A method for predicting thermal coupling vibration of an electric spindle based on dissipative hamiltonian dynamics

Abstract

This invention discloses a method for predicting thermo-mechanically coupled vibration of an electric spindle based on dissipative Hamiltonian dynamics. The method includes: establishing a simplified physical model based on the physical structure of the machine tool electric spindle and defining a state vector containing both mechanical and thermal states; constructing a thermodynamic Hamiltonian neural network, where the input is the state vector and the output is physical parameters; constructing the Hamiltonian function, Rayleigh dissipation function, internal energy function, and entropy function, and obtaining the total energy function; using automatic differentiation to differentiate the total energy function and the entropy function to obtain the energy gradient vector and entropy gradient vector; constructing the reversible and irreversible terms in the thermodynamic uniform evolution equation by combining the degeneracy conditions in the GENERIC architecture, thus obtaining the thermo-mechanically coupled dynamic equation; obtaining the thermodynamic evolution trajectory of the system through numerical integration, extracting the generalized displacement term, and reconstructing the spatial-temporal vibration signal of the electric spindle. Using this invention, the evolution law of thermally induced vibration of the electric spindle can be accurately captured and predicted.

Description

Technical Field

[0001] This invention relates to the fields of digital twin and intelligent manufacturing technology, and in particular to a method for predicting the thermo-mechanical coupling vibration of an electric spindle based on dissipative Hamiltonian dynamics. Background Technology

[0002] The manufacturing industry is currently facing an urgent need to transform towards digitalization, the core of which lies in creating an intelligent, adaptive, and self-aware manufacturing environment. In this context, digital twins play a crucial role as a bridge connecting the physical and digital worlds. The foundation for constructing an effective and physically consistent digital twin lies in establishing a high-fidelity proxy model of the objective object. This requires that designers first conduct a rigorous analysis of the physical object and its underlying physical laws to ensure that the virtual digital entity accurately reflects the performance of the real-world objective object.

[0003] Complex thermo-mechanical coupling effects are prevalent in the operation of high-end equipment such as precision machine tool electric spindles. Mechanical vibrations within the system are often accompanied by dissipation processes such as friction and damping. These irreversible processes generate significant heat, leading to temperature rise. This temperature rise significantly alters the stiffness and damping characteristics of the supporting bearings, subsequently causing thermally induced vibration drift phenomena such as spindle vibration frequency shifts and amplitude fluctuations. Therefore, accurate dynamic analysis of such systems must go beyond the traditional isothermal assumption and explicitly capture the active influence of temperature rise on the system's vibration characteristics to achieve high-precision long-term vibration prediction.

[0004] Currently, constructing complex thermodynamic coupling models that are both physically rigorous and computationally efficient faces significant challenges. Direct analytical methods become difficult to handle due to the nonlinear contact mechanics of bearings; while the finite element method offers high accuracy, its computational cost is too high, making it unsuitable for real-time interaction. Therefore, reduced-order models or various simplified models have become the preferred choice.

[0005] In existing technologies, machine learning methods based on physical information have been attempted for multiphysics modeling. However, while existing frameworks such as dissipative Hamiltonian neural networks (DHNNs) can simulate mechanical energy decay by introducing dissipation terms, they mostly treat dissipation as pure energy loss, failing to explicitly consider the conversion of energy into heat and its reaction on mechanical parameters. This underlying isothermal assumption makes it difficult for the model to track parameter drift caused by internal energy evolution, thus making it difficult to predict the vibration response deviation of the spindle under unsteady thermal conditions. In addition, purely data-driven methods lack physical interpretability and often exhibit insufficient generalization ability and prediction trajectories that violate thermodynamic laws when dealing with the vibration prediction of electric spindles with variable machining conditions. For theoretical frameworks such as irreversible port Hamiltonian systems, there is still a lack of a unified mechanism-data fusion-driven method to directly identify complex, unknown, state-dependent nonlinear thermodynamic coupling relationships from observational data.

[0006] In summary, existing technologies have the following shortcomings in modeling complex thermodynamic coupling: while existing physical information Hamiltonian network frameworks can effectively simulate the decay of mechanical energy, they fail to explicitly track the conversion of dissipated energy into thermal energy and ignore parameter drift phenomena caused by temperature rise actively altering system properties; purely data-driven methods lack physical interpretability and require large amounts of data. Therefore, there is an urgent need for a modeling method that can strictly adhere to the laws of thermodynamics and accurately capture and predict the evolution of thermally induced vibrations of electric principal shafts. Summary of the Invention

[0007] To overcome the shortcomings of existing technologies, this invention provides a method for predicting the thermo-mechanical coupling vibration of electric spindles based on dissipative Hamiltonian dynamics, which can accurately capture and predict the evolution law of thermally induced vibration of electric spindles.

