An equivalent model of friction kinematic pair and its modeling method

Abstract

The present invention belongs to the field of mechanical engineering, and specifically relates to an equivalent model of a friction kinematic pair and a modeling method thereof. The friction test data of the friction pair are statistically analyzed and modeled according to high and low frequency bands and superimposed to construct an equivalent model that can simultaneously reflect the high and low frequency characteristics of the friction pair; for the low-frequency friction characteristics of the friction pair, 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 speed of the friction pair and the positive pressure input; for the high-frequency friction characteristics of the friction pair, positive pressure input information is introduced and combined with the friction self-power spectrum hypothesis in the form of a power function, the implicit relationship between the friction response and the positive pressure of the friction pair is obtained. The equivalent model proposed by the present invention can not only reflect the relationship between friction, positive pressure and relative speed, but also can express the wide-band friction characteristics, and has good applicability.

Description

Technical Field

[0001] The invention belongs to the field of mechanical engineering, and in particular relates to a friction kinematic pair equivalent model and a modeling method thereof. Background Art

[0002] Friction is a natural phenomenon that objects contact each other under normal pressure, and the micro-convexities on the surface fit together. When relative sliding occurs or there is a tendency for relative sliding, resistance to sliding is generated on the contact surface. The resistance is called friction. The mechanism of friction is quite complex, and a large number of studies have been conducted. The characteristics of the friction pair, that is, the relationship between the friction response and the input motion conditions of the friction pair (normal pressure, tangential relative velocity, etc.), also often present a quite complex situation, and are closely related to the properties of the friction pair material, especially the properties of the friction contact surface. Any factors that affect the properties of the contact surface will also affect the friction characteristics. The classic Coulomb friction law believes that the sliding friction force is proportional to the normal pressure, but later studies have shown that the sliding friction force is not always proportional to the normal pressure, and is closely related to the relative motion speed. There are many complex phenomena such as the Stribeck effect, macro sliding hysteresis, and pre-sliding displacement hysteresis. The friction force also often presents a broadband characteristic.

[0003] In view of the complex characteristics of friction pairs and the complex mechanism of friction generation, there is currently no way to effectively complete the theoretical modeling of friction pairs. Therefore, we can only use experimental methods to construct empirical models or semi-empirical models of friction pairs. There are many 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 and some are very complex, but they mainly characterize the low-frequency characteristics of friction and generally do not include high-frequency characteristics.

[0004] In actual engineering, low-frequency vibration excitation causes secondary friction excitation of the friction pair in the mechanical system, which often radiates high-frequency noise, indicating that the high-frequency vibration of the structure is excited. Therefore, the high-frequency characteristics of friction force must be paid attention to. To this end, the present invention provides a new equivalent model of friction kinematic pair, which can characterize the friction force characteristics of a wide frequency band and has a relatively concise form, so as to be used for the prediction, analysis, design and control of the forced vibration response of the local nonlinear mechanical system containing the friction pair. Summary of the invention

[0005] The purpose of the present invention is to provide an equivalent model of a friction kinematic pair and a modeling method thereof, 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 prediction, analysis, design and control of the forced vibration response of a local nonlinear mechanical system containing a friction pair.

[0006] In order to achieve the above-mentioned purpose, the present invention provides an equivalent model of a friction pair and a modeling method thereof, which statistically analyzes and models the friction force test data of the friction pair according to high and low frequency bands and superimposes them to construct an equivalent model that can simultaneously reflect the high and low frequency characteristics of the friction pair;

[0007] 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 and the relative velocity of the friction pair and the positive pressure input.

[0008] According to the high-frequency friction characteristics of the friction pair, the normal pressure input information is introduced and combined with the friction autopower spectrum assumption in the form of a power function, the implicit relationship between the friction response and the normal pressure of the friction pair is obtained.

[0009] Furthermore, the mathematical equation of the equivalent model of the friction pair is as follows:

[0010]

[0011]

[0012]

[0013] f(N)=aN b +c

[0014]

[0015] Among them, 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 an introduced intermediate variable, which is usually a function of time; t represents time and ω represents frequency; s(N,v), f(N), and R(N,ω) are all introduced intermediate functions; r(t) is the time history of friction force in the high-frequency band, and its self-power spectrum density is R(N,ω); the remaining parameters are all unknown coefficients. For a specific friction pair, that is, the material of the friction pair is determined and the properties of the friction contact surface are determined, these unknown parameters should be uniquely determined.

