Research on parameter identification of nonlinear friction on cantilever beam

2.1. Experimental setup

The friction parameter identification is investigated for the sliding process of the moving mass. The experimental setup is shown in Fig. 1. The sketch of the motion is given by Fig. 2.

Fig. 1Picture of the experimental setup

Fig. 2Sketch of the motion

A pneumatic actuator is employed to drive the moving mass which can move along the cantilever beam at a certain speed (5-8 m/s). A high speed camera is used to record the motion with a sampling frequency of 4 kHz [5]. A low-pass filter is used to suppress the high frequency noise in the signal. By analyzing the motion images of the moving mass, the displacement, velocity and acceleration data is obtained. Since the mechanical stress analysis of the moving mass is the main issue, the acceleration data calculated from second order derivative of displacement is needed for further analysis. Based on friction identification model and intelligent optimization algorithm, the parameters of the friction model between the moving mass and cantilever beam will be obtained.

2.2. Tustin model

Based on Stribeck Friction Model, a static friction model can divide system friction into three parts [6]:

(1) Coulomb Friction (CF): The friction during the sliding process of a moving mass.

(2) Static Friction (SF): The friction which prevents a mass from static to dynamic.

(3) Viscous Friction (VF): The friction produced by viscous effect between interface materials.

The Stribeck Friction Model is expressed as:

where $F_f$ is total friction, $F_c$ is CF, $F_v$ is VF, $F_s$ is SF, $V_s$ is Velocity, $\delta$ is an experience parameter.

Generally, when $\delta =$1, we get Tustin Model (TM), which is the most common Stribeck Friction Model. The TM is given by Eq. (2):

2.3. Augmented Tustin model

All the test accelerations in the experiments have sine characteristics, so it is essential to take sine components into account to improve TM, finally to get a more precise system friction model.

In this paper, the friction effect of the horizontal direction can be described by TM. But there are effects of other factors on moving mass, which can be compensated by using sine polynomials through the theory of Fourier transform. Owing to the limitation of numerical calculation efficiency, Augmented Tustin Model can meet the requirements when the exponent number $N$ of the sine polynomials reaches 3. There is a time signal t in the sine polynomial compensation which is also considered as a model parameter. To some extent, ATM can identify the time-varying characteristics of the free section friction on cantilever beam.

The time-varying friction compensation model ATM 1 is:

The three terms in Eq. 3 describe the effect of static friction, sliding friction and viscous friction respectively. In principle, static friction effect cannot be affected by external factors. However, sliding friction and viscous friction are the different cases. Though they are affected by the vibration and impact factors of the cantilever beam mechanism, the effects are not completely consistent. Therefore, sliding friction and viscous friction should be compensated respectively.

The time-varying friction compensation model ATM 2 is:

2.4. Genetic algorithm

Output signal of a time-varying system depends not only on input signal of the system but also on the moment of the input signal, which is different from output signal of a general constant system. The friction identification models in this paper are time-varying systems. Algorithm gain of conventional identification algorithms of a constant system, such as recursive least squares method, Newton iteration algorithm and Hessian gradient algorithm, will be close to 0 with the number of algorithm iterations increasing. Therefore, these algorithms do not have searching and iteration capability for time-varying parameters. Genetic algorithm (GA) is a kind of intelligent algorithm which has a strong iteration capability. GA will be used for the parameter identification in this paper.

Various advantages of GA are as follows:

(1) Self-adaption: the capability to adjust circumstance characteristics and disciplines when they change.

(2) Parallel calculation: decreases calculation expense to improve efficiency.

(3) Fitness function only: just needs a function to estimate individual fitness.

(4) Probabilistic conversion: conversion formula is flexible and probabilistic.

(5) Easily application: results from GA can be applied to solve problems directly.

(6) Several optimal solutions: GA gives a group of feasible solution.

2.5. Concepts about genetic algorithm

(1) Individual: entity with characteristics.

(2) Population: it is made up by several individuals, and individual number in this set is called population size.

(3) Encoding: for initial population of individuals, it must be encoded before computer calculation.

(4) Decoding: process converting coding space to solution space to find out optimal solution.

(5) Fitness: standards to estimate each individual’s fitness in current generation.

(6) Selection: choose better individuals to make up the next generation by certain probability.

(7) Crossing: two individuals are cut at the same position, then connecting crossly which make out two new individuals.

