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Rigid body coupled rotation around no intersecting axes
International Journal of Non-Linear Mechanics 73 (2015) 100–107
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Rigid body coupled rotation around no intersecting axes Ljiljana Veljović a,n, Aleksandar Radaković b, Dragan Milosavljević a, Gordana Bogdanović a a b
Faculty of Engineering, University of Kragujevac, Sestre Janjic 6, 34000 Kragujevac, Serbia The State Universitz of Novi Pazar, Vuka KaradyicaKaradzica b, 36300 Novi Pazar, Serbia
art ic l e i nf o
a b s t r a c t
Article history: Received 18 March 2014 Received in revised form 28 August 2014 Accepted 4 November 2014 Available online 26 November 2014
In this paper rigid body dynamic with coupled rotation around axes that are not intersecting is described by vectors connected to the pole and the axis. These mass moment vectors are defined by K. Hedrih. Dynamic equilibrium of rigid body dynamics with coupled rotations is described by vector equations. Also, they are used for obtaining differential equations to the rotor dynamics. In the case where one component of rotation is programmed by constant angular velocity, the non-linear differential equation of the system dynamics in the gravitational field is obtained and so is the corresponding equation of the phase trajectory. Series of phase trajectory transformations in relation with changes of some parameters of rigid body are presented. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Coupled rotations No intersecting axes of rotation Mass moment vectors Phase trajectory Angle of inclination Graphical presentations
1. Introduction Non-linear oscillation systems are a phenomenon that occurs in all areas of science. Non-linearity exists in the phenomena in physics, chemical reactions and biology, as well as in the economy and medicine and society. Back in 1911 M. Petrovic wrote about founding mathematical theory that would describe phenomena that act similarly although they do not have the same cause [1]. For describing this phenomenon it is not necessary to go into its nature. Some mathematicians, who are not engaged in the experiment, do not have to know the cause of the phenomenon. Before the existence until the arrival of computers, the solution of some differential equations was obtained from analog electrical circuit, velocities and distances were defined by measuring electrical current. The equivalent electrical circuit is obtained by comparing the equations of motion for both systems. Many engineering components consist of vibrating systems that can be modeled using oscillator systems. Every spinning rotor, because it is impossible to make it perfectly mass balanced, has some vibrations. Vibration of gear-pairs, for instance, was studied over the years. Machine tools, gear-pair system, shafts and computer disk drives are some real systems where minimizations, not total eliminations of unwanted vibrations have to be done. Aerospace vehicles, bridges, and automobiles are examples of structures for which many aspects have to be taken into consideration in the design to improve their performance and extend their life. So, it can be said that the
n
Corresponding author. Tel.: þ 381 648626500. E-mail address: [email protected] (L. Veljović).
http://dx.doi.org/10.1016/j.ijnonlinmec.2014.11.001 0020-7462/& 2014 Elsevier Ltd. All rights reserved.
problem of rigid body rotation has been studied over several centuries. With new operating conditions and growing old machines in the industry rotor dynamic research in this area is becoming more and more important. The need for reliable operations of the machines is of worldwide importance. It is important that dynamical questions, regarding the cause of the failures, can be identified and answered. The control of these vibrations in mechanical systems is still a flourishing research field as it enables resistance improvement as well as noise reduction. First studying on rotating machines dates from Rankin's paper [2] in 1869. In 1919, Jeffcott created the idealized rotor model with the disc in the middle of the shaft which includes both vertical and horizontal motion of the rotor [3]. Today, in rotor dynamical systems with multi-degree of freedom rotors modeled as beams are sectioned applying finite elements and standard finite-element procedure. Very old engineering problem which dates from 100 years ago is dynamics of coupled rotations. The dynamics of such systems are interesting and complex. Some essential results on the dynamics of coupled harmonic oscillators are obtained by consideration of two identical free oscillators mutually coupled [4]. The target is to determine when an unexpected vibration and in some cases failure can occur and therefore find out the causes of the problems. So, it is necessary to investigate properties of nonlinear dynamics. The dynamics of coupled rotations is governed by linear and non-linear differential equations. The solution of linear problem is much easier and often it is sufficient for engineering work. The fact is that all systems have some non-linearity. Sometimes the best linear models could not portray all the vibration features that might be obtained from vibration measurements on actual machinery. The behavior of non-linear systems is much
L. Veljović et al. / International Journal of Non-Linear Mechanics 73 (2015) 100–107
richer than that of linear systems. It is difficult to find the exact or closed-form solutions for non-linear problems. A systematic classification of the solutions based on the structure of differential equations is not generally possible. Non-linear analysis of complex systems is one of the most important and complicated tasks, especially in engineering and applied sciences problems. There are many non-linear equations in the study of different branches of science which do not have analytical solutions. So, many analytical and numerical approaches have been investigated. The solution to a problem depends largely on the researchers, their interests and areas of researches. For solving non-linear vibration problems many different methods may be used. For high-speed rotor symmetrically supported on the magneto-hydrodynamic bearing instability regions and amplitude level contours of vertical and horizontal vibrations are obtained in the frequency–amplitude plane [5]. It is a form suitable for engineering applications. Combination of Newton's method and the harmonic balance method presented in [6] led to new technique for solving large amplitude oscillations of a class of conservative single degree-offreedom systems with odd non-linearity. Instead original nonlinear systems, the systems of linear equations are obtained. This technique, basically quite simple, does not require the presence of small perturbation parameter. In the paper [7], the analytical averaging method is applied for analyzing of a system of three coupled differential equations which describe the motion of a shear-building portal plane frame foundation that supports an unbalanced direct current motor with limited power supply. Yia in [8] analyzed the longitudinal coupled vibrations in a flexible shaft with multiple flexible disks. The analysis rotor model consists of multiple flexible disks attached to a flexible shaft with varying annular cross-section, as used for computer storage devices or steam turbines. The effects of disk flexibility on the longitudinal coupled vibrations between the shaft and disks are investigated with varying spindle rotational speed. The mathematical equations which can be used to analyze vibrations in mechanical systems must be obtained primarily. In this paper mass moment vectors are used to present vector method for obtaining non-linear differential equations and the analysis of kinetic parameter of dynamics of coupled rotations. The definitions of mass moment vectors coupled to the pole and the axis are basic for the vector method. Mass moment vectors are presented and analyzed in numerous papers by Hedrih [9–12]. The phenomenon of appearance and disappearance of a trigger of coupled singularities and homoclinic orbits in the form of number eight in the phase portrait in the phase plane is investigated by using example of non-linear dynamics of a pair of coupled gears [13]. That phenomenon is an accompanying phenomenon of loss of stability of the local unique equilibrium position. Based on the introduced vector method and mass moment vectors non-linear phenomena in rotor dynamics were investigated in the series of references and presented in papers [11–16]. These nonlinear phenomena include phase portraits and homoclinic orbits visualization of non-linear dynamics of multiple step multipliers, non-linear dynamics of planetary reductors with turbulent damping, non-linear dynamics of a heavy material particle along a circle which rotates, and homoclinic orbit layering in the coupled rotor non-linear dynamics and the chaotic clock models. The vector expressions of kinetic parameters of a rigid body simple or coupled rotation around no intersecting axes are derived and detailed explained in [9–12].
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! !ðOÞ def !ð n Þ ! !! J ! ; ½ n ; dm, and there is a corresponding D ∭ ½ ρ ρ ¼ O n V
vector of the rotational rigid body mass deviational moment for !ðOÞ ! !ðOÞ ! ! defined by Hedrih [10]. the rotation axis D ! n ¼ n; J n; n !ðOÞ The body mass inertia moment tensor J for point O is determined with six scalar dynamic parameters 2 3 J u Duv Dun ðOÞ ! 6 Jv Dvn 7 ¼ 4 Dvu J 5 Dnu Dnv J n and it is analogous to the stress tensor and strain tensor in the elasticity theory [13]. In the case when rigid body is balanced with !ðOÞ respect to the axis the mass inertia moment vector J ! n is collinear to the axis, axis of rotation is main axis of body inertia. When axis of rotation is not main axis, then the rotor is unbalanced to the axis and the mass inertial moment for the axis !ðOÞ contains deviation part D ! n [11]. The corresponding body mass linear moment of the rigid body ! at point O for the axes oriented by the unit vector n h i !ðOÞ ! !! ! is S ! n ¼ ∭V n ; ρ dm. Here ρ is vector position of mass dm, n is axis unit vector. The rigid body coupled rotations around two no intersecting axes is first presented in [11]. The first axis, fixed positioned, is ! oriented by unit vector n 1 . The second axis is oriented by unit ! vector n 2 which is rotating around fixed axis with angular ! ! velocity ω 1 ¼ ω1 n 1 . Rigid body is positioned on the moving ! rotating axis oriented by unit vector n 2 . Rigid body rotates around ! ! rotating axis with angular velocity ω 2 ¼ ω2 n 2 and around fixed ! ! ! axis oriented by unit vector n 1 with angular velocity ω 1 ¼ ω1 n 1 , ! ! ! so its angular velocity is ω ¼ ω1 n 1 þ ω2 n 2 . The rigid body is screw (inclined) positioned on the self-rotation axis. The angle β is angle of screw position of rigid body to the self-rotation axis. When center C of the mass of rigid body is not on self-rotation axis of rigid body self-rotation, it is said that rigid body is eccentrically positioned in relation to the self-rotation axis. Eccentricity of position is normal distance between body h mass h centeriiC and axis ! ! ! ! ! of self-rotation and it is defined by e ¼ n 2 ; ρ C ; n 2 . Here ρ C is vector position of mass center C with origin in point O2, and position vector of mass center with fixed origin in point O1 is ! ! ! ! ! r C ¼ r O þ ρ C , r 0 ¼ O1 O2 . For detail see Ref. [11] and Fig. 1.
