Two parametric excited nonlinear systems due to electromechanical coupling

Journal of the Brazilian Society of Mechanical Sciences and Engineering manuscript No.(will be inserted by the editor)

Two parametric excited nonlinear systems due toelectromechanical coupling

Roberta Lima · Rubens Sampaio

Received: date / Accepted: date

Abstract This paper analyzes the behavior of two electromechanical systems.In the first part of the work, a simple system is analyzed. It is composed of acart, whose motion is excited by a DC motor (motor with continuous current).The coupling between the motor and the cart is made by a mechanism calledscotch yoke, so that the motor rotational motion is transformed in horizontalcart motion over a rail. The developed model of the system takes into accountthe influence of the DC motor in the dynamic behavior of the system. It isformulated as an initial value problem, in which the coupling torque betweenthe mechanical and electric systems appears as a parametric excitation. Bysimulations, it is verified that the system has a periodic solution with a relation2:1 between the period of rotation of the disk (part of the electromechanicalsystem) and the period of the current in the DC motor. In the second part ofthe work, a pendulum is embarked into the cart. Its suspension point is fixedin cart, so that exists a relative motion between the cart and the pendulum.The excitation of the pendulum comes from the motion of the cart, so it isnot controlled and can vary widely because the pendulum can store energy.The pendulum acts as a hidden parameter of the system. The influence of theembarked pendulum in the dynamic behavior of the system is investigated andit is shown how it can affect the solutions of the dynamic equations.

Keywords Electromechanical system · Parametric excitation · Embarkedsystem · coupled systems · nonlinear dynamics · energy pumping

R. LimaPUC-Rio, Mechanical Engineering DepartmentE-mail: roberta 10 [email protected]

R. SampaioPUC-Rio, Mechanical Engineering DepartmentE-mail: [email protected]

2 Roberta Lima, Rubens Sampaio

1 Introduction

The analysis of electromechanical systems is not a new subject. The interestof analyzing their dynamic behavior is reflected by the increasing amountof research in this area (see for instance [28,24,17]). The mutual interactionbetween electrical and mechanical parts leads us to analyze a very interestingnonlinear dynamical systems [23,15,20,10,13,9,6,5], in which the nonlinearitycomes from the coupling and varies with the coupling conditions. In this paper,two electromechanical systems are analyzed and compared. The first one is asimple system composed by a cart whose motion is driven by a DC motor. Thecoupling between the motor and the cart is made by a mechanism called scotchyoke. In this simple motor-cart system the coupling is a sort of master-slavecondition: the motor drives, the cart is driven, and that is all. The secondsystem has the same two elements of the first and also a pendulum withsuspension point fixed in cart. The pendulum is the embarked system andits motion is driven by the motion of the cart. The pendulum introduces anew feature since the motion of the pendulum acts as a reservoir of energy, i.e.energy from the electrical system is pumped to the pendulum and stored in thependulum motion, changing the characteristics of the mechanical system. Inthe two electromechanical systems analyzed in this paper, the source of energyis the imposed voltage, taken as constant. Beside this, to better see the effectsof coupling, the mechanical system has no dissipation, the only dissipation isin the motor. In [1–3], a vibro-impact electromechanical system with a similarcoupling mechanism to what is used in this paper, the scotch yoke mechanism,was investigated experimentally. In these works, it was considered the existenceof impacts in the mechanical part of the system and the objective was tocharacterize the impact force magnitude. In the present paper, the objectiveis different. We want to analyze the effects of the electromechanical coupling inthe dynamics. Mechanical systems with motion driven by electric motors areusually modeled eliminating the motor and saying that the force between themechanical and electric systems is imposed, so no coupling, and it is harmonicwith frequency given by the nominal frequency of the motor. In this paper,it is shown that this hypothesis is far from true and leads to a completelydifferent dynamics. In the systems we analyse here, the coupling force is notprescribed by a function, it comes from the coupling, varying with the couplingconditions. The dynamics of electromechanical systems is characterized by amutual interaction between the mechanical and electric systems, that is, thedynamics of the motor is heavily influenced by the mechanical system and thedynamics of the mechanical system depends on the dynamics of the motor. Thetime-evolution of the two electromechanical systems analyzed in this paper isformulated as initial value problems, in which the coupling torque appears asa parametric excitation, i.e., a time variation of the system parameters (seefor instance [27,21,25,18,22]). By numerical simulations, it is verified thatthe motor-cart system has a periodic solution with a relation 2:1 betweenthe period of rotation of the disk (part of the electromechanical system) andthe period of the current in the DC motor. This relation 2:1 between periods

