The phase portrait of the vibro-impact dynamics of two mass particle motions along rough circle/Dvieju koncentruotu masiu vibrosmuginio judejimo apskritimine nelygia trajektorija fazinis portretas.

Jovic, S. ; Raicevic, V.

1. Introduction

Nonlinear phenomena in the presence of certain discontinuity represent the area of interest of numerous researchers from all over the world. Theoretical knowledge of vibro-impact systems (see references [1-3]) are of particular importance to engineering practice because of the wide application of vibro-impact effects, used for the realization of the technological process. The analysis of mathematical pendulum with and without "turbulent" attenuation and papers published by Katica (Stevanovic) Hedrih [4,5] related to the heavy mass particle motion along the rough curvilinear routes are the basis of this work. Based on the original works from the area of non-linear mechanics, or vibro-impact systems by the authors: Frantisek Peterka [6-8], Katica (Stevanovic) Hedrih [9], and the others, and the previous works of the authors of this paper [10-15] in which the authors analyzed several variants of vibroimpact system with one degree of freedom, based on the oscillator moving along a rough circle, sliding Coulomb-type friction and limited elongation, in this paper the vibro-impact system with two degrees of freedom, based on forced oscillations of two heavy mass particles, mass [m.sub.1] and m2 moving along rough circle in vertical plane, sliding Coulomb-type friction and limited elongation is studied (Fig. 1).

The elongation limiter is set on the right. The limiter position is determined by the angle [[delta].sub.1], measured from the equilibrium position of the mass particles, i.e. from the vertical line crossing the centre of the circular line. The system consists of two mass particles, [m.sub.1] and [m.sub.2], exposed to the effect of gravity. These mass particles are moving along rough circle in vertical plane on which the two sided impact limiters of elongation (constraints) were placed. The limiter position is determined by the angle o, measured from the equilibrium position of the mass particles, i.e. from the vertical line crossing the centre of the circular line. The limiter set on the right side from the equilibrium position, defined by the angle o1 is stable. The first mass particle is affected by the external periodic force [F.sub.1] (t) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = [F.sub.10] cos[[OMEGA].sub.x]t , where [F.sub.10] is the corresponding force amplitude, and Q1 is the frequency of external force. The angular elongations of the first and second mass particles in an arbitrary moment t were marked by [[phi].sub.1], [phi.sub.2] respectively, and measured from the equilibrium position. At the initial moment of time the material points were on the distances [[phi].sub.10], [[phi].sub.20] from the equilibrium position 0-0, and were given the initial angular velocities, [[phi].sub.l0], [[phi].sub.20].

[FIGURE 1 OMITTED]

The task is to consider the properties of forced oscillation of the first and second mass particles in a circular rough line with limited elongations, so the system becomes vibro-impact with one sided limited angular elongation. The differential equations of motion of the mass particles are requested for each interval of motion from impact to impact, from collision to collision, and the interval of motion when the friction force direction alternation appears associated with the direction alternation of angular velocity of motion of a mass particle, and also velocity alternation as a consequence of the mass particle impact into the angular elongation limiter and mutual impact of the mass particles.

Differential equations are matched to the initial motion conditions, system elongation limitation conditions, the mass particles impact conditions, and alternation conditions of friction force direction. Also, it was necessary to determine the impact conditions of both mass particles separately, the phase trajectory equations in phase planes and the mass particles collision conditions in ideally elastic impacts. Determine after how many impacts the system will stop behaving as vibro-impact system?

2. Differential equation of oscillations of a mass particle moving along rough circle

The observed vibro-impact system has two degrees of freedom, so the corresponding governing nonlinear differential equations of motion presented as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

for [micro] = tg[[alpha].sub.0] is sliding Coulomb-type friction coefficient, [[phi].sub.1], [[phi].sub.2] are generalized coordinates for monitoring motion of the first and second mass particles.

