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Vibro-impact dynamics of two rolling balls along curvilinear trace
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Procedia Engineering 199 (2017) 663–668
X International Conference on Structural Dynamics, EURODYN 2017
Vibro-impact dynamics of two rolling balls along curvilinear trace Katica R. (Stevanović) Hedrih * Mathematical Mathematical Institute Institute of of Serbian Serbian Academy Academy of of Science Science and and Arts Arts (SANU), (SANU), Belgrade, Belgrade, Serbia Serbia Faculty of Mechanical Engineering University of Niš, Niš, Serbia Faculty of Mechanical Engineering University of Niš, Niš, Serbia
Abstract Abstract Using Using this this extended extended theory theory of of impacts impacts and and new new results results of of kinematics kinematics and and dynamics dynamics of of central central collision collision of of two two rolling rolling bodies bodies (balls (balls or or disks) disks) dynamics dynamics of of vibro-impact vibro-impact systems systems is is investigated. investigated. Two Two rolling rolling bodies bodies are are rolling rolling along along curvilinear curvilinear rolling rolling trace. trace. In In considered considered case case rolling rolling trace trace consists consists for for three three part part of of circle circle lines lines with with different different radii. radii. In In dynamics dynamics of of rolling rolling bodies bodies appears appears series series of of the the successive successive central central collisions collisions depending depending of of initial initial kinetic kinetic conditions conditions and and geometrical geometrical parameters parameters of of rolling rolling bodies bodies and and curvilinear curvilinear rolling rolling trace. trace. © 2017 The Authors. Published by Elsevier Ltd. © 2017 2017The TheAuthors. Authors.Published Publishedby byElsevier ElsevierLtd. Ltd. © Peer-review under responsibility of the organizing committee of EURODYN 2017. Peer-review under responsibility of the organizing committeeof ofEURODYN EURODYN 2017. 2017. Peer-review under responsibility of the organizing committee Keywords: Keywords: Central Central collision collision of of two two rolling rolling bodies; bodies; curvilinear curvilinear rolling rolling trace; trace; vibro-impact vibro-impact dynamics. dynamics.
1. Introduction In previous published papers and conference presentations, an extended theory of impacts with new results of kinematics and dynamics of central as well as skew collision of two rolling bodies is founded by author. Using this extended theory of impacts [1] and new results of kinematics and dynamics of central collision of two rolling bodies (balls or disks) dynamics of vibro-impact systems is investigated. Two rolling bodies are in kinetic states of rolling along curvilinear rolling trace. In considered case rolling trace consists for three part of circle lines with different radii. In dynamics of rolling bodies appears series of the successive central collisions depending of initial kinetic conditions and geometrical parameters of rolling bodies and curvilinear rolling trace. Differential equations of motion, and also, corresponding equations of phase trajectories of both rolling bodies are derived. Also, expressions of times of rolling bodies in function of coordinates are defined by series of elliptic
** Corresponding Corresponding author. author. Tel.: Tel.: +381 +381 18 18 32 32 31 31 663; 663; fax: fax: +381 +381 18 18 32 32 31 31 663. 663. E-mail E-mail address: address: [email protected] [email protected] 1877-7058 1877-7058 © © 2017 2017 The The Authors. Authors. Published Published by by Elsevier Elsevier Ltd. Ltd. Peer-review Peer-review under under responsibility responsibility of of the the organizing organizing committee committee of of EURODYN EURODYN 2017. 2017.
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017. 10.1016/j.proeng.2017.09.120
Katica R. (Stevanović) Hedrih et al. / Procedia Engineering 199 (2017) 663–668 Katica R. (Stevanović) Hedrih / Procedia Engineering 00 (2017) 000–000
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integrals. Based on these results positions on the curvilinear rolling trace and times of each of the successive collisions between rolling bodies are defined. Phase portraits of each rolling body are graphically presented. In the phase portraits representative points which correspond to kinetic states and positions of rolling bodies collisions are presented. Use variations of the kinetic and geometric parameters, properties of vibro-impact dynamics of this system are investigated. Velocities and kinetic energies jumps of each rolling body after each collision are visible from phase portraits and portraits of constant energy curves. Using presented approach, theory of collision of two rolling bodies and method of phase plane a complete methodology for investigating other similar vibro-impact system dynamics is founded.