[0008] A method for predicting the thermo-mechanical coupling vibration of an electric principal shaft based on dissipative Hamiltonian dynamics includes: (1) Establish a simplified physical model based on the physical structure of the machine tool electric spindle, and define a state vector that includes mechanical and thermal states. ; (2) Construct a thermodynamic Hamiltonian neural network, the input of which is the current or initial state vector. The output is physical parameters that are affected by temperature; (3) Based on the physical parameters output in step (2), construct the core scalar functions describing the system dynamics, including the Hamiltonian function characterizing the total mechanical energy of the system. Rayleigh dissipation function characterizing the mechanical energy dissipation rate of a system. Internal energy function characterizing the thermal state of a system With entropy function And obtain the total energy function. Represented as ; (4) Use the automatic differentiation to determine the total energy function in step (3). Sum and entropy function Taking the derivative, we obtain the energy gradient vector. and entropy gradient vector By combining the degeneracy conditions in the GENERIC framework, we can construct the invertible and irreversible terms in the thermodynamic uniform evolution equation to obtain the state vector. The time derivative is used to obtain the thermo-coupling kinetic equations; (5) The thermodynamic Hamiltonian neural network described in step (2) is trained using the thermodynamic coupling kinetic equations obtained in step (4) so ​​that the network learns the mapping relationship between the state vector and the physical parameters. After training, in the application stage, the given initial state vector is input into the trained thermodynamic Hamiltonian neural network to predict the physical parameters in the current state, and the thermodynamic coupling kinetic equations are solved by numerical integration to obtain the thermodynamic evolution trajectory of the system. (6) From the predicted trajectory The generalized displacement term is extracted and the spatial-temporal vibration signal of the electric spindle is reconstructed to identify the natural frequency drift phenomenon caused by temperature rise.

[0009] Furthermore, in step (1), the simplified physical model is a two-degree-of-freedom lumped parameter model, and the electric principal axis is regarded as a rigid body supported by two front and rear spring-damped units with thermosensitive characteristics; the state vector Defined as: ; in, This represents the vertical displacement at the center of mass. The generalized momentum generated by the vertical translation. The rotational angular displacement generated by the center of mass. The generalized angular momentum generated by rotation. This refers to the local temperature at the front bearing. This refers to the local temperature at the rear bearing.

[0010] Furthermore, in step (2), the physical parameter values ​​output by the thermodynamic Hamiltonian neural network include: the stiffness coefficient of the front bearing. Stiffness coefficient of the rear bearing Damping coefficient of the front bearing Damping coefficient of the rear bearing .

[0011] Furthermore, in step (2), the thermodynamic Hamiltonian neural network uses the hyperbolic tangent function as the activation function to ensure that the higher-order derivatives of the output physical parameters with respect to the input state exist and are continuous.

[0012] Furthermore, in step (4), the thermo-coupling kinetic equation is: ; in, State vector Time derivative, and These are the reversible and irreversible terms in the thermodynamic uniform evolution equation, respectively. The Poisson matrix is ​​used to describe invertible dynamics. The friction matrix is ​​used to describe irreversible dynamics.

[0013] Furthermore, the Poisson matrix It is a constant antisymmetric matrix, specifically expressed as: .

[0014] Furthermore, the friction matrix Rayleigh dissipation function The non-zero blocks are derived from the damping dissipation terms of the front and rear bearings and the momentum-temperature coupling terms, ensuring that mechanical dissipation energy is converted into internal energy and that the entropy generation rate is non-negative.

[0015] Furthermore, in step (4), the degeneracy condition in the GENERIC architecture is: .

[0016] Compared with the prior art, the present invention has the following beneficial effects: 1. This invention guarantees the thermodynamic consistency of the trained network. Unlike traditional black-box models or models that only consider the conservation of mechanical energy, this invention strictly follows the first and second laws of thermodynamics, ensuring that energy does not appear or disappear out of thin air during long-term prediction, but rather correctly transforms between the mechanical and thermal domains.

[0017] 2. The modeling method of this invention has high fidelity. By explicitly modeling the thermo-mechanical feedback loop, the model can capture the changes in stiffness and damping caused by temperature rise, thereby efficiently and accurately predicting the nonlinear dynamic behavior and parameter drift of the electric spindle.

