it reaches zero, indicating that the components have expanded uniformly. The outer ring and bearing housing reach thermal equilibrium at about 2000 seconds, which means that the temperature of the entire system has become uniform, resulting in a consistent stable state of thermal deformation of all components.

[0293] Fig. 20 (a) shows the temperature rise over time of different components of the bearing assembly under preload and deformation. The ball temperature (Tb), temperature distribution function (Toh), and inner ring temperature (Ti) are plotted over a period of 20,000 seconds. All components initially experience a rapid temperature rise, which is common when a bearing system moves from static to dynamic and friction begins to generate heat. Each bearing component reaches a different stable temperature, which may be due to the different heat capacity, electrical conductivity, and amount of friction in each component. The temperature distribution function Toh stabilizes at the lowest temperature, followed by the ball temperature Tb, and then the inner ring temperature Ti at the highest temperature. The preload and deformation behavior discussed previously can be directly related to these temperature changes. As the components expand with increasing temperature, this can explain the initial preload peak and subsequent stabilization.

[0294] Fig. 20(b) Plots the deformation of various bearing components over time. The curve for the bearing outer ring / housing shows a clear initial negative deformation, indicating that it initially contracts or displaces inward, and then gradually returns to zero deformation, indicating that it stabilizes over time. The rolling elements show an immediate positive deformation, which stabilizes relatively quickly compared to the other components. This may be due to rapid thermal expansion or initial mechanical settling. The inner ring shows a continuous positive deformation, which increases more slowly and stabilizes at a higher value than the rolling elements. This indicates continued expansion, perhaps due to heat buildup. Combining this information with the previous temperature rise data, the deformation trends can be directly related to the temperature distribution of each component. Increased temperature causes expansion, which is described as deformation.

[0295] When the bearing runs at 22000r / min for 1380 seconds, the temperature and deformation of the bearing components are as follows: Fig.21 As shown. Fig.21 (a), when the entire system is unstable, the temperature and thermal deformation of the bearing components show an exponential growth trend, which is typical at higher speeds, resulting in increased friction and heating. The different heating rates of the components may reflect their mass, thermal conductivity, heat capacity, and the amount of friction they experience. The rolling elements (Tb) and inner ring (Ti) heat much faster than the outer ring (To) and temperature distribution function (Toh), which may be due to their smaller mass and closer contact with the heat source. The exponential growth trend of temperature indicates that the system is not in a stable state and may be approaching a critical state, at which the material properties of the components will degrade, the lubricant will fail, or other failure modes will occur.

[0296] according to Fig.21 (b) The inner race shows a significant sharp increase in thermal displacement over time, indicating significant expansion due to heat. The rolling elements show a less pronounced but still significant exponential increase in thermal displacement, which may indicate differences in material composition or thermal properties compared to the inner race. The outer race and housing show relatively small thermal displacements until a sharp increase at the end of the measurement time frame. The exponential growth trend in thermal displacements indicates that the system is experiencing thermal runaway and that heat generated by high speed operation and friction is not being effectively dissipated.

[0297] Fig. 22 The variation of preload during the first 10 seconds and 1380 seconds duration are shown respectively. The variation of preload during the initial 10 seconds shows a linear growth trend, reaching 1200N within 10 seconds. Although the time is short, very rapid thermal expansion may occur due to high speed or friction, resulting in a linear increase in the apparent preload. Subsequently, the preload rises in an almost vertical trend, reaching an infinite value around 1200 seconds. This indicates that a catastrophic failure of the bearing system has occurred, with a dramatic increase in preload due to some form of collapse or blockage.

[0298] 3.2. Verification of the full-dimensional observer

[0299] Fig.23 Display, the observed value T oh is calculated using equation (31), where the outer ring temperature T o and bearing housing temperature T h As input. The original system T oh The difference between the calculated and observed temperatures is used to compensate for systematic errors.

[0300] The calculated T oh As input, when considering temperature error compensation, the temperature state value is as follows Fig.24 A step change in the temperature graph can represent a change in operating conditions, such as an increase in speed, which results in a step increase in temperature as the system adjusts to the new conditions. Including an observer results in a higher temperature state value than would be the case without the addition of a temperature state value. The observer, if included, adjusts the state value and produces a higher temperature reading. This can be critical to system safety and performance, as underestimating the temperature may result in a failure to take necessary cooling or safety measures. Reference Fig.19 , the preload on the bearing is affected by the temperature state value. This can mean that temperature changes cause the bearing components to expand or contract, thus affecting the applied preload. The stability of the preload at around 700N indicates that the system has reached equilibrium once the temperature has stabilized.

