A Method for Mirror Update of Virtual and Real Data of a Digital Twin Model of a Gear Transmission System

Abstract

The present invention discloses a method for mirror updating virtual and real data of a digital twin model of a gear transmission system, including: simplifying the gear transmission system to establish a dynamic differential equation of the gear transmission system using Newton's second law; solving the dynamic differential equation of the gear transmission system to obtain the vibration acceleration response under different working conditions; collecting the vibration acceleration signals on the bearing end cover of the gear transmission system with the same gear parameters, and performing fast Fourier transform processing and analysis on the signals; determining the type and quantity of update parameters, and constructing a multi-parameter optimization objective function for mirror updating of the simulation virtual model of the gear transmission system; using the particle swarm algorithm to solve the optimization objective function and update the simulation virtual model. The present invention has a certain solution accuracy for updating the simulation virtual model, and can also analyze the key parameters of the mirror updating of the virtual and real data between the simulation virtual model and the actual physical model from the solution process and results.

Description

Technical Field

[0001] The present invention belongs to the field of application research on digital twins of rotating machinery, and particularly relates to a method for mirroring and updating virtual and real data of a digital twin model of a gear transmission system. Background Art

[0002] Digital twins can not only be used for system modeling and simulation of system development to assist in designing or verifying system properties, but also support the optimization operation and fault prediction of operation and manufacturing services. The scenario where digital twins are most applied is the transmission system, and the most common one in the transmission system is the gear transmission system, which transmits power through one or more pairs of gear pairs to enable the operation of machines or machine components. If a virtual mirror model of the gear system can be established and the key parameters for updating the virtual model can be found, it will provide an idea for solving methods for actual digital twin cases of transmission systems.

[0003] At present, digital twin technology has great value in industrial production. For offshore wind power, a virtual model with high fidelity can be established for its drivetrain. After measuring the loads of the drivetrain, gear load and load response analysis can be performed simultaneously to obtain real-time operating conditions, thereby detecting problems in a timely manner and even predicting the occurrence of problems (Johansen, Sigrid S., and Nejad, et al. On Digital Twin Condition Monitoring Approach for Drivetrains in Marine Applications[C]. Proceedings of the ASME 2019 38th International Conference on Ocean, Offshore and Arctic Engineering. Volume 10: Ocean Renewable Energy. 2019: V010T09A013.). In large factories, considering the implementation cost and the complexity of digital twins, the digital twin-driven fault prediction and health management method has important application value for monitoring expensive and important equipment in the factory (Fei Tao, Meng Zhang, Yushan Liu, et al. Digital twin driven prognostics and health management for complex equipment[J]. CIRP Annals, 2018, 67(1): 169-172.). It can be seen that digital twins are meaningful when considering high-value assets with difficult-to-access locations. The benefits of overall life detection will help reduce maintenance costs and downtime, saving funds for asset owners (Qinglin Qi, Fei Tao, Tianliang Hu, et al. Enabling technologies and tools for digital twin[J]. Journal of Manufacturing Systems, 2019.). Digital twins can not only be used for system modeling and simulation of system development to help design or verify system properties, but also support the optimization operation and fault prediction of operation and manufacturing services (Schleich, B., N. Anwer, L. Mathieu, et al. Shaping the Digital Twin for Design and Production Engineering[J]. CIRP Annals, 2017, 66(1): 141–144.).Method and system for updating and maintaining digital twin model based on multi-source and multi-modal data (CN202111618605.3) are used to solve the technical problem that the digital twin model updated by the existing digital twin model updating method has a large error from the physical system. However, it does not specifically mention the update of the gear transmission system. It can be seen that the current research on the application of digital twin in rotating machinery is still immature, and the research on gear systems is even less. Summary of the Invention

[0004] The purpose of the present invention is to provide a method for mirror updating virtual and real data of a digital twin model of a gear transmission system in view of the blank of the existing technology. The gear transmission system is simplified into a discrete "mass-spring-damping" system by using the lumped parameter method, the dynamic characteristics of the gear transmission system are solved by using the complex modal analysis method, a multi-parameter optimization objective function of the gear transmission system is constructed, and the PSO algorithm is selected for solution to achieve mirror updating, with a certain solution accuracy and the key parameters for virtual and real mirror updating can be determined.

[0005] The purpose of the present invention is achieved by at least one of the following technical solutions.

[0006] A method for mirror updating virtual and real data of a digital twin model of a gear transmission system includes the following steps:

[0007] S1. Simplify the gear transmission system;

[0008] S2. Establish the dynamic differential equation of the gear transmission system by using Newton's second law;

[0009] S3. Solve the dynamic differential equation of the gear transmission system to obtain the vibration acceleration response under different working conditions;

[0010] S4. Collect the vibration acceleration signals on the bearing end cover of the gear transmission system with the same gear parameters;

[0011] S5. Perform fast Fourier transform processing and analysis on the collected vibration acceleration signals;

[0012] S6. Determine the type and quantity of update parameters, and construct a multi-parameter optimization objective function for mirror updating the simulation virtual model of the gear transmission system;

[0013] S7. Construct an improved particle swarm optimization (PSO) algorithm;

[0014] S8. Determine the parameters of the improved PSO algorithm;

[0015] S9. Initialize each particle in the PSO algorithm;

[0016] S10. Iteratively update the velocity and position of each particle, compare the historical best position of each particle with the current group best position, and solve and update the optimization objective function.