[0018] 3. The overall methodological architecture of this invention has structural interpretability. The neural network is only used to learn unknown constitutive relations, while the overall evolution of the system is controlled by physical equations, enabling the model to maintain good interpretability and generalization ability.

[0019] 4. This method is data efficient. Compared with past purely data-driven methods, this invention significantly reduces the amount of training data required by the neural network by introducing physical information, and can better handle working conditions outside the training distribution. Attached Figure Description

[0020] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0021] Figure 1 This is a cross-sectional view of the electric spindle used in this invention.

[0022] Figure 2 This is a simplified model of the electric spindle spring damping used in this invention example.

[0023] Figure 3 This is a complete flowchart of the invention.

[0024] Figure 4 This invention is used for a comprehensive comparison of various indicators with error bars with other models.

[0025] Figure 5 This is a comparison chart showing the evolution of the prediction results of the qy value between the first set of data from the network of this invention and other benchmark networks over time.

[0026] Figure 6 The graph shows a comparison of the evolution of the prediction results of the method of this invention and other benchmark networks for the py value over time in the first set of data.

[0027] Figure 7 The graph shows a comparison of the evolution of the prediction results of the method of this invention and other benchmark networks for the qθ value over time in the first set of data.

[0028] Figure 8 The graph shows a comparison of the evolution of the prediction results of the method of this invention and other benchmark networks for the pθ value over time in the first set of data.

[0029] Figure 9 This is a comparison of the Hamiltonian error evolution of the network of the present invention with that of other networks in the first set of data.

[0030] Figure 10 This is a comparison of the Hamiltonian error evolution of the network of the present invention with that of other networks in the second set of data.

[0031] Figure 11 The results show the comparison between the method of this invention and five other benchmark networks in terms of inference time and number of model parameters. Detailed Implementation

[0032] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0033] It should be noted that, unless otherwise specified, the features in the following embodiments and implementation methods can be combined with each other.

[0034] This embodiment provides a method for predicting the thermo-mechanical coupling vibration of an electric spindle based on dissipative Hamiltonian dynamics. This method constructs a thermodynamic Hamiltonian neural network using a GENERIC architecture to achieve accurate prediction of the thermo-mechanical coupling dynamics of the electric spindle during operation. Figure 3 As shown, the specific steps include: Step (1): Construct a simplified physical model of the electric spindle and define the state vector.

[0035] Considering the complexity of the electric spindle system, it is first simplified to a 2-DOF (lumped parameter model) model that retains the core dynamic characteristics. For example... Figure 1 As shown, the main shaft is simplified as a rigid body, supported by two spring-damped units at the front and rear.

[0036] Define the system state vector: ;in, This represents the vertical displacement at the center of mass. The generalized momentum generated by the vertical translation. The rotational angular displacement generated by the center of mass. It is the generalized angular momentum generated by rotation. This refers to the local temperature at the front bearing. This represents the local temperature at the rear bearing. This state vector contains the minimum set of variables required to describe the mechanical motion and thermal state of the system.

[0037] Step (2): Construct a thermodynamic Hamiltonian neural network.

[0038] Construct a fully connected neural network whose input is the current or initial state vector. The output consists of temperature-dependent physical parameters. The specific network output... include: ; in, and These are the stiffnesses of the front and rear bearings, respectively. and These are the damping coefficients of the front and rear bearings, respectively.

[0039] The network uses tanh as the activation function to ensure that the output parameters are smooth and differentiable with respect to the input states. Unlike traditional end-to-end learning, the network in this invention only learns constitutive relations, leaving the dynamic evolution to the physical equations.

[0040] Step (3): Construct the core scalar function.

[0041] Based on the parameters output in step (2), construct the core scalar functions describing the system dynamics, including the Hamiltonian function characterizing the total mechanical energy of the system. Rayleigh dissipation function characterizing the mechanical energy dissipation rate of a system. Internal energy function characterizing the thermal state of a system With entropy function .

[0042] First, construct the Hamiltonian function. Let represent the total mechanical energy of the system. Based on the small angle assumption, the system's potential energy function is... for: ; in, and These are the distances from the center of mass to the front and rear bearings, respectively. The kinetic energy function of the system for: ; in, The total mass of the system including the spindle and bearings. and Let be the moments of inertia of the front and rear sections of the main shaft, respectively. Therefore, the total Hamiltonian function of the system is: ; Then, the Rayleigh dissipation function is constructed. To represent the rate of mechanical energy dissipation: ; The generalized velocities are defined as follows: ; Meanwhile, the linear velocities at the front and rear bearings can be calculated as follows: ; For the construction of the internal energy and entropy functions, the current assumption is that the specific heat capacity of the bearing assembly is... Since is a constant, according to thermodynamic relations, the equations for the two are as follows: ; in, The initial entropy of the system, This is the initial temperature of the system.

