MOTION RESULTS OF A SELF-MOBILE SYSTEM WITH MENTIONING THE STOCHASTIC CHARACTERISTIC IN THE FRICTION

International Journal of Mechanical and Production Engineering Research and Development (IJMPERD) ISSN (P): 2249–6890; ISSN (E): 2249–8001 Vol. 10, Issue 3, Jun 2020, 885-894 © TJPRC Pvt. Ltd.

MOTION RESULTS OF A SELF-MOBILE SYSTEM WITH MENTIONING THE STOCHASTIC CHARACTERISTIC IN THE FRICTION FORCE THE-HUNG DUONG & THE-THINHNGUYEN Thai Nguyen University of Technology, Thai Nguyen University, Vietnam ABSTRACT This paper presents and proposes a model of a self-mobile system with mentioning the stochastic characteristic in the friction force. The friction force is assumed as a stationary white noise randomprocess. Implementation is to analyze the stochastic characteristic in the friction force for the motion of the self-mobile system in order to get the responses, such as its displacements and velocities. In this paper, we also have analyzed and evaluated the effect of some model parameters to the motion in the considered system. There are motion results in two models of deterministic and stochastic friction force, showing that when the friction force is deterministic, the progression rate – that is displacement of x2 is always greater than the others. From the given results, engineers could choose the model parameters during monitoring and controlling the motion of a self-mobile system to reach the best progression.

Received: Apr 16, 2020; Accepted: May 06, 2020; Published: Jun 01, 2020; Paper Id.: IJMPERDJUN202079

1. INTRODUCTION A self-mobile system is a self-motion structure, but there are not any wheels or legs attached to it. Some self-mobile systems can move by slide between their bodies and surface around them. They can move because of motors that

Original Article

KEYWORDS: Friction, Monte Carlo, Stochastic & Self-mobile System

mounted and vibrated inside their bodies. Self-mobile systems are very popular in engineering, as per recent researches [1,2,3,4,5]. During researching the motion of self-mobile systems, especially the systems can be applied in different environments such as endoscopic capsules, in pits, pipes, so it is impractical to consider the coefficient of friction forces unchanged. In the previous documents [4,5,6] authors usually assume that the friction coefficient in the friction force is constant during all the time of motion. We can see clearly that if the friction force in the selfmobile system is assumed to be a stochastic process, thenit will be very closer in reality. The models acting on stochastic forces in the self-mobile systems have found in the documents [3,8,9,19]. However, the stochastic characteristic in the friction force when studying self-mobile systems has received little attention in recent studies.

In the document [3], authors have been referred to a self-mobile system acting generally stochastic environment by using a white noise process, but they have not been focus on the friction force is a random process. In the paper [7], authors investigated to define the law of probability distribution of random friction force. From the idea of the paper [7], in this paper, we apply it to study the effect of random friction forces in a self-mobile system with assuming that friction force is a stationary white noise process.

When studying a self-mobile system, the problem is how to reach the maximum progression rate (displacement of x2). In this paper, we will consider the random effect of friction force and adjust model parameters on progression rate to allow the system motion to reach to the best progression rate. This paper also is only referred to the friction force to be a stationary white noise process. In particular, we will focus on the effect of the spectral

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density S0 acting on displacements X1 and X2 in a self-mobile system model.

Contents of this paper include: To present a model of self-mobile system with stochastic friction forces; To analyze the motion of the self-mobile system track; To investigate model parameters (denoted by five coefficients are , , ,  and ) acting on the motion of system such as the displacement X1 and X2 and more different responses in the considered system. The paper has used the Monte Carlo simulation method to obtain displacement and velocity results.

2. MATHEMATICAL MODEL There are many models of self-mobile systems [3, 4, 5, 6], one of them is relatively simple but still contains useful information, as in the previous paper [6] that we have already considered a self-mobile system with the following physical model:

Figure 1: Model of the Self-mobile System [6].