2. Dynamics of the central collision of two rolling balls along curvilinear trace Let’s start with application of mathematical analogy of classical theory of dynamics of collision to the dynanics of collision between two rolling balls, with mass m1 and m2 , and with axial mass inertia moments J P1 and J P 2 for corresponding instanteneous axis of the rolling along trace with pre-impact (arrival) angular velocities ω P1,impact = ω P1 (t0 ) and ω P 2,impact = ω P 2 (t0 ) . Mass centers C1 and C 2 of the balls move transtator with pre-
impact (arrival) velocities vC 1.impact = vC 1 (t0 ) and vC 2.impact = vC 2 (t0 ) . Angular velocities and
ω P 2,impact = ω P 2 (t0 ) we denote as arrival, or
ω P1,impact = ω P1 (t 0 )
impact or pre-impact angular velocietis at the moment t0 (see
Figure 1a). At this moment t0 of the collision start between these rolling balls, contact of these two balls is at point
T12 , in which both balls posses common tangent plane –plane of contact (touch). In the theory of collision, it is proposed that collision takes very shorth time period (t0 ,t0 + τ ) , and that τ tend to zero. After this short period
τ
bodies-two rolling bals in collision separate and outgoing by post-impact-outgoing angular velocities ω P1,outgoing = ω P1 (t 0 + τ ) and ω P 2,outgoing = ω P 2 (t0 + τ ) . Mass centers C1 and C 2 of the balls move transtator with
post-impact (outgoing) translatoor velocities vC 1.outgoing = vC1 (t0 + τ ) and vC 2.outgoing = vC 2 (t0 + τ ) . These translator velocities are possible to express, each by corresponding outgoing post-collision angular velocity and radius of the corresponding ball. Elements of mathematical phenomenology and phenomenological mappings between rolling balls and translator bodies (balls), which are analogous dynamical impact systems (similar as electromechanical analogy between electrical oscillator with one degree of freedom and mechanical oscillator with one degree of freedom; for detail see Refs. [2-5]) are used. On the basis of Petroviċ’s theory [6,7] and qualitative and mathematical analogies considered in previous published author’s papers [1] is possible to introduce hypothesis of conservation of angular momentum sum (moment of impulse for corresponding instantaneous axis of rolling) of two rolling balls pre-collision and postcollision rolling motion in the following form: J P1ω P1 (t0 ) + J P 2ω P 2 (t0 ) = J P1ω P1 (t0 + τ ) + J P 2ω P 2 (t0 + τ ) (1) and the coefficient of the restitution of collision between two rolling balls in the form:
k=
ω r (t0 + τ ) ω P 2 (t0 + τ ) − ω P1 (t0 + τ ) = ω r (t0 ) ω P1 (t0 ) − ω P 2 (t0 )
(2)
as ratio between difference of angular velocities of rolling balls post-collision and pre-collision kinetic states. In present paper coefficient of the restitution k by expression (2) is introduced by angular velocities after and before collision of the two rolling balls and is new definition introduced by author. Also, in analogy with the expressions of post-collision –outgoing body translator velocities is possible to write expressions of post-collision –outgoing rolling balls angular velocities in the following forms:
Katica R. (Stevanović) Hedrih / Procedia Engineering 00 (2017) 000–000 Katica R. (Stevanović) Hedrih et al. / Procedia Engineering 199 (2017) 663–668
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b.1
b.2
b
a
Fig.1.a. Kinematic plan of the velocities in the collision of two balls in rolling motion along curvilinear line; b. Phase portraits and phase trajectories with jumps of the ball’s representative points from one to other phase trajectory in vibro-impact motion along circle line: b.1. for first rolling ball and b.2 for second rolling bal
ω P1 (t 0 + τ ) = ω P1 (t 0 ) −
1+ k (ω P1 (t0 ) − ωP 2 (t0 )) J 1 + P1 J P2
and
ω P 2 (t0 + τ ) = ω P 2 (t0 ) +
1+ k (ω P1 (t0 ) − ωP 2 (t 0 )) J 1 + P2 J P1
(3)
Previous obtained expressions (3) of post-collision –outgoing rolling balls angular velocities are new and original results obtained on the basis of Petrović’s theory of elements of mathematical phenomenology (see Reference [6,7]). Also expression (1) for the hypothesis of conservation of sum of angular momentum (moment of impulse for corresponding instantaneous axis) of impact dynamics of two rolling balls pre-collision and postcollision motion is new introduced relation in impact dynamics as well as expression (2) for the coefficient of restitution in collision of two rolling balls with different size and in central collision. All these results are analytical and can be applicable in other type of collisions. 3. Principal models and equations In the Ref. [8] a model of a moving heavy rigid ball along curvilinear trace in the form of circle in vertical stationary plane is presented. Ordinary differential equations describing dynamics of a heavy rigid ball rolling along curvilinear trace consisting of series of three circle arcs are in the following forms (see Fig. 2.a):
ϕi +
J i2 g g sin ϕi = 0, i = 1,3, and ϕ2 + sin (ϕ 2 + π ) = 0 , where κ i = Ci2 + 1 = Ci2 + 1 Mr r κ i (R − r ) κ 2 (R0 + r )
(4)
where κ i , i = 1,2,3 coefficient of ball (disk) rolling and J Ci axial mass inertia moment for instantaneous axis of rolling Corresponding equations of the phase trajectory branches of a heavy rigid ball rolling along curvilinear trace consisting of series of three circle arcs are in the following forms:
ϕi2 = ϕi2, 0 +
2g
κ i (R − r )
(cos ϕ
i
− cos ϕi , 0 ), i = 1,3, and ϕ 22 = ϕ22,0 +
2g
κ 2 (R0 + r )
(cos ϕ
2
− cos ϕ 2, 0 )
(5)
In Fig. 2.a, a model of a rolling heavy ball (or disk) along curvilinear trace consisting of the series of circle arcs
Katica R. (Stevanović) Hedrih et al. / Procedia Engineering 199 (2017) 663–668 Katica R. (Stevanović) Hedrih / Procedia Engineering 00 (2017) 000–000
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in vertical stationary plane is presented. In Fig.2.b, c and d, different compositions of the phase trajectory portraits of different phase trajectory branches in phase plane (and curves of constant total mechanical energy branches) of a rolling disk in intervals along different arcs of curvilinear traces composed by three coupled circle arcs with different radii are presented [2].
ϕ
ϕ2 r C2
O1O1
R
ωP2
P2 R K1 Mg 0 K 2 ϕ1
ϕ3
C1
ω P1
ω P3 P 3
Mg
a
R
C3 r
r
P1
ϕ
O3
Mg
b
ϕ
ϕ ϕ
c
ϕ
d
Fig. 2.a. A model of a rolling heavy ball (or disk) along curvilinear trace consisting of the series of circle arcs in vertical stationary plane; b, c and d Different compositions of the phase trajectory portraits of different phase trajectory branches in phase plane (and curves of constant total mechanical energy branches) of a rolling disk in intervals along different arcs of curvilinear traces composed by three coupled circle arcs with different radii.
