WO2023125026A1 - 一种摩擦运动副等效模型及其建模方法 - Google Patents

一种摩擦运动副等效模型及其建模方法 Download PDF

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WO2023125026A1
WO2023125026A1 PCT/CN2022/139144 CN2022139144W WO2023125026A1 WO 2023125026 A1 WO2023125026 A1 WO 2023125026A1 CN 2022139144 W CN2022139144 W CN 2022139144W WO 2023125026 A1 WO2023125026 A1 WO 2023125026A1
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friction
pair
frequency
model
equivalent model
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邓兆祥
贺本刚
张宇彪
陈之春
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重庆大学
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    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/27—Design optimisation, verification or simulation using machine learning, e.g. artificial intelligence, neural networks, support vector machines [SVM] or training a model
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N3/00—Computing arrangements based on biological models
    • G06N3/12—Computing arrangements based on biological models using genetic models
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2111/00—Details relating to CAD techniques
    • G06F2111/06—Multi-objective optimisation, e.g. Pareto optimisation using simulated annealing [SA], ant colony algorithms or genetic algorithms [GA]
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces

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  • the invention belongs to the field of mechanical engineering, and in particular relates to an equivalent model of a friction motion pair and a modeling method thereof.
  • Friction is a natural phenomenon in which objects contact each other under the action of normal pressure, and the asperities on the surface fit together. When relative sliding occurs or tends to slide relative to each other, a resistance against sliding is generated on the contact surface. The resistance is called friction.
  • the mechanism of friction is quite complex, and a lot of research has been done; the characteristics of the friction pair, that is, the relationship between the friction response and the input motion conditions (positive pressure, tangential relative velocity, etc.) It is closely related to the properties of the friction pair materials, especially the properties of the friction contact surface, and any factors that affect the properties of the contact surface will also affect the friction characteristics.
  • the empirical model or semi-empirical model of the friction pair can only be constructed by means of experimental methods.
  • empirical or semi-empirical models of friction pairs such as LuGre model, GW model, fractal contact model, Stribeck model, Karnopp model, Dahl model, GMS model, GBM model, etc. Some of them are relatively simple, some It is very complicated, but it mainly characterizes the low-frequency characteristics of friction, and generally does not include high-frequency characteristics.
  • the low-frequency vibration excitation causes the friction pair in the mechanical system to generate secondary friction excitation, which often radiates high-frequency noise, indicating that the high-frequency vibration of the structure is aroused. Therefore, the high-frequency characteristics of the friction force must be Having attention. Therefore, the present invention provides a new equivalent model of friction kinematic pair, which can characterize the friction characteristics of broadband, and has a relatively simple form, so as to be used for the forced vibration response prediction of local nonlinear mechanical systems with friction pairs , analysis, design and control.
  • the purpose of the present invention is to provide an equivalent model of a friction motion pair and its modeling method, which can characterize the friction characteristics of a wide frequency band, and has a relatively simple form, so that it can be used for forcing a local nonlinear mechanical system with a friction pair Vibration response prediction, analysis, design and control.
  • the present invention provides an equivalent model of a friction pair and its modeling method.
  • the friction test data of the friction pair are statistically analyzed and modeled according to the high and low frequency bands and superimposed, and the structure can simultaneously reflect the friction pair high Equivalent models for low-frequency characteristics;
  • the positive pressure input information is introduced and the traditional LuGre model is deformed to obtain the implicit relationship between the friction response and the relative velocity of the friction pair and the positive pressure input;
  • F(N, v) is the friction force output by the equivalent model of the friction pair, which is a function of the normal pressure N of the friction pair and the tangential relative motion velocity v;
  • z is the introduced intermediate variable, which is usually a function of time;
  • t is the time , ⁇ represents the frequency;
  • s(N, v), f(N), R(N, ⁇ ) are all introduced intermediate functions;
  • r(t) is the time history of friction in the high-frequency band, and its self-power spectral density is R (N, ⁇ ); the rest of the parameters are undetermined coefficients.
  • the material of the friction pair is determined and the properties of the friction contact surface are determined.
  • the undetermined parameters of the equivalent model of the friction pair can be identified and determined through the friction test data, so that the equivalent model modeling of a specific friction pair can be completed.
  • the specific steps of its equivalent model modeling include:
  • S1 conducts the friction test of the friction pair sample and records the relevant signals of the friction test.
  • S2 preprocesses the test signal to obtain the friction-relative motion velocity, friction-positive pressure scatter diagram and friction autopower spectrum.
  • S3 is based on the minimum The square method determines the high-frequency friction model parameters, S4 identifies the low-frequency friction model parameters based on the least square method and genetic algorithm, and S5 superimposes to obtain a complete equivalent model.
  • the beneficial effect of the present invention is that: the equivalent model proposed by the present invention can not only reflect the relationship between friction force, normal pressure and relative velocity, but also can express the characteristics of friction force in a wide frequency band, and has good applicability.