This system of double differential non-linear equations is coupled by initial motion conditions:

a) the first mass particle (pellet 1), in further text is marked with subscript 1

[[phi].sub.1(0)] = [[phi].sub.10] and [[phi].sub.1](0) = [[phi].sub.10]; (3)

b) the second mass particle (pellet 2), in further text is marked with subscript 2

[[phi].sub.2(0)] = [[phi].sub.20] and [[phi].sub.2](0) = [[phi].sub.20]; (4)

At the initial moment of motion, the mass particles were given the positive initial angular velocity ([[phi].sub.1] > [[phi].sub.2], > 0) .

For the complete description of the observed vibro-impact system are needed to be set, and also matching of limitation conditions angular elongations and impacts to the elongations limiters.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where k is coefficient of collision (impact), within the interval from k = 0 for ideal plastic collision to k = 1 for ideal elastic collision (impact), and n is the number of impacts until the system is returned into the equilibrium position.

The differential equation of motion of the second pellet (2) can be solved in analytical form, so its first integer is phase trajectory equation in form of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where C is integration constant depending on the initial motion conditions.

For graphic presentations of the phase trajectories in the individual motion intervals of the second mass particle we use software package MathCad 14.

The differential equation of motion of the first heavy mass particle (1) cannot be solved explicitly (in a closed form). For its approximate solution the software package WOLFRAM Mathematica 7 is used. The results are checked by using software package MATLAB R2008a.

3. Motion analysis of the vibro-impact system

The operational system of the mobile angular elongation limiter, which is positioned on the right side from the equilibrium position is based on the fact that the system is pulled by the impact of the pellet, and returned to the initial position by the impact of the pellet into the elongation limiter set on the left side from the equilibrium position. This system creates the motion of:

The first mass particle, mass [m.sub.1], within the interval from the impact to the second mass particle , mass [m.sub.2], to the impact into elongation limiter set on the right side ([[delta].sub.1]), or to the impact into angular elongation limiter set on the left side (2[pi]--[[delta].sub.2]), or, to the first mass particle motion alternation (when it happens). There is a possibility for the first pellet to have an impact into elongation limiter to the left (2[pi]--[[delta].sub.2]), then itreaches the alternation point and hits again into the same elongation limiter.

Motion the second mass particle, mass [m.sub.2] is in the interval from the impact with the first mass particle , mass [m.sub.1], to the impact into angular elongation limiter set on the left ([[delta].sub.2]), or to the impact into angular elongation limiter set on the right ([[delta].sub.1]) i.e. to the second mass particle motion direction alternation (when it happens). There is a possibility that the second mass particle has an impact into elongation limiter to the left ([[delta].sub.2]) reaches the alternation point, and hits again into the same elongation limiter.

For the determination of phase portrait branches of the first and second heavy mass particle individually, the motion of the heavy mass particles along rough circle line is divided into corresponding motion intervals and subintervals.

The first pellet--the first motion interval represents the interval from the initial time until the first impact of the pellet 1 into the angular elongation limiter on the right side.

The first motion interval of the first mass particle corresponds to the differential Eq. (1) of motion for [[??].sub.1] ft >0.

The impact conditions are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[FIGURE 2 OMITTED]

Phase trajectory [[??].sub.11] = / ([[phi].sub.1]) in the first motion interval (that will be used for the determination of the velocity of the mass particle impact into the angular elongation limiter) defined by using the software package Wolfram Mathematica 7 (also used for all other graphic presentations) is presented in Fig. 2.

The parameter values are: [[delta].sub.1] = [pi]/4 [rad],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The angular velocity of the first mass particle into elongation limiter [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is read from the phase trajectory, presented in Fig. 2, a. The time interval of the first heavy mass particle impact into elongation limiter ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) is determined by using software package MATLAB R2008a. Both values ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) are taken for the angle where the elongation limiter is positioned.

The second pellet--the first motion interval represents the interval from the initial moment until the first collision of the pellet 2 to the pellet 1.