Two parametric excited nonlinear systems due to electromechanical coupling 3

Fig. 1 Electrical motor DC.

is a common phenomenon of parametric excited systems. Without simplifyinghypothesis about the terms, the behavior of the system variables are analyzed,as the motor current over time, angular displacement of the motor shaft andcoupling force and torque. The influence of the pendulum embarked in thecart is investigated and it is shown the changes it causes in the dynamics ofthe second system with respect to the first. This paper is organized as follows.Sect. 2 describes the two coupled electromechanical systems analyzed. Sect. 3presents the results of the simulations developed to each one of them. Theinfluence of the attached mass and the amplitude of its motion are discussed.In Sect. 4, it is presented a configuration of the coupled system that leads tothe phenomenon of energy pumping and causes revolution, i. e. the inversionof the master-slave relation.

2 Dynamics of the coupled systems

2.1 Electrical system: motor DC

The mathematical modeling of DC motors is based on the Kirchhoff’s law [14].It is written as

lc(t) + r c(t) + keα(t) = ν , (1)

jmα(t) + bmα(t) − ke c(t) = −τ(t) , (2)

where t is the time, ν is the source voltage, c is the electric current, α isthe angular speed of the motor, l is the electric inductance, jm is the inertiamoment of the motor, bm is the damping ratio in the transmission of thetorque generated by the motor to drive the coupled mechanical system, keis the motor electromagnetic force constant and r is the electrical resistance.Figure 1 shows a sketch of the DC motor. The available torque deliveredto the coupled mechanical system is represented by τ , that is the componentof the torque vector τ in the z-direction shown in Fig. 1. Assuming that τand ν are constant in time, the motor achieves a steady state in which theelectric current and the angular speed become constant in time. By Eqs. (1)and (2), the angular speed of the motor shaft and the current in steady state,


Fig. 2 Cart-motor system.

respectively αsteady and csteady, are written as

αsteady =−τ r + ke v

bm r + k2e, csteady =

v

r− ke

r

(−τ r + ke v

bm r + k2e

). (3)

When τ is not constant in time, the angular speed of the motor shaft and thecurrent do not reach a constant value. This kind of situation happens when,for example, a mechanical system is coupled to the motor. In this case, α andc variate in time in a way that the dynamics of the motor will be influencedby the coupled mechanical system.

2.2 Cart-motor system: a master-slave relation

As described in the introduction, the system is composed by a cart whosemotion is driven by the DC motor. The motor is coupled to the cart through apin that slides into a slot machined in an acrylic plate that is attached to thecart, as shown in Fig. 2. The off-center pin is fixed on the disc at distance d ofthe motor shaft, so that the motor rotational motion is transformed into a carthorizontal movement. It is noticed that with this configuration, the center ofmass of the mechanical system is always located in the center of mass of thecart, so its position does not change. To model the coupling between the motorand the mechanical system, the motor shaft is assumed to be rigid. Thus, theavailable torque vector to the coupled mechanical system, τ , can be writtenas

τ (t) = d(t) × f(t) , (4)

where d = (d cosα(t), d sinα(t), 0) is the vector related to the eccentricity ofthe pin, and where f is the coupling force between the DC motor and the cart.Assuming that there is no friction between the pin and the slot, the vector fonly has a horizontal component, f (the horizontal force that the DC motorexerts in the cart). The available torque τ is written as

τ(t) = −f(t) d sinα(t) . (5)

Due to constraints, the cart is not allowed to move in the vertical direction.The mass of the mechanical system, m, is equal the cart mass, mc, and the


Fig. 3 Cart-motor-pendulum system.

horizontal cart displacement is represented by x. Since the cart is modeled asa particle, it satisfies the equation

m x = f(t) . (6)

