CN114282441A - An equivalent model of friction kinematic pair and its modeling method - Google Patents

Abstract

The invention belongs to the field of mechanical engineering, and particularly relates to a friction kinematic pair equivalent model and a modeling method thereof, wherein friction force test data of a friction pair are subjected to statistical analysis modeling and superposition respectively according to high and low frequency bands to construct an equivalent model capable of reflecting the high and low frequency characteristics of the friction pair; aiming at the low-frequency friction characteristic of the friction pair, introducing positive pressure input information and deforming a traditional LuGre model to obtain an implicit relation between the friction response and the relative speed of the friction pair and the positive pressure input; aiming at the high-frequency section friction force characteristic of the friction pair, positive pressure input information is introduced and the friction force self-power spectrum assumption in a power function form is combined to obtain the implicit relation between the friction force response and the positive pressure of the friction pair. The equivalent model provided by the invention not only can reflect the relation between the friction force and the positive pressure and the relative speed, but also can express the friction force characteristic of a wide frequency band, and has good applicability.

Description

An equivalent model of friction kinematic pair and its modeling method

technical field

The invention belongs to the field of mechanical engineering, and particularly relates to an equivalent model of a friction kinematic pair and a modeling method thereof.

Background technique

Friction is a natural phenomenon in which objects contact each other under the action of normal pressure, and the surface asperities are fitted with each other. When relative sliding occurs, or there is a tendency to relative sliding, resistance against sliding is generated on the contact surface. The resistance is called frictional force. The mechanism of friction is quite complex, and there have been a lot of studies; the characteristics of the friction pair, that is, the relationship between the friction force response and the input motion conditions of the friction pair (positive pressure, tangential relative velocity, etc.) It is closely related to the material properties of the friction pair, especially the properties of the friction contact surface. All factors that affect the properties of the contact surface will also affect the friction characteristics. The classical Coulomb friction law holds that the sliding friction force is proportional to the positive pressure, but later studies have shown that the sliding friction force is not always proportional to the positive pressure, and is closely related to the relative motion speed, and there are Stribeck effects and macroscopic sliding time delays. , pre-sliding displacement hysteresis and other complex phenomena, the friction force often presents the characteristics of broadband.

In view of the complex characteristics of the friction pair and the complex mechanism of friction force, there is no way to effectively complete the theoretical modeling of the friction pair. Therefore, the empirical or semi-empirical model of the friction pair can only be constructed by means of experimental methods. There are many empirical or semi-empirical models for friction pairs, such as LuGre model, GW model, fractal contact model, Stribeck model, Karnopp model, Dahl model, GMS model, GBM model, etc. It is very complicated, but it mainly characterizes the low-frequency characteristics of friction, and generally does not include high-frequency characteristics.

In practical engineering, 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. To this end, the present invention provides a new equivalent model of friction kinematic pairs, which can characterize the friction force characteristics in a wide frequency band and has a relatively concise form, so as to be used for the forced vibration response prediction of a local nonlinear mechanical system with friction pairs. , analysis, design and control.

SUMMARY OF THE INVENTION

The purpose of the present invention is to provide an equivalent model of friction kinematic pair and its modeling method, which can characterize the characteristics of friction force in a wide frequency band, and has a relatively concise form, so as to be used for the forcing of a local nonlinear mechanical system containing friction pairs Vibration response prediction, analysis, design and control.

In order to achieve the above purpose, the present invention provides an equivalent model of friction motion pair and a modeling method thereof. The friction force test data of the friction pair is statistically analyzed and modeled according to high and low frequency bands and superimposed, and the structure can simultaneously reflect the high and low frequency of the friction pair. Equivalent model of low frequency characteristics;

Aiming at the low-frequency friction characteristics of the friction pair, the positive pressure input information is introduced and the traditional LuGre model is deformed to obtain the implicit relationship between the friction response, the relative velocity of the friction pair and the positive pressure input;

Aiming at the high-frequency friction characteristics of the friction pair, the positive pressure input information and the assumption of the friction self-power spectrum in the form of a power function are combined to obtain the implicit relationship between the friction response and the positive pressure of the friction pair.