4. Vibro-impact dynamics of multiple collisions of two different rolling heavy balls along circle trace in vertical plane Let us to consider vibro-impact dynamics [3,4,5] of two rolling heavy balls along curvilinear line consisting by three circle arches in stationary vertical plane. Using listed, in previous part of the paper, the ordinary differential equations of ball rolling and the equations of the phase trajectory branches of two separate rolling balls along curvilinear line consisting of three circle arches, we can consider as dynamics of two rolling balls in vibro-impact dynamics. Taking into account that these equations, which are valid for the balls dynamics between two successive collisions of balls. Central angle coordinates of positions of the balls in state of the collisions are in the following relation: ϕ 2,impact ,k = ϕ1,impact ,k + β , where angle β depend of geometrical parameters of circle line radius R , and of the both balls radiuses: r1 and r2 and is defined by expression in the form:
β = arccos
where
λ1 =
(R − r1 )2 + (R − r2 )2 − (r1 + r2 )2 = arccos λ1 (λ1 − λ2 − 1) − λ2 2(R − r1 )(R − r2 ) (λ1 − λ2 )(λ1 − 1)
(6)
R r and λ2 = 2 . r1 r1
In Fig.1b., phase portraits and phase trajectories with jumps of each of the balls’ representative points from one to other phase trajectory (1.b.1. for first rolling ball and 1.b.2 for second rolling bal) in vibro-impact motion in the rolling along one circle line are presented. In Fig. 1b. is possible to see basic elements of the vibro-impact methodology using phase trajectory portraits of two rolling balls. Using phase portraits for the case when curvilinear trace consists
Katica R. (Stevanović) Hedrih et al. / Procedia Engineering 199 (2017) 663–668 Katica R. (Stevanović) Hedrih / Procedia Engineering 00 (2017) 000–000
667 5
of three circle arches, and taking into account that two balls are rolling along same curvilinear trace, we can applied presented methodology for research on a same way as it was applied in the case when curvilinear trace is one circle. Main problem, in research of the vibro-impact dynamics of two rolling balls (or disks) rolling along curvilinear trace consists of three circle arches, is determination of the time and positions of the first and next successive collisions. This task must be solved numerically taking into account analytical expressions of kinetic parameters of each ball and corresponding approximate solutions of elliptic integrals and roots of nonlinear and transcendent equations. Relation between duration of time of ball rolling between two angle positions and angle coordinate is in the form of elliptic integral presented in the form: ϕ1
t=
dϕ1
∫ ϕ
ϕ12,0 +ω 21 (cosϕ1 − cosϕ1, 0
1,0
ϕ1
dϕ1
) ϕ∫ =
ϕ + 2ω sin 2 1, 0
1,0
2 1
2
ϕ1,0 2
(7) 2 1
− 2ω sin
2
ϕ1 2
and it is necessary to take into account corresponding geometrical data of each circle arch. For obtaining the coordinates of ball’s positions at configuration of first collision between rolling heavy balls at first circle line in vertical plane, it is necessary to obtain time timpact ,1 of first collision at which both balls are in the configuration of first collision. If we propose that mass center C1,impact ,1 of first ball is in position defined by
(
)
angle coordinate ϕ1 timpact ,1 = ϕ1,impact ,1 , then coordinate of mass center C2 ,impact ,1 of second ball is defined by angle coordinate:
ϕ 2 (timpacr ,1 ) = ϕ1 (tompact ,1 ) + β , where angle β
is defined by expression (6). Using approximate
expressions (7) for time timpact ,1 duration of balls motions from corresponding initial positions, positions,
ϕ1,impact ,1
and
and
ϕ 2, 0 , to the
ϕ 2,impact ,1 , of the first collision between rolling balls we can write the following: sin
timpact,1 = t1,impact,1
ϕ1, 0
ϕ1, impact , 1
2 2 1 2 3 1⋅ 3 4 5 ≈ k 1u + k 1u u + ω0,1 2⋅3 2⋅4⋅6 sin ϕ1, 0
ϕ1, impact , 1
+
(8)
2
2
sin
sin
2 2 1 3 1 2 5 1⋅ 3 4 7 + u + k 1u + 2 k 1u ω0,1 2 ⋅ 3 4 ⋅5 2 ⋅4⋅7 sin ϕ1, 0
ϕ1, impact , 1
2 2 2 1 ⋅ 3 5 1⋅ 3 1 1 ⋅ 3 4 9 + u + 2 k 12u 7 + k 1u ϕ1, 0 ω0,1 2 ⋅ 4 ⋅ 5 2 ⋅4⋅7 9 2⋅4 sin sin
timpact ,1 = t 2,impact ,1
2 ϕ1 ,impact ,1 + β
2 1 2 3 1⋅ 3 4 5 ≈ k 2u + k 2u u + ω0, 2 2⋅3 2⋅4⋅6 sin ϕ2 ,0
2
2
2 2 1⋅ 3 5 1 ⋅ 3 2 7 1 1 ⋅ 3 4 9 u + 2 k 1u + k 1u + ω0,1 2 ⋅ 4 ⋅ 5 2 ⋅4⋅7 9 2⋅ 4
sin
sin
sin
ϕ1,impact ,1 + β
2 1 3 1 2 5 1⋅ 3 u + k 2u + 2 k 24u 7 + ω0, 2 2 ⋅ 3 4 ⋅5 2 ⋅4⋅7 sinϕ 2, 0 2
2
+
(9)
ϕ1,impact ,1 + β 2
ϕ2 , 0 2
Taking into account, that both balls start from initial positions,
ϕ1, 0
and
ϕ1,impact ,1
and
ϕ 2,impact ,1 , expressions (8) and (9) have to
the first position of first collision, defined by coordinates
ϕ 2, 0 , must
arrive at configuration of
be equal. As result of this condition we have a nonlinear transcendent equation with respect to unknown angle coordinate ϕ1 timpact ,1 = ϕ1,impact ,1 of the mass centre position of first ball at position of first collision..