3. Vector forms of the derivatives of linear and angular momentum of rigid body coupled rotations around two no intersecting axes Vector expressions for linear and angular momentum are expressed in Ref. [12]. By using vector method based on the mass moment vectors and vector rotators for a rigid body coupled rotations around no intersecting axes, the vector expressions of linear and angular momentum are presented in the following form: !ðO2 Þ !ðO2 Þ ! h! ! i ! K ¼ ω 1 ; r 0 M þ ω1 S ! n þ ω2 S n 1
2. Model of a rigid body coupled rotations around two no intersecting axes !ðOÞ The principal vector is J ! n of the body mass inertia mom! ent at point O for the axis oriented by the unit vector n as
ð1Þ
2
h h ii ðO2 Þ ! ! ! ! ! !ðO2 Þ ! !! L O1 ¼ ω1 r O ; n 1 ; r O M þ ω2 r O ; S ! n þ ω1 r O ; S n 2
1
h
h ii !ðO2 Þ !ðO2 Þ ! ! ! ! þ ω1 ρ C ; n 1 ; r O M þ ω1 J ! n þ ω2 J n : 1
2
ð2Þ
102
L. Veljović et al. / International Journal of Non-Linear Mechanics 73 (2015) 100–107
2Þ !n !ðO2 Þ !n !ðO2 Þ ! _ ! !ðO ! þ R 20 D ! þ R 10 D ! ω n ; J þ þ n 1 1 1 n 1 n 2 n1 i
ðO2 Þ ðO2 Þ ðO2 Þ h ! ! ! ! ! ! ! ! n 2; J ! n 1; n 2 þ ω1 ω2 n þ n 1; J n þ J 1
2
ð5Þ ! ! ! ! ! ! 2 2 _ 1 u 1 þ ω1 v 1 and R 2 ¼ ω _ 1 u 2 þ ω1 v 2 Vectors rotators: R 1 ¼ ω are introduced in the following vector forms: 2 3 !ðO2 Þ !ðO2 Þ ! ! D n D n 7 6 ! 2 6! 2! 1 1 7 _ ! _ 1 ðO R 1 ¼ω ! 2 Þ þ ω1 4 n 1 ; !ðO2 Þ 5 ¼ ω1 u 1 þ ω1 v 1 D ! D ! n 1 n 1 2 3 ðO Þ ! 2 !ðO2 Þ ! ! D D 6 ! n 2 n 2 7 2 6! 2! 7 _ ! _ 2 ðO R 2 ¼ω ! 2 Þ þ ω2 4 n 2 ; !ðO2 Þ 5 ¼ ω2 u 2 þ ω2 v 2 : D ! D ! n 2 n 2 Another vector rotator is defined by ðO2 Þ ! !! n 1; J n 2 ! ¼ 2ω1 ω2 ! u 12 : R 12 ¼ 2ω1 ω2 ðO2 Þ ! ! n 1; J ! n2
Fig. 1. Arbitrary position of rigid body coupled rotations around two no intersecting axes. System is with two degrees of mobility with φ1 as rheonomic and φ2 as generalized coordinates.
Corresponding derivative of linear momentum (1) is presented in the following form: ! ! !ðO2 Þ ! !ðO2 Þ d K ! h! ! i ! ¼ R 01 n 1 ; r 0 M þ R 011 S ! S þ R 022 n 2 n 1 dt ! !ðO2 Þ þ 2 ω1 ω2 n 1 ; S ! n
ð3Þ
2
From the structure of linear momentum derivative terms, it is visible that the expression contains terms with product of the pure ! ! kinematic vectors rotators R 01 (and R 0ii ) and corresponding mass moments vectors coupled for pole and axis. In vector expression (3) there are three vectors rotators introduced in the following form: 2 3 !ðO2 Þ !ðO2 Þ ! 7 S! S 6 ! ! ni 7 _ i n i þ ω2i 6 ð4Þ R 0ii ¼ ω 4 n i ; !ðO2 Þ 5; i ¼ 1; 2 !ðO2 Þ S! S! n i n i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ 2i þ ω4i Intensities of vectors which are in the form R0ii ¼ ω i ¼ 1; 2 depend on corresponding component angular velocity and angular accelerations. The first vector component is orthogonal to the corresponding axis of component rotation, and the second is in directions depending of mass linear moment vector coupled for corresponding axis of component rotation and for pole at selfrotation axis [14]. The derivative of angular momentum (2) is presented in the form ! h h ih i ih i d L ðO1 Þ ! ! ! ! ! ! ! ! ¼ M n 1 ; r 0 r 0 ; R 01 þM n 1 ; r 0 ρ C ; R 01 dt ðO Þ h ! 2 ! ! i !ðO2 Þ h! ! i r 0; R 1 þ S ! r 0; R 2 þ þ S ! n 1 n 1 h ! !ðO2 Þ 2Þ ! i ! ! !ðO 2ω1 ω2 ! ! _ þ n r ; R ω n ; S þ n 1 ; S ! 0 21 2 2 2 n n 2
2
The derivatives of angular momentum written in a form with pure kinematic vectors depending on angular velocities and angular accelerations of component angular velocities of coupled rotations in the form with vector rotators is very suitable for analyzing kinetic pressure. It is shown that different pure kinematical vectors rotators rotate around first and second axes of component axes of rotation [16–18].
4. Non-linear dynamics parameters of a rigid body coupled rotation around two orthogonal no intersecting axes and with one degree of freedom. By using theorems of changes of angular momentum with respect to time ! h i d L ðO1 Þ ! ! ! ! ! ¼ L ðO1 Þ þ ω L ðO1 Þ ¼ ∑ ρ i F i ð6Þ dt i vector equations of dynamic equilibrium of rigid body coupled multi-rotations about axes without intersection can be written in the following form: ðO1 Þ ðO1 Þ !ðO1 Þ !ðO1 Þ 2 ! !! 2 ! !! _2 J ! ω_ 1 J ! n 1 þ ω1 n 1 ; J n 1 þ ω n 2 þ ω2 n 2 ; J n 2
ðO1 Þ 1Þ !ðO1 Þ h! ! i ! !ðO ! !! þ ω1 ω2 n 1; J ! ; J n ; n þ n þ J ¼ 2 2 1 n2 n1 82 2 3 ! < h i h i> ! ! 6! _ 6! h r 0 i7 ! ! ¼ ∑ ρ i; F i n 2; r 0 4 ρ C ; ω 2 4 n 2 ; ! ! 5 > : i n 2; r 0 2 2 3339 > ! = r 0 6! 6! 777 ð7Þ þ ω22 4 n 2 ; 4 n 2 ; h! ! i555 M > ; n 2; r 0 In general case the equations are non-linear and coupled by generalized coordinates and their derivatives. For axes that are perpendicular some terms in equations are equal to zero but nonlinearity still remains. These differential equations are coupled by generalized coordinates and their derivatives, also. Expression (7) can be written in a form with vector rotators, similar with expression for derivative of linear momentum of rigid body coupled rotations (4) in a form 1 !n !ðO1 Þ ! _ ! !O! n 1 ω1 n 1 ; J n 1 þ R 10 D ! n 1
L. Veljović et al. / International Journal of Non-Linear Mechanics 73 (2015) 100–107
103
ðO1 Þ !n !ðO1 Þ ! _ ! !! þ n 2ω þ R 20 D ! 2 n 2; J n n 2
2
h i
ðO1 Þ 1Þ ! !! ! !ðO ðO1 Þ ! ! þ n þ J n 1; J ! ; J n ; n þ ω1 ω2 2 2 1 n n 2
1
i h ih i ! ! ! ! ! ! ¼ ∑ ρ i ; F i n 2 ; r 0 ρ C ; R 01 M h
ð8Þ
i
It is visible that the expression contains terms with product of ! pure kinematics vectors rotators R i0 and corresponding mass moments vectors coupled for pole and axis. The vector rotators in previous vector expression are introduced and analyzed in [16–19]. They are expressed in the following form: 2 3 h i !ðO1 Þ !ðO1 Þ ! ! ! ! n i; ρ C S n S 6 7 ! ! n i i 7 _ i h _ i þ ω2i 6 i R i0 ¼ ω 4 n i ; !ðO1 Þ 5 ¼ ω ! ! !ðO1 Þ n i; ρ C S! S! n i n i 2 h i3 ! ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ; ρ i C 6! 7 ! _ 22 þ ω42 ; i ¼ 1; 2 ð9Þ þ ω2i 4 n i ; h! ! i5; R i0 ¼ ω n i; ρ C For that special case when axes are perpendicular, Eq. (8) is ! ! multiplied by unit vector n 1 and after that by unit vector n 2 . Two new equations are obtained and they are in a form with vector rotators as it is explained 2Þ !ðO2 Þ ! ! !ðO2 Þ ! !ðO ! ! ω_ 1 ! n 1; J ! n 1; J ! n ; þ R 2 ; n 1 D n þ 2ω1 ω2 n ; n1 1
2
!
! ! ! _2 ∑ MO F 1 ; n 1 ω
¼
i
þ 2ω1 ω2
2
2Þ ! ! !ðO2 Þ ! !ðO ! D n 2; J ! ; þ R ; n 1 2 n n 2
2Þ ! !ðO ! n 1; J ! ; n ¼ 2 n 2
! ! ! ∑ MO F 1 ; n 2
ð10Þ
i
u1n1
þ 2Meaω21 sin β cos φ2 :
ðO2 Þ ðO2 Þ ðO2 Þ are J ¼ Mr 2 =4, J ! ¼ Jð1 þ ε sin β Þ J ! ! ¼ 0 and J ! ! ¼ n2 u1n1 v1n1 2 Jð1 ε sin β Þ sin ϕ2 cos ϕ2 . So, Eq. (11) can be written in a form
φ€ 2 þ Ω2 ðλ cos φ2 Þ sin φ2 Ω2 ψ cos φ2 ¼ 0;
2 ðO2 Þ 2Þ 2Þ € 2 J ðO φ€ 2 J ðO ! þ ω1 J ! ! φ ! ! ¼ Mge sin β sin φ2
v1n1
products of inertia, the deviational moment of the rigid body for a ! ! ! couple of normal axis oriented by the unit vectors n 1 , v 1 and u 1 . It is considered an eccentric disc (eccentricity is e), with mass M and radius r, which is inclined to the axis of its own self-rotation by the angle β (Fig. 2). In this case mass inertia moment intensities 2
1
!
These two vector equations are equations of rigid body rotation around fixed point and special case of precesion with angular ! velocity ω 1 and self-angular velocity. A special case of the heavy rigid body with coupled rotations about two orthogonal no intersecting axes with one degree of freedom, and in the gravitation field is considered. This example is studied during some years of investigation [18]. For this case generalized coordinate φ2 is independent, and coordinate ϕ1 is programmed. In that case, it is said that coordinate φ1 is rheonomic coordinate and system is with kinematical excitation, programmed by forced support rotation by constant angular velocity. When the angular velocity of shaft support axis is _ 1 ¼ ω1 ¼ const; it is obvious that rheonomic coordinate constant, φ is linear function of time, φ1 ¼ ω1 t þ φ10 and angular acceleration _ 1 ¼ 0. around fixed axis is equal to zero ω Special case is when the support shaft axis is vertical and the gyro-rotor shaft axis is horizontal, and all time in horizontal plane, and when axes are without intersection with normal distance a between. The angle of self-rotation around moveable self-rotation ! axis oriented by the unit vector n 2 is φ2 whose angular velocity is ω2 ¼ φ_ 2 . Differential equation of the heavy gyro rotor-disk selfrotation of reviewed model in Fig. 2, for the case coupled rotations about two orthogonal axes by using (10) in the case when axes are orthogonal, the support shaft axis is vertical and the gyro-rotor shaft axis is horizontal, and when axes are without intersection ! (detail in [18]) after multiplying scalar by n 2 , and taking into account orthogonally between axes of coupled rotations, is obtained in the following form: n2
Fig. 2. Model of heavy gyro rotor with two component coupled rotations around orthogonal axes without intersections.