Due to the system geometry, x(t) and α(t) are related by the following con-straint

x(t) = d cos (α(t)) . (7)

Substituting Eqs. (4) to (7) into Eqs. (1) and (2), we obtain the initial valueproblem for the motor-cart system that is written as follows. Given a constantsource voltage ν, find (α, c) such that, for all t > 0,

l c(t) + r c(t) + ke α(t) = ν , (8)

α(t)[jm +md2(sinα(t))

2]+α

[bm +md2α(t) cosα(t) sinα(t)

]−ke c(t) = 0 ,

(9)with the initial conditions,

α(0) = 0 , α(0) = 0 , c(0) =ν

r. (10)

Comparing Eq. (9) with Eq. (2), it is seen that the mechanical system in-fluences the motor in a parametric way, [16]. The coupling torque, τ , thatappears in the left side of Eq. (2), appears now as a time variation of thesystem parameters.

2.3 Cart-motor-pendulum system: introduction of a mechanical energyreservoir

The second electromechanical system analyzed in this paper has the same twoelements of the first system and a pendulum that is embarked into the cart,as shown in Fig. 3. Its suspension point is fixed in the cart, hence moves withit. The main point here is that the pendulum can have a relative motion withrespect to the cart. The pendulum is modeled as a mathematical pendulum(bar without mass and particle of mass mp at the end). Its length is noted


as lp and the pendulum angular displacement as θ. The equation of the cart-pendulum are

mpl2pθ(t) +mplpx(t) cos θ(t) +mpglp sin θ(t) = 0 , (11)

(mp +mc)x(t) +mplpθ(t) cos θ(t) −mplpθ2(t) sin θ(t) = f(t) , (12)

where, again, f represents the horizontal coupling force between the DC mo-tor and the cart, g is the gravity and the horizontal cart displacement is x.The mass of the mechanical system, m, is equal the cart mass plus the pen-dulum mass, mc +mp. The relative motion of the embarked pendulum causesa variation in the position of the center of mass of the mechanical system.As in the first coupled system, the cart is not allowed to move in the verticaldirection. Due to the problem geometry, x(t) and α(t) are related by Eq. (7).Once again, it is assumed that the motor shaft is rigid and that there is nofriction between the pin and the slot. Thus, the available torque to the cou-pled mechanical system, τ , is written as Eq. (4). Substituting the Eq. (5), (7),(11) and (12) into Eqs. (1) and (2), we obtain the initial value problem forthe motor-cart-pendulum system that is written as follows. Given a constantsource voltage ν, find (α, c, θ) such that, for all t > 0,

lc(t) +rc(t) + keα(t) = v ,

α(t)[jm + (mc +mp)d2(sinα(t))

2]

+ kt c(t)

+ α(t)[bm + (mc +mp)d2α(t) cosα(t) sinα(t)

]− θ(t) [mplp cos θ(t)d sinα(t)] + θ(t)

[mp lpθ(t) sin θ(t)d sinα(t)

]= 0 ,

θ(t)[mp l

2p

]− α(t) [mp lp cos θ(t)d sinα(t)]

− α(t) [mp lp cos θ(t)d cosα(t)α(t)] + mp g lp sin θ(t) = 0 ,(13)

with the initial conditions,

α(0) = 0 , α(0) = 0 , θ(0) = 0 , θ(0) = 0 , c(0) =ν

r. (14)

Observing Eq. (13), it is verified that the motor-pendulum system influencesthe motor in a parametric way.

3 Numerical simulations of the dynamics of the electromechanicalsystems

3.1 Simulations of the motor-cart system

Looking at Eqs. (8) to (10), it can be observed that if the nominal eccentricityof the pin, d, is small, the initial value problem of the motor-cart system tendsto the linear system equations of the DC motor, Eq. (1) and (2), in caseof no load. But as the eccentricity grows, the non-linearities become more


pronounced. The nonlinearity also increases with the attached mass, m. Tounderstand the influence of d and m in the dynamic behavior of the motor-cart system, a parametric excited system, simulations with different valuesto these system parameters were performed. The objective was to observethe graphs of the system variables, as the motor current over time, angulardisplacement of the motor shaft and coupling force. For computation, theinitial value problem defined by Eqs. (8) to (10) was integrated in a range of[0.0, 2.0] seconds. The 4th-order Runge-Kutta method is used for the timeintegration scheme with a time-step equal to 10−4. The motor parameters usedin all simulations are listed in Table 1. The source voltage is assumed to beconstant in time and equal to 2.4 V. To observe the influence of the eccentricity

