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Experimental realization and simulations a self-sustained Macro ElectroMechanical System
Mechanics Research Communications 37 (2010) 106–110
Contents lists available at ScienceDirect
Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom
Experimental realization and simulations a self-sustained Macro ElectroMechanical System C.A. Kitio Kwuimy *, P. Woafo Laboratory of Modelling and Simulation in Engineering and Biological Physics, University of Yaounde I, Box 812, Yaounde, Cameroon
a r t i c l e
i n f o
Article history: Received 7 August 2009 Received in revised form 19 October 2009 Available online 10 November 2009 Keywords: Self-sustained MaEMS Experimental simulations Bifurcations and chaos
a b s t r a c t This paper deals with experimental realization and simulations of a self-sustained electromechanical system made up of an electrical implementation of a van der Pol-Duffing type oscillator driving a macro scale mass–spring–damper linear oscillator. Using fundamental laws such as Newton and Kirchhoff laws, the parts of the system are characterized and constraints for experimental realization of a prototype are defined. Modelling equations are derived that show the possibility of self-oscillations. Experimental simulations reveal that the proposed device exhibits periodic oscillations and complex dynamics. These results are consistent with published theoretical results. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Nonlinear systems may exhibit considerably complex dynamic behavior such as change in stability of response, quenching, quasiperiodic motion and chaotic motion. Both predictability and stability of engineering systems are rather important, which explains why research in the area of dynamics, synchronization and control of nonlinear systems has received a great deal of attention in the past two decades. An important class of engineering systems is magnetically actuated. In this class of systems, nonlinear terms can arise from a mechanical part (material, geometric or inertial nonlinearities), from an electrical circuit (nonlinear self, nonlinear condenser, nonlinear resistance) and from the coupling (coupling between the electromagnets, saturation, hysteresis, nonlinear magnetic force, time delay). See for details Hayashi (1964), Ji (2003), Ji et al. (2008), Nayfeh and Mook (1979) and Preumont (2006). One consequence of nonlinearity is self-sustained oscillation. Self-sustained engineering systems can run without a permanent time varying excitation, thus they have some economic advantages. The earliest mathematical model associated to self-excited systems, which is now called the van der Pol equation, was derived by van der Pol (van der Pol and van der Mark, 1927) to describe a vacuum tube circuit. The electrical model of the van der Pol equation is made up of negative resistance and it has been shown that this model presents complex phenomena (Ueda and Akamatsu, 1981).
* Corresponding author. Tel.: +27 733 14 59 08. E-mail address: [email protected] (C.A. Kitio Kwuimy). URLs: http://users.aims.ac.za/~cedrick/ (C.A. Kitio Kwuimy), http://www. lamsebp.org (C.A. Kitio Kwuimy, P. Woafo). 0093-6413/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2009.10.007
Many researchers have presented contributions to understand and apply self-oscillation in engineering. D’Acunto (2006) recently proposed a modified van der Pol equation to describe a self-excited body sliding on a periodic potential, Hamdi and Belhaq (2009) examined the self-excitated vibration of a simply-supported beam subjected to an axially high frequency excitation. In the area of Macro ElectroMechanical Systems (MaEMS), Ji (2003). Kitio Kwuimy and Woafo (2007, 2008) and Yamapi and Woafo (2009) presented some models of MaEMS with self-sustained oscillations. A model of self-sustained MEMS was recently presented by Kitio Kwuimy and Woafo (2009). Extensions have been made on synchronization and control of nonlinear phenomena. All the results and references mentioned above show that selfsustained MaEMS may really exhibit remarkable dynamical behavior. Since these results were based on mathematical analysis and numerical simulations, two fundamental questions arise: (1) How to implement experimentally a self-sustained magnetically actuated MaEMS? This interrogation will define experimental constraints. (2) Can nonlinear phenomena be observed experimentally in such MaEMS? A positive response of these questions will motivate extension to synchronization and control. In this paper, experimental realization and simulation of a van der Pol-Duffing (vdPD) oscillator driving a mass spring damping mechanical arm is considered. Descriptions of the parts of the device are given. In order to achieve experimental constraints an audio amplifier is integrated into the device. The modelling equations of the device are then derived showing a self-sustained oscillator coupled to a mass–spring–damped linear oscillator. Based on theoretical results published on these types of systems, attention is focussed on experimental simulations. The paper is presented as follows. Section 2 deals with the characterization of the device
