THE NONLINEAR DYNAMICS OF A VIBRATING SYSTEM … · THE NONLINEAR DYNAMICS OF A VIBRATING SYSTEM...

THE NONLINEAR DYNAMICS OF A VIBRATING SYSTEM MODELLED BY A INVERTEDPENDULUM, WITH AN ELECTRODYNAMIC SHAKER

Ana Rosa Klinke1, José Manoel Balthazar2, Jorge Luiz Palacios Felix3, Fábio Roberto Chavarette4

1,2UNESP: Univ Estadual Paulista, Department of Statistics Applied Mathematics and Computation, PO Box 178, 13500-230, Rio Claro -SP, Brazil, [email protected], [email protected]

3Regional University of Northwestern of Rio Grande do Sul - Department of Physics Mathematical and Statistics, São Geraldo, Brazil,[email protected]

4UNESP: Univ Estadual Paulista, Department of Mathematics, Ilha Soltlteira, Brazil, [email protected]

Abstract: The technology of macro-electro-mechanicalsystems has many applications on Engineering Sciences.However, the design of such mechanical systems can be quitechallenging due to nonlinear effects, which may strongly af-fected the dynamics behavior. The physical model studiedis a macro-electro-mechanical coupling with an electrody-namic shaker, by using numerical simulations.

Keywords: Macro-Electro-Mechanical Systems, NonlinearDynamics, Chaos.

1. INTRODUCTION

Micro-electro-mechanical systems (MEMS) are smallintegrated devices or systems those combine electricaland mechanical components. Examples of MEMS de-vices applications are emerging as the existing technol-ogy is applied to the miniaturization and integration ofconventional devices. Examples of MEMS devices ap-plications,include inertial sensors, as an example. Theyare used as an example in airbag-deployment sensors,in automobiles, and as tilt or shock sensors Some de-tails of (MEMS), may be easly find on the web ad-dress (http://en.wikipedia.org/wiki/Microelectromechanicalsystems). 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 toexhibit chaotic behavior, as well a periodic oscillations. Achaotic system is a nonlinear deterministic system, which itis very sensitive to small perturbations on its initial condi-tions, and its long time behavior it is unpredictable . It is wellknown that the classical tuning-fork (MEMS) gyroscope,contains a pair of masses that are driven to oscillate withequal amplitude but in opposite directions. When rotated,

the Coriolis force creates an orthogonal vibration that canbe sensed by a variety of mechanisms. The Draper Lab gyro-scope shown in Figure 1, uses a comb-type structures to drivethe tuning fork into resonance. Rotation causes the proofmasses to vibrate out of plane, and this motion is sensed ca-pacitively with a custom CMOS ASIC. The micro-tuning-fork gyroscope,presented in Figure 1, may be mathemati-cally modeled(by a Macro Model)by being two in- vertedpendulums, which their motions, opposing set in support ofmass vertical and horizontal movement [1, 2]. We remarkedthat and, include, the effects of the basis of excitation of thependulum (parametric excitation), used an electro-dynamicsshaker [3, 4]. Its nonlinear dynamical behavior was stud-ied by [3, 4] basead on a Macro model. In this paper, weconsider two Macro models. They also presented rich non-linear dynamical behavior [4]. Next we will discuss new andinteresting models, separately.

Figure 1 – A tuning fork gyroscope with comb drive for com-mercial 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

The simplified Macro-Mechanical-Mathematical ModelI, considered, consists of one inverted pendulum fixed in sus-pension mass, showed in Figure 2. The pendulum has a massm connected by a rod of length l. And are θ the angle of pen-dulum and stabilized by torsion springs, with strongly non-linear, kθ = k + hθ2. The suspension with mass M , andspring constant Kx for X direction [4]. Then the variablesof the problem are the displacement and the suspension ofmass in the horizontal respectively (two degree of Freedom).

Figure 2 – Simple model of the Macro-Electro-MechanicalModel I

The governing equations of motion, for the Macro-Electro-Mechanical model, defined by Figure 2, are

ml2θ + cl2θ + k1θ +

hθ3 = mlx cos θ +ml(y + g) sin θ (1)Mtx+ cxx+ kxx−ml(θ cos θ − θ2 sin θ) = 0 (2)

Where Mt = M + m and g is the acceleration of grav-ity. They may be transformed into dimensionless equations,making the following changes of variables:

X = xlc

, τ = ωet, ωe =1√LC0

, ωx =√

kx

Mtωe

2,

F1 = m1l1Mtlc

, δ1 = k1

m1l12ωe2 , λ1 = h1

m1l12ωe2 ,

µ1 = c1m1l12ωe

, η1 = lcl1

, g1 = gωe

2lc, µ4 = cx

Mtωe.