This self-mobile system includes two masses m1 and m2 with horizontal displacements are X1 and X2; a cubic spring Fspr and damping force with a coefficient c. The system is acted by a periodic impulse force F with amplitude A and frequency Ω that can be expressed in a form of the Heaviside function. Because of a friction force Ff it could be moved backward or forward. In order to consider this system by the probabilistic method [8, 9, 10], in this paper the force Ff is assumed a stochastic process that can consist of two parts: Ff = F f + Fst

(1)

Where, F f is a deterministic part as same as the previous document [6]:

 dX  F f = FC sign  2  , Fc = f . ( m1 + m2 ) g  dt 

(2)

andFstis a stochastic part thatisassumed by the form:

 dX Fst = Fng FC sign  2  dt

  , Fng =  (t ) 

(3)

Where, d()/dt denotes differentiation with respect to time, t; sign() denotes the sign of velocity (dX2/dt); fis the Impact Factor (JCC): 8.8746

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friction coefficient between the frame body and the environment; g is the gravitational acceleration;  with 0<<<1 is a determininistic constant; (t) is a stochastic process with mean to be zero and that is assumed a stationary white noise process with the intensity V and the power spectral S0 is constant (V and S0 are expressed by the relationship S0=V/2 [9]).

The equations describing the motion of the mobile system can be written as [6]:

 d 2 X1 dX   dX = F − Fspr − c  1 − 2  m1 2 dt  dt  dt   d2X dX  dX 2 = − F + Fspr + c  1 − 2 m2 2 dt dt dt     A,sin ( t )  0 F =    0,sin ( t )  0  3  Fspr = k1 ( X 1 − X 2 ) + k2 ( X 1 − X 2 )

  − Ff 

(4)

The equation (4) would be expressed by the dimensionless form as following

  H − ( x1 − x2 ) −  ( x1 − x2 )3 −  2  d x1    2 =   dx1 dx2  d    −2  d − d             − H + ( x − x ) +  ( x − x )3 +   1 2 1 2    2   d x dx dx  1  2 2  = +2  − / −  d 2   d d        dx2    − (1 +  ( ) ) sign      d    

(5)

where, we call coefficients describing below are the model parameters of the system:

  x 1 = k1 X 1 ; x 2 = k1 X 2 ; 0 = k1 ; = 0t ;  FC FC m1  m2   ; H = heaviside ( sin ( ) ) ;  = ;  = m1 0  2    = c ;  = A ;  = k2  FC  ;    2m10 FC k1  k1   Set v1 = x1 =

(6)

dv dv d 2 x1 d 2 x2 dx1 dx , v2 = x2 = 2 , v1 = 1 = and v2 = 2 = then Eq.(5) yields as an one-order d d d d d d

differential equation:

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 x1 = v1   H − ( x1 − x2 ) −  ( x1 − x2 )3 −    v = 1   −2 ( v1 − v2 )   x = v 2  2      − H + ( x − x ) +  ( x − x )3 +  1 2 1 2    v =  +2 ( v − v ) − / 2 1 2        dx2  − 1 +  (  ) sign )     (  d      

(7)

The displacement x1, x2 and velocity v1, v2will use in the next sections below. The Eq. (7) includes non-smooth, therefore the Monte Carlo simulation is used to analyz ebehavior of the system.

3. RESULTS AND DISCUSSIONS In the following sections, we will call the “Deterministic Model” (DM) that its results are calculated in the previous paper [6] and the “Stochastic Model” (SM) that is implemented in this paper, considering the stochastic characteristic of the friction force. When varying for a parameter during calculation, except some particular cases, we will usually assign other parameters as following =1.0, =0.025, =8, =-3.75, =2.0 and =0.02. When implementing the Monte Carlo simulation, we will usually assigna t values of S0=1, S0=5, S0=10 and S0=15; And amount of samples will be changing around 100-300 realizations to be enough by the law of large number and depending on the specific circumstances.

In this section, contents include, by order is as follows: Section 1 deals with the random force part and calculation of the progression rate x2whenchanging and S0; Section 2 deals with time histories of displacements of x1 and x2; Section 3 deals with phase trajectories of relative motion of displacement x 1-x2 and velocity v1-v2; and the diagrams of the Joint Probability Density Functions PDF(x,v); from Section 4 to 6. deal with investigating responses of the displacement x2whenchanging one of the model parameters (five coefficients are , , ,  and ) at the time =400.