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and the modelling equations. Section 3 presents a brief background on theoretical studies. In Section 4, results of experimental simulations are presented. Section 5 is for the conclusion and hints of applications. 2. Experimental set-up of the self-sustained MaMES Fig. 1 shows a schematic representation of the self-sustained MaEMS to be investigated. It consists of a mechanical part and a Both parts are coupled through a magnetic field of magnitude Bf = 16 103 T. The coil is made of copper wire of length l = 3.768 m and has a resistance r0 = 2 X and an inductance L0 = 1.43 109 H. 2.1. The mechanical part The mechanical part is made of a rigid rod and a linear spring of elasticity constant K = 7.98 N/m. The total mass of the mechanical elements is m = 40 g and its dissipative coefficient k = 1.13 104 Kg s1 determined from the measurement of the logarithm decrement. Using Newton’s laws, the dynamics of the mechanical part is modelled by the following second order differential equation:
d Y dY þk þ KY ¼ FðtÞ; ds2 ds
contrasts with the frequency of MEMS and NEMS whose values range from Megahertz to Terahertz. 2.2. The electrical part The electrical part is a van der Pol-Duffing (vdPD) oscillator made up of resistances, inductor, capacitance, operational amplifier (JR42AF) as linear components and diodes (IN 4001) as non linear components (see Fig. 2). The operational amplifiers are polarized with a Direct Current (DC) voltage of 12 V using a DCregulated power of type PS23023 (Hight Quality). The diode voltage–current characteristic is given by the following relation (Tooley, 2006):
i ¼ i0 ðexp ðV=V 0 Þ 1Þ;
ð2Þ 4
where i0 is the reverse saturation current equal to 10 A and V0 = 26 103 V at the room temperature. Kirchhoff’s laws lead to (in the absence of the mechanical part):
2 d V dV Li0 V # 1 cos h þ VN ds2 ds V0 V 0# V þ i0 ðR þ RL Þ sin h V0
LC
2
m
where FðtÞ ¼ Bf li, i is the current through the coupling zone and Y is the displacement of the mechanical arm (see Fig. 1). The natural freqffiffiffi K ¼ 14:12 Hz. This quency of the mechanical part is given by x1 ¼ m
ð1Þ
¼ 0; where # ¼
ð3Þ h
i
LR36 CR3M R26 ðRþRL Þ R3M R26
ðRþRL Þ and N ¼ 1 R36 , should be positive R3M R26
real numbers and V is the voltage across the diodes. Eq. (3) can be viewed as a generalized form of the van der Pol equation, by expanding sin h and cos h. 2.3. Constrains for experimental realization As the mechanical arm moves under the action of a vertical force at least equal to the weight of the mechanical elements, one finds that the Laplace force should be at least 0.4 N. In addition, when activating the mechanical arm with the proposed electronic device, the maximum delivery force (the Laplace one) is around 1 mN which is less than the required force to set the mechanical arm into motion. Furthermore, the natural frequency of the electronic oscillator is about 3 KHz which is very high compared to that of the mechanical arm (x1 = 14.12 Hz). These two facts constitute limitations to the functioning of the device. A generalization can be made with many macro scale electromechanical system. There is thus a necessity to overcome the limitations. It is quite difficult to increase the frequency of the mechanical arm (or to reduce one
Fig. 1. Synoptic of the self-sustained Macro ElectroMechanical System.
Fig. 2. Synoptic of the van der Pol oscillator.
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of the electrical circuits) to obtain resonance, therefore, the appropriate solution is increased the intensity of the Laplace force. This can be done using an electromagnet for the coupling or an amplifier as is generally done for magnetic bearing systems (Ji, 2003; Ji et al., 2008; Preumont, 2006). An electromagnet runs with a permanent time varying voltage. Here a large band amplifier made up of TDA of type 2050 (http://www.datasheetarchive.com, 2008) is used for this task. See Fig. 3. With a DC voltage of 40 V, the TDA presents a very low distortion output signal and a high output power. The parameters of the amplifier are listed in Table 1. The amplified oscillator is obtained by connecting the vdPD oscillator to the input of the amplifier delivery through a resistance r0 = 2 X. The resulting equation is given by
2 d V dV i0 L V cos h LC 2 # 1 þ VN ds ds V0 V 0 # Ri0 ðR8 þ RL Þ V ¼ 0; þ sin h L V0
Table 1 Physical properties of the audio amplifier. Components
Symbol
Values
Unity
Resistance
R1 R2 R3 R4 R5 R6
22,000 22,000 22,000 10,000 330 10
X X X X X X
Capacitance
Polarization
C1
47 106
F
C2
1 106
F
C3
3:3 103
F
C4
47 106
F
C5
108
F
V cc
40
V
ð4Þ
where
L R8 # ¼ a0 þ C ðR8 þ RL Þ ; R R R RL ¼ 8 a0 are positive real numbers; N R R==ðR8 =a0 Þ R8 ¼ R==R1 ==ðZ e =A12 Þ; r0 ½1 þ ðR4 þ R3 ==Z s Þ=R2 þ R3 ==Z s ð1 þ R4 =R2 Þ ; D D ¼ R3 ==Z s ½1 þ R4 =ðZ e ==R2 Þ þ r 0 ½1 þ Av ðR3 ==Z s Þ=Zs
A12 ¼
Fig. 4. Amplification process.
