Journal of System Design and Dynamics

Transcription

1 Vol. 5, No. 6, Effects of Stable Nonlinear Normal Modes on Self-Synchronized Phenomena* Hiroki MORI**, Takuo NAGAMINE**, Yukihiro AKAMATSU** and Yuichi SATO** ** Deartment of Mechanical Engineering, Saitama University, Shimo-Okubo 55, Sakura-ku, Saitama-shi, Saitama , Jaan Abstract This aer exerimentally examines effects of stable nonlinear normal modes on self-synchronized henomena by using oscillators that are excited by intermittent electromagnetic ower. The oscillator behaves as a endulum-tye oscillator when the alied voltage is low whereas as a rotor-tye oscillator when the voltage is high. Stable nonlinear normal modes are obtained numerically by modified shooting method. Exerimental and analytical results demonstrate that self-synchronized henomena of endulum-tye oscillators as well as rotor-tye oscillators can be exlained well by using stable nonlinear normal modes of the system. Key words: Self-Synchronized Phenomenon, Self-Excited Vibration, Nonlinear Normal Mode, Shooting Method. Introduction *Received May, (No. -8) [DOI:.99/jsdd.5.337] Coyright by JSME In a system where nonlinear self-excited oscillators are couled, sometimes self-synchronized henomenon occurs due to interaction between oscillators (). This henomenon is well known from old times in mechanics. Inoue et al. roosed a vibration screen, crushing machine and vibrating lift using self-synchronized henomena of rotor-tye oscillators ()-(4). M. Paz et al. discussed hase difference of oscillators and condition for stability by using Hamilton s Princile (5). They also alied their analysis to the described vibrating lift. Kurita et al. roosed effective vibration transortation using self-synchronized henomena of vibratory machines that are excited by seed feedback control (6). Bonkobara et al. alied self-synchronized henomena of rotor-tye oscillators to hand-held vibrating tools (7). Self-synchronized henomenon is currently being reviewed from the standoint of dynamical systems theory and comlex systems science (8). So far, it has been clarified numerically that fundamental characteristics of self-synchronized henomena of rotor-tye oscillators can be exlained by using nonlinear normal modes of the system (9). Here, the nonlinear normal modes reresent the eriodic solutions of the equations of motion in which the energy suly and energy dissiation terms are omitted, that is, a conservative system. In a revious reort (), we addressed self-synchronized henomena of two endulum-tye oscillators. The nonlinearity of the oscillators were relatively small: Amlitudes of the oscillators were less than 3 deg. Since nonlinearity is a key factor of the mechanism of self-synchronized henomena, characteristics of self-synchronized henomena of endulum-tye oscillators with large amlitudes are also imortant. Nonlinear normal modes have the otential for being useful tools to investigate such characteristics. Thus, this aer exerimentally examines effects of stable nonlinear 337

2 Vol. 5, No. 6, normal modes on self-synchronized henomena by using oscillators that are excited by intermittent electromagnetic ower. The oscillator behaves as a endulum-tye oscillator when the alied voltage is low whereas as a rotor-tye oscillator when the voltage is high. Exerimental and analytical results demonstrate that self-synchronized henomena of endulum-tye oscillators as well as rotor-tye oscillators can be exlained well by using stable nonlinear normal modes of the system.. Exeriment. Exerimental aaratus Figure shows a schematic of an oscillator. The oscillator is suorted by an acrylic frame through a bearing. As shown in Fig., a magnet is attached to the ti of the oscillator. The frame contains two coils which are connected to a transistor. When the oscillator ti asses above the coils, variation of magnetic field roduces electric current in one of the coils. The electric current is amlified by the transistor and then alternatively flows into the other coil, which roduces a force to excite the oscillator. The oscillator behaves as a self-excited endulum-tye oscillator when the sulied voltage is low whereas as a self-excited rotor-tye oscillator when the voltage is high. Linear natural frequencies of Oscillator and Oscillator are measured as f n =.545 Hz and f n =.558 Hz, resectively. Figure shows frequencies of Oscillator and Oscillator as a function of alied voltage. As shown in the figure, oscillators are not comletely identical but have little difference. As alied voltage increases, frequency of endulum motion decreases due to its soft sring characteristic. As to rotational motion, frequency increases with alied voltage. Figure 3 shows a schematic of an exerimental aaratus. Oscillator and Oscillator Bearing Magnet 4 55 Transistor Coils Driver 47 5 (a) Front view (b) Side view Fig. Oscillator and suort frame. Frequency [Hz] 6 4 Oscillator Oscillator (Rotational motion) (Pendulum motion) Alied Voltage [V] Fig. Frequency of Oscillator and Oscillator. 338

