Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

53, 3, pp. 653-664, Warsaw 2015
DOI: 10.15632/jtam-pl.53.3.653

ON SUPPRESSION OF CHAOTIC MOTIONS OF A PORTAL FRAME

STRUCTURE UNDER NON-IDEAL LOADING USING

A MAGNETO-RHEOLOGICAL DAMPER

Angelo Marcelo Tusset, Vinicius Piccirillo

Federal University of Technology-Paraná – UTFPR, Ponta Grossa, PR, Brazil

e-mail: tusset@utfpr.edu.br; piccirillo@utfpr.edu.br

José Manoel Balthazar

University Paulista State – UNESP, Rio Claro, SP, Brazil; e-mail: jmbaltha@rc.unesp.br

Reyolando Manoel Lopes Rebello da Fonseca Brasil

Federal University of ABC – UFABC, Santo Andre, SP, Brasil; e-mail: reyolando.brasil@ufabc.edu.br

Weconsider chaoticmotions of aportal frame structureundernon-ideal loading.To suppress
this chaotic behavior, a controlling scheme is implemented. The control strategy involves
application of two control signals and nonlinear feedforward control to maintain a desired
periodic orbit, and state feedback control to bring the system trajectory into the desired
periodic orbit. Additionally, the control strategy includes an active magneto-rheological
damper to actuate the system. The control force of the damper is a function of the voltage
applied in the coil of the damper that is based on the force given by the controller.

Keywords: feedback control, feedforward control, MR damper

1. Introduction

The study of non-ideal vibrating systems, that is, when the excitation is influenced by the
response of the system, has been considered a major challenge in theoretical and practical
engineering research (as examples, among others, see Bolla et al. (2007), Castão et al. (2010),
Samantaray et al. (2010), Djanan et al. (2013). In this work, we observed chaotic vibrations
of a portal frame structural system mathematical model under non-ideal loading. We intend
not only to suppress large amplitude oscillations but also to reduce them to a periodic orbit.
The suppression of chaos and keeping the oscillations into a desired periodic orbit was obtained
using active control considering theOptimal Linear Feedback (OLFC) proposed byRafikov and
Balthazar (2004) and the use of a magnetorheological (MR) damper modeled considering the
hysteresis phenomenon.

TheMRdamper uses anMRfluidwhich is basically composed ofmicrometer-sized particles
of iron suspended in an oil base. TheMR response ofMR fluids is a result of polarization indu-
ced in the suspended particles by application of an external field. The interaction between the
resulting induced dipoles makes the particles form columnar structures parallel to the applied
field, increasing the viscous characteristics of the device (Kasemi et al., 2012;Dutta andChakra-
borty, 2014). These magnetic properties permit its use as a damper controlled by an electrical
current (Tusset andBalthazar, 2013; Tusset et al., 2012, 2013).When usingMRdamper control
for suppression of unwanted oscillations, the viscosity of the internal fluid varies according to a
variable electrical current or voltage (Tusset et al., 2009; Piccirillo et al., 2014). According to
Cetin et al. (2011), there are two main approaches in the literature to describe the hysteresis
dynamic behavior of MR dampers. One of them is the Bouc-Wen hysteresis model proposed



654 A.M. Tusset et al.

by Spencer et al. (1997), and other one is the LuGre hysteresis model that has been obtained
from the nonlinear friction model proposed in (Jimenez and Alvarez, 2002, 2005; Terasawa et
al., 2004; Sakai et al., 2003). In Dyke et al. (1996), the Bouc-Wen model of anMR damper was
implemented with successful results to a three-story frame model. Cetin et al. (2011) used the
LuGre frictionmodel of anMR damper to a six-story framemodel. Although Bouc-Wen model
can accurately predictMR damper dynamics for active or semi-active control, it is too complex
and difficult to implement (Cetin et al., 2011). On the other hand, Cetin et al. (2011) observed
that theLuGreModel ismore versatile than theBouc-Wenmodel in semi-active control systems,
therefore, in this work, the LuGre friction model is preferred over the Bouc-Wen model.