ð11Þ
ðO2 Þ ðO2 Þ 2Þ Here J ðO ! is axial mass inertia moment and J ! ! and J ! ! are n2 v1n1 u1n1
ð12Þ
where corresponding coefficients, in case when rigid body is disk, are e 2 ε sin 2 β 1 g ðε 1Þ sin β 2 2 ; ε ¼ 1 þ 4 ; Ω ¼ ω1 ; λ¼ 2 r ε sin β þ 1 eω21 ε sin 2 β 1 2ea sin β : ψ¼ er ε sin 2 β 1
ð13Þ
This differential equation is quite the same in the form as differential equation given in [20] obtained for heavy material point rotating around circle. So, dynamics of a rotor defined in this way can be studied by applying of a differential equation which is used for describing the dynamics of a heavy material point on the circle with coupled rotations. Thus it is confirmed that the different physical problems can be described with the same differential equations and analyzed on similar way. The derived equation is a non-linear one. There is not a general method to solve non-linear equations like the one that is employed to obtain the motions of the several types of harmonic oscillators. In general, the motion of the oscillator in phase space has to be obtained by means of one or more of the numerical methods that have been developed to solve differential equations. Although there are analytical tools to deal with non-linear systems, computational methods play an important role in the study of non-linear dynamical systems. Many non-linear effects, especially in control systems, are best approximated by segmented characteristics. Effects of this type of non-linearity on the system response are analyzed using the phase plane. The motion in each segmented region is describing with a linear differential equation. For examining non-linear dynamics of rotor, it is necessary to investigate its properties, phase portraits, structures of homoclinic orbits and its bifurcations.
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L. Veljović et al. / International Journal of Non-Linear Mechanics 73 (2015) 100–107
For the non-linear system that can be written in theory well known form that is x€ þ f ðxÞ ¼ 0, upon integrating the following R form can be obtained ð1=2Þx_ 2 þF ðxÞ ¼ h; F ðxÞ ¼ f ðxÞdx. This mathematical equation has been obtained by many researches in different areas of investigations. For mathematicians it is only second kind differential equation with function f ðxÞ which may be linear or non-linear. There are many solutions depending on expression f ðxÞ. In most university books variable x is displacement, but it may be angle, or electrical current, or flux, or share. It is the interaction between different processes and phenomenology [1]. For the example here analyzed the differential equation (12) can be written in previous mentioned form. For relative coordinate ϕ2 that form is Z
_ 22 þ F φ2 ¼ h; F φ2 ¼ ð1=2Þφ ð14Þ f φ2 dφ2 : Here is another example of the phenomenon that was written by M. Petrovic. For analyzing this equation that may be non-linear it is not necessary to go into nature of terms. But, in physics, this represents that the sum of the kinetic energy and potential energy, the total energy, is constant for an idealized system with unit mass and with one degree of motion. Kinetic energy of thus defined _ 22 =2, while potential energy is determined by the system is φ
expression F φ2 . The system analyzed in this paper is rheonomic. The expression which, with analogy, can be here named as potential energy is in relation to the relative coordinate. It is potential of conservative system which is equivalent to the real rheonomic system.
Ω2 β ; ε; ω2
cos 2 φ2 2λ β; ε; e; ω2 Ep φ2 ¼ F φ2 ¼ 2
ð15Þ cos φ2 2ψ β; e; a; r sin φ2 1 The relative equilibrium positions can be determined from this expression. Also, from the above equation, the phase portrait or the trajectories for different energy level may be plotted and study qualitatively about the response of the system. The dynamical equilibrium positions of the gyro-rotor can be determined for ðλ cos φ2 Þ sin φ2 ψ cos φ2 ¼ 0. This equation has to be solved numerically as it is presented in [21] and all solutions depend on system parameters. In Fig. 3, transformation of a potential trajectory of the heavy gyro-rotor-disk with rotating axes that are without intersection for different values of disk inclination angle β to the axis of self _ 0 ¼ 0 rad=s is rotation and for initial conditions φ0 ¼ π =2 rad ; φ presented. For different angles of screw position of rigid body to the selfrotation axis the optimum values of potential energy changes. In Fig. 3 characteristic potential energy curves for different parameter values of the basis system correspond to the rigid body dynamic model are presented. It can be seen that for increasing the angle β of disk inclination to the proper shaft axis rotation some max of potential energy curves disappeared. Now by taking different energy level h, the relation between _ 2 and angular displacement φ2 may be the angular velocity φ finding as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
φ_ 2 ¼ 7 2 h F φ_ 2 , where F φ2 is determined by the
2 expression F φ2 ¼ ðΩ =2Þ cos 2 φ2 2λ cos φ2 2ψ sin φ2 1 , and total energy h is _ 220 Ω2 2λ cos φ20 þ sin 2 φ20 þ 2ψ sin φ20 : h¼φ By plotting the phase portrait it can be finding that the trajectory which clearly depicts that the motion corresponding to maximum potential energy is unstable and the bifurcation point
is of saddle-node type. The motion corresponding to the minimum potential energy is stable center type. In Fig. 4, transformation of a phase trajectory of the heavy gyrorotor-disk with rotating axes that are without intersection for different values of initial conditions and for disk inclination angle β to the axis of self-rotation β ¼ 0:01π rad is presented. Also potential well with its maximum and minimum corresponding to saddle point and center is presented. Total energy, as it can be seen, is depending not only on initial conditions but on parameters of rigid body: angle of screw position of rigid body to the self-rotation axis β, eccentricity e and normal distance between axes a. Thus, the following figure shows transformation of phase trajectory depending on the angle of inclination to the self-rotation axis β . It can be seen that there are various forms of trajectory that means that oscillations may be of various types. Also progressive, non-oscillatory movement is possible. Some equilibrium positions may be stable or unstable. With changing angle of inclination there is a possibility to make unstable equilibrium position stable. There are some experimental results that confirm this statement. In Fig. 5, transformation of a phase trajectory of the heavy gyrorotor-disk with rotating axes that are without intersection for
Fig. 3. Potential energy curves analog of the heavy gyro rotor with rotating axis that are without intersection for different values of the angle β of disk inclination to the proper shaft axis rotation and for initial conditions φ0 ¼ π and φ_ 0 ¼ 0.
Fig. 4. Potential well and phase portrait showing saddle point and center corresponding to maximum and to minimum potential energy.
L. Veljović et al. / International Journal of Non-Linear Mechanics 73 (2015) 100–107
different values of disk inclination angle β to the axis of self- _ 0 ¼ π =12 rad=s rotation and for initial conditions φ0 ¼ π =4 rad ; φ is presented. It is possible to see appearance of the homoclinic orbits in the form number eight and coupled singularities. A closed homoclinic trajectory in the form of number eight and a trigger of coupled singularities composed of three singular points (one unstable saddle node and two stable centers) appears. It is possible by transformation of the phase trajectory to indicate that disk inclination angle β to the axis of self-rotation has bifurcation values at which homoclinic phase trajectory appears. In Fig. 6 transformations of a phase trajectory in a case when gyro rotor is rotating around axes without intersection with distance between axes is a ¼ 2 mm and for different initial conditions and different values of angle of inclination β are presented. Fig. 6a shows transformation of a phase trajectory in a case when gyro rotor is rotating around axes that are in intersection and when angle of inclination is β ¼ 0:218 π . In Fig. 6b transformation of a phase trajectory for the angle of inclination β ¼ π =3 is presented. This type of trajectories is quite different. Fig. 6c shows transformation of a phase trajectory for the case when gyro rotor is rotating around axes that are in intersection and when angle of inclination is β ¼ 0:1π . When angle of inclination is β ¼ 0:01π transformation of a phase trajectory is presented in Fig. 6d. The particular energy levels h are determined for different values of initial conditions, primarily by changing the initial
Fig. 5. Transformation of phase trajectory for different values of the angle β of screw positioned rigid body.
105
angular velocity while the angle of inclination stayed generally the same. Of course, with increasing initial angular velocity the stability disrupt at low angles of inclination (Fig. 6b). For example, if angle of inclination is β ¼ π =3, the initial angular velocity of 1:54 s 1 produced progressive movement, while in a case when the angle of inclination is β ¼ 0:01π the initial angular velocity that produces progressive movement is 2:53 s 1 . Hence, for particular energy level h, the system will be under oscillation, if the potential energy Ep ðφ2 Þ is less than the total energy h.
5. Concluding remarks In this paper some results of the rigid body coupled rotations are presented. A numerical experiment with the use of derived analytical expressions and MathCAD program was used to create a visualization of phase portraits of non-linear dynamics of rigid body coupled rotations and layering of homoclinic orbits with respect to the system parameter of disk inclination angle β to the axis of self-rotation change different values. The phase plane used to study the effects of this type of non-linearity on the system response was one of the original contributions to the phase plane method in control, but it is very useful in engineering, too. The rigid body is the eccentrically, skew positioned disc on the shaft rotor support in the field without turbulent damping. In that case the transformation of the graphical presentation of the potential energy for different values of the angle of the disc inclination to the proper shaft axis rotation is graphically presented. Transformation of the phase trajectory for different values of the angle β of disk inclination to the proper shaft axis rotation is presented graphically also. Special attention is focused to the angle of inclination, as possible cause of the imbalance, as well as to the phase trajectory portrait for the case of the heavy gyrorotor disc coupled rotations about two axes without intersection. Mass moment vectors in our opinion can be new open way for applications in this area. Transformation of the phase trajectory of the rigid body non-linear dynamics with coupled rotations around no intersecting axes contains or not contains a trigger of coupled singularities. Relative non-linear dynamics of the heavy gyro-
Fig. 6. Transformation of a phase trajectory of the heavy gyro-rotor with rotating axis that is without intersection for different values of disk inclination angle β to the axis of self-rotation and for different initial conditions: ðaÞ β ¼ 0; 218π, ðbÞ β ¼ π=3, ðcÞ β ¼ 0; 1π, ðdÞ β ¼ 0; 01π Examples of the trigger of the coupled singularities and homoclinic trajectories in the form of the number eight.