[0043] The total energy function of the system can then be written as: ; Step (4): Working thermo-dynamic coupling kinetic equations.

[0044] The thermodynamic evolution of the system follows the GENERIC equations: .

[0045] The equation consists of two sub-terms, the first of which... The second term is used to describe the change in the conservative energy field of the system. This is used to describe the irreversible part and provide a detailed expression of the thermo-coupling relationship. Therefore, the equation can also be written as: .

[0046] in, Represents the reversible process of the system. This represents the irreversible process represented by the damping dissipation and heat transfer of the system.

[0047] The invertible terms in the equation are represented by a Poisson matrix. To describe the symplectic geometric structure in conservative Hamiltonian dynamics, the matrix is ​​specifically expressed as: ; The first term in the GENERIC equation is calculated by substituting this matrix. The undamped Hamiltonian equation of motion for a recoverable system is written as: ; For non-invertible terms, a friction matrix needs to be constructed. The construction of this matrix must satisfy the degeneracy condition. This ensures that total energy is conserved, meaning the loss of mechanical energy equals the increase in internal energy. The friction matrix derived in this invention... It has the following block format: ; Each of these submatrices can be assigned a physical meaning.

[0048] matrix submatrix in For the degenerate matrix and submatrix during the translation process Let be the degenerate matrix and submatrix during the rotation process. The thermodynamic coupling matrix during translation, and its submatrix The thermo-mechanical coupling matrix during rotation, and its submatrix These are thermally non-invertible matrices. The specific form of each matrix is: ; This invention explicitly constructs the mechanism for the conversion of mechanical energy and internal energy during thermo-coupling in the friction matrix, so that the irreversible expression of the system comprises two parts: one is the damping force acting on momentum. Secondly, the temperature rise term acting on temperature. Simultaneously, the thermal coupling matrix and and thermally invertible matrices The structure ensures that: ; That is, the energy dissipated through damping is completely converted into heat energy, expressed as: ; Finally, the thermo-coupling state-space equation of this invention can be written as: ; Step (5): Network training and numerical integral prediction.

[0049] The specific architecture of the entire network is as follows: Figure 2 As shown.

[0050] For this network design, the loss function during training is first defined: ; in, For Hamiltonian dynamics error term loss, For the loss of thermal error term, These are the coefficients for the thermal error loss term. Specifically, the two loss terms can be written as: ; When training the network using actual or simulated data, if the network's loss function is less than 1e-6, then the thermodynamic Hamiltonian neural network training is complete. After training, in the application phase, the given initial state vector is input into the trained thermodynamic Hamiltonian neural network to predict the physical parameters in the current state. The parameters are then substituted into the derived equations, and the thermodynamic coupling dynamic equations are solved through numerical integration to obtain the system's thermodynamic evolution trajectory for subsequent prediction. Step (6): Vibration signal reconstruction and performance prediction.

[0051] After completing the network training in step (5), the thermodynamic Hamiltonian neural network is used to predict the dynamic response of the electric spindle under continuous machining conditions, and the system evolution trajectory obtained by numerical integration in step (5) is used. Extracting the generalized displacement term and This represents the instantaneous vibration signal of the electric principal shaft's centroid under thermo-mechanical coupling. The evolution trajectory obtained based on this method... This is the vibration prediction trajectory achieved by this method.

[0052] The prediction steps of this model are as follows: A. Given an initial state Determine the start time of prediction and set the initial state vector. The specific values ​​should include the initial mechanical displacement and momentum of the spindle, as well as the initial temperature distribution of the front and rear bearings.

[0053] B. Network Predicts and Outputs Current Parameters: The initial state vector is input into the trained network model, and the network outputs the predicted constitutive parameters at the current temperature. .

[0054] C. Calculate the gradient using a computational graph. Utilizing automatic differentiation techniques, a total energy function operator is first constructed within the computational graph. and total entropy function operator Then, based on the current state and the parameters predicted in step (2), the specific values ​​of total energy and total entropy are calculated. By calculating the partial derivatives of each operator with respect to the state vector, the energy gradient vector and entropy gradient vector are finally obtained.