The first motion interval of the second mass particle corresponds to the differential Eq. (2) of motion for [[??].sub.2]> 0, matching the initial conditions (2).

The phase trajectory <[[??].sub.21] = /([[phi].sub.2]) in the first motion interval is presented in Fig. 3.

The parameter values are: [[delta].sub.20] = [pi]/12[rad],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The second pellet's phase trajectory, shown in Fig. 3, b, points on the periodic motion of the second pellet.

For the further study it was necessary to define the position of the pellet 2 when the pellet 1 reaches the elongation limiter. The position of the second pellet [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in time [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is determined by using MATLAB-u R2008a.

After the definition of the position [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] angular velocity of pellet 2 at the moment when from pellet 1 reaches the elongation limiter can be read from graphic presentation of the phase trajectory [[??].sub.21] = f([[phi].sub.2]) (presented in Fig. 3, a).

The mass particles have an impact in the second motion interval of the first mass particle and in the first motion interval of the second mass particle.

The first mass particle--the second motion interval is an interval from the first impact into elongation limiter to the first heavy mass particles collision.

The second motion interval of the first mass particle corresponds to the differential equation of motion in form of (1) for [[??].sub.1] < 0, matched to the initial conditions of motion [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[FIGURE 3 OMITTED]

Phase trajectory [[??].sub.12] = f([[delta].sub.1]) in the second motion interval is presented in Fig. 4.

Further analysis is based to the definition of time interval in which the first collision occurs. After the definition of the moment of the first impact [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the angle [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is determined as a basis for further motion analysis of the pellets.

[FIGURE 4 OMITTED]

The initial data for the determination of the time interval [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are:

a) for the first pellet [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

b) for the second pellet [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

NOTE: In the previous expressions the indexes ij present: i - pellets number; j - motion interval number. The condition for the time tsud1 definition is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Time [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is determined from the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

The accelerations [[??].sub.12] and [[??].sub.21] were approximately determined (with sufficient accuracy) as the median value of the average accelerations in the sub-intervals of the observed interval. In this case the interval [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is divided into six equal sub-intervals.

For the obtained value of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]with the corresponding program files from MATLAB R2008a for the second motion interval of the first pellet and the first motion interval of the second pellet (values must match) the angle of the first impact [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is determined.

Values for the angle [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be used for the determination of angular velocities of the pellets 1 and 2 immediately before the first impact from the phase trajectories for the second motion interval of the first pellet (Fig. 4, a) and the first motion interval of the second pellet (Fig. 3, a) i.e. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The mass centres of particles are positioned on the rough circle line, i.e. the impact centres are positioned on the same axes. This is about central impact.

The expressions for explicit definition of the angular velocities immediately after the impacts with using Law of momentum and Newton's hypothesis about the relation of relative angular velocities of the mass particles are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The generalized coordinate [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where the first impact appears and velocities of the pellets immediately after the collision [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the initial conditions of motion of the pellets in the following motion intervals.

The motion analysis of the observed vibro-impact system is conducted up to the twelfth impact of pellets 1 and 2. It should be mentioned that until the fourth impact of the pellets, the pellets are moving in zone from [[delta].sub.1] to [[delta].sub.2]. From the fourth to the twelfth impact, the pellets are moving in zone from [[delta].sub.1] to I[pi] _[[delta].sub.2]. After the twelfth impact of the pellets, the motion zone is divided, so the first pellet is moving in zone ([[delta].sub.1])--(2[pi]- [[delta].sub.2]), and the other pellet is moving within the zone ([[delta].sub.1])--([[delta].sub.2]). The first pellet influenced by the external single frequency force after the eighth impact into elongation limiter at the coordinate (2[pi] -[[delta].sub.2]) does not have a strength to cross the limit [pi], i.e. the alternation point is positioned on the distance [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], that points out that the pellet will be still after several impacts at the coordinate 2[pi]- [[delta].sub.2]. After the alternation point in zone ([[delta].sub.1]) - ([[delta].sub.11]), the second pellet has only two impacts into mobile elongation limiter set at the coordinate [[delta].sub.1] which is not pulled inside at those moments. After the second impact, the second pellet continues to move without impacts and in several motion intervals returns into equilibrium position [[delta].sub.2] = 0. The second pellet completed thirteen impacts into elongation limiter, three of them into stable, and ten of them into mobile elongation limiter.