Further, the mathematical equation of the equivalent model of its friction kinematic pair is as follows:

f(N)=aN ^b +c

Among them, F(N,v) is the friction force output by the equivalent model of the friction pair, which is a function of the positive pressure N of the friction pair and the tangential relative motion velocity v; z is the introduced intermediate variable, usually a function of time; t represents time , ω represents the frequency; s(N,v), f(N), R(N,ω) 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, ω); the remaining parameters are all undetermined coefficients. For a specific friction pair, that is, the determination of the material of the friction pair and the determination of the properties of the friction contact surface, these undetermined parameters should be uniquely determined.

Further, 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. For any specific friction pair, the specific steps for modeling its equivalent model include:

S1 performs the friction test of the friction pair sample and records the relevant signals of the friction force test, S2 preprocesses the test signal, and obtains the friction force-relative motion speed, friction force-positive pressure scattergram and friction force self-power 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 squares method and genetic algorithm, and S5 superimposes to obtain a complete equivalent model.

The beneficial effect of the invention is that the equivalent model proposed by the invention can not only reflect the relationship between the friction force, the positive pressure and the relative velocity, but also express the friction force characteristics of a wide frequency band, and has good applicability.

Description of drawings

Figure 1. Photo of typical friction pair material standard sample;

Fig. 2 Typical measured friction force signal and relative displacement signal of the friction pair test of the standard sample;

Figure 3. The self-power spectrum of the high-frequency part of the friction force under the excitation of the triangular displacement wave with a positive pressure of 30N and a velocity of 3mm/s for the friction pair PC/ABS-PC/ABS;

Fig.4 Fitting effect of friction force-velocity curve and friction force-positive pressure curve under triangular displacement wave excitation of friction pair PC/ABS-PC/ABS;

Figure 5. Iterative effect of low-frequency partial friction time history model of friction pair PC/ABS-PC/ABS under the excitation of triangular displacement wave with positive pressure of 30N and velocity of 1mm/s;

Figure 6. The prediction effect of the equivalent model of friction time history under the excitation of the friction pair PC/ABS-PC/ABS triangular displacement wave;

FIG. 7 is a comparison diagram of the time history of the friction force predicted by the model of the equivalent model of the present invention and the measured value of other typical friction pairs.

Detailed ways

The following is further described in detail through specific embodiments.

Based on the collection and analysis of friction force test signals of a large number of friction pair standard samples, we have carried out statistical analysis and observation on the friction force characteristics. The power function feature, that is, the self-power spectrum of the friction force decays according to the nonlinear power function law as the frequency increases. Generally, the friction force in the high frequency band increases with the increase of the positive pressure, but the correlation with the relative motion speed is not obvious. Therefore, the present invention proposes an equivalent model that superimposes low-frequency and high-frequency characteristics, that is, the model structure is divided into low-frequency and high-frequency, and the corresponding frequency band of the experimental data is also used to filter the signal in the identification of model parameters. For the low-frequency model, it is considered that the traditional LuGre model has a clear physical meaning and a simpler form, that is, fewer model parameters to be identified, and the friction characteristics that can be characterized include that the maximum static friction force is greater than the sliding friction force. , pre-sliding displacement lag, Stribeck effect and sliding friction time lag, so the traditional LuGre model is used, and the positive pressure relationship is introduced to modify the model structure appropriately to characterize more complete friction characteristics in the low frequency band; for the model in the high frequency band, the Directly construct the friction self-power spectral power function model including the positive pressure input; considering the randomness of the high-frequency friction signal, the time history of the high-frequency friction can be further obtained by combining the random phase, so the low-frequency frequency is directly superimposed in the time domain. The friction force, the complete friction force output of the whole frequency band is obtained.