[0055] D. Supplement the invertible partial Poisson matrix and the irreversible friction matrix Assemble the Poisson matrix The matrix is ​​a constant antisymmetric matrix, and its structure is determined by the symplectic geometric properties of the system and does not change with the state. The friction matrix is ​​supplemented using the damping coefficient predicted in step (2) and the gradient information calculated in step (3). ; Ensure the matrix Satisfying the degeneracy condition of the first law of thermodynamics, i.e., constructing the matrix. The thermodynamic coupling term in the equation allows the rate of dissipation of mechanical energy to be precisely equal to the rate of generation of internal energy.

[0056] E. Calculate the state derivative Calculate the reversible dynamics term and irreversible terms The total time derivative of the system is obtained by adding the two terms. .

[0057] F. Time step integration: Using Python, the initial value problem is solved through the Solve_ivp function in the Scipy library, and the predicted thermodynamic trajectory is finally obtained.

[0058] G. The full-cycle thermodynamic state trajectory obtained from the previous step. The corresponding generalized displacement components are extracted to reconstruct the spatial-temporal vibration signal of the electric spindle. Based on the reconstructed vibration information, the vibration performance of the electric spindle can be predicted and evaluated.

[0059] To verify the effectiveness of the invention, two sets of synthetic datasets with strong thermo-coupling effects were used for testing. The first set of data represents an ideal spring-damped system, while the second set represents a high-frequency heat-sensitive spindle system to simulate the high stiffness and high thermal sensitivity under real-world operating conditions.

[0060] Meanwhile, this invention was compared with the following four benchmark models: a) Neural ODE; b) Long Short-Term Memory Network (LSTM); c) Hamiltonian Neural Network (HNN); d) Dissipative Hamiltonian Neural Network (DHNN).

[0061] By calculating the mean squared error of the six state-space dimensions between the predicted and actual values ​​of each model, as well as the Hamiltonian evolution error of each system, it was found that in long-term prediction tasks, the purely data-driven models Neural ODE and LSTM failed to maintain physical constraints, leading to energy divergence or non-physical oscillations in the prediction results. While HNN and DHNN, which incorporate Hamiltonian information, exhibit a certain degree of stability, their isothermal assumption prevents them from capturing stiffness changes caused by frictional heating, resulting in significant phase and amplitude deviations in the predicted trajectory. Comparison results with other networks are shown below. Figures 4 to 8 As shown.

[0062] Figure 4 The results show a comparison of the method of this invention with other benchmark networks using error bars in the first set of data. The error of the method of this invention is significantly lower than that of other benchmark models. Figure 5 The results show the evolution of the prediction results of the method of the present invention and other benchmark networks for the qy value over time in the first set of data. The method of the present invention maintains the lowest prediction error and shows a continuous convergence trend. Figure 6 The graph shows the evolution of the prediction results of the method of this invention and other benchmark networks for the py value over time in the first set of data. Figure 7 The graph shows a comparison of the evolution of the prediction results of the method of this invention and other benchmark networks for the qθ value over time in the first set of data. Figure 8 The graph shows a comparison of the evolution of the prediction results of the method of this invention and other benchmark networks for the pθ value over time in the first set of data. Figure 9 The results show the evolution of Hamiltonian prediction errors of the method of this invention with four other benchmark networks over time in the first set of data. Figure 10 The results show the evolution of Hamiltonian prediction errors of the method of this invention with four other benchmark networks over time in the second set of data. Figure 11The results show the comparison between the method of the present invention and five other benchmark networks in terms of inference time and number of model parameters. The method of the present invention shows significant advantages in terms of number of parameters and various error indicators.

[0063] The proposed method for modeling and predicting the thermodynamic coupling performance of electric spindles not only accurately predicts the mechanical trajectory but also successfully reproduces the system's temperature evolution curve. In Hamiltonian energy error analysis, this invention demonstrates the correct energy decay trend, while other models exhibit energy drift or systematic errors.

[0064] In summary, this invention solves the problem of describing the thermodynamic coupling in digital twin modeling of electric principal axes by embedding non-equilibrium thermodynamic structures in neural networks, and achieves high-precision thermodynamic parameter identification and state prediction.

