# the nonlinear dynamics of a vibrating system modelled by a inverted

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the nonlinear dynamics of a vibrating system modelled by a inverted

THE NONLINEAR DYNAMICS OF A VIBRATING SYSTEM MODELLED BY A INVERTED PENDULUM, WITH AN ELECTRODYNAMIC SHAKER Ana Rosa Klinke1 , José Manoel Balthazar2 , Jorge Luiz Palacios Felix3 , Fábio Roberto Chavarette4 1,2 UNESP: Univ Estadual Paulista, Department of Statistics Applied Mathematics and Computation, PO Box 178, 13500-230, Rio Claro SP, Brazil, [email protected], [email protected] 3 Regional University of Northwestern of Rio Grande do Sul - Department of Physics Mathematical and Statistics, São Geraldo, Brazil, [email protected] 4 UNESP: Univ Estadual Paulista, Department of Mathematics, Ilha Soltlteira, Brazil, [email protected] Abstract: The technology of macro-electro-mechanical systems has many applications on Engineering Sciences. However, the design of such mechanical systems can be quite challenging due to nonlinear effects, which may strongly affected the dynamics behavior. The physical model studied is a macro-electro-mechanical coupling with an electrodynamic shaker, by using numerical simulations. Keywords: Macro-Electro-Mechanical Systems, Nonlinear Dynamics, Chaos. 1. INTRODUCTION Micro-electro-mechanical systems (MEMS) are small integrated devices or systems those combine electrical and mechanical components. Examples of MEMS devices applications are emerging as the existing technology is applied to the miniaturization and integration of conventional devices. Examples of MEMS devices applications,include inertial sensors, as an example. They are used as an example in airbag-deployment sensors, in automobiles, and as tilt or shock sensors Some details of (MEMS), may be easly find on the web address (http://en.wikipedia.org/wiki/Microelectromechanical systems). However, the design of such dynamical systems, may be quite challenging, due to their nonlinear effects, which may strongly affect their dynamics behavior. It is known that nonlinear systems, have been shown to exhibit chaotic behavior, as well a periodic oscillations. A chaotic system is a nonlinear deterministic system, which it is very sensitive to small perturbations on its initial conditions, and its long time behavior it is unpredictable . It is well known that the classical tuning-fork (MEMS) gyroscope, contains a pair of masses that are driven to oscillate with equal amplitude but in opposite directions. When rotated, the Coriolis force creates an orthogonal vibration that can be sensed by a variety of mechanisms. The Draper Lab gyroscope shown in Figure 1, uses a comb-type structures to drive the tuning fork into resonance. Rotation causes the proof masses to vibrate out of plane, and this motion is sensed capacitively with a custom CMOS ASIC. The micro-tuningfork gyroscope,presented in Figure 1, may be mathematically modeled(by a Macro Model)by being two in- verted pendulums, which their motions, opposing set in support of mass vertical and horizontal movement [1, 2]. We remarked that and, include, the effects of the basis of excitation of the pendulum (parametric excitation), used an electro-dynamics shaker [3, 4]. Its nonlinear dynamical behavior was studied by [3, 4] basead on a Macro model. In this paper, we consider two Macro models. They also presented rich nonlinear dynamical behavior [4]. Next we will discuss new and interesting models, separately. Figure 