- Email: [email protected]
Energy–vector method in mechanical oscillations
Chaos, Solitons and Fractals 39 (2009) 1684–1697 www.elsevier.com/locate/chaos
Energy–vector method in mechanical oscillations Artur Dabrowski Division of Dynamics, Technical University of Lodz, ul. Stefanowskiego 1/15 Lodz, Poland Accepted 18 June 2007 Communicated by Prof. G. Iovane
Abstract The new conception of the space, and its construction are introduced. The space allows for a geometrical view on energy changes in mechanical vibrating systems. It posses all the possibilities of the system dynamics analysis that are used in the traditional phase space but it also shows an amount of the energy accumulated in the system, an energy changes, flow, dissipation, synchronization, an energy attractors. Thus it increases our knowledge and intuition on energy changes in vibrating systems. Dynamics of the chaotic system is studied using the energy space. New types of maps are introduced. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction An energy flow modelling still arouses interests in the scientific world. Different methods are applied to solve problems connected with energy flow: Statistical Energy Analysis [1, 2, 3], Finite Element Method [4, 5]. But these methods do not allow for a special kind of a geometrical view on energy changes, which could develop our intuitional knowledge on energy flow phenomenon. This intuition is very important especially in modelling the systems, where we still can not measure energy flow, such as bioenergy, observed and applied in medicine diagnostics thanks to GDV method based on Kirlian effect [6, 7, 8], telepathy – the problem of an energy flow between two bioelectromagnetic systems, and so on. Nowadays one of the most important tools used in dynamical systems investigations is the phase space. Many aspects of the systems dynamics can be concluded using this space, but it also allows for the intuitional, geometrical view on an existing dynamical phenomena. The new conception of the space is proposed in [9, 11, 12]. It allows for a geometrical view on energy changes in mechanical vibrating systems. This space posses all the advantages of the phase space but it also shows an amount of the energy accumulated in the system, an energy changes, flow, dissipation, synchronization, an energy attractors. Thus it increases our knowledge and intuition on energy changes in vibrating systems. In this article an application of the energy space in the mechanical vibrating system dynamics investigations is shown. Some aspects of motion, energy flow, synchronization are considered, new types of maps are introduced. 2. Introduction to the energy–vector analysis – Energy flow and synchronization in the resonance state The energy space is constructed in a way, that the trajectory of the system, which accumulates a constant amount of energy, lays on a surface of the multidimensional sphere. The radius of this sphere is equal to the square root of the total E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.06.096
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energy accumulated in the system. From the image of the trajectory projection onto suitably chosen energy planes, one can see some aspects of an energy flow. The radius of this projection shows the total energy accumulated in considered part of the system. Moreover, one can conclude about all aspects of the motion, like from the phase plane. It is possible thanks to the fact that the kinetic energy is the function of velocity and the potential one – the function of displacement. Consider the vibrating system with the physical model shown in Fig. 1. The system consists of two oscillators. The external harmonic force excites the oscillator l. This oscillator is joined with the second oscillator l1. The mathematical model of the system is given by four differential equations of the first order: 8 x_ ¼ y; > > > < y_ ¼ ðF sin gs cy c1 ðy y Þ rx r1 ðx x1 ÞÞ 1 ; 1 l ð1Þ _ x ¼ y ; > 1 1 > > : y_ 1 ¼ ðc1 ðy 1 yÞ r1 ðx1 xÞÞ l11 ; where l, l1 is the masses of the oscillators, r, r1 is the stiffness coefficients of the springs, c, c1 is the damping coefficients, F is the amplitude of the external excitation force, x is the frequency of the external excitation force, rffiffiffi x r g¼ ; a¼ : a l The phase vector in the standard phase space R4 is represented by four components: x; y; x1 ; y 1 ; x2 ; y 2 : Transform the phase space of the system using the given function f: R4 ! R4: 8 pffiffir x ¼ ze ; > > > plffiffi2 > < y ¼ ye; 2 f ðx; y; x1 ; y 1 Þ ¼ prffiffiffi ffi > > 21 ðx1 xÞ ¼ z1e ; > > : plffiffiffi1ffi y ¼ y 1e : 2 1
ð2Þ
After this transformation the mathematical model of the system in the energy space is given by the following differential equations:
Fig. 1. The physical model of the system.
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qffiffi 8 > z_ e ¼ rly e ; > > > > qffiffiffiffi qffiffi > > 1 > y rlze þ rl1 z1e þ Fl sin gs; < y_ e ¼ lc y e cl1 y e þ pcffiffiffiffiffi ll1 1e qffiffiffiffi qffiffiffiffi > > z_ 1e ¼ rl11 y 1e rl1 y e ; > > > > qffiffiffiffi > > : y_ 1e ¼ r1 z1e c1 y þ pcffiffiffiffiffi 1 y: l1
l1
1e
ll1
ð3Þ
e
To simplify the description of the transformation note, that in the energy space there are square roots of the potential or kinetic energies on each axis. Note also, that function f transforms energy into vector form. 2.1. Energy flow and synchronization Some special advantages brings consideration of the energy planes. These subspaces are the equivalents of the phase planes of the phase space. In our case there are four interesting planes. The first one (ze : ye) – the energy plane of the oscillator l system. This plane is obtained after transformation of the plane (x : y). This transformation is linear. One can find eigenvectors of the transformation as orthogonal basis vectors [1, 0] and [0, 1], of the space (x : y). Eigenvalues of the transformation are equal: rffiffiffi rffiffiffi r l k1 ¼ ; k2 ¼ respectively: 2 2 Thus depending on r and l the phase space is just only squeezed or stretched in directions of the eigenvectors. In Fig. 2a plane (ze : ye) can be seen. The norm of the vector projection on that plane shows the energy accumulated in the oscillator l system. The value of ze shows the potential energy of the spring r, the value of ye – the kinetic energy of the mass l, and by means of these coordinates these energies can be calculated. One can see the changes from the potential energy into the kinetic one and vice versa. Note that total amount of the energy accumulated in the oscillator l system is not constant. There exists an energy flow between the oscillators l and l1. This flow will be analysed further. The second interesting plane is (z1e : y1e) – the energy plane of the oscillator l1 system. There exist transformations of two kinds which were made on the phase plane (x1 : y1) to obtain this energy plane. The first one: instead of the state variable x1 we have the spring r1 deflection: z1 = x1 x. The second one is similar to the considered transformation of the plane (x : y): transformation is linear and one can find eigenvectors of the transformation as orthogonal basis vectors [1, 0] and [0, 1]. Eigenvalues of the transformation are equal: rffiffiffiffiffi rffiffiffiffiffi r1 l1 ; k2 ¼ respectively: k1 ¼ 2 2 Thus in time of this part of transformation, depending on r1 and l1, the phase space is just only squeezed or stretched in directions of the given eigenvectors. In Fig. 2b plane (z1e : y1e) can be seen. The projection of the vector on that energy plane shows the energy accumulated in the oscillator l1 system. The value of z1e shows the potential energy of the spring r1 and the value of y1e – the kinetic energy of the mass l1, and by means of these coordinates the energies can be calculated. The position of the vector projections on the energy planes (ze : ye) and (z1e : y1e) at the same moment of time is marked by a small circle. See that at the same moment the spring r and spring r1 have the extremes of the potential energy, and the same concerns kinetic energies. Note also that the maximum of the potential energy of the oscillator l system equals the maximum of the kinetic energy of the oscillator l1 system and vice versa. The energy flow between the main mass l and the dynamical absorber l1 can be seen if you compare Fig. 1c and d, which show the combined energy planes. Note, that these trajectory projections are regular circles. It means, that energy accumulated in that parts of the system is constant. It shows some interesting aspects of the continuous energy flow between oscillator l and l1. All the energy, that flows out the oscillator l is intercepted by oscillator l1, and vice versa. The potential energy of the spring r transforms into kinetic energy of the oscillator l, and vice versa (Fig. 2c). The same situation takes place with the potential energy of the spring r1 and the kinetic energy of the oscillator l (Fig. 2d). This energy flow synchronization manifests also the next circles, that can be seen in Fig. 2e and f, where: dze dye dz1e
is the difference of the dze in ds interval, is the difference of the dye in ds interval, is the difference of the dz1e in ds interval,
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c
d
e
f
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Fig. 2. (a) Energy plane of the oscillator l system; (b) Energy plane of the oscillator l1 system; (c) and (d) Combined energy planes; (e) and (f) Differences of the variables ze and ye in ds intervals n = 0.85, l = 1 kg, l1 = 0.1 kg, r = 100 N/m, r1 = 100 N/m, c = 2 Ns/m, c1 = 0.001 Ns/m, F = 10 N.