[0301] exist Fig.25 In (a), the change in bearing preload when temperature error compensation with the full-dimensional observer is considered is illustrated. When temperature error compensation is considered, the bearing preload is larger than when there is no state observer, and the preload eventually stabilizes at around 700N. The preload trajectory (blue) shows higher values ​​when the state observer is used than when there is no observer. This shows that the correction of temperature error by the full-dimensional observer leads to a higher estimate of preload. The preload trajectory (red) without the observer shows lower preload values, which may indicate that temperature error may cause the preload to be underestimated. Both trajectories appear to reach a stable point, but the preload with the observer is always higher and stabilizes at around 700N. Both trajectories show step changes in preload. These steps may represent operational changes, such as speed adjustments that lead to corresponding changes in preload.

[0302] Fig.25(b) shows that the inner ring has the largest deformation, followed by the outer ring / housing deformation, and the ball has the smallest deformation. When the speed is 4000r / min, the thermal deformation of the inner ring is about 20μm due to thermal mechanical interaction. The inner ring of the bearing shows obvious deformation, with several steps indicating an incremental increase and stabilization at around 20μm. This indicates that the inner ring is most affected by thermal expansion, which is consistent with high-speed operation (4000r / min) where heat is generated due to friction. The rolling elements have less deformation compared to the inner ring. The outer ring / bearing housing of the bearing has the largest deformation, which may be due to its larger size.

[0303] 3.3 Proportional Integral (PI) Observer Verification

[0304] Fig.26 shows the change in temperature state when a PI observer is added, which is used to adjust the state according to the difference between the simulated value and the actual test data. Since the simulated value in Section 4.1 is lower than the test data, this means that the change from the thermomechanical interaction system to Figure 6 The heat transfer to the transfer process is shown to be reduced. Therefore, to compensate for this, the heat transfer from the bearing system to the transfer process should be increased. Therefore, the temperature state values ​​when the full-dimensional observer is added should be higher than when the observer is not added. The role of the PI observer is to correct the predictions of the model to better match the observed data, especially when there are differences in heat transfer rates. This is done by adjusting the parameters in the PI observer to increase or decrease the estimated heat transfer until the simulated temperatures are closer to the test data. When the PI observer is added, the temperature state values ​​are indeed higher (as shown in the blue trace), indicating that the PI observer compensates for the previously underestimated heat transfer in the thermo-mechanical interaction system. The temperature trace shows a step change, indicating that the system is under different operating conditions at these points in time. The full-dimensional observer model is unable to capture all of the heat transferred within the system, so the PI observer is needed to correct this underestimate.

[0305] exist Fig. 27In , the change in bearing preload when the observer is added is plotted, which is greater than when the PI observer is not added. The bearing preload eventually stabilizes at around 625N. The PI observer may correct for the underestimated preload. The observer may consider additional variables or more accurately model the system dynamics, resulting in a higher estimate of the preload. The lower preload value indicates that the no-observer case may not fully account for all forces acting on the bearing or may not correctly model the behavior of the system over time. Without observer-based corrections, an underestimate of the actual forces on the bearing may result. The preload eventually reaches a stable point for both the PI observer and the no-observer case, with the PI observer case stabilizing at a higher preload value. This suggests that the PI observer may be better at capturing the long-term equilibrium state of the system. A full-dimensional observer, which generally refers to an observer that considers the complete state space of the bearing system, may provide a more comprehensive correction than a reduced-order observer. This suggests that compared to the no-observer case, Fig.25 The full-dimensional observer shown in (a) produces a higher preload estimate, which may indicate a more complete explanation of the bearing system dynamics.

[0306] Fig.28 The variation of the nonlinear term for heat generation corresponding to the integral term is shown. The nonlinear term increases with the operating time. This term may be related to the integral term in the PI observer, indicating the cumulative effect of heat generation in the bearing system. This increasing trend indicates that the heat in the system is not dissipated at the same rate as it is generated, resulting in a continuous increase in the internal energy of the bearing.