3 Vol. 5, No. 6, are mounted on a horizontal late that is hung by four thin metal wires of length l. Linear natural frequency f of the late is adjusted by the wire length l. The wires are inclined from the vertical so that the late moves only in the horizontal direction as shown in Fig. 3. θ, θ and θ denote angular dislacements of Oscillator, Oscillator and the late. Motions of the two oscillators are monitored by a video camera and dislacement of the late is measured by a laser sensor. Hereafter, the alied voltages of Oscillator and Oscillator are denoted by e and e, resectively.. Synchronized attern Figure 4 shows atterns of self-synchronized oscillations observed when 7V is Wire Oscillator Oscillator l θ Plate θ 6 θ (a) Front view Fig. 3 Exerimental aaratus. Sensor 5 (b) Side view θ, θ [deg] 8 9 f =.96 Hz f s =.7 Hz 9 θ θ θ 8 Time [s] (a) Pattern θ [deg] θ, θ [deg] 8 9 f =.96 Hz f s =.5 Hz 9 θ θ θ 8 Time [s] (b) Pattern θ [deg] θ, θ [deg] 8 9 f =.4 Hz f s =. Hz 9 θ θ θ 8 Time [s] (c) Pattern 3 Fig. 4 Time histories for Pattern, Pattern and Pattern 3 ( e = e = 7V). θ [deg] 339

4 θ, θ [deg] 8 9 f =.96 Hz f s =5.38 Hz 9 8 Time [s] Fig. 5 Time history for Pattern 4 ( e = e = 8 V ). θ [deg] Vol. 5, No. 6, alied to each oscillator. In these figures, f s denotes synchronized frequency. Left-side and right-side ordinates indicate values of θ, θ and θ, resectively. Figures 4(a), 4(b) and 4(c) corresond to the first, second and third natural oscillation modes of endulum motion, that is, the two oscillators and the late are all in-hase in Fig. 4(a), the oscillators are out-of-hase with each other and the late is almost stationary in Fig. 4(b), the oscillators are in-hase with each other and are out-of-hase with the late in Fig. 4(c). In the following, these oscillation atterns are called Pattern, Patten and Pattern 3, resectively. We note that results of Figs. 4(a) and 4(b) are obtained from the same arameters. Oscillation atterns are determined from the initial condition. Values of f in Pattern and Pattern 3 are determined based on the results in the revious reort () : fni < f ( i =,) in Pattern and f ni > f in Patten 3. Figure 5 shows rotational synchronized attern observed when 8 V voltage is alied to each oscillator. As shown in Fig. 5, hase difference of rotating oscillators is out-of-hase. In the following, this oscillation attern is called Pattern 4. Figures 6 and 7 show exerimentally obtained frequency ratio fs / f of the synchronized frequency to the natural frequency of the late, angular amlitude ratio θ / θ of the late to Oscillator, and hase lag φ of Oscillator from Oscillator, for each oscillation attern. In Fig. 7 angular amlitude of Oscillator is set to θ = 36deg. From Fig. 6 we see that existence region of Patterns, and Pattern 3 are searated by natural frequency of the late. We note that hase lag φ for Pattern in Fig. 6 does not have trivial value φ 8 deg but searates above and below φ = 8 deg. f s /f [ ] θ θ..5, : Pattern, ( f =.96 Hz) : Pattern3 ( f =.4 Hz) φ [deg] Alied Voltage [V] Fig.6 Exerimental results for Patterns, and 3 ( e = e ). 34