þðR4 þ R3 ==Z s Þ=ðR2 ==Z e Þ; RR36 Ri Rj 1; and Ri ==Rj ¼ ; a0 ¼ R3M R26 Ri þ Rj
points corresponding to the founding target signal on the electrical circuit.
Ze = 500 KX is the input impedance of the audio amplifier and Zs = 4 X the output impedance, Av is the voltage gain of the composite in audio amplifier given as (Tooley, 2006): V cc Ze Av ¼ K 0 e V ; Ze þ Zs
K 0 ¼ 875;
and V ¼ 20 V:
ð5Þ
The output voltage of the original VdPD oscillator (Fig. 2) is thus amplified by the device of Fig. 3 with a factor almost equal to 27 (where 27 is the voltage gain of the amplifier that can be obtained using electronic laws; Tooley, 2006). Fig. 4 shows the amplification process. The vertical scale for the amplified curve is the half of the corresponding non-amplified curve. Observe that the experimental voltage gain is almost 27. The signal from the electronic circuit was observed using a two-input probes oscilloscope of type HM303-6 (Hameg Instrument). The inputs were connected to the
2.4. The modified self-sustained MaEMS Connecting the mechanical arm at the output of the modified VdPD oscillator, that is replacing the resistance r0 by the mechanical arm, yields the following expression of the actuation force:
FðtÞ ¼ Bf lA22 V L ;
ð6Þ
with A22 ¼ ðR3 Zs Þ=Ze þAv ðR3D==Zs Þð1þR4 =R2 Þ=Zs and V L ¼ a0 V þ Ri0 sin hðVV0 Þþ is the voltage across the mechanical arm. In Eq. (6), it was asRC dV ds sumed that the inductance effects in the coupling branch are negligible compared to the effects of the inner resistance of the coupling . The electromechanical zone and Lenz electromotive voltage Bf i dY ds system is then modelled by a system of two differential equations (Eqs. (1) and (4)) with elastic (with nonlinear term) and dissipative coupling as shown in Eq. (6). A theoretical study of a similar system
Fig. 3. The audio amplifier: (a) Synoptic of the audio amplifier. (b) Experimental realization of the audio amplifier.
C.A. Kitio Kwuimy, P. Woafo / Mechanics Research Communications 37 (2010) 106–110
109
Fig. 5. Chaotic phase portrait of the system. (a) Chaotic phase portrait of the electric circuit. (b) Chaotic phase portrait of the mechanical arm.
is done in Kitio Kwuimy and Woafo (2007). In the next section, results from theoretical studies are briefly mentioned.
the experimental simulation of the behaviors of the MaEMS considered.
