A fast analysis method for local nonlinearity of gear transmission system

Abstract

The present invention discloses a method for rapid local nonlinear analysis of a gear transmission system, which relates to the field of dynamic calculation of a gear transmission system and includes the following steps: treating the bearings of the gear transmission system as linear elastic supports to construct the dynamic equations of the gear transmission system; utilizing nonlinear interface separation technology to separate the linear and nonlinear parts of the gear transmission system; performing model condensation and dimensionality reduction on the linear part of the gear transmission system, and calculating the nonlinear load and comprehensive equivalent load for the nonlinear part; and utilizing the Newmark‑β method to perform numerical iterative solution on the dynamic equations after condensation and dimensionality reduction to obtain the dynamic response of the gear transmission system. The present invention has significant static deformation characteristics and is very suitable for dynamic solution of gear transmission systems. It can meet the accuracy requirements by using a lower truncation order and has higher computational efficiency than the traditional modal superposition method. It can serve the service performance evaluation of gear transmission systems in major equipment.

Description

Technical Field

[0001] The present invention relates to the field of dynamic calculation of gear transmission systems, and in particular to a local nonlinear rapid analysis method for gear transmission systems. Background Art

[0002] The dynamic modeling of gear transmission systems is developing towards flexibility, multi-level and complexity, which puts higher requirements on the efficiency of solving dynamic models. Xu Huachao et al. [1] used the substructure modal synthesis method to reduce the super unit of the main reducer of a helicopter and analyzed the influence of the flexibility of the box on the dynamic characteristics of the main reducer. Cappellini et al. [2] proposed a parameterized modal dimension reduction method to solve the contact problem in gear meshing, and compared this method with the traditional modal dimension reduction method and the finite element method. The results showed that this method can obtain higher calculation accuracy while maintaining calculation efficiency. Huangfu et al. [3] used the modal superposition method to reduce the dimension of the flexible gear-rotor system, and verified the model using simulation and experimental cases. On this basis, the excitation mechanism of the axial excitation of the helical gear on the pitch diameter vibration of the flexible body of the helical gear was analyzed.

[0003] Summarizing the existing public technologies, we can draw the following two deficiencies:

[0004] (i) Currently, most dimensionality reduction algorithms for gear transmission system dynamic models are based on linearization assumptions. However, the gear and bearing interfaces in gear transmission systems are typically nonlinear. Currently, there is a lack of fast-response calculation methods that consider the nonlinear interfaces of gear transmission systems.

[0005] (ii) Model dimensionality reduction methods are often used to improve computational efficiency. A common model reduction method is the modal superposition method. However, the modal superposition method only considers the system's vibration mode shape and does not take into account the deformation information experienced by the system, resulting in a slow convergence rate.

[0006] An efficient and high-precision dynamic model is the prerequisite for dynamic damage evolution analysis. Full life cycle damage evolution analysis requires iterative solution of the dynamic response, that is, the dynamic response needs to be repeatedly calculated at different degradation stages, which places high demands on the computational efficiency of the dynamic model.

[0007] Therefore, those skilled in the art are committed to developing a fast analysis method for local nonlinearity of a gear transmission system.

[0008] References:

[0009] [1] Xu Huachao, Qin Datong, Liu Changzhao, et al. Analysis of vibration characteristics of helicopter main reducer considering structural flexibility[J]. Journal of Aerospace Power, 2019, 34(5): 1020-1028.

[0010] [2]Cappellini N,Tamarozzi T,Blockmans B,et al.,Semi-analytic contacttechnique in anon-linear parametric model order reduction method for gearsimulations[J].Meccanica,2018,53:49-75.

[0011] [3]Huangfu YF, Zeng J, Ma H, et al. A flexible-helical-geared rotordynamic model based on hybrid beam-shell elements[J]. Journal of Sound andVibration, 2021,511:116361.1-21. Summary of the Invention

[0012] In view of the above-mentioned defects of the prior art, the technical problem to be solved by the present invention is that the current dimensionality reduction algorithms of the gear transmission system dynamics model are mostly based on linearization assumptions and cannot take into account the nonlinear characteristics of the gear and bearing interface in the gear transmission system.

[0013] To achieve the above object, the present invention provides a method for rapid local nonlinear analysis of a gear transmission system, characterized in that the method comprises the following steps:

[0014] S101: Considering the bearing of the gear transmission system as a linear elastic support, constructing a first dynamic equation of the gear transmission system;

[0015] S103: Separating the linear part and the nonlinear part of the gear transmission system by using a nonlinear interface separation technology;

[0016] S105: performing model condensation and dimensionality reduction on the linear part of the gear transmission system;

[0017] S107: Calculating a nonlinear load and a comprehensive equivalent load for the nonlinear part in the gear transmission system;

[0018] S109: performing numerical iteration on the second dynamic equation after condensation and dimensionality reduction using the Newmark-β method to obtain a dynamic response of the gear transmission system;

[0019] S111: The effectiveness of the fast analysis method for gear transmission system dynamics and local nonlinearity is verified through comparative analysis of dynamic responses between simulation and experiment.

[0020] Furthermore, in S101, the first dynamic equation of the gear transmission system is:

[0021]

[0022] Where M is the overall mass matrix, C(t) is the overall damping matrix, K(t) is the time-varying overall stiffness matrix, and F T is the static torque, F E (t) is the load vector corresponding to the displacement excitation, X is the displacement variable, and t is the time variable;

[0023] The overall damping matrix C(t) is the sum of proportional damping and meshing damping:

[0024] C(t)=α D M+β D K(t)+C m (t),

[0025]

[0026] Where, α D is the mass damping coefficient, β D is the stiffness damping coefficient, C m (t) is the meshing damping matrix, f n1 、f n2 are the first and second order natural frequencies, ξ1 and ξ2 are the first and second order modal damping ratios, c(t) is the time-varying meshing damping, T m is the transformation matrix of the meshing element, V m is the meshing projection vector;

[0027] The time-varying overall stiffness matrix K(t) is:

[0028] K(t)=K b +K shaft +K m (t),

[0029]

[0030] V m =[-sinψ,cosψ,0,0,0,-r b1 ,sinψ,-cosψ,0,0,0,-r b2 ],

[0031] Where K b is the bearing stiffness matrix, Kshaft is the stiffness matrix of the shaft, K m (t) is the meshing stiffness matrix, k(t) is the time-varying integrated meshing stiffness, ψ is the angle between the meshing plane and the y-axis, r b1 is the base circle radius of the driving wheel, r b2 is the base circle radius of the driven wheel;

[0032] Displacement excitation F E The load vector corresponding to (t) is:

[0033]

[0034] Where, e(t) is the no-load transfer error, is the derivative of the no-load transfer error.