Considering the simplified motion in the horizontal direc-tion and a excitation force on Y direction F = A cosΩτ , theequation of motion of the simplified system is given by:

θ + δ1θ + λ1θ3 + µ1θ − η1(X cos θ + g1 sin θ) = 0 (3)

X + µ4X + ωx2X − F1(θ cos θ − θ2 sin θ) = F (4)

Isolating θ and X:

θ − η1X cos θ = −δ1θ − λ1θ3 − µ1θ + η1g1 sin θ ≡ f1 (5)

X − F1θ cos θ = F − µ4X − ωx2X − F1θ

2 ≡ f2 (6)

are obtained

θ =1

∆(f1 + η1f2 cos θ) (7)

X =1

∆(F1f1 cos θ + f2 + F ) (8)

where

∆ = 1− F1η1cos θ2 (9)

f1 = −δ1θ − λ1θ3 − µ1θ + η1g1 sin θ (10)

f2 = −µ4X − ωx2X − F1θ

2 sin θ (11)F = A cosΩτ (12)

Making x1 = θ, x2 = θ, x3 = X , x4 = X , has a systemof first order in the state variables

x1 = x2 (13)

x2 =1

∆(f1 + η1f2 cosx1) (14)

x3 = x4 (15)

x4 =1

∆(F1f1 cosx1 + f2 + F ) (16)

where

∆ = 1− F1η1cosx12 (17)

f1 = −δ1x1 − λ1x13 − µ1x2 + η1g1 sinx1 (18)

f2 = −µ4x4 − ωx2x3 − F1x2

2 sinx1 (19)F = A cosΩτ (20)

2.1. NUMERICAL SIMULATION RESULTS

The numerical simulations, were carried out, by us-ing Matlabo, taken as the numerical integrator the Runge-Kutta method of fourth order algorithm with variable timesteplength. The following is presented an analysis of the be-havior of each component of the proposed models, makingan approach on the nonlinear behavior.

The initial conditions of (θ, θ,X, X), were fixed as[0.12, 0.01, 0.0025, 0.01].The equations parameters for theoscillator are shown in Table 1 [1, 2, 5], are:

The numerical results of the equations 13, 14, 15, 16 isillustrated through Figures 3, 4, 5.

The Figure 3(a) shows the phase portrait of (θ, θ) and Fig-ure 3(b) shows the phase motion of (X, X). The time historyof angular displacement of θ in the time range 0 ≤ t ≤ 4000is showed in Figure 4(a) and the time history of X is showedby Figure 4(b)


Table 1 – Parameters for the chaotic oscillator.

Parameters ValuesA 0.2Ω 1.4F1 0.1µ4 0µ1 0.01ωx 1.0η1 0.9λ1 0.75g1 1.0δ1 1.0

Figure 3: (a) Phase Portrait of θ, (b) Phase Portrait of X .

Figure 4: (a) Time History of θ, (b)Time History of X .

Figure 5(a) illustrates the frequency spectrum of angulardisplacement θ and Figure 5(b) the frequency spectrum ofmotion X . According to the angular movement of the pen-dulum and the movement in the X direction ,we can see anonlinear interaction between the inverted pendulum and theforce excitation, evidenced by the three peaks, occurred inthe frequency spectrum.

Figure 5: (a)Frequency Spectrum of θ, (b) Frequency Spectrum of X .

2.2. PARAMETERS ANALYSIS

The following results, were obtained in order to observethe effect of parameter bifurcation in equations 13, 14, 15,16. The bifurcation parameter, change in the invervalo 0 ≤Parameter ≤ 6 and other parameters of the equations, wereheld constant.

Figure 6 shows the bifurcation diagram of the parame-ter µ4, with the purpose of observing the damping effectsof the suspension mass on the oscillation of the pendulums(X-direction). The analysis of results shows that when theparameter µ4 is very close to 0, causing an instability in theconsidered system.

Figure 6: Bifurcation Diagram of parameter µ4

Figure 7 shows the bifurcation diagram of the parameterωx, this parameter represents the natural excitement of thesystem in the direction X. When this parameter adquive avalue between 1.2 and 1.4 the system becomes unstable.