  • Figure 1 is a photo of a standard sample of a typical friction pair material
  • Figure 2 is a typical measured friction signal and relative displacement signal diagram of the friction pair test of the standard sample
  • Fig. 3 is the self-power spectrum diagram of the high-frequency part of the friction force under the triangular displacement wave excitation of the friction pair PC/ABS-PC/ABS with a positive pressure of 30N and a velocity of 3mm/s;
  • Fig. 4 is the friction force-velocity curve and the friction force-normal pressure curve fitting effect diagram under the triangular displacement wave excitation of the friction pair PC/ABS-PC/ABS;
  • Fig. 5 is an iterative effect diagram of the friction time history model of the friction pair PC/ABS-PC/ABS under the excitation of the friction pair PC/ABS-PC/ABS with a positive pressure of 30N and a velocity of 1mm/s triangular displacement wave;
  • Fig. 6 is the prediction effect diagram of the equivalent model of friction force time history under the triangular displacement wave excitation of friction pair PC/ABS-PC/ABS;
  • Fig. 7 is a comparison chart of the time history of the modeling predicted friction force and the measured value of other typical friction pairs using the equivalent model of the present invention.
  • the characteristics of the low frequency band are basically consistent with the existing research conclusions, and the characteristics of the high frequency band are presented
  • the power function feature that is, the self-power spectrum of the friction force decays according to the nonlinear power function law with the increase of the frequency.
  • the friction force in the high frequency band increases with the increase of the positive pressure, but the relationship with the relative motion speed is not obvious.
  • the present invention proposes an equivalent model that superimposes low-frequency band characteristics and high-frequency band characteristics, that is, the model structure is divided into low-frequency bands and high-frequency bands, and in the identification of model parameters, the corresponding frequency band filtering signals of test data are also used.
  • the traditional LuGre model has a clear physical meaning and a simpler form, that is, there are fewer model parameters to be identified, and the friction characteristics that can be characterized include that the maximum static friction is greater than the sliding friction , pre-sliding displacement hysteresis, Stribeck effect and sliding friction time delay, so the traditional LuGre model is adopted, and the positive pressure relationship is introduced to modify the model structure appropriately to characterize the more complete friction characteristics of the low frequency band; for the high frequency band model, then Directly construct the power function model of friction power spectrum including positive pressure input; considering the randomness of high-frequency friction signal, the friction time history of high-frequency band can be further obtained by combining random phase, so the low-frequency band can be superimposed directly in the time domain Friction, you get a full friction output across the frequency band.
  • the present invention proposes a new broadband friction equivalent model of the friction kinematic pair as follows.
  • Equation (1) expresses the friction force F(N, v) output by the equivalent model of the friction pair from the low frequency part and the high-frequency part r(t) are directly superimposed in the time domain, which is an implicit function of the friction pair normal pressure N and the tangential relative motion velocity v; where z is the introduced intermediate variable, usually a function of time t.
  • the friction force in the low frequency band is given by equations (2) ⁇ (4), equation (4) expresses the power function relationship between friction force f(N) and positive pressure, equation (3) expresses friction force s(N, v) relationship with positive pressure and relative motion velocity, N 0 is the reference positive pressure, equation (2) provides the differential equation that the intermediate variable z satisfies, obviously, for given N, v, by equation (2) can By solving z and its first order derivative, the time history of friction in the low frequency band is determined.
  • the friction force of the high frequency part is given by Equation (5), where ⁇ represents the frequency, ⁇ 0 is the boundary frequency between the high and low frequency bands, and R(N, ⁇ ) is the self-power spectral density function of the friction force of the high frequency part.
  • the random nature of the friction force in the frequency band is assumed to have a random phase spectrum, so the time history r(t) of the friction force in the high frequency band can be obtained by inverse Fourier transform.
  • the remaining parameters in equations (1) to (5) are undetermined coefficients of the equivalent model. For a specific friction pair, that is, if the material of the friction pair is determined and the properties of the friction contact surface are determined, these undetermined parameters should be uniquely determined.
  • equations (1) to (5) the model structure provided by the present invention is shown in equations (1) to (5), in which the undetermined coefficients can be determined through experimental identification, so as to complete the modeling of the equivalent model of the friction kinematic pair.
  • the specific steps of the method are as follows.
  • S1 Carry out the friction test of the friction pair sample and record the relevant signals of the friction test.
  • a standard friction pair test sample should be prepared for the target friction pair, and its material and contact surface properties (surface hardness, roughness, etc.) are consistent with the target friction pair.
  • the friction test should simultaneously record the normal pressure, friction force and tangential relative motion displacement or velocity signals of the friction pair, and should include the pre-slip state and macro-slide state of the friction pair.
