An electric cylinder vibration response time-varying reliability analysis method

Abstract

The application discloses a kind of electric cylinder vibration response time-varying reliability analysis method, it is related to the dynamic reliability analysis technical field of electromechanical system.Solve the problem that existing fixed time analysis cannot reflect whole course risk, cannot accurately identify most unfavorable moment and high cost of calculation.The method comprises the following steps: a coupled dynamic model considering the position change of moving member, support constraint and nonlinear contact is established;Hammersley sequence is used to generate samples and solve the full time history, and limit state function is constructed at each discrete time;Based on unified index comparison, Kriging, polynomial chaos expansion and limit learning machine surrogate model are compared and optimized;Large-scale random propagation is carried out using the optimized model, and the failure probability and time-varying reliability are calculated at each time, to identify the most unfavorable moment and risk interval.The application can characterize the whole course vibration reliability evolution of electric cylinder and reduce cost.

Description

Technical Field

[0001] This application relates to the field of dynamic reliability analysis of electromechanical systems, and in particular to a time-varying reliability analysis method for vibration response of electric cylinders. Background Technology

[0002] An electric cylinder is an electromechanical actuator that converts the rotary motion of a servo motor into the linear motion of a piston rod. It features high transmission efficiency, high positioning accuracy, and fast response speed, and is widely used in CNC machine tools, robots, aerospace equipment, and precision loading devices. During operation, the lead screw drives the nut and piston rod to move axially, with sliding bearings, anti-rotation pins, and paired angular contact ball bearings providing support and constraint. As the piston rod extends and retracts, the positions of the nut, sliding bearings, and anti-rotation pins constantly change, causing the structural support span, mass distribution, and dynamic stiffness to exhibit time-varying characteristics. Simultaneously, the lead screw-nut pair and support bearings have nonlinear contact, the assembly preload and geometric dimensions are random, and the external load changes over time, resulting in the electric cylinder's vibration response exhibiting characteristics such as moving boundaries, time-varying parameters, multi-component coupling, and random nonlinearity.

[0003] Existing reliability analyses of electric cylinders often focus on static displacement, steady-state amplitude, maximum response, or responses at pre-selected fixed times, making it difficult to describe the evolution of vibration failure risk throughout the entire operation. When random parameters cause shifts in the magnitude and timing of vibration peaks, fixed-time analyses may miss the most unfavorable stage, thus failing to accurately identify the most unfavorable moment and its surrounding risk time intervals. While direct Monte Carlo simulations can handle complex nonlinear limit states, they require repeated solutions to transient dynamic equations on a large number of random samples, resulting in high computational costs. First-order and second-order reliability methods rely on local approximations, making it difficult to guarantee analytical accuracy under conditions of contact opening and closing, moving boundaries, and response peak migration. Furthermore, existing surrogate model methods typically employ a single model and verify accuracy only with a small number of characteristic responses or at a single time point. Different surrogate models exhibit varying adaptability to sample size, response smoothness, and local nonlinearities, making it difficult to maintain stable predictive performance across all time points.

[0004] Therefore, this application provides a time-varying reliability analysis method for the vibration response of an electric cylinder. Summary of the Invention

[0005] This application provides a time-varying reliability analysis method for the vibration response of an electric cylinder, which solves the technical problems of existing fixed-time reliability analysis, which cannot reflect the vibration failure risk of the electric cylinder throughout the entire operation process, cannot accurately identify the most unfavorable moment with the lowest reliability, and has a high computational cost for direct large-sample transient dynamic analysis.

[0006] To achieve the above objectives, this application adopts the following technical solution: Firstly, a time-varying reliability analysis method for the vibration response of an electric cylinder is provided, including: A coupled dynamic model is established to determine the random parameters and their probability distribution, the analysis time interval, and the allowable threshold for vibration response. The coupled dynamic model is based on the positional changes of the moving component of the electric cylinder, support constraints, and nonlinear contact effects. Random parameter samples are generated using low-bias sequences, and the initial sample points are transformed into an input sample set that conforms to the original probability distribution of each random parameter through the inverse function of the cumulative distribution function of each random parameter. Each random parameter sample in the input sample set is input into the coupled dynamic model to obtain the full-time history of the vibration response corresponding to each sample; each random parameter sample and its corresponding full-time history of vibration response are combined into a sample pair to construct a training sample set and a test sample set; and the full-time history of vibration response at each discrete moment and the allowable threshold of vibration response are used to establish a time-by-time limit state function at each discrete moment. Multiple proxy models are constructed, and the optimal proxy model is determined according to a unified full-time history accuracy index; The preferred surrogate model is used to establish a transient vibration response surrogate model for each discrete moment. The failure probability and instantaneous time-varying reliability are calculated using random parameter samples to form an instantaneous time-varying reliability curve. Candidate worst-case moments are determined based on instantaneous time-varying reliability curves. The time step of the neighborhood is densified to determine the worst-case moment, minimum reliability, maximum failure probability, and risk time interval, and the sensitivity of key random parameters is output.

[0007] Based on the above technical solution, the time-varying reliability analysis method for vibration response of an electric cylinder provided in this application takes the full-time history of vibration response under random parameters as the analysis object. A limit state function is established at each discrete moment, and the failure probability is calculated, overcoming the deficiency of traditional fixed-time reliability analysis in describing the evolution of vibration failure risk during the entire operation process. Through the instantaneous time-varying reliability curve, the most unfavorable moment with the lowest reliability can be directly identified, avoiding the problem of missing the real dangerous stage due to the vibration peak time shift caused by random parameters. Based on this, the risk time interval is determined, providing a clear basis for dynamic monitoring and control during the operation of the electric cylinder. Simultaneously, the Hammersley low-bias sequence is used to improve the spatial coverage efficiency of random parameters, obtaining uniformly distributed training data under conditions of fewer dynamic samples. By uniformly comparing and optimizing three types of surrogate models—Kriging, multinomial chaotic expansion, and extreme learning machine—under the same training data, test data, and evaluation time nodes, the subjectivity of selecting a single surrogate model is effectively reduced. By directly applying the preferred surrogate model to model the transient vibration response at each discrete moment, replacing the repeated transient dynamic solution under large-scale random samples, the accuracy of time-by-time reliability evaluation is guaranteed, and the computational cost of time-varying reliability analysis is significantly reduced. This forms a complete technical closed loop from dynamic modeling, sample generation, surrogate model selection to the identification of the most unfavorable moment, providing an efficient and stable analysis method for vibration reliability assessment and structural optimization design of electric cylinders throughout their entire life cycle.