[0017] Further, in step S1, the gear transmission system is simplified into a discrete "mass-spring-damper" system. The motion relationship δ of a certain order of gear pair on the meshing line ij :

[0018] δ ij =|x i -x j |sinα + |y i -y j |cosα + θ i r bi -θ j r bj +e(t);

[0019] where i and j represent the input shaft gear and the output shaft gear respectively; x, y, and θ represent the displacements of the gear rotation center in the x and y directions and the rotation angle respectively; α is the pressure angle of the gear; r b is the base circle radius; e(t) is the comprehensive meshing error of the gear teeth.

[0020] Further, in step S2, according to the relative motion relationship of a certain order of gear pair in the meshing line direction, under a stable speed input and a certain load, the Newton's second law is used to write the system motion differential equation.

[0021] Substitute the motion relationship δ ij on the meshing line into the system motion differential equation. After calculation and arrangement, the system motion differential equation in matrix form can be obtained:

[0022]

[0023] where q is the system coordinate column matrix; and represent the system coordinate velocity column matrix and the system coordinate acceleration column matrix respectively; M is the system mass matrix; C is the system damping matrix; K is the system stiffness matrix; T is the system torque matrix; E is the system error excitation matrix.

[0024] Further, in step S3, considering the parts of the stiffness parameter and the damping parameter that change with time in the system, assuming that the gear meshing state is ideal, that is, the comprehensive meshing error of the gear teeth is set to zero, the motion differential equation is as follows:

[0025]

[0026] where q is the system coordinate column matrix; and respectively represent the system coordinate velocity array and the system coordinate acceleration array; M is the system mass matrix; C is the system damping matrix; K is the system stiffness matrix; T is the system torque matrix.

[0027] Write a program to solve this kinematic differential equation using the ode15s solver. Introduce auxiliary equations and state vectors, transform the kinematic differential equation, and substitute it into the solver. Set the length of the solution time, set the relative error tolerance and absolute error tolerance of the solver, and default the initial values of all independent variables to zero. By setting different input rotational speeds and load torques, run the program to solve the kinematic differential equation and obtain the dynamic response signals of the gear transmission system under different working conditions.

[0028] Further, step S4 specifically includes the following steps:

[0029] S4.1. Install sensors. Install piezoelectric acceleration sensors above the bearing end caps of the input and output shafts at all levels to measure the vibration acceleration signals in the x and y directions. The x and y directions correspond to the x and y directions of the simplified gear transmission system in step S1; correctly connect the sensors, computer, and data acquisition system.

[0030] S4.2. Set data acquisition parameters and collect signals. Set the total sampling duration to 1 to 10 s, set the sampling frequency to 12800 Hz to 51200 Hz, set the rotational speed of the input shaft and the load torque of the output shaft, and collect vibration signals.

[0031] Further, in step S5, signal processing and analysis. Use the fast Fourier transform to convert the continuous vibration response signal in the time domain into a discrete signal in the frequency domain, and analyze whether the main frequency components in the signal correspond to the meshing frequencies of the gear teeth at all levels under the set working conditions.

[0032] Further, in step S6, the multi-parameter optimization objective function for mirror updating of the gear transmission system simulation virtual model:

[0033] R(η 1 ,η 2 ,...,η n )=f D (η 1 ,η 2 ,...,η n )-f M (η 1 ,η 2 ,...,η n );

[0034] Among them, η 1 ,η 2 ,...,ηn For the update parameter, f D and f M are the response signals related to the update parameter of the simulation virtual model and the actual physical model respectively.

[0035] Then, regard the process of mirror update of the simulation virtual model as the process of solving the optimization problem of multi-parameter, and correctly list the objective function of the optimization problem:

[0036]

[0037] Among them, H is the column vector of variables selected as update parameters, H min and H max are the column vectors of the minimum and maximum values that the selected variables can take respectively, g i (H) is the inequality constraint condition, h j (H) is the equality constraint condition, i and j are used to distinguish different inequality constraint conditions and different equality constraint conditions respectively, k and m represent the total numbers of inequality constraint conditions and equality constraint conditions respectively, and R(H) is the multi-parameter optimization objective function.

[0038] Furthermore, in step S7, the PSO algorithm after performance improvement:

[0039]

[0040] Among them, i = 1, 2,..., N, and N is the total number of particles in the particle swarm. x i refers to the current position of the particle, and the dimension is n in S6. v i refers to the velocity of the particle, and each item corresponds to the velocity of the corresponding item in x i . Each particle contains its current position and velocity value; pbest is the optimal position of each particle in its own search history, and gbest is the optimal position that the group can search so far. c 1 is the weight coefficient for the particle to track its own historical optimal value, c 2 is the weight coefficient for the particle to track the group optimal value. rand() is a random number between 0 and 1. ω is the inertia weight factor and is a positive value.

[0041] Furthermore, in step S8, determine the parameters of the PSO algorithm after performance improvement. The parameters of the PSO algorithm after performance improvement include the setting of the number of independent variables of the objective function, that is, the determination of the dimension of the particle, the value range of each independent variable of the objective function; the inertia weight factor ω, the weight coefficient c 1 for the particle to track its own historical optimal value, the weight coefficient c 2 for the particle to track the group optimal value, the number of individuals in the particle swarm, and the number of iterations.