The organization of this paper is as follows: in Section 2, the mathematical model of the
portal frame under a non-ideal excitation is described and numerical simulations necessary to
analyze the dynamics of the systemare performed. In Section 3, the control of chaoticmotion by
the application of the OLFC method is presented. Section 4 presents the mathematical model
of the MR damper as a function of the applied voltage in its coil and numerical simulations
necessary to analyze the dynamics of the controlled system. In Section 5, the robustness of the
control techniques is tested by including parameters uncertainties on the control signals. The
final remarks and the acknowledgments are in Sections 6.

2. Formulation of the engineering problem

Here, we will consider the horizontal motion of a portal frame under a non-ideal excitation (see
Fig. 1a) and the approximated schematicmodel of the system, represented by coupled oscillators
(see Fig. 1b).

Fig. 1. (a) Non-ideal portal frame; (b) schematic of a non-ideal oscillator

Theparameters of this coupled dynamical system consist ofm0,m1,k1,knl,c1,x1,ϕ,J,r,d,s,
the mass, unbalanced mass, linear stiffness, non-linear stiffness, linear damping, displacement,
angular displacement, inertia moment, eccentricity of the unbalanced mass. d is related to the
voltage applied across the armature of the DC motor and s is a constant for each model of
the DC motor considered. The resulting mathematical model of the structure is a Duffing-like
equation

(m1+m0)ẍ+ bẋ−klx+knlx
3 =m0r(ϕ̈sinϕ+ ϕ̇

2cosϕ)

(J+r2m0)ϕ̈−rm0ẍsinϕ=L(ϕ̇)= d−sϕ̇
(2.1)

Next,we renderEqs. (2.1) dimensionless in terms of newvariables definedby: τ =ωt,x1 =x/x
∗,

andx3 =ϕ/ϕ
∗, wherex∗ andϕ∗ are constant characteristics. Equations (2.1) canbe represented

in a state space, in dimensionless form, as

x′1 =x2 x
′

2 =−αx2+β1x1−β3x
3
1+ δ1 sin(ϕ

∗x3)x
′

4+ δ1cos(ϕ
∗x3)x

2
4

x′3 =x4 x
′

4 = ρ1 sin(x3)x
′

2−ρ3x4+ρ2
(2.2)



On suppression of chaotic motions of a portal frame structure... 655

where

α=
b

(m1+m0)ω
ω=

√
k1

m1+m0
β1 =

kl
(m1+m0)ω2

β3 =
knlx

∗2

(m1+m0)ω
2

δ1 =
m0rϕ

∗

(m1+m0)x
∗

ρ2 =
d

(J+r2m0)ω
2ϕ∗

δ2 =
m0rϕ

∗2

(m1+m0)x∗
ρ1 =

rm0x
∗

(J+r2m0)ϕ∗
ρ3 =

sωϕ∗

(J+r2m0)ω2ϕ∗

2.1. Numerical simulations

Numerical simulations are performed using Matlab RO ode45 integrator with h = 0.01 and
considering parameters:α= 0.1, β1 = 1, β3 = 2, δ1 = 8.373, ρ1 = 0.05, ρ2 = 100 and ρ3 = 200
(Tusset et al., 2013). System (2.2) displays chaotic motions, as shown in Fig. 2.

Fig. 2. (a) Displacement; (b) phase diagram; (c) Lyapunov exponent; (d) frequency spectrum

Computation of the first Lyapunov exponent λ1 = 0.075 confirms chaotic behavior for the
parameters used.