Parameter Valuel 1.880 × 10−4 Hjm 1.210 × 10−4 Kg m2

bm 1.545 × 10−4 Nm/(rad/s)r 0.307 Ωke 5.330 × 10−2 V/(rad/s)

Table 1 Values of the motor parameters used in simulations.

of the pin in the behavior of the system, the mass was fixed to 5 Kg and theresults of simulations with two values of d were compared. The selected valuesare d = 0.001 m and d = 0.01 m. For d = 0.001 m, Figs. 4(a) and 4(b)displays α as function of time and the Fast Fourier Transform (FFT) of thecart displacement, x. It can be noted that the angular speed of the motorshaft oscillates with a small amplitude around 7 Hz and the FFT graph of xpresents only one peak at this frequency. In contrast to this, when d is bigger,as d = 0.01 m, observing Figs. 5(a) and 5(b), it is verified that the amplitudeof the oscillations of α grows and, due to the non-linearity effects, the FFTgraph of x presents more than one peak. The first one of them is at 6.56 Hz and,the following are at odd multiples of this value. As said in the introduction ofthis paper, normally problems of coupled systems are modeled as uncoupledsaying that the force is imposed, and it is harmonic with frequency givenby the nominal frequency of the motor. The dynamic of the motor is notconsidered. The graphs of Fig. 4(a) and 5(a) confirm that this hypothesisdoes not correspond to reality. As d increases, increases the nonlinearity of theproblem, and the hypothesis of harmonic force is completely inadequate sinceit falsifies the dynamics. Even when d is small, the angular speed of the motorshaft do not reach a constant value. After a transient it achieves a periodicstate. It oscillates around a mean value and these oscillations are periodic. Toenrich the analysis in the frequency domain, the Fast Fourier Transform of thecurrent over time, c, was computed for the two values of d. The results areshown in Fig. 6(a) and 6(b). It can be observed that in both cases, the FFTgraph of c presents a peak at a frequency that is twice the peak frequency ofthe FFT x indicating the parametric excitation, [16]. In the following analysis


(a) (b)

Fig. 4 Motor-cart system with d = 0.001 m: (a) angular speed of the motor shaft over timeand (b) Fast Fourier Transform of the cart displacement.

(a) (b)

Fig. 5 Motor-cart system with d = 0.01 m: (a) angular speed of the motor shaft over timeand (b) Fast Fourier Transform of the cart displacement.

of the motor-cart system, the nominal eccentricity of the pin was consider tobe 0.01 m. This value was selected to highlight the non-linearity effects. Theresults obtained to the cart displacement and current in motor over time areobserved in Fig. 7(a) and Fig. 7(b). The behavior found for the current overtime is similar to the behavior found for the angular speed of the motor shaft,Fig. 5(a). It achieves a periodic state after a transient phase. Other graphs tobe analyzed are the f(t) and τ(t) variation during one cart movement cycle inthe periodic state, phase portraits of the system, as it is shown in Figs. 8(a) and8(b). Observing the f graph and remembering the constrain x(t) = d cosα(t),it is verified that the horizontal force presents its maximum value when x(t) =−d and its minimum value when x(t) = d. Besides this, the coupling forcechanges it sign twice. Observing the τ graph, it is verified that the torquepresents four points of sign change. Two of them occur when x(t) = −d andx(t) = d, corresponding respectively to α multiple of π and α multiple of


(a) (b)

Fig. 6 Motor-cart system: Fast Fourier Transform of the current (a) when d = 0.001 [m]and (b) when d = 0.01 [m].

(a) (b)

Fig. 7 Motor-cart system with d = 0.01 [m]: (a) cart displacement and (b) motor currentover time.

