Resonance, stability and chaotic vibration of a quarter-car vehicle model with time-delay feedback

Commun Nonlinear Sci Numer Simulat 16 (2011) 3397–3410

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Commun Nonlinear Sci Numer Simulat

journal homepage: www.elsevier .com/locate /cnsns

Resonance, stability and chaotic vibration of a quarter-car vehiclemodel with time-delay feedback

Raghavendra D. Naik ⇑, Pravin M. SingruDepartment of Mechanical Engineering, Birla Institute of Technology and Science, Pilani, K. K. Birla Goa Campus, Zuarinagar, Goa 403726, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 2 July 2009Received in revised form 9 November 2010Accepted 10 November 2010Available online 24 December 2010

Keywords:ResonanceQuarter-carMelnikov chaosControl

1007-5704/$ - see front matter � 2010 Elsevier B.Vdoi:10.1016/j.cnsns.2010.11.006

⇑ Corresponding author. Tel.: +91 832 2580256; fE-mail addresses: [email protected], raghavchao

This paper examines dynamical behavior of a nonlinear oscillator with a symmetric poten-tial that models a quarter-car forced by the road profile. The primary, superharmonic andsubharmonic resonances of a harmonically excited nonlinear quarter-car model with lineartime delayed active control are investigated. The method of multiple scales is utilized toobtain first order approximation of response. We focus on the influence of delay in the sys-tem. This naturally gives rise to a delay deferential equation (DDE) model of the system.The effect of time delay and feedback gains of the steady state responses of primary, super-harmonic and subharmonic resonances are investigated. By means of Melnikov technique,necessary condition for onset of chaos resulting from homoclinic bifurcation is derivedanalytically. We describe a method to identify the critical forcing function and time delayabove which the system becomes unstable. It is found that proper selection of time-delayshows optimum dynamical behavior. The accuracy of the method is obtained from the frac-tal basin boundaries.

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1. Introduction

The important feature of magneto-rheological fluid is its ability to reversibly change states from a viscous fluid to a semi-solid or even solid with controlled yielding strength within a few milliseconds when it is subjected to controlled magneticfield. The dislocation movement and plastic slipping among molecular chains or crystal lattices consume energy such thatthe restoring force of a magneto-rheological fluid damper always delays the input displacement or velocity. This phenom-enon of energy dissipation is known as hysteresis. Multivalued and non-smooth hysteresis will lead to many complicatedbehaviors such as bifurcation and chaos [4]. Recently, many new applications of active and semi-active control proceduresto minimize vehicle vibrations have been developed by Guo et al. [1], Liu et al. [2], Lauwerys et al. [3]. Chaos and bifurcationin nonlinear vehicle model have been studied by Li et al. [4], Zhu and Mitsuaki [5], Litak et al. [6]. Resonance, stability andchaotic motion of a quarter-car model excited by a road surface profile have been investigated by Siewe Siewe [7].

Recently attention is being paid to the research on active control of various mechanical systems. All controllers exhibit acertain time-delay during operation. Most controlled mechanical systems with time-delay, can be modeled as dynamic sys-tems, described by a set of ordinary differential equations. These differential equations are known as delayed differentialequations. Many researchers have studied the behavior of delayed differential equations. The chaotic behavior of delayeddynamic systems have been studied extensively by Vander Hieden and Walther [8], Nana Nbendjo et al. [9], Wang et al.[10], Xu and Yu [11], Nana Nbendjo et al. [12], Sun et al. [13]. Primary, superharmonic and subharmonic resonances of aharmonically excited nonlinear quarter-car model have been studied by Ji and Leung [14].