dy1e ds
is the difference of the dy1e in ds interval, is the 1/1000 of the excitation force period.
Note that these circles are the consequences of circles shown in Fig. 2c and d, but also constant frequency of the vector rotation. From the whole considered figures one can conclude, that radius of the trajectory is constant, what means, that the trajectory lays on a surface of the four-dimensional sphere. The radius of this sphere is equal to the square root of the energy accumulated in the whole system. Note that it means constant amount of energy accumulated in the system.
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Energy flows only between the l and l1 systems and the excitation energy is dissipated in the dampers c and c1.This energy flow synchronization is possible just only for specially selected parameters. In our case the excitation frequency is equal to the resonance frequency of the system.
3. Energy flow in the chaotic systems Consider the system shown in Fig. 1 with nonlinear spring r. The mathematical model of the system is given by four differential equations of the first order: 8 x_ ¼ y; > > > < y_ ¼ ðF sin gs cy c1 ðy y Þ rx3 r1 ðx x1 ÞÞ 1 ; 1 l ð4Þ x_ 1 ¼ y 1 ; > > > : y_ 1 ¼ ðc1 ðy y 1 Þ þ r1 ðx x1 ÞÞ l11 : Because of nonlinearity of the spring r the transformation of the phase space, to obtain the energy space, in direction of x is nonlinear. In the considered case we have to show the transformation of the space, as function f: R4 ! R4, that transforms the phase space the following way: 8 prffiffi 2 x signðxÞ ¼ ze ; > 2 > > > < plffiffiy ¼ y ; e 2 f ðx; y; x1 ; y 1 Þ ¼ prffiffiffi ð5Þ ffi 1 > ðx xÞ ¼ z1e ; > 1 2 > > : plffiffiffi1ffi y ¼ y 1e : 2 1 To simplify the description of the transformation note, that in the energy space there are square roots of the potential or kinetic energies on each axis. Note also, that function f transforms energy into vector form. Similarly to the model given in paragraph 2 one can obtain the mathematical model of the system after this transformation, but for the nonlinear system it is easier to integrate the mathematical model (4) and afterwards use the function f to calculate vector coordinates in the energy space. Both methods were applied to examine the system dynamics, and the same results were obtained. There exists just only one unsolved problem with the critical point for the transformed mathematical model, thus these results are not presented.
Fig. 3. Bifurcation diagram of z1 (in grey, left); The largest Lyapunov exponent k max (in black, right); l = 1 kg, l1 = 0.07 kg, r = 1 N/m, r1 = 0.07 N/m, c = 0.1 Ns/m, c1 = 0.1 Ns/m, F = 2.5 N.
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Dependence of the system dynamics on the excitation force frequency is shown in the Fig. 3. The bifurcation diagrams of z1 variable in grey, and the largest Lyapunov exponent kmax in black can be seen in it. The value of largest Lyapunov exponent was estimated using Stefanski method. It is based on the phenomenon of synchronization of identical systems [13]. Different types of dynamics and bifurcations leading to the chaotic state can be identified in Fig. 3. Chaotic energy flow for g = 0.6025 is shown in Fig. 4a and b. Intersection of the attractor using Poincare plane was applied. For comparison the same intersection of the attractor in the phase space was shown in Fig. 1c and d. One can see the same type of dynamics in these charts. What can not be seen in the phase space is, that the oscillator l accumulates much more energy than oscillator l1. For the same system parameters new kind of maps is presented in Fig. 5. The system energy state on two energy levels can be seen in it. From the geometrical viewpoint it shows projections of the special attractor intersection. Attractor is cut using the sphere with the radius R matched up with the concerned energy level. Intersections by small Fig. 5a and b and big Fig. 5c and d spheres are presented. The scales of the charts are matched up with the intersection radius R. From these figures one can conclude, that the attractor is flat in directions of z3 and z4. Especially the intersection by the big sphere shows that the attractor cuts the sphere in the ‘‘equator’’ plane, what is the reason of the circles, that can be seen in Fig. 5a and d. 3.1. Energy flow In Fig. 6 one can see the resonance diagram for the selected system parameters. Energy flow for the special point (g = 1) was analysed. From Fig. 5 one can see, that for g = 1 energy accumulated in the spring r is close to zero. It means, that whole energy accumulated in the oscillator l system is close to zero, although this part of the system is forced by external force. It is the case, when the oscillator l1 works as the dynamical damper, that is for the excitation frequency equal to the system l1 free vibrations frequency. Energy flow for the considered case is shown in Fig. 7. In Fig. 7a and b one can see energy planes of the oscillators l and l1, respectively. Note, that difference between these
ye
0
y1e
z1e
c
d
Fig. 4. (a) Energy plane of the oscillator l system; (b) Energy plane of the oscillator l1 system; (c) Phase plane of the oscillator l system; (d) Phase plane of the oscillator l1 system; l = 1 kg, l1 = 0.07 kg, r = 1 N/m, r1 = 0.07 N/m, c = 0.1 Ns/m, c1 = 0.1 Ns/m, F = 2.5 N.
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1/2
y1e
[J ]
-0.3 -0.3
0.3
z1e
c
d
1/2
[J ]
y1e
0
-1.6
z1e Fig. 5. Sphere cut radius: (a) and (b) R = 0.3 [J1/2], (c) and (d) R = 0.3 [J1/2]; (a) Energy plane of the oscillator l system; (b) Energy plane of the oscillator l1 system; (c) Phase plane of the oscillator l system; (d) Phase plane of the oscillator l1 system; l = 1 kg, l1 = 0.07 kg, r = 1 N/m, r1 = 0.07 N/m, c = 0.1 Ns/m, c1 = 0.1 Ns/m, F = 2.5 N.