  • the reciprocating sliding test of the friction pair can be carried out, and the one-way sliding test of the friction pair can also be carried out. Choosing different friction test methods often requires different friction pair friction test devices or equipment.
  • the invention recommends the triangular wave displacement excitation of the friction pair reciprocating sliding friction test, which can quickly carry out the test and obtain the friction force and other signals of the required friction state, and can obtain the correlation of the friction pair pre-sliding stage in the reversing section of the relative reciprocating sliding In the rest of the stable sliding section, the relevant information of the macroscopic sliding phase of the friction pair at a constant speed can be obtained, which is convenient for modeling.
  • the relative sliding test of the friction pair is carried out under a given positive pressure. Each test should include a stable sliding state at a given speed and a transition state from static to a given speed. Adjusting the normal pressure and speed can obtain the required model. Friction test signals for various states.
  • S2 preprocesses the test signal to obtain friction force-relative motion velocity, friction force-normal pressure scatter diagram and friction force autopower spectrum.
  • the friction signal measured in the test is filtered to obtain the low frequency band friction signal and the high frequency band friction signal respectively.
  • the relative motion displacement signal and velocity signal (if only one signal is recorded, the other signal can be obtained by its differentiation or integration) and the low-frequency friction signal are cross-referenced, so that it is easy to distinguish the friction force in the macro-sliding and pre-sliding stages, and also It is easy to distinguish the frictional force of the constant speed sliding section.
  • the friction-relative motion velocity scatter diagram can be obtained; using the friction signals of a given speed sliding section under different positive pressures, it can be obtained Friction-Normal Pressure Scatterplot.
  • high-frequency friction-normal pressure scatter diagrams can also be obtained by using high-frequency friction signals, and its self-power spectrum can be obtained by performing frequency spectrum analysis on high-frequency friction signals under a given reference positive pressure.
  • S3 determines the high-frequency friction model parameters based on the least square method.
  • the undetermined parameters in equation (5) can be determined, thereby determining the spectrum model of high-frequency friction.
  • S4 identifies low-frequency friction model parameters based on the least squares method and genetic algorithm.
  • the undetermined parameters in equations (4) and (3) can be determined respectively. Then use the low-frequency friction time history and the corresponding relative motion speed time history, as well as the identified intermediate functions s(N, v) and f(N), based on the genetic algorithm (of course, you can also choose other mature optimization algorithm), the undetermined parameters ⁇ 0 and ⁇ 1 in equations (1) and (2) can be obtained.
  • Figure 1 shows the physical photos of typical material standard samples that constitute the friction pair.
  • a self-developed reciprocating friction testing machine is used to carry out a friction test to obtain relevant measured signals, and then an equivalent model of the friction pair is established according to the method of the present invention, and the friction predicted by the equivalent model The force is compared with the measured friction force to illustrate the validity of the equivalent model.
  • the specific implementation process and results are as follows.
  • Step 1 Install the standard sample of the friction pair on a special friction testing machine, load different positive pressures, set different speeds for triangular wave displacement excitation, make the friction pair sample tangential reciprocating relative motion, record the friction force and relative
  • the time history of displacement, typical friction force and relative displacement measured signals are shown in Fig. 2.
  • the ambient temperature is 20 ⁇ 5°C
  • the humidity is 75% to 85%.
  • the positive pressure and relative velocity selected for the test conditions are shown in Table 1.
  • Step 2 Select the test data of different speeds under 30N pressure, perform 2Hz low-pass filtering on the friction time history, compare the relative displacement and speed time history, and calculate the average value of the friction force for the friction signal in the constant speed sliding stage, and obtain Friction-velocity scatter diagram; select the test data of 3mm/s speed under different positive pressure, perform 2Hz low-pass filtering on the friction time history, and obtain the friction-positive pressure scatter diagram by similar processing; select 30N, 3mm/s In order to remove the influence of the fundamental wave of reciprocating motion, the time history of friction force in one direction of reciprocating motion is intercepted, the DC component is subtracted, and the power spectral density is estimated, that is, the power spectral density curve of friction force in the high frequency band is obtained ; It is also possible to perform high-pass filtering on the friction time history (the cut-off frequency is 2 Hz), and then carry out spectral analysis on the filtered signal to obtain the friction power spectral density curve of the high-frequency part. The results of the two methods are
  • Step 3 Establish a model of friction in the high frequency band. That is, based on the least squares method, according to the measured high-frequency partial friction power spectral density curve, the undetermined parameters in the self-power spectrum model (that is, equation (5)) are obtained.
  • Figure 3 shows the high-frequency partial friction self-power Comparing the curve calculated by the spectral model with the measured curve, and assuming a random phase spectrum (uniform distribution), with the help of Fourier inverse transform, the friction time history of the high frequency part is obtained.
  • the reference positive pressure is 30N.