[0035] Furthermore, in said S103, the meshing stiffness matrix and the meshing damping matrix are decomposed into a constant part and a fluctuating part according to the time-varying characteristics of the gear interface, wherein:

[0036]

[0037] Where, is the constant part of the meshing stiffness matrix, is the fluctuating part of the meshing stiffness matrix; is the constant part of the meshing damping matrix, is the fluctuating part of the meshing damping matrix.

[0038] Furthermore, after separating the linear part and the nonlinear part in the gear transmission system, the equivalent dynamic equation of the gear transmission system is:

[0039]

[0040] Among them, C eff is the equivalent damping matrix, which no longer contains the wave component:

[0041]

[0042] K eff is the equivalent stiffness matrix, which no longer contains the wave component:

[0043]

[0044] N eff is the equivalent load:

[0045]

[0046] Where, F NL (t) is the load vector representing the nonlinear bearing force.

[0047] Furthermore, the S105 includes the following steps:

[0048] S1051: Use the modal superposition method to build an overall framework for dimensionality reduction. Using the modal shape matrix as the basis, the physical coordinates are converted to modal coordinates to achieve model dimensionality reduction.

[0049] S1052: Obtain a static load Ritz basis matrix, where the static load Ritz basis matrix can achieve diagonalization of the mass matrix and the stiffness matrix;

[0050] S1053: Using the static Ritz basis matrix as a basis, perform condensation and dimensionality reduction.

[0051] Furthermore, in S1051, the modal vibration matrix is ​​obtained by performing generalized eigenvalue analysis on the equivalent stiffness matrix and the overall mass matrix, and then the second dynamic equation after model reduction and dimensionality reduction is obtained:

[0052]

[0053] Where q is the modal displacement in modal coordinates, Modal velocity in modal coordinates, is the modal acceleration in modal coordinates is the mode shape matrix;

[0054] M r is the reduced mass matrix in modal coordinates,

[0055] C r is the reduced damping matrix in modal coordinates,

[0056] K r is the reduced stiffness matrix in modal coordinates,

[0057] F r is the equivalent load vector in modal coordinates,

[0058] Furthermore, in the step S1052, the static load Ritz basis matrix Φ R for:

[0059] Φ R =∑Z,

[0060] Where Z is the eigenvector matrix, Z=[z1 z2 … z n ], z1, z2, z n is the eigenvector;

[0061] ∑ is the static deformation matrix of the system after loading, ∑=[σ1 σ2 … σ n ];

[0062] σ1, σ2, σ n is the static deformation of the system after loading,

[0063] Furthermore, the S1052 further includes the following steps:

[0064] S10521: Calculate the static deformation of the gear transmission system under the action of static torque, and use the static deformation as the first-order Ritz vector;

[0065] S10522: Generate inertial force using the first-order Ritz vector to obtain an n-th-order Ritz vector;

[0066] S10523: Perform a diagonal transformation on the mass matrix to obtain the static load Ritz basis matrix.

[0067] Furthermore, in S1053, the modal vibration matrix is ​​replaced by the static load Ritz basis matrix to achieve model dimensionality reduction, wherein,

[0068] Reduced mass matrix M r for:

[0069] Reduced damping matrix C r for:

[0070] Reduced stiffness matrix K r for:

[0071] Equivalent load vector F r for:

[0072] Furthermore, in S109, the second dynamic equation after dimensionality reduction is numerically iteratively solved using the Newmark-β method. After obtaining the modal coordinates, the static load Ritz basis matrix is ​​used to transform back to the physical coordinates to obtain the dynamic response of the gear transmission system:

[0073]

[0074] Where: X is the displacement in physical coordinates, is the velocity in physical coordinates, is the acceleration in physical coordinates.

[0075] In a preferred embodiment of the present invention, compared with the prior art, the present invention has the following beneficial effects:

[0076] 1. The present invention adopts nonlinear interface separation technology to separate the linear and nonlinear parts in the gear transmission system, and forms an equivalent nonlinear load from the nonlinear part caused by the time-varying parameter excitation of the gear interface and the nonlinear bearing force. The linear part is subjected to dimensionality reduction by condensation, which reduces the dimension of the model and greatly improves the computational efficiency.

[0077] 2. The traditional modal superposition method only considers the system's vibration mode, not the deformation information it undergoes. Therefore, a very high modal truncation order is required to meet accuracy requirements. The present invention uses a static Ritz basis to replace the modal vibration matrix to achieve model dimensionality reduction. This not only includes the dynamic characteristics of the system, but also supplements the static deformation information of the system after being loaded. When the truncation order is small, the error of the modal superposition method is significantly greater than that of the static Ritz basis method, and the static Ritz basis converges faster than the modal superposition method.

[0078] 3. The calculation efficiency of the existing gear transmission system dynamic response is low, especially after considering complex nonlinear factors. The present invention adopts nonlinear interface separation technology and static load Ritz method to greatly improve the calculation speed of nonlinear dynamic response.

[0079] The concept, specific structure and technical effects of the present invention will be further described below in conjunction with the accompanying drawings to fully understand the purpose, characteristics and effects of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0080] Figure 1 1 is a schematic diagram of the overall flow of the method according to an embodiment of the present invention;

[0081] Figure 2 is a schematic diagram of a gear rotor dynamics model according to an embodiment of the present invention;

[0082] Figure 3 is a flowchart of a local nonlinear fast algorithm according to an embodiment of the present invention;

[0083] Figure 4 2. It is a schematic diagram of the dimensional parameters of the shaft system of the gear transmission system according to an embodiment of the present invention;

[0084] Figure 5 1 is a schematic diagram of the mode superposition method of an embodiment of the present invention and the convergence analysis of this method;

[0085] Figure 6 2. It is a schematic diagram of static deformation of a shaft system of a gear transmission system according to an embodiment of the present invention;

[0086] Figure 7 2 is a schematic diagram of the time domain waveform of the x-direction displacement of the driving wheel node according to an embodiment of the present invention;

[0087] Figure 8 is a schematic diagram of the input shaft deformation analysis of an embodiment of the present invention;

[0088] Figure 9 1 is a schematic diagram comparing the calculation times of different algorithms according to an embodiment of the present invention;

[0089] Figure 10 1 is a schematic diagram of a time domain waveform of a nonlinear bearing force under zero initial conditions according to an embodiment of the present invention;

[0090] Figure 11 1 is a schematic diagram of a time domain waveform of a nonlinear bearing force under static displacement initial conditions according to an embodiment of the present invention;

[0091] Figure 12 Schematic diagram of a parallel shaft gear transmission system test bench according to an embodiment of the present invention;

[0092] Figure 13 1 is a schematic diagram comparing simulation and experimental spectra of an embodiment of the present invention.