[0016] Furthermore, the undetermined parameters of the friction pair equivalent model can be identified and determined through friction test data, so that the equivalent model of a specific friction pair can be built. For any specific friction pair, the specific steps of its equivalent model building include:

[0017] S1 conducts friction test on friction pair samples and records friction test related signals, S2 preprocesses the test signals to obtain friction-relative motion speed, friction-normal pressure scatter plots and friction autopower spectrum, S3 determines high-frequency friction model parameters based on least squares method, S4 identifies low-frequency friction model parameters based on least squares method and genetic algorithm, and S5 superimposes to obtain a complete equivalent model.

[0018] 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, normal pressure and relative speed, but also express the friction characteristics of a wide frequency band and has good applicability. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1 Photos of standard samples of typical friction pair materials;

[0020] Figure 2 Typical measured friction force signal and relative displacement signal of standard sample friction pair test;

[0021] Figure 3 The auto-power spectrum of the high-frequency part of the friction force of the friction pair PC / ABS-PC / ABS under the excitation of triangular displacement wave with a positive pressure of 30N and a speed of 3mm / s;

[0022] Figure 4 The fitting effect of friction force-velocity curve and friction force-normal pressure curve under triangular displacement wave excitation of friction pair PC / ABS-PC / ABS;

[0023] Figure 5 Iterative effect of the low-frequency partial friction force time history model of the friction pair PC / ABS-PC / ABS under the excitation of triangular displacement wave with positive pressure of 30N and speed of 1mm / s;

[0024] Figure 6 Prediction effect of equivalent model of friction time history under triangular displacement wave excitation of friction pair PC / ABS-PC / ABS;

[0025] Figure 7 A comparison chart of the friction time history predicted by the equivalent model of the present invention for other typical friction pairs and the measured values. DETAILED DESCRIPTION

[0026] The following is further explained in detail through specific implementation methods.

[0027] Based on the collection and analysis of friction test signals of a large number of standard friction pair samples, we have conducted statistical analysis and observation of the friction characteristics. Its low-frequency characteristics are basically consistent with the existing research conclusions, and its high-frequency characteristics show power function characteristics, that is, the friction self-power spectrum decays according to the nonlinear power function law as the frequency increases. Usually, the friction in the high-frequency band will increase with the increase of normal pressure, but the correlation with the relative motion speed is not obvious. Therefore, the present invention proposes an equivalent model that superimposes low-frequency characteristics and high-frequency characteristics, that is, it is divided into low-frequency and high-frequency bands in the model structure, and the corresponding frequency band filtering signal of the test data is also used in the identification of model parameters. For the low-frequency model, the traditional LuGre model is used, taking into account that it has clear physical meaning and a simpler form, that is, fewer model parameters that need to be identified, and the friction characteristics that can be characterized include the maximum static friction force being greater than the sliding friction force, pre-sliding displacement hysteresis, Stribeck effect and sliding friction lag. Therefore, the traditional LuGre model is used, and the positive pressure relationship is introduced to modify the model structure appropriately to characterize a more complete friction force characteristic in the low-frequency band; for the high-frequency model, a friction self-power spectrum power function model containing positive pressure input is directly constructed; considering the randomness of the high-frequency friction signal, the friction time history of the high-frequency band can be further obtained in combination with the random phase, so the low-frequency friction force is directly superimposed in the time domain to obtain the complete friction force output of the entire frequency band.

[0028] According to the above ideas, the present invention proposes a new broadband friction equivalent model of the friction pair as shown below.

[0029]

[0030]

[0031]

[0032] f(N)=aN b +c (4)

[0033]

[0034] Equation (1) expresses that the friction force F(N,v) output by the friction pair equivalent model is composed of the low-frequency part The high frequency part r(t) is directly superimposed in the time domain, which is an implicit function of the normal 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 part is given by equations (2) to (4). Equation (4) expresses the power function relationship between the friction force f(N) and the normal pressure, and equation (3) expresses the relationship between the friction force s(N,v) and the normal pressure and the relative motion velocity. N0 is the reference normal pressure. Equation (2) gives the differential equation satisfied by the intermediate variable z. Obviously, for given N and v, equation (2) can be used to solve z and its first-order derivative, thereby determining the time history of the friction force in the low frequency part. The friction force in the high frequency band is given by equation (5), where ω represents the frequency, ω0 is the boundary frequency between the high and low frequency bands, and R(N,ω) is the self-power spectrum density function of the friction force in the high frequency band. Considering the random nature of the friction force in the high frequency band, it is assumed that it has a random phase spectrum, so the time history of the friction force in the high frequency band r(t) can be obtained by inverse Fourier transform. The remaining parameters in equations (1) to (5) are all undetermined coefficients of the equivalent model. For a specific friction pair, that is, 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.