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rotor-disk around self-rotation shaft axis is possible to present by means of the phase portrait method. Forms of phase trajectories and their transformations by changes of initial conditions, and for different cases of angle of inclination may present character of non-linear oscillations. The particular energy levels h which were determined for different values of initial conditions, primarily by changing the initial angular velocity confirms that statement. The example of rigid body coupled rotations once again proved the trigger of coupled singularities theorem and the existence and non-existence of homoclinic orbits in the form of number eight which is formulated and detailed analyzed in [14]. The theorem on the trigger of coupled singularities speaks of the conditions existence, or not, the coupled equilibrium position, and could speak on coupled dynamical states of relative rest. Small changes of initial conditions may change kinetic parameters of the system and cause the appearance or disappearance trigger of coupled singularities. Two physical phenomena, essentially different, can have the same mathematical model. If one of these phenomena is well known, the theory and analysis discussed for the former can be applied equally well to the latter. On the basis of phenomenological mapping and mathematical analogy, analysis for one of the system non-linear dynamics is possible to apply for qualitative analysis of non-linear phenomena appeared in dynamics of other disparate model or nature system, in Economy, in Ecology, dynamics of web cities in internet, in Chemical engineering, Cryptography, in Computer Science, Electronic and Communication Technologies, etc. Analogy between the expression of change of angular momentum with respect to time (6) which is used for describing the movement of a rigid body with a fixed point and the relation between the efforts in the cross section of a bar, having as axis a curve in the three-dimensional space and the exterior load on the bar unit length is noticed. This analogy includes analogies between ! the angular momentum L ðO1 Þ and the resultant of the axial and ! shearing effects on the transversal section R , and between the ! ! angular velocity vector ω and the Darboux vector Ω [22]. In [22] an analogy between two completely different physical phenomena: the rectilinear displacement in the relativistic frame under a constant force and the large deformations of a straight bar for a constant bending moment and constant rigidity is emphasized. The vector method, based on mass moment vectors coupled for pole and oriented axes defined by K. Hedrih, is used for investigation of the rigid body coupled rotations. The possibilities to establish the phenomenological analogy of different physical model dynamics expressed by vectors connected to the pole and the axis and the influence of such possibilities to applications allow researchers and scientists to obtain larger views within their specialization fields. The !ðOÞ analogy between the vector J ! n of the body mass inertia moment at ! the observed point for the axis oriented by the unit vector n and the !ðOÞ vector p ! of the total stress at a certain body point for the plain with n !ðOÞ ! the normal oriented by the unit vector n and the vector δ ! n of the total strain of the line element drawn from the observed point in the ! direction of the unit vector n is presented. The non-linear differential equation (12) putted under a common form of depending on parameters λ, Ω and ψ , (13), has been analyzed by K. Hedrih in numerous papers during investigations in non-linear dynamics of a heavy mass particle rolling along rotating circle about axis inclined to the vertical direction with constant angular velocity. Parameters are in the same form but they are of different nature. The obtained differential equation allows getting equations for many different cases of dynamical systems. There are analogies between non-linear phenomena in coupled rotor nonlinear dynamics, in non-linear dynamics of multiple step reductor, in non-linear dynamics of a car model and the non-linear dynamics of a heavy material particle along a circle with coupled rotations.
Mechanical systems, electrical systems, liquid-level systems, thermal systems can be presented and studied on the same way. The series of trigger of coupled singularities in the phase plane here presented are identified in the movement of heavy material point by rotating rough curve line. Examples of the trigger of the coupled singularities and coupled triggers of the coupled singularities for different parameter values of the basic system correspond to the rigid body dynamic model are equivalent to the corresponding one obtained in the analyzing non-linear dynamics of the car dynamic model [23]. The series of triggers of coupled singularities in the phase plane, and the trigger of coupled singularities are identified in non-linear electrical circuit, elliptical-ring plate loaded by concentrated forces. The theorem on the trigger of coupled singularities speaks of the conditions existence, or not, the coupled equilibrium position, and could speak on coupled dynamical states of relative rest. On the basis of the type of the trigger of singularities conclusion on the behavior of the dynamical processes in the society can be made. If there is one of homoclinic orbit in the form of number eight and trigger of singularities stable–unstable–stable stability of social dynamical system exists. If there is no homoclinic orbit in the form of number eight and there is only trigger of singularities unstable–stable– unstable there are great social changes [24]. Examples of coupled singularities can be found exactly in nonlinear dynamical systems disparate nature, for example, in technical, in biological, in economic and in social. It can be shown that mathematic used in explaining phenomena of an inverted pendulum [25] relates to other systems that are completely different but are fundamentally similar [26]. The dynamics of ecosystem, catalytic reactions, biochemical processes in the cells of living organism, economics, traffic, always show the same universal features. Because of some symmetry in living organisms' classic Lagrange equations are applied to simulate oscillation in amino-acid chains. The dynamics of different types of the system without its nature is always the same. Mechanical phenomena, physical phenomena, chemical, physiological and social, mutually disparate, are one and the same analytic problem, which consists in the integration, discussion and interpretation of some of the same equation which makes the corresponding phenomenon among them analytically equivalent. Analogy may lead under some circumstances to new discoveries. It is well known that an analogy between the corpuscular and the undulate features of light and the features of electron De Brogliy discovered theoretically the undulate features of the electron. Analyses in very different fields of science, from physics to medicine show although appearances have different causes, they appear in a similar way and they are analytically equivalent. The motion of a shear-building portal plane frame foundation that supports an unbalanced direct current motor with limited power supply, the longitudinal coupled vibrations in a flexible shaft with multiple flexible disks non-linear oscillation bearings of centrifugal pumps, elliptical-ring plate loaded by concentrated forces, nonlinear electrical circuit, the large deformations of a straight bar for a constant bending moment and constant rigidity, etc. – many phenomena, mutually disparate, have its analytically equivalency. All that plays a role in the concrete loses its specific meaning. Remains only the functions described in the most general possible way. One whatever process physics, chemical, physiological, geological, social entered into a dynamic stationary phase from the moment when the intensity of the process, measured by the size change of selected elements become unchangeable [1].
Acknowledgments Parts of this research were supported by the Ministry of Sciences and Technology of Republic of Serbia through
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Mathematical Institute SANU Belgrade Grant ON174001 Dynamics of hybrid systems with complex structures, Mechanics of materials and Faculty of Mechanical Engineering University of Niš and Faculty of Mechanical Engineering, University of Kragujevac. Special thanks to Professor Katica Hedrih for the unselfish help and constant encouragement in the work. References [1] M. Petrovic, Elementi matematičke fenomenologije (elements of mathematical phenomenology), Srpska kraljevska akademija, Beogr. (1911) str.789. [2] A.A. Andonov, A.A. Vitt, S.E. Haykin, Teoriya kolebaniy, Nauka, Moskva (1981) 568. [3] E. Butikov, Inertial rotation of a rigid body, Eur. J. Phys. 27 (2006) 913–922. [4] D. Rašković, Mehanika III – Dinamika (mechanic III – dynamics), Naučna (knjiga, 1972) 424. [5] J. Awrejcewicz, L.P. Dzyubak, 2-dof non-linear dynamic of a rotor suspended in the magneto-hydrodynamic field in the case of soft and rigid magnetic materials, Int. J. Non-linear Mech. 45 (2010) 919–930. [6] B.S. Wu, W.P. Sun, C.W. Lim, An analytical approximate technique for a class of strongly non-linear oscillators, Int. J. Non-linear Mech. 41 (2006) 766–774. [7] P. Felix, et al., On saturation of a non-ideal vibrating portal frame foundation type shear-building, J. Vib. Control 11 (2005) 121–136. [8] H.S. Jia, S.B. Chun, C.W. Lee, Evaluation of the longitudinal coupled vibrations in rotating; flexible disks/spindle systems, J. Sound Vib. 208 (2) (1997) 157–187. [9] K. Hedrih (Stevanović), Some vectorial interpretations of the kinetic parameters of solid material lines, ZAMM. Angew. Math. Mech. 73 (4–5) (1993) T153–T156. [10] Hedrih Stevanović, The mass moment vectors at n-dimensional coordinate system, Tensor, Japan 54 (1993) 83–87. [11] K. Hedrih Stevanović, Vector method of the heavy rotor kinetic parameter analysis and nonlinear dynamics (YU ISBN 86-7181-046-1), Monograph, University of Niš (2001) 252 (in English). [12] K. Hedrih Stevanović, Vectors of the body mass moments, monograph paper, topics from mathematics and mechanics, mathematical institute SANU, Belgrade, Proceedings, vol. 8, 199845–104 (Published in 1999. (In English), (Zentralblatt Review). [13] K. Hedrih (Stevanović), Analogy between models of stress state, strain state and state of moment inertia mass of body, Facta Universitatis, Series Mechanics, Automatic Control and Robotics, Vol 1, Nis (1991) 105–120.