  • the genetic algorithm is used to take the error standard deviation of the friction time history calculated by the low-frequency friction model and the 2Hz low-pass filter time history of the experimental friction under the working conditions of 30N and 3mm/s as the objective function, and iteratively obtains the optimal objective function.
  • the parameters ⁇ 0 and ⁇ 1 of the hour determine the partial friction model of the low frequency band.
  • Figure 5 shows the comparison of the time history of the low frequency partial friction model prediction and the actual measurement.
  • Step 5 Directly superimpose the friction force of the low-frequency band and the high-frequency band in the time domain to obtain the total friction force of the equivalent model of the friction pair, that is, the friction force equivalent model equation (1)-(5) All the undetermined parameters have been determined, and the modeling of the equivalent model of the friction pair has been completed.
  • the model input that is, the time history of positive pressure and relative velocity (including the case of being constant)
  • the Each intermediate function and intermediate variable in equations (1) to (5) and the final equivalent model output required for calculation can be calculated, that is, the friction time history.
  • Figure 6 shows the comparison effect between the measured time history of friction force and the time history of friction force predicted by the equivalent model under the typical working conditions of friction test of PC/ABS-PC/ABS material vice sample.
  • the PC/ABS-PC/ABS material friction pair equivalent model established by the method of the present invention can accurately predict the friction time history. In fact, it has similar effects for other typical material friction pairs.
  • Figure 7 shows some typical Comparing the time history of the measured friction force with the time history predicted by the equivalent model, the size and change trend of the two are basically consistent, indicating that the equivalent model well characterizes the dynamic characteristics of the friction pair.

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Abstract