[0008] In conjunction with the first aspect above, in one possible implementation, the process of constructing the coupled dynamics model includes: The lead screw, piston rod, and cylinder are discretized using beam elements. A time-varying position function is used to describe the positional changes of the lead screw-nut pair, sliding bearing, and anti-rotation pin relative to their respective load-bearing components. The lengths of the beam elements on both sides of the moving component are updated based on this position function, and a mass matrix varying with time and random parameters is obtained. Damping matrix and stiffness matrix ; The system generalized coordinate vectors corresponding to the mass matrix, damping matrix, and stiffness matrix. Including the generalized coordinate vector of the lead screw Generalized coordinate vector of the piston rod and the generalized coordinate vector of the cylinder block ; ; Based on the quality matrix Damping matrix and stiffness matrix Construct the coupled dynamic equations of the electric cylinder: ; in, It is a random parameter vector; External load vector; A collection of supporting and connecting components; Displacement mapping matrix of the r-th type of support or connection component; Let be the force vector of the r-th type of support or connection component in its local coordinate system; The first derivative of the generalized coordinate vector with respect to time t; is the second derivative of the generalized coordinate vector with respect to time t; r is the index of the support or connection component type; and These represent the relative displacement and relative velocity on both sides of the r-th type of connecting component, respectively. The supporting and connecting components include at least a paired angular contact ball bearing, a lead screw-nut pair, a sliding bearing, and an anti-rotation pin; wherein, the paired angular contact ball bearing and the lead screw-nut pair adopt a Hertzian nonlinear contact model with a contact opening and closing criterion, the ball position of the lead screw-nut pair is determined by equal perimeter channel mapping, the sliding bearing is modeled using the Winkler foundation model and the slicing method, and the anti-rotation pin is modeled using a linear spring damping model; The vibration response output at a specified position of the piston rod under given random parameters is defined by the observation vector h. .

[0009] In conjunction with the first aspect above, in one possible implementation, the damping matrix... In addition to structural damping, it also includes velocity-related additional terms caused by time-varying coordinate mapping and moving connection positions; the force vector Defined according to the direction acting on the connecting positive and negative nodes, its reaction force is determined by the mapping matrix. The transposed part is automatically assembled to the other node.

[0010] In conjunction with the first aspect above, in one possible implementation, the process of determining the most unfavorable moment and the lowest reliability includes: Discretize the analysis time interval into: ,in, For time step, For discrete time intervals, use indexes. ; This represents the total number of time steps. In the Establish the limit state function at each discrete time point. : ; in, For a moment The allowable threshold for vibration response, For vibration response amplitude operator; when When indicates a reliable state, This indicates a failure state; Failure probability at each time step : ; in, Indicates the probability of the random event within the parentheses occurring; Instantaneous time-varying reliability is obtained based on the time-by-time failure probability at each discrete moment. : ; Among them, instantaneous time-varying reliability is used to characterize the marginal reliability state at each discrete moment; interval task reliability is used to characterize the probability that no vibration exceeds the limit from the initial moment to the given moment. The two are used for locating dangerous moments and assessing cumulative task risk, respectively. The instantaneous time-varying reliability curve is obtained based on the instantaneous time-varying reliability, and the most unfavorable moment is determined. and minimum reliability : ; ; After initially obtaining the discrete time node with the lowest reliability, the time step size of its adjacent time intervals is reduced and recalculated until the difference between two adjacent most unfavorable moments is less than the preset time error limit.

[0011] In conjunction with the first aspect above, in one possible implementation, the method for obtaining the full-time history of the vibration response corresponding to each sample includes: Generate the i-th Hammersley low-bias sequence point within a d-dimensional unit hypercube. The probability distribution mapping is completed through the inverse function of the cumulative distribution function of each random parameter: ; Where j is the index of the random parameter, j=1,2,...,d, and d is a positive integer; Let be the cumulative distribution function of the j-th random parameter; i = 1, 2, ..., N; where i is the sample index of the random parameter, i = 1, 2, ..., N, and N is the total number of random parameter samples; Let j be the j-th dimension component of the i-th sample point; The random parameter samples are input into the coupled dynamic model, and the vibration response amplitude is calculated at all discrete time points. And construct the full-time history response matrix. : ; in, ; Each row of the full-time history response matrix represents the response time history corresponding to a set of random parameter samples, and each column represents a random vibration response sample at a discrete moment; the random parameters include the ball diameter of the lead screw-nut pair. Angular contact ball bearing ball diameter Preload of lead screw-nut pair Preload of paired angular contact ball bearings Lead screw Lead screw shaft diameter Piston rod diameter and cylinder diameter .

[0012] In conjunction with the first aspect above, in one possible implementation, the construction of multiple proxy models includes: Kriging surrogate model, chaotic polynomial expansion surrogate model and extreme learning machine neural network surrogate model were constructed using the same training samples, test samples and evaluation time points, respectively. The Kriging surrogate model consists of a regression term and a zero-mean Gaussian stochastic process; The chaotic polynomial expansion proxy model uses an orthogonal polynomial basis that matches the probability distribution of random parameters for truncated expansion. The extreme learning machine neural network surrogate model adopts a single hidden layer feedforward structure. The input weights and hidden layer biases are randomly generated and then fixed. The output weights are obtained through the generalized inverse or regularized least squares solution of the hidden layer output matrix.

[0013] In conjunction with the first aspect above, in one possible implementation, the method for determining the preferred proxy model includes: Calculate the coefficient of determination at each evaluation time point. and normalized root mean square error And construct a comprehensive evaluation function for the entire time history: ; Where m is the index of the proxy model; , and For pre-defined non-negative weights, and ; This is the set of time indices used in the accuracy evaluation of the surrogate model. This represents the number of evaluation time points included in the set. The evaluation time point set does not include time points where the response variance is lower than the preset tolerance. The denominator of the normalized error uses the response range and positive tolerance. The larger value in; Will The surrogate model that has the smallest accuracy and meets the preset accuracy requirement is determined as the preferred surrogate model. When none of the three surrogate models meet the requirements, additional training samples are added and the training and evaluation are repeated.