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[14] K. Hedrih Stevanović, Trigger of coupled singularities, Invited Plenary Lecture, in: J. Awrejcwicz, J. Grabski, J. Nowakowski (Eds.), Proceedings International Symposium 6th Conference Dynamical Systems-Theory and Applications, Lodz, 2001, pp. 51–78 (Poland on December 10–12). [15] K. Hedrih Stevanović, L.J. Veljović, The kinetic pressure of the gyrorotor eigen shaft bearings and rotators. In: Proceedings of The Third International Conference Nonlinear Dynamics, 2010, pp. 78–83. [16] K. Hedrih Stevanović, LJ. Veljović, Nonlinear dynamics of the heavy gyro-rotor with two skew rotating axes, Journal of Physics: Conference Series, 96, IOP Publishing (2008) 012221. http://dx.doi.org/10.1088/1742-6596/96/1/012221. [17] K. Hedrih Stevanović, LJ. Veljović, Vector Rotators of a Rigid Body Dynamics with Coupled Rotations around Axes without Intersection, Mathematical Problems in Engineering, ISSN 1024-123x, vol. 2011, Article ID 351269, 26 pages, 10.11552011/ 351269. 〈http://www.hindawi.com/journals/mpe/contents〉. [18] LJ. Veljović, Nelinearne oscilacije giro-rotora (nonlinear oscillations of gyrorotor), Doktorska disertacija, Mašinski Fakultet, Niš (2011). [19] K. Hedrih Stevanović, D. Milosavljević, LJ. Veljović, Multi-parameter analysis of a rigid body nonlinear coupled rotations around no intersecting axes based on the vector method, Adv. Theor. Appl. Mech., ISSN 1313-65506 (No. 2) (2013) 49–70. [20] K. Hedrih Stevanović, Nonlinear dynamics of a heavy material particle along a circle which rotates and optimal control. In: UITAM Symposium on Chaotic Dynamic and Control of Systems and Processes in Mechanics, 2005, vol. 122, pp. 37–45. [21] K. Hedrih Stevanović, LJ. Veljović, New vector description of kinetic pressures on shaft bearings of a rigid body nonlinear dynamics with coupled rotations around no intersecting axes, Acta Polytech. Hung. 10 (7) (2013) 151–170 (ISSN 185-8860). [22] Radu P. Voinea, On some analogies, facta universitatis, series mechanics, Autom. Control Robot., 3, Nis (1993) 785–790. [23] K. Hedrih Stevanović, J. Simonović, Nonlinear phenomena in dynamics of car model, facta universitatis, series mechanics, Autom. Control Robot., 3, Nis (2003) 865–879. [24] Stevanović K. Hedrih, Mihailo Petrovic Alas, 1868–1943. Mathematical phenomenology. In: The Sixth International Symposium on Nonlinear Mechanics Nonlinear Science and Applications, Booklet of Abstracts, pp. 27–33. [25] J.A. Blackbum, H.J.T. Smith, N. Gronbech-Jensen, Stability and Hopf bifurcation in an inverted pendulum, Am. J. Phys. 60 (10) (1992) 903–908. [26] M. Petrovic, Fenomenološko preslikavanje (Phenomenological mapp), Srpska Kraljevska akademija, Beograd (1933) str.33.
Contents lists available at ScienceDirect
International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm
Rigid body coupled rotation around no intersecting axes Ljiljana Veljović a,n, Aleksandar Radaković b, Dragan Milosavljević a, Gordana Bogdanović a a b
Faculty of Engineering, University of Kragujevac, Sestre Janjic 6, 34000 Kragujevac, Serbia The State Universitz of Novi Pazar, Vuka KaradyicaKaradzica b, 36300 Novi Pazar, Serbia
art ic l e i nf o
a b s t r a c t
Article history: Received 18 March 2014 Received in revised form 28 August 2014 Accepted 4 November 2014 Available online 26 November 2014
In this paper rigid body dynamic with coupled rotation around axes that are not intersecting is described by vectors connected to the pole and the axis. These mass moment vectors are defined by K. Hedrih. Dynamic equilibrium of rigid body dynamics with coupled rotations is described by vector equations. Also, they are used for obtaining differential equations to the rotor dynamics. In the case where one component of rotation is programmed by constant angular velocity, the non-linear differential equation of the system dynamics in the gravitational field is obtained and so is the corresponding equation of the phase trajectory. Series of phase trajectory transformations in relation with changes of some parameters of rigid body are presented. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Coupled rotations No intersecting axes of rotation Mass moment vectors Phase trajectory Angle of inclination Graphical presentations
1. Introduction Non-linear oscillation systems are a phenomenon that occurs in all areas of science. Non-linearity exists in the phenomena in physics, chemical reactions and biology, as well as in the economy and medicine and society. Back in 1911 M. Petrovic wrote about founding mathematical theory that would describe phenomena that act similarly although they do not have the same cause [1]. For describing this phenomenon it is not necessary to go into its nature. Some mathematicians, who are not engaged in the experiment, do not have to know the cause of the phenomenon. Before the existence until the arrival of computers, the solution of some differential equations was obtained from analog electrical circuit, velocities and distances were defined by measuring electrical current. The equivalent electrical circuit is obtained by comparing the equations of motion for both systems. Many engineering components consist of vibrating systems that can be modeled using oscillator systems. Every spinning rotor, because it is impossible to make it perfectly mass balanced, has some vibrations. Vibration of gear-pairs, for instance, was studied over the years. Machine tools, gear-pair system, shafts and computer disk drives are some real systems where minimizations, not total eliminations of unwanted vibrations have to be done. Aerospace vehicles, bridges, and automobiles are examples of structures for which many aspects have to be taken into consideration in the design to improve their performance and extend their life. So, it can be said that the
n
Corresponding author. Tel.: þ 381 648626500. E-mail address: [email protected] (L. Veljović).
http://dx.doi.org/10.1016/j.ijnonlinmec.2014.11.001 0020-7462/& 2014 Elsevier Ltd. All rights reserved.
problem of rigid body rotation has been studied over several centuries. With new operating conditions and growing old machines in the industry rotor dynamic research in this area is becoming more and more important. The need for reliable operations of the machines is of worldwide importance. It is important that dynamical questions, regarding the cause of the failures, can be identified and answered. The control of these vibrations in mechanical systems is still a flourishing research field as it enables resistance improvement as well as noise reduction. First studying on rotating machines dates from Rankin's paper [2] in 1869. In 1919, Jeffcott created the idealized rotor model with the disc in the middle of the shaft which includes both vertical and horizontal motion of the rotor [3]. Today, in rotor dynamical systems with multi-degree of freedom rotors modeled as beams are sectioned applying finite elements and standard finite-element procedure. Very old engineering problem which dates from 100 years ago is dynamics of coupled rotations. The dynamics of such systems are interesting and complex. Some essential results on the dynamics of coupled harmonic oscillators are obtained by consideration of two identical free oscillators mutually coupled [4]. The target is to determine when an unexpected vibration and in some cases failure can occur and therefore find out the causes of the problems. So, it is necessary to investigate properties of nonlinear dynamics. The dynamics of coupled rotations is governed by linear and non-linear differential equations. The solution of linear problem is much easier and often it is sufficient for engineering work. The fact is that all systems have some non-linearity. Sometimes the best linear models could not portray all the vibration features that might be obtained from vibration measurements on actual machinery. The behavior of non-linear systems is much
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richer than that of linear systems. It is difficult to find the exact or closed-form solutions for non-linear problems. A systematic classification of the solutions based on the structure of differential equations is not generally possible. Non-linear analysis of complex systems is one of the most important and complicated tasks, especially in engineering and applied sciences problems. There are many non-linear equations in the study of different branches of science which do not have analytical solutions. So, many analytical and numerical approaches have been investigated. The solution to a problem depends largely on the researchers, their interests and areas of researches. For solving non-linear vibration problems many different methods may be used. For high-speed rotor symmetrically supported on the magneto-hydrodynamic bearing instability regions and amplitude level contours of vertical and horizontal vibrations are obtained in the frequency–amplitude plane [5]. It is a form suitable for engineering applications. Combination of Newton's method and the harmonic balance method presented in [6] led to new technique for solving large amplitude oscillations of a class of conservative single degree-offreedom systems with odd non-linearity. Instead original nonlinear systems, the systems of linear equations are obtained. This technique, basically quite simple, does not require the presence of small perturbation parameter. In the paper [7], the analytical averaging method is applied for analyzing of a system of three coupled differential equations which describe the motion of a shear-building portal plane frame foundation that supports an unbalanced direct current motor with limited power supply. Yia in [8] analyzed the longitudinal coupled vibrations in a flexible shaft with multiple flexible disks. The analysis rotor model consists of multiple flexible disks attached to a flexible shaft with varying annular cross-section, as used for computer storage devices or steam turbines. The effects of disk flexibility on the longitudinal coupled vibrations between the shaft and disks are investigated with varying spindle rotational speed. The mathematical equations which can be used to analyze vibrations in mechanical systems must be obtained primarily. In this paper mass moment vectors are used to present vector method for obtaining non-linear differential equations and the analysis of kinetic parameter of dynamics of coupled rotations. The definitions of mass moment vectors coupled to the pole and the axis are basic for the vector method. Mass moment vectors are presented and analyzed in numerous papers by Hedrih [9–12]. The phenomenon of appearance and disappearance of a trigger of coupled singularities and homoclinic orbits in the form of number eight in the phase portrait in the phase plane is investigated by using example of non-linear dynamics of a pair of coupled gears [13]. That phenomenon is an accompanying phenomenon of loss of stability of the local unique equilibrium position. Based on the introduced vector method and mass moment vectors non-linear phenomena in rotor dynamics were investigated in the series of references and presented in papers [11–16]. These nonlinear phenomena include phase portraits and homoclinic orbits visualization of non-linear dynamics of multiple step multipliers, non-linear dynamics of planetary reductors with turbulent damping, non-linear dynamics of a heavy material particle along a circle which rotates, and homoclinic orbit layering in the coupled rotor non-linear dynamics and the chaotic clock models. The vector expressions of kinetic parameters of a rigid body simple or coupled rotation around no intersecting axes are derived and detailed explained in [9–12].
101
! !ðOÞ def !ð n Þ ! !! J ! ; ½ n ; dm, and there is a corresponding D ∭ ½ ρ ρ ¼ O n V
vector of the rotational rigid body mass deviational moment for !ðOÞ ! !ðOÞ ! ! defined by Hedrih [10]. the rotation axis D ! n ¼ n; J n; n !ðOÞ The body mass inertia moment tensor J for point O is determined with six scalar dynamic parameters 2 3 J u Duv Dun ðOÞ ! 6 Jv Dvn 7 ¼ 4 Dvu J 5 Dnu Dnv J n and it is analogous to the stress tensor and strain tensor in the elasticity theory [13]. In the case when rigid body is balanced with !ðOÞ respect to the axis the mass inertia moment vector J ! n is collinear to the axis, axis of rotation is main axis of body inertia. When axis of rotation is not main axis, then the rotor is unbalanced to the axis and the mass inertial moment for the axis !ðOÞ contains deviation part D ! n [11]. The corresponding body mass linear moment of the rigid body ! at point O for the axes oriented by the unit vector n h i !ðOÞ ! !! ! is S ! n ¼ ∭V n ; ρ dm. Here ρ is vector position of mass dm, n is axis unit vector. The rigid body coupled rotations around two no intersecting axes is first presented in [11]. The first axis, fixed positioned, is ! oriented by unit vector n 1 . The second axis is oriented by unit ! vector n 2 which is rotating around fixed axis with angular ! ! velocity ω 1 ¼ ω1 n 1 . Rigid body is positioned on the moving ! rotating axis oriented by unit vector n 2 . Rigid body rotates around ! ! rotating axis with angular velocity ω 2 ¼ ω2 n 2 and around fixed ! ! ! axis oriented by unit vector n 1 with angular velocity ω 1 ¼ ω1 n 1 , ! ! ! so its angular velocity is ω ¼ ω1 n 1 þ ω2 n 2 . The rigid body is screw (inclined) positioned on the self-rotation axis. The angle β is angle of screw position of rigid body to the self-rotation axis. When center C of the mass of rigid body is not on self-rotation axis of rigid body self-rotation, it is said that rigid body is eccentrically positioned in relation to the self-rotation axis. Eccentricity of position is normal distance between body h mass h centeriiC and axis ! ! ! ! ! of self-rotation and it is defined by e ¼ n 2 ; ρ C ; n 2 . Here ρ C is vector position of mass center C with origin in point O2, and position vector of mass center with fixed origin in point O1 is ! ! ! ! ! r C ¼ r O þ ρ C , r 0 ¼ O1 O2 . For detail see Ref. [11] and Fig. 1.