η Fig. 6. Resonance diagram of z1; l = 1 kg, l1 = 0.07 kg, r = 1 N/m, r1 = 0.07 N/m, c = 0.1 Ns/m, c1 = 0.0001 Ns/m, F = 2.5 N.
oscillators is one time of magnitude order. Note also from the marked small circles, that system l1 oscillates in antiphase to the system l. External force, that excites oscillator l is in the same phase as this oscillator. Thus all the energy of the excitation just only flows through the l system and is intercepted by l1 system. What is interesting, the energy of
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Fig. 7. (a) The energy plane of the oscillator l system; (b) Energy plane of the oscillator l1 system; (c) and (d) Differences of the variables in ds intervals; (e) Radius of the attractor. g = 1, l = 1 kg, l1 = 0.07 kg, r = 1 N/m, r1 = 0.07 N/m, c = 0.1 Ns/m, c1 = 0.0001 Ns/m, F = 2.5 N.
the l1 system is constant, what can be seen in Fig. 7b and d. The energy accumulated in the system flows only between the masses l and l1 systems and the excitation energy is dissipated in the dampers c and c1. The energy of the whole system is almost constant, what can be seen in Fig. 7f. 3.2. Application of the energy–vector analysis in the system with impacts Consider the system shown in Fig. 8. For some ranges of the system parameters, it works as an impact damper of the motion of the oscillator l (Refs. [9, 10]). The system consists of three oscillators. The external harmonic force excites the oscillator l. It is joined with the classical dynamical absorber l1. This absorber is allowed to collide with the third oscillator l2. In the periods between the impacts the mathematical model of the system is given by six differential equations of the first order:
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Fig. 8. The physical model of the system.
8 x_ ¼ y; > > > > > y_ ¼ ðF sin gs cy c1 ðy y 1 Þ rx r1 ðx x1 ÞÞ l1 ; > > > > < x_ 1 ¼ y ; 1 1 _ y ¼ ðc > 1 1 ðy 1 yÞ r1 ðx1 xÞÞ l ; > 1 > > > > x_ 2 ¼ y 2 ; > > > : y_ ¼ ðc y r x Þ 1 ; 2 2 2 2 2 l2
ð6Þ
where l, l1, l2 is the masses, r, r1, r2 is the stiffness coefficients of the springs, c, c1, c2 is the damping coefficients, F is the amplitude of the external excitation force, x is the frequency of the external excitation force, rffiffiffi x r : g ¼ ; s ¼ at; a ¼ a l
ð7Þ
The impact between the dynamical and impact absorbers is put into the mathematical model over the restitution coefficient r. The phase vector in the standard phase space R6 is represented by six components: ð8Þ
x; y; x1 ; y 1 ; x2 ; y 2 : Transform the phase space as follows: 1. Instead of the displacement coordinates x, x1, x2, take the deflections of the springs. z ¼ x;
z1 ¼ x1 x;
z2 ¼ x2 :
ð9Þ
2. Depending on the coefficients ri and li squeeze and stretch the space in directions of zi and yi. In these two steps a new energy space can be obtained. In the first step, one obtains a new space V with the basis E: E ¼ ðez ; ey ; e1z ; e1y ; e2z ; e2y Þ;
ð10Þ
where: ex ¼ ½1; 0; 0; 0; 0; 0T ; e1x ¼ ½0; 0; 1; 0; 0; 0T ; T
e2x ¼ ½0; 0; 0; 0; 1; 0 ;
ey ¼ ½0; 1; 0; 0; 0; 0T ; e1y ¼ ½0; 0; 0; 1; 0; 0T ; T
e2y ¼ ½0; 0; 0; 0; 0; 1 :
ð11Þ
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Change the basis vectors of the space V. Then, the new energy basis EN of this space is: EN ¼ ðbz ; by ; b1z ; b1y ; b2z ; b2y Þ;
ð12Þ
where: hpffiffiffi iT h pffiffiffi iT 1 1 r ; 0; 0; 0; 0; 0 ; by ¼ 0; l ; 0; 0; 0; 0 ; h iT h iT pffiffiffiffiffi1 pffiffiffiffiffi 1 b1z ¼ 0; 0; ð r1 Þ ; 0; 0; 0 ; b1y ¼ 0; 0; 0; l1 ; 0; 0 ; h h pffiffiffiffiffi1 iT pffiffiffiffiffi 1 iT b2z ¼ 0; 0; 0; 0; ð r2 Þ ; 0 ; b2y ¼ 0; 0; 0; 0; 0; l2 : bz ¼
The transition matrix AEN E from the basis E to the basis EN of the energy space takes the form: 3 2 pffiffiffi r 0 0 0 0 0 6 0 pffiffiffi l 0 0 0 0 7 7 6 7 6 pffiffiffiffiffi 6 0 r1 0 0 0 0 7 7: AEN E ¼ 6 pffiffiffiffiffi 6 0 0 0 7 0 0 l1 7 6 7 6 pffiffiffiffiffi 4 0 0 0 0 r2 0 5 pffiffiffiffiffi 0 0 0 0 0 l2
ð13Þ
ð14Þ
The coordinates of the vector ve ¼ ½ze ; y e ; z1e ; y 1e ; z2e ; y 2e TEN with respect to the energy basis EN can be obtained from the vector v with respect to the basis E, using the transition matrix: ve ¼ AEN
E v:
ð15Þ
Then ve ¼
pffiffiffi pffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi T rz; ly; r1 z1 ; l1 y 1 ; r2 z2 ; l2 y 2 EN :
The norm of the vector ve in the energy space with the energy product is as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi 2 pffiffiffiffiffi 2 pffiffiffiffiffi 2 pffiffiffiffiffi 2 i 1 hpffiffiffi 2 pffiffiffi 2 rðzÞ þ ly þ ð r1 z1 Þ þ l1 y 1 þ ð r2 z2 Þ þ l2 y 2 jve j ¼ hve ; ve i ¼ 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rz2 ly 2 r1 z21 l1 y 21 r2 z22 l2 y 22 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ þ þ þ þ þ ¼ Ep þ Ek þ Ep1 þ Ek1 þ Ep2 þ Ek2 ; 2 2 2 2 2 2
ð16Þ
ð17Þ
where: Ep is the potential energy accumulated in the spring r, Ek is the kinetic energy of the mass l, Ep1 is the potential energy accumulated in the spring r1, Ek1 is the kinetic energy of the mass l1, Ep2 is the potential energy accumulated in the spring r2, Ek2 is the kinetic energy of the mass l2. The mathematical model of the system in the transformed space is given by following differential equations: 8 qffiffi > z_ e ¼ rly e ; > > > > qffiffiffiffi qffiffi > > > 1 > y_ e ¼ lc y e cl1 y e þ pcffiffiffiffiffi y rlze þ rl1 z1e þ Fl sin gs; > ll1 1e > > > qffiffiffiffi qffiffiffiffi > > > < z_ 1e ¼ rl11 y 1e rl1 y e ; qffiffiffiffi > 1 > y; y_ 1e ¼ rl11 z1e lc11 y 1e þ pcffiffiffiffiffi > ll1 e > > > ffiffiffi ffi q > > > > z_ 2e ¼ rl22 y 2e ; > > > > qffiffiffiffi > > : y_ 2e ¼ r2 z2e c2 y 2e ; l2 l2
ð18Þ
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where: pffiffiffiffiffi ze ¼ signðxÞ Ep ; pffiffiffiffiffi y e ¼ signðyÞ Ek ; pffiffiffiffiffiffiffi z1e ¼ signðx1 Þ Ep1 ; pffiffiffiffiffiffiffi y 1e ¼ signðy 1 Þ Ek1 ; pffiffiffiffiffiffiffi z2e ¼ signðx2 Þ Ep2 ; pffiffiffiffiffiffiffi y 2e ¼ signðy 2 Þ Ek2 :
ð19Þ
3.3. Energy flow and dissipation during the impact – Impact map Let p be the plane determined by the basis vectors (Fig. 9): h iT pffiffiffiffiffi1 b1y ¼ 0; 0; 0; l1 ; 0; 0 ; h pffiffiffiffiffi1 iT b2y ¼ 0; 0; 0; 0; 0; l2 :
ð20Þ ð21Þ
The vectors y1e and y2e correspond to the kinetic energies of the masses l1 and l2, respectively. The position of the vector is marked on this plane just before and after impact. Thus, this plane is a special kind of the impact map. In order to make this map clearer, note that it represents only the points for different types of impact. As the choice criterion, y 1e < y 2e has been applied. The consideration of the position and norm of the vector on this map allows one to conclude about the energy flow between the dynamical and impact absorbers and also about the energy dissipation during each collision. The energy dissipation is included into the mathematical model of the system over the restitution coefficient r. Let vep be the projection of the ve on the plane p just before the impact, and v0ep after it.