[0307] 3.4 Digital Twin System Verification

[0308] In order to verify the digital twin system for bearing thermomechanical condition monitoring, a preload test device was designed. The test principle is as follows: when rotating at a low speed, tension and pressure are applied to the spindle bearing system, and a high-precision displacement sensor is used to measure the thermal deformation of the spindle under the influence of tension and compression. The displacement is then processed to determine the bearing preload.

[0309] The purpose of this experiment is to test the initial preload and the preload at thermal equilibrium. The test procedure is outlined as follows:

[0310] (1) Measure the initial preload of the bearing under static conditions.

[0311] (2) The speed increases from 1000r / min to 5000r / min.

[0312] (3) When the temperature is stable, test the bearing preload in thermal equilibrium.

[0313] The initial preload is measured and determined to be 400 N. At a spindle speed of 1000 r / min, the test preload in thermal equilibrium is 410 N. The corresponding test temperature is Fig.29 All components heat up over time, with the inner ring (Ti) heating up the most significantly. This suggests that the inner ring is not dissipating heat as efficiently as the other components. The temperature stabilizes after a period of time, indicating that the system has reached thermal equilibrium. In this state, the test preload of 410N shows that the slight increase in preload from the initial value of 400N is likely due to thermal expansion of the components. The different heating rates and different stabilization points of each component indicate their different thermal behaviors, such as thermal conductivity and specific heat capacity, as well as the different heat they are exposed to during operation. Considering that the preload only increases slightly at thermal equilibrium, the design of the bearing system appears to be effective in reducing the thermal expansion effects of the preload. This is important to maintain bearing performance and life, as excessive preload can lead to increased wear and energy consumption.

[0314] Fig.30 (a) and 31(a) show that the PI observer exhibits higher tracking accuracy compared to the full-dimensional observer model and the model without an observer. The temperature profile of the PI observer is very close to the experimental data, but it seems to underestimate the temperature slightly compared to the actual measured values. This may be due to the PI observer's reliance on a feedback loop, which may not explain all the dynamics of the system or may have a lag response to temperature changes. The PI observer seems to track the temperature more closely than the full-dimensional observer. This suggests that it uses a more detailed model of the system and can accurately predict the temperature dynamics of the temperature distribution function Toh. The scenario without an observer shows the least accurate estimate, with a larger deviation from the experimental data. This shows the importance of the observer in predicting the behavior of the system.

[0315] Fig.30 (b) and 31(b) provide a comparison and analysis of the dynamic preload changes obtained by the models without observer, PI observer, and full-dimensional observer. The peak preload value obtained by the full-dimensional state observer is the largest and eventually stabilizes at about 327N, which is lower than the initial preload force of 400N. The simulated steady-state preload obtained by the PI observer is 387N, while the simulated steady-state preload without observer is 355N. According to Fig.31 (b.

[0316] Table 4 shows the steady-state simulated and tested preloads at speeds of 1000 r / min and 5000 r / min. At 1000 r / min, for the scenario without an observer, the simulated preload is 355 N, while the tested preload is 410 N, resulting in a relative error of 13.4%. For the scenario with a full-dimensional observer: the simulated preload is 327 N and the tested preload is 410 N, resulting in a relative error of 20.2%. For the scenario with a PI observer, the simulated preload is 387 N, while the tested preload is 410 N, resulting in a minimum relative error of 5.6%. At 5000 r / min, for the scenario without an observer, the simulated preload is 629.6 N and the tested preload is 660 N, resulting in a relative error of 4.61%. For the scenario with a full-dimensional observer, the simulated preload is 725.8 N, significantly higher than the tested preload of 660 N, resulting in a relative error of 9.97%. For the scenario with the PI observer, the simulated preload is very close to the tested preload (665.9N vs. 660N) with a very low relative error of 0.89%. The data shows that the PI observer tends to be more accurate in estimating the steady-state preload, especially at high speeds (5000r / min). The lower relative error of the PI observer may be due to several factors, including a potentially better tuned feedback mechanism that is closely aligned with the dynamics of the system under steady-state conditions. The full-dimensional observer, while providing a more comprehensive model of the system, shows higher relative errors at both speeds, which may be due to overfitting or complexity in the model that does not accurately translate to the steady-state conditions being tested. No observer produces larger errors at low speeds but performs better at high speeds, which may indicate that the dynamics of the system are more predictable or linear at higher operating speeds without the influence of the observer. Overall, the relative error below 21% shows that the digital twin system with a full-dimensional observer developed for bearing thermal-mechanical interaction condition monitoring is reliable. The relative error of the digital twin system with the PI observer is less than 6%. Without adding an observer, the relative error of the digital twin system is below 14%. The performance of the PI observer in particular suggests that it may be a robust choice for applications where high fidelity under steady-state conditions is critical.