3.1 The part of Random Friction Forces and Values of x2 Figure 2a denotes realizations of the part of random friction forces Fng =  ( ) at [0,25] corresponding to four cases of S0=1, S0=5, S0=10 and S0=15. The mean of () equals to zero. This figure described by the naked eyes how much the random part of friction forces willact to the self-mobile, and then how this part will be measured by two parameters S0 and . The figures 2b. and 2c. with the horizontal axis to be S 0,  depict the progression rate of x2 (at =400) on the vertical axis. Clearly, the larger or S0 gets, the smaller x2 receives. And when comparing with DM (in the Figure 2b and 2c to be depicted by the value of S0=0 or =0), we clearly see that the values of x2 in DM are always larger than in SM.

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(a) 28 26.25

26

Displacement, x2 (=400)

Displacement, x2 (=400)

25.00

24 22 20 18

=0 =0.01 =0.02 =0.03 =0.04

16 14

2

3

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20.00

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10 11 12 13 14 15

15.00 0.012 0.015 0.018 0.021 0.024 0.027 0.030 0.033 0.036

Density spectral, S0

Coefficient, 

(b)

(c) Figure 2: The Random Force Part and x2.

3.2 Observating Time Histories of Relative Displacement x1-x2 and Absolute Displacement x2 We can see the progression during time histories in the Figure 3 for the relative displacement x1-x2 and in the Figure 4 for the absolute displacement x2. In these two figures, there are blue lines and red lines corresponding to DM and SM, respectively. The Figure 3a and Figure 4a is shown at [0,400] and the Figure 3b and Figure 4b is shown at [0,25]; From the Figure 4, it is shown that the red lines always tend to go down; it proves that the values of x2 in SM always are smaller than in DM, and here, we have assigned S0=5. This problem is also verified in the next sections.

(a) (b) Figure 3: Time Histories of the Relative Displacement x1-x2 when (a) =0400 and (b) =025

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(a) (b) Figure 4: Time Histories of the Absolute Displacement x2 when (a) =0400 and (b) =025 3.3Phase Trajectories and PDF(x,v) Figures 5a1 and 5a2 depict phase trajectories of relative motion at =3.75 and =1.0, in which the horizontal axis is the relative displacement x1-x2 and the vertical axis is the relative velocity v 1-v2. In these two figures, the red lines denote following to DM and the blue lines to SM, and then shown that the blue lines are oscillated clearly around the red lines.

From the phase trajectories in Figure 5a1 and 5a2, we can construct the joint Probabilistic Density Functions PDF(x,v) of the relative displacement and velocity, as shown in the Figure 5b1 and 5b2. Additionally, we can see clearly and compare with the PDFs(x,v) in the documents [3] about appearing the P-bifurcation phenomenon, corresponding to values of =3.75 and =1.0. Based on the phase trajectories and the PDF(x,v) shown in Figure 5, engineers could evaluate and control the motion of the self-mobile system, and subsequently choose the best model parameters.

(a2)

(a1)

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(b1)

(b2) Figure 5: Phase Trajectories and PDF(x,v).

3.4 Effect of Changing Frequency  and Amplitude  to Value of x2 The system response results (the progression rate of displacement x2) when changing frequency [0.1÷4.3] are shown on the Figure. 6. This figure depicts mean values of displacement x 2 at =400 corresponding to four cases of S0=1, S0=5, S0=10 and S0=15. We already have calculated the Monte Carlo simulation for each case with around 300 samples and the results are compared with DM that is denoted by S0=0. After getting those results, it shows that the degree of disturbance at each value of to x2 is different. We can see the progression rate x2is changed not completely proportional to the value of S0, these are shown x2 to reach maximum are at =1.9 and =2.8 (corresponding to the value of S0=1) and at =1.9 and =3.1 (corresponding to the values of S0=5, S0=10 and S0=15). At =1.3 the value of x2 is inverse proportional to S0 and vice versa at around =3.75 then x2 is clearly proportional to S0. Hence, during controlling the system by frequencies we need choose them very thoughtfully.

As same as  shown in Figure 6, Figure 7 depicts the mean value of x2 (at =400) when changing the amplitude of excited forces [-4.3÷-0.1] corresponding to S0=1, S0=5, S0=10 and S0=15 afterward comparing with DM denoted by S0=0. The results that changing amplitude  are evaluated nearly as similar as . However, the rule of variation of x 2 can see clearly or proportional or inverse proportional related to increase or decrease values of S 0 at each value of .