y2=-5*y 1/3 -
-
Fig. 9. The impact map. n = 1.165, l = 1 kg, l1 = l2 = 0.1 kg, r = 1 N/m, r1 = 0.1 N/m, r2 = 0.1 N/m, c = 0.04 Ns/m, c1 = 0.01 Ns/ m, c2 = 0.02 Ns/m, F = 0.002 N, d = 0.0216 m, r = 0.5.
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The transformation of the vector vep during each impact is given by the matrix AEN : v0ep ¼ AEN vep ; where:
" AEN ¼
l¼
lr lþ1
1þr lþ1
lþlr lþ1
1lr lþ1
# ;
ð22Þ
l1 ; l2
r is the restitution coefficient. In the case l = 1 the eigenvalues of the AEN matrix are: k1 ¼ r;
k2 ¼ 1
ð23Þ
and the eigenvectors are: w1 ¼ ½1; 1T ;
w2 ¼ ½1; 1T ;
ð24Þ
respectively. The directions of the eigenvectors are shown in Fig. 9. Note that during impact the vector vep is transformed only in the direction given by the eigenvector w1. The energy dissipation in time of each collision can be found from the change of the norm of the energy space vector: jvep j jv0ep j:
ð25Þ
The maximum dissipation of the energy takes place when the vector vep has the same direction as the eigenvector w1. Then v0ep ¼ k1 vep : Taking in the consideration that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jvep j ¼ EK 1 þ EK 2 ;
ð26Þ
ð27Þ
where: EK 1 is the kinetic energy of the mass l1 before impact, EK 2 is the kinetic energy of the mass l2 before impact, and jv0ep j ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E0K 1 þ E0K 2 ;
ð28Þ
where: E0K 1 is the kinetic energy of the mass l1 after impact, E0K 2 is the kinetic energy of the mass l2 after impact, one can find easily that the energy relation after and before impact assumes the form: E0K 1 þ E0K 2 ¼ k21 ¼ r2 : EK 1 þ E K 2
ð29Þ
The closer the direction of the vector vep is to the second eigenvector w2, the less energy dissipation occurs. In the case when the directions of vep and w2 are almost the same, there is almost no energy dissipation during the collision. The velocities of the oscillators l1 and l2 are almost equal then, and in practice we do not know if impact occurs or not. It is so called grazing collision and it causes chaotic motion of the system. These special points can be seen in Fig. 9 as common points of the before and after impact attractors. The transformation matrix AEN allows one also to divide the impact map into two kinds of fields: the first one for the case when the energy flows during impact from the dynamical to impact absorber and the second when the energy flows in the opposite direction.
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Consider the matrix AEN in the case l1 = l2: " # 1r 2 1þr 2
AEN ¼
1þr 2 1r 2
ð30Þ
:
Let: vep ¼ ½v1 ; v2 T
and
v0ep ¼ ½v01 ; v02 T :
ð31Þ
Then v0ep ¼ AEN vep ; so
v01
"
v02
¼
1r 2 1þr 2
ð32Þ
1þr 2 1r 2
#
v1 : v2
ð33Þ
The energy flow from the dynamical to impact absorber is given by the condition: v1 6 v01 6 v1 :
ð34Þ
Consider: 1. v01 ¼ v1 The condition is satisfied when the directions of vep and the eigenvector w2 are the same. 2. v01 ¼ v1 then v1 ¼
1r 1þr v1 þ v2 2 2
and we obtain v2 ¼
3r v1 : 1þr
ð35Þ
For the case shown in Fig. 4 r = 0.5 and then 5 v2 ¼ v1 : 3
ð36Þ
As a result, the energy flow from the dynamical to the impact absorber occurs:if v1 > 0 5 v1 6 v2 6 v1 ; 3
ð37Þ
if v1 < 0 5 v1 6 v2 6 v1 : 3
ð38Þ
The field of energy flow in this direction is marked in Fig. 9 by the grey colour. Note that it is just for y1e< y2e, which was the criterion of the impact choice. It can be seen that during impacts in the considered case energy flows in both directions, but definitely more often from the dynamical to the impact absorber. 4. Conclusions The way of transformation of the phase space to obtain the energy space has been shown. It has been proved that this new kind of space allows for concluding about the energy state of a vibrating system. The norm of the vector in that space is equal to the square root of the total energy accumulated in the system. The projection of the vector space on energy subspaces show the amount of the energy that accumulates in some parts of the system. It has been shown that using this kind of spaces, all aspects of the kind of motion can be concluded about, like from the phase space and, moreover, the energy state, accumulation, flow and dissipation can be observed. Different types of the energy flow synchronization were shown. New kind of maps was introduced. It was shown, that the energy space allows for a new, geometrical view on energy changes in vibrating systems.
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References [1] Maidanik G, Becker KJ. Dependence of the induced loss factor on the coupling forms and coupling strengths: Linear analysis. J Sound Vib 2003;266:15–32. [2] Kishimoto Y, Bernstein DS, Hall SR. Energy flow modeling of interconnected structures: A deterministic foundation for statistical energy analysis. J Sound Vib 1995;183(3):407–45. [3] Mace BR. Power flow between two continuous one-dimensional subsystems: A wave solution. J Sound Vib 1992;154:289–320. [4] Xu HD, Lee HP, Lu C. Numerical study on energy transmission for rotating hard disk systems by structural intensity technique. Int J Tech Sci 2004;46:639–52. [5] Jang HH, Lee SH. Free vibration analysis of a spinning flexible disk-spindle system supported by ball bearing and flexible shaft using the FEM and substructure synthesis. J Sound Vib 2002;251:59–78. [6] Opalinski J. Kirlian type images and the transport of thin-film materials in high voltage corona discharges. J Appl Phys 1979;50(1):498–504. [7] Bankovskiv NG, Korotkov KG, Petrov NN. Physical image-forming processes in gas-discharge visualisation (the Kirlian effect review). Sov J Commun Technol 1986;31:29–45. [8] Korotkov KG. Human Energy Field: Study with GDV Bioelectrography’’. USA: Backbone Publishing Company; 2002. [9] Da˛browski A, Kapitaniak T. Using chaos to reduce oscillations. Nonlin Phenom Complex Syst 2001;4(2):206–11. [10] Dabrovski A. New design of the impact damper. Mech Mech Eng 2000;4(2):191–6. [11] Dabrovski A. The construction of the energy space. Chaos, Solitons & Fractals 2005;26:1277–92. [12] Da˛browski A. The energy space, energy flow, and synchronization. Mech Mech Eng 2000;8(1):191–6. [13] Stefanski A, Dabrowski A, Kapitaniak T. Evaluation of the largest Lyapunov exponent in dynamical systems with time delay. Chaos, Solitons & Fractals 2005;23:1651–9.