[0035] For any friction kinematic pair, the model structure provided by the present invention is shown in equations (1) to (5), wherein the unknown coefficients can be determined by experimental identification, thereby completing the modeling of the equivalent model of the friction kinematic pair. The specific steps of the method are as follows.

[0036] S1 performs friction test on friction pair samples and records friction force test related signals.

[0037] This step should prepare a standard friction pair test specimen for the target friction pair, whose material and contact surface properties (surface hardness, roughness, etc.) are consistent with those of the target friction pair. The friction test should synchronously record the positive pressure, friction force and tangential relative motion displacement or velocity signal of the friction pair, and should include the pre-sliding state and macro sliding state of the friction pair. The friction pair reciprocating sliding test can be carried out, and the friction pair unidirectional sliding test can also be carried out. Different friction test methods often require different friction pair friction force test devices or equipment. The present invention recommends the use of a triangular wave displacement excitation friction pair reciprocating sliding friction test, which can quickly conduct the test and obtain the friction force and other signals of the required friction state. In the reversing section of relative reciprocating sliding, the relevant information of the pre-sliding stage of the friction pair can be obtained, and in the remaining stable sliding sections, the relevant information of the macro sliding stage of the friction pair at a constant speed can be obtained, which is convenient for modeling. Usually, 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. By adjusting the positive pressure and speed, the friction test signals of various states required for modeling can be obtained.

[0038] S2 preprocesses the test signal to obtain the friction force-relative motion speed, friction force-normal pressure scatter plot and friction force autopower spectrum.

[0039] The friction force signals measured in the test are filtered to obtain low-frequency and high-frequency friction force signals. By referring to 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) with the low-frequency friction force signal, it is easy to distinguish the friction force in the macro sliding and pre-sliding stages, and it is also easy to distinguish the friction force in the constant speed sliding section. The friction force signals of different constant speed sliding sections under a given positive pressure can be used to obtain a friction force-relative motion speed scatter plot; the friction force signals of the constant speed sliding section at a given speed under different positive pressures can be used to obtain a friction force-positive pressure scatter plot. Similarly, the friction force signals of the high-frequency band can also be used to obtain a high-frequency friction force-positive pressure scatter plot. The high-frequency band friction force signal under a given reference positive pressure is subjected to spectrum analysis to obtain its autopower spectrum.

[0040] S3 determines the high-frequency friction model parameters based on the least squares method.

[0041] Using the high-frequency friction-normal pressure scatter diagram and the high-frequency friction self-power spectrum, the unknown parameters in equation (5) can be determined based on the least squares method, thereby determining the spectrum model of the high-frequency friction.

[0042] S4 identifies the low-frequency friction model parameters based on the least squares method and genetic algorithm.

[0043] Using the friction force-normal pressure scatter plot and the friction force-relative motion velocity scatter plot in the low frequency band, the various unknown parameters in equations (4) and (3) can be determined based on the least squares method. Then, using the friction force time history and the corresponding relative motion velocity time history in the low frequency band, as well as the identified intermediate functions s(N,v) and f(N), based on the genetic algorithm (of course, other mature optimization algorithms can also be selected), the unknown parameters σ0 and σ1 in equations (1) and (2) can be obtained.

[0044] S5 was superimposed to obtain a complete equivalent model.

[0045] Considering the random nature of the high-frequency friction force, it is assumed that it has a random phase spectrum. Then, using the identified high-frequency friction force auto-power spectrum density function, that is, formula (5), its time history can be obtained by inverse Fourier transform, that is, the high-frequency friction force r(t), and then compared with the low-frequency friction force obtained in step S4. By superimposing, the complete friction force output can be obtained. At this point, all the undetermined parameters and functions in the equivalent model equations (1) to (5) have been determined, and the complete equivalent model of the target friction kinematic pair has been obtained.