(8) Mutation: in each generation, individuals probably mutate to become a new one by certain regularity.

GA is a stochastic optimization technique inspired by genetic evolution mechanism [7, 8, 9]. It starts the search process from a set of randomly generated initial solutions which is called population. Each individual in the population is a solution to the problem, called chromosome. These chromosomes will evolve in subsequent iterations, and this process is heredity. GA is achieved by three aspects: crossover, mutation and selection. The operations of crossover or mutation will generate the next generation of chromosomes which are called offspring. The standard used to evaluate the quality of chromosome is called fitness, according to which a certain amount of individuals from current generation and offspring are chosen as the next generation of population to continue evolution. After several generations, the algorithm converges to the best chromosome, which is the optimal or suboptimal solution of a problem [10].

The iterative process of genetic algorithm is shown in Fig. 3.

Fig. 3Genetic algorithm flowchart

3.1. Analysis of TM

According to Eq. 2, there are four parameters to identify: $F_c$, $F_s$, $F_v$ and $V_s$. In the experiment, pressure of 2 MPa and 3 MPa is applied respectively to produce the motion at two different speeds, which are low-speed group (4-5 m/s) and high-speed group (7-8 m/s). MGA is used for parameter identification. The identification results are as follows:

(1) Low-speed group (4-5 m/s):

(2) High-speed group (7-8 m/s):

The test acceleration dithers obviously with certain degree of sine characteristics which is shown in Fig. 4 and Fig. 5. For high speed motion while friction is not significant, TM finds out easily that the friction or the acceleration is linear, as shown in Fig. 6 and Fig. 7. It is not ideal enough for current identification in real circumstance. Observing from the dithery test acceleration, we can learn that traditional TM is unsuitable for current system modeling. So existing classical models should be improved.

Fig. 4Low-speed acceleration trace

Fig. 5High-speed acceleration trace

Fig. 6Result of low-speed group

Fig. 7Result of high-speed group

Test acceleration curves of experiment are drawn in Fig. 8. It shows that all the test acceleration curves have sine characteristics, which means that the vertical vibration of the cantilever beam and the collision between the cantilever beam and the moving mass affect the motion of the moving mass in the horizontal direction. So it is reasonable to introduce sine polynomial compensation into TM.

3.2. Parameter identification of ATM

Pressure of 2 MPa and 3 MPa is applied respectively to produce the motion at two different speeds, which are low-speed group (5 m/s) and high-speed group (8 m/s). For the two friction compensation models ATM 1 and ATM 2, to distinguish the performance more accurately, the compensation orders of both methods are set to 2.

Fig. 8Test acceleration curves of experiment

Fig. 9Low-speed acceleration curve

Fig. 10High-speed acceleration curve

For ATM 1, there are 10 parameters to be identified from Eq. 3 when $n=$2. Table 1, Table 2, Fig. 9 and Fig. 10 show the results of identification for both the low-speed group and high-speed group.

The error of the low-speed group is 4.22×10³, while the result for the high-speed group is 8.06×10².

For ATM 2, there are 12 parameters to be identified from Eq. 4 when $n=$2. Table 3, Table 4, Fig. 11 and Fig. 12 show the results of identification for both the low-speed group and high-speed group.

The error of the low-speed group is 1.37×10³, while the result for the high-speed group is 3.42×10². So ATM 2 for both the low-speed and high-speed group has a higher degree of accuracy than ATM 1.

Fig. 11Low-speed acceleration curve

Fig. 12High-speed acceleration curve

It can be concluded from both the errors of low-speed groups and the errors of high-speed groups that, under the same experimental parameters and exponent number $N$, ATM 2’s fitting degree of the actual friction process is higher than ATM 1’s.

Based on the theory of Fourier series, when the exponent number $N$ of the sine polynomials increases high enough, the compensation term can help to process almost any signal precisely. That is, the errors of both ATM 1 and ATM 2 will be reduced with the exponent number $N$ increasing, and the results will be more accurate. In order to validate above analysis, two simulations under low speed condition were performed in which $N$ is 2 and 3 respectively.