Further reading [14] Griffel DH. Linear algebra and its applications vol. 2 more advanced. 1989. Ellis Horwood Limited. [15] Kapitaniak T. Chaos for engineers. Berlin Heidelberg: Springer-Verlag; 1998.
Energy–vector method in mechanical oscillations Artur Dabrowski Division of Dynamics, Technical University of Lodz, ul. Stefanowskiego 1/15 Lodz, Poland Accepted 18 June 2007 Communicated by Prof. G. Iovane
Abstract The new conception of the space, and its construction are introduced. The space allows for a geometrical view on energy changes in mechanical vibrating systems. It posses all the possibilities of the system dynamics analysis that are used in the traditional phase space but it also shows an amount of the energy accumulated in the system, an energy changes, flow, dissipation, synchronization, an energy attractors. Thus it increases our knowledge and intuition on energy changes in vibrating systems. Dynamics of the chaotic system is studied using the energy space. New types of maps are introduced. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction An energy flow modelling still arouses interests in the scientific world. Different methods are applied to solve problems connected with energy flow: Statistical Energy Analysis [1, 2, 3], Finite Element Method [4, 5]. But these methods do not allow for a special kind of a geometrical view on energy changes, which could develop our intuitional knowledge on energy flow phenomenon. This intuition is very important especially in modelling the systems, where we still can not measure energy flow, such as bioenergy, observed and applied in medicine diagnostics thanks to GDV method based on Kirlian effect [6, 7, 8], telepathy – the problem of an energy flow between two bioelectromagnetic systems, and so on. Nowadays one of the most important tools used in dynamical systems investigations is the phase space. Many aspects of the systems dynamics can be concluded using this space, but it also allows for the intuitional, geometrical view on an existing dynamical phenomena. The new conception of the space is proposed in [9, 11, 12]. It allows for a geometrical view on energy changes in mechanical vibrating systems. This space posses all the advantages of the phase space but it also shows an amount of the energy accumulated in the system, an energy changes, flow, dissipation, synchronization, an energy attractors. Thus it increases our knowledge and intuition on energy changes in vibrating systems. In this article an application of the energy space in the mechanical vibrating system dynamics investigations is shown. Some aspects of motion, energy flow, synchronization are considered, new types of maps are introduced. 2. Introduction to the energy–vector analysis – Energy flow and synchronization in the resonance state The energy space is constructed in a way, that the trajectory of the system, which accumulates a constant amount of energy, lays on a surface of the multidimensional sphere. The radius of this sphere is equal to the square root of the total E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.06.096
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energy accumulated in the system. From the image of the trajectory projection onto suitably chosen energy planes, one can see some aspects of an energy flow. The radius of this projection shows the total energy accumulated in considered part of the system. Moreover, one can conclude about all aspects of the motion, like from the phase plane. It is possible thanks to the fact that the kinetic energy is the function of velocity and the potential one – the function of displacement. Consider the vibrating system with the physical model shown in Fig. 1. The system consists of two oscillators. The external harmonic force excites the oscillator l. This oscillator is joined with the second oscillator l1. The mathematical model of the system is given by four differential equations of the first order: 8 x_ ¼ y; > > > < y_ ¼ ðF sin gs cy c1 ðy y Þ rx r1 ðx x1 ÞÞ 1 ; 1 l ð1Þ _ x ¼ y ; > 1 1 > > : y_ 1 ¼ ðc1 ðy 1 yÞ r1 ðx1 xÞÞ l11 ; where l, l1 is the masses of the oscillators, r, r1 is the stiffness coefficients of the springs, c, c1 is the damping coefficients, F is the amplitude of the external excitation force, x is the frequency of the external excitation force, rffiffiffi x r g¼ ; a¼ : a l The phase vector in the standard phase space R4 is represented by four components: x; y; x1 ; y 1 ; x2 ; y 2 : Transform the phase space of the system using the given function f: R4 ! R4: 8 pffiffir x ¼ ze ; > > > plffiffi2 > < y ¼ ye; 2 f ðx; y; x1 ; y 1 Þ ¼ prffiffiffi ffi > > 21 ðx1 xÞ ¼ z1e ; > > : plffiffiffi1ffi y ¼ y 1e : 2 1
ð2Þ
After this transformation the mathematical model of the system in the energy space is given by the following differential equations:
Fig. 1. The physical model of the system.
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qffiffi 8 > z_ e ¼ rly e ; > > > > qffiffiffiffi qffiffi > > 1 > y rlze þ rl1 z1e þ Fl sin gs; < y_ e ¼ lc y e cl1 y e þ pcffiffiffiffiffi ll1 1e qffiffiffiffi qffiffiffiffi > > z_ 1e ¼ rl11 y 1e rl1 y e ; > > > > qffiffiffiffi > > : y_ 1e ¼ r1 z1e c1 y þ pcffiffiffiffiffi 1 y: l1
l1
1e
ll1
ð3Þ
e
To simplify the description of the transformation note, that in the energy space there are square roots of the potential or kinetic energies on each axis. Note also, that function f transforms energy into vector form. 2.1. Energy flow and synchronization Some special advantages brings consideration of the energy planes. These subspaces are the equivalents of the phase planes of the phase space. In our case there are four interesting planes. The first one (ze : ye) – the energy plane of the oscillator l system. This plane is obtained after transformation of the plane (x : y). This transformation is linear. One can find eigenvectors of the transformation as orthogonal basis vectors [1, 0] and [0, 1], of the space (x : y). Eigenvalues of the transformation are equal: rffiffiffi rffiffiffi r l k1 ¼ ; k2 ¼ respectively: 2 2 Thus depending on r and l the phase space is just only squeezed or stretched in directions of the eigenvectors. In Fig. 2a plane (ze : ye) can be seen. The norm of the vector projection on that plane shows the energy accumulated in the oscillator l system. The value of ze shows the potential energy of the spring r, the value of ye – the kinetic energy of the mass l, and by means of these coordinates these energies can be calculated. One can see the changes from the potential energy into the kinetic one and vice versa. Note that total amount of the energy accumulated in the oscillator l system is not constant. There exists an energy flow between the oscillators l and l1. This flow will be analysed further. The second interesting plane is (z1e : y1e) – the energy plane of the oscillator l1 system. There exist transformations of two kinds which were made on the phase plane (x1 : y1) to obtain this energy plane. The first one: instead of the state variable x1 we have the spring r1 deflection: z1 = x1 x. The second one is similar to the considered transformation of the plane (x : y): transformation is linear and one can find eigenvectors of the transformation as orthogonal basis vectors [1, 0] and [0, 1]. Eigenvalues of the transformation are equal: rffiffiffiffiffi rffiffiffiffiffi r1 l1 ; k2 ¼ respectively: k1 ¼ 2 2 Thus in time of this part of transformation, depending on r1 and l1, the phase space is just only squeezed or stretched in directions