[0093] The parts in the figure are numbered as follows:

[0094] 1-Input shaft, 2-Output shaft, 3-Driving wheel, 4-Driven wheel, 5-Bearing 1, 6-Bearing 2, 7-Bearing 3, 8-Bearing 4, 9-Three-axis acceleration sensor, 10-Data acquisition device, 11-Drive motor, 12-Gearbox, 13-Magnetic powder brake;

[0095] 21-beam element, 31-driving wheel node, 41-driven wheel node. DETAILED DESCRIPTION

[0096] The following describes several preferred embodiments of the present invention with reference to the accompanying drawings to make its technical content clearer and easier to understand. The present invention can be embodied in many different forms of embodiments, and the scope of protection of the present invention is not limited to the embodiments mentioned herein.

[0097] In the drawings, components with identical structures are denoted by the same reference numerals, and components with similar structures or functions are denoted by similar reference numerals. The size and thickness of each component shown in the drawings are arbitrary and are not limited by the present invention. For clarity, the thickness of components in some places in the drawings is appropriately exaggerated.

[0098] Currently, most algorithms for reducing the dimensionality of gear transmission system dynamic models are based on linearization assumptions and fail to account for the nonlinear characteristics of the gear and bearing interfaces within the gear transmission system. However, these interfaces exhibit typical nonlinear characteristics, while the integrated housing and drive shaft can be considered linear. Separating the linear and nonlinear components of a gear transmission system allows for model reduction and dimensionality reduction of the linear component, allowing the nonlinear effects to be applied to the reduced linear component in the form of nonlinear loads.

[0099] In view of the fact that most of the gear transmission system is linear and only the local interface position needs to consider nonlinearity, this paper proposes a local nonlinear fast algorithm to improve the computational efficiency of the dynamic response. The gear meshing interface load contact analysis model and the nonlinear bearing interface model are used to model the two major local nonlinear sources (gear interface and bearing interface) in the gear transmission system respectively. The local nonlinear interface separation technology is used to equate the nonlinearity introduced by the gear and bearing interface to a nonlinear load applied to the system. The characteristic that the vibration of the gear transmission system contains a significant static component is fully utilized. The static load Ritz basis is selected for model condensation and dimensionality reduction. Finally, the effectiveness of the algorithm will be verified through simulation and experimental cases.

[0100] In the present invention, the basic assumptions for the local nonlinear rapid analysis of the gear transmission system can be summarized as follows:

[0101] (i) The nonlinear parts in the gear transmission system are the gear meshing interface and the bearing interface;

[0102] (ii) Except for the gear and bearing interfaces, the rest are linear.

[0103] like Figure 1 As shown, the local nonlinear rapid analysis method of the gear transmission system provided by the embodiment of the present invention includes the following steps:

[0104] S1: Consider the bearings of the gear transmission system as linear elastic supports and construct the dynamic equations of the gear transmission system.

[0105] The dynamic equation of the constructed gear transmission system is:

[0106]

[0107] Where M is the overall mass matrix, C(t) is the overall damping matrix, K(t) is the time-varying overall stiffness matrix, and F T is the static torque, F E (t) is the load vector corresponding to the displacement excitation, X is the displacement variable, and t is the time variable;

[0108] The overall damping matrix C(t) is the sum of proportional damping and meshing damping:

[0109] C(t)=α D M+β D K(t)+C m (t),

[0110]

[0111]

[0112] Where, α D is the mass damping coefficient, β D is the stiffness damping coefficient, C m (t) is the meshing damping matrix, f n1 、f n2 are the first and second order natural frequencies, ξ1 and ξ2 are the first and second order modal damping ratios, c(t) is the time-varying meshing damping, T m is the transformation matrix of the meshing element, V m is the meshing projection vector;

[0113] The time-varying overall stiffness matrix K(t) is:

[0114] K(t)=K b +K shaft +K m (t),

[0115]

[0116] V m =[-sinψ,cosψ,0,0,0,-r b1 ,sinψ,-cosψ,0,0,0,-r b2 ],

[0117] Where K b is the bearing stiffness matrix, K shaft is the stiffness matrix of the shaft, K m (t) is the meshing stiffness matrix, k(t) is the time-varying integrated meshing stiffness, ψ is the angle between the meshing plane and the y-axis, r b1 is the base circle radius of the driving wheel, r b2 is the base circle radius of the driven wheel;

[0118] Displacement excitation F E The load vector corresponding to (t) is:

[0119]

[0120] Where, e(t) is the no-load transfer error, is the derivative of the no-load transfer error.

[0121] S2: Use nonlinear interface separation technology to separate the linear and nonlinear parts of the gear transmission system.

[0122] like Figure 3 As shown in Figure 2, according to the time-varying characteristics of the gear interface, the meshing stiffness matrix and meshing damping matrix are decomposed into a constant part and a fluctuating part, where:

[0123]

[0124] Where, is the constant part of the meshing stiffness matrix, is the fluctuating part of the meshing stiffness matrix; is the constant part of the meshing damping matrix, is the fluctuating part of the meshing damping matrix.

[0125] After separating the linear part and the nonlinear part in the gear transmission system, the equivalent dynamic equation of the gear transmission system is:

[0126]

[0127] Among them, C eff is the equivalent damping matrix, which no longer contains the wave component:

[0128]

[0129] K eff is the equivalent stiffness matrix, which no longer contains the wave component:

[0130]

[0131] N eff is the equivalent load:

[0132]

[0133] Where, F NL (T) is the load vector representing the nonlinear bearing force.

[0134] S3: For the linear part of the gear transmission system, perform model dimensionality reduction.

[0135] For models with a large number of degrees of freedom, the dynamic equations in physical coordinates have large dimensions, which places a heavy computational burden on the numerical iterative solution of the equations. Model reduction methods are often used to improve computational efficiency. One common model reduction method is the modal superposition method.