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Nomenclature

M the mass of the body, kgX the body’s vertical displacement, mk1 suspension stiffness, Nm�1

X0 excitation frequency, HzX0 the road excitation, ms time delayC2 nonlinear damping coefficient, Ns3m�3

k2 nonlinear stiffness, Nm�3

C1 linear damping coefficient, Nsm�1

x natural frequency, rads�1

b1 and b2 control gain parameters

3398 R.D. Naik, P.M. Singru / Commun Nonlinear Sci Numer Simulat 16 (2011) 3397–3410

This paper deals with study of dynamics of quarter-car model with hysteretic nonlinear suspension system with both de-layed displacement and velocity feedback i.e. with active control. The stability analysis of the linearized system is studied.Then the problem of suppressing smale horseshoe chaos in the model with time-delay is also studied. The accuracy of themethod is obtained from the fractal basin boundaries. Finally, analysis of non-linear dynamical behavior of the system underprimary, superharmonic and subharmonic resonance conditions is performed.

2. The quarter-car model

The one degree of freedom quarter-car model with hysteretic nonlinear damping is studied as shown in Fig. 1. The equa-tion of motion of the system is given by,

M€xþ k1ðx� x0Þ þ Fh ¼ 0 ð1Þ

where Fh is hysteretic nonlinear damping force of magnetorheological fluid damper, and one of its hysteresis loop is shown inFig. 2. Asymmetries in the hysteresis loops correspond with the sign of cubic stiffness k2 [15].

where x0 ¼ ASinX0T and Fh ¼ k2ðx� x0Þ3 þ C1ð _x� _x0Þ þ C2ð _x� _x0Þ3

Defining a new variable for the relative vertical displacement as y = x � x0, Eq. (1) can be written as

€yþx2yþ B1y3 þ B2 _yþ B3 _y3 ¼ F 0 SinX0t ð2Þ

where x2 ¼ k1M ;B1 ¼ k2

M ;B2 ¼ C1M ; B3 ¼ C2

M ; F0 ¼ AX02

The corresponding dimensionless equation of motion can be written for a scaled time variable s = xt as

€yþ yþ ky3 þ a _yþ b _y3 ¼ F0 Sin ðXsÞ ð3Þ

Fig. 1. The quarter-car model (a) semi-active control (b) active control.

Fig. 2. Hysteresis loop for X = 0.8.

R.D. Naik, P.M. Singru / Commun Nonlinear Sci Numer Simulat 16 (2011) 3397–3410 3399

where F0 ¼ AX2; k ¼ B1

x2 ¼k2

k1; a ¼ B2

x¼ C1ffiffiffiffiffiffiffiffiffiffi

k1Mp ; b ¼ B3x ¼

C2

M

ffiffiffiffiffik1

M

r; X ¼ X0

xð4Þ

Here, the values of M, k1, k2, c1 and c2 are chosen from [1].

M ¼ 240 Kg; k1 ¼ 160; 000 Nm�1; k2 ¼ �300;000 Nm�3; C1 ¼ �250 Nsm�1; C2 ¼ 25 Ns3m�3

Introducing active control into Eq. (3) we obtain

€yþ yþ ky3 þ a _yþ b _y3 ¼ U þ F0 Sin ðXtÞ ð5Þ

where ‘U’ is the active damping force. U=ðb1yðt � sÞ þ b2 _yðt � sÞÞ, b1 and b2 are control gain parameters and s is time-delay.Let e = a, substituting in Eq. (5) we obtain

€yþ yþ ky3 þ eð _yþ c _y3 � 1aðb1yðT � sÞ þ b2 _yðT � sÞÞ � f Sin ðXtÞÞ ¼ 0 ð6Þ

Eq. (6) in state space form can be written as,

_y1 ¼ y2

_y2 ¼ �y1 � ky31 þ e �y2 � cy3

2 þ1aðb1y1ðt � sÞ þ b2y2ðt � sÞÞ þ f Sin ðXtÞ

� �ð7Þ

where c ¼ c2x2

c1; b ¼ Mx

c1; f ¼ MxAX2

c1¼ MxF0

c1ð8Þ

3. Stability analysis of the linearized system

The autonomous system is obtained for F0 = 0. By applying Lyapunov concept for examining fundamental solution, ekt, thecharacteristic equation is given by,

k2 þ akþ 1 ¼ ðb1e�kt þ b2k�kte Þ ð9Þ

To obtain the stability boundary in the plane of the control gain parameters (b1,b2) we use D-subdivision method [16].According to this method, the stability boundary in the plane (b1,b2), are determined by the points that yield either to a rootk = 0 or a pair of pure imaginary roots of Eq. (9).