of the given eigenvectors. In Fig. 2b plane (z1e : y1e) can be seen. The projection of the vector on that energy plane shows the energy accumulated in the oscillator l1 system. The value of z1e shows the potential energy of the spring r1 and the value of y1e – the kinetic energy of the mass l1, and by means of these coordinates the energies can be calculated. The position of the vector projections on the energy planes (ze : ye) and (z1e : y1e) at the same moment of time is marked by a small circle. See that at the same moment the spring r and spring r1 have the extremes of the potential energy, and the same concerns kinetic energies. Note also that the maximum of the potential energy of the oscillator l system equals the maximum of the kinetic energy of the oscillator l1 system and vice versa. The energy flow between the main mass l and the dynamical absorber l1 can be seen if you compare Fig. 1c and d, which show the combined energy planes. Note, that these trajectory projections are regular circles. It means, that energy accumulated in that parts of the system is constant. It shows some interesting aspects of the continuous energy flow between oscillator l and l1. All the energy, that flows out the oscillator l is intercepted by oscillator l1, and vice versa. The potential energy of the spring r transforms into kinetic energy of the oscillator l, and vice versa (Fig. 2c). The same situation takes place with the potential energy of the spring r1 and the kinetic energy of the oscillator l (Fig. 2d). This energy flow synchronization manifests also the next circles, that can be seen in Fig. 2e and f, where: dze dye dz1e
is the difference of the dze in ds interval, is the difference of the dye in ds interval, is the difference of the dz1e in ds interval,
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c
d
e
f
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Fig. 2. (a) Energy plane of the oscillator l system; (b) Energy plane of the oscillator l1 system; (c) and (d) Combined energy planes; (e) and (f) Differences of the variables ze and ye in ds intervals n = 0.85, l = 1 kg, l1 = 0.1 kg, r = 100 N/m, r1 = 100 N/m, c = 2 Ns/m, c1 = 0.001 Ns/m, F = 10 N.
dy1e ds
is the difference of the dy1e in ds interval, is the 1/1000 of the excitation force period.
Note that these circles are the consequences of circles shown in Fig. 2c and d, but also constant frequency of the vector rotation. From the whole considered figures one can conclude, that radius of the trajectory is constant, what means, that the trajectory lays on a surface of the four-dimensional sphere. The radius of this sphere is equal to the square root of the energy accumulated in the whole system. Note that it means constant amount of energy accumulated in the system.
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Energy flows only between the l and l1 systems and the excitation energy is dissipated in the dampers c and c1.This energy flow synchronization is possible just only for specially selected parameters. In our case the excitation frequency is equal to the resonance frequency of the system.
3. Energy flow in the chaotic systems Consider the system shown in Fig. 1 with nonlinear spring r. The mathematical model of the system is given by four differential equations of the first order: 8 x_ ¼ y; > > > < y_ ¼ ðF sin gs cy c1 ðy y Þ rx3 r1 ðx x1 ÞÞ 1 ; 1 l ð4Þ x_ 1 ¼ y 1 ; > > > : y_ 1 ¼ ðc1 ðy y 1 Þ þ r1 ðx x1 ÞÞ l11 : Because of nonlinearity of the spring r the transformation of the phase space, to obtain the energy space, in direction of x is nonlinear. In the considered case we have to show the transformation of the space, as function f: R4 ! R4, that transforms the phase space the following way: 8 prffiffi 2 x signðxÞ ¼ ze ; > 2 > > > < plffiffiy ¼ y ; e 2 f ðx; y; x1 ; y 1 Þ ¼ prffiffiffi ð5Þ ffi 1 > ðx xÞ ¼ z1e ; > 1 2 > > : plffiffiffi1ffi y ¼ y 1e : 2 1 To simplify the description of the transformation note, that in the energy space there are square roots of the potential or kinetic energies on each axis. Note also, that function f transforms energy into vector form. Similarly to the model given in paragraph 2 one can obtain the mathematical model of the system after this transformation, but for the nonlinear system it is easier to integrate the mathematical model (4) and afterwards use the function f to calculate vector coordinates in the energy space. Both methods were applied to examine the system dynamics, and the same results were obtained. There exists just only one unsolved problem with the critical point for the transformed mathematical model, thus these results are not presented.
Fig. 3. Bifurcation diagram of z1 (in grey, left); The largest Lyapunov exponent k max (in black, right); l = 1 kg, l1 = 0.07 kg, r = 1 N/m, r1 = 0.07 N/m, c = 0.1 Ns/m, c1 = 0.1 Ns/m, F = 2.5 N.
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Dependence of the system dynamics on the excitation force frequency is shown in the Fig. 3. The bifurcation diagrams of z1 variable in grey, and the largest Lyapunov exponent kmax in black can be seen in it. The value of largest Lyapunov exponent was estimated using Stefanski method. It is based on the phenomenon of synchronization of identical systems [13]. Different types of dynamics and bifurcations leading to the chaotic state can be identified in Fig. 3. Chaotic energy flow for g = 0.6025 is shown in Fig. 4a and b. Intersection of the attractor using Poincare plane was applied. For comparison the same intersection of the attractor in the phase space was shown in Fig. 1c and d. One can see the same type of dynamics in these charts. What can not be seen in the phase space is, that the oscillator l accumulates much more energy than oscillator l1. For the same system parameters new kind of maps is presented in Fig. 5. The system energy state on two energy levels can be seen in it. From the geometrical viewpoint it shows projections of the special attractor intersection. Attractor is cut using the sphere with the radius R matched up with the concerned energy level. Intersections by small Fig. 5a and b and big Fig. 5c and d spheres are presented. The scales of the charts are matched up with the intersection radius R. From these figures one can conclude, that the attractor is flat in directions of z3 and z4. Especially the intersection by the big sphere shows that the attractor cuts the sphere in the ‘‘equator’’ plane, what is the reason of the circles, that can be seen in Fig. 5a and d. 3.1. Energy flow In Fig. 6 one can see the resonance diagram for the selected system parameters. Energy flow for the special point (g = 1) was analysed. From Fig. 5 one can see, that for g = 1 energy accumulated in the spring r is close to zero. It means, that whole energy accumulated in the oscillator l system is close to zero, although this part of the system is forced by external force. It is the case, when the oscillator l1 works as the dynamical damper, that is for the excitation frequency equal to the system l1 free vibrations frequency. Energy flow for the considered case is shown in Fig. 7. In Fig. 7a and b one can see energy planes of the oscillators l and l1, respectively. Note, that difference between these
ye
0
y1e
z1e
c
d
Fig. 4. (a) Energy plane of the oscillator l system; (b) Energy plane of the oscillator l1 system; (c) Phase plane of the oscillator l system; (d) Phase plane of the oscillator l1 system; l = 1 kg, l1 = 0.07 kg, r = 1 N/m, r1 = 0.07 N/m, c = 0.1 Ns/m, c1 = 0.1 Ns/m, F = 2.5 N.