[0042] Further, in step S9, each particle in the PSO algorithm is initialized. Within the range of the independent variables of the objective function, the position and velocity values of each particle are randomly generated and substituted into the objective function respectively to calculate the initialized objective function value. The best position in the initialized particle swarm is obtained by comparing the objective function values and recorded.

[0043] Further, in step 10, the velocity and position of each particle are iteratively updated to solve the optimization objective function. The velocity of each particle is updated, and the particles whose velocities exceed the range are corrected; then the position of each particle is updated according to the algorithm formula, and the particles whose positions exceed the position range are corrected. Subsequently, the objective function value of each particle after the position update is calculated, and the historical best position of each particle itself and the current best position of the group are compared. Finally, the number of iterative updates is reached.

[0044] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0045] (1) A method for mirror updating virtual and real data of a digital twin model of a gear transmission system proposed by the present invention can not only realize the mirror updating of virtual and real data of the digital twin model of the gear transmission system, but also analyze the key parameters of the mirror updating of virtual and real data.

[0046] (2) Compared with the existing methods for mirror updating virtual and real data of a digital twin model, the present invention has a certain solution accuracy for the updating of the simulation virtual model.

[0047] (3) Compared with the existing methods for mirror updating virtual and real data of a digital twin model, the present invention can also analyze the key parameters of the mirror updating of virtual and real data between the simulation virtual model and the actual physical model from the solution process and solution results. Description of the Drawings

[0048] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required in the embodiments. The drawings form a part of this application, but are only non-limiting examples reflecting the inventive concept and are not used for any limitation.

[0049] Figure 1 is the flow chart of the method for mirror updating virtual and real data of the digital twin model of the gear transmission system in the method embodiment of the present invention;

[0050] Figure 2 is the solution flow chart of the PSO algorithm in the method of the present invention;

[0051] Figure 3 is the lumped parameter model diagram of the single-stage fixed-axis gear transmission system in Embodiment 1 of the present invention;

[0052] Figure 4 It is the dynamic response signal diagram under the input rotational speed of 1000 rpm and the output load of 150 N·m in Embodiment 1 of the present invention;

[0053] Figure 5 It is the vibration signal collected by the data acquisition system in Embodiment 1 of the present invention;

[0054] Figure 6 It is the diagram of converting the vibration signal collected by the data acquisition system into a frequency-domain signal in Embodiment 1 of the present invention;

[0055] Figure 7 It is the comparison diagram of the vibration responses before and after the update of the simulation virtual model and the vibration response signal of the actual physical model in Embodiment 1 of the present invention;

[0056] Figure 8 It is the comparison diagram of the vibration responses before and after the update of the simulation virtual model and the vibration response signal of the actual physical model in Embodiment 2 of the present invention;

[0057] Figure 9 It is the comparison diagram of the vibration responses before and after the update of the simulation virtual model and the vibration response signal of the actual physical model in Embodiment 3 of the present invention. Specific Embodiments

[0058] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, rather than all of the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0059] Embodiment 1

[0060] A method for mirror update of virtual and real data of a digital twin model of a gear transmission system in this embodiment is mainly used for updating parameters, where the input rotational speed accounts for 50% and the load torque accounts for 50%

[0061] Taking a single-stage fixed-axis gear transmission system in a normal state as an example, since the vibration characteristics of a single-stage fixed-axis gear transmission system in a normal state are jointly determined by the input rotational speed and the load torque, the percentage of the updated parameters in the objective function is divided into three groups. The first group is that the input rotational speed accounts for 50% and the load torque accounts for 50%. The second group is that the input rotational speed accounts for 80% and the load torque accounts for 20%. The third group is that the input rotational speed accounts for 20% and the load torque accounts for 80%. The key parameters of the virtual-real mirror update are analyzed by observing the iterative optimization speed of the optimization solution process and the deviation from the actual physical system working conditions after the optimization solution. The specific steps are as follows:

[0062] Step S1. Simplify the gear transmission system into a discrete "mass-spring-damper" system using the lumped parameter method, as shown in Figure 3 . The motion relationship δ of the gear pair on the meshing line is 12 :

[0063] δ 12 =(x 1 -x 2 )sinα+(y i -y j )cosα+θ 1 r b1 -θ 2 r b2 +e(t);

[0064] Among them, the subscripts 1 and 2 represent the input shaft gear and the output shaft gear respectively; x, y, and θ represent the displacements of the gear rotation center in the x and y directions and the rotation angle respectively; α is the pressure angle of the gear; r b is the base circle radius; e(t) is the comprehensive meshing error of the gear teeth.

[0065] Step S2. According to the relative motion relationship of a certain order of gear pair in the meshing line direction, under a stable rotational speed input and a certain load, use Newton's second law to write the system motion differential equation. For the input shaft gear:

[0066]

[0067] For the output shaft gear:

[0068]

[0069] Among them, m 1 , m 2 are the equivalent masses after simplification of the input shaft and the input shaft gear, and the output shaft and the output shaft gear respectively; I 1 , I 2 are the equivalent moments of inertia around the rotation axis after simplification of the input shaft and the input shaft gear, and the output shaft and the output shaft gear respectively; T in , T out are the input torque and the output torque respectively; k 12 and c 12 are the meshing stiffness and the meshing damping of the gear pair respectively; for the convenience of analysis, it is assumed that the stiffnesses and dampings of the input and output shaft bearings in the two directions are the same, which are k 1 and c 1 , k 2 and c 2 .