[0014] In conjunction with the first aspect above, in one possible implementation, the calculation process of the time-by-time failure probability further includes: M sets of random parameter samples are generated for reliability analysis and input into the transient vibration response surrogate model at each discrete time to obtain the predicted response. And estimate the time using the failure indication function. failure probability : ; in, For the index of the random parameter sample group, , It is a positive integer; This is a failure indication function; This is the vibration response amplitude operator.

[0015] In conjunction with the first aspect above, in one possible implementation, the instantaneous time-varying reliability further includes: ; in, For a moment The time-varying reliability estimate; Based on the preset reliability threshold Determine the risk time range When it is necessary to evaluate the probability that vibration exceeding the limit will not occur within a specified operating range, an interval limit state is established. It also calculates the reliability of the interval task; when it is necessary to identify key random parameters, it calculates the local sensitivity of the instantaneous time-varying reliability to the corresponding distributed parameters by perturbing the mean or standard deviation of the random parameter distribution; in, From the initial time to The interval limit state; This is an index for discrete time intervals within the interval.

[0016] In conjunction with the first aspect above, in one possible implementation, the full-time history of the vibration response is the vibration response at each discrete moment within the complete analysis time interval at a specified position of the piston rod; the allowable threshold for the vibration response is a pre-set boundary value used to determine whether the vibration response exceeds the limit.

[0017] This application provides a time-varying reliability analysis method for the vibration response of an electric cylinder. It takes the entire time history of the vibration response under random parameters as the analysis object, establishes a limit state function at each discrete moment, and calculates the failure probability time-by-time. This overcomes the shortcomings of traditional fixed-time reliability analysis, which cannot describe the evolution of vibration failure risk during the entire operation process, and comprehensively reveals the vibration reliability change trend of the electric cylinder throughout its entire stroke. It can directly identify the most unfavorable moment with the lowest reliability based on the instantaneous time-varying reliability curve, avoiding the omission of the true dangerous stage due to the vibration peak time shift caused by random parameters. Furthermore, it delineates the risk time interval, providing a clear temporal positioning basis for dynamic monitoring, early warning, and control during the operation of the electric cylinder. Simultaneously, it improves the spatial coverage efficiency of random parameters by using Hammersley low-bias sequences, obtaining uniformly distributed training data even with fewer dynamic samples. Training data effectively controls the computational cost of initial samples. Furthermore, by uniformly comparing and optimizing three types of surrogate models—Kriging, polynomial chaotic expansion, and extreme learning machine—under the same training data, test data, and evaluation time points, the subjectivity of selecting a single surrogate model is reduced. The optimized surrogate model is directly used for transient vibration response modeling at each discrete time point, replacing the repeated transient dynamics solution under large-scale random samples. This ensures the accuracy of time-by-time reliability evaluation while significantly reducing the computational cost of time-varying reliability analysis. It can also identify key random parameters affecting time-varying reliability through distributed parameter sensitivity analysis, forming a complete technical closed loop from dynamic modeling, sample generation, surrogate model optimization, worst-case moment identification to sensitivity analysis. This provides an efficient and stable analytical tool for vibration reliability assessment, structural parameter optimization, and control strategy design throughout the entire life cycle of electric cylinders.

[0018] It should be understood that the descriptions of technical features, technical solutions, beneficial effects, or similar language in this application do not imply that all features and advantages can be achieved in any single embodiment. Rather, it is understood that the description of a feature or beneficial effect means that a specific technical feature, technical solution, or beneficial effect is included in at least one embodiment. Therefore, the descriptions of technical features, technical solutions, or beneficial effects in this specification do not necessarily refer to the same embodiment. Furthermore, the technical features, technical solutions, and beneficial effects described in this embodiment can be combined in any suitable manner. Those skilled in the art will understand that embodiments can be implemented without one or more specific technical features, technical solutions, or beneficial effects of a particular embodiment. In other embodiments, additional technical features and beneficial effects may be identified in specific embodiments that do not embody all embodiments. Attached Figure Description

[0019] Figure 1 A flowchart illustrating a time-varying reliability analysis method for the vibration response of an electric cylinder, provided as an embodiment of this application; Figure 2 A schematic diagram of a time-varying dynamics model of an electric cylinder provided in an embodiment of this application; Figure 3 A schematic diagram showing the predicted and actual results of the Kriging model at representative evaluation time points in an embodiment of this application. Figure 4 A schematic diagram illustrating the relative error of the Kriging model on the validation set, provided in an embodiment of this application; Figure 5 A schematic diagram showing the predicted and actual results of the PCE model provided in this application at representative evaluation time points for test samples; Figure 6 A schematic diagram illustrating the relative error of the PCE model on the validation set provided in this application embodiment; Figure 7 A schematic diagram showing the test sample prediction results and actual results of the ELM model provided in the embodiments of this application at representative evaluation time points; Figure 8 A schematic diagram illustrating the relative error of the ELM model on the validation set provided in this application embodiment; Figure 9 A schematic diagram illustrating the small sample statistical results of the finite element method and the large sample statistical results generated by the preferred surrogate model in the embodiments of this application; Figure 10 A schematic diagram showing the instantaneous time-varying reliability curve, reliability threshold, most unfavorable moment, and risk time interval provided for embodiments of this application; Figure 11 A schematic diagram illustrating the time-varying reliability sensitivity of the mean of random parameters provided in the embodiments of this application; Figure 12 A schematic diagram of time-varying reliability sensitivity of standard deviation provided for embodiments of this application. Detailed Implementation

[0020] The present invention will be further described below with reference to the accompanying drawings and specific embodiments. The following embodiments are used to illustrate the technical concept of the present invention, and not to limit the scope of protection of the present invention; equivalent substitutions made to the dynamic model discretization method, random parameter type, surrogate model parameters, reliability threshold, and probability estimation method without departing from the core concept of the present invention are all within the scope of protection of the present invention. The method provided by the present invention will be further illustrated below through an embodiment.