(
)
To find first real root of this transcendent equation, is not possible to solve analytically and it is necessary to use some of numerical methods as well as some commercial software tool. In this paper we deal with ideas and analytical approach to defined task and methodology of vibro-impact dynamics. Then, we suppose that we have angle coordinate ϕ1 timpact,1 = ϕ1,impact ,1 of mass centre C1,impact ,1 of first ball at position of the first collision, and also
(
)
Katica R. (Stevanović) Hedrih et al. / Procedia Engineering (2017) 663–668 Katica R. (Stevanović) Hedrih / Procedia Engineering 00 (2017)199 000–000
668 6
(
)
(
)
angle coordinate ϕ 2 timpacr ,1 = ϕ1 t ompact ,1 + β of mass centre C2 ,impact ,1 of the second ball at position of the first collision. With this assumption it is possible to compose pre-first collision impact angular velocities
ω P 2,impact ,1
ω P1,impact ,1
and
of the heavy rolling balls. For obtaining post-first-collision outgoing angular velocities
ω P1 (tu ,1 + τ ) = ω P1,outgoing ,1 and ω P 2 (t 0 + τ ) = ω P 2,outgoing ,1 of the rolling balls along circle line, at same position of the first collision balls, determined by generalized coordinates
ϕ1,impact ,1 and ϕ 2,impact ,1 , we use expressions (3) and we
obtain initial conditions for next branches of phase trajectory for rolling each ball. To obtain kinetic parameters of the successive collisions same methodology [3,4,5] have to be applied.
4. Concluding remarks At the end, it is useful to conclude that presented kinetic parameters of dynamics of the central collision of two rolling balls is possible to use in study of the skew collision of two rolling balls that role along two straight as well as curvilinear line trace with intersection as well as parallel at distance smaller them sum of the balls radiuses. Aim of this paper is not to present a possibility about generalization of the all results in area of the vibroimpact dynamics in system containing the collision of two rolling balls with different dimension and collisions. The paper is focused to central collision of two rolling rigid and heavy smooth balls and to using elements of mathematical phenomenology and phenomenological mapping to obtain corresponding new expressions for the post-collision and outgoing angular velocity of each ball and to apply these results for investigation of the vibroimpact dynamics of two rolling balls along circle trace. This task is fully analytical solved and obtained analytical results are original and new! Also, these results can be fundamental for next development and investigation of the special class of vibroimpact dynamical systems with collision of the rigid and/or deformable balls and also in application in different area of engineering systems with coupled rotations (in rolling bearings, rolling vibro-impact dampers - mechanisms for dynamic absorption of torsional vibrations, or other). Acknowledgements Parts of this research were supported by the Ministry of Sciences and Technology of Republic of Serbia through Mathematical Institute SASA, Belgrade Grant ON174001 “Dynamics of hybrid systems with complex structures.”, Mechanics of materials and Faculty of Mechanical Engineering University of Niš.
References [1]
[2]
[3] [4]
[5] [6]
[7] [8]
K. R. Hedrih (Stevanović), Dynamics of Impacts and Collisions of the Rolling Balls, Dynamical Systems: Theoretical and Experimental Analysis, Springer Proceedings in Mathematics & Statistics, Volume Number: 182, Chapter 13, pp. 157-168. Springer ISBN 978-3-31942407-1 DOI 10.1007/978-3-319-42408-8. K. R. Hedrih (Stevanović), Nonlinear Dynamics of a Heavy Material Particle Along Circle which Rotates and Optimal Control, Chaotic Dynamics and Control of Systems and Processes in Mechanics (Eds: G. Rega, and F. Vestroni), p. 37-45. IUTAM Book, in Series Solid Mechanics and Its Applications, Editerd by G.M.L. Gladwell, Springer. 2005, XXVI, 504 p., Hardcover ISBN: 1-4020-3267-6. K. R. Hedrih (Stevanović), Vibro-impact dynamics in system with trigger of coupled three singular points: Collision of two rolling bodies, Extended abstract in the form short paper on CD, IUTAM ICTAM Montreal 2016. K. R. Hedrih (Stevanović), V. Raičević and S. Jović., Phase Trajectory Portrait of the Vibro-impact Forced Dynamics of Two Heavy Mass Particles Motions along Rough Circle, Communications in Nonlinear Science and Numerical Simulations, 2011 16 (12):4745-4755, DOI 10.1016/j.cnsns.2011.05.027. K. R. Hedrih (Stevanović), V. Raičević, S. Jović, Vibro-impact of a Heavy Mass Particle Moving along a Rough Circle with Two Impact Limiters, ©Freund Publishing House Ltd., International Journal of Nonlinear Sciences & Numerical Simulation 10(11): 1713-1726, 2009. M. Petrović, Elementi matematičke fenomenologije (Elements of mathematical phenomenology), Srpska kraljevska akademija, Beograd, 1911. str. 89. http://elibrary.matf.bg.ac.rs/handle/123456789/476?locale-attribute=sr M. Petrović, Fenomenološko preslikavanje (Phenomenological mapping), Srpska kraljevska akademija, Beograd, 1933. str. 33.http://elibrary.matf.bg.ac.rs/handle/123456789/475 D. P. Rašković, Teorija oscilacija (Theory of oscillations), Naučna knjiga, 1952.