According to the above thinking, the present invention proposes a new broadband friction equivalent model of the friction motion pair as shown below.

f(N)=aN ^b +c (4)

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)摩擦力s(N,v)与正压力及相对运动速度的关系，N₀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，ω₀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。Equation (1) expresses that the friction force F(N,v) output by the equivalent model of the friction pair is determined by the low frequency part

It is directly superimposed with the high-frequency part r(t) in the time domain, which is an implicit function of the positive pressure N of the friction pair and the tangential relative motion velocity v; where z is an 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 the friction force f(N) and the normal pressure. Equation (3) friction force s(N,v) The relationship with the positive pressure and relative motion speed, N ₀ is the reference positive pressure, equation (2) gives the differential equation satisfied by the intermediate variable z, obviously, for the given N and v, equation (2) can be solved by z and its first derivative, thereby determining the time history of friction in the low frequency band. The friction force in the high frequency band is given by equation (5), where ω represents the frequency, ω ₀ is the boundary frequency between the high and low frequency bands, and R(N,ω) is the self-power spectral density function of the friction force in the high frequency band. 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 through the 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, the determination of the material of the friction pair and the determination of the properties of the friction contact surface, these undetermined parameters should be uniquely determined.

For any friction moving pair, the model structure given by the present invention is shown in equations (1) to (5), and the undetermined coefficients can be identified through test identification, thereby completing the modeling of the equivalent model of the friction moving pair. The specific steps of the method are as follows.

S1 performs the friction test of the friction pair sample and records the relevant signals of the friction force test.

In this step, a standard friction pair test sample should be prepared for the target friction moving pair, and its material and contact surface properties (surface hardness, roughness, etc.) are consistent with the target friction moving pair. The friction test should simultaneously record the positive pressure, friction force and tangential relative motion displacement or velocity signal of the friction pair, and should include the pre-slip state and macro-slip 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 reciprocating sliding friction test of the friction pair excited by the triangular wave displacement, which can quickly conduct the test and obtain the friction force and other signals of the required friction state. In the remaining stable sliding sections, relevant information about the macroscopic sliding stage of the friction pair under constant speed can be obtained, which is convenient for modeling. The relative sliding test of the friction pair is usually carried out under a given positive pressure. Each test should include a stable sliding state at a given speed and a transition state from rest to a given speed. Adjust the positive pressure and speed to obtain the required modeling requirements. Friction test signals in various states.

S2 preprocesses the test signal to obtain friction force-relative motion velocity, friction force-positive pressure scattergram and friction force self-power spectrum.

The frictional force signal measured in the experiment is filtered, and the low-frequency and high-frequency frictional force signals are obtained 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 force signal are cross-referenced, it is easy to distinguish the friction force in the macro-slip and pre-slip stages, and also The friction of the constant speed sliding section is easily distinguished. Using the friction force signals of different constant-speed sliding sections under a given positive pressure, the friction force-relative motion velocity scatter diagram can be obtained; Friction-positive pressure scatterplot. Similarly, the high-frequency friction-positive pressure scattergram can also be obtained by using the high-frequency friction signal, and the self-power spectrum can be obtained by spectrum analysis of the high-frequency friction signal under a given reference positive pressure.

S3 determines the parameters of the high-frequency friction model based on the least squares method.

Using the high-frequency friction-positive pressure scattergram and the high-frequency friction self-power spectrum, based on the least squares method, the undetermined parameters in equation (5) can be determined, thereby determining the high-frequency friction spectrum model.

S4 identifies low-frequency friction model parameters based on least squares method and genetic algorithm.

Using the friction force-positive pressure scattergram and friction force-relative motion velocity scattergram in the low frequency band, based on the least squares method, the undetermined parameters in equations (4) and (3) can be determined respectively. Then use the friction time history of the low frequency band 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, other mature can also be selected. optimization algorithm), the undetermined parameters σ ₀ and σ ₁ in equations (1) and (2) can be obtained.