3. Background
4. Experimental simulations
Many theoretical works have been devoted to mathematical analysis and numerical simulation of a device modelled by equations of the type (1) and (4). Averaging, multi-scale method and harmonic balance are some of the mathematical tools used to characterize periodic or quasi periodic behavior. Lyapunov exponents, Poincare maps and bifurcation diagrams associated to the fourth order Runge Kutta algorithm are helpful to investigate nonlinear phenomena. Amongst useful nonlinear phenomena found, there are quenching, bifurcation, chaos, sub and super harmonics, and hysteresis. See for details Kitio Kwuimy and Woafo (2007, 2008, 2009) and Yamapi and Woafo (2009). Since the physical variables are all mixed with the dimensionless coefficients, it appears too complicated to vary a coefficient while maintain the others constant as currently done in theoretical studies. Moreover, imperfections in the experimental model and various approximations will generate a slight difference in theoretical and experimental physical values. Thus, in what follows, attention is focussed on
4.1. The readout principle
Table 2 Physical properties of the VdPD oscillator for various bifurcations. Components
Symbol
Values
Unity
Resistance
R3M R36 R26 R
1200 1.5 232 0.6
X X X X
Capacitance
C
32 1012
F
Inductance
RL L
289 1020
X mH
The best way to measure the dynamical characteristic of an oscillating body is to use either a motion detector or an accelerometer. Here the video analyzer function of a motion detector of type Looger Pro 3.6.1 is used. Data are extracted from the video of the displacement of the mechanical arm. An experimental model of nonlinear MaEMS, shown in Fig. 1, has been designed and fabricated. A photographic image of the parts of the device is shown in Fig. 5. The following subsections give results from experiments compared with those of numerical simulations. 4.2. Experimental bifurcation and chaos In order to test whether the device can experimentally achieve bifurcation and chaos, the physical values of Table 2 are used with R3M as a control parameter. It is observed that, by increasing R3M from zero, there is no displacement of the mechanical arm for R3M < 700 X. At this value, the device enters into chaotic dynamics. A chaotic phase portrait for R3M ¼ 1 kX is plotted in the plane ðV; VLÞ in Fig. 5a for the electrical circuit and in Fig. 5b in the phase plane for the mechanical arm. The next bifurcation point is at R3M ¼ 1:7 kX. By increasing R3M , the system goes from a chaotic dynamics to an almost bi-periodic dynamics. Fig. 6a and b (respectively for the electrical and mechanical parts) show the phase portrait for R3M ¼ 1:9 kX, with maximum amplitude of 0.9 and 1.45 cm for the mechanical arm and 0.41 and 0.49 V for the VdPD oscillator. The values of the mechanical displacement are obtained with an appropriate ruler. The experiments
Fig. 6. periodic phase portrait. (a) Periodic phase portrait of the electric circuit. (b) Periodic phase portrait of the mechanical arm.
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Fig. 7. Death and birth amplitude. (a) Displacement of the electric circuit. (b) Displacement of the mechanical arm.
show a possibility of death and birth oscillation for large values of the control parameter ðR3M > 3 kXÞ. The phenomenon is illustrated in Fig. 7. This result was not obtained numerically from Eqs. (1) and (4). 5. Conclusion In this paper, a self-sustained Macro ElectroMechanical System (MaEMS) was considered. The device is made up of an electrical implementation of a van der Pol-Duffing oscillator driving a mass spring damping oscillator. An audio amplifier has been integrated in order to avoid experimental constraints. Using Newton and Kirchhoff laws, equations of the devices have been derived showing a self-sustained oscillator coupled to a mass–spring–damper linear oscillator. An experimental model was fabricated and by tuning some physical parameters some interesting phenomena have been found. Results of the work are relevant to a broad variety of applications including actuation, mixing and energy autonomy. The device shows a possibility of generating harmonic and complex dynamics, such as bifurcations and chaotic behavior, by using a low power DC voltage. These bifurcations and chaotic behavior can be used in the optimization process such as shaking, mixing and manufacturing chains. This paper opens the way to experimental synchronization of two or more self-sustained electromechanical devices both in their regular and chaotic dynamics as reported in theoretical studies. Acknowledgements This work is supported by the Academy of Sciences for the Developing World (TWAS) under Research Grant No. N.03-322
RG/PHYS/AF/AC. C.A. Kitio Kwuimy thanks Dr. Nana Bonaventure for the prolific discussions. References D’Acunto, M., 2006. Determination of limit cycles for a modified van der Pol oscillator. Mechanics Research Communications 33, 93–98. Hamdi, M., Belhaq, M., 2009. Self-excited vibration control for axially fast excited beam by a time delay state feedback. Chaos Solitons and Fractals 41, 521–532. Hayashi, C., 1964. Nonlinear Oscillations in Physical Systems. McGraw-Hill, New York., 2008. Ji, J.C., 2003. Stability and bifurcation in an electromechanical system with time delay. Mechanics Research Communications 30, 217–225. Ji, J.C., Hansen, C.H., Zander, A.C., 2008. Nonlinear dynamics of magnetic bearing systems. Journal of Intelligent Material System and Structures 12, 1471–1491. Kitio Kwuimy, C.A., Woafo, P., 2007. Dynamics of a self-sustained electromechanical system with flexible arm and cubic coupling. Communications in Nonlinear Science and Numerical Simulation 12 (2), 1504–1517. Kitio Kwuimy, C.A., Woafo, P., 2008. Dynamics, chaos and synchronization of selfsustained electromechanical system with clamped-free flexible arm. Nonlinear Dynamics 53, 201–208. Kitio Kwuimy, C.A., Woafo, P., 2009. Modelling and dynamics of a self-sustained electrostatic Macro ElectroMechanical System. Journal of Computational Nonlinear Dynamics. Nayfeh, A.H., Mook, D.T., 1979. Nonlinear Oscillations. Wiley-Interscience, New York. Preumont, A., 2006. Mechatronics, Dynamics of Electromechanical and Piezoelectric Systems. Springer, Dordrecht. Tooley, M., 2006. Electronic Circuits: Fundamentals and Applications, third ed. Elsevier, Oxford. Ueda, Y., Akamatsu, N., 1981. Chaotically transitional phenomena in the forced negative resistance oscillator. IEEE Transaction of Circuits and Systems 28, 217– 223. van der Pol, B., van der Mark, J., 1927. Frequency demultiplication. Nature 120, 363– 364. Yamapi, R., Woafo, P., 2009. Nonlinear Electromechanical Devices: Dynamics and Synchronization, Mechanical Vibrations: Measurement, Effects and Control. Nova Science Publishers.