3. Proposed active control

Consider now the introduction of a controllable damper in the system, as shown in Fig. 3.
Introduction of the control U leads system (2.2) to

x′1 =x2 x
′

2 =−αx2+β1x1−β3x
3
1+ δ1 sin(x3)x

′

4+ δ1cos(x3)x
2
4+U

x′3 =x4 x
′

4 = ρ1 sin(x3)x
′

2−ρ3x4+ρ2
(3.1)

where

U =u∗+u (3.2)

u∗ will be the feedforward control and u the feedback control.



656 A.M. Tusset et al.

Fig. 3. (a) Portal frame with active control; (b) schematic oscillator with active control

Since the objective of this work is to control x1 and x2, the variables x3 and x4 will be
considered only as disturbances of the system. Thus

u∗ =x∗2
′
+αx∗2−β1x

∗

1+β3x
∗

1
3
− δ1 sin(x3)x

′

4− δ1cos(x3)x
2
4 (3.3)

Substituting (3.3) into (3.1) and defining the deviation of the desired trajectory as

y=
[
x1−x

∗

1 x2−x
∗

2

]T
(3.4)

we rewrite system (3.1) in matrix form

ẏ=Ay+G(y,x∗)+Bu (3.5)

where

A=

[
0 1
β1 −α

]
B=

[
0
1

]
G(y,x∗)=

[
0

−β3(y1+x
∗

1)
3+β3x

∗

1
3

]

The feedback control u can be found solving Eq. (3.6)

u=−R−1BTPy (3.6)

According toRafikov et al. (2008), if there arematricesQ andR (Q symmetric positive definite)
such that

Q
∗ =Q−GT(y,x∗)P−PG(y,x∗) (3.7)

is positive definite, the matrix G restricted, then the control u is optimal and transfers the
non-linear systems from any initial state to the final state y(∞) =0

J =

∞∫

0

(yTQ∗y+uTRu) dt (3.8)

The symmetric matrixP can be found from the Riccati algebraic equation

PA+ATP−PBR−1BTP+Q=0 (3.9)

For the optimal control verification (3.6), function (3.7) is numerically calculated using

L(t)=yTQ̃y (3.10)

The sufficient criterion to guarantee that control (3.8) is optimal is thatL(t) is positive definite
(Rafikov et al., 2008).



On suppression of chaotic motions of a portal frame structure... 657

3.1. Numerical simulations

Let us define the desired trajectory as being a periodic orbit x∗1 = 0.01cos(πτ). As can be
seen, in Fig. 2d, the choice of this orbit allows the system to keep out of the resonance region
and with a low amplitude value of the displacement. Definingmatrices

Q=

[
1000 0
0 1000

]
R= [0.001]

and solving the Riccati algebraic equation (3.9), we obtain the optimal feedback control (3.6)

u=−1001(x1−x
∗

1)−1001.9(x2 −x
∗

2) (3.11)

Substituting feedforward control (3.3) and the optimal feedback control (3.11) into (3.1), wewill
obtain trajectories shown in Fig. 4.

Fig. 4. (a) Displacement with active control; (b) phase portrait with active control; (c)L(t) calculated
in the optimal trajectory

As can be observed in Figs. 4a and 4b, the control is effective in reducing the displacement
amplitude and frequency desired, and is optimal as shown in Fig. 4c.

4. Proposed control by MR damper with hysteresis

The MR damper has hysteresis effects due to a nonlinear friction mechanism. Many research
efforts have beendevoted to themodeling of this nonlinear behavior.Analternative is theLuGre
friction model (Sakai et al., 2003) which was originally developed to describe nonlinear friction
phenomena (Jimenez and Alvarez, 2002).
The frictionmechanism is a phenomenon inwhich two surfacesmake contact at a number of

asperities at the microscopic level. In themodified LuGre friction model (Jimenez and Alvarez,
2002), thismechanism is expressed by the average behavior of the bristles. In (Sakai et al., 2003;
Cetin et al., 2011), another MR dampermodel based on the LuGremodel is described as

F =σaz+σ0zv+σ1ż+σ2ẋ+σbẋv

ż= ẋ−σ0a0|ẋ|z
(4.1)



658 A.M. Tusset et al.

whereF is the damping force, v is the input voltage, z(t) is the internal state variable [m], ẋ is
the velocity of the damper piston [m/s],σ0 is the stiffness of z(t) influenced by v [N/(mV)], σ1 is
the damping coefficient of z(t) [Ns/m], σ2 is the viscous damping coefficient [Ns/m], σa is the
stiffness of z(t) [N/m], σb is the viscous damping coefficient influenced by v [Ns/(mV)], a0 is the
constant value [V/N].