3. Vector forms of the derivatives of linear and angular momentum of rigid body coupled rotations around two no intersecting axes Vector expressions for linear and angular momentum are expressed in Ref. [12]. By using vector method based on the mass moment vectors and vector rotators for a rigid body coupled rotations around no intersecting axes, the vector expressions of linear and angular momentum are presented in the following form: !ðO2 Þ !ðO2 Þ ! h! ! i ! K ¼ ω 1 ; r 0 M þ ω1 S ! n þ ω2 S n 1
2. Model of a rigid body coupled rotations around two no intersecting axes !ðOÞ The principal vector is J ! n of the body mass inertia mom! ent at point O for the axis oriented by the unit vector n as
ð1Þ
2
h h ii ðO2 Þ ! ! ! ! ! !ðO2 Þ ! !! L O1 ¼ ω1 r O ; n 1 ; r O M þ ω2 r O ; S ! n þ ω1 r O ; S n 2
1
h
h ii !ðO2 Þ !ðO2 Þ ! ! ! ! þ ω1 ρ C ; n 1 ; r O M þ ω1 J ! n þ ω2 J n : 1
2
ð2Þ
102
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2Þ !n !ðO2 Þ !n !ðO2 Þ ! _ ! !ðO ! þ R 20 D ! þ R 10 D ! ω n ; J þ þ n 1 1 1 n 1 n 2 n1 i
ðO2 Þ ðO2 Þ ðO2 Þ h ! ! ! ! ! ! ! ! n 2; J ! n 1; n 2 þ ω1 ω2 n þ n 1; J n þ J 1
2
ð5Þ ! ! ! ! ! ! 2 2 _ 1 u 1 þ ω1 v 1 and R 2 ¼ ω _ 1 u 2 þ ω1 v 2 Vectors rotators: R 1 ¼ ω are introduced in the following vector forms: 2 3 !ðO2 Þ !ðO2 Þ ! ! D n D n 7 6 ! 2 6! 2! 1 1 7 _ ! _ 1 ðO R 1 ¼ω ! 2 Þ þ ω1 4 n 1 ; !ðO2 Þ 5 ¼ ω1 u 1 þ ω1 v 1 D ! D ! n 1 n 1 2 3 ðO Þ ! 2 !ðO2 Þ ! ! D D 6 ! n 2 n 2 7 2 6! 2! 7 _ ! _ 2 ðO R 2 ¼ω ! 2 Þ þ ω2 4 n 2 ; !ðO2 Þ 5 ¼ ω2 u 2 þ ω2 v 2 : D ! D ! n 2 n 2 Another vector rotator is defined by ðO2 Þ ! !! n 1; J n 2 ! ¼ 2ω1 ω2 ! u 12 : R 12 ¼ 2ω1 ω2 ðO2 Þ ! ! n 1; J ! n2
Fig. 1. Arbitrary position of rigid body coupled rotations around two no intersecting axes. System is with two degrees of mobility with φ1 as rheonomic and φ2 as generalized coordinates.
Corresponding derivative of linear momentum (1) is presented in the following form: ! ! !ðO2 Þ ! !ðO2 Þ d K ! h! ! i ! ¼ R 01 n 1 ; r 0 M þ R 011 S ! S þ R 022 n 2 n 1 dt ! !ðO2 Þ þ 2 ω1 ω2 n 1 ; S ! n
ð3Þ
2
From the structure of linear momentum derivative terms, it is visible that the expression contains terms with product of the pure ! ! kinematic vectors rotators R 01 (and R 0ii ) and corresponding mass moments vectors coupled for pole and axis. In vector expression (3) there are three vectors rotators introduced in the following form: 2 3 !ðO2 Þ !ðO2 Þ ! 7 S! S 6 ! ! ni 7 _ i n i þ ω2i 6 ð4Þ R 0ii ¼ ω 4 n i ; !ðO2 Þ 5; i ¼ 1; 2 !ðO2 Þ S! S! n i n i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ 2i þ ω4i Intensities of vectors which are in the form R0ii ¼ ω i ¼ 1; 2 depend on corresponding component angular velocity and angular accelerations. The first vector component is orthogonal to the corresponding axis of component rotation, and the second is in directions depending of mass linear moment vector coupled for corresponding axis of component rotation and for pole at selfrotation axis [14]. The derivative of angular momentum (2) is presented in the form ! h h ih i ih i d L ðO1 Þ ! ! ! ! ! ! ! ! ¼ M n 1 ; r 0 r 0 ; R 01 þM n 1 ; r 0 ρ C ; R 01 dt ðO Þ h ! 2 ! ! i !ðO2 Þ h! ! i r 0; R 1 þ S ! r 0; R 2 þ þ S ! n 1 n 1 h ! !ðO2 Þ 2Þ ! i ! ! !ðO 2ω1 ω2 ! ! _ þ n r ; R ω n ; S þ n 1 ; S ! 0 21 2 2 2 n n 2
2
The derivatives of angular momentum written in a form with pure kinematic vectors depending on angular velocities and angular accelerations of component angular velocities of coupled rotations in the form with vector rotators is very suitable for analyzing kinetic pressure. It is shown that different pure kinematical vectors rotators rotate around first and second axes of component axes of rotation [16–18].
4. Non-linear dynamics parameters of a rigid body coupled rotation around two orthogonal no intersecting axes and with one degree of freedom. By using theorems of changes of angular momentum with respect to time ! h i d L ðO1 Þ ! ! ! ! ! ¼ L ðO1 Þ þ ω L ðO1 Þ ¼ ∑ ρ i F i ð6Þ dt i vector equations of dynamic equilibrium of rigid body coupled multi-rotations about axes without intersection can be written in the following form: ðO1 Þ ðO1 Þ !ðO1 Þ !ðO1 Þ 2 ! !! 2 ! !! _2 J ! ω_ 1 J ! n 1 þ ω1 n 1 ; J n 1 þ ω n 2 þ ω2 n 2 ; J n 2
ðO1 Þ 1Þ !ðO1 Þ h! ! i ! !ðO ! !! þ ω1 ω2 n 1; J ! ; J n ; n þ n þ J ¼ 2 2 1 n2 n1 82 2 3 ! < h i h i> ! ! 6! _ 6! h r 0 i7 ! ! ¼ ∑ ρ i; F i n 2; r 0 4 ρ C ; ω 2 4 n 2 ; ! ! 5 > : i n 2; r 0 2 2 3339 > ! = r 0 6! 6! 777 ð7Þ þ ω22 4 n 2 ; 4 n 2 ; h! ! i555 M > ; n 2; r 0 In general case the equations are non-linear and coupled by generalized coordinates and their derivatives. For axes that are perpendicular some terms in equations are equal to zero but nonlinearity still remains. These differential equations are coupled by generalized coordinates and their derivatives, also. Expression (7) can be written in a form with vector rotators, similar with expression for derivative of linear momentum of rigid body coupled rotations (4) in a form 1 !n !ðO1 Þ ! _ ! !O! n 1 ω1 n 1 ; J n 1 þ R 10 D ! n 1
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103
ðO1 Þ !n !ðO1 Þ ! _ ! !! þ n 2ω þ R 20 D ! 2 n 2; J n n 2
2
h i
ðO1 Þ 1Þ ! !! ! !ðO ðO1 Þ ! ! þ n þ J n 1; J ! ; J n ; n þ ω1 ω2 2 2 1 n n 2
1
i h ih i ! ! ! ! ! ! ¼ ∑ ρ i ; F i n 2 ; r 0 ρ C ; R 01 M h
ð8Þ
i
It is visible that the expression contains terms with product of ! pure kinematics vectors rotators R i0 and corresponding mass moments vectors coupled for pole and axis. The vector rotators in previous vector expression are introduced and analyzed in [16–19]. They are expressed in the following form: 2 3 h i !ðO1 Þ !ðO1 Þ ! ! ! ! n i; ρ C S n S 6 7 ! ! n i i 7 _ i h _ i þ ω2i 6 i R i0 ¼ ω 4 n i ; !ðO1 Þ 5 ¼ ω ! ! !ðO1 Þ n i; ρ C S! S! n i n i 2 h i3 ! ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ; ρ i C 6! 7 ! _ 22 þ ω42 ; i ¼ 1; 2 ð9Þ þ ω2i 4 n i ; h! ! i5; R i0 ¼ ω n i; ρ C For that special case when axes are perpendicular, Eq. (8) is ! ! multiplied by unit vector n 1 and after that by unit vector n 2 . Two new equations are obtained and they are in a form with vector rotators as it is explained 2Þ !ðO2 Þ ! ! !ðO2 Þ ! !ðO ! ! ω_ 1 ! n 1; J ! n 1; J ! n ; þ R 2 ; n 1 D n þ 2ω1 ω2 n ; n1 1
2
!
! ! ! _2 ∑ MO F 1 ; n 1 ω
¼
i
þ 2ω1 ω2
2
2Þ ! ! !ðO2 Þ ! !ðO ! D n 2; J ! ; þ R ; n 1 2 n n 2
2Þ ! !ðO ! n 1; J ! ; n ¼ 2 n 2
! ! ! ∑ MO F 1 ; n 2
ð10Þ
i
u1n1
þ 2Meaω21 sin β cos φ2 :
ðO2 Þ ðO2 Þ ðO2 Þ are J ¼ Mr 2 =4, J ! ¼ Jð1 þ ε sin β Þ J ! ! ¼ 0 and J ! ! ¼ n2 u1n1 v1n1 2 Jð1 ε sin β Þ sin ϕ2 cos ϕ2 . So, Eq. (11) can be written in a form
φ€ 2 þ Ω2 ðλ cos φ2 Þ sin φ2 Ω2 ψ cos φ2 ¼ 0;
2 ðO2 Þ 2Þ 2Þ € 2 J ðO φ€ 2 J ðO ! þ ω1 J ! ! φ ! ! ¼ Mge sin β sin φ2
v1n1
products of inertia, the deviational moment of the rigid body for a ! ! ! couple of normal axis oriented by the unit vectors n 1 , v 1 and u 1 . It is considered an eccentric disc (eccentricity is e), with mass M and radius r, which is inclined to the axis of its own self-rotation by the angle β (Fig. 2). In this case mass inertia moment intensities 2
1
!