ScienceDirect Available online at www.sciencedirect.com Procedia Procedia Engineering Engineering 00 00 (2017) (2017) 000–000 000–000
ScienceDirect
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
Procedia Engineering 199 (2017) 663–668
X International Conference on Structural Dynamics, EURODYN 2017
Vibro-impact dynamics of two rolling balls along curvilinear trace Katica R. (Stevanović) Hedrih * Mathematical Mathematical Institute Institute of of Serbian Serbian Academy Academy of of Science Science and and Arts Arts (SANU), (SANU), Belgrade, Belgrade, Serbia Serbia Faculty of Mechanical Engineering University of Niš, Niš, Serbia Faculty of Mechanical Engineering University of Niš, Niš, Serbia
Abstract Abstract Using Using this this extended extended theory theory of of impacts impacts and and new new results results of of kinematics kinematics and and dynamics dynamics of of central central collision collision of of two two rolling rolling bodies bodies (balls (balls or or disks) disks) dynamics dynamics of of vibro-impact vibro-impact systems systems is is investigated. investigated. Two Two rolling rolling bodies bodies are are rolling rolling along along curvilinear curvilinear rolling rolling trace. trace. In In considered considered case case rolling rolling trace trace consists consists for for three three part part of of circle circle lines lines with with different different radii. radii. In In dynamics dynamics of of rolling rolling bodies bodies appears appears series series of of the the successive successive central central collisions collisions depending depending of of initial initial kinetic kinetic conditions conditions and and geometrical geometrical parameters parameters of of rolling rolling bodies bodies and and curvilinear curvilinear rolling rolling trace. trace. © 2017 The Authors. Published by Elsevier Ltd. © 2017 2017The TheAuthors. Authors.Published Publishedby byElsevier ElsevierLtd. Ltd. © Peer-review under responsibility of the organizing committee of EURODYN 2017. Peer-review under responsibility of the organizing committeeof ofEURODYN EURODYN 2017. 2017. Peer-review under responsibility of the organizing committee Keywords: Keywords: Central Central collision collision of of two two rolling rolling bodies; bodies; curvilinear curvilinear rolling rolling trace; trace; vibro-impact vibro-impact dynamics. dynamics.
1. Introduction In previous published papers and conference presentations, an extended theory of impacts with new results of kinematics and dynamics of central as well as skew collision of two rolling bodies is founded by author. Using this extended theory of impacts [1] and new results of kinematics and dynamics of central collision of two rolling bodies (balls or disks) dynamics of vibro-impact systems is investigated. Two rolling bodies are in kinetic states of rolling along curvilinear rolling trace. In considered case rolling trace consists for three part of circle lines with different radii. In dynamics of rolling bodies appears series of the successive central collisions depending of initial kinetic conditions and geometrical parameters of rolling bodies and curvilinear rolling trace. Differential equations of motion, and also, corresponding equations of phase trajectories of both rolling bodies are derived. Also, expressions of times of rolling bodies in function of coordinates are defined by series of elliptic
** Corresponding Corresponding author. author. Tel.: Tel.: +381 +381 18 18 32 32 31 31 663; 663; fax: fax: +381 +381 18 18 32 32 31 31 663. 663. E-mail E-mail address: address: [email protected] [email protected] 1877-7058 1877-7058 © © 2017 2017 The The Authors. Authors. Published Published by by Elsevier Elsevier Ltd. Ltd. Peer-review Peer-review under under responsibility responsibility of of the the organizing organizing committee committee of of EURODYN EURODYN 2017. 2017.
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017. 10.1016/j.proeng.2017.09.120
Katica R. (Stevanović) Hedrih et al. / Procedia Engineering 199 (2017) 663–668 Katica R. (Stevanović) Hedrih / Procedia Engineering 00 (2017) 000–000
664 2
integrals. Based on these results positions on the curvilinear rolling trace and times of each of the successive collisions between rolling bodies are defined. Phase portraits of each rolling body are graphically presented. In the phase portraits representative points which correspond to kinetic states and positions of rolling bodies collisions are presented. Use variations of the kinetic and geometric parameters, properties of vibro-impact dynamics of this system are investigated. Velocities and kinetic energies jumps of each rolling body after each collision are visible from phase portraits and portraits of constant energy curves. Using presented approach, theory of collision of two rolling bodies and method of phase plane a complete methodology for investigating other similar vibro-impact system dynamics is founded.