[0317] Table 4 Steady-state preload comparison

[0318]

[0319] 4. Conclusion

[0320] In the study of this embodiment, a digital twin system for bearing thermomechanical interaction state monitoring was developed to meet the challenges of bearing thermomechanical interaction. The process involves proposing a bearing heating model and calculating the thermally induced preload. The study deeply studied the heat transfer and heat dissipation mechanism of the bearing, obtained the deformation of the bearing components, and constructed a bearing thermomechanical interaction state monitoring model. Finally, the model was integrated into the digital twin system, using the tested bearing outer ring and bearing housing temperatures as input to achieve real-time preload monitoring. The key conclusions are:

[0321] (1) A digital twin system for monitoring the thermal-mechanical interaction of bearings was comprehensively constructed. The modeling method and operation mechanism based on the physical system and supported by multi-sensor fusion were described. The virtual system is used as the engine, and multi-dimensional modeling is used to implement core functions such as bearing preload monitoring and bearing condition monitoring. The physical model is implemented using MATLAB and Python to accurately reflect the actual situation of bearing temperature and preload. A behavioral model that reflects the dynamic changes of boundary conditions is also established. The modeling results are intuitively presented to the user through graphics on the software interface. Driven by dual data, the physical data is fused into a dual database based on MySQL to ensure consistency between virtual and real spaces. The connection system acts as an artery, realizing the dynamic and real-time interconnection of virtual and real spaces.

[0322] (2) A thermomechanical interaction state space model was constructed and used to achieve the online preload monitoring objective with the tested bearing outer ring temperature and speed as input. Then, the bearing preload was predicted using models without state observer, full-dimensional observer, and PI observer. The thermomechanical interaction state monitoring model was used to predict bearing thermal instability, and the PI observer showed the highest tracking accuracy, followed by the model without state observer design. Among the mentioned models, the full-dimensional observer had the lowest tracking accuracy.

[0323] (3) The state observer design takes into account temperature error compensation and achieves accurate online preload monitoring. When the speed is 1000 r / min, the relative errors between the steady-state simulation and test preload for the model without added observer, full-dimensional observer and PI observer are 13.4%, 20.2% and 5.6%, respectively. When the speed is 5000 r / min, the relative errors between the steady-state simulation and test preload for the model without added observer, full-dimensional observer and PI observer are 4.61%, 9.97% and 0.89%, respectively. Overall, the relative error below 21% indicates that the digital twin system with full-dimensional observer developed for bearing thermal-mechanical interaction state monitoring is reliable. The relative error of the digital twin system with PI observer is less than 6%. Without adding observer, the relative error of the digital twin system is less than 14%. In particular, the performance of PI observer shows that it may be a robust choice for applications where high fidelity under steady-state conditions is critical.

[0324] The above-described embodiments are only preferred embodiments for fully illustrating the present invention, and the protection scope of the present invention is not limited thereto. Equivalent substitutions or changes made by those skilled in the art based on the present invention are within the protection scope of the present invention. The protection scope of the present invention shall be subject to the claims.