Figure 7: Bifurcation Diagram of parameter ωx

Next,we present the results of the numerical simulations,those were carried out, is done to observe the influence of theparameter, λ1, of the nonlinearity of stiffness of the pendu-lum. The bifurcation diagram showed the interaction of thisparameter, with the system illustrated in Figure 8. As ex-


pected, with increasing of the cubic nonlinearity coefficientλ1, the degree of instability of the system increases consid-erably. Increasing of instability is caused by the fact that thisparameter is a nonlinear coefficient of the system.

Figure 8: Bifurcation Diagram of parameter λ1

The bifurcation diagram of the parameter w, shown inFigure 9, illustrates the effect of parameter oscillation fre-quency of the excitation force, F = A cosΩτ , in the system.We can observe the occurrence of instabilidade of the system,when the parameter ranging from 0.4 to 1.3.

Figure 9: Bifurcation Diagram of parameter Ω

The study of the amplitude A shows the intensity of theexcitation force F = A cosΩτ . The bifurcation diagramof Figure 10 shows the interaction between this parameterand the system. It is possible to observe that with increas-ing value of A, there is an instability occurring in the sys-tem, which increasing along with increasing A. This resultis in complete agreement with the proposed model, becauseof higher the level of excitement of the pendulum and thesuspension of mass those suffering the greater the instabilitygenerated in the system.

Figure 10: Bifurcation Diagram of parameter A

3. SIMPLIFIED MATHEMATICAL MODEL II

Figure 11 – Macro-Electro-Mechanical Model II, coupling withan electrodynamic shaker

The Macro-Mechanical-Mathematical Model II, consid-ered here, was based on initial experimental work on the ba-sis of excitation of a pendulum (parametric excitation) by us-ing electrodynamic shaker made by [3, 4]. The mechanismis studied here, was implemented as excited at its base byan electrodynamic shaker as shown in Figure 11.The electro-dynamic shaker acts on the suspension of mass M. The cou-pling between both parties is achieved by the electromagneticforce due to the permanent magnet Fem = Kq (Fem = lcB,where lc e B are the length of the conductor and the magneticfield, respectively). This generates a force of Laplace on themechanics and the Lenz electromotive voltage in the power[1, 2, 5]. The electrical system consists of resistor R, an in-ductor L, a capacitor C and a source of sinusoidal voltagee(t) = e0 cosΩt(e0 and Ω are the amplitude and frequencyrespectively; t is the time), all connected in series. In thismodel, the voltage of the capacitor is a nonlinear function ofinstantaneous electric charge q, defined as:


Vc =1

C0q + α3q

3 (21)

where C0 is the linear value of C and α3 is the nonlinearcoefficient depending on the type of capacitor to be used [1,2, 5].

The kinetic energy and potential energy of the system aregiven by

T =1

2Mtx

2 +1

2Mty

2 −

ml(xθ cos θ + yθ sin θ) +1

2ml2θ2 +

1

2Lq2 (22)

V = Mtgy +mg(l cos θ + y) +ky2y2 +

kx2x2 +

Kqy +1

2kθ2 +

1

4hθ2 +

1

2C0q2 +

α3

4q4 (23)

The dissipation energy function of Rayleigh’s is given by

D =1

2c(lθ)

2+

1

2Rq2 +

1

2cxx

2 +1

2cy y

2 (24)

The motion equation of the system is given by:

ml2θ + cl2θ + k1θ +

hθ3 = mlx cos θ +ml(y + g) sin θ (25)Mtx+ cxx+ kxx−ml(θ cos θ − θ2 sin θ) = 0 (26)

Mty + cy y + kyy −ml(θ sin θ + θ2 cos θ)−Kq = 0 (27)

Lq +Rq +1

C0q + α3q

3 +Ky = E0 cos (Ωt) (28)

Where Mt = M +m and g is the acceleration of gravity.Making the following changes of variables:

X = xlc

, Y = ylc

, Q = qq0

, τ = ωet,

ωe =1√LC0

, ωx =√

kx

Mtωe2 , ωy =

√ky

Mtωe2 ,

F1 = mlMtlc

, δ1 = k1

ml2ωe2 , λ1 = h1

ml2ωe2 ,

λ3 = α3q02

Lωe2 , µ1 = c1

ml2ωe, η1 = lc

l , g1 = gωe

2lc,

µ4 = cxMtωe

, µ3 = RLωe

, µ5 =cy

Mtωe,

γ1 = Kq0Mtωelc

, γ2 = KlcLωeq0

, E0 = e0Lωe

2q0, ω = Ω

ωe.