本发明属于机械工程领域，具体涉及一种摩擦运动副等效模型及其建模方法，对摩擦副的摩擦力试验数据按高低频段分别进行统计分析建模并进行叠加，构造能同时反映摩擦副高低频特性的等效模型；针对摩擦副的低频段摩擦力特性，引入正压力输入信息并对传统的LuGre模型进行变形得到摩擦力响应与摩擦副相对速度及正压力输入的隐式关系；针对摩擦副的高频段摩擦力特性，引入正压力输入信息并结合幂函数形式的摩擦力自功率谱假设，得到摩擦力响应与摩擦副正压力的隐式关系。本发明提出的等效模型不但能反映了摩擦力与正压力和相对速度的关系，而且可以表达宽频带的摩擦力特性，具有良好的适用性。。

Description

一种摩擦运动副等效模型及其建模方法
本申请要求于2021年12月27日提交中国专利局、申请号为202111616885.4、发明名称为“一种摩擦运动副等效模型及其建模方法”的中国专利申请的优先权，其全部内容通过引用结合在本申请中。
技术领域
本发明属于机械工程领域，具体涉及一种摩擦运动副等效模型及其建模方法。
背景技术
摩擦是物体在法向压力作用下相互接触，表面微凸体间相互嵌合，在发生相对滑动，或有相对滑动的趋势时，在接触表面上产生抵抗滑动的阻力的一种自然现象，其中的阻力被称为摩擦力。摩擦的机制相当复杂，已经有大量的研究；摩擦副的特性，即摩擦力响应与摩擦副输入运动条件(正压力、切向相对速度等)的关系，也往往呈现出相当复杂的状况，且与摩擦副材料性质、尤其是摩擦接触表面的性质紧密相关，凡是影响接触表面性质的因素也都会影响摩擦特性。经典的库仑摩擦定律认为，滑动摩擦力与正压力成正比，但后来的研究表明，滑动摩擦力与正压力不总是成正比，且与相对运动速度密切相关，存在Stribeck效应、宏观滑动时滞、预滑动位移迟滞等多种复杂现象，摩擦力也往往呈现出宽频带的特征。
鉴于摩擦副的复杂特性以及产生摩擦力的复杂机制，目前尚没有办法有效地完成摩擦副的理论建模，因此，只能借助试验方法构建摩擦副的经验模型或半经验模型。关于摩擦副的经验或半经验模型，已有许多种，例如LuGre模型、GW模型、分形接触模型、Stribeck模型、Karnopp模型、Dahl模型、GMS模型、GBM模型等，它们有的比较简洁、有的很复杂，但主要表征的都是摩擦力的低频特性，一般都不包含高频特性。
在实际工程中，低频的振动激励引起机械系统中的摩擦副产生二次摩擦激励，往往都会辐射出高频的噪声，说明激起了结构的高频振动，因此，摩擦力的高频特性必然受到关注。为此，本发明提供一种新的摩擦运动副等效模型， 能够表征宽频带的摩擦力特性，且具有比较简洁的形式，以便用于含摩擦副的局部非线性机械系统的强迫振动响应预测、分析、设计以及控制。
发明内容
本发明的目的在于提供一种摩擦运动副等效模型及其建模方法，能够表征宽频带的摩擦力特性，且具有比较简洁的形式，以便用于含摩擦副的局部非线性机械系统的强迫振动响应预测、分析、设计以及控制。
为了达到上述目的，本发明提供一种摩擦运动副等效模型及其建模方法，对摩擦副的摩擦力试验数据按高低频段分别进行统计分析建模并进行叠加，构造能同时反映摩擦副高低频特性的等效模型；
针对摩擦副的低频段摩擦力特性，引入正压力输入信息并对传统的LuGre模型进行变形得到摩擦力响应与摩擦副相对速度及正压力输入的隐式关系；
针对摩擦副的高频段摩擦力特性，引入正压力输入信息并结合幂函数形式的摩擦力自功率谱假设，得到摩擦力响应与摩擦副正压力的隐式关系。
进一步，其摩擦运动副等效模型的数学方程如下：
Figure PCTCN2022139144-appb-000001
Figure PCTCN2022139144-appb-000002
Figure PCTCN2022139144-appb-000003
f(N)＝aN b+c
Figure PCTCN2022139144-appb-000004
其中F(N，v)为摩擦副等效模型输出的摩擦力，是摩擦副正压力N和切向相对运动速度v的函数；z为引入的中间变量，通常是时间的函数；t表示时间，ω表示频率；s(N，v)、f(N)、R(N，ω)均为引入的中间函数；r(t)是高频段的摩擦力时间历程，其自功率谱密度即R(N，ω)；其余参数均为待定系数，针对特定的摩擦副，即摩擦副的材料确定、摩擦接触表面性质确定，这些待定参数就应该是唯一确定的。
进一步，摩擦副等效模型的待定参数，可经由摩擦试验数据来辨识确定， 从而可完成特定摩擦副的等效模型建模。针对任何特定的摩擦副，其等效模型建模的具体步骤包括：
S1进行摩擦副样件的摩擦试验并记录摩擦力试验相关信号，S2对试验信号预处理，获取摩擦力-相对运动速度、摩擦力-正压力散点图和摩擦力自功率谱，S3基于最小二乘法确定高频段摩擦力模型参数，S4基于最小二乘法和遗传算法识别低频段摩擦力模型参数，S5叠加获得完整等效模型。
本发明的有益效果在于：本发明提出的等效模型不但能反映了摩擦力与正压力和相对速度的关系，而且可以表达宽频带的摩擦力特性，具有良好的适用性。
附图说明
图1为典型摩擦副材料标准样件照片；
图2为标准样件摩擦副试验的典型实测摩擦力信号及相对位移信号图；