2π. This changes were expected from Eq. (5). The others two changes occurexactly in the same cart positions that we have the sign of f changing. In eachcart movement cycle, the horizontal force f and the torque τ follow once thepaths shown in Fig. 8(a) and 8(b). Figures 9(a) and 9(b) show the phaseportraits graphs of the current variation during one cart movement cycle andthe torque variation in function of the current. In the left graph, it is notedthat the current presents four points of sign change in each cart movementcycle. Observing the right graph, it is verified that the current follows twotimes the path shown in Fig. 9(b). Thus, there is a relation 2:1 between theperiod of rotation of the disk (part of the electromechanical system) and theperiod of the current in the DC motor. This relation 2:1 between periods is acommon phenomenon of parametric excited systems. Others phase portraitsgraphs are shown in Figs. 10(a), 10(b), 11(a) and 11(b). Due to the couplingmechanism, the coupling torque, τ , variates in time. Thus, the angular speed


(a) (b)

Fig. 8 Motor-cart system with d = 0.01 m: (a) horizontal force f and (b) torque τ duringone cycle of the cart movement.

(a) (b)

Fig. 9 Motor-cart system with d = 0.01 m: (a) current variation during one cart movementcycle and (b) torque variation as function of the current.

of the motor shaft and the current are not constant values after the transient.As the motor-cart system does not have any mechanism of storing energy, afterthe transient the dynamics achieves a periodic state. Therefore, to compare theresponse of the coupled systems for different values of d and m, the duration ofone cart movement cycle were computed in the periodic state. Figures 12(a)and 12(b) show the graphs of the computed periods as function of d and m. Inboth graphs it is observed that, the bigger d or m is, the bigger is the periodof the cart movement cycle in the periodic state. It is noted too that thisincrement is more pronounced in relation to d. This result is compatible withthe results presented in [7], in which a similar electromechanical motor-cartsystem was analyzed and the existence and asymptotic stability of a periodicorbit to this system were obtained in a mathematically rigorous way. In theexpression presented in [7] to the period, it is verified that the period growsproportionally to m2 d4 and so, the growing of the period is faster in relationto d than to m.


(a) (b)

Fig. 10 Motor-cart system with d = 0.01 m: (a) angular velocity of the motor shaft duringone cart movement cycle and (b) current variation as function of the angular velocity of themotor shaft.

(a) (b)

Fig. 11 Motor-cart system d = 0.01 m: (a) torque variation as function of the horizontalforce f and (b) horizontal force variation as function of the angular velocity of the motorshaft.

3.2 Simulations of the motor-cart-pendulum system

A similar analysis to the one made to the motor-cart system, based on theresults of numerical simulations, was developed for the motor-cart-pendulumsystem. Looking at the initial value problem Eqs. (13) to (14), it is observedthat if the nominal eccentricity of the pin, d, is small and the angle θ(t) isnear zero, Eq. (13) tends to a linear system. But as the eccentricity grows,the nonlinearities become more pronounced. To understand the influence ofd in the dynamic behavior of the motor-cart system, simulations with twodifferent values to this system parameters were performed. The selected valuesare d = 0.001 m and d = 0.01 m. In these simulations, the cart and thependulum masses were mc = 0.0 Kg and mp = 5.0 Kg, so that the totalmass, m = mc + mp = 5.0 Kg, is equal to the embarked mass. Although


(a) (b)

Fig. 12 Motor-cart system: period of one cart movement cycle (a) as function of d withm = 5.0 Kg and (b) as function of m with d = 0.005 m.

the masses are equal, this configuration contrasts with the one of the motor-cart system used in the previous simulations. In spite of having the samemasses, the pendulum has a relative motion with respect to the cart, and thismakes a huge difference. The pendulum length was assumed to be 0.075 m.For d = 0.001 m, Figs. 13(a), 13(b), 14(a) and 14(b) show the graphsof the angular velocity of the motor shaft, current, pendulum displacementand cart displacement over time. These results reveal that when d is small,the angular speed of the motor shaft oscillates over time with a very smallamplitude around 7 Hz, the current also oscillates with a small amplitudearound 0.13 A, and the angular displacement of pendulum is near zero. The

(a) (b)

Fig. 13 Motor-cart-pendulum system with d = 0.001 m: (a) angular velocity of the motorshaft and (b) current over time.