40 S0=0

S0=1

S0=1

Displacement, x2 (=400)

S0=10 S0=15

0

S0=0

-20

-40

20

Displacement, x2 (=400)

20

S0=5

S0=5 S0=10 S0=15

0

-20

-40

-60

-60 -80

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

-4.0

Figure 6: Value of Progression x2 when Changing  www.tjprc.org

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-3.2

-2.8

-2.4

-2.0

-1.6

-1.2

-0.8

-0.4

Coefficient, 

frequency, 

Figure 7: Value of Progression x2 when Changing 

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3.5Effect of Changing Parameter  to Value of x2 Figure 8 depicts values of x2 (at =400) when changing [0÷28] at S0=1, S0=5, S0=10 and S0=15; and also comparing with DM denoted by S0=0. The rule of variation of values of x2 following to  is same as to , namely at =16 reveal that the value of x2 is not proportional or inverse proportional to increase or decrease corresponding to the value of S 0.

40

S0=10 S0=1

Displacement, x2 (=400)

20

S0=15 S0=0

0

S0=5

-20

-40

-60

-80 0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

Coefficient, 

Figure 8: Progression of x2 when Changing .

3.6 Effect of Changing Parameters  and  to x2 Figure 9 shows the value of x2 (at =400) when changing mass coefficient [0.98 ÷ 1.08] with at S0=1, S0=5, S0=10 and S0=15; and compared to DM denoted by S0=0. Except the rule of variation of values of x 2 to be not smooth in DM, the values of x2 in SM are very smooth and decreasing clearly to S 0. For the values of x2 changed corresponding to the viscous damping coefficient [0.005÷0.205] that shown in Figure 10 has the same rule with changing mass coefficient .

36

28 S0=0

27

34

S0=1 S0=5 S0=10

Displacement, x2 (=400)

S0=15

25 24 23 22 21 20

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18 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08

12 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

Coefficient, 

Coefficient, 

Figure 9: Progression of x2 when changing 

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Figure 10: Progression of x2 when changing 

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4. CONCLUSIONS This paper has already proposed a self-mobile system with stochastic characteristics of friction force. Compared with the calculation following the determinicstic model, the model with the random characteristic of friction force could reflect closer work to reality. The dependence of the progression rate (displacement x2) has been assessed by model parameters (, , ,  and ) and the noise level of friction force (white noise intensity S 0 or ) as shown in the following characteristics: •

Observating phase trajectories of relative motion and the diagrams of PDF(x,v) will sense the motion track in the self-mobile system.

•

In DM the results of values x2ishigherthan in SM. Namely, the higher S0 (or ) is, the lower x2 receives.

•

We can divide the results into three types, as following: (1) with the parameters  and β– at some particular values, the values of x2 are changed following or not following to proportional to them; (2) with the parameter α, the values of x2 are changed completely following proportional or inverse proportional to them; and (3) with  and , the values of x2 are changed following inverse proportional to them.

•

Motion results of a self-mobile system in this paper are the beginning for implementing further researches, to choose and control optimal parameters for the progression rate x2 to reach maximum.

ACKNOWLEDGEMENTS This research is funded by Thai Nguyen University of Technology (TNUT) for supporting to publish new researches.

REFERENCES 1.

A. Shukla and H. Karki, Application of robotics in on shoreoil and gas industry – a review Part I, Robot. Auton. Syst. 75, 490507, (2016) DOI:10.1016/j.robot.2015.09.012.

2.

A. Shukla and H. Karki, Application of robotics in onshoreoil and gas industry – a review Part II, Robot. Auton. Syst. 75, pp508-524, (2016) DOI:10.1016/j.robot.2015.09.013.

3.

X. D. Gu · Z. CH. Deng. Dynamical analysis of vibro-impact capsule system with Hertzian contact model and random perturbation excitations. Nonlinear Dynamics, 2018, Vol 92, Issue 4, pp 1781–1789; doi: 10.1007/s11071-018-4161-x.

4.