[0136] When using the modal superposition method, the modal vibration matrix is ​​used as the basis to convert the physical coordinates into modal coordinates to achieve model dimensionality reduction. In theory, such a basis is not unique, just as mathematically, spatial coordinates can be described based on different orthogonal bases. The modal superposition method only considers the vibration mode of the system, and does not consider the deformation information borne by the system. The static load Ritz basis is a basis that considers static load information. It not only includes the dynamic characteristics of the system, but also supplements the static deformation information of the system after being loaded. The static load Ritz basis can be used to replace the modal vibration matrix to achieve model dimensionality reduction, such as Figure 3 shown.

[0137] When performing dimensionality reduction on a model, the following steps are included:

[0138] S31: The modal superposition method is used to construct an overall framework for dimensionality reduction. The modal vibration matrix is ​​used as the basis to convert the physical coordinates into modal coordinates to achieve model dimensionality reduction.

[0139] By performing generalized eigenvalue analysis on the equivalent stiffness matrix and the overall mass matrix, the modal vibration matrix is ​​obtained, and then the dynamic equation after model dimensionality reduction is obtained:

[0140]

[0141] Where q is the modal displacement in modal coordinates, Modal velocity in modal coordinates, is the modal acceleration in modal coordinates is the mode shape matrix;

[0142] M r is the reduced mass matrix in modal coordinates,

[0143] C r is the reduced damping matrix in modal coordinates,

[0144] K r is the reduced stiffness matrix in modal coordinates,

[0145] F r is the equivalent load vector in modal coordinates,

[0146] S32: Obtain the static load Ritz basis matrix, which can realize the diagonalization of the mass matrix and the stiffness matrix.

[0147] When obtaining the static load Ritz basis matrix, the following steps are included:

[0148] S321: Solve the static deformation of the gear transmission system under static torque and use the static deformation as the first-order Ritz vector;

[0149] S322: Generate inertial force using the first-order Ritz vector to obtain the n-th-order Ritz vector;

[0150] S323: Perform diagonal transformation on the mass matrix to obtain the static load Ritz basis matrix.

[0151] Through the above steps, the static load Ritz basis matrix Φ is obtained R for:

[0152] Φ R =ΣZ,

[0153] Where Z is the eigenvector matrix, Z=[z1 z2 … z n ], z1, z2, z n is the eigenvector;

[0154] ∑ is the static deformation matrix of the system after loading, ∑=[σ1 σ2 … σ n ];

[0155] σ1, σ2, σ n is the static deformation of the system after loading,

[0156] S33: Use the static Ritz basis matrix as the basis to perform condensation and dimensionality reduction.

[0157] The modal vibration matrix is ​​replaced by the obtained static load Ritz basis matrix to achieve model dimensionality reduction, where:

[0158] Reduced mass matrix M r for:

[0159] Reduced damping matrix C r for:

[0160] Reduced stiffness matrix K r for:

[0161] Equivalent load vector F r for:

[0162] S4: Calculate the nonlinear load and comprehensive equivalent load for the nonlinear part of the gear transmission system.

[0163] Comprehensive equivalent load N eff for:

[0164]

[0165] Among them, F T is the static torque, F E (t) is the load vector corresponding to the displacement excitation, is the fluctuating part of the meshing stiffness matrix (i.e., the nonlinear load generated by the gear meshing interface), is the fluctuation part of the meshing damping matrix (i.e., the nonlinear load generated by the gear meshing interface), K b is the bearing stiffness matrix, F NL (t) is the load vector representing the nonlinear bearing force.

[0166] S5: The dynamic equation after dimensionality reduction is numerically solved using the Newmark-β method to obtain the dynamic response of the gear transmission system.

[0167] By solving the following dynamic equations after model dimensionality reduction, the dynamic response of the gear transmission system can be obtained:

[0168]

[0169] Where q is the modal displacement in modal coordinates, Modal velocity in modal coordinates, is the modal acceleration in modal coordinates is the mode shape matrix.

[0170] After obtaining the modal coordinates by numerical iteration using the Newmark-β method, the static load Ritz basis matrix is ​​used to transform back to the physical coordinates to obtain the dynamic response of the gear transmission system:

[0171]

[0172] Where: X is the displacement in physical coordinates, is the velocity in physical coordinates, is the acceleration in physical coordinates.

[0173] S6: The effectiveness of the fast analysis method for gear transmission system dynamics and local nonlinearity is verified through comparative analysis of dynamic responses between simulation and experiment.

[0174] Compared with the prior art, the method for rapid local nonlinear analysis of a gear transmission system provided by the embodiment of the present invention has the following technical effects:

[0175] 1. Most of the current dimensionality reduction algorithms for the dynamic models of gear transmission systems are based on linearization assumptions and cannot take into account the nonlinear characteristics of the gear and bearing interfaces in the gear transmission system. The present invention uses nonlinear interface separation technology to separate the linear and nonlinear parts in the gear transmission system, and uses dimensionality reduction on the linear part to greatly improve the computational efficiency. By targeting the time-varying characteristics of the gear interface, the damping stiffness matrix and the meshing damping matrix are decomposed into a constant part and a fluctuating part. The nonlinear part caused by the time-varying parameter excitation of the gear interface and the nonlinear bearing force is moved to the right side of the dynamic equation to form an equivalent nonlinear load. The equivalent mass, equivalent damping and equivalent stiffness matrices on the left side of the dynamic equation no longer have time-varying characteristics and can be regarded as linear parts. In this way, the linear part can be dimensionality reduced to reduce the dimension of the model.

[0176] 2. The traditional modal superposition method only considers the vibration mode of the system, but does not consider the deformation information of the system. Therefore, a very high modal truncation order is required to meet the accuracy requirements. The present invention uses the static load Ritz basis to replace the modal vibration mode matrix to achieve model dimensionality reduction, which not only includes the dynamic characteristics of the system, but also supplements the static deformation information of the system after being loaded; the traditional modal superposition method uses the modal vibration mode matrix as the basis to convert the physical coordinates into modal coordinates to achieve model dimensionality reduction. In theory, such a basis is not unique, just as mathematically, spatial coordinates can be described based on different orthogonal bases. When the truncation order is small, the error of the modal superposition method is significantly greater than that of the static load Ritz basis method, and the static load Ritz basis converges faster than the modal superposition method (it can also meet the accuracy at a lower truncation order).