S5 superposition obtains the full equivalent model.

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。Considering the random nature of the friction force in the high frequency band, it is assumed that it has a random phase spectrum, and then the self-power spectral density function of the friction force in the high frequency band that has been identified and obtained, that is, equation (5), can be obtained by inverse Fourier transform. The time history, that is, the friction force r(t) in the high frequency band, and the friction force in the low frequency band obtained in step S4

Superimpose to get the full friction output. So far, all undetermined parameters and functions in the equivalent model equations (1) to (5) have been determined, that is, a complete equivalent model of the target friction kinematic pair is obtained.

Example:

Figure 1 shows a real photo of a typical material standard sample that constitutes a friction pair. For a specific friction pair standard sample, a self-developed reciprocating friction testing machine is used to carry out the 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 is used. 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 friction pair standard sample 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 Figure 2. During the test, the ambient temperature was 20±5°C, and the humidity was 75% to 85%. The positive pressure and relative velocity selected for the test conditions are shown in Table 1.

Table 1 Test conditions

Step 2: Select the test data of different speeds under 30N pressure, perform 2Hz low-pass filtering on the time history of friction force, compare the relative displacement and speed time history, and find the average value of the friction force for the friction force signal in the constant speed sliding stage, and obtain Friction force-velocity scatter diagram; select the test data of 3mm/s speed under different positive pressures, perform 2Hz low-pass filtering on the time history of friction force, and obtain the friction force-positive pressure scattergram by similar processing; select 30N, 3mm/s In order to remove the fundamental wave effect of the reciprocating motion, intercept the friction time history in one direction of the reciprocating motion, subtract the DC component, and estimate the power spectral density of it, that is, the power spectral density curve of the 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 perform spectral analysis on the filtered signal to obtain the friction power spectral density curve in the high-frequency part. The results of the two methods are basically the same.

Step 3: Build a model of the high-frequency part of the friction force. That is, based on the least squares method, according to the measured power spectral density curve of the high-frequency part of the friction force, the undetermined parameters in the self-power spectrum model (ie equation (5)) are obtained. Figure 3 shows the high-frequency part of the friction force self-power Comparing the calculated curve of the spectral model with the measured curve, and assuming a random phase spectrum (uniform distribution), with the help of inverse Fourier transform, the friction time history of the high-frequency part is obtained.

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、σ₂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σ₀、σ₁，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。Step 4: Establish a low-frequency partial friction model. Firstly, based on the least squares method, the friction force-positive pressure scattergram and the friction force-velocity scattergram obtained by the analysis of the measured signal are used, respectively, by fitting f(N)=aN ^b +c friction force-positive pressure curve and

The friction force-velocity curve, and the undetermined parameters a, b, c, μ _c , μ _s , _vc , σ ₂ and δ _s are obtained. Figure 4 shows the friction of the PC/ABS-PC/ABS material pair sample Fitting effect of friction force-velocity scatter plot and friction force-positive pressure scatter plot of the experiment. Here, the reference positive pressure is 30N. Then, using the genetic algorithm, the error standard deviation between the time history of friction force calculated by the low-frequency friction force model and the time history of the friction force measured at 2Hz low-pass filter under the working conditions of 30N and 3mm/s is used as the objective function, and iteratively obtains the most optimal objective function. The hourly parameters σ ₀ , σ ₁ determine the low-frequency partial friction model, and Fig. 5 shows the comparison of the predicted and measured time history of the low-frequency partial friction model.

Step 5: The friction force of the low frequency band and the friction force of the high frequency band are directly superimposed in the time domain, and the total friction force of the equivalent model of the friction pair is obtained, that is, the friction force equivalent model equations (1)-(5). All undetermined parameters have been determined, and the modeling of the equivalent model of the friction pair has been completed. Using this equivalent model, as long as the model input is given, that is, the time history of positive pressure and relative velocity (including the case of constant constant), The intermediate functions and intermediate variables in equations (1) to (5) can be calculated, as well as the final equivalent model output that needs to be calculated, that is, the time history of frictional force. Fig. 6 shows the comparison effect of 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 the friction test of PC/ABS-PC/ABS material sub-samples.