2. Dynamics of the central collision of two rolling balls along curvilinear trace Let’s start with application of mathematical analogy of classical theory of dynamics of collision to the dynanics of collision between two rolling balls, with mass m1 and m2 , and with axial mass inertia moments J P1 and J P 2 for corresponding instanteneous axis of the rolling along trace with pre-impact (arrival) angular velocities ω P1,impact = ω P1 (t0 ) and ω P 2,impact = ω P 2 (t0 ) . Mass centers C1 and C 2 of the balls move transtator with pre-
impact (arrival) velocities vC 1.impact = vC 1 (t0 ) and vC 2.impact = vC 2 (t0 ) . Angular velocities and
ω P 2,impact = ω P 2 (t0 ) we denote as arrival, or
ω P1,impact = ω P1 (t 0 )
impact or pre-impact angular velocietis at the moment t0 (see
Figure 1a). At this moment t0 of the collision start between these rolling balls, contact of these two balls is at point
T12 , in which both balls posses common tangent plane –plane of contact (touch). In the theory of collision, it is proposed that collision takes very shorth time period (t0 ,t0 + τ ) , and that τ tend to zero. After this short period
τ
bodies-two rolling bals in collision separate and outgoing by post-impact-outgoing angular velocities ω P1,outgoing = ω P1 (t 0 + τ ) and ω P 2,outgoing = ω P 2 (t0 + τ ) . Mass centers C1 and C 2 of the balls move transtator with
post-impact (outgoing) translatoor velocities vC 1.outgoing = vC1 (t0 + τ ) and vC 2.outgoing = vC 2 (t0 + τ ) . These translator velocities are possible to express, each by corresponding outgoing post-collision angular velocity and radius of the corresponding ball. Elements of mathematical phenomenology and phenomenological mappings between rolling balls and translator bodies (balls), which are analogous dynamical impact systems (similar as electromechanical analogy between electrical oscillator with one degree of freedom and mechanical oscillator with one degree of freedom; for detail see Refs. [2-5]) are used. On the basis of Petroviċ’s theory [6,7] and qualitative and mathematical analogies considered in previous published author’s papers [1] is possible to introduce hypothesis of conservation of angular momentum sum (moment of impulse for corresponding instantaneous axis of rolling) of two rolling balls pre-collision and postcollision rolling motion in the following form: J P1ω P1 (t0 ) + J P 2ω P 2 (t0 ) = J P1ω P1 (t0 + τ ) + J P 2ω P 2 (t0 + τ ) (1) and the coefficient of the restitution of collision between two rolling balls in the form:
k=
ω r (t0 + τ ) ω P 2 (t0 + τ ) − ω P1 (t0 + τ ) = ω r (t0 ) ω P1 (t0 ) − ω P 2 (t0 )
(2)
as ratio between difference of angular velocities of rolling balls post-collision and pre-collision kinetic states. In present paper coefficient of the restitution k by expression (2) is introduced by angular velocities after and before collision of the two rolling balls and is new definition introduced by author. Also, in analogy with the expressions of post-collision –outgoing body translator velocities is possible to write expressions of post-collision –outgoing rolling balls angular velocities in the following forms:
Katica R. (Stevanović) Hedrih / Procedia Engineering 00 (2017) 000–000 Katica R. (Stevanović) Hedrih et al. / Procedia Engineering 199 (2017) 663–668
3 665
b.1
b.2
b
a
Fig.1.a. Kinematic plan of the velocities in the collision of two balls in rolling motion along curvilinear line; b. Phase portraits and phase trajectories with jumps of the ball’s representative points from one to other phase trajectory in vibro-impact motion along circle line: b.1. for first rolling ball and b.2 for second rolling bal
ω P1 (t 0 + τ ) = ω P1 (t 0 ) −
1+ k (ω P1 (t0 ) − ωP 2 (t0 )) J 1 + P1 J P2
and
ω P 2 (t0 + τ ) = ω P 2 (t0 ) +
1+ k (ω P1 (t0 ) − ωP 2 (t 0 )) J 1 + P2 J P1
(3)
Previous obtained expressions (3) of post-collision –outgoing rolling balls angular velocities are new and original results obtained on the basis of Petrović’s theory of elements of mathematical phenomenology (see Reference [6,7]). Also expression (1) for the hypothesis of conservation of sum of angular momentum (moment of impulse for corresponding instantaneous axis) of impact dynamics of two rolling balls pre-collision and postcollision motion is new introduced relation in impact dynamics as well as expression (2) for the coefficient of restitution in collision of two rolling balls with different size and in central collision. All these results are analytical and can be applicable in other type of collisions. 3. Principal models and equations In the Ref. [8] a model of a moving heavy rigid ball along curvilinear trace in the form of circle in vertical stationary plane is presented. Ordinary differential equations describing dynamics of a heavy rigid ball rolling along curvilinear trace consisting of series of three circle arcs are in the following forms (see Fig. 2.a):
ϕi +
J i2 g g sin ϕi = 0, i = 1,3, and ϕ2 + sin (ϕ 2 + π ) = 0 , where κ i = Ci2 + 1 = Ci2 + 1 Mr r κ i (R − r ) κ 2 (R0 + r )
(4)
where κ i , i = 1,2,3 coefficient of ball (disk) rolling and J Ci axial mass inertia moment for instantaneous axis of rolling Corresponding equations of the phase trajectory branches of a heavy rigid ball rolling along curvilinear trace consisting of series of three circle arcs are in the following forms:
ϕi2 = ϕi2, 0 +
2g
κ i (R − r )
(cos ϕ
i
− cos ϕi , 0 ), i = 1,3, and ϕ 22 = ϕ22,0 +
2g
κ 2 (R0 + r )
(cos ϕ
2
− cos ϕ 2, 0 )
(5)
In Fig. 2.a, a model of a rolling heavy ball (or disk) along curvilinear trace consisting of the series of circle arcs
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in vertical stationary plane is presented. In Fig.2.b, c and d, different compositions of the phase trajectory portraits of different phase trajectory branches in phase plane (and curves of constant total mechanical energy branches) of a rolling disk in intervals along different arcs of curvilinear traces composed by three coupled circle arcs with different radii are presented [2].
ϕ
ϕ2 r C2
O1O1
R
ωP2
P2 R K1 Mg 0 K 2 ϕ1
ϕ3
C1
ω P1
ω P3 P 3
Mg
a
R
C3 r
r
P1
ϕ
O3
Mg
b
ϕ
ϕ ϕ
c
ϕ
d
Fig. 2.a. A model of a rolling heavy ball (or disk) along curvilinear trace consisting of the series of circle arcs in vertical stationary plane; b, c and d Different compositions of the phase trajectory portraits of different phase trajectory branches in phase plane (and curves of constant total mechanical energy branches) of a rolling disk in intervals along different arcs of curvilinear traces composed by three coupled circle arcs with different radii.