5 Vol. 5, No. 6, f s /f [ ] : Pattern4 ( f =.96 Hz). θ θ φ [deg] Alied Voltage [V] Fig. 7 Exerimental results for Pattern 4 ( e = e ). 3. Results and Discussion Figure 8 shows an analytical model to calculate nonlinear normal modes. Two oscillators are mounted on the horizontal late that is hung by wires. The oscillators are modeled as hysical endulums of mass m and length l. The late is modeled as a oint mass endulum of mass m and length l. The oscillators can rotate freely around their suort axes. J and J denote moments of inertia of Oscillator and Oscillator, resectively. By omitting electromagnetic and daming forces, we obtain equations of motion J mg l ml l i, () iθi + sin θi + { θ cos( θi θ) + θ sin( θi θ)} = ( =, ) θ θ θi θi θ θi θi θ i= ( m + m) l + ( m + m) g l sin + ml l{ cos( ) sin( )} =, () where subscrit i ( =, ) reresents hysical quantities associated with Oscillator i. In addition, arameters ω = π f = ml g / J, ω = π f = g / l, γ = m/ m (3) i i i are introduced. Table shows arameters used for calculation, which are obtained from θ l Wire Oscillator Oscillator Plate m θ l G θ l G m, J m, J Fig. 8 Analytical model. 34

6 Vol. 5, No. 6, the exerimental aaratus in Fig. 3. Shooting method is emloyed to calculate nonlinear normal modes. Since Eqs. () and () have an infinite number of eriodic solutions, we use the aroriately modified shooting method (9). First, we consider the case of e = e. Figure 9 shows comarison between exerimental results (in Fig 6) and stable nonlinear normal modes for (a) Pattern, (b) Pattern and (c) Pattern 3. We note that Fig. 9 shows only nonlinear normal modes corresonding to each synchronized oscillation mode. Figure shows comarison for Pattern 4 (in Fig 7). Real line denotes stable nonlinear normal modes. Since alied voltages e and e are not considered in the calculation of nonlinear normal modes, Table System arameters ω [rad/s].545 π l [mm] 7 ω [rad/s].558 π γ [ ]. f s /f [ ]. : Ex. θ θ.5 φ [deg] (a) Pattern ( f =.96 Hz ) f s /f [ ]. : Ex. θ θ.5 φ [deg] (b) Pattern ( f =.96 Hz ) Fig. 9 Comarison of exerimental results and nonlinear normal modes (Pendulum-tye). 34

7 Vol. 5, No. 6, f s /f [ ]. : Ex. θ θ.5 φ [deg] 8 Fig (c) Pattern 3 ( f =.4 Hz ) Comarison of exerimental results and nonlinear normal modes (Pendulum-tye). f s /f [ ] : Ex.. θ θ φ [deg] Fig. Comarison of exerimental results and nonlinear normal modes (Pattern 4, f =.96 Hz ). mechanical energy E normalized by m g l is used as the abscissa, which is obtained as the average value of E. Here, E is given by E = T + U, (4) where T = { mll θθ i cos( θi θ)} + J iθi + ( m+ m) l θ i i = =. (5) U = mg ( lcosθi + l cos θ) m l cosθ g i= In the exeriment, dimensionless mechanical energy increases with alied voltages e and e. 343