[0046] Example:

[0047] Figure 1 The following is a photo of a typical material standard sample of a friction pair. For a specific friction pair standard sample, a friction test is carried out using a self-developed reciprocating friction tester to obtain relevant measured signals. Then, an equivalent model of the friction pair is established according to the method described in the present invention, and the friction force predicted by the equivalent model is compared with the measured friction force to illustrate the effectiveness of the equivalent model. The specific implementation process and results are as follows.

[0048] The first step is to install the friction pair standard sample on a special friction tester, load different positive pressures, set different speeds for triangular wave displacement excitation, make the friction pair sample undergo tangential reciprocating relative motion, and record the time history of friction force and relative displacement. Typical friction force and relative displacement measured signals are as follows: Figure 2 During the test, the ambient temperature was 20±5℃ and the humidity was 75%~85%. The positive pressure and relative speed selected for the test conditions are shown in Table 1.

[0049] Table 1 Test conditions

[0050]

[0051] 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 friction value for the friction signal in the constant speed sliding stage to obtain the friction-speed scatter plot; select the test data of 3mm / s speed under different positive pressures, perform 2Hz low-pass filtering on the friction time history, and obtain the friction-positive pressure scatter plot by similar processing; select the test data of 30N and 3mm / s, in order to remove the fundamental wave influence of reciprocating motion, intercept the friction time history in one direction of reciprocating motion, subtract the DC component, and perform power spectral density estimation on it, that is, obtain the power spectral density curve of friction in the high-frequency band; you can also perform high-pass filtering on the friction time history (cutoff frequency is 2Hz), and then perform spectral analysis on the filtered signal to obtain the power spectral density curve of friction in the high-frequency band. The results of the two methods are basically the same.

[0052] Step 3: Establish a model of the friction force in the high-frequency band. That is, based on the least squares method, according to the measured power spectrum density curve of the friction force in the high-frequency band, find the unknown parameters in its power spectrum model (i.e., equation (5)). Figure 3 The figure shows the comparison between the calculated curve of the friction force in the high-frequency band using the power spectrum model and the measured curve. Assuming a random phase spectrum (uniform distribution), the time history of the friction force in the high-frequency band is obtained using the inverse Fourier transform.

[0053] Step 4: Establish a low-frequency friction model. First, based on the least squares method, the friction-normal pressure scatter plot and friction-velocity scatter plot obtained by analyzing the measured signal are respectively fitted with f(N)=aN b +c Friction-normal pressure curve and Friction-speed curve, find the unknown parameters a, b, c, μ c , μ s 、v c , σ2 and δ s , Figure 4 The figure shows the fitting effect of the friction-velocity scatter plot and the friction-normal pressure scatter plot of the PC / ABS-PC / ABS material sub-sample friction test. Here, the reference normal pressure is 30N. Then, a 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 actual friction force under the conditions of 30N and 3mm / s as the objective function, and iterate the parameters σ0 and σ1 when the objective function is minimized, that is, the low-frequency partial friction model is determined. Figure 5 The figure shows the time history comparison between the model prediction and the measured friction force in the low-frequency band.

[0054] Step 5: Directly superimpose the low-frequency friction force and the high-frequency friction force in the time domain to obtain the total friction force of the friction pair equivalent model. That is, all the undetermined parameters in the friction equivalent model equations (1) to (5) have been determined, and the modeling of the friction pair equivalent model has been completed. Using this equivalent model, as long as the model input is given, that is, the time history of the normal pressure and the relative velocity (including the case where they are constants), the various intermediate functions and intermediate variables in equations (1) to (5) and the final equivalent model output that needs to be calculated, that is, the friction time history, can be calculated. Figure 6 The comparison between the measured friction time history under typical working conditions of PC / ABS-PC / ABS material pair sample friction test and the friction time history predicted by the equivalent model is given.

[0055] Obviously, the equivalent model of PC / ABS-PC / ABS material friction pair established by the method of the present invention can accurately predict the time history of friction force. In fact, similar effects are also achieved for other typical material friction pairs. Figure 7 Some typical comparisons of the measured friction force time histories and the time histories predicted by the equivalent model are given. The magnitudes and changing trends of the two are basically consistent, indicating that the equivalent model can well characterize the dynamic characteristics of the friction pair.