[0021] This embodiment provides a time-varying reliability analysis method for the vibration response of an electric cylinder. The overall calculation sequence is as follows: Figure 1As shown: First, a dynamic model is established and random parameters, analysis time intervals, and allowable thresholds are determined; then, low-bias samples are generated and the full-time history response of each sample is solved to establish the time-by-time limit state; then, three types of surrogate models are compared and the preferred model form is determined. If the accuracy is not satisfied, supplementary training samples are returned; finally, large-sample propagation is performed, instantaneous time-varying reliability is calculated, local encryption is used to identify the most unfavorable moment, and the results are output.

[0022] Step S1: Establish the time-varying dynamics model of the electric cylinder: like Figure 2 As shown, the electric cylinder includes a lead screw, a lead screw-nut pair, a piston rod, a cylinder body, a paired angular contact ball bearing, a sliding bearing, and an anti-rotation pin. Figure 2 The left side shows the structural configuration of the electric cylinder in different operating positions, and the right side shows the changes in the corresponding beam unit nodes and support positions. From t=0 to subsequent times, the positions of the nut, sliding bearing and anti-rotation pin change with the movement of the piston rod, so the length of the beam unit on both sides and the span of the structural support need to be updated step by step.

[0023] A servo motor drives the leadscrew to rotate, and the leadscrew-nut pair converts the rotational motion of the leadscrew into linear motion of the nut and piston rod. An anti-rotation pin restricts the nut from rotating around the leadscrew axis, a sliding bearing provides radial support to the piston rod, and paired angular contact ball bearings provide axial and radial support to the ends of the leadscrew.

[0024] The lead screw, piston rod, and cylinder are discretized using Timoshenko beam elements, with each beam element node having three translational degrees of freedom and three rotational degrees of freedom. The generalized coordinates of the nodes of the lead screw, piston rod, and cylinder are denoted as follows: , , ;in, It is a random parameter vector; The system's generalized coordinate vector is uniformly represented as: ; For any moving member r, its axial position Represented as:

[0025] In the formula, The initial position of the moving component, This refers to movement speed.

[0026] If the left and right sides of the movable component are respectively divided into and For each beam element, the lengths of the elements on both sides of the moving member are expressed as follows: ; ; in, and These are the left and right boundaries of the moving interval, respectively.

[0027] The above expressions are used to describe the movement of the nut on the lead screw, the movement of the sliding bearing relative to the piston rod, and the movement of the anti-rotation pin relative to the cylinder body, respectively. Figure 2 The changes in node position at different running times are determined by the aforementioned position function and length update relationship.

[0028] No. e The mass matrix, damping matrix, and stiffness matrix of each beam element are expressed as follows: 、 、 ;in, Let be the element length of the e-th beam element at time t, and its value varies with the position of the moving component.

[0029] The system's fundamental matrix is ​​obtained by assembling the individual element matrices: ; ; ; in, Let e ​​be the assembly matrix of the e-th beam element. ], It is a positive integer; An assembly of support and connecting components consisting of paired angular contact ball bearings, lead screw-nut pairs, sliding bearings, and anti-rotation pins. Represented as:

[0030] in, mb Indicates a paired angular contact ball bearing. sn This indicates a leadscrew-nut assembly. sb Indicates a sliding bearing. pin This indicates that the product is intended to prevent resale.

[0031] The coupled dynamic equations are expressed as follows: ; in, For the first r The relative displacement mapping matrix of the connected components; and The first r The relative displacement and relative velocity on both sides of the connecting component. ; Because of B ( tThe relative speed should include the changes in the mobile connection location. item, The first derivative of the relative displacement mapping matrix with respect to time; External load vector; The forces acting in the local coordinate system of the connecting components.

[0032] In this manner, each support and connecting component is coupled with the system dynamics equations using the same displacement mapping and force feedback method.

[0033] The normal contact force of the j-th ball in a mating angular contact ball bearing is adopted using the Hertzian nonlinear contact model. Represented as: ; in, , Called the positive part function or Macaulay brackets, it is used to indicate that contact force is generated only when the contact compressibility is positive; For the first in a paired angular contact ball bearing j The normal contact deformation of each ball, The contact coefficient is Hertz. The contact damping coefficient is... It is the first derivative of the normal contact deformation of the j-th ball with respect to time, i.e., the normal contact deformation rate.

[0034] when When the ball disengages from the raceway, the ball contact force is zero; when At that time, the balls are in contact.

[0035] No. j The contact deformation of each ball is determined by the distance between the centers of curvature of the inner and outer ring grooves, the bearing preload deformation, and the relative displacement of corresponding nodes on the screw and cylinder. After local contact direction matrix transformation, the forces acting on each ball are mapped by shape functions to adjacent nodes on the screw and cylinder, forming the forces of the paired angular contact ball bearings. .

[0036] The ball bearing channel of the lead screw-nut assembly includes a load-bearing helical raceway and a non-load-bearing return channel. The composite channel is equivalent to a closed channel of the same total length, denoted as [equation missing]. ; No. j The arc length coordinates of each ball in the equivalent channel Represented as: ; Corresponding position phase for: ; in, The tangential velocity of the ball in the channel.

[0037] Based on arc length coordinates Determine whether the ball is located in the bearing raceway or the return path, and define the contact activation function. : ; No. j Normal contact force of each ball Represented as: ; in, For the first in the lead screw-nut pair j The normal contact deformation of each ball, For the first j Hertzian normal contact stiffness coefficient of each ball. Normal contact damping coefficient, This represents the rate of change of the normal contact deformation.

[0038] The forces exerted by each ball are mapped to adjacent nodes of the lead screw and piston rod via the contact direction matrix and beam element shape function, forming a lead screw-nut side effect force. .

[0039] The sliding bearing is divided circumferentially into Each slice is equivalent to a massless spring-damped unit.

[0040] No. Normal contact force of each slice Represented as: ; in, For the first The equivalent area of ​​each slice Equivalent stiffness per unit area For equivalent damping per unit area, For the piston rod and sliding bearing in the first Normal contact deformation along each slice direction Let be the first derivative of the normal contact deformation amount with respect to time along the γ-th slice direction, i.e., the normal contact deformation rate.