Contents lists available at ScienceDirect
Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom
Experimental realization and simulations a self-sustained Macro ElectroMechanical System C.A. Kitio Kwuimy *, P. Woafo Laboratory of Modelling and Simulation in Engineering and Biological Physics, University of Yaounde I, Box 812, Yaounde, Cameroon
a r t i c l e
i n f o
Article history: Received 7 August 2009 Received in revised form 19 October 2009 Available online 10 November 2009 Keywords: Self-sustained MaEMS Experimental simulations Bifurcations and chaos
a b s t r a c t This paper deals with experimental realization and simulations of a self-sustained electromechanical system made up of an electrical implementation of a van der Pol-Duffing type oscillator driving a macro scale mass–spring–damper linear oscillator. Using fundamental laws such as Newton and Kirchhoff laws, the parts of the system are characterized and constraints for experimental realization of a prototype are defined. Modelling equations are derived that show the possibility of self-oscillations. Experimental simulations reveal that the proposed device exhibits periodic oscillations and complex dynamics. These results are consistent with published theoretical results. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Nonlinear systems may exhibit considerably complex dynamic behavior such as change in stability of response, quenching, quasiperiodic motion and chaotic motion. Both predictability and stability of engineering systems are rather important, which explains why research in the area of dynamics, synchronization and control of nonlinear systems has received a great deal of attention in the past two decades. An important class of engineering systems is magnetically actuated. In this class of systems, nonlinear terms can arise from a mechanical part (material, geometric or inertial nonlinearities), from an electrical circuit (nonlinear self, nonlinear condenser, nonlinear resistance) and from the coupling (coupling between the electromagnets, saturation, hysteresis, nonlinear magnetic force, time delay). See for details Hayashi (1964), Ji (2003), Ji et al. (2008), Nayfeh and Mook (1979) and Preumont (2006). One consequence of nonlinearity is self-sustained oscillation. Self-sustained engineering systems can run without a permanent time varying excitation, thus they have some economic advantages. The earliest mathematical model associated to self-excited systems, which is now called the van der Pol equation, was derived by van der Pol (van der Pol and van der Mark, 1927) to describe a vacuum tube circuit. The electrical model of the van der Pol equation is made up of negative resistance and it has been shown that this model presents complex phenomena (Ueda and Akamatsu, 1981).