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1/2
y1e
[J ]
-0.3 -0.3
0.3
z1e
c
d
1/2
[J ]
y1e
0
-1.6
z1e Fig. 5. Sphere cut radius: (a) and (b) R = 0.3 [J1/2], (c) and (d) R = 0.3 [J1/2]; (a) Energy plane of the oscillator l system; (b) Energy plane of the oscillator l1 system; (c) Phase plane of the oscillator l system; (d) Phase plane of the oscillator l1 system; l = 1 kg, l1 = 0.07 kg, r = 1 N/m, r1 = 0.07 N/m, c = 0.1 Ns/m, c1 = 0.1 Ns/m, F = 2.5 N.
η Fig. 6. Resonance diagram of z1; l = 1 kg, l1 = 0.07 kg, r = 1 N/m, r1 = 0.07 N/m, c = 0.1 Ns/m, c1 = 0.0001 Ns/m, F = 2.5 N.
oscillators is one time of magnitude order. Note also from the marked small circles, that system l1 oscillates in antiphase to the system l. External force, that excites oscillator l is in the same phase as this oscillator. Thus all the energy of the excitation just only flows through the l system and is intercepted by l1 system. What is interesting, the energy of
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Fig. 7. (a) The energy plane of the oscillator l system; (b) Energy plane of the oscillator l1 system; (c) and (d) Differences of the variables in ds intervals; (e) Radius of the attractor. g = 1, l = 1 kg, l1 = 0.07 kg, r = 1 N/m, r1 = 0.07 N/m, c = 0.1 Ns/m, c1 = 0.0001 Ns/m, F = 2.5 N.
the l1 system is constant, what can be seen in Fig. 7b and d. The energy accumulated in the system flows only between the masses l and l1 systems and the excitation energy is dissipated in the dampers c and c1. The energy of the whole system is almost constant, what can be seen in Fig. 7f. 3.2. Application of the energy–vector analysis in the system with impacts Consider the system shown in Fig. 8. For some ranges of the system parameters, it works as an impact damper of the motion of the oscillator l (Refs. [9, 10]). The system consists of three oscillators. The external harmonic force excites the oscillator l. It is joined with the classical dynamical absorber l1. This absorber is allowed to collide with the third oscillator l2. In the periods between the impacts the mathematical model of the system is given by six differential equations of the first order:
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Fig. 8. The physical model of the system.
8 x_ ¼ y; > > > > > y_ ¼ ðF sin gs cy c1 ðy y 1 Þ rx r1 ðx x1 ÞÞ l1 ; > > > > < x_ 1 ¼ y ; 1 1 _ y ¼ ðc > 1 1 ðy 1 yÞ r1 ðx1 xÞÞ l ; > 1 > > > > x_ 2 ¼ y 2 ; > > > : y_ ¼ ðc y r x Þ 1 ; 2 2 2 2 2 l2
ð6Þ
where l, l1, l2 is the masses, r, r1, r2 is the stiffness coefficients of the springs, c, c1, c2 is the damping coefficients, F is the amplitude of the external excitation force, x is the frequency of the external excitation force, rffiffiffi x r : g ¼ ; s ¼ at; a ¼ a l
ð7Þ
The impact between the dynamical and impact absorbers is put into the mathematical model over the restitution coefficient r. The phase vector in the standard phase space R6 is represented by six components: ð8Þ
x; y; x1 ; y 1 ; x2 ; y 2 : Transform the phase space as follows: 1. Instead of the displacement coordinates x, x1, x2, take the deflections of the springs. z ¼ x;
z1 ¼ x1 x;
z2 ¼ x2 :
ð9Þ
2. Depending on the coefficients ri and li squeeze and stretch the space in directions of zi and yi. In these two steps a new energy space can be obtained. In the first step, one obtains a new space V with the basis E: E ¼ ðez ; ey ; e1z ; e1y ; e2z ; e2y Þ;
ð10Þ
where: ex ¼ ½1; 0; 0; 0; 0; 0T ; e1x ¼ ½0; 0; 1; 0; 0; 0T ; T
e2x ¼ ½0; 0; 0; 0; 1; 0 ;
ey ¼ ½0; 1; 0; 0; 0; 0T ; e1y ¼ ½0; 0; 0; 1; 0; 0T ; T
e2y ¼ ½0; 0; 0; 0; 0; 1 :
ð11Þ
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Change the basis vectors of the space V. Then, the new energy basis EN of this space is: EN ¼ ðbz ; by ; b1z ; b1y ; b2z ; b2y Þ;
ð12Þ
where: hpffiffiffi iT h pffiffiffi iT 1 1 r ; 0; 0; 0; 0; 0 ; by ¼ 0; l ; 0; 0; 0; 0 ; h iT h iT pffiffiffiffiffi1 pffiffiffiffiffi 1 b1z ¼ 0; 0; ð r1 Þ ; 0; 0; 0 ; b1y ¼ 0; 0; 0; l1 ; 0; 0 ; h h pffiffiffiffiffi1 iT pffiffiffiffiffi 1 iT b2z ¼ 0; 0; 0; 0; ð r2 Þ ; 0 ; b2y ¼ 0; 0; 0; 0; 0; l2 : bz ¼
The transition matrix AEN E from the basis E to the basis EN of the energy space takes the form: 3 2 pffiffiffi r 0 0 0 0 0 6 0 pffiffiffi l 0 0 0 0 7 7 6 7 6 pffiffiffiffiffi 6 0 r1 0 0 0 0 7 7: AEN E ¼ 6 pffiffiffiffiffi 6 0 0 0 7 0 0 l1 7 6 7 6 pffiffiffiffiffi 4 0 0 0 0 r2 0 5 pffiffiffiffiffi 0 0 0 0 0 l2
ð13Þ
ð14Þ
The coordinates of the vector ve ¼ ½ze ; y e ; z1e ; y 1e ; z2e ; y 2e TEN with respect to the energy basis EN can be obtained from the vector v with respect to the basis E, using the transition matrix: ve ¼ AEN
E v:
ð15Þ
Then ve ¼
pffiffiffi pffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi T rz; ly; r1 z1 ; l1 y 1 ; r2 z2 ; l2 y 2 EN :
The norm of the vector ve in the energy space with the energy product is as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi 2 pffiffiffiffiffi 2 pffiffiffiffiffi 2 pffiffiffiffiffi 2 i 1 hpffiffiffi 2 pffiffiffi 2 rðzÞ þ ly þ ð r1 z1 Þ þ l1 y 1 þ ð r2 z2 Þ þ l2 y 2 jve j ¼ hve ; ve i ¼ 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rz2 ly 2 r1 z21 l1 y 21 r2 z22 l2 y 22 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ þ þ þ þ þ ¼ Ep þ Ek þ Ep1 þ Ek1 þ Ep2 þ Ek2 ; 2 2 2 2 2 2
ð16Þ
ð17Þ
where: Ep is the potential energy accumulated in the spring r, Ek is the kinetic energy of the mass l, Ep1 is the potential energy accumulated in the spring r1, Ek1 is the kinetic energy of the mass l1, Ep2 is the potential energy accumulated in the spring r2, Ek2 is the kinetic energy of the mass l2. The mathematical model of the system in the transformed space is given by following differential equations: 8 qffiffi > z_ e ¼ rly e ; > > > > qffiffiffiffi qffiffi > > > 1 > y_ e ¼ lc y e cl1 y e þ pcffiffiffiffiffi y rlze þ rl1 z1e þ Fl sin gs; > ll1 1e > > > qffiffiffiffi qffiffiffiffi > > > < z_ 1e ¼ rl11 y 1e rl1 y e ; qffiffiffiffi > 1 > y; y_ 1e ¼ rl11 z1e lc11 y 1e þ pcffiffiffiffiffi > ll1 e > > > ffiffiffi ffi q > > > > z_ 2e ¼ rl22 y 2e ; > > > > qffiffiffiffi > > : y_ 2e ¼ r2 z2e c2 y 2e ; l2 l2
ð18Þ
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where: pffiffiffiffiffi ze ¼ signðxÞ Ep ; pffiffiffiffiffi y e ¼ signðyÞ Ek ; pffiffiffiffiffiffiffi z1e ¼ signðx1 Þ Ep1 ; pffiffiffiffiffiffiffi y 1e ¼ signðy 1 Þ Ek1 ; pffiffiffiffiffiffiffi z2e ¼ signðx2 Þ Ep2 ; pffiffiffiffiffiffiffi y 2e ¼ signðy 2 Þ Ek2 :
ð19Þ
3.3. Energy flow and dissipation during the impact – Impact map Let p be the plane determined by the basis vectors (Fig. 9): h iT pffiffiffiffiffi1 b1y ¼ 0; 0; 0; l1 ; 0; 0 ; h pffiffiffiffiffi1 iT b2y ¼ 0; 0; 0; 0; 0; l2 :
ð20Þ ð21Þ
The vectors y1e and y2e correspond to the kinetic energies of the masses l1 and l2, respectively. The position of the vector is marked on this plane just before and after impact. Thus, this plane is a special kind of the impact map. In order to make this map clearer, note that it represents only the points for different types of impact. As the choice criterion, y 1e < y 2e has been applied. The consideration of the position and norm of the vector on this map allows one to conclude about the energy flow between the dynamical and impact absorbers and also about the energy dissipation during each collision. The energy dissipation is included into the mathematical model of the system over the restitution coefficient r. Let vep be the projection of the ve on the plane p just before the impact, and v0ep after it.
