External models of frictional interaction dynamics

This investigation suggests a method used to determine the evolution of metallic wear and friction by sliding. The friction of steel moving over brass was taken as an example. The problem of external dynamics friction is investigated through the definition of the dynamic characteristics such as damping factor and natural frequency. Some certain automatic control methods were applied for sliding friction contact, including parametric identification, ARX simulation and Newton’s dynamic equation. The suggested approach allows using amplitude-frequency characteristics to assess the dynamic factors (coefficients) under friction interaction. The research findings indicate that the proposed method allows monitoring the evolution of metallic wear and friction.


Introduction
The article investigates some issues referred to the friction study based on input-output models. The main aspect of this approach is to identify damping and vibration properties of a tribological system. The input-output model is widely used to solve a range of experimental problems in automatic control theory [1]. The use of well-known approaches to triboprocess stimulation and the application of multiorder equations allow analyzing friction processes with subject to their surface and intermediate layers [2,3].
It is important to predict some variations in the phase of a relative face slip in the majority of practical systems where metals, fiber, composite alloys are exposed to sliding with metallic materials. These measurements can be implemented differently. Some of them are vibration, noise monitoring, and temperature control of the friction surface. At present, however, the problem of accurate forecast based on the friction model has not been solved yet, due to the complicacy of the processes during friction interaction. The aim of this work is to study internal dynamic processes, friction evolution and determine the adaptive sliding model [4,5].

Fundamental principles
It is convenient to consider the tribological system with regard to its input-output. The input parameters of this system are dynamic characteristics like sliding velocity, rolling speed, force and pressure. Vibration and power dissipation (heating) belong to its output parameters. Under some external actions on this system it is possible to keep track of its reaction on this impact. Upon the results of the process identification This model is represented in time and frequency domains. Moreover, this model reproduces and imitates the adequate object behavior. The operation mode of the Tribal 1(a). This device is intended to determine the monitoring of the process. The main distinction of this device from similar friction and wear test machines is in the free movement a Figure 1. а -Schematic diagram of the device and its components: (t), a (t) -velocity (speed) and acceleration of the given law of motion b displacement curves, where е dt is the phase shift The phase shift curve of displacement curves is shown in Figure 1(b). The lower forward, while the upper one y(t) is delayed movement. There is some relative device operates in this mode.

Experimental unit
The experimental tribological unit Tribal to determine the tribological and mechanical properties of materials. The device for wear testing, which description is given in [2], was used as an analog. The idea samples relative to each other was implemented in this unit [ Figure 2: Tribometer Tribal-T appearance and its main components: 1 sensor; 3.4load sensors in normal and radial directions; the system response are its resistance to friction, surface deterioration, temerature rise on the friction surface and adherance of surfaces. [6].
Upon the results of the process identification we can create the mathematical model This model is represented in time and frequency domains. Moreover, this model reproduces and behavior. The operation mode of the Tribal-T device is shown in Figure  1(a). This device is intended to determine the friction factor (coefficient) and perform a real monitoring of the process. The main distinction of this device from similar friction and wear test machines is in the free movement of the upper sample relative to a lower one.
b Schematic diagram of the device and its components: Р-load, сvelocity (speed) and acceleration of the given law of motion b -The basic mode of е dt is the phase shift, Ax, Ay are the amplitudes of oscillations.
The phase shift curve of displacement curves is shown in Figure 1(b). The lower forward, while the upper one y(t) is delayed due to spring elasticity and friction forces affecting its ome relative motion of the samples with a typical sliding friction tribological unit Tribal-T ( Fig. 2) was developed on that basis. This unit is intended tribological and mechanical properties of materials. The device for wear testing, ], was used as an analog. The idea [7,8] of reciprocating motion of test was implemented in this unit [3].
T appearance and its main components: 1samples; load sensors in normal and radial directions; 5-precision linear sliding bearings ; 6.7 deterioration, temerature rise on the mathematical model of the object. This model is represented in time and frequency domains. Moreover, this model reproduces and T device is shown in Figure  and perform a real-time monitoring of the process. The main distinction of this device from similar friction and wear test b variable rigidity, v The basic mode of es of oscillations.
The phase shift curve of displacement curves is shown in Figure 1(b). The lower sample x(t) moves due to spring elasticity and friction forces affecting its with a typical sliding friction when the . This unit is intended tribological and mechanical properties of materials. Lower samples execute cyclically reciprocating movements driven by a cyclic linear drive (8). Upper samples are in motion due to the impact of the friction force. Displacement and pressure sensors allow tracking the absolute displacement of samples and generate a digital signal in a real time. The measured signals are inputs and outputs of the investigated system. The proposed approach provides for using the Tribal-T tribometer to study sliding (rubbing) friction.