4. Vibro-impact dynamics of multiple collisions of two different rolling heavy balls along circle trace in vertical plane Let us to consider vibro-impact dynamics [3,4,5] of two rolling heavy balls along curvilinear line consisting by three circle arches in stationary vertical plane. Using listed, in previous part of the paper, the ordinary differential equations of ball rolling and the equations of the phase trajectory branches of two separate rolling balls along curvilinear line consisting of three circle arches, we can consider as dynamics of two rolling balls in vibro-impact dynamics. Taking into account that these equations, which are valid for the balls dynamics between two successive collisions of balls. Central angle coordinates of positions of the balls in state of the collisions are in the following relation: ϕ 2,impact ,k = ϕ1,impact ,k + β , where angle β depend of geometrical parameters of circle line radius R , and of the both balls radiuses: r1 and r2 and is defined by expression in the form:
β = arccos
where
λ1 =
(R − r1 )2 + (R − r2 )2 − (r1 + r2 )2 = arccos λ1 (λ1 − λ2 − 1) − λ2 2(R − r1 )(R − r2 ) (λ1 − λ2 )(λ1 − 1)
(6)
R r and λ2 = 2 . r1 r1
In Fig.1b., phase portraits and phase trajectories with jumps of each of the balls’ representative points from one to other phase trajectory (1.b.1. for first rolling ball and 1.b.2 for second rolling bal) in vibro-impact motion in the rolling along one circle line are presented. In Fig. 1b. is possible to see basic elements of the vibro-impact methodology using phase trajectory portraits of two rolling balls. Using phase portraits for the case when curvilinear trace consists
Katica R. (Stevanović) Hedrih et al. / Procedia Engineering 199 (2017) 663–668 Katica R. (Stevanović) Hedrih / Procedia Engineering 00 (2017) 000–000
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of three circle arches, and taking into account that two balls are rolling along same curvilinear trace, we can applied presented methodology for research on a same way as it was applied in the case when curvilinear trace is one circle. Main problem, in research of the vibro-impact dynamics of two rolling balls (or disks) rolling along curvilinear trace consists of three circle arches, is determination of the time and positions of the first and next successive collisions. This task must be solved numerically taking into account analytical expressions of kinetic parameters of each ball and corresponding approximate solutions of elliptic integrals and roots of nonlinear and transcendent equations. Relation between duration of time of ball rolling between two angle positions and angle coordinate is in the form of elliptic integral presented in the form: ϕ1
t=
dϕ1
∫ ϕ
ϕ12,0 +ω 21 (cosϕ1 − cosϕ1, 0
1,0
ϕ1
dϕ1
) ϕ∫ =
ϕ + 2ω sin 2 1, 0
1,0
2 1
2
ϕ1,0 2
(7) 2 1
− 2ω sin
2
ϕ1 2
and it is necessary to take into account corresponding geometrical data of each circle arch. For obtaining the coordinates of ball’s positions at configuration of first collision between rolling heavy balls at first circle line in vertical plane, it is necessary to obtain time timpact ,1 of first collision at which both balls are in the configuration of first collision. If we propose that mass center C1,impact ,1 of first ball is in position defined by
(
)
angle coordinate ϕ1 timpact ,1 = ϕ1,impact ,1 , then coordinate of mass center C2 ,impact ,1 of second ball is defined by angle coordinate:
ϕ 2 (timpacr ,1 ) = ϕ1 (tompact ,1 ) + β , where angle β
is defined by expression (6). Using approximate
expressions (7) for time timpact ,1 duration of balls motions from corresponding initial positions, positions,
ϕ1,impact ,1
and
and
ϕ 2, 0 , to the
ϕ 2,impact ,1 , of the first collision between rolling balls we can write the following: sin
timpact,1 = t1,impact,1
ϕ1, 0
ϕ1, impact , 1
2 2 1 2 3 1⋅ 3 4 5 ≈ k 1u + k 1u u + ω0,1 2⋅3 2⋅4⋅6 sin ϕ1, 0
ϕ1, impact , 1
+
(8)
2
2
sin
sin
2 2 1 3 1 2 5 1⋅ 3 4 7 + u + k 1u + 2 k 1u ω0,1 2 ⋅ 3 4 ⋅5 2 ⋅4⋅7 sin ϕ1, 0
ϕ1, impact , 1
2 2 2 1 ⋅ 3 5 1⋅ 3 1 1 ⋅ 3 4 9 + u + 2 k 12u 7 + k 1u ϕ1, 0 ω0,1 2 ⋅ 4 ⋅ 5 2 ⋅4⋅7 9 2⋅4 sin sin
timpact ,1 = t 2,impact ,1
2 ϕ1 ,impact ,1 + β
2 1 2 3 1⋅ 3 4 5 ≈ k 2u + k 2u u + ω0, 2 2⋅3 2⋅4⋅6 sin ϕ2 ,0
2
2
2 2 1⋅ 3 5 1 ⋅ 3 2 7 1 1 ⋅ 3 4 9 u + 2 k 1u + k 1u + ω0,1 2 ⋅ 4 ⋅ 5 2 ⋅4⋅7 9 2⋅ 4
sin
sin
sin
ϕ1,impact ,1 + β
2 1 3 1 2 5 1⋅ 3 u + k 2u + 2 k 24u 7 + ω0, 2 2 ⋅ 3 4 ⋅5 2 ⋅4⋅7 sinϕ 2, 0 2
2
+
(9)
ϕ1,impact ,1 + β 2
ϕ2 , 0 2
Taking into account, that both balls start from initial positions,
ϕ1, 0
and
ϕ1,impact ,1
and
ϕ 2,impact ,1 , expressions (8) and (9) have to
the first position of first collision, defined by coordinates
ϕ 2, 0 , must
arrive at configuration of
be equal. As result of this condition we have a nonlinear transcendent equation with respect to unknown angle coordinate ϕ1 timpact ,1 = ϕ1,impact ,1 of the mass centre position of first ball at position of first collision..