Fast Fourier Transform was computed to the cart and pendulum displacementsand to the current for this small value of d. The obtained graphs are shownin Figs. 15(a) and 15(b). The FFT graph of x, shown in Fig. 15(a), presents


(a) (b)

Fig. 14 Motor-cart-pendulum system with d = 0.001 m: (a) pendulum displacement and(b) cart displacement over time.

only one peak at the frequency at 7.04 Hz. This peak was expected, since thisis close to the angular speed of the motor shaft. The FFT graph of θ presentstwo peaks. One of them coincides with the x peak and the other one is atthe natural frequency of the pendulum, i.e., ωn =

√g/lp/(2π) = 1.82 Hz.

The FFT graph of c presents three peaks. The first one is at 5.22 Hz, thesecond on is at 8.86 Hz and the third one at 14.08 Hz, that is twice the peakfrequency of the FFT x. This relation 2:1 between the peak frequency of cand x indicates the parametric excitation. Figures 16(a), 16(b), 17(a) and

(a) (b)

Fig. 15 Motor-cart-pendulum system with d = 0.001 m: Fast Fourier Transform of (a) cartand pendulum displacements and (b) of current.

17(b) show the graphs of angular speed of the motor, current, pendulum andcart displacement over time when d = 0.01 m. Comparing these graphs withFigs. 13(a), 13(b), 14(a) and 14(b), It is verified that with a bigger d, the


amplitude of the oscillations of α, c, and θ in the steady state will be alsobigger. As done to the results with small d, the FFT was computed to the

(a) (b)


(a) (b)

Fig. 17 Motor-cart-pendulum system with d = 0.01 m: (a) pendulum and (b) cart dis-placement over time.

cart and pendulum displacements, angular speed of the motor shaft and to thecurrent for the new value of d. Figures 18(a), 18(b), 18(a), 18(b) display theobtained graphs. Observing Figs. 18(a) and 18(b), it can be noted that x and

θ present peaks at the same frequencies. The first of them is at 1.61 Hz andthe following are at odd multiples of this value. Comparing x with d = 0.001 mand with d = 0.01 m, it is verified that the peak at the natural frequency ofthe pendulum, ωn = 1.82 Hz, vanished when d grows. This result certifies thatthe system behavior is far from linear. Regarding Figs. 19(a) and 19(b), itcan be noted that the FFT graphs of the current and of the angular speedof the motor shaft over time also present peaks at the same frequencies. The


(a) (b)

Fig. 18 Motor-cart-pendulum system with d = 0.01 m: Fast Fourier Transform of (a)pendulum and (b) cart displacements.

first of them is at 3.21 Hz and the following are at multiples of this value.Comparing Figs. 19(a) and 18(b), it can be verified that the frequencies inwhich c presents peaks are twice the frequencies in which x presents peaks.

(a) (b)

Fig. 19 Motor-cart-pendulum system with d = 0.01 m: Fast Fourier Transform of (a)current and (b) angular speed of the motor shaft over time.

4 Pumping Leads To Revolution

In the previous section of the paper, in the analysis developed to the themotor-cart-pendulum system, the cart mass was considered to be zero and thependulum mass 5.0 Kg. Next, it is presented an analysis of the behavior of thissystem with a different mass configuration. The cart mass is kept as 0.0 Kg (alimit case) and a smaller value is selected to the pendulum mass, mp = 4.0 Kg,


so that the total mass, mc +mp = 4.0 Kg, is still equal to the embarked mass.Figures 20(a), 20(b), 21(a) and 21(b) show the graphs of the angular speed ofthe motor shaft, current and cart and pendulum displacements over time forthis new mass configuration when d = 0.01 m. Regarding these graphs, it canbe observed that after the transient state, the dynamics achieves a periodicstate, in which α takes negatives values. With this new mass configuration,the mechanical system pumps energy from the motor and the amplitude of thependulum grows reaching a point where the mechanical system starts to drivethe motion, [12,11,8]. This is seen observing that α takes negatives values,indicating that the motor shaft changes its motion direction sometimes. Whenthe angular speed of the motor shaft is positive, it is considered that the motordrives the cart motion, the cart is driven. But when it is negative, the motorlooses the control over the cart and drives it no more, it is now driven by themechanical system. In these situations, it will be said that the relation master-slave is reversed. To understand the sign changing of the angular speed of the