Substituting k = 0 in Eq. (9) one obtains,

b1 ¼ 1 ð10Þ

Setting k = is (where s is a real constant) in characteristic Eq. (9) one finds the systems of equations are,

�b1 sinðssÞ þ b2s cosðssÞ ¼ as

b1 cosðssÞ þ b2s sinðssÞ ¼ �s2 þ 1 ð11Þ


which leads to

b1 ¼D1

Dand b2 ¼

D2

Dð12Þ

where

D ¼� sinðssÞ s cosðssÞcosðssÞ s sinðssÞ

��; D1 ¼

as s cosðssÞð1� s2Þ s sinðssÞ

��

D2 ¼� sinðssÞ as

cosðssÞ ð1� s2Þ

��

The stability boundary in the plane (b1,b2) can be found from bifurcation curve defined by the Eq. (12), and the bifurcationline defined by Eq. (10).

The stability boundaries are plotted in Fig. 3. Thus the time-delay increases, the stability area decreases.

4. Melnikov method and fractal basin boundaries

The Melnikov method is a procedure that identifies parameter ranges of a system in which chaos occurs. It is applicable toconservative systems which include a separatrix loop, and which are perturbed by small forcing and damping. Melnikov’scriterion generally yields lower bounds on the parameters that admit chaotic solutions [17]. There exists an intersectionof the stable and unstable manifolds of an equilibrium point in a two-dimensional Poincaré map. This implies that thereis a horseshoe in the map, which in turn implies that there exist periodic motions of all periods, as well as motions whichare not periodic. The horseshoe exhibits sensitive dependence on initial conditions. It is required to find how the parametersand the time delay affect the Melnikov criterion for chaos.

When e = 0, Eq. (6) becomes a two-dimensional conservative system. As k > 0, there is only one fixed point O(0,0), whilefor k < 0 three fixed points appear: O(0,0), A �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=� k

p;0

� �; B

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=� k

p;0

� �. In the phase portrait, point O is centre; point A

and B are saddle points. Since phase trajectories cannot cross the centre, the unperturbed system shows a stable periodicmotion. Chaotic motion in the sense of smale horseshoe cannot exist in the system when k > 0. Hence k < 0 is investigatedonly. The phase trajectories are shown in Fig. 4. The Hamiltonian function of this system for e = 0 is

H ¼ y22

2þ y2

1

2þ ky4

1

4¼ � 1

4k

and the heteroclinic orbits, which form a cycle, lie in the locus of H = �1/4k can be obtained as follows:

Y0 ¼y2ðtÞy1ðtÞ

� �¼

�ffiffiffiffiffi1�k

qtanh

ffiffi12

qt

� ��

ffiffiffiffiffiffi1�2k

qsec h2

ffiffi12

qt

� �0B@

1CA

The Melnikov functions of the heteroclinic orbits forming the heteroclinic cycle are

M�ðt0Þ ¼Z þ1

�1f ðY0ðt � t0ÞÞ ^ gðY0ðt � t0Þ; tÞdt

¼Z þ1

�1y2ðtÞ �y2ðtÞ � cy3

2ðtÞ þ f sin Xðt þ t0Þ þ1aðb1y1ðt � sÞ þ b2y2ðt � sÞÞ

� dt

Fig. 3. Stability boundary in the space (b1,b2) for various values of time-delay.

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

y1

y 2

Fig. 4. Phase space trajectories.


Integrating the above equation one obtains

M�ðt0Þ ¼2ffiffiffi2p

3k� 8

ffiffiffi2p

c

35k2 � Bþ C ð13Þ

where

B ¼fp

ffiffiffiffiffi1�k

qX sin Xt0ffiffiffi

2p and C ¼ �

ffiffiffiffiffiffiffiffi1

2k2

s1a

b1D1 þ1

2k1a

b2D2

!