(
)
To find first real root of this transcendent equation, is not possible to solve analytically and it is necessary to use some of numerical methods as well as some commercial software tool. In this paper we deal with ideas and analytical approach to defined task and methodology of vibro-impact dynamics. Then, we suppose that we have angle coordinate ϕ1 timpact,1 = ϕ1,impact ,1 of mass centre C1,impact ,1 of first ball at position of the first collision, and also
(
)
Katica R. (Stevanović) Hedrih et al. / Procedia Engineering (2017) 663–668 Katica R. (Stevanović) Hedrih / Procedia Engineering 00 (2017)199 000–000
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(
)
(
)
angle coordinate ϕ 2 timpacr ,1 = ϕ1 t ompact ,1 + β of mass centre C2 ,impact ,1 of the second ball at position of the first collision. With this assumption it is possible to compose pre-first collision impact angular velocities
ω P 2,impact ,1
ω P1,impact ,1
and
of the heavy rolling balls. For obtaining post-first-collision outgoing angular velocities
ω P1 (tu ,1 + τ ) = ω P1,outgoing ,1 and ω P 2 (t 0 + τ ) = ω P 2,outgoing ,1 of the rolling balls along circle line, at same position of the first collision balls, determined by generalized coordinates
ϕ1,impact ,1 and ϕ 2,impact ,1 , we use expressions (3) and we
obtain initial conditions for next branches of phase trajectory for rolling each ball. To obtain kinetic parameters of the successive collisions same methodology [3,4,5] have to be applied.
4. Concluding remarks At the end, it is useful to conclude that presented kinetic parameters of dynamics of the central collision of two rolling balls is possible to use in study of the skew collision of two rolling balls that role along two straight as well as curvilinear line trace with intersection as well as parallel at distance smaller them sum of the balls radiuses. Aim of this paper is not to present a possibility about generalization of the all results in area of the vibroimpact dynamics in system containing the collision of two rolling balls with different dimension and collisions. The paper is focused to central collision of two rolling rigid and heavy smooth balls and to using elements of mathematical phenomenology and phenomenological mapping to obtain corresponding new expressions for the post-collision and outgoing angular velocity of each ball and to apply these results for investigation of the vibroimpact dynamics of two rolling balls along circle trace. This task is fully analytical solved and obtained analytical results are original and new! Also, these results can be fundamental for next development and investigation of the special class of vibroimpact dynamical systems with collision of the rigid and/or deformable balls and also in application in different area of engineering systems with coupled rotations (in rolling bearings, rolling vibro-impact dampers - mechanisms for dynamic absorption of torsional vibrations, or other). Acknowledgements Parts of this research were supported by the Ministry of Sciences and Technology of Republic of Serbia through Mathematical Institute SASA, Belgrade Grant ON174001 “Dynamics of hybrid systems with complex structures.”, Mechanics of materials and Faculty of Mechanical Engineering University of Niš.
References [1]
[2]
[3] [4]
[5] [6]
[7] [8]
K. R. Hedrih (Stevanović), Dynamics of Impacts and Collisions of the Rolling Balls, Dynamical Systems: Theoretical and Experimental Analysis, Springer Proceedings in Mathematics & Statistics, Volume Number: 182, Chapter 13, pp. 157-168. Springer ISBN 978-3-31942407-1 DOI 10.1007/978-3-319-42408-8. K. R. Hedrih (Stevanović), Nonlinear Dynamics of a Heavy Material Particle Along Circle which Rotates and Optimal Control, Chaotic Dynamics and Control of Systems and Processes in Mechanics (Eds: G. Rega, and F. Vestroni), p. 37-45. IUTAM Book, in Series Solid Mechanics and Its Applications, Editerd by G.M.L. Gladwell, Springer. 2005, XXVI, 504 p., Hardcover ISBN: 1-4020-3267-6. K. R. Hedrih (Stevanović), Vibro-impact dynamics in system with trigger of coupled three singular points: Collision of two rolling bodies, Extended abstract in the form short paper on CD, IUTAM ICTAM Montreal 2016. K. R. Hedrih (Stevanović), V. Raičević and S. Jović., Phase Trajectory Portrait of the Vibro-impact Forced Dynamics of Two Heavy Mass Particles Motions along Rough Circle, Communications in Nonlinear Science and Numerical Simulations, 2011 16 (12):4745-4755, DOI 10.1016/j.cnsns.2011.05.027. K. R. Hedrih (Stevanović), V. Raičević, S. Jović, Vibro-impact of a Heavy Mass Particle Moving along a Rough Circle with Two Impact Limiters, ©Freund Publishing House Ltd., International Journal of Nonlinear Sciences & Numerical Simulation 10(11): 1713-1726, 2009. M. Petrović, Elementi matematičke fenomenologije (Elements of mathematical phenomenology), Srpska kraljevska akademija, Beograd, 1911. str. 89. http://elibrary.matf.bg.ac.rs/handle/123456789/476?locale-attribute=sr M. Petrović, Fenomenološko preslikavanje (Phenomenological mapping), Srpska kraljevska akademija, Beograd, 1933. str. 33.http://elibrary.matf.bg.ac.rs/handle/123456789/475 D. P. Rašković, Teorija oscilacija (Theory of oscillations), Naučna knjiga, 1952.