Obviously, the PC/ABS-PC/ABS material friction pair equivalent model established by the method of the present invention can accurately predict the time history of friction force. In fact, it has similar effects for other typical material friction pairs. Figure 7 shows some typical The comparison between the measured friction time history and the time history predicted by the equivalent model shows that the magnitude and change trend of the two are basically the same, indicating that the equivalent model can well characterize the dynamic characteristics of the friction pair.

It should be noted in advance that in the present invention, unless otherwise expressly specified and limited, terms such as "installation", "connection", "connection" and "fixation" should be understood in a broad sense, for example, it may be a fixed connection, It can also be a detachable connection, or an integral connection; it can be directly connected, or indirectly connected through an intermediate medium, and it can be internal communication between two components. For those of ordinary skill in the art, the specific meanings of the above terms in the present invention can be understood according to specific situations.

The above descriptions are only embodiments of the present invention, and common knowledge such as well-known specific structures and characteristics in the solution are not described too much here. It should be pointed out that for those skilled in the art, some modifications and improvements can be made without departing from the structure of the present invention. These should also be regarded as the protection scope of the present invention, and these will not affect the implementation of the present invention. Effectiveness and utility of patents. The scope of protection claimed in this application shall be based on the content of the claims, and the descriptions of the specific implementation manners in the description can be used to interpret the content of the claims.

Claims (3)

1. A friction kinematic pair equivalent model and its modeling method, characterized by that, carry on the statistical analysis modeling and superpose separately according to the high-low frequency band to the friction test data of the friction pair, construct the equivalent model that can reflect the high-low frequency characteristic of the friction pair at the same time;

aiming at the low-frequency friction characteristic of the friction pair, introducing positive pressure input information and deforming a traditional LuGre model to obtain an implicit relation between the friction response and the relative speed of the friction pair and the positive pressure input;

aiming at the high-frequency section friction force characteristic of the friction pair, positive pressure input information is introduced and the friction force self-power spectrum assumption in a power function form is combined to obtain the implicit relation between the friction force response and the positive pressure of the friction pair.

2. The friction kinematic pair equivalent model and the modeling method thereof according to claim 1, characterized in that the mathematical equation of the friction kinematic pair equivalent model is as follows:

f(N)＝aN^b+c

f (N, v) is the friction force output by the friction pair equivalent model and is a function of the positive pressure N and the tangential relative movement speed v of the friction pair; z is an intermediate variable introduced, typically as a function of time; t represents time, ω represents frequency; s (N, v), f (N), R (N, omega) are all introduced intermediate functions; r (t) is the time history of the friction force in the high frequency band, its self-power spectral density is R (N, omega); the other parameters are all pending coefficients, and the pending parameters are determined uniquely for specific friction pairs, namely the material determination of the friction pairs and the surface property determination of the friction contact.

3. The friction kinematic pair equivalent model and the modeling method thereof as claimed in claim 2, wherein the undetermined parameters of the friction kinematic pair equivalent model can be identified and determined through friction test data, so as to complete the modeling of the equivalent model of a specific friction pair. For any specific friction pair, the specific steps of modeling the equivalent model thereof comprise:

s1 friction test of the friction pair sample piece is carried out, friction force test related signals are recorded, S2 preprocessing of the test signals is carried out, friction force-relative movement speed, a friction force-positive pressure scatter diagram and a friction force self-power spectrum are obtained, S3 determines high-frequency-band friction force model parameters based on a least square method, S4 identifies low-frequency-band friction force model parameters based on a least square method and a genetic algorithm, and S5 is overlapped to obtain a complete equivalent model.

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