[0070] Substitute the motion relationship δ on the meshing line ijSubstituting it into the differential equations of motion of the system, after calculation and arrangement, the differential equations of motion of the system in matrix form can be obtained:

[0071]

[0072] where q is the column matrix of system coordinates; and represent the column matrix of system coordinate velocities and the column matrix of system coordinate accelerations respectively; M is the system mass matrix; C is the system damping matrix; K is the system stiffness matrix; T is the system torque matrix; E is the system error excitation matrix.

[0073] Step S3: Considering the parts of the stiffness parameters and damping parameters in the system that vary with time, assuming the gear meshing state is ideal, i.e., the comprehensive meshing error of gear teeth is set to zero, the differential equations of motion are as follows:

[0074]

[0075] Using MATLAB and the ode15s solver to solve this differential equation of motion. Introduce auxiliary equations and state vectors, and transform the differential equation of motion and substitute it into the solver. Set the solution time length to 0.6 s, and the relative error tolerance and absolute error tolerance of the solver to the default values, and default all initial values of independent variables to zero. By setting different input rotational speeds and load torques, run the program to solve the differential equation of motion, and obtain the dynamic response signals of the gear transmission system under different working conditions. The dynamic response signals at an input rotational speed of 1000 rpm and an output load of 150 N·m are as shown in Figure 4 a of Figure 4 and b of Figure 4 a of Figure 4 is the time-domain image of the response signal in the first 0.2 s. There is a transient response with a large amplitude before 0.1 s, and a gradually stable periodic signal after 0.1 s.

[0076] Step S4: Specifically includes the following steps:

[0077] S4.1: Install sensors. Install piezoelectric acceleration sensors above the bearing end caps of the input and output shafts of the single-stage fixed-axis gearbox respectively to measure the vibration acceleration signals in the x and y directions. The x and y directions correspond to the x and y directions of the simplified gear transmission system in Step S1; correctly connect the sensors, computer, and data acquisition system.

[0078] S4.2. Set data acquisition parameters and acquire signals. The total sampling duration is set to 10 s, the sampling frequency is set to 51200 Hz, the rotational speed of the input shaft and the load torque of the output shaft are set, and vibration signal acquisition is performed. As Figure 5 shown.

[0079] Step S5. Use the fast Fourier transform to convert the continuous vibration response signal in the time domain into a discrete signal in the frequency domain, and analyze whether the main frequency components in the signal correspond to the meshing frequencies of each stage of gear teeth under the set working conditions. As Figure 6 shown.

[0080] Step S6. Determine the type and quantity of update parameters, and establish the multi-parameter optimization objective function for mirror update of the gear transmission system simulation virtual model:

[0081] R(n,T) = f D (n,T) - f M (n,T);

[0082] where n is the input rotational speed, T is the load torque, both n and T are used as update parameters, f D and f M are the response signals related to the update parameters of the simulation virtual model and the actual physical model respectively.

[0083] Then, regard the process of mirror update of the simulation virtual model as a process of solving the multi-parameter optimization problem, and correctly write out the objective function of the optimization problem:

[0084]

[0085] where H is the column vector of variables selected as update parameters, H min and H max are the column vectors of the minimum and maximum values that the selected variables can take respectively, g i (H) is the inequality constraint condition, h j (H) is the equality constraint condition, i and j are used to distinguish different inequality constraint conditions and different equality constraint conditions respectively, k and m represent the total number of inequality constraint conditions and equality constraint conditions respectively, and R(H) is the multi-parameter optimization objective function.

[0086] Step S7. Construct the PSO algorithm with improved performance. The algorithm form is:

[0087]

[0088] where i = 1, 2,..., N, and N is the total number of particles in the particle swarm. x i refers to the current position of the particle, the dimension is n in S6, v irefers to the velocity of the particle, where each term corresponds to the velocity of the corresponding term in x i The velocity of the corresponding term in i . Each particle contains its current position and velocity value, that is, the values of n and T themselves and the change rates of n and T. pbest is the optimal position of each particle in its own search history, and gbest is the optimal position that the population can search so far. c 1 is the weight coefficient for the particle to track its own historical optimal value, c 2 is the weight coefficient for the particle to track the optimal value of the population. rand() is a random number between 0 and 1. ω is the inertia weight factor and is a positive value.

[0089] Step S8: Determine the parameters of the PSO algorithm after performance improvement. This includes setting the number of independent variables of the objective function, that is, determining the dimension of the particle; determining the value range of each independent variable of the objective function; determining ω, c 1 and c 2 ; determining the number of individuals in the particle swarm and the number of iterations required for the algorithm program. The algorithm parameter settings are shown in Table 4.