D1 ¼Z 1

�1sec2 h

ffiffiffi12

rt

!tanh

ffiffiffi12

rðt � sÞ

!dt

D2 ¼Z 1

�1sec2 h

ffiffiffi12

rt

!sec2 h

ffiffiffi12

rðt � sÞ

!dt

The functions M±(t0) vanish when the stable and unstable manifolds of the 2p/X map associated with the heteroclinic cycleof Eq. (6) intersect, and since zeros are simple, intersections are transverse, which implies horseshoe chaos [17]. A more re-cent result shows that chaos is achieved if only one branch of the cycle has its tangent intersection perturbed into transverseintersections [18]. Using Melnikov criterion, the possible chaotic region resulting from homoclinic bifurcation, in terms ofparameters and time delay, of the system (5) is given by

�

ffiffiffiffiffiffiffiffi1

2k2

s1a

b1D1 þ1

2k1a

b2D2 �fp

ffiffiffiffiffi1�k

qX csc h pX

2ffiffi2p

� �sec h pX

2ffiffi2p

� �ffiffiffi2p 6

8ffiffiffi2p

c

35k2 �2ffiffiffi2p

3k

!ð14Þ

The integrals given by D1 and D2 are evaluated analytically for different values of time-delay and is shown in Table 1. FromEq. (14) the effect of the control gain parameters and time delay on the critical value of F0 is discussed. Fig. 5 shows thedomain in the space parameters (b1,b2), where chaos is suppressed for F0 = 0.255 and X = 0.8. It corresponds to area I.The effect of time-delay s, on the critical value of F0c for onset of chaos is shown in Fig. 6. It is clear that the presence of timedelay enlarges the possible chaotic domain. To validate the analytical results, the Eq. (5) is numerically simulated to analyzefractality in the basins of attraction.

Table 1Integral table.

Time delay D1 D2

0 0 1.8850.2 �0.266 1.8700.4 �0.527 1.8260.6 �0.781 1.7550.8 �1.023 1.6611 �1.25 1.549

Area I

−2.4

−1.9

−1.4

−0.9

−0.4

0.1

0.6

1.1

1.6

2.1

−1 −0.5 0 0.5 1β2

β 1

Area II

Fig. 5. Region in the plane (b1,b2) with s = 0.4. Horseshoe does not occur in area I.

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0 0.2 0.4 0.6 0.8 1τ

Foc

Fig. 6. Critical forcing amplitude F0cv/s time delay s.


Fig. 7 shows fractal basin boundary for uncontrolled system, fractal dimension is 1.869, the fractality appears and fingersoccur in the boundary of safe basin. Figs. 8–11 shows safe basin for controlled system. For fractal dimension�2 the boundaryof the safe basin is smooth. For s = 0.1, increase in F0 causes the safe basin to erode, the fractal dimension is reduced from1.986 for F0 = 0.264–1.978 for F0 = 0.66. When s = 0.4, the fractal dimension suddenly reduces from 1.983 for F0 = 0.264 and1.976 for F0 = 0.66. This implies that for increase in forcing function F0 and time delay, the erosion of safe basin is aggravatedsince the fractal dimension is reduced. Fig. 12 indicates that for delayed state feedback, fractality appears with fractal dimen-sion of 1.976 and for delayed velocity feedback safe basin boundary is small with fractal dimension 1.953 and fingers occur in

Fig. 7. Fractal basin boundary for uncontrolled system b1 = 0, b2 = 0, F0 = 0.2624 fractal dimension = 1.869.

Fig. 8. Fractal basin boundaries for b1 = 0.3, b2 = 0.8, (a) F0 = 0.2624 (b) F0 = 0.4 and s = 0.1. (a) Fractal dimension = 1.986, (b) Fractal dimension = 1.985.



the boundary, which suggests the fractal behavior for the same parameters. Therefore the effect of the delayed state feedbackis best compared to delayed velocity feedback for smaller time delay.