[0041] The contact forces of each slice are summarized into the sliding bearing force after direction transformation and shape function mapping. .

[0042] The anti-rotation pin employs a linear spring-damped model in its constrained degrees of freedom, excluding the piston rod feed direction, and its force... Represented as: ; in, , These are the equivalent stiffness matrix and damping matrix of the anti-rotation pin, respectively. To prevent the relative displacement vector on the constrained degrees of freedom from being converted, This is the corresponding relative velocity vector.

[0043] Using observation vectors Extracting the vibration response at a specified position of the piston rod from the system's generalized coordinates: ; When evaluating unidirectional vibration displacement This represents the displacement component in that direction; when evaluating the radial resultant displacement, the response amplitude can be... Defined as: ; in, and These are the vibration displacement components of the piston rod at a specified position along two mutually orthogonal radial directions.

[0044] Using the implicit Newmark-β method, generalized The system dynamic equations are solved using other transient integration methods, and the moving position, time-varying element matrix, ball position and contact state are updated in each integration step to obtain the full time history of the vibration response.

[0045] Step S2: Determine the time-by-time limit states and reliability indicators: Let the analysis time interval for the electric cylinder be: ; Discretize the time interval into: , ; in, For time step, This represents the total number of time steps.

[0046] In the k Establish the limit state function at each discrete time node: ; When the allowable threshold for vibration response remains constant throughout the entire analysis interval, then ; when When, it indicates that the electric cylinder is at a certain time. In a reliable state; when When, it indicates that the electric cylinder is at a certain time. If the vibration response exceeds the limit, the system is in a state of failure.

[0047] time Instantaneous failure probability : ; The corresponding instantaneous time-varying reliability is: ; After traversing all discrete time points, we obtain , ,..., And form a time-varying reliability curve of the vibration response of the electric cylinder.

[0048] Will This is called instantaneous time-varying reliability, used to describe the marginal failure risk at each discrete moment; Most unfavorable moment Defined as: ; ; Minimum reliability at the worst-case scenario for: ; Maximum failure probability at the most unfavorable moment for: ; The most unfavorable moment is determined through time-by-time probability analysis, without being pre-limited to the peak moment of the vibration response under nominal parameter conditions, thus allowing for shifts in the peak time of different random parameter samples. To reduce the positioning error caused by the discrete time step, candidate most unfavorable discrete moments are first determined based on the full time history calculation results, and then the time step is reduced and recalculated in adjacent time intervals until the difference between two adjacent most unfavorable moments is less than the preset time error limit.

[0049] Step S3: Construct the random sample and full-time response matrix: In this embodiment, typical structural parameters of an electric cylinder are selected to form a random parameter vector. Among them, random parameters include the diameter of the ball bearings in the lead screw-nut pair. Angular contact ball bearing ball diameter Preload of lead screw-nut pair Preload of paired angular contact ball bearings Lead screw Lead screw shaft diameter Piston rod diameter and cylinder diameter .

[0050] ; In this embodiment, each random parameter is independent of the others and adopts a normal distribution determined by the mean and coefficient of variation. Example parameters are shown in Table 1. If there is a correlation between the parameters, the Nataf transformation or Rosenblatt transformation is used to process the correlation before the distribution mapping.

[0051] Table 1 Typical structural parameters of electric cylinders

[0052] Unit hypercube samples are generated using Hammersley low-bias sequences. The i-th Hammersley low-bias sequence sample point is located at... d 1D unit hypercube Inside; ; in, They are distinct prime numbers. For prime numbers p The root inverse function of the basis. This sample point is used to subsequently map to random parameter samples that conform to the actual probability distribution through the inverse cumulative distribution function.

[0053] Initial sample points within a unit hypercube are generated using Hammersley low-bias sequences.

[0054] Specifically, let N be the total number of samples required, and d be the total number of random parameters. For the i-th sample, construct... d dimensional vector ;in, They are distinct prime numbers. For prime numbers p The root inverse function with basis is used to map integer i to a uniformly distributed number in the interval [0,1).

[0055] The resulting N sample points are uniformly distributed within a d-dimensional unit hypercube. Inside. Then, the quantity... Substituting each dimension into the inverse cumulative distribution function of the corresponding random parameter yields a sample of physical parameters that conforms to the actual probability distribution of each parameter. ;in, , Let be the inverse cumulative distribution function of the j-th random parameter; Let be the j-th dimension component of the i-th sample point.

[0056] It should be noted that the above physical parameter samples are used for subsequent batch simulation calculations of the coupled dynamics model. This sampling method can effectively reduce the number of samples required and lower computational costs while ensuring uniform coverage of the parameter space.

[0057] Each set of random parameter samples is input into the coupled dynamics model of the electric cylinder, and the output is... The response sequence arranged in chronological order is the full-time history of the vibration response corresponding to the i-th sample.

[0058] And construct the vibration response matrix over the entire time history: ; in, ; Each row of the response matrix corresponds to the complete time history of a set of random parameter samples, and each column corresponds to a random vibration response sample at a discrete time.

[0059] This embodiment can generate 100 sets of random parameter samples, and after randomly shuffling the sample order, divide them into training and test sets; of which, 80 sets are used to train the surrogate model, and 20 sets are used to test the surrogate model; the test samples do not participate in the training of the surrogate model or the selection of hyperparameters. The number of samples and the division ratio can be adjusted according to the dimension of the random parameters, the computational cost of the dynamic model, and the convergence of the surrogate model.

[0060] Step S4: Construct three types of proxy models and determine the preferred proxy model form: To improve the computational efficiency of time-by-time large sample reliability analysis, Kriging surrogate model, chaotic polynomial expansion surrogate model and extreme learning machine neural network surrogate model were established using the same training and test samples, respectively.