图3为摩擦副PC/ABS-PC/ABS正压力30N、速度3mm/s三角位移波激励下的摩擦力高频部分自功率谱图；
图4为摩擦副PC/ABS-PC/ABS三角位移波激励下的摩擦力-速度曲线和摩擦力-正压力曲线拟合效果图；
图5为摩擦副PC/ABS-PC/ABS正压力30N、速度1mm/s三角位移波激励下的低频段部分摩擦力时间历程模型迭代效果图；
图6为摩擦副PC/ABS-PC/ABS三角位移波激励下的摩擦力时间历程等效模型预测效果图；
图7为其它典型摩擦副采用本发明等效模型的建模预测摩擦力时间历程与实测值对比图。
具体实施方式
下面通过具体实施方式进一步详细说明。
在大量摩擦副标准样件摩擦力试验信号采集分析基础上，我们对摩擦力特性进行了统计分析与观察，其低频段的特性与已有的研究结论基本一致，其高频段的特性则呈现出幂函数特征，即摩擦力自功率谱随频率增加而按非线性幂函数规律衰减，通常，高频段的摩擦力会随正压力增加而增加，但与相对运动 速度的关联不明显。因此，本发明提出叠加低频段特性和高频段特性的等效模型，即在模型结构上区分为低频段和高频段，在模型参数的辨识上，也采用试验数据的相应频段滤波信号。针对低频段的模型，则考虑到传统的LuGre模型具有明确的物理意义和较简洁的形式，即较少的需辨识的模型参数，以及所能表征的摩擦特性包括了最大静摩擦力大于滑动摩擦力、预滑动位移迟滞、Stribeck效应和滑动摩擦时滞，因此采用传统的LuGre模型，并引入正压力关系进行模型结构适当修改，以表征低频段更完整的摩擦力特性；针对高频段的模型，则直接构建包含正压力输入的摩擦力自功率谱幂函数模型；考虑到高频摩擦力信号的随机性，就可以结合随机相位进一步获得高频段的摩擦力时间历程，于是在时域直接叠加低频段摩擦力，就获得整个频段的完整摩擦力输出。
按照上述思路，本发明提出摩擦运动副新的宽频带摩擦力等效模型如下所示。
Figure PCTCN2022139144-appb-000005
Figure PCTCN2022139144-appb-000006
Figure PCTCN2022139144-appb-000007
f(N)＝aN b+c  (4)
Figure PCTCN2022139144-appb-000008
方程(1)表达了摩擦副等效模型输出的摩擦力F(N，v)由低频段部分
Figure PCTCN2022139144-appb-000009
Figure PCTCN2022139144-appb-000010
和高频段部分r(t)在时域直接叠加而成，是摩擦副正压力N和切向相对运动速度v的隐函数；其中z为引入的中间变量，通常是时间t的函数。低频段部分的摩擦力由方程(2)～(4)给出，方程(4)表达了摩擦力f(N)与正压力的幂函数关系，方程(3)表达了摩擦力s(N，v)与正压力及相对运动速度的关系，N 0为参考正压力，方程(2)给出了中间变量z满足的微分方程，显然， 针对给定的N、v，由方程(2)可解得z及其一阶导数，从而确定了低频段部分摩擦力时间历程。高频段部分的摩擦力由方程(5)给出，其中ω表示频率，ω 0为高、低频段分界频率，R(N，ω)为高频段部分摩擦力的自功率谱密度函数，考虑高频段部分摩擦力的随机性质，假设具有随机相位谱，于是通过傅里叶逆变换可求出高频段摩擦力时间历程r(t)。方程(1)～(5)中的其余参数均为等效模型的待定系数，针对特定的摩擦副，即摩擦副的材料确定、摩擦接触表面性质确定，这些待定参数就应该是唯一确定的。
针对任何摩擦运动副，本发明给出的模型结构如方程(1)～(5)所示，其中的待定系数可通过试验辨识确定，从而完成摩擦运动副等效模型的建模。该方法的具体步骤如下。
S1进行摩擦副样件的摩擦试验并记录摩擦力试验相关信号。
该步骤应该针对目标摩擦运动副制备标准的摩擦副试验样件，其材料及接触表面性质(表面硬度、粗糙度等)与目标摩擦运动副一致。摩擦试验应同步记录摩擦副的正压力、摩擦力和切向相对运动位移或速度信号，应包含摩擦副预滑动状态和宏观滑动状态。可以进行摩擦副往复滑动试验，也可以进行摩擦副单向滑动试验。选择不同的摩擦试验方式，往往需要不同的摩擦副摩擦力试验装置或设备。本发明推荐采用三角波位移激励的摩擦副往复滑动摩擦试验，可以快速进行试验并获得所需摩擦状态的摩擦力等信号，在相对往复滑动的换向区段，可获得摩擦副预滑动阶段的相关信息，在其余稳定滑动区段，可获得恒速下的摩擦副宏观滑动阶段的相关信息，便于建模使用。通常都在给定正压力下进行摩擦副相对滑动试验，每次试验应包含给定速度的稳定滑动状态和由静止到给定速度的过渡状态，调整正压力和速度，可得到建模所需各种状态的摩擦试验信号。
S2对试验信号预处理，获取摩擦力-相对运动速度、摩擦力-正压力散点图和摩擦力自功率谱。
对试验实测摩擦力信号进行滤波处理，分别获得低频段摩擦力信号和高频段摩擦力信号。将相对运动位移信号和速度信号(若仅记录了一种信号，另一 种信号可由其微分或积分得到)与低频段摩擦力信号相互参照，容易区分宏观滑动与预滑动阶段的摩擦力，也容易区分恒定速度滑动区段的摩擦力。利用给定正压力下不同恒定速度滑动区段的摩擦力信号，可得到摩擦力-相对运动速度散点图；利用不同正压力下给定速度的恒速滑动区段的摩擦力信号，可得到摩擦力-正压力散点图。类似地，利用高频段的摩擦力信号，也可以获得高频段摩擦力-正压力散点图，对给定参考正压力下的高频段摩擦力信号进行频谱分析，可获得其自功率谱图。