5. Method of multiple scales

The local bifurcation analysis is performed using method of multiple scales [19]. It can be assumed that the uniform solu-tion of Eq. (5) can be represented in the form

yðt; eÞ ¼ y0ðT0; T1Þ þ ey1ðT0; T1Þ þ . . . ð15Þ

where T0 = t and T1 = et. Here we take two time scales T0 and T1. Suppose k, a and b to be small parameters that is k = ek;a = ea and b = eb where e is a small perturbation parameter, then the differential operator become

ddt¼ D0 þ eD1 þ . . . ;

d2

dt2 ¼ D20 þ 2eD0D1 þ . . . ð16Þ

where Dn ¼ @@Tn; Tn ¼ en; n ¼ 0;1:




5.1. The primary resonance case (X = 1 + e r)

To find the amplitude of the oscillations at the primary resonance, we assume that F0 = eF0. Substituting Eqs. (15) and (16)into Eq. (5) and equating coefficients of like powers of e leads to the following differential equations.

D20y0 þ y0 ¼ 0 ð17Þ

D20y1 þ y1 ¼ �2D0D1y0 � ky3

0 � aD0y0 � bðD0y0Þ3 þ b1yðt � sÞ þ b2 _yðt � sÞ þ F0 sinðT0 þ rT1Þ ð18Þ

The general solution of Eq. (17) is given by:

y0 ¼ AðT1Þ expðjT0Þ þ c:c: ð19Þ

where c.c. stands for complex conjugates.Substituting Eq. (19) into Eq. (18) and eliminating secular terms, one obtains

�2j _A� jaA� 3bj Aj j2A� 3kjAj2Aþ b1A expð�jsÞ þ b2Aj expð�jsÞ þ 12

F0 exp j rT1 �p2

� �¼ 0 ð20Þ

Fig. 12. Evolution of safe basin for (a)b1 = 0.3, b2 = 0, F0 = 0.364, s = 0.4 and (b) b1 = 0, b2 = 0.3, F0 = 0.364 and s = 0.4. (a) Fractal dimension = 1.976, (b) Fractaldimension = 1.953.


It is convenient to put A(T1) in the polar form

AðT1Þ ¼12

rðT1Þ expðjUðT1ÞÞ ð21Þ

where r(T1) and h(T1) are the amplitudes and phases. Substituting Eq. (21) into Eq. (20) and separating the real and imaginaryparts, following set of differential equations that govern the amplitude r(T1) and the phase h(T1) are obtained.

_r ¼ �a2

r � 3b8

r3 � b1

2r sinðsÞ þ b2

2r cosðsÞ þ 1

2F0 sinðhÞ

r _h ¼ rr� 3k8

r3 þ b1

2r cosðsÞ þ b2

2r sinðsÞ þ 1

2F0 cosðhÞ

ð22Þ

where hðT1Þ ¼ rT1 �U� p2. Then eliminating h from Eq. (22) the following nonlinear algebraic equation for the steady state

at the primary resonance is obtained.

r0r�3k8

r30 þ

b1

2r0 cosðsÞ þ b2

2r0 sinðsÞ

� �2

þ �a2

r0 �3b8

r30 �

b1

2r0 sinðsÞ þ b2

2r0 cosðsÞ

� �2

¼ 14

F20 ð23Þ

Let ae ¼ � a2�

b12 sinðsÞ þ b2

2 cosðsÞ; re ¼ rþ b12 cosðsÞ þ b2

2 sinðsÞ.The above Eq. (23) reduces to

r0re �3k8

r30

� �2

þ aer0 �3b8

r30

� �2

¼ 14

F20 ð24Þ

There can be either one or three solutions of the above amplitude equation. Three real solutions exist between two points ofvertical tangents i.e. saddle node bifurcation, which are determined by differentiating Eq. (23) with respect to r2. This leads tothe condition

r2e �

3k2

r2re þ a2e þ

3b8

aer2 þ 27b2

64þ 27k2

64

!r4 ¼ 0

with solutions

r�e ¼3k4

r2 � 9k2

64� 27b2

64

!r4 � 3b

2aer2 � a2

e

!1=2

For 9k2

64 �27b2

64

� �r4 � 3b

2 aer2 > a2e there exists an interval in which three real and positive solutions of Eq. (23) exist. In the

limit, 9k2

64 �27b2

64

� �r4 � 3b

2 aer2 ¼ a2e , this interval shrinks to the point re ¼ 3k

4 r2. The critical force amplitude obtained from

Eq. (23) is


F2critical ¼

9k2

16� 27b2

16

!r6

c þ 3baer4c þ 4a2

e r2c ð25Þ

where rc ¼3b4

ae �9k2

16þ 9b2

16

!a2

e

4

!1=2

:

For F0 < Fcritical there is only one solution while for F0 > Fcritical there are three. The stability of the solutions is determinedby the eigenvalues of the corresponding Jacobian matrix of Eq. (22). The corresponding eigenvalues are the roots of

k2 þ Pkþ Q ¼ 0

where

P ¼ 2ae þ 3br20

�and Q ¼ ae þ

9b8

r20

� ae þ

3b8

r20

� þ re �

3k8

r20

� re �

9k8

r20

�

If P < 0, at least one of the two eigenvalues will have positive real parts and the system will be unstable. If P = 0 a pair ofpurely imaginary eigenvalues and hence a Hopf bifurcation may occur. The above two cases should be avoided. Whereasfor, P > 0 the sum of the two eigenvalues is always negative. The other eigenvalue is zero when Q = 0 where a saddle-nodebifurcation occurs. When Q > 0 no unstable solutions exist. The feedback should be implemented at least in such a way thatae > 0. Thus, the appropriate feedback gains and time-delays can enhance the control performance. Fig. 13 indicates an in-crease of the peak of the resonance as the time delay is increased. Fig. 14 indicates that the presence of time delay reduces

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

−5 −3 −1 1 3σ

r

Fig. 13. Frequency response curve for different time delay.s = 0.01 (solid line), s = 0.4 (dotted line).

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8τ

Fc

Fig. 14. Critical force v/s time delay for primary resonance.


the critical force while amplitude attains a larger value. This leads to poor control performance. Fig. 15 indicates that optimalcontrol can be achieved if the time delay is implemented in the region 0 < s < 0.4.

5.2. Superharmonic resonance (3X = 1 + er)

D20y0 þ y0 ¼ F0 sin Xt ð26Þ

D20y1 þ y1 ¼ �2D0D1y0 � ky3

0 � aD0y0 � bðD0y0Þ3 þ b1yðt � sÞ þ b2 _yðt � sÞ ð27Þ

The general solution of Eq. (26) is given by:

y0 ¼ AðT1Þ expðJT0Þ þK exp j XT0 �p2

� �þ c:c: ð28Þ

where c.c. stands for complex conjugates, and K ¼ F0

2ð1�X2Þ. Substituting Eq. (28) into Eq. (27), one obtains

D20y1 þ y1 ¼ ð�2j _A� jaA� 3kjAj2AÞ exp jXT0 � ð6kAK2 þ 3jbjAj2AÞ exp jT0 � ð6jbAX2K2Þ exp jT0

� ðKaX� 6KjkjAj2 � 3jkK3Þ exp jXT0 � ð6bAXKA2 þ 3bX3K3Þ exp jXT0

� 3AK2ðk� jbX2Þ exp jðð1þ 2XÞT0 � pÞ þ A3ð�kþ jbÞ exp 3jT0 �K3ðjkþ bX3Þ exp 3jXT0

� 3AK2ðk� jbX2Þ exp jðð1� 2XÞT0 þ pÞ � 3A2Kðjk� bXÞ exp jð2þXÞT0

� 3A2Kðjk� bXÞ exp jð2�XÞT0 � 3A2ðjk� bXÞ exp jð1þXÞT0 þ b1A expð�jsÞ þ b2Aj expð�jsÞ þ c:c:

ð29Þ

For the superharmonic state X ¼ 13, then the term proportional to ej3XT0 will be near-resonant. Substituting Eq. (21) into Eq.