As can be seen in equation (4.1)1, the LuGre model has a parameter that represents the
voltage applied to the coil of the damper v. This parameter allows one to control the force of
theMRdamper through the control of the voltage vmaking the LuGremodel themost suitable
for the active control system.

4.1. Control of oscillations using an active MR damper

For numerical simulations we will consider ẋ in equations (4.1) as ẋ=−0.01π sin(πτ), and
the parameters:σ0 =8·10

5,σ1 =1.6·10
3,σ2 =1.5·10

2,σa =4·10
5,σb =8·10

2 anda0 =3·10
−3

(Sakai et al., 2003). In Fig. 5, one can observe the force the damperMR (4.1)1.

Fig. 5. Characteristics of theMR damper as a function of voltage: (a) force vs. velocity, (b) force vs.
displacement, (c) force vs. time

TheMRdamper semi-active system and the damping force can be controlled by controlling
the applied voltage in the damper coil. According to Tusset et al. (2013), one can numerically
determine the voltage v required for the force of the MR damper F (4.1)1 to coincide with the
desired control forceU, (3.2), obtained from the control strategy.

Considering F = U, the voltage to be applied can be determined by solving the following
function

Γ(v)=σaz+σ0zv+σ1ż+σ2ẋ+σbẋv−U (4.2)

Through Eq. (4.2), we can determine the voltage being applied to the control system con-
sidering the force estimated by the control method, such as control proposed in this paper. In
Fig. 6, we can see the force applied to control the oscillations.



On suppression of chaotic motions of a portal frame structure... 659

Fig. 6. Force used to control the non-ideal system

Considering forces U (Fig. 6) and F (Fig. 5) normalized, the voltage to be applied is deter-
mined by solving numerically function (4.2). In Fig. 7, we see the values of z, ż and ẋ used in
equation (4.2).

Fig. 7. (a) Internal state variable z(τ); (b) derivative of internal state variable ż(τ); (c) velocity of the
piston of the damper ẋ(τ)

In Fig. 8, we can see the voltage control estimated considering Eq. (4.2) and values from
Fig. 7.
Considering these results, we can observe that the proposedmethodology allowed the control

of the voltage (Fig. 8) considering the forceU in Fig. 6 and Eq. (4.2).

4.2. Control of oscillations using a passive MR damper

The objective of introducing a passive MR damper to the portal frame structure shown in
Fig. 9 is to control the displacement of the portal frame in the same scale considered in the
proposed active control.
Next, we will consider the introduction of a passiveMR damperUp into system (2.2)

x′1 =x2 x
′

2 =−αx2+β1x1−β3x
3
1+ δ1 sin(x3)x

′

4+ δ1cos(x3)x
2
4−Up

x′3 =x4 x
′

4 = ρ1 sin(x3)x
′

2−ρ3x4+ρ2
(4.3)



660 A.M. Tusset et al.

Fig. 8. Voltage used to control the non-ideal system

Fig. 9. (a) Portal frame with passive control; (b) schematic oscillator with passive control

where

Up =σaz+σ0zv+σ1ż+σ2ẋ+σbẋv (4.4)

and z is obtained by solving Eq. (4.1)2.
In Figs. 10-13, we can observe motions of the portal frame for different constant voltages v

applied to theMR damper, (4.4).