1 – A tuning fork gyroscope with comb drive for commercial applications by Draper Lab Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications Serra Negra, SP - ISSN 2178-3667 1179 2. MATHEMATICAL MODEL I are obtained The simplified Macro-Mechanical-Mathematical Model I, considered, consists of one inverted pendulum fixed in suspension mass, showed in Figure 2. The pendulum has a mass m connected by a rod of length l. And are θ the angle of pendulum and stabilized by torsion springs, with strongly nonlinear, kθ = k + hθ2 . The suspension with mass M , and spring constant Kx for X direction [4]. Then the variables of the problem are the displacement and the suspension of mass in the horizontal respectively (two degree of Freedom). 1 (f1 + η1 f2 cos θ) ∆ (7) 1 (F1 f1 cos θ + f2 + F ) ∆ (8) θ̈ = Ẍ = where ∆ = 1 − F1 η1 cos θ2 f1 = −δ1 θ − λ1 θ3 − µ1 θ̇ + η1 g1 sin θ f2 = −µ4 Ẋ − ωx 2 X − F1 θ̇2 sin θ F = A cos Ωτ (9) (10) (11) (12) Making x1 = θ, x2 = θ̇, x3 = X, x4 = Ẋ, has a system of first order in the state variables x˙1 = x2 x˙2 = 1 x˙4 = (F1 f1 cos x1 + f2 + F ) ∆ Figure 2 – Simple model of the Macro-Electro-Mechanical Model I The governing equations of motion, for the MacroElectro-Mechanical model, defined by Figure 2, are ml2 θ̈ + cl2 θ̇ + k1 θ + hθ3 = mlẍ cos θ + ml(ÿ + g) sin θ (1) Mt ẍ + cx ẋ + kx x − ml(θ̈ cos θ − θ̇2 sin θ) = 0 (2) Where Mt = M + m and g is the acceleration of gravity. They may be transformed into dimensionless equations, making the following changes of variables: kx 1 X = lxc , τ = ωe t, ωe = √LC , ωx = M ωe 2 , t 0 k1 h1 1 l1 F1 = m M t lc , δ 1 = m 1 l1 2 ω e 2 , λ 1 = m 1 l 1 2 ω e 2 , µ1 = m1 lc112 ωe , η1 = ll1c , g1 = ωeg2 lc , µ4 = Mctxωe . Considering the simplified motion in the horizontal direction and a excitation force on Y direction F = A cos Ωτ , the equation of motion of the simplified system is given by: θ̈ + δ1 θ + λ1 θ3 + µ1 θ̇ − η1 (Ẍ cos θ + g1 sin θ) = 0 2 2 Ẍ + µ4 Ẋ + ωx X − F1 (θ̈ cos θ − θ̇ sin θ) = F (3) (4) Isolating θ̈ and Ẍ: θ̈ − η1 Ẍ cos θ = −δ1 θ − λ1 θ3 − µ1 θ̇ + η1 g1 sin θ ≡ f1 (5) Ẍ − F1 θ̈ cos θ = F − µ4 Ẋ − ωx 2 X − F1 θ̇2 ≡ f2 (6) 1 (f1 + η1 f2 cos x1 ) ∆ x˙3 = x4 (13) (14) (15) (16) where ∆ = 1 − F1 η1 cos x1 2 f1 = −δ1 x1 − λ1 x1 3 − µ1 x2 + η1 g1 sin x1 f2 = −µ4 x4 − ωx 2 x3 − F1 x˙2 2 sin x1 F = A cos Ωτ (17) (18) (19) (20) 2.1. NUMERICAL SIMULATION RESULTS The numerical simulations, were carried out, by using Matlabő, taken as the numerical integrator the RungeKutta method of fourth order algorithm with variable time steplength. The following is presented an analysis of the behavior of each component of the proposed models, making an approach on the nonlinear behavior. The initial conditions of (θ, θ̇, X, Ẋ), were fixed as [0.12, 0.01, 0.0025, 0.01].The equations parameters for the oscillator are shown in Table 1 [1, 2, 5], are: The numerical results of the equations 13, 14, 15, 16 is illustrated through Figures 3, 4, 5. The Figure 3(a) shows the phase portrait of (θ, θ̇) and Figure 3(b) shows the phase motion of (X, Ẋ). The time history of angular displacement of θ in the time range 0 ≤ t ≤ 4000 is showed in Figure 4(a) and the time history of X is showed by Figure 4(b) Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications Serra Negra, SP - ISSN 2178-3667 1180 Table 1 – Parameters for the chaotic oscillator. Parameters Values A 0.2 Ω 1.4 F1 0.1 µ4 0 µ1 0.01 ωx 1.0 η1 0.9 λ1 0.75 g1 1.0 δ1 1.0 2.2. PARAMETERS ANALYSIS The following results, were