Mathematical representation of system identification
In the general case , system " input "and "output" can be described by the n-th order differential equation, where the right side is" input ", the parameters , , … , are selected for a given input signal, and , , … , are considered being specified.
where x ( ) = d x/dt under the assumption that n > . However, to solve this equation it is necessary, first of all to take into account the order of differentiation, which provides physical interpretation. When = 2 we will get second-order differential motion equation: x -the input value, n -damping factor (depends on loading rate); f(t) -external action on the system; w -natural-vibration frequency. Second-order differential motion equation, where n=2, complies with the model of the system investigated on the basis of the Tribal-T tribometric system, where the relative displacement of two surfaces takes place. The dynamics of mchanical oscillating system can be represented by Newton equation [9,10].
Disturbing forces are located on the right side of the equation, while the system response to the external action is on the left side of this equation. To integrate the equation we have created the performance (characteristic) equation and evaluated its roots: Damped frequency: The period of damped oscillations is the time interval between two successive passes through the point "0" in the propagation (traveling) direction: Where: T = 2π/w is the natural period of vibration. Hence, oscillation damping period lasts much longer than its natural period of vibration.
The period of damped oscillations exposed to low resistance can be equal to natural oscillation. Damping is very fast, even at low resistance. Thus, low frequency components have the main influence on resistance during free oscillations. The laws of movements, described by the secondorder differential equation, are the most understable from the viewpoint of classicalal mechanics [11].

Computational solution of the system in a state space
Let us consider the solution of simultaneous equations (1) for the second-order dynamic characteristics. The solution of this equations is based on identificationof a nonlinear autoregression model (ARX). The vector-matrix form of the first-order differential equation system, also called the Applying the Matlab application package model (for the second-order differential equation where: z is the matrix of input and output discrete signals «2» is the order of the differential equation triboprocess; «1» is the number of inputs per one channel «0» means that the delay is left out of account; «0» is zero-initial condition. Further solution of this equation allows determining the parameters of the system. These parameters have oscillating nature such as damping coefficient and self-resonant (natural) frequency The obtained parameters such as system eigenval and natural (undamped) frequency allow describing the system features and changes [6]. Besides, the dynamic friction coefficient can be determined with regard to the rate: where: A(ω)= A(out)/ A(in) is the ratio of the amplitude of input and output signals

Practical application of the method and discussions
This part presents the testing of samples made from the constructional bearing steel (CBS analogs) has been investigated. lower samples Fig. (1). In the course of the experiment roughness (Ra 0.25). To achieve the desired result f These time intervals lasted 30 min., 60 min., 90 min. zero till reaching the time domain. measured, the model of a system was constructed and calculated. At each stage of the experi dynamic system identification. a the basis of the mathematical model of a multidimensional system in the time The equation of state fully describes the control object, taking into account The model of friction (rubbing) state of the tribometer Tribal-T is as follows: are correlated stochastic processes. The effective recursive Kalman filter stationarity. Matlab application package system for the solution of the system we create a discrete order differential equation), where th = canstart (z, 2,1, [0,1,0]); here: z is the matrix of input and output discrete signals; «2» is the order of the differential equation, which is selected depending on the type of «1» is the number of inputs per one channel; delay is left out of account; solution of this equation allows determining the transfer function and characteristic These parameters have oscillating nature such as damping coefficient and [12,13]. parameters such as system eigenvalue (characteristic constant), frequency allow describing the system features and predicting their ]. Besides, the dynamic friction coefficient can be determined with regard to the is the ratio of the amplitude of input and output signals of the method and discussions of the above mentioned mathematical apparatus. Sliding constructional bearing steel (CBS15) and L90 brass (C22000, CuZn10 The input data were the displacements (movements) of upper and n the course of the experiment we studied 10 samples with t To achieve the desired result five test runs with different time domains (intervals) were carried out. These time intervals lasted 30 min., 60 min., 90 min. The experiments were carried out from the time o till reaching the time domain. After each time interval the profiles of the s the model of a system was constructed and the dynamic parameters of the system were At each stage of the experiment we recorded the input u(t) and output data y(t)for further b the basis of the mathematical model of a multidimensional system in the time , taking into account the change of its (7) he effective recursive Kalman filter is The variation in surface roughness the intensification of the oscillating nature of process by th diagram (Fig. 3). It is confirmed by Impulse and Step Responses illustrated in the diagram (Fig. 4 from right to the left). The system input is exposed to single step impact of impulse characteristic with specified length. Both systems are equivalent if the oscillation phase is not considered. In accordance with the specified task of external dynamics the following parameters were determined: ndamping coefficient (rate), w concluded that for given experimental conditions frequency have differently directed competitive dynamics

Relative unit
Response to step impulse action at different time domains depending on frequency: 1 surface roughness of samples was not registered. We determined the tendency of the intensification of the oscillating nature of process by the increase of the resonance spike in Bode 3). It is confirmed by Impulse and Step Responses illustrated in the diagram (Fig. 4 from ight to the left). The system input is exposed to single step impact of impulse characteristic with length. Both systems are equivalent if the oscillation phase is not considered.
The diagrams show the variation in damping coefficient and natural vibration frequency (total duration of the experiment is 90 minutes) In accordance with the specified task of external dynamics the following parameters were damping coefficient (rate), w -natural (undamped) frequency for given experimental conditions the damping coefficient and natural frequency have differently directed competitive dynamics. We determined the tendency of e increase of the resonance spike in Bode 3). It is confirmed by Impulse and Step Responses illustrated in the diagram (Fig. 4 from ight to the left). The system input is exposed to single step impact of impulse characteristic with length. Both systems are equivalent if the oscillation phase is not considered.
The diagrams show the variation in damping coefficient and natural vibration frequency (total In accordance with the specified task of external dynamics the following parameters were frequency (Fig. 5). It can be coefficient and natural-vibration