we will obtain

θ + δ1θ + λ1θ3 +

µ1θ − η1(X cos θ + (Y + g1) sin θ) = 0 (29)X + µ4X + ωx

2X −F1(θ cos θ − θ2 sin θ) = 0 (30)

Y + µ5Y + ωy2Y −

F1(θ sin θ + θ2 cos θ) + γ1Q = 0 (31)Q+ µ3Q+Q+ λ3Q

3 + γ2Y = E0 cos (ωτ) (32)

Isolating θ, X , Y and Q:

θ − η1(X cos θ − Y sin θ) =

−δ1θ − λ1θ3 − µ1θ + η1g1 sin θ ≡ f1 (33)

X − F1θ cos θ =

−µ4X − ωx2X − F1θ

2 sin θ ≡ f2 (34)Y − F1θ sin θ =

−µ5Y − ωy2Y + F1θ

2 cos θ + γ1Q ≡ f3 (35)

Q = −µ3Q−Q− λ3Q3 − γ2Y + E0 cos (ωτ) (36)

Making x1 = θ, x2 = θ, x3 = X , x4 = X , x5 = Y ,x6 = Y , x7 = Q, x8 = Q, we will obtain the system of firstorder in the state variables

x1 = x2 (37)

x2 =1

∆(f1 + η1f2 cosx1 + η1f3 sinx1) (38)

x3 = x4 (39)

x4 =1

∆(F1f1 cosx1 + f2) (40)

x5 = x6 (41)

x6 =1

∆(F1f1 sinx1 + f3) (42)

x7 = x8 (43)x8 = −µ3x8 − x7 − λ3x7

3 − γ2x6 + E0 cos (ωτ) (44)

where

∆ = 1− F1η1cosx12 − F1η1sinx1

2 (45)f1 = −δ1x1 − λ1x1

3 − µ1x2 + η1g1 sinx1 (46)f2 = −µ4x4 − ωx

2x3 − F1x22 sinx1 (47)

f3 = −µ5x4 − ωy2x5 + F1x2

2 cosx1 + γ1x8 (48)(49)

3.1. NUMERICAL SIMULATION RESULTS

The initial conditions of (θ, θ,X, X, Y, Y , Q, Q) fixedas being (0.12, 0.01, 0.025, 0.01, 0.1, 0.01, 0.3, 0.01) respec-tively. The equations parameters for the oscillator, which wasfabricated using standard silicon on insulator processing de-vice shown in Table 1 [1, 2, 5], are:

Table 2 – Parameters for the chaotic oscillator, coupling with anelectrodynamic shaker.

Parameters Values Parameters ValuesF1 0.1 µ4, µ5 0µ1 0.01 µ3 0.05ωx 1.0 ωy 1.0η1 0.9 γ2 0.2λ1 0.75 ω 1.5g1 1.0 ωe 1.0δ1 1.0 E0 0.24λ3 1.0 γ1 0.4


The numerical results of the equations 37, 38, 39, 40, 41,42, 43, 44 are illustrated by Figures 12 to 19. These resultsare all givwen is in the time range 0 ≤ t ≤ 5000.

The Figure 12(a) shows the phase portrait (θ, θ) and Fig-ure 12(b) shows the frequency spectrum of θ. The time his-tory of angular displacement of θ is shows by Figure 13(a)and the time history of θ is shows by Figure 13(b). In thissystem, when we consider the action of the electrodynamicshaker, we obtain that the angular movement of the invertedpendulum is stabilized. This is evidenced by the frequencyspectrum , which show only one peak.

Figure 12: (a)Phase portrait, (b)Frequency Spectrum

Figure 13: (a)Time History of θ, (b)Time History of θ.

The Figure 14(a) shows the phase portrait (X, X) andFigure 14(b) shows the frequency spectrum of X . The timehistory of X is shows by Figure 15(a) and the time history ofX is shows by Figure 15(b). In the motion X has a periodicbehavior, as illustrated by the phase portrait.

Figure 14: (a)Phase Portrait of θ, (b)Frequency Spectrum of X .

Figure 15: (a)Time History of X , (b)Time History of X .

In the Figure 16(a) and 16(b) shows the phase portait(Y, Y ) and the frequency spectrum of Y , respectively. TheFigure 17(a) and 17(b) illustrates the time history of Y andthe time history of Y , respectively. The Y movement has anonlinear behavior (shown by the frequency spectrum con-taining multiple peaks).

Figure 16: (a)Phase Portrait of Y , (b)Frequency Spectrum of Y .