[0065] The embodiments described above provide a detailed explanation of the technical solutions and beneficial effects of the present invention. It should be understood that the above descriptions are merely specific embodiments of the present invention and are not intended to limit the present invention. Any modifications, additions, and equivalent substitutions made within the scope of the principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for predicting the thermo-mechanical coupling vibration of an electric principal shaft based on dissipative Hamiltonian dynamics, characterized in that, include: (1) Establish a simplified physical model based on the physical structure of the machine tool electric spindle, and define a state vector that includes mechanical and thermal states. ; (2) Construct a thermodynamic Hamiltonian neural network, the input of which is the current or initial state vector. The output is physical parameters; (3) Based on the physical parameters output in step (2), construct the core scalar functions describing the system dynamics, including the Hamiltonian function characterizing the total mechanical energy of the system. Rayleigh dissipation function characterizing the mechanical energy dissipation rate of a system. Internal energy function characterizing the thermal state of a system With entropy function And obtain the total energy function. Represented as ; (4) Use the automatic differentiation to determine the total energy function in step (3). Sum and entropy function Taking the derivative, we obtain the energy gradient vector. and entropy gradient vector By combining the degeneracy conditions in the GENERIC framework, we can construct the invertible and irreversible terms in the thermodynamic uniform evolution equation to obtain the state vector. The time derivative is used to obtain the thermo-coupling kinetic equations; (5) Use the thermodynamic coupling kinetic equations obtained in step (4) to train the thermodynamic Hamiltonian neural network described in step (2) so that the network learns the mapping relationship between the state vector and the physical parameters. After training, in the application phase, the given initial state vector is input into the trained thermodynamic Hamiltonian neural network to predict the physical parameters in the current state. The thermodynamic coupling kinetic equations are then solved by numerical integration to obtain the thermodynamic evolution trajectory of the system. ; (6) From the predicted trajectory The generalized displacement term is extracted and the spatial-temporal vibration signal of the electric spindle is reconstructed to identify the natural frequency drift phenomenon caused by temperature rise.

2. The method for predicting the thermo-mechanical coupling vibration of an electric principal shaft based on dissipative Hamiltonian dynamics according to claim 1, characterized in that, In step (1), the simplified physical model is a two-degree-of-freedom lumped parameter model, and the electric principal axis is regarded as a rigid body supported by two front and rear spring-damped units with thermosensitive properties; the state vector Defined as: ； in, This represents the vertical displacement at the center of mass. The generalized momentum generated by the vertical translation. The rotational angular displacement generated by the center of mass. The generalized angular momentum generated by rotation. This refers to the local temperature at the front bearing. This refers to the local temperature at the rear bearing.

3. The method for predicting the thermo-mechanical coupling vibration of an electric principal shaft based on dissipative Hamiltonian dynamics according to claim 1, characterized in that, In step (2), the physical parameter values ​​output by the thermodynamic Hamiltonian neural network include: the stiffness coefficient of the front bearing. stiffness coefficient of the rear bearing Damping coefficient of the front bearing Damping coefficient of the rear bearing .

4. The method for predicting the thermo-mechanical coupling vibration of an electric principal shaft based on dissipative Hamiltonian dynamics according to claim 1, characterized in that, In step (2), the thermodynamic Hamiltonian neural network uses the hyperbolic tangent function as the activation function to ensure that the higher-order derivatives of the output physical parameters with respect to the input state exist and are continuous.

5. The method for predicting the thermo-mechanical coupling vibration of an electric principal shaft based on dissipative Hamiltonian dynamics according to claim 1, characterized in that, In step (4), the thermo-coupling kinetic equation is: ； in, State vector Time derivative, and These are the reversible and irreversible terms in the thermodynamic uniform evolution equation, respectively. The Poisson matrix is ​​used to describe invertible dynamics. The friction matrix is ​​used to describe irreversible dynamics.

6. The method for predicting the thermo-mechanical coupling vibration of an electric principal shaft based on dissipative Hamiltonian dynamics according to claim 5, characterized in that, Poisson matrix It is a constant antisymmetric matrix, specifically expressed as: 。 7. The method for predicting thermo-mechanical coupling vibration of an electric principal shaft based on dissipative Hamiltonian dynamics according to claim 5, characterized in that, Friction Matrix Rayleigh dissipation function The non-zero blocks are derived from the damping dissipation terms of the front and rear bearings and the momentum-temperature coupling terms, ensuring that mechanical dissipation energy is converted into internal energy and that the entropy generation rate is non-negative.

8. The method for predicting thermo-mechanical coupling vibration of an electric principal shaft based on dissipative Hamiltonian dynamics according to claim 5, characterized in that, In step (4), the degeneracy condition in the GENERIC architecture is: .

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