[0061] For the first k At each discrete time point, the input to the surrogate model is a random parameter vector. The output is the transient vibration response: ; The Kriging proxy model is represented as follows: ; in, ; For the regression matrix, ; To train the correlation matrix of the samples, ; To train the response vector; Let be the correlation vector between the point to be predicted and each training sample. For the point to be predicted The regression basis function vector calculated at that point. This is a deterministic trend term; This is the vector of coefficients for the regression trend term. denoted as the number of training samples; p represents the number of regression basis functions.

[0062] When using the Gaussian correlation function, any two training samples and The correlation function between them is expressed as: ; in, These are the relevant hyperparameters determined through maximum likelihood estimation.

[0063] The multinomial chaotic expansion surrogate model transforms mutually independent original random parameters into standard random variables; if the original parameters are correlated, an equal-probability independence transformation is first performed: ; in, Let be a vector of d-dimensional independent standard random variables obtained through equal probability transformation.

[0064] When the original random parameters follow a normal distribution, Hermite polynomials orthogonal to the standard normal probability measure are used as one-dimensional basis functions, and multidimensional orthogonal polynomial bases are constructed through the tensor product of each basis function.

[0065] The chaotic polynomial expansion model is represented as: ; in, , For multiple indicators, ; For a set of multiple indices whose total order does not exceed c, is the expansion factor.

[0066] Construct a design matrix based on training samples. The expansion coefficients are solved using the regularized least squares method: ; in, is the regularization coefficient; when When the above equation degenerates into an ordinary least squares solution; The coefficient vector of the chaotic polynomial expansion; Let A be the identity matrix of order |A(d,c)|.

[0067] Extreme Learning Machine (ELM) neural networks employ a single hidden layer feedforward structure, and their prediction function... Represented as: ; In the formula, For the index of the hidden layer, =1,2,..., L ; L This represents the number of neurons in the hidden layer. For the input layer to the first The connection weights of each hidden layer node; For the first The bias of each hidden layer node; For activation functions; These are the connection weights from the hidden layer to the output layer.

[0068] For the training samples, construct the hidden layer output matrix. : ; Input weights and bias After being randomly generated and then fixed, the output layer weights are solved using the Moore-Penrose generalized inverse: ; in, The Moore-Penrose generalized inverse of the hidden layer output matrix H; This is the weight vector of the ELM output layer.

[0069] Regularization can also be used: ; For the m Candidate agent model in the first k The coefficient of determination for the test results at each evaluation time point is: ; The normalized root mean square error is: ; in, For the number of test samples, For the first k The average value of the test response at each evaluation time. The true response of the i-th test sample at the k-th evaluation time is calculated by the original dynamic model. Let m be the predicted response of the agent model for this test sample.

[0070] If the response variance at a certain evaluation moment is less than the preset tolerance, that moment is not used for calculation. The normalized denominator of NRMSE uses ;in, To ensure positive tolerance, in order to avoid division by zero at the initial rest time or in the zero response interval. and These are the maximum and minimum values ​​of the actual response of the test sample at the k-th evaluation time, respectively.

[0071] To avoid selecting a surrogate model based solely on accuracy evaluation results at a single moment, a set of evaluation time points covering the entire analysis period is chosen. And construct a comprehensive evaluation function: ; in, For interval task reliability, it is used to describe the reliability from the initial time to... The probability that vibration exceeding the limit has never occurred. , and For pre-defined non-negative weights, and ; The first term is used to evaluate the coefficient of determination of the surrogate model over the entire time history, the second term is used to evaluate the average prediction error, and the third term is used to limit the occurrence of excessive prediction errors at local time points.

[0072] The comprehensive evaluation function The candidate model that has the smallest minimum determination coefficient and the maximum normalized root mean square error is selected as the preferred surrogate model form. ; Using a preferred surrogate model approach, a transient vibration response surrogate model is established for each discrete time node: ; The full-time history proxy model family consists of proxy models for all time nodes: ; Figures 3-8 The prediction results of the Kriging, PCE, and ELM models at representative evaluation time points are presented respectively. The left side of each figure compares the calculated values ​​of the dynamic model with the predicted values ​​of the surrogate model, while the right side observes the relative error distribution; finally, the results for all evaluation time points are presented. NRMSE and comprehensive evaluation function Determine the optimal model. If none of the three models meet the preset accuracy requirements, supplement the training samples, calculate the full-time history response of the new samples, and repeat the training and evaluation.

[0073] Step S5: Calculate the time-varying reliability of the vibration response step by step: M sets of large-scale random parameter samples for reliability analysis were generated using the Hammersley low-bias sequence and probability distribution mapping method. ; This embodiment is acceptable. ; For the k At each discrete time point, inputting the large-scale random parameter samples into the corresponding transient vibration response surrogate model yields: ; No. l The sample at the th k The predicted limit state value at time t is: ; in, For a moment The allowable threshold for vibration response; Define the failure indication function: ; No. k The estimated failure probability at time t is: ; The corresponding time-varying reliability estimate is: ; Calculate in sequence This will allow you to obtain the complete time-varying failure probability curve. and complete time-varying reliability curve .

[0074] In this embodiment, the surrogate model is only used to replace the repeated solution of transient dynamics under large-scale random sample conditions. The vibration response corresponding to the training sample and the test sample is still calculated by the original electric cylinder coupled dynamics model.

[0075] Step S6: Most unfavorable moment, risk time interval, and extended reliability assessment: Determine the most unfavorable moment based on the time-varying reliability curve: ; The corresponding minimum reliability is: ; The corresponding maximum failure probability is: ; Output the vibration response probability density, cumulative distribution function, limit state distribution, and the influence of key random parameters at the most unfavorable moment.

[0076] Let the preset reliability threshold be... Risk time interval Defined as: ; Consecutive risk time points are combined into one or more risk time periods to characterize the time range during electric cylinder operation where key monitoring or adjustment of control parameters is required.

[0077] When the evaluation requirement is that the electric cylinder must not experience vibration response exceeding the limit throughout the entire operating range from the initial moment to the current moment, the interval limit state is established: ; For the l A random sample, interval limit state estimate for: ; Interval task reliability estimate for: ; in, ; Instantaneous time-varying reliability is used to determine the degree of danger at a certain moment; interval task reliability is used to evaluate the probability that no vibration exceeding the limit has occurred from the start of operation to the current moment.