[0056] It should be noted in advance that, in the present invention, unless otherwise clearly specified and limited, the terms "installed", "connected", "connected", "fixed" and the like should be understood in a broad sense, for example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a direct connection, or an indirect connection through an intermediate medium, or it can be the internal communication of two elements. For ordinary technicians in this field, the specific meanings of the above terms in the present invention can be understood according to specific circumstances.

[0057] The above is only an embodiment of the present invention, and the common knowledge such as the known specific structure and characteristics in the scheme is not described in detail here. It should be pointed out that for those skilled in the art, several deformations and improvements can be made without departing from the structure of the present invention, which should also be regarded as the protection scope of the present invention, and these will not affect the effect of the implementation of the present invention and the practicality of the patent. The scope of protection required by this application shall be based on the content of its claims, and the specific implementation methods and other records in the specification can be used to interpret the content of the claims.

Claims

1. A friction kinematic pair equivalent model and a modeling method thereof, characterized in that: The friction test data of the friction pair are statistically analyzed and modeled according to the high and low frequency bands, and then superimposed to construct an equivalent model that can simultaneously reflect the high and low frequency characteristics of the friction pair. The mathematical equation of the equivalent model is as follows: f(N)=aN b +c (4) Where F(N,v) is the friction force output by the friction pair equivalent model, which is composed of the low-frequency part The high-frequency part r(t) is directly superimposed in the time domain, which is a function of the normal pressure N of the friction pair and the tangential relative motion speed v; z is an introduced intermediate variable, which is a function of time; the friction force in the low-frequency part is given by equations (2) to (4). Equation (4) expresses the power function relationship between the friction force f(N) and the normal pressure, and equation (3) expresses the relationship between the friction force s(N,v) and the normal pressure and the relative motion speed. N0 is the reference normal pressure. Equation (2) gives the differential equation satisfied by the intermediate variable z. For given N and v, equation (2) can be used to solve z and its first-order derivative, thereby determining the time history of the friction force in the low-frequency band; the friction force in the high-frequency band is given by equation (5), where ω represents the frequency, ω0 is the boundary frequency between the high-frequency band and the low-frequency band, and R(N,ω) is the self-power spectral density function of the friction force in the high-frequency band. Considering the random nature of the friction force in the high-frequency band, assuming that it has a random phase spectrum, the time history of the friction force in the high-frequency band r(t) can be obtained by inverse Fourier transform; σ0, σ1, μ c , μ s , μ c ,σ2,v s , δ, h0, β, -α are all the unknown parameters of the equivalent model, among which, the friction-normal pressure scatter plot and friction-relative motion velocity scatter plot in the low frequency band are used to determine the unknown parameters in equations (4) and (3) based on the least squares method. The time history of the friction in the low frequency band and the corresponding time history of the relative motion velocity, as well as the identified intermediate functions s(N,v) and f(N), are used to obtain the unknown parameters σ0 and σ1 in equations (1) and (2) based on the genetic algorithm. Considering the random nature of the friction in the high frequency band, it is assumed that it has a random phase spectrum, and the self-power spectrum density function of the friction in the high frequency band, i.e., equation (5), is used to obtain its time history, i.e., the friction in the high frequency band r(t), through inverse Fourier transform, and then superimposed with the low frequency band friction obtained in step S4 to obtain the complete friction output. 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 and the relative velocity of the friction pair and the positive pressure input. Aiming at the high-frequency friction characteristics of the friction pair, the normal pressure input information is introduced and combined with the power function form of the friction autopower spectrum assumption, the implicit relationship between the friction response and the normal pressure of the friction pair is obtained. The undetermined parameters of the equivalent model are identified and determined through friction test data, thereby completing the equivalent model building of a specific friction pair; for any specific friction pair, the specific steps of its equivalent model building include: S1 conducts friction test on friction pair samples and records friction test related signals, S2 preprocesses the test signals to obtain friction-relative motion speed, friction-normal pressure scatter plots and friction autopower spectrum, S3 determines high-frequency friction model parameters based on least squares method, S4 identifies low-frequency friction model parameters based on least squares method and genetic algorithm, and S5 superimposes to obtain a complete equivalent model.

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