y2=-5*y 1/3 -
-
Fig. 9. The impact map. n = 1.165, l = 1 kg, l1 = l2 = 0.1 kg, r = 1 N/m, r1 = 0.1 N/m, r2 = 0.1 N/m, c = 0.04 Ns/m, c1 = 0.01 Ns/ m, c2 = 0.02 Ns/m, F = 0.002 N, d = 0.0216 m, r = 0.5.
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The transformation of the vector vep during each impact is given by the matrix AEN : v0ep ¼ AEN vep ; where:
" AEN ¼
l¼
lr lþ1
1þr lþ1
lþlr lþ1
1lr lþ1
# ;
ð22Þ
l1 ; l2
r is the restitution coefficient. In the case l = 1 the eigenvalues of the AEN matrix are: k1 ¼ r;
k2 ¼ 1
ð23Þ
and the eigenvectors are: w1 ¼ ½1; 1T ;
w2 ¼ ½1; 1T ;
ð24Þ
respectively. The directions of the eigenvectors are shown in Fig. 9. Note that during impact the vector vep is transformed only in the direction given by the eigenvector w1. The energy dissipation in time of each collision can be found from the change of the norm of the energy space vector: jvep j jv0ep j:
ð25Þ
The maximum dissipation of the energy takes place when the vector vep has the same direction as the eigenvector w1. Then v0ep ¼ k1 vep : Taking in the consideration that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jvep j ¼ EK 1 þ EK 2 ;
ð26Þ
ð27Þ
where: EK 1 is the kinetic energy of the mass l1 before impact, EK 2 is the kinetic energy of the mass l2 before impact, and jv0ep j ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E0K 1 þ E0K 2 ;
ð28Þ
where: E0K 1 is the kinetic energy of the mass l1 after impact, E0K 2 is the kinetic energy of the mass l2 after impact, one can find easily that the energy relation after and before impact assumes the form: E0K 1 þ E0K 2 ¼ k21 ¼ r2 : EK 1 þ E K 2
ð29Þ
The closer the direction of the vector vep is to the second eigenvector w2, the less energy dissipation occurs. In the case when the directions of vep and w2 are almost the same, there is almost no energy dissipation during the collision. The velocities of the oscillators l1 and l2 are almost equal then, and in practice we do not know if impact occurs or not. It is so called grazing collision and it causes chaotic motion of the system. These special points can be seen in Fig. 9 as common points of the before and after impact attractors. The transformation matrix AEN allows one also to divide the impact map into two kinds of fields: the first one for the case when the energy flows during impact from the dynamical to impact absorber and the second when the energy flows in the opposite direction.
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A. Dabrowski / Chaos, Solitons and Fractals 39 (2009) 1684–1697
Consider the matrix AEN in the case l1 = l2: " # 1r 2 1þr 2
AEN ¼
1þr 2 1r 2
ð30Þ
:
Let: vep ¼ ½v1 ; v2 T
and
v0ep ¼ ½v01 ; v02 T :
ð31Þ
Then v0ep ¼ AEN vep ; so
v01
"
v02
¼
1r 2 1þr 2
ð32Þ
1þr 2 1r 2
#
v1 : v2
ð33Þ
The energy flow from the dynamical to impact absorber is given by the condition: v1 6 v01 6 v1 :
ð34Þ
Consider: 1. v01 ¼ v1 The condition is satisfied when the directions of vep and the eigenvector w2 are the same. 2. v01 ¼ v1 then v1 ¼
1r 1þr v1 þ v2 2 2
and we obtain v2 ¼
3r v1 : 1þr
ð35Þ
For the case shown in Fig. 4 r = 0.5 and then 5 v2 ¼ v1 : 3
ð36Þ
As a result, the energy flow from the dynamical to the impact absorber occurs:if v1 > 0 5 v1 6 v2 6 v1 ; 3
ð37Þ
if v1 < 0 5 v1 6 v2 6 v1 : 3
ð38Þ
The field of energy flow in this direction is marked in Fig. 9 by the grey colour. Note that it is just for y1e< y2e, which was the criterion of the impact choice. It can be seen that during impacts in the considered case energy flows in both directions, but definitely more often from the dynamical to the impact absorber. 4. Conclusions The way of transformation of the phase space to obtain the energy space has been shown. It has been proved that this new kind of space allows for concluding about the energy state of a vibrating system. The norm of the vector in that space is equal to the square root of the total energy accumulated in the system. The projection of the vector space on energy subspaces show the amount of the energy that accumulates in some parts of the system. It has been shown that using this kind of spaces, all aspects of the kind of motion can be concluded about, like from the phase space and, moreover, the energy state, accumulation, flow and dissipation can be observed. Different types of the energy flow synchronization were shown. New kind of maps was introduced. It was shown, that the energy space allows for a new, geometrical view on energy changes in vibrating systems.
A. Dabrowski / Chaos, Solitons and Fractals 39 (2009) 1684–1697
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Further reading [14] Griffel DH. Linear algebra and its applications vol. 2 more advanced. 1989. Ellis Horwood Limited. [15] Kapitaniak T. Chaos for engineers. Berlin Heidelberg: Springer-Verlag; 1998.