(a) (b)


motor shaft, some phase portrait graphs were plotted. Figures 22(a) and 22(b)show the α graph as function of α and the α graph as function of x duringone movement cycle. It is verified that when α(t) turns negative, the motorshaft has a negative acceleration. After a short period of time, its accelerationbecomes positive and brakes the motor shaft motion. This causes other signchanging in α(t) and consequently, it turns positive again. Thus, the motorrecovers the control over the cart motion. Looking at Fig. 22(b), it is noted thatthis reversion in the relation master-slave occurs two times in each movementcycle. The position and angular speed of the pendulum, at the moment of thereversion, can be observed by the graphs of θ as function of α and θ in functionof α, shown in Figs. 23(a) and 23(b). It is verified that when the the motorlooses the control over the cart by the sign changing of α, the pendulum angleis around 21.6o or around −21.6o. When the motor recovers the control, the


(a) (b)

Fig. 21 Motor-cart-pendulum system with d = 0.01 m: (a) pendulum and (b) cart dis-placement over time.

(a) (b)

Fig. 22 Motor-cart-pendulum system with d = 0.01 m: portrait graphs of (a) α graph asfunction of α and (b) α graph as function of x.

pendulum angle is around 6.0o or around −6.0o. It is also noted that, duringthe period of reversion, the pendulum does not change its direction of motionin spite of its angular speed presents a change of behavior. In the beginning ofthe reversion the modulus of θ grows, but when it achieves the value 2.95 Hz, itstarts to decrease. This change occurs due to the sign changing in the tangentangular acceleration of the pendulum, as can be observed in Fig. 24(a). Thegraph of the torque variation in function of the angular speed of the motor shaftshows that the maximum torque is achieved during the period of reversion.

5 Conclusions

In this paper, the formulation and analysis of the dynamics of two electrome-chanical systems have been presented. The developed models to these systemsrevealed that the electromechanical systems are parametric excited system, in


(a) (b)

Fig. 23 Motor-cart-pendulum system with d = 0.01 m: portrait graphs of (a) θ graph asfunction of dotα and (b) θ as function of dotα.

(a) (b)

Fig. 24 Motor-cart-pendulum system with d = 0.01 m: portrait graphs of (a) tangent θgraph as function of dotα and (b) τ as function of dotα.

which the coupling torque appears as a time variation of the system param-eters. Simulations of these systems were performed for different values of dand m and the results of these numerical simulations, as the graphs the sys-tems variables over time, graphs of the FFT of systems variables and phaseportraits graphs were analyzed. By these graphs, a typical phenomenon ofparametric excited systems was observed: the existence of a periodic solutionwith a relation 2:1 between the period of rotation of the disk and the periodof the current. This result is compatible with earlier numerical findings in [19]and it is compatible with the results presented in [7], in which the existenceand asymptotic stability of a periodic orbit to an electromechanical systemare obtained in a mathematically rigorous way. The nominal eccentricity ofthe pin of the motor, d, was characterized as a parameter that controls thenonlinearities of the equations of motion of both systems, and its influencein the dynamic equations was analyzed by the Fast Fourier Transform. The


influence of a embarked mass was demonstrated and it was shown the changesit causes in the solutions of the dynamic equations. The motor-cart systemhas no capacity to pump energy from the motor, it is a master-slave system:the motor drives the cart motion, the cart is driven. The only interesting fea-ture is how the nonlinearity changes with d and m, the mass of the cart. Themotor-cart-pendulum system has a new feature, the capacity to store energyin the motion of the pendulum. With this, the mechanical system can pumpenergy from the motor and, in certain cases, revert the relation master-slave,that is the mechanical system can be itself the master stopping the motor andreversing its motion.

Acknowledgements This work was supported by the Brazilian Agencies CNPQ, CAPESand Faperj.

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Two parametric excited nonlinear systems due to electromechanical coupling

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