V.D. Nguyen, K.C. Woo and E. Pavlovskaia, Experimental study and Mathematical Modelling of a New of Vibro-impact molingdevice, Int. J. Nonlinear. Mech. 43 (6), pp542-550, (2008) DOI:10.1016/j.ijnonlinmec.2007.10.003;

5.

V-D. Nguyen, T-H. Duong, N-H. Chu and Q-H. Ngo, The effect of inertial mass and excitation frequency on a Duffingvibroimpact drifting system, Int. J. Mech. Sci., 124-125, pp. 9-21 (2017), DOI: 10.1016/j.ijmecsci.2017.02.023.

6.

Duong The Hung, Nguyen Van Dự, La Ngọc Tuan (2017). “A new autogenous mobile system driven by vibration without impacts, excited by an impulse periodic force”, International Conferenceon Engineering Vibration 9/ 2017, ICOEV 2017, MATEC Web of Conferences 148, 04005 (2018), DOI: 10.1051/matecconf/201814804005.

7.

Won Seok Kang, Chan Kyu Choi and Hong HeeYoo (2015). Stochasticmodeling of friction force and vibration analysis of a mechanical system using the model. Journal of Mechanical Science and Technology 29 (9) (2015), pp 3645-3652. DOI 10.1007/s12206-015-0808-4.

8.

Adhikari S. Structural Dynamic Analysis with Generalized DampingModels: Analysis. John Wiley& Sons, Inc, 2014.

9.

Soong, T.T. and Grigoriu, M. Random Vibration of Mechanical and Structural Systems. Prentice-Hall, Inc., Englewood Cliffs,

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The-Hung Duong & The-Thinhnguyen New Jersey, 1993; 10. Shinozuka M., and Wen Y.K., Monte Carlo Solution of Nonlinear Vibrations, AIAA Journal, 1972, Vol. 10, No. 1, pp. 37-40.

11. Birendra Kumar Singh & Anand Kumar Sinha, « Friction Factor in Open Channel », InternationalJournal of Civil Engineering (IJCE), Vol. 3, Issue 3, pp. 19-38 12. Pradip R. Bodade, Nishant Jogi, Mahesh Gorde, Ravi Paropte & Ranjan Waghchore , « Use of Internal Threads of Different Pitches to Enhance Heat Transfer in a Circular Channel « , BEST: International Journal of Management, Information Technology and Engineering (BEST: IJMITE), Vol. 3, Issue 1, pp. 1-4 13. Mohammed Yunus & Mohammad S.Alsoufi, « A Statistical Analysis of Joint Strength of Dissimilar Aluminium Alloys Formed

by Friction Stir Welding Using Taguchi Design Approach, Anova for the Optimization of Process Parameters », IMPACT: International Journal of Research in Engineering & Technology (IMPACT: IJRET), Vol. 3, Issue 7, pp. 63-70 14. M. El-Shennawy, Adel A. Omar & M. Ayad, « Similar And Dissimilar Friction Stir Welding of AA7075« International Journal of Mechanical Engineering (IJME), Vol. 3, Issue 4, pp. 69-86

AUTHOR’S PROFILE

The-Hung Duong received his M.S. and Ph.D. in Mechanics from Vietnam National University of Civil Engineering, Hanoi, Vietnam, in 2000 and 2011, respectively. He is currently a lecturer and researcher at Thainguyen University of Technology, Thainguyen, Vietnam. His research interests include mechanical vibration, computational mechanics, dynamics of vibro-impact structures, fracture mechnics, structural health monitoring, and stochastic mechanics. He has teached subjects of dynamics of structures in civil and mechanical engineering. His reseach area is Fracture mechanics, Mechanical vibration, Structural dynamics, Vibration and Vibro-impact analysis, Random process simulation, Structural health monitoring, Computational structural mechanics, Smart materials. He has 20 papers in journals and proceedings of conferences.

N.T. Thinh received his B.S and M.S. in Civil Engineering from Thainguyen University of Technology, Thainguyen, Vietnam, in 2010 and 2017, respectively. He is currently a lecturer at Thainguyen University of Technology, Thainguyen, Vietnam. His research interests include mechanical vibration, computational mechanics, structural health monitoring, and stochastic mechanics.

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