[0177] 3. The calculation efficiency of the existing gear transmission system dynamic response is low, especially after considering complex nonlinear factors. The present invention adopts nonlinear interface separation technology and static load Ritz method to greatly improve the calculation speed of nonlinear dynamic response. Considering that the vast majority of the gear transmission system is linear, only the local interface position needs to consider nonlinearity, and the vibration of the gear rotor can be regarded as a small-amplitude vibration superimposed on a larger static deformation. Through condensation and dimensionality reduction, the calculation efficiency is greatly improved. The calculation efficiency of the proposed method is higher than that of the complete method and the modal superposition method.

[0178] The method proposed in the present invention is very suitable for dynamic solution of gear transmission system because of its significant static deformation characteristics. This method is more computationally efficient than the traditional modal superposition method, and can meet the accuracy requirements by using a lower truncation order through the static load Ritz basis method. In terms of the calculation time of the dynamic response of the gear transmission system, the method proposed in the present invention is superior to the modal superposition method and the complete method in the prior art. The present invention can obtain accurate and efficient dynamic response solutions, laying the foundation for the subsequent full life cycle damage evolution analysis based on dynamic loads, and can serve the service performance evaluation of gear transmission systems in major equipment. Typical application industries include: machinery, ships, aerospace, etc.

[0179] The present invention will be described in detail below in conjunction with the preferred embodiments of the present invention.

[0180] Most of the current reduction and dimensionality reduction algorithms for the dynamic models of gear transmission systems are based on linearization assumptions and cannot consider the nonlinear characteristics of the gear and bearing interfaces in the gear transmission system. The preferred embodiment of the present invention provides a fast analysis method for local nonlinearity of a gear transmission system, models the two main local nonlinear sources (gear interface and bearing interface) in the gear transmission system, and uses local nonlinear interface separation technology to equate the nonlinearity introduced by the gear and bearing interface to a nonlinear load applied to the system. The characteristic that the vibration of the gear transmission system contains a significant static component is fully utilized, and the static load Ritz basis is selected for model reduction and dimensionality reduction. Finally, the effectiveness of the algorithm will be verified through simulation and experimental cases.

[0181] The method provided in the preferred embodiment of the present invention specifically includes the following steps:

[0182] Step 1: Construct the dynamic equations of the gear transmission system.

[0183] If the bearings of the gear rotor system are regarded as linear elastic supports, the dynamic equation of the gear transmission system is:

[0184]

[0185] Where M is the overall mass matrix, C(t) is the overall damping matrix, K(t) is the time-varying overall stiffness matrix, and F T is the static torque, F E (t) is the load vector corresponding to the displacement excitation, X is the displacement variable, and t is the time variable.

[0186] The time-varying overall stiffness matrix K(t) is:

[0187] K(t)=K b +K shaft +K m (t), (2)

[0188] Where K b is the bearing stiffness matrix, K shaft is the stiffness matrix of the shaft, K m (t) is the time-varying meshing stiffness matrix, and the axis adopts Timoshenko beam element (such as Figure 2 The beam element in 21) is used for modeling.

[0189] Meshing stiffness matrix K m (t) is:

[0190]

[0191] Where k(t) is the time-varying comprehensive meshing stiffness, T m is the transformation matrix of the meshing element, V m is the meshing projection vector.

[0192] Meshing projection vector V m :

[0193] V m =[-sinψ,cosψ,0,0,0,-r b1 ,sinψ,-cosψ,0,0,0,-r b2 ], (4)

[0194] Where ψ is the angle between the meshing plane and the y-axis (e.g. Figure 2 shown), r b1 is the base circle radius of the driving wheel 3, r b2 is the base circle radius of the driven wheel 4.

[0195] The overall damping matrix is ​​the sum of proportional damping and meshing damping:

[0196] C(t)=α D M+β D K(t)+C m (t), (4)

[0197] Where, α D is the mass damping coefficient, β D is the stiffness damping coefficient, C m (t) is the meshing damping matrix;

[0198]

[0199] Where, f n1 、f n2 are the first and second order natural frequencies (excluding rigid body modes), ξ1 and ξ2 are the first and second order modal damping ratios, respectively.

[0200] The time-varying meshing damping matrix is:

[0201]

[0202] Where c(t) is the time-varying mesh damping.

[0203] The load vector corresponding to the displacement excitation is:

[0204]

[0205] Where, e(t) and represent the no-load transfer error and the derivative of the no-load transfer error, respectively.

[0206] Step 2: Separate the linear and nonlinear parts using nonlinear interface separation technology.

[0207] Step 2.1: Separate the constant and fluctuating parts

[0208] In order to facilitate the subsequent model reduction and dimensionality reduction, the nonlinear interface separation technology is used to separate the linear and nonlinear parts of the gear transmission system. According to the time-varying characteristics of the gear interface, the damping stiffness matrix and the meshing damping matrix are decomposed into a constant part and a fluctuating part:

[0209]

[0210] The dynamic equation of the gear transmission system can be rearranged as follows:

[0211]

[0212] The equivalent stiffness matrix and equivalent damping matrix on the left side of the dynamic equation no longer contain fluctuation components:

[0213]

[0214] Step 2.2: Obtain the equivalent kinetic equation

[0215] The parametric excitations due to the time-varying mesh stiffness and damping are moved to the right side of the dynamic equations and become part of the equivalent loads:

[0216]

[0217] If the bearings of the gear system are considered as nonlinear supports, the dynamic equations are:

[0218]

[0219] Where F NL (t) is the load vector representing the nonlinear bearing force. The dynamic system in the above equation does not have sufficient support and relies solely on the nonlinear bearing force to maintain equilibrium, which is very unfavorable for the convergence of the equation. In order to avoid the singularity of the stiffness matrix, K is added to both ends of the equation.b X:

[0220]

[0221] Theoretically, any stiffness value can be applied, but applying the linear stiffness of the bearing (the equivalent linear stiffness of the nonlinear bearing under this load condition) is optimal for rapid convergence of the equation.

[0222] The difference between the linear and nonlinear bearing forces is expressed as the following vector:

[0223] E NL =K b XF NL (t), (17)

[0224] The equivalent load under nonlinear bearing conditions is defined as:

[0225]

[0226] Then the equivalent kinetic equation can be expressed as:

[0227]

[0228] Step 3: Reduce the model’s dimensionality

[0229] Step 3.1: Build an overall framework for dimensionality reduction

[0230] For models with a large number of degrees of freedom, the dynamic equations in physical coordinates have large dimensions, which places a heavy computational burden on the numerical iterative solution of the equations. Model reduction methods are often used to improve computational efficiency. One common model reduction method is the modal superposition method.