These two vector equations are equations of rigid body rotation around fixed point and special case of precesion with angular ! velocity ω 1 and self-angular velocity. A special case of the heavy rigid body with coupled rotations about two orthogonal no intersecting axes with one degree of freedom, and in the gravitation field is considered. This example is studied during some years of investigation [18]. For this case generalized coordinate φ2 is independent, and coordinate ϕ1 is programmed. In that case, it is said that coordinate φ1 is rheonomic coordinate and system is with kinematical excitation, programmed by forced support rotation by constant angular velocity. When the angular velocity of shaft support axis is _ 1 ¼ ω1 ¼ const; it is obvious that rheonomic coordinate constant, φ is linear function of time, φ1 ¼ ω1 t þ φ10 and angular acceleration _ 1 ¼ 0. around fixed axis is equal to zero ω Special case is when the support shaft axis is vertical and the gyro-rotor shaft axis is horizontal, and all time in horizontal plane, and when axes are without intersection with normal distance a between. The angle of self-rotation around moveable self-rotation ! axis oriented by the unit vector n 2 is φ2 whose angular velocity is ω2 ¼ φ_ 2 . Differential equation of the heavy gyro rotor-disk selfrotation of reviewed model in Fig. 2, for the case coupled rotations about two orthogonal axes by using (10) in the case when axes are orthogonal, the support shaft axis is vertical and the gyro-rotor shaft axis is horizontal, and when axes are without intersection ! (detail in [18]) after multiplying scalar by n 2 , and taking into account orthogonally between axes of coupled rotations, is obtained in the following form: n2
Fig. 2. Model of heavy gyro rotor with two component coupled rotations around orthogonal axes without intersections.
ð11Þ
ðO2 Þ ðO2 Þ 2Þ Here J ðO ! is axial mass inertia moment and J ! ! and J ! ! are n2 v1n1 u1n1
ð12Þ
where corresponding coefficients, in case when rigid body is disk, are e 2 ε sin 2 β 1 g ðε 1Þ sin β 2 2 ; ε ¼ 1 þ 4 ; Ω ¼ ω1 ; λ¼ 2 r ε sin β þ 1 eω21 ε sin 2 β 1 2ea sin β : ψ¼ er ε sin 2 β 1
ð13Þ
This differential equation is quite the same in the form as differential equation given in [20] obtained for heavy material point rotating around circle. So, dynamics of a rotor defined in this way can be studied by applying of a differential equation which is used for describing the dynamics of a heavy material point on the circle with coupled rotations. Thus it is confirmed that the different physical problems can be described with the same differential equations and analyzed on similar way. The derived equation is a non-linear one. There is not a general method to solve non-linear equations like the one that is employed to obtain the motions of the several types of harmonic oscillators. In general, the motion of the oscillator in phase space has to be obtained by means of one or more of the numerical methods that have been developed to solve differential equations. Although there are analytical tools to deal with non-linear systems, computational methods play an important role in the study of non-linear dynamical systems. Many non-linear effects, especially in control systems, are best approximated by segmented characteristics. Effects of this type of non-linearity on the system response are analyzed using the phase plane. The motion in each segmented region is describing with a linear differential equation. For examining non-linear dynamics of rotor, it is necessary to investigate its properties, phase portraits, structures of homoclinic orbits and its bifurcations.
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For the non-linear system that can be written in theory well known form that is x€ þ f ðxÞ ¼ 0, upon integrating the following R form can be obtained ð1=2Þx_ 2 þF ðxÞ ¼ h; F ðxÞ ¼ f ðxÞdx. This mathematical equation has been obtained by many researches in different areas of investigations. For mathematicians it is only second kind differential equation with function f ðxÞ which may be linear or non-linear. There are many solutions depending on expression f ðxÞ. In most university books variable x is displacement, but it may be angle, or electrical current, or flux, or share. It is the interaction between different processes and phenomenology [1]. For the example here analyzed the differential equation (12) can be written in previous mentioned form. For relative coordinate ϕ2 that form is Z
_ 22 þ F φ2 ¼ h; F φ2 ¼ ð1=2Þφ ð14Þ f φ2 dφ2 : Here is another example of the phenomenon that was written by M. Petrovic. For analyzing this equation that may be non-linear it is not necessary to go into nature of terms. But, in physics, this represents that the sum of the kinetic energy and potential energy, the total energy, is constant for an idealized system with unit mass and with one degree of motion. Kinetic energy of thus defined _ 22 =2, while potential energy is determined by the system is φ
expression F φ2 . The system analyzed in this paper is rheonomic. The expression which, with analogy, can be here named as potential energy is in relation to the relative coordinate. It is potential of conservative system which is equivalent to the real rheonomic system.
Ω2 β ; ε; ω2
cos 2 φ2 2λ β; ε; e; ω2 Ep φ2 ¼ F φ2 ¼ 2
ð15Þ cos φ2 2ψ β; e; a; r sin φ2 1 The relative equilibrium positions can be determined from this expression. Also, from the above equation, the phase portrait or the trajectories for different energy level may be plotted and study qualitatively about the response of the system. The dynamical equilibrium positions of the gyro-rotor can be determined for ðλ cos φ2 Þ sin φ2 ψ cos φ2 ¼ 0. This equation has to be solved numerically as it is presented in [21] and all solutions depend on system parameters. In Fig. 3, transformation of a potential trajectory of the heavy gyro-rotor-disk with rotating axes that are without intersection for different values of disk inclination angle β to the axis of self _ 0 ¼ 0 rad=s is rotation and for initial conditions φ0 ¼ π =2 rad ; φ presented. For different angles of screw position of rigid body to the selfrotation axis the optimum values of potential energy changes. In Fig. 3 characteristic potential energy curves for different parameter values of the basis system correspond to the rigid body dynamic model are presented. It can be seen that for increasing the angle β of disk inclination to the proper shaft axis rotation some max of potential energy curves disappeared. Now by taking different energy level h, the relation between _ 2 and angular displacement φ2 may be the angular velocity φ finding as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
φ_ 2 ¼ 7 2 h F φ_ 2 , where F φ2 is determined by the
2 expression F φ2 ¼ ðΩ =2Þ cos 2 φ2 2λ cos φ2 2ψ sin φ2 1 , and total energy h is _ 220 Ω2 2λ cos φ20 þ sin 2 φ20 þ 2ψ sin φ20 : h¼φ By plotting the phase portrait it can be finding that the trajectory which clearly depicts that the motion corresponding to maximum potential energy is unstable and the bifurcation point
is of saddle-node type. The motion corresponding to the minimum potential energy is stable center type. In Fig. 4, transformation of a phase trajectory of the heavy gyrorotor-disk with rotating axes that are without intersection for different values of initial conditions and for disk inclination angle β to the axis of self-rotation β ¼ 0:01π rad is presented. Also potential well with its maximum and minimum corresponding to saddle point and center is presented. Total energy, as it can be seen, is depending not only on initial conditions but on parameters of rigid body: angle of screw position of rigid body to the self-rotation axis β, eccentricity e and normal distance between axes a. Thus, the following figure shows transformation of phase trajectory depending on the angle of inclination to the self-rotation axis β . It can be seen that there are various forms of trajectory that means that oscillations may be of various types. Also progressive, non-oscillatory movement is possible. Some equilibrium positions may be stable or unstable. With changing angle of inclination there is a possibility to make unstable equilibrium position stable. There are some experimental results that confirm this statement. In Fig. 5, transformation of a phase trajectory of the heavy gyrorotor-disk with rotating axes that are without intersection for
Fig. 3. Potential energy curves analog of the heavy gyro rotor with rotating axis that are without intersection for different values of the angle β of disk inclination to the proper shaft axis rotation and for initial conditions φ0 ¼ π and φ_ 0 ¼ 0.
Fig. 4. Potential well and phase portrait showing saddle point and center corresponding to maximum and to minimum potential energy.
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different values of disk inclination angle β to the axis of self- _ 0 ¼ π =12 rad=s rotation and for initial conditions φ0 ¼ π =4 rad ; φ is presented. It is possible to see appearance of the homoclinic orbits in the form number eight and coupled singularities. A closed homoclinic trajectory in the form of number eight and a trigger of coupled singularities composed of three singular points (one unstable saddle node and two stable centers) appears. It is possible by transformation of the phase trajectory to indicate that disk inclination angle β to the axis of self-rotation has bifurcation values at which homoclinic phase trajectory appears. In Fig. 6 transformations of a phase trajectory in a case when gyro rotor is rotating around axes without intersection with distance between axes is a ¼ 2 mm and for different initial conditions and different values of angle of inclination β are presented. Fig. 6a shows transformation of a phase trajectory in a case when gyro rotor is rotating around axes that are in intersection and when angle of inclination is β ¼ 0:218 π . In Fig. 6b transformation of a phase trajectory for the angle of inclination β ¼ π =3 is presented. This type of trajectories is quite different. Fig. 6c shows transformation of a phase trajectory for the case when gyro rotor is rotating around axes that are in intersection and when angle of inclination is β ¼ 0:1π . When angle of inclination is β ¼ 0:01π transformation of a phase trajectory is presented in Fig. 6d. The particular energy levels h are determined for different values of initial conditions, primarily by changing the initial
Fig. 5. Transformation of phase trajectory for different values of the angle β of screw positioned rigid body.
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angular velocity while the angle of inclination stayed generally the same. Of course, with increasing initial angular velocity the stability disrupt at low angles of inclination (Fig. 6b). For example, if angle of inclination is β ¼ π =3, the initial angular velocity of 1:54 s 1 produced progressive movement, while in a case when the angle of inclination is β ¼ 0:01π the initial angular velocity that produces progressive movement is 2:53 s 1 . Hence, for particular energy level h, the system will be under oscillation, if the potential energy Ep ðφ2 Þ is less than the total energy h.