[0090] Table 4 Algorithm Parameter Settings

[0091] Algorithm parameters Number of parameters Number of particles Number of iterations <![CDATA[c 1 > <![CDATA[c 2 > ω Set value 2 8 18 2 2 0.5

[0092] Step S9: Initialize each particle in the PSO algorithm. Within the range of the independent variables of the objective function, write a program to randomly generate the position and velocity values of each particle, substitute them into the objective function respectively to calculate the initialized objective function value, and find the one with the best position in the initialized particle swarm by comparing the objective function values, and record it.

[0093] Step S10: Iteratively update the velocity and position of each particle to solve the optimization objective function. Update the velocity of each particle according to the algorithm formula, and correct the particles whose velocities exceed the range; then update the position of each particle according to the algorithm formula, and correct the particles whose positions exceed the position range. Subsequently, calculate the objective function value of each particle after updating the position, and compare to update the best position in each particle's own history and the current best position of the population. Finally, when the number of iterations is reached, complete the solution process. The comparison of the vibration responses before and after the update of the simulation virtual model and the vibration response signal of the actual physical model is as Figure 7 shown in a of Figure 7 and b of Figure 7 In a of Figure 7 before the update, the comparison between the actual signal and the simulation virtual model signal shows that there is a large difference in their states.

[0094] Example 2: In the updated parameters, the input rotational speed accounts for 80% and the load torque accounts for 20%

[0095] Taking the single-pole fixed-axis gear transmission system in the normal state as an example, when the input speed in the updated parameters accounts for 80% and the load torque accounts for 20%.

[0096] Step S1: Simplify the gear transmission system into a discrete "mass-spring-damper" system, the same as in Embodiment 1.

[0097] Step S2: According to the relative motion relationship of a certain pair of gears in the meshing line direction, under a stable speed input and a certain load, use Newton's second law to write the system motion differential equation. The input shaft and the output shaft are the same as in Embodiment 1.

[0098] Substitute the motion relationship δ ij into the system motion differential equation. After calculation and arrangement, the system motion differential equation in matrix form can be obtained:

[0099]

[0100] where q is the system coordinate column matrix; and respectively represent the system coordinate velocity column matrix and the system coordinate acceleration column matrix; M is the system mass matrix; C is the system damping matrix; K is the system stiffness matrix; T is the system torque matrix; E is the system error excitation matrix. The same as in Embodiment 1.

[0101] In Step S3, considering the parts of the stiffness parameters and damping parameters in the system that change with time, assuming the gear meshing state is ideal, that is, the comprehensive meshing error of the gear teeth is set to zero, the motion differential equation is as follows:

[0102]

[0103] Use MATLAB to write a program and use the ode15s solver to solve this motion differential equation. Introduce auxiliary equations and state vectors, and transform the motion differential equation and substitute it into the solver. Set the solution time length to 0.6 s, and the relative error tolerance and absolute error tolerance of the solver are the default values. Default all initial values of independent variables to zero. By setting different input speeds and load torques, run the program to solve the motion differential equation to obtain the dynamic response signals of the gear transmission system under different working conditions. The dynamic response signals under an input speed of 1000 rpm and an output load of 150 N·m are the same as in Embodiment 1.

[0104] Step S4: Specifically includes the following steps:

[0105] S4.1. Install sensors. Install piezoelectric acceleration sensors above the bearing end caps of the input and output shafts of the single-pole fixed-axis gearbox respectively, which are used to measure the vibration acceleration signals in the x and y directions. The x and y directions correspond to the x and y directions of the simplified gear transmission system in step S1. Correctly connect the sensors, computer and data acquisition system.

[0106] S4.2. Set data acquisition parameters and collect signals. Set the total sampling duration to 1 s to 10 s, set the sampling frequency to 51,200 Hz, set the rotational speed of the input shaft and the load torque of the output shaft, and collect vibration signals. The same as in Embodiment 1.

[0107] Step S5. Use the fast Fourier transform to convert the continuous vibration response signal in the time domain into a discrete signal in the frequency domain, and analyze whether the main frequency components in the signal correspond to the meshing frequencies of each stage of gear teeth under the set working conditions. The same as in Embodiment 1.

[0108] Step S6. Determine the type and quantity of update parameters, and establish an optimization objective function with multiple parameters for mirror update of the gear transmission system simulation virtual model. The same as in Embodiment 1.

[0109] Then regard the process of mirror update of the simulation virtual model as a process of solving an optimization problem with multiple parameters, and correctly write out the objective function of the optimization problem. The same as in Embodiment 1.

[0110] Step S7. Construct a PSO algorithm with improved performance. The algorithm form is:

[0111]

[0112] where \(i = 1, 2, \cdots, N\), and \(N\) is the total number of particles in the particle swarm. \(x\) i refers to the current position of the particle, and the dimension is \(n\) in S6. \(v\) i refers to the velocity of the particle, and each item corresponds to the velocity of the corresponding item in \(x\) i Each particle contains its current position and velocity value, that is, the values of \(n\) and \(T\) themselves and the change rates of \(n\) and \(T\). \(pbest\) is the optimal position of each particle in its own search history, and \(gbest\) is the optimal position that the population can search so far. \(c\) 1 is the weight coefficient for the particle to track its own historical optimal value, and \(c\) 2 is the weight coefficient for the particle to track the optimal value of the population. \(rand()\) is a random number between 0 and 1. \(\omega\) is the inertia weight factor and is a positive value.