[0231] K eff Perform generalized eigenvalue analysis on the M matrix to obtain the eigenvector matrix (i.e., modal vibration matrix) Φ M The solution of the eigenvector matrix can be carried out according to existing mature technologies, such as the block Lanczos method and the subspace iteration method. Let Φ = Φ M , that is, using the modal vibration matrix as the basis, coordinate transformation is performed to convert the physical coordinates into modal coordinates. The reduced mass matrix, reduced damping matrix, and reduced stiffness matrix in modal coordinates are:

[0232] M r =Φ T MΦ, (20)

[0233] C r =Φ T C eff Φ, (21)

[0234] K r =Φ T K eff Φ, (22)

[0235] The equivalent load vector in modal coordinates is:

[0236] F r =Φ T N eff , (twenty three)

[0237] The dynamic equation after model reduction and dimensionality reduction is:

[0238]

[0239] Compared with the dynamic equation without dimensionality reduction (Equation (20)), the dimension of Equation (25) is greatly reduced. The dynamic equation after dimensionality reduction is numerically solved using the Newmark-β method. and modal acceleration Then, use the Φ matrix to transform back to physical coordinates:

[0240]

[0241] Step 3.2: Obtain a statically loaded Ritz base.

[0242] When using the modal superposition method, the modal vibration matrix is ​​used as the basis (Φ=Φ M ), convert the physical coordinates to modal coordinates to achieve model dimensionality reduction. Theoretically, such a basis is not unique, just as mathematically, space coordinates can be described based on different orthogonal bases. The modal superposition method only considers the vibration mode of the system, without considering the deformation information borne by the system. The static load Ritz basis is a basis that considers static load information, which not only includes the dynamic characteristics of the system, but also supplements the static deformation information of the system after being loaded. The static load Ritz basis can be used to replace the modal vibration mode matrix to achieve model dimensionality reduction. The flowchart of the local nonlinear fast algorithm is as follows: Figure 3 shown.

[0243] Step 3.2.1: Get the first-order Ritz vector

[0244] Solve for the static deformation of the gear system under static torque as the first-order Ritz vector (not normalized):

[0245]

[0246] Solve the above linear equations and normalize the mass matrix to obtain the normalized first-order Ritz vector:

[0247]

[0248] Step 3.2.2: Obtain the nth-order Ritz vector (n≥2)

[0249] When obtaining the second-order Ritz vector, generate the inertia force using the first-order Ritz vector, solve the following linear equations, and obtain the displacement θ2 under the inertia force:

[0250] K eff θ2 = Mσ1, (28)

[0251] θ2 contains a part linearly related to σ1:

[0252]

[0253] Multiply both sides by We can get:

[0254]

[0255] According to the orthogonality of the Ritz vectors, we can get:

[0256]

[0257] Substitute equation (32) into equation (30), we can get:

[0258]

[0259] After normalizing the mass matrix, we can get:

[0260]

[0261] Similar to calculating the second-order Ritz vector, when calculating the nth-order Ritz vector (n≥2), generate the inertia force using the (n - 1)th-order Ritz vector and obtain the displacement under the inertia force:

[0262] K eff θ n = Mσ n-1 , (34)

[0263] According to whether it is linearly related to the previous (n - 1)th-order Ritz vectors, this displacement can be decomposed into two parts:

[0264]

[0265] Multiply both sides by (i < n), we can get:

[0266]

[0267] According to the orthogonality of the Ritz vectors, we can get:

[0268]

[0269] Further we can get:

[0270]

[0271] Normalize the mass matrix of the nth-order Ritz vector:

[0272]

[0273] Form the Ritz basis matrix:

[0274] ∑=[σ1 σ2 … σ n ], (40)

[0275] Formula (39) can be rewritten as:

[0276]

[0277] Step 3.2.3: Diagonalization

[0278] The initial Ritz basis matrix Σ thus formed can be used for model reduction, but it only diagonalizes the mass matrix. In order to also diagonalize the stiffness matrix of the reduced model, further transformations are required.

[0279] Solve the generalized eigenvalue problem in the following equation:

[0280] ∑ T K eff ∑z=λ∑ T M∑z, (42)

[0281] According to the diagonalization property of the mass matrix, the generalized eigenvalue problem can be simplified to T K eff Eigenvalue problem of ∑ matrix:

[0282] ∑ T K eff ∑z=λz, (43)

[0283] Where λ is the eigenvalue and z is the eigenvector.

[0284] Get the eigenvector matrix:

[0285] Z=[z1 z1 … z n ], (44)

[0286] Finally, the Ritz basis matrix that can realize the diagonalization of the stiffness matrix is ​​obtained:

[0287] Φ R=∑Z, (45)

[0288] Step 3.3 Constructing the polycondensation model

[0289] Let Φ = Φ R , that is, the static load Ritz basis matrix is ​​used as the basis to perform dimensionality reduction. The reduced mass matrix, reduced damping matrix, and reduced stiffness matrix are:

[0290] M r =Φ T MΦ, (46)

[0291] C r =Φ T C eff Φ, (47)

[0292] K r =Φ T K eff Φ, (48)

[0293] The equivalent load vector in modal coordinates is:

[0294] F r =Φ T N eff , (49)

[0295] The dynamic equation after model reduction and dimensionality reduction is:

[0296]

[0297] For the dynamic equation after dimension reduction, the Newmark-β method is used as the numerical iteration method to perform time history analysis and calculate the dynamic response. and modal acceleration Then, use the Φ matrix to transform back to physical coordinates:

[0298]

[0299] Step 4: Nonlinear dynamic characteristics analysis of gear transmission system

[0300] Based on the gear transmission system dynamics model and local nonlinear rapid analysis method, the characteristics of the gear transmission system meshing interface and nonlinear bearing interface are analyzed. The effectiveness of the gear transmission system dynamics and local nonlinear rapid analysis method is verified by comparing the dynamic response of simulation and experiment.

[0301] Step 4.1: Interface nonlinear characteristics analysis

[0302] For example Figure 4For the single-stage parallel shaft gear transmission system shown in FIG, a dynamic model of the gear rotor system is established, and the bearing interface of the gear rotor system is simulated as a nonlinear interface.

[0303] The gear pair parameters are shown in Table 1. The input shaft 1 and output shaft 2 are supported by single-row cylindrical roller bearings N304 and N207, respectively. The specific parameters are shown in Table 2.