The graphic visualization of the motion analysis, performed for the observed vibro-impact system based on oscillator moving along rough circle line, composed of two ideally smooth pellets is shown in Fig. 5 and Fig. 6. The phase portrait of the pellet 1 is shown in Fig. 5, and phase portrait of the pellet 2 is shown in Fig. 6.

4. Conclusions

Non-linearity of the observed vibro-impact system is due to the discontinuity of angular velocities of the mass particles moving along rough circle line. The discontinuities of angular velocities occur at the moment of impact of mass particle 1 into angular elongation limiters at the coordinate [[delta].sub.1] and (2[pi] - [[delta].sub.11]), at the moment of direction alternation of motion of the mass particles 1 and 2 (when it happens), causing the alternation of angular velocity direction and friction force alternation, and at the moment of impact (collision) of mass particles. This nonlinearity is described for both mass particles by the system of regular non-linear differential equations, particularly by the second member, representing angular velocity square of the generalized coordinate [[??].sup.2.sub.1], [[??].sup.2.sub.2]. That corresponds to the case of turbulent attenuation. It should be mentioned that in the observed vibro-impact system with two degrees of freedom we have trigger constrained singularities, i.e. we have bifurcation phenomena of the equilibrium positions due to the influence of the sliding Coulomb's friction force and the alternations of angular velocities direction of the mass particles.

For the individual motion intervals of the mass particles the differential equations of motion with matched initial motion conditions are written in this paper, related to the positions of the mass particles at the moment of impact into elongation limiters, at the moment of motion direction alternation and at the moment of collision of the mass particles.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

It should be noted, that with mobile elongation limiter set in positions "on" and "off, in the coordinate o1, after the twelfth impact of the pellets their zones of motion are separated. The first mass particle is calmed down at the position defined by the coordinate (2[pi]-[[delta].sub.2]), and the second mass particle is back into the equilibrium position ([[delta].sub.2] = 0).

It should be noticed the methodology of the determination of time and position of the mass particles in the moment of collision. The outgoing velocities of the mass particles after the impacts are determined analytically and time of the impact, the position of the mass particles at the moment of collision, and the ingoing velocities are determined numerically. By the numerical solutions of the differential equations of motion (MATLAB R20008a and Wolfram Mathematica 7) by using the initial motion conditions, the graphic visualization of oscillations of the mass particles in the observed vibro-impact system with two degrees of freedom is given.

The phase portraits of mass particle 1 and mass particle 2 are obtained by the combination of analytical and numerical results in the procedure of producing of graphic interpretations of phase trajectories in the individual motion intervals of the mass particles, with the application of software programs MATLAB R20008a and Corel Draw 12. In these phase portraits there the phenomena of non-linearity of vibro-impact system with two degrees of freedom are clearly visible.

10.5755/j01.mech.18.6.3159

Acknowledgment

Parts of this research were supported by the Ministry of Sciences and Technology of Republic of Serbia through Mathematical Institute SANU Belgrade Grant ON174001 Dynamics of hybrid systems with complex structures. Mechanics of materials and Faculty of Technical Sciences University of Pristina residing in Kosovska Mitrovica.

Received August 01, 2011 Accepted November 15, 2012

References

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S. Jovic, V. Raicevic

Faculty of Technical Sciences, Kosovska Mitrovica, University of Pristina, Ul. Kralja Petra I br. 149/12, Kosovska

Mitrovica, Serbia, E-mail: jovic003@gmail.com