[0113] Step S8. Determine the parameters of the PSO algorithm with improved performance. It includes setting the number of independent variables of the objective function, that is, determining the dimension of the particle; determining the value range of each independent variable of the objective function; determining \(\omega\), \(c\)1 , c 2 ; Determine the number of individuals in the particle swarm and the number of iterations required for the algorithm program. The same as in Embodiment 1.

[0114] Step S9: Initialize each particle in the PSO algorithm. Within the range of the independent variables of the objective function, write a program to randomly generate the position and velocity values of each particle, substitute them into the objective function respectively to calculate the initialized objective function value, and obtain the particle with the best position in the initialized particle swarm by comparing the objective function values, and record it.

[0115] Step S10: Iteratively update the velocity and position of each particle to solve the optimization objective function. Update the velocity of each particle according to the algorithm formula, and correct the particles whose velocity exceeds the range; then update the position of each particle according to the algorithm formula, and correct the particles whose position exceeds the position range. Subsequently, calculate the objective function value of each particle after updating the position, and compare to update the historical best position of each particle itself and the current best position of the group. Finally, when the number of iterative updates is reached, the solution process is completed.

[0116] Iteratively update the velocity and position of each particle to solve the optimization objective function: Update the velocity of each particle, and correct the particles whose velocity exceeds the range; then update the position of each particle, and correct the particles whose position exceeds the position range. Subsequently, calculate the objective function value of each particle after updating the position, and compare to update the historical best position of each particle itself and the current best position of the group. Finally, reach the number of iterative updates.

[0117] Comparison of the vibration responses before and after the virtual model update with the vibration response signals of the actual physical model, as Figure 8 shown in a of Figure 8 and b of Figure 8 a of Figure 8 is the comparison between the actual signal and the simulation virtual model signal before the update. It can be seen that the states of the two are quite different.

[0118] Embodiment 3: The input rotational speed accounts for 20% and the load torque accounts for 80% in the updated parameters

[0119] Taking the single-pole fixed-axis gear transmission system in the normal state as an example, when the input rotational speed accounts for 20% and the load torque accounts for 80% in the updated parameters.

[0120] Step S1: Simplify the gear transmission system into a discrete "mass-spring-damper" system. The same as in Embodiment 1.

[0121] Step S2: Based on the relative motion relationship of a certain pair of gears in the direction of the meshing line, under a stable rotational speed input and a certain load, apply Newton's second law to write the differential equation of the system's motion. The input shaft and the output shaft are the same as in Embodiment 1.

[0122] Substitute the motion relationship δ on the meshing line ij into the differential equation of the system's motion. After calculation and arrangement, the differential equation of the system's motion in matrix form can be obtained:

[0123]

[0124] where q is the system coordinate column matrix; and represent the system coordinate velocity column matrix and the system coordinate acceleration column matrix respectively; M is the system mass matrix; C is the system damping matrix; K is the system stiffness matrix; T is the system torque matrix; E is the system error excitation matrix. The same as in Embodiment 1.

[0125] Step S3: Consider the parts of the stiffness parameters and damping parameters in the system that vary with time. Assume the gear meshing state is ideal, that is, the comprehensive meshing error of the gear teeth is set to zero. The differential equation of the motion is as follows:

[0126]

[0127] Use MATLAB to write a program and use the ode15s solver to solve this differential equation of the motion. Introduce auxiliary equations and state vectors, and transform the differential equation of the motion and substitute it into the solver. Set the solution time length to 0.6 s, and the relative error tolerance and absolute error tolerance of the solver to the default values. Default all initial values of the independent variables to zero. By setting different input rotational speeds and load torques, run the program to solve the differential equation of the motion to obtain the dynamic response signals of the gear transmission system under different working conditions. The dynamic response signals at an input rotational speed of 1000 rpm and an output load of 150 N·m are the same as in Embodiment 1.

[0128] Step S4: Specifically includes the following steps:

[0129] S4.1: Install sensors. Install piezoelectric acceleration sensors above the bearing end caps of the input and output shafts of the single-pole fixed-axis gearbox respectively to measure the vibration acceleration signals in the x and y directions. The x and y directions correspond to the x and y directions of the simplified gear transmission system in Step S1; correctly connect the sensors, the computer and the data acquisition system.

[0130] S4.2. Set data acquisition parameters and acquire signals. The total sampling duration is set to 10 s, the sampling frequency is set to 12,800 Hz to 51,200 Hz. Set the rotational speed of the input shaft and the load torque of the output shaft, and then acquire vibration signals. The same as in Embodiment 1.

[0131] Step S5. Use the fast Fourier transform to convert the continuous vibration response signals in the time domain into discrete signals in the frequency domain, and analyze whether the main frequency components in the signals correspond to the meshing frequencies of each stage of gear teeth under the set working conditions. The same as in Embodiment 1.

[0132] Step S6. Determine the types and quantities of the update parameters, and establish a multi-parameter optimization objective function for the mirror update of the gear transmission system simulation virtual model. The same as in Embodiment 1.

[0133] Then regard the process of mirror updating the simulation virtual model as a process of solving a multi-parameter optimization problem, and correctly write out the objective function of the optimization problem. The same as in Embodiment 1.