[0078] When it is necessary to identify the most unfavorable moment and key random parameters within the risk time interval, for the first... j The mean or standard deviation of a random parameter distribution Apply positive and negative perturbations.

[0079] Dimensionless local reliability sensitivity is calculated using central difference: ; In the formula, This represents the disturbance of the distributed parameters.

[0080] To reduce random sampling fluctuations, the same standard low-biased sample is used under the condition of positive and negative perturbation of the distribution parameters, and the perturbed random parameter sample is obtained by different distribution mappings.

[0081] according to The size of the random parameters is sorted to identify structural or assembly parameters that have a significant impact on reliability at the worst-case scenario.

[0082] In an exemplary electric cylinder model, 100 sets of dynamic samples are generated using the random parameters and probability distribution shown in Table 1, of which 80 sets are used as training samples and 20 sets are used as test samples. Three types of surrogate models are established using the same training samples, test samples and evaluation time nodes, and the preferred model form is determined according to the comprehensive evaluation function.

[0083] Figure 9 The statistical results of small samples from the finite element method (FEM) were compared with those of large samples generated by the optimized surrogate model. The bar or scatter distribution represents the empirical probability density obtained from the finite element samples, while the continuous curve represents the probability density and cumulative distribution obtained from the large samples of the surrogate model. The consistent distribution trends of both indicate that the optimized surrogate model can maintain the statistical characteristics of vibration displacement.

[0084] exist Figure 9 In the example shown, the allowable vibration displacement threshold is set to 90.5 μm. At a representative evaluation time, the cumulative distribution function at this threshold is approximately 0.822, representing the instantaneous reliability at that time. This value is only used to illustrate the probability reading method at a single moment; the main evaluation result of this invention is... Figure 10 The instantaneous time-varying reliability curve, reliability threshold, worst-case moment, and risk time interval are shown. Figure 11 and Figure 12 Time-varying reliability sensitivity for the mean and standard deviation of random parameters is presented. The results show that the lowest reliability point is located in the neighborhood of the vibration response peak; key parameters should be sorted according to the absolute value of sensitivity.

[0085] Under the example conditions, the local minimum of time-varying reliability is usually located near the peak of the vibration response. However, since the random parameters of the structure affect the magnitude and timing of the peak, the moment of lowest reliability does not necessarily coincide perfectly with the moment of maximum vibration displacement in the nominal parameter model. Therefore, the true worst-case scenario should be identified based on the reliability results for all discrete time periods.

[0086] The numerical examples above are only used to illustrate the implementation process of the method of the present invention and do not constitute a limitation on the random parameter type, sample size, surrogate model parameters, vibration response threshold and reliability calculation results.

[0087] The above are merely preferred embodiments of the present invention. For those skilled in the art, any equivalent changes or substitutions made to the beam element type, transient solution method, random sampling method, surrogate model type, response amplitude operator, allowable threshold form, and failure probability estimation method without departing from the core concept of the present invention—full-time history vibration response, time-by-time limit state, time-varying reliability, and identification of the most unfavorable moment—should be included within the scope of protection of the present invention.

[0088] Some of the data in the above formula are calculated by removing dimensions and taking their numerical values. The formula is the closest to the real situation obtained by software simulation of a large amount of collected data. The preset parameters and preset thresholds in the formula are set by those skilled in the art according to the actual situation or obtained through simulation of a large amount of data.

Claims

1. A time-varying reliability analysis method for the vibration response of an electric cylinder, characterized in that, include: A coupled dynamic model is established to determine the random parameters and their probability distribution, the analysis time interval, and the allowable threshold for vibration response. The coupled dynamic model is based on the positional changes of the moving component of the electric cylinder, support constraints, and nonlinear contact effects. Random parameter samples are generated using low-biased sequences, and the initial sample points are transformed into an input sample set that conforms to the original probability distribution of each random parameter through the inverse function of the cumulative distribution function of each random parameter. Each random parameter sample in the input sample set is input into the coupled dynamic model to obtain the full-time history of the vibration response corresponding to each sample; each random parameter sample and its corresponding full-time history of vibration response are combined into a sample pair to construct a training sample set and a test sample set; and the full-time history of vibration response at each discrete moment and the allowable threshold of vibration response are used to establish a time-by-time limit state function at each discrete moment. Multiple proxy models are constructed, and the optimal proxy model is determined according to a unified full-time history accuracy index; The preferred surrogate model is used to establish a transient vibration response surrogate model for each discrete moment. The failure probability and instantaneous time-varying reliability are calculated using random parameter samples to form an instantaneous time-varying reliability curve. Candidate worst-case moments are determined based on instantaneous time-varying reliability curves. The time step of the neighborhood is densified to determine the worst-case moment, minimum reliability, maximum failure probability, and risk time interval, and the sensitivity of key random parameters is output.

2. The method for time-varying reliability analysis of vibration response of an electric cylinder according to claim 1, characterized in that, The process of constructing the coupled dynamics model includes: The lead screw, piston rod, and cylinder are discretized using beam elements. A time-varying position function is used to describe the positional changes of the lead screw-nut pair, sliding bearing, and anti-rotation pin relative to their respective load-bearing components. The lengths of the beam elements on both sides of the moving component are updated based on this position function, and a mass matrix varying with time and random parameters is obtained. Damping matrix and stiffness matrix ; The system generalized coordinate vectors corresponding to the mass matrix, damping matrix, and stiffness matrix. Including the generalized coordinate vector of the lead screw Generalized coordinate vector of the piston rod and the generalized coordinate vector of the cylinder block ; ； Based on the quality matrix Damping matrix and stiffness matrix Construct the coupled dynamic equations of the electric cylinder: ； in, It is a random parameter vector; External load vector; A collection of supporting and connecting components; Displacement mapping matrix of the r-th type of support or connection component; Let be the force vector of the r-th type of support or connection component in its local coordinate system; The first derivative of the generalized coordinate vector with respect to time t; is the second derivative of the generalized coordinate vector with respect to time t; r is the index of the support or connection component type; and These represent the relative displacement and relative velocity on both sides of the r-th type of connecting component, respectively. The supporting and connecting components include at least a paired angular contact ball bearing, a lead screw-nut pair, a sliding bearing, and an anti-rotation pin; wherein, the paired angular contact ball bearing and the lead screw-nut pair adopt a Hertzian nonlinear contact model with a contact opening and closing criterion, the ball position of the lead screw-nut pair is determined by equal perimeter channel mapping, the sliding bearing is modeled using the Winkler foundation model and the slicing method, and the anti-rotation pin is modeled using a linear spring damping model; The vibration response output at a specified position of the piston rod under given random parameters is defined by the observation vector h. .