obtained in order to observe the effect of parameter bifurcation in equations 13, 14, 15, 16. The bifurcation parameter, change in the invervalo 0 ≤ P arameter ≤ 6 and other parameters of the equations, were held constant. Figure 6 shows the bifurcation diagram of the parameter µ4 , with the purpose of observing the damping effects of the suspension mass on the oscillation of the pendulums (X-direction). The analysis of results shows that when the parameter µ4 is very close to 0, causing an instability in the considered system. Figure 3: (a) Phase Portrait of θ, (b) Phase Portrait of X. Figure 6: Bifurcation Diagram of parameter µ4 Figure 7 shows the bifurcation diagram of the parameter ωx , this parameter represents the natural excitement of the system in the direction X. When this parameter adquive a value between 1.2 and 1.4 the system becomes unstable. Figure 4: (a) Time History of θ, (b)Time History of X. Figure 5(a) illustrates the frequency spectrum of angular displacement θ and Figure 5(b) the frequency spectrum of motion X. According to the angular movement of the pendulum and the movement in the X direction ,we can see a nonlinear interaction between the inverted pendulum and the force excitation, evidenced by the three peaks, occurred in the frequency spectrum. Figure 7: Bifurcation Diagram of parameter ωx Figure 5: (a)Frequency Spectrum of θ, (b) Frequency Spectrum of X. Next,we present the results of the numerical simulations, those were carried out, is done to observe the influence of the parameter, λ1 , of the nonlinearity of stiffness of the pendulum. The bifurcation diagram showed the interaction of this parameter, with the system illustrated in Figure 8. As ex- Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications Serra Negra, SP - ISSN 2178-3667 1181 pected, with increasing of the cubic nonlinearity coefficient λ1 , the degree of instability of the system increases considerably. Increasing of instability is caused by the fact that this parameter is a nonlinear coefficient of the system. Figure 10: Bifurcation Diagram of parameter A 3. SIMPLIFIED MATHEMATICAL MODEL II Figure 8: Bifurcation Diagram of parameter λ1 The bifurcation diagram of the parameter w, shown in Figure 9, illustrates the effect of parameter oscillation frequency of the excitation force, F = A cos Ωτ , in the system. We can observe the occurrence of instabilidade of the system, when the parameter ranging from 0.4 to 1.3. Figure 11 – Macro-Electro-Mechanical Model II, coupling with an electrodynamic shaker Figure 9: Bifurcation Diagram of parameter Ω The study of the amplitude A shows the intensity of the excitation force F = A cos Ωτ . The bifurcation diagram of Figure 10 shows the interaction between this parameter and the system. It is possible to observe that with increasing value of A, there is an instability occurring in the system, which increasing along with increasing A. This result is in complete agreement with the proposed model, because of higher the level of excitement of the pendulum and the suspension of mass those suffering the greater the instability generated in the system. The Macro-Mechanical-Mathematical Model II, considered here, was based on initial experimental work on the basis of excitation of a pendulum (parametric excitation) by using electrodynamic shaker made by [3, 4]. The mechanism is studied here, was implemented as excited at its base by an electrodynamic shaker as shown in Figure 11.The electrodynamic