Claims

1. A digital twin system for monitoring the thermal interaction operating status of high-speed precision machine tool bearings, characterized by: Includes the physical world, virtual world, twin database and twin server; The material world includes a physical entity, a control system, a cooling system and a data acquisition system of a spindle bearing system; the spindle bearing system includes a spindle and bearings located at both ends of the spindle; the data acquisition system includes a device for collecting real-time data during the operation of the spindle bearing system; The virtual world includes a virtual model of the spindle bearing system and a bearing thermal interaction state monitoring model embedded in the virtual model, wherein the bearing thermal interaction state monitoring model includes a bearing heating model for describing the heating of bearing components and a state space equation group for describing the thermal interaction behavior of bearing components; The virtual model is used to capture and describe the operating state and behavior of the spindle bearing system, which is expressed as: AND=(G v ,P v ,B v ,R v ) Among them: Geometric model G v It is a three-dimensional representation, including the physical geometry, component dimensions and assembly relationships of the spindle bearing system; the thermal interaction model P v Used to describe multi-dimensional physical properties and practical constraints; behavioral model B v It is used to illustrate the real-time response at various time and space scales, and to describe the time-varying behavior of key boundary conditions in thermal modeling, taking into account external environmental factors and internal mechanisms; knowledge-based model R v Capturing the operating rules, experience and continuous learning of virtual and actual spindle bearing systems, so that the digital twin system can make judgments, assessments and predictions based on evolving knowledge and experience; The twin database stores the twin data DD of the material world and the virtual world data: DD=(D p ,D v ,D s ,D k ,D f ) Wherein: Dp represents data related to the physical entity of the spindle bearing system, including basic information and operation data of the spindle bearing system, wherein the basic information of the spindle bearing system includes the model, specification, function and performance of the spindle bearing system, and the operation data of the spindle bearing system includes operation status data and environmental status variables; D v Relevant data representing the virtual model of the spindle bearing system, including G v Related data, physical model P v Related data, behavior model B v Related data and rule model B v Related data; D s Includes data related to bearing preload monitoring services and bearing operating condition monitoring services k Including data related to functional services f It represents the field knowledge related to the modeling principle of thermal interaction, integrates the real-time sensor data, virtual spindle bearing system simulation data, historical data and rule data, and is the most important factor in processing D p , D v , D s and D k The fusion of data provides accurate and comprehensive information for digital twins; A state observer is built in the twin server, and the state observer is used for steady-state simulation and prediction of dynamic changes of bearing preload; In the material world, the data acquisition system includes a device for collecting real-time data during the operation of the physical entity of the spindle bearing system, and transmitting the real-time data to the virtual model, the twin database and the twin server; In the virtual world, the bearing heating model obtains the heating output of the bearing components according to the real-time data; the state space equation group obtains the thermal interaction state between the bearing components according to the real-time data; The state observer performs a steady-state simulation on the spindle bearing system according to the twin data DD and predicts the predicted bearing preload.

2. The digital twin system for monitoring the operating status of thermal interaction of high-speed precision machine tool bearings according to claim 1 is characterized in that: The bearing heating model is expressed as: H total =1.047×10 -4 ×[n·(M v +M gy )+n s ·M s +n·f1·(F β +0.9F pt ·cotα)·d m ] Where: M v 、M s and M gy represent the viscous friction torque, rotational friction torque and gyroscopic torque respectively; n and n s Respectively represent the bearing speed and rotation speed; n and n s Respectively represent bearing speed and rotation speed; F β represents the equivalent external load; f1 represents the load factor; F pt represents the thermally induced preload; α represents the contact angle; d m Indicates the bearing pitch diameter.

3. The digital twin system for monitoring the operating status of thermal interaction of high-speed precision machine tool bearings according to claim 2 is characterized in that: Assumption: The thermal deformation of the inner and outer rings are represented by ε3 and ε2 respectively, and the center of the rolling element is from C b Move to C bn , and the center of curvature of the inner circle is from C i Transition to C in ; The center of curvature of the outer ring is from C o Move to C on ; The contact angle changes from α0 to α; The sphere radius changes from r b Change to r b ′, the change of interference fit in the contact direction is given by If it means: Total interference fit ε t is the rolling element thermal deformation ε1 and the initial interference fit ε 10 The superposition of is expressed as: Among them: ε3 represents the deformation of the inner ring; ε2 represents the deformation of the outer ring; The thermal deformation of the inner ring and rolling element is a linear function of temperature. The deformation of the rolling element is expressed as: e1=a d ·r b ·T b Where: α d 、r b and T b They represent the thermal expansion coefficient, ball radius and rolling element temperature respectively; The inner ring deformation is expressed as: e3=a d ·r i ·T i Where: r i and T i represent the inner circle radius and temperature respectively; The outer ring / bearing housing deformation ε2 is related to the thickness of the bearing housing and is expressed as: Where: T(r) represents the outer ring / bearing housing temperature; v represents the Poisson’s ratio; T o and T h Respectively represent the outer ring and bearing housing temperature; r o and r h Respectively represent the outer ring and bearing housing diameters; By interference fit t The contact pressure change caused by the change is determined as: Where: k t represents the normal contact stiffness; β represents the exponent; The change in thermally induced preload is expressed as: F pt =F t ·sinα·N Where: N represents the number of steel balls.