[0304] Table 1 Main parameters of spur gear pairs

[0305]

[0306] Table 2 Main parameters of bearings

[0307]

[0308] The flexibility of the gear shaft is simulated by Timoshenko beam element. The time-varying meshing stiffness is used as the parameter excitation source. The meshing stiffness is introduced into the dynamic model. The nonlinear dynamic model is solved by the local nonlinear fast algorithm to analyze the nonlinear dynamic behavior.

[0309] The convergence speed of the local nonlinear fast algorithm and the traditional mode superposition method is Figure 5 The difference between the two methods is that the modal superposition method uses the modal matrix for transformation, while the local nonlinear fast algorithm uses the static load Ritz basis for transformation. Figure 5 The error in refers to the percentage error between the mean dynamic transmission errors obtained by the two dimensionality reduction algorithms and the complete method. When the truncation order is small, the error of the modal superposition method is significantly greater than that of the static load Ritz basis method. If 0.1% is used as the error tolerance, the minimum truncation order required for the modal superposition method and the static load Ritz basis method is 48 and 21, respectively. The truncation order determines the dimension of the dynamic equation after condensation. A lower model dimension will help improve the efficiency of the numerical iteration solution. The following will explain why the static load Ritz basis has better convergence characteristics through the analysis of the system deformation of the gear rotor.

[0310] Unlike most rotating machines (such as turbines, generators, motors, etc.) that have only one concentric rotor, the gear rotor system consists of at least two shafts. Taking the single-stage parallel shaft gear transmission system in this embodiment as an example, the two shafts are coupled and transmit torque through the gear meshing interface. After being subjected to torque, the gear rotor will produce obvious static deformation. Under the action of the meshing force, the rotor undergoes bow deformation, such as Figure 6 shown.

[0311] The time domain waveform of the displacement of the driving wheel node 31 in the x direction is as follows: Figure 7 As shown by Figure 7It can be seen that under different torques, the mean part of the displacement response is much larger than the fluctuation part, that is, the vibration of the gear rotor can be regarded as a small-amplitude vibration superimposed on a large static deformation.

[0312] The deformation of each node on the axis is decomposed into static deformation components and fluctuation components, such as Figure 8 As shown. The height of the vertical error bar represents the peak-to-peak value of the dynamic deformation component. For different positions, the static deformation part is several times the fluctuating part. When the torque increases from 7.3N·m to 14.6N·m, the shape of the shaft deformation curve is almost completely consistent, and the deformation of the shaft is approximately twice the original amount. This also shows from one aspect that although the bearing has nonlinear characteristics, the overall characteristics of the gear rotor system are still mainly linear. Under different torques, the vibration response of the gear system contains a significant static deformation component. In other words, the vibration of the gear rotor system is dominated by static deformation.

[0313] In the static-load Ritz basis, the first column of the transformation matrix contains the static deformation component. However, the modal superposition method uses the mode matrix as the transformation matrix and does not reflect any information related to the static deformation of the gear rotor system. In a sense, the static deformation information is equivalent to the high-order modal information in the modal superposition method. The natural frequency corresponding to the high-order modes of the system is very high, much greater than the excitation frequency of the system. When the excitation force is far away from the resonance peak, the stiffness characteristics of the system are reflected (the contributions of damping and mass can be ignored), that is, the static characteristics. Therefore, the modal superposition method requires a higher truncation order to effectively reflect the vibration characteristics of the gear rotor system: that is, a small-amplitude vibration is superimposed on the static deformation.

[0314] Static Ritz basis is very suitable for solving the dynamics of gear transmission systems because they have significant static deformation characteristics. However, since the static Ritz basis uses the static deformation as the first-order Ritz vector, it is not suitable for mechanical systems without significant static deformation.

[0315] The calculation time of different dynamic solution algorithms is as follows Figure 9 As shown in Figure 2, the calculation time of the traditional method (complete method) is significantly higher than that of the modal superposition method and the static load Ritz basis, and the calculation time of the complete method increases sharply with the increase in the number of degrees of freedom. By fitting the calculation time of different algorithms with polynomials, we can obtain:

[0316]

[0317] The static load Ritz basis method is more efficient than the modal superposition method because the static load Ritz basis method can meet the accuracy requirements with a lower truncation order.

[0318] Under zero initial conditions (X0=0), the iterative process of nonlinear bearing force is as follows: Figure 10 As shown. Figure 10 In (a), the nonlinear bearing force gradually reaches a steady state after a period of oscillation. The steady-state vibration includes a significant DC component, which is the static bearing force. The unsteady oscillation phase includes both the attenuation of the free vibration and the transition from linear to nonlinear bearing forces. Figure 10 (b) shows the process of nonlinear bearing force gradually replacing linear bearing force. NL represents the difference between the nonlinear and linear bearing forces in the x-direction for bearing 5. Initially, the gear rotor system is fully supported by the linear bearing force. As the iterations progress, the difference between the nonlinear and linear bearing forces decreases, eventually reaching a steady state. This demonstrates that the nonlinear bearing force gradually replaces the linear bearing force.

[0319] When the initial displacement condition is static displacement The iterative process of nonlinear bearing force is as follows Figure 11 As shown in Figure 2, it can be seen that the non-steady oscillation phase of the nonlinear bearing force under the static displacement initial condition is significantly shorter than that under the zero initial condition. This indicates that the static displacement initial condition is more conducive to the convergence of the nonlinear bearing force.

[0320] Step 4.2: Experimental verification of dynamic response

[0321] Based on the Houde ZDY80 parallel shaft gear test bench, the experimental verification of dynamic response is carried out, such as Figure 12 As shown. The drive motor 11 serves as the torque input to the gearbox 12, and the magnetic powder brake 13 serves as the load. Sensor measurement points are located at the bearing seat. The three-axis acceleration sensor 9 is a Donghua DH339E three-axis acceleration sensor. The data acquisition device 10 is a DH8303. The response test sampling time is 30 seconds and the sampling frequency is 10,000 Hz. The device is equipped with an input shaft 1, an output shaft 2, a driving pulley 3, a driven pulley 4, and multiple bearings, including bearing 1 5, bearing 2 6, bearing 3 7, and bearing 4 8.

[0322] Under the working conditions of 8.5N·m and 2398.7rpm, the acceleration response of bearing 5 in the x direction obtained by simulation and experiment is compared. Figure 13 shown.