shaker acts on the suspension of mass M. The coupling between both parties is achieved by the electromagnetic force due to the permanent magnet Fem = K q̇ (Fem = lc B, where lc e B are the length of the conductor and the magnetic field, respectively). This generates a force of Laplace on the mechanics and the Lenz electromotive voltage in the power [1, 2, 5]. The electrical system consists of resistor R, an inductor L, a capacitor C and a source of sinusoidal voltage e(t) = e0 cos Ωt(e0 and Ω are the amplitude and frequency respectively; t is the time), all connected in series. In this model, the voltage of the capacitor is a nonlinear function of instantaneous electric charge q, defined as: Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications Serra Negra, SP - ISSN 2178-3667 1182 Vc = 1 q + α3 q 3 C0 (21) θ̈ − η1 (Ẍ cos θ − Ÿ sin θ) = −δ1 θ − λ1 θ3 − µ1 θ̇ + η1 g1 sin θ ≡ f1 where C0 is the linear value of C and α3 is the nonlinear coefficient depending on the type of capacitor to be used [1, 2, 5]. The kinetic energy and potential energy of the system are given by Ẍ − F1 θ̈ cos θ = −µ4 Ẋ − ωx 2 X − F1 θ̇2 sin θ ≡ f2 Ÿ − F1 θ̈ sin θ = −µ5 Ẏ − ωy 2 Y + F1 θ̇2 cos θ + γ1 Q̇ ≡ f3 3 1 1 T = Mt ẋ2 + Mt ẏ 2 − 2 2 1 2 2 1 2 ml(ẋθ̇ cos θ + ẏ θ̇ sin θ) + ml θ̇ + Lq̇ 2 2 ky 2 kx 2 V = Mt gy + mg(l cos θ + y) + y + x + 2 2 1 2 1 2 1 2 α3 4 q K q̇y + kθ + hθ + q + 2 4 2C0 4 Q̈ = −µ3 Q̇ − Q − λ3 Q − γ2 Ẏ + E0 cos (ωτ ) (22) 2 1 1 1 1 c(lθ̇) + Rq̇ 2 + cx ẋ2 + cy ẏ 2 2 2 2 2 x˙1 = x2 1 x˙2 = (f1 + η1 f2 cos x1 + η1 f3 sin x1 ) ∆ x˙3 = x4 1 x˙4 = (F1 f1 cos x1 + f2 ) ∆ x˙5 = x6 1 x˙6 = (F1 f1 sin x1 + f3 ) ∆ x˙7 = x8 (23) (24) The motion equation of the system is given by: 2 2 ml θ̈ + cl θ̇ + k1 θ + hθ3 = mlẍ cos θ + ml(ÿ + g) sin θ (25) Mt ẍ + cx ẋ + kx x − ml(θ̈ cos θ − θ̇2 sin θ) = 0 (26) Mt ÿ + cy ẏ + ky y − ml(θ̈ sin θ + θ̇2 cos θ) − K q̇ = 0 (27) 1 Lq̈ + Rq̇ + q + α3 q 3 + K ẏ = E0 cos (Ωt) (28) C0 Where Mt = M + m and g is the acceleration of gravity. Making the following changes of variables: X= x lc , Y = y lc , Q= √ 1 , ωx = LC0 1 F1 = Mml , δ1 = mlk2 1ωe 2 , λ1 = mlh2 ω 2, t lc e 2 α3 q0 c1 lc λ3 = Lωe 2 , µ1 = ml2 ωe , η1 = l , g1 = ωeg2 lc , c R , µ5 = Mtyωe , µ4 = Mctxωe , µ3 = Lω e Klc 0 γ1 = MKq , γ2 = Lω , E0 = Lωee02 q0 , ω = ωΩe . t ω e lc e q0 ωe = 3 x˙8 = −µ3 x8 − x7 − λ3 x7 − γ2 x6 + E0 cos (ωτ ) (35) (36) (37) (38) (39) (40) (41) (42) (43) (44) where ∆ = 1 − F1 η1 cos x1 2 − F1 η1 sin x1 2 f1 = −δ1 x1 − λ1 x1 3 − µ1 x2 + η1 g1 sin x1 f2 = −µ4 x4 − ωx 2 x3 − F1 x2 2 sin x1 2 2 f3 = −µ5 x4 − ωy x5 + F1 x2 cos x1 + γ1 x8 q q0 , τ = ωe t, ky kx , ω = 2 y Mt ω e Mt ω e 2 , (34) Making x1 = θ, x2 = θ̇, x3 = X, x4 = Ẋ, x5 = Y , x6 = Ẏ , x7 = Q, x8 = Q̇, we will obtain the system of first order in the state variables The dissipation energy function of Rayleigh’s is given by D= (33) (45) (46) (47) (48) (49) 3.1. NUMERICAL SIMULATION RESULTS The initial conditions of (θ, θ̇, X, Ẋ, Y, Ẏ , Q, Q̇) fixed as being (0.12, 0.01, 0.025, 0.01, 0.1, 0.01, 0.3, 0.01) respectively. The equations parameters for the oscillator, which was fabricated using standard silicon on insulator processing device shown in Table 1 [1, 2, 5], are: we will obtain θ̈ + δ1 θ + λ1 θ3 + µ1 θ̇ − η1 (Ẍ cos θ + (Ÿ + g1 ) sin θ) = 0 Ẍ + µ4 Ẋ + ωx 2 X − Table 2 – Parameters for the chaotic oscillator, coupling with an electrodynamic shaker. (29) F1 (θ̈ cos θ − θ̇2 sin θ) = 0 (30) F1 (θ̈ sin θ + θ̇2 cos θ) + γ1 Q̇ = 0 Q̈ + µ3 Q̇ + Q + λ3 Q3 + γ2 Ẏ = E0 cos (ωτ ) (31) (32) 2 Ÿ + µ5 Ẏ + ωy Y − Isolating θ̈, Ẍ, Ÿ and Q̈: Parameters Values Parameters Values F1 0.1 µ4 , µ5 0 µ1 0.01 µ3 0.05 ωx 1.0 ωy 1.0 η1 0.9 γ2 0.2 λ1 0.75 ω 1.5 g1 1.0 ωe 1.0 δ1 1.0 E0 0.24 λ3 1.0 γ1 0.4 Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications Serra Negra, SP - ISSN 2178-3667 1183 The numerical results of the equations 37, 38, 39, 40, 41, 42, 43, 44 are illustrated by Figures 12 to 19. These results are all givwen is in the time range 0 ≤ t ≤ 5000. The Figure 12(a) shows the phase portrait (θ, θ̇) and Figure 12(b) shows the frequency spectrum of θ. The time history of angular displacement of θ̇ is shows by Figure 13(a) and the time history of θ is shows by Figure 13(b). In this system, when we consider the action of the electrodynamic shaker, we obtain that the angular movement of the inverted pendulum is stabilized. This is evidenced by the frequency spectrum , which show only one peak. Figure 15: (a)Time History of Ẋ, (b)Time History of X. In the Figure 16(a) and 16(b) shows the phase portait (Y, Ẏ ) and the frequency spectrum of Y , respectively. The Figure 17(a) and 17(b) illustrates the time history of Ẏ and the time history of Y , respectively. The Y movement has a nonlinear behavior (shown by the frequency spectrum containing multiple peaks). Figure 12: (a)Phase portrait, (b)Frequency Spectrum Figure 16: (a)Phase Portrait of Y , (b)Frequency Spectrum of Y . Figure 13: (a)Time History of θ̇, (b)Time History of θ. The Figure 14(a) shows the phase portrait (X, Ẋ) and Figure 14(b) shows the frequency spectrum of X. The time history of Ẋ is shows by Figure 15(a) and the time history of X is shows by Figure 15(b). In the motion X has a periodic behavior, as illustrated by the phase portrait. Figure 17: (a)Time History of Ẏ , (b)Time History of Y . The behavior of the electrodynamic shaker it is illustrated in the phase portrait (Q, Q̇), Figure 18(a). The Figure 18(b) shows the frequency spectrum of Q. The Figure 19(a) and 19(b) illustrates the time history of Q̇ and the time history of Q, respectively. Figure 14: (a)Phase Portrait of θ, (b)Frequency Spectrum of X. Figure 18: (a)Phase Portrait of Q, (b)Frequency Spectrum of Q. Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications Serra Negra, SP - ISSN 2178-3667 1184 Figure 19: (a)Time History of Q̇, (b)Time History of Q. 3.2. PARAMETERS ANALYSIS The next results of numerical simulations, were obtained, in order to observe the effect of parameter bifurcation in this equations 37, 38, 39, 40, 41, 42, 43, 44. The bifurcation parameter change in the time range 0 ≤ P arameter ≤ 5 and other parameters of the equations were held constant. Figure 20 shows the bifurcation diagram of the parameter µ4 , with the purpose of observing the damping effects of the suspension mass on the oscillation of the pendulums (Xdirection) and the interaction with the eletrodynamic shaker. The analysis of results shows that when the parameter µ4 is very close to 0 causes an instability in the system. This result resembles the results were obtained with the MacroElectro-Mechanical Model I. We conclude that the damping effect on both systems cause instability to values close to 0, this is because the damping effect of the oscillation decreases in directions X and Y . Figure 21: Bifurcation Diagram of parameter ωx Figure 22 shows the bifurcation diagram of the parameter ωy , which represents the natural excitement of the system in the Y direction, analyzing the