(29), the amplitudes can be obtained as

r0re �3k8

r30 � 3kK2r0

� �2

þ aer0 þ3b8

r30 þ 3bX2K2r0

� �2

¼ K3bX3� �

þ ðK3kÞ ð30Þ

where

ae ¼ �a2� b1

2sinðsÞ þ b2

2cosðsÞ; re ¼

b1

2cosðsÞ þ b2

2sinðsÞ

The critical force amplitude obtained from Eq. (30) is

F2critical½K

3bX3 þK3k� ¼ 9k2

16� 27b2

16

!r6

c þ 3baer4c þ 4a2

e r2c ð31Þ

where

rc ¼3b4

ae �9k2

16þ 9b2

16

!a2

e

4

!1=2

:

For F0 < Fcritical there is only one solution while for F0 > Fcritical there are three. The stability of the solutions is determinedby the eigenvalues of the corresponding Jacobian matrix. The corresponding eigenvalues are the roots of

k2 þ Pkþ Q ¼ 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.1 0.4 0.7 1 1.3 1.6 1.9 2.2 2.5 2.8

τ

r

Fig. 15. Amplitude v/s time delay for primary resonance. r = 0.425 (solid line), r = 0.03 (dotted line).


where

P ¼ 2ae þ3b2

r20

� �and Q ¼ ae þ

9b8

r20

� ae þ

3b8

r20

� � re �

3k8

r20

� re �

9k8

r20

�

The steadystate motions are stable only when two inequalities P > 0 and Q > 0, simultaneously hold. The occurrence ofsaddle-node bifurcation can be delayed or eliminated by selecting appropriate feedback gains and time-delay.

Fig. 16 indicates a shift of frequency response from left to right with the increase of the peak of the resonance as the timedelay is increased. Fig. 17 indicates that the presence of time delay reduces the critical force while amplitude attains a largervalue. This leads to poor control performance. Usually, ae should have larger value in order to improve the control perfor-mance. From Fig. 17 it is concluded that the time delay should be implemented in a region such that larger Fcritical and smalleramplitudes are obtained.

5.3. Subharmonic resonance (X = 3 + er)

For the subharmonic state X = 3, then the term proportional to ejðX�2ÞT0 will be near-resonant. We seek a solution of Eq.(29) and by considering X = 3 + er, the amplitudes are obtained as

r0re �3k8

r20 � 3kK2

� �2

þ aer0 þ3b8

r20 þ 3bX2K2

� �2

¼ 3bKX4

r0

� �2

þ 3Kk4

r0

� �2

ð32Þ

where

ae ¼ �a2� b1

2sinðsÞ þ b2

2cosðsÞ; re ¼

b1

2cosðsÞ þ b2

2sinðsÞ

The roots of the above equations are given by

r20 ¼ p� ðp2 � qÞ1=2

0.020.03

0.040.05

0.060.070.08

0.090.1

0.110.12

−4 −3 −2 −1 0 1 2 3 4

σ

r

Fig. 16. Frequency response curve for different time delay. s = 0.01 (solid line), s = 0.4 (dotted line).

4

5

6

7

8

9

10

11

0 0.2 0.4 0.6 0.8 1 1.2

τ

Fc

Fig. 17. Critical force v/s time delay for superharmonic resonance.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.605 0.625 0.645 0.665 0.685 0.705 0.725

σ

r

Fig. 18. Frequency response curve for delay. s = 1.52.


where p ¼ �3b8

ae þ 3bX2K2� �

þ 3k8ðre � 3kK2Þ þ 3bKX

16r0

� �2

þ 3Kk16

r0

� �2

and

q ¼ p2

4� 9b

64þ 9k

64

� �re � 3kK2� �2

þ ae � 3bX2K2� �2

� 1=2

the non-trival free-oscillation amplitudes occurs only when p > 0 and p2 P q as q is always positive. The steady state solu-tions of subharmonic resonance response is determined by the eigenvalues of the characteristic equation, which are theroots of

k2 þ Pkþ Q ¼ 0

where

P ¼ 2ae and Q ¼ ae �3b8

r20

� ae þ

3b8

r20

� þ re �

3k8

r20

� re �

3k8

r20

�

It can be found that if the feedback control is appropriately implemented, violating one of the two inequalities i.e. p > 0 andp2 P q, the system will not exhibit subharmonic resonance.