* Corresponding author. Tel.: +27 733 14 59 08. E-mail address: [email protected] (C.A. Kitio Kwuimy). URLs: http://users.aims.ac.za/~cedrick/ (C.A. Kitio Kwuimy), http://www. lamsebp.org (C.A. Kitio Kwuimy, P. Woafo). 0093-6413/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2009.10.007
Many researchers have presented contributions to understand and apply self-oscillation in engineering. D’Acunto (2006) recently proposed a modified van der Pol equation to describe a self-excited body sliding on a periodic potential, Hamdi and Belhaq (2009) examined the self-excitated vibration of a simply-supported beam subjected to an axially high frequency excitation. In the area of Macro ElectroMechanical Systems (MaEMS), Ji (2003). Kitio Kwuimy and Woafo (2007, 2008) and Yamapi and Woafo (2009) presented some models of MaEMS with self-sustained oscillations. A model of self-sustained MEMS was recently presented by Kitio Kwuimy and Woafo (2009). Extensions have been made on synchronization and control of nonlinear phenomena. All the results and references mentioned above show that selfsustained MaEMS may really exhibit remarkable dynamical behavior. Since these results were based on mathematical analysis and numerical simulations, two fundamental questions arise: (1) How to implement experimentally a self-sustained magnetically actuated MaEMS? This interrogation will define experimental constraints. (2) Can nonlinear phenomena be observed experimentally in such MaEMS? A positive response of these questions will motivate extension to synchronization and control. In this paper, experimental realization and simulation of a van der Pol-Duffing (vdPD) oscillator driving a mass spring damping mechanical arm is considered. Descriptions of the parts of the device are given. In order to achieve experimental constraints an audio amplifier is integrated into the device. The modelling equations of the device are then derived showing a self-sustained oscillator coupled to a mass–spring–damped linear oscillator. Based on theoretical results published on these types of systems, attention is focussed on experimental simulations. The paper is presented as follows. Section 2 deals with the characterization of the device
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and the modelling equations. Section 3 presents a brief background on theoretical studies. In Section 4, results of experimental simulations are presented. Section 5 is for the conclusion and hints of applications. 2. Experimental set-up of the self-sustained MaMES Fig. 1 shows a schematic representation of the self-sustained MaEMS to be investigated. It consists of a mechanical part and a Both parts are coupled through a magnetic field of magnitude Bf = 16 103 T. The coil is made of copper wire of length l = 3.768 m and has a resistance r0 = 2 X and an inductance L0 = 1.43 109 H. 2.1. The mechanical part The mechanical part is made of a rigid rod and a linear spring of elasticity constant K = 7.98 N/m. The total mass of the mechanical elements is m = 40 g and its dissipative coefficient k = 1.13 104 Kg s1 determined from the measurement of the logarithm decrement. Using Newton’s laws, the dynamics of the mechanical part is modelled by the following second order differential equation:
d Y dY þk þ KY ¼ FðtÞ; ds2 ds
contrasts with the frequency of MEMS and NEMS whose values range from Megahertz to Terahertz. 2.2. The electrical part The electrical part is a van der Pol-Duffing (vdPD) oscillator made up of resistances, inductor, capacitance, operational amplifier (JR42AF) as linear components and diodes (IN 4001) as non linear components (see Fig. 2). The operational amplifiers are polarized with a Direct Current (DC) voltage of 12 V using a DCregulated power of type PS23023 (Hight Quality). The diode voltage–current characteristic is given by the following relation (Tooley, 2006):
i ¼ i0 ðexp ðV=V 0 Þ 1Þ;
ð2Þ 4
where i0 is the reverse saturation current equal to 10 A and V0 = 26 103 V at the room temperature. Kirchhoff’s laws lead to (in the absence of the mechanical part):
2 d V dV Li0 V # 1 cos h þ VN ds2 ds V0 V 0# V þ i0 ðR þ RL Þ sin h V0
LC
2
m
where FðtÞ ¼ Bf li, i is the current through the coupling zone and Y is the displacement of the mechanical arm (see Fig. 1). The natural freqffiffiffi K ¼ 14:12 Hz. This quency of the mechanical part is given by x1 ¼ m
ð1Þ
¼ 0; where # ¼
ð3Þ h
i
LR36 CR3M R26 ðRþRL Þ R3M R26
ðRþRL Þ and N ¼ 1 R36 , should be positive R3M R26
real numbers and V is the voltage across the diodes. Eq. (3) can be viewed as a generalized form of the van der Pol equation, by expanding sin h and cos h. 2.3. Constrains for experimental realization As the mechanical arm moves under the action of a vertical force at least equal to the weight of the mechanical elements, one finds that the Laplace force should be at least 0.4 N. In addition, when activating the mechanical arm with the proposed electronic device, the maximum delivery force (the Laplace one) is around 1 mN which is less than the required force to set the mechanical arm into motion. Furthermore, the natural frequency of the electronic oscillator is about 3 KHz which is very high compared to that of the mechanical arm (x1 = 14.12 Hz). These two facts constitute limitations to the functioning of the device. A generalization can be made with many macro scale electromechanical system. There is thus a necessity to overcome the limitations. It is quite difficult to increase the frequency of the mechanical arm (or to reduce one
Fig. 1. Synoptic of the self-sustained Macro ElectroMechanical System.
Fig. 2. Synoptic of the van der Pol oscillator.