results we have obtained that the system becomes unstable for values of ωy greater than 1. Figure 22: Bifurcation Diagram of parameter ωy Figure 20: Bifurcation Diagram of parameter µ4 The parameter λ1 represents the nonlinear coefficient of the pendulum, and the bifurcation diagram shows the interaction of this parameter, the system is illustrated in Figure 23. Analyzing the results, we noted that the system behaves unstable, when the coefficient of cubic nonlinearity reached values above 4.5. Figure 21 shows the bifurcation diagram of the parameter ωx , which represents the natural excitement of the system in the X direction. The system becomes unstable in the range of values from 0 to 2. Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications Serra Negra, SP - ISSN 2178-3667 1185 chanical Systems and Signal Processing, pp. 1146–1156, 2008. [3] X. Xu, E. Pavlovskaia, M. wiercigroch, F. Romeo, and S. Lenci., “Dynamic interactions between parametric pendulum and electo-dynamical shaker,” ZAMM Z. Angew. Math. Mech., pp. 172–186, 2007. [4] J. M. Balthazar, J. L. P. Felix, and M. L. R. F. Reyolando, “On an energy transfer and nonlinear, nonideal and chaotic behavior of a macro tuning fork beam(tfb), under an electro-dynamical shaker excitation(eds).,” Proceedings of 11th Pan-American Congress of Applied Mechanics, 2010. Figure 23: Bifurcation Diagram of parameter λ1 [5] J. L. P. Felix and J. M. Balthazar, “Comments on a nonlinear and non-ideal electromechanical damping vibration absorber. sommerfeld effect and energy transfer,” Nonlinear Dynamics, pp. 1–11, 2009. 4. CONCLUSION In this paper, the study of an macro-electro-mechanical, firstly modeled by one inverted pendulum fixed in a suspension mass (Macro-Mechanical-Mathematical Model I), was done by numerical simulations. With the analysis of results obtained in 2.1 Session, we obtained a nonlinear interaction between the inverted pendulum and the force excitation, evidenced by the three peaks, occurred in the frequency spectrum, Figure 5. The results of the bifurcation diagrams, Session 2.2, showed that the parameters λ1 , A, Ω, ωx generated instability in the system, as illustrated by Figures 8, 9, 7, 10, respectively. The second system, macro-electro-mechanical (MacroMechanical-Mathematical Model II), showed in the results obtained with numerical simulations that the interaction of electrodynamic shaker with the pendulum has generated a stability in the angular motion of the pendulum, as illustrated by Figure 12, 13. The behavior of electrodynamic shaker is periodic motion, as illustrated by Figure 18, 19. The analysis of nonlinear effects are of great importance for macroelectro-mechanical systems, because they can strongly affect the dynamics. We found the interdependence of inverted pendulums and eletrodynamic shaker, and provide evidence of the influence of motion of the pendulum about the vertical excitation provided by eletrodynamic shaker. Through this interaction was detected the presence of nonlinear phenomena illustrated through Figure 14(b). ACKNOWLEDGMENTS The authors acknowledge the financial support by FAPESP. REFERENCES [1] Y. Lee and Z. C. Feng, “Feasibility study of parametric excitations of a tuning fork microgyroscope,” Proceedings of IMECE, 2004. [2] Y. Lee, P. F. Pai, and Z. C. Feng, “Nonlinear complex response of a parametrically excited tuning fork,” Me- Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications Serra Negra, SP - ISSN 2178-3667 1186