3. The method for time-varying reliability analysis of vibration response of an electric cylinder according to claim 2, characterized in that, The damping matrix In addition to structural damping, it also includes velocity-related additional terms caused by time-varying coordinate mapping and moving connection positions; the force vector Defined according to the direction acting on the connecting positive and negative nodes, its reaction force is determined by the mapping matrix. The transposed part is automatically assembled to the other node.

4. The time-varying reliability analysis method for vibration response of an electric cylinder according to claim 1, characterized in that, The process of determining the most unfavorable moment and the lowest reliability includes: Discretize the analysis time interval into: ,in, For time step, For discrete time intervals, use indexes. ; This represents the total number of time steps. In the Establish the limit state function at each discrete time point. : ； in, For a moment The allowable threshold for vibration response, For vibration response amplitude operator; when When indicates a reliable state, This indicates a failure state; It is a random parameter vector; Failure probability at each time step : ； in, Indicates the probability of the random event within the parentheses occurring; Instantaneous time-varying reliability is obtained based on the time-by-time failure probability at each discrete moment. : ； The instantaneous time-varying reliability curve is obtained based on the instantaneous time-varying reliability, and the most unfavorable moment is determined. and minimum reliability : ； ； After initially obtaining the discrete time node with the lowest reliability, the time step size of its adjacent time intervals is reduced and recalculated until the difference between two adjacent most unfavorable moments is less than the preset time error limit.

5. The time-varying reliability analysis method for vibration response of an electric cylinder according to claim 1, characterized in that, The method for obtaining the full-time history of the vibration response corresponding to each sample includes: Generate the i-th Hammersley low-bias sequence point within a d-dimensional unit hypercube. The probability distribution mapping is completed through the inverse function of the cumulative distribution function of each random parameter: ； Where j is the index of the random parameter, j=1,2,...,d, and d is a positive integer; Let be the cumulative distribution function of the j-th random parameter; i = 1, 2, ..., N; where i is the sample index of the random parameter, i = 1, 2, ..., N, and N is the total number of random parameter samples; Let j be the j-th dimension component of the i-th sample point; The random parameter samples are input into the coupled dynamic model, and the vibration response amplitude is calculated at all discrete time points. And construct the full-time history response matrix. : ； in, ; Each row of the full-time history response matrix represents the response time history corresponding to a set of random parameter samples, and each column represents a random vibration response sample at a discrete moment; the random parameters include the ball diameter of the lead screw-nut pair. Angular contact ball bearing ball diameter Preload of lead screw-nut pair Preload of paired angular contact ball bearings Lead screw Lead screw shaft diameter Piston rod diameter and cylinder diameter .

6. The time-varying reliability analysis method for vibration response of an electric cylinder according to claim 1, characterized in that, The construction of multiple proxy models includes: Kriging surrogate model, chaotic polynomial expansion surrogate model and extreme learning machine neural network surrogate model were constructed using the same training samples, test samples and evaluation time points, respectively. The Kriging surrogate model consists of a regression term and a zero-mean Gaussian stochastic process; The chaotic polynomial expansion proxy model uses an orthogonal polynomial basis that matches the probability distribution of random parameters for truncated expansion. The extreme learning machine neural network surrogate model adopts a single hidden layer feedforward structure. The input weights and hidden layer biases are randomly generated and then fixed. The output weights are obtained through the generalized inverse or regularized least squares solution of the hidden layer output matrix.

7. The method for time-varying reliability analysis of vibration response of an electric cylinder according to claim 6, characterized in that, The method for determining the preferred proxy model includes: Calculate the coefficient of determination at each evaluation time point. and normalized root mean square error And construct a comprehensive evaluation function for the entire time history: ； Where m is the index of the proxy model; , and For pre-defined non-negative weights, and ; For discrete time intervals, use indexes. This is the set of time indices used in the accuracy evaluation of the surrogate model. This represents the number of evaluation time points included in the set. The evaluation time point set does not include time points where the response variance is lower than the preset tolerance. The denominator of the normalized error uses the response range and positive tolerance. The larger value in; Will The surrogate model that is smallest and meets the preset accuracy requirement is determined as the preferred surrogate model. When none of the three surrogate models meet the requirements, additional training samples are added and the training and evaluation are repeated.

8. The time-varying reliability analysis method for vibration response of an electric cylinder according to claim 4, characterized in that, The calculation process for the time-by-time failure probability also includes: M sets of random parameter samples are generated for reliability analysis and input into the transient vibration response surrogate model at each discrete time to obtain the predicted response. And estimate the time using the failure indication function. failure probability : ； in, For the index of the random parameter sample group, , It is a positive integer; This is a failure indication function; This is the vibration response amplitude operator.

9. The time-varying reliability analysis method for vibration response of an electric cylinder according to claim 4, characterized in that, The instantaneous time-varying reliability also includes: ； in, For a moment The time-varying reliability estimate; Based on the preset reliability threshold Determine the risk time range When it is necessary to evaluate the probability that vibration exceeding the limit will not occur within a specified operating range, an interval limit state is established. It also calculates the reliability of the interval task; when it is necessary to identify key random parameters, it calculates the local sensitivity of the instantaneous time-varying reliability to the corresponding distributed parameters by perturbing the mean or standard deviation of the random parameter distribution; in, From the initial time to The interval limit state; It serves as an index for discrete time intervals within the interval.

10. The method for time-varying reliability analysis of vibration response of an electric cylinder according to claim 1, characterized in that, The full-time history of the vibration response is the vibration response at each discrete moment within the complete analysis time interval at a specified position of the piston rod; the allowable threshold for the vibration response is a pre-set boundary value used to determine whether the vibration response exceeds the limit.

Images