S3基于最小二乘法确定高频段摩擦力模型参数。
利用高频段摩擦力-正压力散点图和高频段摩擦力自功率谱图，基于最小二乘法，可确定方程(5)中的待定参数，从而确定高频段摩擦力的频谱模型。
S4基于最小二乘法和遗传算法识别低频段摩擦力模型参数。
利用低频段的摩擦力-正压力散点图和摩擦力-相对运动速度散点图，基于最小二乘法，可分别确定方程(4)和(3)中的各个待定参数。再利用低频段的摩擦力时间历程和对应的相对运动速度时间历程，以及已辨识求出的中间函数s(N，v)及f(N)，基于遗传算法(当然，也可以选择其它成熟的优化算法)，可以求出方程(1)、(2)中的待定参数σ 0和σ 1。
S5叠加获得完整等效模型。
考虑到高频段摩擦力的随机性质，假设它具有随机相位谱，再利用已辨识求出的高频段摩擦力自功率谱密度函数，即(5)式，可通过傅里叶逆变换求出其时间历程，即高频段的摩擦力r(t)，再与S4步求出的低频段摩擦力
Figure PCTCN2022139144-appb-000011
Figure PCTCN2022139144-appb-000012
叠加，即可获得完整的摩擦力输出。至此，等效模型方程(1)～(5)中的所有待定参数及函数都已确定，即获得了目标摩擦运动副完整的等效模型。
实施例：
图1所示为构成摩擦副的典型材料标准样件实物照片。针对特定摩擦副标准样件，采用自研的往复式摩擦试验机开展摩擦试验，获取相关实测信号，然后按照本发明所述方法建立该摩擦副的等效模型，并将等效模型预测的摩擦力 与实测摩擦力进行对比，以说明等效模型的有效性。具体的实施过程和结果如下。
第一步：将摩擦副标准样件安装于专门的摩擦试验机，加载不同正压力，设定不同的速度进行三角波位移激励，使摩擦副样件发生切向往复相对运动，记录摩擦力和相对位移的时间历程，典型的摩擦力及相对位移实测信号如图2所示。试验期间，环境温度在20±5℃，湿度在75％～85％。试验工况选择的正压力和相对速度如表1所示。
表1试验工况
Figure PCTCN2022139144-appb-000013
第二步：选择30N压力下不同速度的试验数据，对摩擦力时间历程进行2Hz低通滤波，对照相对位移及速度时间历程，针对恒速滑动阶段的摩擦力信号，求摩擦力平均值，获得摩擦力-速度散点图；选择不同正压力下3mm/s速度的试验数据，对摩擦力时间历程进行2Hz低通滤波，类似处理得到摩擦力-正压力散点图；选取30N，3mm/s的试验数据，为了去除往复运动的基波影响，截取往复运动的一个方向上的摩擦力时间历程，减去直流分量，对其进行功率谱密度估计，即得到高频段部分摩擦力功率谱密度曲线；也可以对摩擦力时间历程进行高通滤波(截止频率取2Hz)，然后再对滤波信号进行谱分析求出高频段部分的摩擦力功率谱密度曲线，两种做法的结果基本相同。
第三步：建立高频段部分摩擦力的模型。即基于最小二乘法，根据实测的高频段部分摩擦力功率谱密度曲线，求出其自功率谱模型(即方程(5))中的待定参数，图3所示为高频段部分摩擦力自功率谱模型计算曲线与实测曲线的对比，再假设有随机相位谱(均匀分布)，借助傅里叶逆变换，求出高频段部分的摩擦力时间历程。
第四步：建立低频段部分摩擦力模型。首先基于最小二乘法，分别利用实测信号分析得到的摩擦力-正压力散点图和摩擦力-速度散点图，分别通过拟合 f(N)＝aN b+c摩擦力-正压力曲线和
Figure PCTCN2022139144-appb-000014
摩擦力-速度曲线，求出其中的待定参数a、b、c、μ c、μ s、v c、σ 2和δ s，图4所示为PC/ABS-PC/ABS材料副样件摩擦试验的摩擦力-速度散点图和摩擦力-正压力散点图的拟合效果。此处，参考正压力取的是30N。然后采用遗传算法，以低频段摩擦力模型计算出的摩擦力时间历程与30N、3mm/s工况下试验实测摩擦力2Hz低通滤波时间历程的误差标准差为目标函数，迭代出目标函数最小时的参数σ 0、σ 1，即确定了低频段部分摩擦力模型，图5所示为低频段部分摩擦力模型预测与实测的时间历程对比。
第五步：将低频段部分摩擦力与高频段部分摩擦力在时间域直接叠加，就得到摩擦副等效模型总的摩擦力，即摩擦力等效模型方程(1)～(5)中的所有待定参数都已确定，完成了该摩擦副等效模型的建模，利用该等效模型，只要给定模型输入，即正压力与相对速度的时间历程(包括恒为常量的情形)，就可以计算出方程(1)～(5)中的各个中间函数、中间变量以及最终所需计算的等效模型输出，即摩擦力时间历程。图6给出了PC/ABS-PC/ABS材料副样件摩擦试验典型工况下的摩擦力实测时间历程与等效模型预测摩擦力时间历程的对比效果。
显然，按本发明方法建立的PC/ABS-PC/ABS材料摩擦副等效模型能够准确地预测摩擦力时间历程，事实上，针对其它典型材料摩擦副也有类似效果，图7给出了一些典型的实测摩擦力时间历程与等效模型预测时间历程的对比，两者的大小和变化趋势基本一致，说明等效模型很好地表征了摩擦副的动力学特性。