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C.A. Kitio Kwuimy, P. Woafo / Mechanics Research Communications 37 (2010) 106–110
of the electrical circuits) to obtain resonance, therefore, the appropriate solution is increased the intensity of the Laplace force. This can be done using an electromagnet for the coupling or an amplifier as is generally done for magnetic bearing systems (Ji, 2003; Ji et al., 2008; Preumont, 2006). An electromagnet runs with a permanent time varying voltage. Here a large band amplifier made up of TDA of type 2050 (http://www.datasheetarchive.com, 2008) is used for this task. See Fig. 3. With a DC voltage of 40 V, the TDA presents a very low distortion output signal and a high output power. The parameters of the amplifier are listed in Table 1. The amplified oscillator is obtained by connecting the vdPD oscillator to the input of the amplifier delivery through a resistance r0 = 2 X. The resulting equation is given by
2 d V dV i0 L V cos h LC 2 # 1 þ VN ds ds V0 V 0 # Ri0 ðR8 þ RL Þ V ¼ 0; þ sin h L V0
Table 1 Physical properties of the audio amplifier. Components
Symbol
Values
Unity
Resistance
R1 R2 R3 R4 R5 R6
22,000 22,000 22,000 10,000 330 10
X X X X X X
Capacitance
Polarization
C1
47 106
F
C2
1 106
F
C3
3:3 103
F
C4
47 106
F
C5
108
F
V cc
40
V
ð4Þ
where
L R8 # ¼ a0 þ C ðR8 þ RL Þ ; R R R RL ¼ 8 a0 are positive real numbers; N R R==ðR8 =a0 Þ R8 ¼ R==R1 ==ðZ e =A12 Þ; r0 ½1 þ ðR4 þ R3 ==Z s Þ=R2 þ R3 ==Z s ð1 þ R4 =R2 Þ ; D D ¼ R3 ==Z s ½1 þ R4 =ðZ e ==R2 Þ þ r 0 ½1 þ Av ðR3 ==Z s Þ=Zs
A12 ¼
Fig. 4. Amplification process.
þðR4 þ R3 ==Z s Þ=ðR2 ==Z e Þ; RR36 Ri Rj 1; and Ri ==Rj ¼ ; a0 ¼ R3M R26 Ri þ Rj
points corresponding to the founding target signal on the electrical circuit.
Ze = 500 KX is the input impedance of the audio amplifier and Zs = 4 X the output impedance, Av is the voltage gain of the composite in audio amplifier given as (Tooley, 2006): V cc Ze Av ¼ K 0 e V ; Ze þ Zs
K 0 ¼ 875;
and V ¼ 20 V:
ð5Þ
The output voltage of the original VdPD oscillator (Fig. 2) is thus amplified by the device of Fig. 3 with a factor almost equal to 27 (where 27 is the voltage gain of the amplifier that can be obtained using electronic laws; Tooley, 2006). Fig. 4 shows the amplification process. The vertical scale for the amplified curve is the half of the corresponding non-amplified curve. Observe that the experimental voltage gain is almost 27. The signal from the electronic circuit was observed using a two-input probes oscilloscope of type HM303-6 (Hameg Instrument). The inputs were connected to the
2.4. The modified self-sustained MaEMS Connecting the mechanical arm at the output of the modified VdPD oscillator, that is replacing the resistance r0 by the mechanical arm, yields the following expression of the actuation force:
FðtÞ ¼ Bf lA22 V L ;
ð6Þ
with A22 ¼ ðR3 Zs Þ=Ze þAv ðR3D==Zs Þð1þR4 =R2 Þ=Zs and V L ¼ a0 V þ Ri0 sin hðVV0 Þþ is the voltage across the mechanical arm. In Eq. (6), it was asRC dV ds sumed that the inductance effects in the coupling branch are negligible compared to the effects of the inner resistance of the coupling . The electromechanical zone and Lenz electromotive voltage Bf i dY ds system is then modelled by a system of two differential equations (Eqs. (1) and (4)) with elastic (with nonlinear term) and dissipative coupling as shown in Eq. (6). A theoretical study of a similar system
Fig. 3. The audio amplifier: (a) Synoptic of the audio amplifier. (b) Experimental realization of the audio amplifier.
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Fig. 5. Chaotic phase portrait of the system. (a) Chaotic phase portrait of the electric circuit. (b) Chaotic phase portrait of the mechanical arm.
is done in Kitio Kwuimy and Woafo (2007). In the next section, results from theoretical studies are briefly mentioned.
the experimental simulation of the behaviors of the MaEMS considered.