[0134] Step S7. Construct a PSO algorithm with improved performance, in the form of:

[0135]

[0136] where \(i = 1, 2, \cdots, N\), and \(N\) is the total number of particles in the particle swarm. \(x\) i refers to the current position of the particle, and the dimension is \(n\) in S6. \(v\) i refers to the velocity of the particle, and each term corresponds to the velocity of the corresponding term in \(x\) i . Each particle contains its current position and velocity value, that is, the values of \(n\) and \(T\) themselves and the change rates of \(n\) and \(T\). \(pbest\) is the optimal position of each particle in its own search history, and \(gbest\) is the optimal position that the swarm can search so far. \(c\) 1 is the weight coefficient for the particle to track its own historical optimal value, and \(c\) 2 is the weight coefficient for the particle to track the optimal value of the swarm. \(rand()\) is a random number between 0 and 1. \(\omega\) is the inertia weight factor and is a positive value.

[0137] Step S8. Determine the parameters of the PSO algorithm with improved performance. This includes setting the number of independent variables of the objective function, that is, determining the dimension of the particle; determining the value range of each independent variable of the objective function; determining \(\omega\), \(c\) 1 , \(c\) 2 ; determining the number of individuals in the particle swarm and the number of iterations that the algorithm program needs to perform. The same as in Embodiment 1.

[0138] Step S9: Initialize each particle in the PSO algorithm. Within the range of the independent variables of the objective function, write a program to randomly generate the position and velocity values of each particle, substitute them into the objective function respectively to calculate the initialized objective function value, obtain the particle with the best position in the initialized particle swarm by comparing the objective function values, and record it.

[0139] Step S10: Iteratively update the velocity and position of each particle to solve the optimization objective function. Update the velocity of each particle according to the algorithm formula, and correct the particles whose velocities exceed the velocity range; then update the position of each particle according to the algorithm formula, and correct the particles whose positions exceed the position range. Subsequently, calculate the objective function value of each particle after updating the position, and update each particle's own historical best position and the current swarm best position by comparison. Finally, when the number of iterative updates is reached, the solution process is completed. Compare the vibration responses before and after the update of the simulation virtual model with the vibration response signals of the actual physical model, as Figure 9 shown in a of Figure 9 shown in b of Figure 9 a before the update is the comparison between the actual signal and the simulation virtual model signal, and it can be seen that the states of the two are quite different. Figure 9 b after the update is the comparison between the actual signal and the simulation virtual model signal, and it can be seen that the states of the two are relatively close.

[0140] Put together the results of three groups with different percentages of the updated parameters in the objective function for comparison. Analyze the key parameters of the virtual-real mirror image update by observing the deviation between the optimized solution and the actual physical system working conditions. As shown in Table 5.

[0141] Table 5 Record of Deviation between Simulation and Actual Working Conditions

[0142] Parameter Solution result Actual deviation First group of input speeds 975 rpm 2.5％ First group of load torques 159 N·m 6.0％ Second group of input speeds 990 rpm 1.0％ Second group of load torques 181 N·m 20.7％ Third group of input speeds 990 rpm 1.0％ Third group of load torques 184 N·m 22.7％

[0143] As can be seen from the above table, the proportion of the updated parameters in the objective function affects the solution result. The group with the smallest error can be controlled within 10%. Comparing the three groups, the load torque has a greater impact. It can be said that both the input speed and the load torque are the key parameters for the virtual-real mirror image update of the single-stage fixed-axis gear transmission system, and the proportion of the two key parameters in the objective function will affect the solution result. At the same time, the simulation results also show that this method has a certain solution accuracy.

[0144] The above is only a preferred embodiment of the present invention, but the protection scope of the present invention is not limited thereto. Any person skilled in the art within the scope disclosed by the present invention, according to the technical solution and inventive concept of the present invention, makes equivalent substitutions or changes, all belong to the protection scope of the present invention.