Fig. 10. Voltage ν=0V and ν=0.5V: (a) displacement, (b) phase portrait

We can observe that as we increase the value of voltage, the amplitude of the displacement
reduces and shifts the equilibrium point of origin. We also observe that for ν = 2.5V the
amplitudes are smaller than those obtained with the active damper. For ν =2.3808V we have
the same displacement amplitude for the active damper and passive damping, as can be seen in
Fig. 14.
As it can be seen inFig. 14, it is possible tomaintain the system on periodic orbits using the

MR damper energized with ν = 2.3808V. One factor which compromises this control strategy
is that the system stabilizes after a long time τ > 113.5, and it is not recommended to keep
the coil of theMR damper energized continuously for long periods, as required by the proposed
active control (Fig. 8).



On suppression of chaotic motions of a portal frame structure... 661

Fig. 11. Voltage ν =1V and ν=1.5V: (a) displacement, (b) hase portrait

Fig. 12. Voltage ν =2V: (a) displacement, (b) phase portrait

Fig. 13. Voltage ν=2.5V: (a) displacement for 0¬ τ ¬ 1000, (b) displacement for 800¬ τ ¬ 1000,
(c) phase portrait



662 A.M. Tusset et al.

Fig. 14. Voltage ν=2.3808V: (a) displacement for 0¬ τ ¬ 1000, (b) displacement for 800¬ τ ¬ 1000,
(c) phase portrait

5. Active control with uncertainties

The parameters used in the control strategy are obtained from a certain data set. The data set
provides parametric errors due to measurement errors or model uncertainties. To consider the
effect of parameter uncertainties on the performance of the controller, the parameters used in
the proposed control will be considered having a randomerror of±20% (Balthazar et al., 2014).

Inorder to consider the effect of parameter uncertainties on theperformanceof the controller,
the real unknown parameters of the system are supposed to be as follows: α̂=0.08+0.04r(t),
β̂1 = 0.8 + 0.4r(t), β̂3 = 1.6 + 0.8r(t), δ̂1 = 6.6984 + 3.3492r(t), ρ̂1 = 0.04 + 0.02r(t),
ρ̂2 = 80+ 40r(t) and ρ̂3 = 160+ 80r(t), where r(t) are normally distributed random func-
tions. To analyze the sensitivity of the error, we consider

[
e1
e2

]
=

[
x1− x̂1
x2− x̂2

]
(5.1)

where xi is obtained for the control without parametric errors and x̂i is obtained for the control
with the parametric error (i=1,2).

Fig. 15. Error of uncertainty in parameters: (a) x1− x̂1, (b) x2− x̂2



On suppression of chaotic motions of a portal frame structure... 663

In Fig. 15, we can observe the sensitivity of the proposed active control with parametric
errors.

As can be observe inFig. 15, the proposed active control is somewhat sensitive to parametric
errors being under 10−4.

6. Conclusions

As can be seen in Figs. 4a and 4b, with the proposed association of feedforward control u∗

(3.3) and feedback u (3.11), it is possible to control the oscillations of the studied portal frame
structure under non-ideal loading into periodic orbits, and the control is optimal (Fig. 4c). Our
model of theMR damper force (4.2) allows us to determine the electrical current to be applied
to the dampers, as shown in Fig. 8. Regarding the use of the passive control, it is possible to
observe that to keep the system in a periodic displaced orbit, the originmust be changed. Thus,
the active control is more indicated, as it keeps the oscillations around the origin.With respect
to the parametric sensitivity, the active control proved to be less sensitive than the passive
control.

With the results obtained, we can conclude that the proposed active control using a MR
damper is more appropriate than the use of a passive MR damper.

Acknowledgments

The authors acknowledge the financial support by FAPESP and CNPq (grant: 420026/2013-4 and

grant: 484729/2013-6), both Brazilian research funding agencies.

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Manuscript received May 9, 2014; accepted for print February 10, 2015