Figs. 13-16 show that when the exponent number $N$ of the sine polynomials increases from 2 to 3, the identification accuracy of ATM 1 raise by about 9 %, which is much less than 300 %, the improved accuracy by ATM 2. It is because the compensation methods of ATM 1 is a merged compensation, where all other additional factors are merged together to form a compensation. The advantage of ATM 1 is that the exact physical meaning of compensation term does not need to be known a priori. But the flaw of merge compensation lies in adjustment and optimization flexibility of compensation terms, thus prevents further improvement of the identification accuracy.

Fig. 13Results of ATM 1 (N= 2)

Fig. 14Results of ATM 1 (N= 3)

Fig. 15Results of ATM 2 (N= 2)

Fig. 16Results of ATM 2 (N= 3)

As mentioned before, the friction effect is divided into three parts: static friction, sliding friction and viscous friction. The effect of vibration and impact on these three parts is merged into a united expression in ATM 1. Static friction effect cannot be affected by external factors. However, sliding friction and viscous friction are the different cases. Though they are affected by the vibration and impact factors of the cantilever beam mechanism, the effects are not completely consistent. Accordingly the merged compensation model of ATM 1 will arise some identification error. Therefore, we proposed ATM 2, where sliding friction and viscous friction are compensated separately. Sliding friction and viscous friction are added by weight factors composed of sine terms. ATM 2 is more in line with the friction process description, which can make the accuracy better than that of ATM 1.

However, the exponent number $N$ of the sine polynomials increases by one, identification parameters will gain two at least. For ATM 2, if the exponent numbers of sliding friction and viscous friction both increase by one, four new parameters will arise. Too many identification parameters will significantly affect the performance of the algorithm. Accordingly, for the parameter identification of friction compensation model, it is important to select an appropriate compensation order number to ensure the balance of computation precision and efficiency.

4.1. Analysis of MGA

In order to compare algorithm efficiency and capability to avoid local optima of GA and MGA, a trigonometric function of single target and multiple local extremum is adopted as a verification example, which is:

The size of the population is 40, while the code length is 20. The iteration is 200 generations and the fitness function is ordering distribution. A method of stochastic universal sampling is utilized for searching. The function diagram is given by Fig. 17.

Fig. 17Target value curve of the example function

Simulation results (MATLAB) say that the maximum target value $y$ is 3.8503 and the corresponding $x$ is 1.8505. Because GA itself is an intelligent optimization algorithm with stochastic and probabilistic characteristics, 50 simulations were carried out for GA and MGA separately, among which the code length of MGA was a fixed number 20. The average value of the numbers, which represent the times of iteration to find the optimal solution, is chosen as a measure of the optimal iterative efficiency of the algorithm. Meanwhile, the execution time of each simulation is recorded to obtain the average time of these 50 simulations, which is also a measure of the calculation efficiency of the algorithm. The simulation results are shown in Fig. 18 and Fig. 19, in which the horizontal axis represents experiment amount and the vertical axis represents the time to obtain the optimal solution. Table 5 gives the efficiency of the algorithms.

Fig. 18Iteration times of GA

Fig. 19Iteration times of MGA

The MGA introduces a self-adaptive method for crossover and mutation, which increases the algorithm complexity compared with GA. The average execution time is slightly longer (+0.4 %), while a stronger iterative capability can be obtained (the number of iterations decreases by 12.64 %).

Using the trigonometric function of Eq. 7, three coding ways, i.e. 10 bits binary code, 20 bits binary code and self-adaptive binary code, are adopted to test the algorithm performance in the iterative environment of 200 generations. The parameters of the self-adaptive binary code are: $L_{const}=\text{10}$, $s=\text{50}$, $d=\text{0.2}$ and $G_{max}=\text{200}$. Execution performances of these three algorithms with different binary codes are given in Table 6.

Table 6 reflects that the self-adaptive binary code maintains the global optimal. It not only has the same accuracy with the fixed number binary codes, but also has higher execution efficiency than the fixed number binary codes. Therefore, the self-adaptive coding strategy contributes to effective balance of the execution efficiency and accurate search capability.

The proposed MGA is an optimized algorithm. In order to test whether it has optimized performance, 50 contrast experiments in which the traditional GA and the MGA with the self-adaptive binary code were utilized were conducted. The main algorithm performance comparisons are listed in Table 7.

It declares that for the friction process of the free motion on cantilever beam, introducing multiple self-adaptive strategies into optimize GA guarantees higher algorithm efficiency based on high identification precision.