Claims

1. A method for mirror updating virtual and real data of a digital twin model of a gear transmission system, characterized in that, it includes the following steps: S1. Simplify the gear transmission system; S2. Use Newton's second law to establish the dynamic differential equation of the gear transmission system; S3. Solve the dynamic differential equation of the gear transmission system to obtain the vibration acceleration response under different working conditions; S4. Collect the vibration acceleration signals on the bearing end covers of the gear transmission system with the same gear parameters; S5. Perform fast Fourier transform processing and analysis on the collected vibration acceleration signals; use the fast Fourier transform to convert the continuous vibration response signals in the time domain into discrete signals in the frequency domain, and analyze whether the frequency components in the signals correspond to the meshing frequencies of each stage of gear teeth under the set working conditions; S6. Determine the type and quantity of update parameters, and construct a multi-parameter optimization objective function for mirror updating of the simulation virtual model of the gear transmission system; the multi-parameter optimization objective function R for mirror updating of the simulation virtual model of the gear transmission system is: R(η 1 ,η 2 ,...,η n ) = f D (η 1 ,η 2 ,...,η n ) - f M (η 1 ,η 2 ,...,η n ); Among them, η 1 , η 2 ,..., η n are update parameters, f D and f M are the response signals related to the update parameters of the simulation virtual model and the actual physical model respectively; Then regard the process of mirror updating the simulation virtual model as a process of solving a multi-parameter optimization problem, and obtain the objective function of the optimization problem: where, H is a column vector of variables selected as update parameters, H min and H max are column vectors of the minimum and maximum values that the selected variables can take, respectively, g i (H) is the inequality constraint condition, h j (H) is the equality constraint condition, i and j are used to distinguish different inequality constraint conditions and different equality constraint conditions, respectively, k and m represent the total numbers of inequality constraint conditions and equality constraint conditions, respectively, and R(H) is the multi-parameter optimization objective function; S7. Construct the improved particle swarm optimization algorithm PSO: where \(i = 1, 2, \cdots, N\), and \(N\) is the total number of particles in the particle swarm; \(x\) i refers to the current position of the particle, with the dimension being \(n\) in \(S6\), and \(v\) i refers to the velocity of the particle, where each term corresponds to the velocity of the corresponding term in \(x\) i Each particle contains its current position and velocity value; pbest is the optimal position of each particle in its own search history, and gbest is the optimal position that the swarm has been able to search for so far. \(c\) 1 is the weight coefficient for the particle to track its own historical optimal value, and \(c\) 2 is the weight coefficient for the particle to track the swarm's optimal value. rand() is a random number between 0 and 1, and \(\omega\) is the inertia weight factor, which is a positive value; S8. Determine the parameters of the improved PSO algorithm; S9. Initialize each particle in the PSO algorithm; S10. Iteratively update the velocity and position of each particle, compare the best position in the history of each particle itself with the current best position of the group, and solve and update the optimization objective function.

2. The method for mirror updating virtual and real data of a digital twin model of a gear transmission system according to claim 1, characterized in that, In step S1, the gear transmission system is simplified into a discrete mass-spring-damper system by using the lumped parameter method; the motion relationship δ of a certain order gear pair on the meshing line ij : δ ij = |x i - x j | sinα + |y i - y j | cosα + θ i r bi - θ j r bj + e(t); where, i and j represent the input shaft gear and the output shaft gear respectively; x, y and θ represent the displacements of the gear rotation center in the x and y directions and the rotation angle; α is the pressure angle of the gear; r b is the base circle radius; e(t) is the comprehensive meshing error of the gear teeth.

3. The method for mirror updating virtual and real data of a digital twin model of a gear transmission system according to claim 1, characterized in that, In step S2, according to the relative motion relationship of a certain stage of gear pair in the meshing line direction, under a stable rotational speed input and load, use Newton's second law to establish the system motion differential equation; Substitute the motion relationship δ ij into the system's differential equation of motion to obtain the system's differential equation of motion in matrix form: where q is the system coordinate array; and represent the system coordinate velocity array and the system coordinate acceleration array respectively; M is the system mass matrix; C is the system damping matrix; K is the system stiffness matrix; T is the system torque matrix; E is the system error excitation matrix.

4. The method for mirror updating virtual and real data of a digital twin model of a gear transmission system according to claim 1, characterized in that, In step S3, consider the part of the stiffness parameter and damping parameter in the system that changes with time, assume that the comprehensive meshing error of gear teeth is set to zero, and the motion differential equation is as follows: where q is the system coordinate array; and respectively represent the system coordinate velocity array and the system coordinate acceleration array; M is the system mass matrix; C is the system damping matrix; K is the system stiffness matrix; T is the system torque matrix; Use the ode15s solver to solve the differential equation of motion. Introduce the auxiliary equation and the state vector, transform the differential equation of motion, and substitute it into the solver; set the length of the solution time, set the relative error tolerance and absolute error tolerance of the solver, and default the initial values of all independent variables to zero; by setting different input rotational speeds and load torques, run the program to solve the differential equation of motion, and obtain the dynamic response signal f of the gear transmission system under different working conditions M .

5. The method for mirror updating virtual and real data of a digital twin model of a gear transmission system according to claim 2, characterized in that, Step S4 specifically includes the following steps: S4.

1. Install piezoelectric acceleration sensors above the bearing end covers of each input and output shaft respectively, for measuring the vibration acceleration signals in the x and y directions, and the x and y directions correspond to the x and y directions of the simplified gear transmission system in step S1; connect the sensors, computer and data acquisition system; S4.

2. Set the total sampling duration and sampling frequency, set the rotational speed of the input shaft and the load torque of the output shaft, and perform vibration signal acquisition.

6. The method for mirror updating virtual and real data of a digital twin model of a gear transmission system according to claim 1, characterized in that, In step S8, determine the parameters of the PSO algorithm after performance improvement. The parameters of the PSO algorithm after performance improvement include the setting of the number of independent variables of the objective function, that is, the determination of the particle dimension and the value range of each independent variable of the objective function; the inertia weight factor ω, the weight coefficient c for the particle to track its own historical optimal value 1 , the weight coefficient c for the particle to track the global optimal value of the population 2 , the number of individuals in the particle swarm and the number of iterations.

7. A method for mirroring and updating virtual and real data of a digital twin model of a gear transmission system according to any one of claims 1 to 6, characterized in that, in step S9, each particle in the PSO algorithm is initialized: within the range of the independent variable of the objective function, the position and velocity values of each particle are randomly generated and substituted into the objective function respectively to calculate the initialized objective function value. The best position in the initialized particle swarm is obtained by comparing the objective function values and recorded.

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