8 Vol. 5, No. 6, From the viewoint of nonlinear dynamics, it is imortant to clarify the bifurcation mechanism of nonlinear normal modes. However, analysis of bifurcations in this system is difficult since the system is conservative and fn fn. Therefore, we do not discuss bifurcations of nonlinear normal modes here. From Figs. 9 and, we see that stable nonlinear normal modes corresond well with each exerimental oscillation attern. In articular, the imortance is that non-trivial hase difference φ 8 deg of Pattern is obtained numerically as well as exerimentally in Fig. 9(b). Next, we discuss shortly the case of e e. In this case difference between the two oscillators is larger than the case of e = e. Figure shows comarison of exerimentally observed synchronized oscillation and calculated stable nonlinear normal modes. The exeriments are made only for endulum motion, that is, for Pattern, Pattern and Pattern 3. In each exeriment, constant voltage e is alied to Oscillator, which is described in cation of each figure. On the other hand, the alied voltage e of Oscillator is varied. Dimensionless mechanical energy increases with the alied voltage e. We note that nonlinear normal modes in Fig. are the same as those in Fig. f s /f [ ]. : Ex. θ θ.5 φ [deg] (a) Pattern ( f =.96 Hz, e =.5 V ) f s /f [ ]. : Ex. θ θ.5 φ [deg] (b) Pattern ( f =.96 Hz, e =. V ) Fig. Comarison of exerimental results and nonlinear normal modes (Pendulum-tye). 344

9 Vol. 5, No. 6, f s /f [ ]. : Ex. θ θ.5 φ [deg] 8 Fig (c) Pattern 3 ( f =.4 Hz, e = 3. V ) Comarison of exerimental results and nonlinear normal modes (Pendulum-tye). 9, since the voltages e and e are not included in Eqs. () and (). From Fig., we see that exerimental results agree well with the stable nonlinear normal modes. The imortance is that characteristics of self-synchronized henomena can be exlained well by nonlinear normal modes even when e e. The fact that exerimental results aear only above φ = 8 deg in Fig. (b) is may be due to initial conditions. 4. Summary Characteristics of self-synchronized henomena of endulum-tye oscillators as well as rotor-tye oscillators can be exlained by nonlinear normal modes of the system. We showed that out-of-hase endulum-tye synchronized oscillation has non-trivial oscillation modes when energy of oscillation is large. Nonlinear normal modes also exlain well these non-trivial oscillation modes. In addition, we showed that characteristics of self-synchronized henomena of oscillators with relatively large difference are similar to those of self-synchronized henomena of oscillators with small difference in the range of exeriment. References () I. I. Blekhman, Synchronization in Science and Technology, ASME Press, 988. () J. Inoue et al., On the Self-Synchronization of Vibrators (st Reort, Rotation of an Unbalanced Rotor with a Harmonically Oscillating Shaft), Transactions of the Jaan Society of Mechanical Engineers, Volume 3(34), 966, (in Jaanese) (3) J. Inoue et al., On the Self-Synchronization of Vibrators (5th Reort, Imact Vibrator), Transactions of the Jaan Society of Mechanical Engineers, Volume 4(353), 976, (in Jaanese) (4) J. Inoue et al., Study on the Interaction between Power Source and Machine (3rd Reort, Vibrating Lift Using Self-Synchronization), Transactions of the Jaan Society of Mechanical Engineers, Volume 49(437), 983, (in Jaanese) (5) M. Paz and J. D. Cole, Self-Synchronization of two Unbalanced Rotors, Transactions of the American Society of Mechanical Engineers, Journal of Vibration and Acoustics, Volume 4, 99,

10 Vol. 5, No. 6, (6) Y. Kurita et al., Vibration Transortation by Cooeration of Decentralized Self-Excited Vibratory Machines, Transactions of the Jaan Society of Mechanical Engineers, Series C, Volume 69(68), 3, (in Jaanese) (7) Y. Bonkobara et al., Develoment of Generation Mechanism of Synchronous Vibration Suitable for Hand-Held Vibrating Tools (Investigation for Imact Model with Two Oscillators), Transactions of the Jaan Society of Mechanical Engineers, Series C, Volume 76(77),, (in Jaanese) (8) A. Pikovsky et al., Synchronization (A Universal Concet in Nonlinear Sciences), Cambridge University Press,. (9) H. Mori et al., Self-Synchronized Phenomena Generated in Rotor-tye Oscillators: Investigation Using Nonlinear Normal Modes, Journal of System, Volume (3), 8, () T. Nagamine et al., Couled Vibration of Oscillators and their Suort, Transactions of the Jaan Society of Mechanical Engineers, Series C, Volume 74(739), 8, (in Jaanese) 346