[0323] The amplitude and frequency components obtained by simulation and experiment are very similar: the meshing frequency and its harmonics are the main frequency components. In addition, due to the tooth profile error of the driving wheel and the driven wheel, the meshing frequency f m The rotation frequency of the driving wheel and the driven wheel (f s1 and f s2) modulation phenomenon occurs. The spectrum of both simulation and experiment has the highest amplitude near the 3rd order meshing frequency. This is because the 10th order natural frequency (f n10 =2616.5Hz) is close to the third-order meshing frequency of the gear, which shows that the simulation model accurately simulates the dynamic characteristics of the test bench.

[0324] The above describes in detail the preferred embodiments of the present invention. It should be understood that those skilled in the art can make numerous modifications and variations based on the concepts of the present invention without inventive effort. Therefore, any technical solutions that can be derived by those skilled in the art through logical analysis, reasoning, or limited experimentation based on the concepts of the present invention and the prior art should be within the scope of protection defined by the claims.

Claims

1. A method for rapid local nonlinear analysis of a gear transmission system, characterized in that: The method comprises the following steps: S101: Considering the bearing of the gear transmission system as a linear elastic support, constructing a first dynamic equation of the gear transmission system; S103: Separating the linear part and the nonlinear part of the gear transmission system by using a nonlinear interface separation technology; S105: performing model condensation and dimensionality reduction on the linear part of the gear transmission system; S107: Calculating a nonlinear load and a comprehensive equivalent load for the nonlinear part in the gear transmission system; S109: performing numerical iteration on the second dynamic equation after condensation and dimensionality reduction using the Newmark-β method to obtain a dynamic response of the gear transmission system; S111: Verify the effectiveness of the fast analysis method for gear transmission system dynamics and local nonlinearity through comparative analysis of dynamic responses between simulation and experiment; in, In step S103, the meshing stiffness matrix and the meshing damping matrix are decomposed into a constant part and a fluctuating part according to the time-varying characteristics of the gear interface, wherein: Where, is the constant part of the meshing stiffness matrix, is the fluctuating part of the meshing stiffness matrix; is the constant part of the meshing damping matrix, is the fluctuation part of the meshing damping matrix; The step S105 includes the following steps: S1051: Use the modal superposition method to build an overall framework for dimensionality reduction. Using the modal shape matrix as the basis, the physical coordinates are converted to modal coordinates to achieve model dimensionality reduction. S1052: Obtain a static load Ritz basis matrix, where the static load Ritz basis matrix can achieve diagonalization of the mass matrix and the stiffness matrix; S1053: Using the static Ritz basis matrix as a basis, performing condensation and dimensionality reduction; In step S109, the Newmark-β method is used to perform numerical iteration to solve the second dynamic equation after dimensionality reduction. After obtaining the modal coordinates, the static load Ritz basis matrix is ​​used to transform back to the physical coordinates to obtain the dynamic response of the gear transmission system: Where: X is the displacement in physical coordinates, is the velocity in physical coordinates, is the acceleration in physical coordinates.

2. The method according to claim 1, wherein In the S101, the first dynamic equation of the gear transmission system is: Where M is the overall mass matrix, C(t) is the overall damping matrix, K(t) is the time-varying overall stiffness matrix, and F T is the static torque, F E (t) is the load vector corresponding to the displacement excitation, X is the displacement variable, and t is the time variable; The overall damping matrix C(t) is the sum of proportional damping and meshing damping: C(t)=α D M+β D K(t)+C m (t), Where, α D is the mass damping coefficient, β D is the stiffness damping coefficient, C m (t) is the meshing damping matrix, f n1 、f n2 are the first and second order natural frequencies, ξ1 and ξ2 are the first and second order modal damping ratios, c(t) is the time-varying meshing damping, T m is the transformation matrix of the meshing element, V m is the meshing projection vector; The time-varying overall stiffness matrix K(t) is: K(t)=K b +K shaft +K m (t), V m =[-sinψ,cosψ,0,0,0,-r b1 ,sinψ,-cosψ,0,0,0,-r b2 ], Where K b is the bearing stiffness matrix, K shaft is the stiffness matrix of the shaft, K m (t) is the meshing stiffness matrix, k(t) is the time-varying integrated meshing stiffness, ψ is the angle between the meshing plane and the y-axis, r b1 is the base circle radius of the driving wheel, r b2 is the base circle radius of the driven wheel; Displacement excitation F E The load vector corresponding to (t) is: Where, e(t) is the no-load transfer error, is the derivative of the no-load transfer error.

3. The method according to claim 2, wherein After separating the linear part and the nonlinear part in the gear transmission system, the equivalent dynamic equation of the gear transmission system is: Among them, C ef is the equivalent damping matrix, which no longer contains the wave component: K ef is the equivalent stiffness matrix, which no longer contains the wave component: N ef is the equivalent load: Where, F NL (t) is the load vector representing the nonlinear bearing force.

4. The method according to claim 3, wherein In S1051, the modal vibration matrix is ​​obtained by performing generalized eigenvalue analysis on the equivalent stiffness matrix and the overall mass matrix, and then the second dynamic equation after model reduction and dimensionality reduction is obtained: Where q is the modal displacement in modal coordinates, Modal velocity in modal coordinates, is the modal acceleration in modal coordinates is the mode shape matrix; M r is the reduced mass matrix in modal coordinates, C r is the reduced damping matrix in modal coordinates, K r is the reduced stiffness matrix in modal coordinates, F r is the equivalent load vector in modal coordinates, 5. The method according to claim 4, wherein In the S1052, the static load Ritz basis matrix Φ R for: F R =∑Z, Where Z is the eigenvector matrix, Z=[z1 z2 … z n ], z1, z2, z n is the eigenvector; ∑ is the static deformation matrix of the system after loading, ∑=[σ1 σ2 … σ n ]; σ1, σ2, σ n is the static deformation of the system after loading, 6. The method according to claim 5, wherein The S1052 further includes the following steps: S10521: Calculate the static deformation of the gear transmission system under the action of static torque, and use the static deformation as the first-order Ritz vector; S10522: Generate inertial force using the first-order Ritz vector to obtain an n-th-order Ritz vector; S10523: Perform a diagonal transformation on the mass matrix to obtain the static load Ritz basis matrix.

7. The method according to claim 6, wherein In S1053, the modal vibration matrix is ​​replaced by the static load Ritz basis matrix to achieve model dimensionality reduction, wherein, Reduced mass matrix M r for: Reduced damping matrix C r for: Reduced stiffness matrix K r for: Equivalent load vector F r for:

Images