The amplitude response for F0 = 0.2624 is plotted in Fig. 18. In Fig. 18 the isolation of limit cycles is generated in a saddle-node bifurcation near the detuning parameter of r = 0.620.

6. Results and discussion

To obtain the stability boundary in the plane of the control gain parameters D-subdivision method is used. Fig. 3 indicatesthat as the time-delay increases the stability area decreases. Fig. 5 shows the domain in the space parameters (b1,b2), wherechaos is suppressed for F0 = 0.255 and X = 0.8. It corresponds to area I. Fig. 6 indicates that the presence of time delay lowersthe threshold and enlarges the possible chaotic domain. To validate the analytical results, the Eq. (5) is numerically simu-lated to analyze fractality in the basins of attraction. Fig. 7 shows fractal basin boundary for uncontrolled system, fractaldimension is 1.869, the fractality appears and fingers occur in the boundary of safe basin. For s = 0.1, increase in F0 causesthe safe basin to erode, the fractal dimension is reduced from 1.986 for F0 = 0.264 to 1.978 for F0 = 0.66. When s = 0.4, thefractal dimension suddenly reduces from 1.983 for F0 = 0.264 and 1.976 for F0 = 0.66. This implies that for increase in forcingfunction F0 and time-delay, the erosion of safe basin is aggravated since the fractal dimension is reduced. The increase intime-delay reduces the distance of the saddles, which results in shrinking of the safe basin. It attributes to the heteroclinictangle between the manifolds of the two saddles, which produces a basin whose boundary is fractal. Fig. 12 indicates that fordelayed state feedback, fractality appears with fractal dimension of 1.976. For delayed velocity feedback fingers occur at thesafe basin boundary with fractal dimension 1.953, which suggests the fractal behavior for the same parameters. Thereforethe effect of the delayed state feedback is the best compared to delayed velocity feedback for smaller time-delay in suppress-ing horseshoe chaos. Fig. 13 indicates an increase of the peak of the resonance as the time-delay is increased. Fig. 14 indicatesthat the presence of time-delay reduces the critical force while amplitude attains a larger value. This leads to poor controlperformance. Fig. 15 indicates that optimal control can be achieved if the time-delay is implemented in the region 0 < s < 0.4.Usually, ae should have larger value in order to improve the control performance. From Fig. 17 it is concluded that the time-delay should be implemented in a region such that larger Fcritical and smaller amplitudes are obtained. It can be found that ifthe feedback control is appropriately implemented, violating one of the two inequalities i.e.p > 0 and p2 P q, the system willnot exhibit subharmonic resonance. In Fig. 18 the isolation of limit cycles is generated in a saddle-node bifurcation near thedetuning parameter of r = 0.620.


7. Conclusion

This study addresses dynamics and homoclinic bifurcations of a forced nonlinear oscillator which models a single degreeof freedom quatercar with both delayed displacement and velocity feedback. Using Melnikov technique, condition for chaos,resulting from homoclinic bifurcation, is derived analytically. The effect of control gain parameters on the chaotic motion isstudied in detail. The effect of time-delay on the critical forces, for which Melnikov chaos appears, is important and should betaken into account for designing vehicle suspension system. The accuracy of the method is obtained by basin of attraction. Itis found that erosion of safe basin boundaries can be reduced when the time- delay is short. Compared to delayed velocityfeedback, delayed state feedback is the best strategy to reduce the erosion of safe basins at small values of time-delay. Itimplies that delayed feedbacks prevent the stable and unstable manifolds from tangling. The method of multiple scales isused to find the amplitude at different resonances. The effect of time-delay on both resonance and chaotic motion of thevehicle system is studied. The results reveal that the increase in time-delay reduces the critical force which leads to poorcontrol performance. It is found that an appropriate feedback and time-delay can enhance the control performance in caseof primary and superharmonic resonances. For subharmonic resonance, an adequate feedback can eliminate the occurrenceof subharmonic resonance response.

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Resonance, stability and chaotic vibration of a quarter-car vehicle model with time-delay feedback

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