5. Concluding remarks In this paper some results of the rigid body coupled rotations are presented. A numerical experiment with the use of derived analytical expressions and MathCAD program was used to create a visualization of phase portraits of non-linear dynamics of rigid body coupled rotations and layering of homoclinic orbits with respect to the system parameter of disk inclination angle β to the axis of self-rotation change different values. The phase plane used to study the effects of this type of non-linearity on the system response was one of the original contributions to the phase plane method in control, but it is very useful in engineering, too. The rigid body is the eccentrically, skew positioned disc on the shaft rotor support in the field without turbulent damping. In that case the transformation of the graphical presentation of the potential energy for different values of the angle of the disc inclination to the proper shaft axis rotation is graphically presented. Transformation of the phase trajectory for different values of the angle β of disk inclination to the proper shaft axis rotation is presented graphically also. Special attention is focused to the angle of inclination, as possible cause of the imbalance, as well as to the phase trajectory portrait for the case of the heavy gyrorotor disc coupled rotations about two axes without intersection. Mass moment vectors in our opinion can be new open way for applications in this area. Transformation of the phase trajectory of the rigid body non-linear dynamics with coupled rotations around no intersecting axes contains or not contains a trigger of coupled singularities. Relative non-linear dynamics of the heavy gyro-
Fig. 6. Transformation of a phase trajectory of the heavy gyro-rotor with rotating axis that is without intersection for different values of disk inclination angle β to the axis of self-rotation and for different initial conditions: ðaÞ β ¼ 0; 218π, ðbÞ β ¼ π=3, ðcÞ β ¼ 0; 1π, ðdÞ β ¼ 0; 01π Examples of the trigger of the coupled singularities and homoclinic trajectories in the form of the number eight.
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rotor-disk around self-rotation shaft axis is possible to present by means of the phase portrait method. Forms of phase trajectories and their transformations by changes of initial conditions, and for different cases of angle of inclination may present character of non-linear oscillations. The particular energy levels h which were determined for different values of initial conditions, primarily by changing the initial angular velocity confirms that statement. The example of rigid body coupled rotations once again proved the trigger of coupled singularities theorem and the existence and non-existence of homoclinic orbits in the form of number eight which is formulated and detailed analyzed in [14]. The theorem on the trigger of coupled singularities speaks of the conditions existence, or not, the coupled equilibrium position, and could speak on coupled dynamical states of relative rest. Small changes of initial conditions may change kinetic parameters of the system and cause the appearance or disappearance trigger of coupled singularities. Two physical phenomena, essentially different, can have the same mathematical model. If one of these phenomena is well known, the theory and analysis discussed for the former can be applied equally well to the latter. On the basis of phenomenological mapping and mathematical analogy, analysis for one of the system non-linear dynamics is possible to apply for qualitative analysis of non-linear phenomena appeared in dynamics of other disparate model or nature system, in Economy, in Ecology, dynamics of web cities in internet, in Chemical engineering, Cryptography, in Computer Science, Electronic and Communication Technologies, etc. Analogy between the expression of change of angular momentum with respect to time (6) which is used for describing the movement of a rigid body with a fixed point and the relation between the efforts in the cross section of a bar, having as axis a curve in the three-dimensional space and the exterior load on the bar unit length is noticed. This analogy includes analogies between ! the angular momentum L ðO1 Þ and the resultant of the axial and ! shearing effects on the transversal section R , and between the ! ! angular velocity vector ω and the Darboux vector Ω [22]. In [22] an analogy between two completely different physical phenomena: the rectilinear displacement in the relativistic frame under a constant force and the large deformations of a straight bar for a constant bending moment and constant rigidity is emphasized. The vector method, based on mass moment vectors coupled for pole and oriented axes defined by K. Hedrih, is used for investigation of the rigid body coupled rotations. The possibilities to establish the phenomenological analogy of different physical model dynamics expressed by vectors connected to the pole and the axis and the influence of such possibilities to applications allow researchers and scientists to obtain larger views within their specialization fields. The !ðOÞ analogy between the vector J ! n of the body mass inertia moment at ! the observed point for the axis oriented by the unit vector n and the !ðOÞ vector p ! of the total stress at a certain body point for the plain with n !ðOÞ ! the normal oriented by the unit vector n and the vector δ ! n of the total strain of the line element drawn from the observed point in the ! direction of the unit vector n is presented. The non-linear differential equation (12) putted under a common form of depending on parameters λ, Ω and ψ , (13), has been analyzed by K. Hedrih in numerous papers during investigations in non-linear dynamics of a heavy mass particle rolling along rotating circle about axis inclined to the vertical direction with constant angular velocity. Parameters are in the same form but they are of different nature. The obtained differential equation allows getting equations for many different cases of dynamical systems. There are analogies between non-linear phenomena in coupled rotor nonlinear dynamics, in non-linear dynamics of multiple step reductor, in non-linear dynamics of a car model and the non-linear dynamics of a heavy material particle along a circle with coupled rotations.
Mechanical systems, electrical systems, liquid-level systems, thermal systems can be presented and studied on the same way. The series of trigger of coupled singularities in the phase plane here presented are identified in the movement of heavy material point by rotating rough curve line. Examples of the trigger of the coupled singularities and coupled triggers of the coupled singularities for different parameter values of the basic system correspond to the rigid body dynamic model are equivalent to the corresponding one obtained in the analyzing non-linear dynamics of the car dynamic model [23]. The series of triggers of coupled singularities in the phase plane, and the trigger of coupled singularities are identified in non-linear electrical circuit, elliptical-ring plate loaded by concentrated forces. The theorem on the trigger of coupled singularities speaks of the conditions existence, or not, the coupled equilibrium position, and could speak on coupled dynamical states of relative rest. On the basis of the type of the trigger of singularities conclusion on the behavior of the dynamical processes in the society can be made. If there is one of homoclinic orbit in the form of number eight and trigger of singularities stable–unstable–stable stability of social dynamical system exists. If there is no homoclinic orbit in the form of number eight and there is only trigger of singularities unstable–stable– unstable there are great social changes [24]. Examples of coupled singularities can be found exactly in nonlinear dynamical systems disparate nature, for example, in technical, in biological, in economic and in social. It can be shown that mathematic used in explaining phenomena of an inverted pendulum [25] relates to other systems that are completely different but are fundamentally similar [26]. The dynamics of ecosystem, catalytic reactions, biochemical processes in the cells of living organism, economics, traffic, always show the same universal features. Because of some symmetry in living organisms' classic Lagrange equations are applied to simulate oscillation in amino-acid chains. The dynamics of different types of the system without its nature is always the same. Mechanical phenomena, physical phenomena, chemical, physiological and social, mutually disparate, are one and the same analytic problem, which consists in the integration, discussion and interpretation of some of the same equation which makes the corresponding phenomenon among them analytically equivalent. Analogy may lead under some circumstances to new discoveries. It is well known that an analogy between the corpuscular and the undulate features of light and the features of electron De Brogliy discovered theoretically the undulate features of the electron. Analyses in very different fields of science, from physics to medicine show although appearances have different causes, they appear in a similar way and they are analytically equivalent. The motion of a shear-building portal plane frame foundation that supports an unbalanced direct current motor with limited power supply, the longitudinal coupled vibrations in a flexible shaft with multiple flexible disks non-linear oscillation bearings of centrifugal pumps, elliptical-ring plate loaded by concentrated forces, nonlinear electrical circuit, the large deformations of a straight bar for a constant bending moment and constant rigidity, etc. – many phenomena, mutually disparate, have its analytically equivalency. All that plays a role in the concrete loses its specific meaning. Remains only the functions described in the most general possible way. One whatever process physics, chemical, physiological, geological, social entered into a dynamic stationary phase from the moment when the intensity of the process, measured by the size change of selected elements become unchangeable [1].
Acknowledgments Parts of this research were supported by the Ministry of Sciences and Technology of Republic of Serbia through
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Mathematical Institute SANU Belgrade Grant ON174001 Dynamics of hybrid systems with complex structures, Mechanics of materials and Faculty of Mechanical Engineering University of Niš and Faculty of Mechanical Engineering, University of Kragujevac. Special thanks to Professor Katica Hedrih for the unselfish help and constant encouragement in the work. References [1] M. Petrovic, Elementi matematičke fenomenologije (elements of mathematical phenomenology), Srpska kraljevska akademija, Beogr. (1911) str.789. [2] A.A. Andonov, A.A. Vitt, S.E. Haykin, Teoriya kolebaniy, Nauka, Moskva (1981) 568. [3] E. Butikov, Inertial rotation of a rigid body, Eur. J. Phys. 27 (2006) 913–922. [4] D. Rašković, Mehanika III – Dinamika (mechanic III – dynamics), Naučna (knjiga, 1972) 424. [5] J. Awrejcewicz, L.P. Dzyubak, 2-dof non-linear dynamic of a rotor suspended in the magneto-hydrodynamic field in the case of soft and rigid magnetic materials, Int. J. Non-linear Mech. 45 (2010) 919–930. [6] B.S. Wu, W.P. Sun, C.W. Lim, An analytical approximate technique for a class of strongly non-linear oscillators, Int. J. Non-linear Mech. 41 (2006) 766–774. [7] P. Felix, et al., On saturation of a non-ideal vibrating portal frame foundation type shear-building, J. Vib. Control 11 (2005) 121–136. [8] H.S. Jia, S.B. Chun, C.W. Lee, Evaluation of the longitudinal coupled vibrations in rotating; flexible disks/spindle systems, J. Sound Vib. 208 (2) (1997) 157–187. [9] K. Hedrih (Stevanović), Some vectorial interpretations of the kinetic parameters of solid material lines, ZAMM. Angew. Math. Mech. 73 (4–5) (1993) T153–T156. [10] Hedrih Stevanović, The mass moment vectors at n-dimensional coordinate system, Tensor, Japan 54 (1993) 83–87. [11] K. Hedrih Stevanović, Vector method of the heavy rotor kinetic parameter analysis and nonlinear dynamics (YU ISBN 86-7181-046-1), Monograph, University of Niš (2001) 252 (in English). [12] K. Hedrih Stevanović, Vectors of the body mass moments, monograph paper, topics from mathematics and mechanics, mathematical institute SANU, Belgrade, Proceedings, vol. 8, 199845–104 (Published in 1999. (In English), (Zentralblatt Review). [13] K. Hedrih (Stevanović), Analogy between models of stress state, strain state and state of moment inertia mass of body, Facta Universitatis, Series Mechanics, Automatic Control and Robotics, Vol 1, Nis (1991) 105–120.
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