需要提前说明的是，在本发明中，除非另有明确的规定和限定，术语“安装”、“相连”、“连接”、“固定”等术语应做广义理解，例如，可以是固定连接，也可以是可拆卸连接，或一体地连接；可以是直接相连，也可以通过中间媒介间接相连，可以是两个元件内部的连通。对于本领域的普通技术人员而言，可以根据具体情况理解上述术语在本发明中的具体含义。
以上所述的仅是本发明的实施例，方案中公知的具体结构及特性等常识在此未作过多描述。应当指出，对于本领域的技术人员来说，在不脱离本发明结构的前提下，还可以作出若干变形和改进，这些也应该视为本发明的保护范围，这些都不会影响本发明实施的效果和专利的实用性。本申请要求的保护范围应当以其权利要求的内容为准，说明书中的具体实施方式等记载可以用于解释权利要求的内容。

Claims (4)

  1. 一种摩擦运动副等效模型及其建模方法，其特征在于，对摩擦副的摩擦力试验数据按高低频段分别进行统计分析建模并进行叠加，构造能同时反映摩擦副高低频特性的等效模型；
    针对摩擦副的低频段摩擦力特性，引入正压力输入信息并对传统的LuGre模型进行变形得到摩擦力响应与摩擦副相对速度及正压力输入的隐式关系；
    针对摩擦副的高频段摩擦力特性，引入正压力输入信息并结合幂函数形式的摩擦力自功率谱假设，得到摩擦力响应与摩擦副正压力的隐式关系。
  2. 根据权利要求1所述的摩擦运动副等效模型及其建模方法，其特征在于，其摩擦运动副等效模型的数学方程如下：
    Figure PCTCN2022139144-appb-100001
    Figure PCTCN2022139144-appb-100002
    Figure PCTCN2022139144-appb-100003
    f(N)＝aN b+c
    Figure PCTCN2022139144-appb-100004
    其中F(N，v)为摩擦副等效模型输出的摩擦力，是摩擦副正压力N和切向相对运动速度v的函数；z为引入的中间变量，通常是时间的函数；t表示时间、ω表示频率；s(N，v)、f(N)、R(N，ω)均为引入的中间函数；r(t)是高频段的摩擦力时间历程，其自功率谱密度即R(N，ω)；其余参数均为待定系数，针对特定的摩擦副，即摩擦副的材料确定、摩擦接触表面性质确定，这些待定参数就应该是唯一确定的，N 0为参考正压力，ω 0为高、低频段分界频率。
  3. 根据权利要求2所述的摩擦运动副等效模型及其建模方法，其特征在于，摩擦副等效模型的待定参数，可经由摩擦试验数据来辨识确定，从而可完成特定摩擦副的等效模型建模。针对任何特定的摩擦副，其等效模型建模的具体步骤包括：
    S1进行摩擦副样件的摩擦试验并记录摩擦力试验相关信号，S2对试验信号预处理，获取摩擦力-相对运动速度、摩擦力-正压力散点图和摩擦力自功率谱，S3基于最小二乘法确定高频段摩擦力模型参数，S4基于最小二乘法和遗传算法识别低频段摩擦力模型参数，S5叠加获得完整等效模型。
  4. 一种摩擦运动副等效模型及其建模方法，其特征在于，所述摩擦运动副等效模型为：
    Figure PCTCN2022139144-appb-100005
    Figure PCTCN2022139144-appb-100006
    Figure PCTCN2022139144-appb-100007
    f(N)＝aN b+c；
    Figure PCTCN2022139144-appb-100008
    其中，F(N，v)为摩擦副等效模型输出的总摩擦力，N为正压力，v为切向相对运动速度；z为引入的中间变量，为关于时间的函数；t表示时间，ω表示频率；s(N，v)为摩擦力因数，f(N)为摩擦力的正压力因数，R(N，ω)均为引入的中间函数；r(t)是高频段的摩擦力时间历程，R(N，ω)为r(t)的自功率谱密度；σ 0、σ 1、σ 2、μ s、μ c、δ、a、b、c、h 0、α、β和ω 0均为待定系数，N 0为参考正压力，v s为参考切向相对运动速度，ω 0为高、低频段分界频率；
    所述摩擦运动副等效模型的建模过程具体包括：
    进行摩擦副样件的摩擦试验，获取摩擦试验信号；所述摩擦试验信号是关于所述正压力、所述总摩擦力和所述切向相对运动速度的信号；
    对所述摩擦试验信号进行预处理，得到高频段摩擦力自功率谱图、低频段摩擦力-正压力散点图和低频段摩擦力-相对运动速度散点图；
    根据所述高频段摩擦力自功率谱图，基于最小二乘法，确定h 0、α、β和ω 0，从而确定高频段摩擦力的频谱模型；
    根据所述低频段摩擦力-正压力散点图和所述低频段摩擦力-相对运动速度散点图，基于最小二乘法，确定σ 2、μ s、μ c、δ、a、b和c，从而确定s(N,v)和f(N)；
    根据s(N,v)和f(N)，基于遗传算法，确定σ 0和σ 1，从而确定低频段摩擦力的频谱模型；
    根据所述高频段摩擦力的频谱模型和所述低频段摩擦力的频谱模型确定所述摩擦运动副等效模型。
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