Figure 17: (a)Time History of Y , (b)Time History of Y .

The behavior of the electrodynamic shaker it is illustratedin the phase portrait (Q, Q), Figure 18(a). The Figure 18(b)shows the frequency spectrum of Q. The Figure 19(a) and19(b) illustrates the time history of Q and the time history ofQ, respectively.

Figure 18: (a)Phase Portrait of Q, (b)Frequency Spectrum of Q.


Figure 19: (a)Time History of Q, (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 thisequations 37, 38, 39, 40, 41, 42, 43, 44. The bifurcationparameter change in the time range 0 ≤ Parameter ≤ 5and other parameters of the equations were held constant.

Figure 20 shows the bifurcation diagram of the parame-ter µ4, with the purpose of observing the damping effects ofthe suspension mass on the oscillation of the pendulums (X-direction) 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. Thisresult resembles the results were obtained with the Macro-Electro-Mechanical Model I. We conclude that the dampingeffect on both systems cause instability to values close to 0,this is because the damping effect of the oscillation decreasesin directions X and Y .

Figure 20: Bifurcation Diagram of parameter µ4

Figure 21 shows the bifurcation diagram of the parameterωx, which represents the natural excitement of the system inthe X direction. The system becomes unstable in the rangeof values from 0 to 2.

Figure 21: Bifurcation Diagram of parameter ωx

Figure 22 shows the bifurcation diagram of the parameterωy , which represents the natural excitement of the system inthe Y direction, analyzing the results we have obtained thatthe system becomes unstable for values of ωy greater than 1.

Figure 22: Bifurcation Diagram of parameter ωy

The parameter λ1 represents the nonlinear coefficient ofthe pendulum, and the bifurcation diagram shows the interac-tion of this parameter, the system is illustrated in Figure 23.Analyzing the results, we noted that the system behaves un-stable, when the coefficient of cubic nonlinearity reached val-ues above 4.5.


Figure 23: Bifurcation Diagram of parameter λ1

4. CONCLUSION

In this paper, the study of an macro-electro-mechanical,firstly modeled by one inverted pendulum fixed in a suspen-sion mass (Macro-Mechanical-Mathematical Model I), wasdone by numerical simulations. With the analysis of resultsobtained in 2.1 Session, we obtained a nonlinear interactionbetween the inverted pendulum and the force excitation, ev-idenced by the three peaks, occurred in the frequency spec-trum, Figure 5. The results of the bifurcation diagrams, Ses-sion 2.2, showed that the parameters λ1, A, Ω, ωx generatedinstability in the system, as illustrated by Figures 8, 9, 7, 10,respectively.

The second system, macro-electro-mechanical (Macro-Mechanical-Mathematical Model II), showed in the resultsobtained with numerical simulations that the interaction ofelectrodynamic shaker with the pendulum has generated astability in the angular motion of the pendulum, as illustratedby Figure 12, 13. The behavior of electrodynamic shaker isperiodic motion, as illustrated by Figure 18, 19. The anal-ysis of nonlinear effects are of great importance for macro-electro-mechanical systems, because they can strongly affectthe dynamics. We found the interdependence of inverted pen-dulums and eletrodynamic shaker, and provide evidence ofthe influence of motion of the pendulum about the verticalexcitation provided by eletrodynamic shaker. Through thisinteraction was detected the presence of nonlinear phenom-ena illustrated through Figure 14(b).

ACKNOWLEDGMENTS

The authors acknowledge the financial support byFAPESP.

REFERENCES

[1] Y. Lee and Z. C. Feng, “Feasibility study of parametricexcitations of a tuning fork microgyroscope,” Proceed-ings of IMECE, 2004.

[2] Y. Lee, P. F. Pai, and Z. C. Feng, “Nonlinear complexresponse of a parametrically excited tuning fork,” Me-

chanical Systems and Signal Processing, pp. 1146–1156,2008.

[3] X. Xu, E. Pavlovskaia, M. wiercigroch, F. Romeo,and S. Lenci., “Dynamic interactions between paramet-ric 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 andchaotic behavior of a macro tuning fork beam(tfb), un-der an electro-dynamical shaker excitation(eds).,” Pro-ceedings of 11th Pan-American Congress of Applied Me-chanics, 2010.

[5] J. L. P. Felix and J. M. Balthazar, “Comments on a non-linear and non-ideal electromechanical damping vibra-tion absorber. sommerfeld effect and energy transfer,”Nonlinear Dynamics, pp. 1–11, 2009.


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