4. The digital twin system for monitoring the operating status of thermal interaction of high-speed precision machine tool bearings according to claim 1 is characterized in that: The state space equation is expressed as: Where: C oh represents heat capacity; T oh represents the temperature distribution function; T o represents the temperature of the control volume Ω; T b Indicates rolling element temperature; T i Indicates the inner ring temperature; R ib Indicates the TCR of the inner ring / ball interface; R bo Indicates the TCR of the outer ring / ball interface; R is Indicates the TCR of the inner ring / shaft interface; R oh Indicates the TCR of the outer ring / bearing housing interface; h ∞ is the convection coefficient between the bearing housing and the air; h c is the convection coefficient r between the rolling element and the lubricant h Indicates the shell diameter; r o Indicates the outer circle radius; h c represents the natural convection coefficient; h ∞ represents the convection coefficient; A0 represents the outer surface area; A h Indicates the surface area of ​​the bearing box; A b is the surface area of ​​the sphere; A i is the surface area of ​​the inner ring of the bearing; η o Indicates the heat absorption coefficient of the outer ring; η b Indicates the heat absorption coefficient of the ball; η i Indicates the heat absorption coefficient of the inner ring; Q f represents the total heat generation; n represents the bearing speed; F β Represents the equivalent external load; F pt represents the change of thermal induced preload; N represents the number of steel balls; b represents steel balls; ρ is the material density; m b Indicates the mass of the steel ball; c b Represents the specific heat capacity of the steel ball; m i Indicates the quality of the inner ring; c i Indicates the specific heat capacity of the inner ring.

5. The digital twin system for monitoring the operating status of thermal interaction of high-speed precision machine tool bearings according to claim 4 is characterized in that: The state observer adopts a stateless observer; Speed ​​n and temperature T of the bearing outer ring o They are denoted as u1 and u2 respectively and used as system input; the state variables are x1, x2 and x3 respectively to represent T oh (r, t), T b and T i ; Heat-induced preload F pt is the output, the state variable vector is represented by X = [x1, x2, x3], and the initial preload function is F p0 , then: f(X,u)=[f1(x,u),f2(x,u),f3(x,u)], the state space model structure is: Where: u represents the system input vector, z represents the system output, z=F pt =p(x,F p0 ); The state space model is: Among them: A, B and C represent coefficient matrices; The coefficient matrix A is of 3×3 dimensions, the matrix B is of 3×1 dimensions, and the matrix C also has 3×1 dimensions; these coefficient matrices in the state-space model are expressed as: A 21 ＝0； A 31 ＝0； 6. The digital twin system for monitoring the thermal interaction operating status of high-speed precision machine tool bearings according to claim 4 is characterized in that: The state observer adopts a full-dimensional observer; Taking into account the uncertainty of heat generation calculation, Δq(u, z) is incorporated into the bearing thermal model: z=p(x,F p0 ) Where: Δq(u,z) represents the bearing heat value adjusted for temperature error compensation; Introduce the difference between observed and experimental values: The observer dynamic error is: Among them: δq(e) represents a higher-order infinitesimal; Defined as: Where: x = [x1, x2, ..., x n ]; q(x)=[q1(x), q2(x),…, q n (x)].

7. The digital twin system for monitoring the operating status of thermal interaction of high-speed precision machine tool bearings according to claim 4 is characterized in that: The state observer adopts a proportional integral observer; The bearing heating process and the additional heat-induced preload generation process are both linear processes. Considering the nonlinear factors such as lubrication, bearing component sliding and wear during the heating process, an integral term is introduced to improve the steady-state tracking accuracy of the observer: Where: D1 represents the coefficient matrix; K p and K I represents the matrix of the corresponding dimension, and K p and K I denote the proportional gain and integral gain of the proportional observer and the integral observer respectively; get: in: f∈R m , represents the introduced integral action; The state estimation error is defined as but: in:

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