3. Background
4. Experimental simulations
Many theoretical works have been devoted to mathematical analysis and numerical simulation of a device modelled by equations of the type (1) and (4). Averaging, multi-scale method and harmonic balance are some of the mathematical tools used to characterize periodic or quasi periodic behavior. Lyapunov exponents, Poincare maps and bifurcation diagrams associated to the fourth order Runge Kutta algorithm are helpful to investigate nonlinear phenomena. Amongst useful nonlinear phenomena found, there are quenching, bifurcation, chaos, sub and super harmonics, and hysteresis. See for details Kitio Kwuimy and Woafo (2007, 2008, 2009) and Yamapi and Woafo (2009). Since the physical variables are all mixed with the dimensionless coefficients, it appears too complicated to vary a coefficient while maintain the others constant as currently done in theoretical studies. Moreover, imperfections in the experimental model and various approximations will generate a slight difference in theoretical and experimental physical values. Thus, in what follows, attention is focussed on
4.1. The readout principle
Table 2 Physical properties of the VdPD oscillator for various bifurcations. Components
Symbol
Values
Unity
Resistance
R3M R36 R26 R
1200 1.5 232 0.6
X X X X
Capacitance
C
32 1012
F
Inductance
RL L
289 1020
X mH
The best way to measure the dynamical characteristic of an oscillating body is to use either a motion detector or an accelerometer. Here the video analyzer function of a motion detector of type Looger Pro 3.6.1 is used. Data are extracted from the video of the displacement of the mechanical arm. An experimental model of nonlinear MaEMS, shown in Fig. 1, has been designed and fabricated. A photographic image of the parts of the device is shown in Fig. 5. The following subsections give results from experiments compared with those of numerical simulations. 4.2. Experimental bifurcation and chaos In order to test whether the device can experimentally achieve bifurcation and chaos, the physical values of Table 2 are used with R3M as a control parameter. It is observed that, by increasing R3M from zero, there is no displacement of the mechanical arm for R3M < 700 X. At this value, the device enters into chaotic dynamics. A chaotic phase portrait for R3M ¼ 1 kX is plotted in the plane ðV; VLÞ in Fig. 5a for the electrical circuit and in Fig. 5b in the phase plane for the mechanical arm. The next bifurcation point is at R3M ¼ 1:7 kX. By increasing R3M , the system goes from a chaotic dynamics to an almost bi-periodic dynamics. Fig. 6a and b (respectively for the electrical and mechanical parts) show the phase portrait for R3M ¼ 1:9 kX, with maximum amplitude of 0.9 and 1.45 cm for the mechanical arm and 0.41 and 0.49 V for the VdPD oscillator. The values of the mechanical displacement are obtained with an appropriate ruler. The experiments
Fig. 6. periodic phase portrait. (a) Periodic phase portrait of the electric circuit. (b) Periodic phase portrait of the mechanical arm.
110
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Fig. 7. Death and birth amplitude. (a) Displacement of the electric circuit. (b) Displacement of the mechanical arm.
show a possibility of death and birth oscillation for large values of the control parameter ðR3M > 3 kXÞ. The phenomenon is illustrated in Fig. 7. This result was not obtained numerically from Eqs. (1) and (4). 5. Conclusion In this paper, a self-sustained Macro ElectroMechanical System (MaEMS) was considered. The device is made up of an electrical implementation of a van der Pol-Duffing oscillator driving a mass spring damping oscillator. An audio amplifier has been integrated in order to avoid experimental constraints. Using Newton and Kirchhoff laws, equations of the devices have been derived showing a self-sustained oscillator coupled to a mass–spring–damper linear oscillator. An experimental model was fabricated and by tuning some physical parameters some interesting phenomena have been found. Results of the work are relevant to a broad variety of applications including actuation, mixing and energy autonomy. The device shows a possibility of generating harmonic and complex dynamics, such as bifurcations and chaotic behavior, by using a low power DC voltage. These bifurcations and chaotic behavior can be used in the optimization process such as shaking, mixing and manufacturing chains. This paper opens the way to experimental synchronization of two or more self-sustained electromechanical devices both in their regular and chaotic dynamics as reported in theoretical studies. Acknowledgements This work is supported by the Academy of Sciences for the Developing World (TWAS) under Research Grant No. N.03-322
RG/PHYS/AF/AC. C.A. Kitio Kwuimy thanks Dr. Nana Bonaventure for the prolific discussions. References D’Acunto, M., 2006. Determination of limit cycles for a modified van der Pol oscillator. Mechanics Research Communications 33, 93–98. Hamdi, M., Belhaq, M., 2009. Self-excited vibration control for axially fast excited beam by a time delay state feedback. Chaos Solitons and Fractals 41, 521–532. Hayashi, C., 1964. Nonlinear Oscillations in Physical Systems. McGraw-Hill, New York.