Delayed feedback for controlling the nature of bifurcations in friction-induced vibrations

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Journal of Sound and Vibration

Journal of Sound and Vibration ] (]]]]) ]]]–]]]

0022-46

doi:10.1

� Cor

E-m

Pleasfricti

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

Delayed feedback for controlling the nature of bifurcations infriction-induced vibrations

Ashesh Saha, Pankaj Wahi �

Mechanical Engineering Department, Indian Institute of Technology, Kanpur 208016, India

a r t i c l e i n f o

Article history:

Received 20 February 2011

Received in revised form

18 July 2011

Accepted 22 July 2011

Handling Editor: D.J. Waggbifurcation to supercritical along with increasing the stability boundaries. A nonlinear

0X/$ - see front matter & 2011 Elsevier Ltd.

016/j.jsv.2011.07.032

responding author. Tel.: þ91 512 2596092; f

ail address: [email protected] (P. Wahi).

e cite this article as: A. Saha, &on-induced vibrations, Journal of Sou

a b s t r a c t

We analyse the control of friction-induced vibrations using time-delayed displacement

feedback. We have used the exponential model for the drooping characteristics of the

friction force for which the bifurcation is subcritical in nature. With an appropriate

choice of the control parameters we have managed to change the nature of the

controller is required when the control force is applied in a direction parallel to the

friction force. In contrast, a linear time-delayed displacement feedback applied in a

direction normal to the friction force achieves our dual objective of controlling the

nature of the bifurcation as well as quenching the vibrations. We also consider a

dynamic friction model (the LuGre model) and observe that the qualitative change in

the nature of the bifurcation is independent of the complexity considered in modeling

the friction force.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

In this paper, we use time-delayed position feedback to control friction-induced vibrations and the nature of theassociated bifurcation. We use both linear and nonlinear controller for this purpose. A nonlinear controller is required forcontrolling the nature of the bifurcation when the control force is applied in a direction parallel to the friction force.However, a linear controller is sufficient when the control force is applied in a direction normal to the friction force. Somepreliminary results from this work were also presented in [1].

Friction-induced vibrations is a type of self-excited oscillation [2,3] which is frequently encountered in manyengineering systems. Some of the examples are: brake-squeal, train curve squeal, clutch chatter, machine-tool chatter,vibrations in systems with controlled positioning like signal receivers and radars. These vibrations are a major concern inmany engineering systems due to their detrimental effects like high levels of noise, excessive wear of components, surfacedamage and fatigue failure.

Research in friction-induced vibrations can be broadly divided into three categories: (i) modeling and analysis of thestructure with frictional contact (the instability mechanisms), (ii) modeling the friction phenomenon itself, and (iii) controlstrategies for quenching the resulting vibrations. There are several instability mechanisms that have been associated withfrictional systems. The major mechanisms which are widely studied in the literature are: (i) the drooping characteristics ofthe friction force in the low relative velocity regime [4–14], (ii) the mode coupling instability [15–21], and (iii) the sprag-slip instability [22,23]. The last two instability mechanisms do not require a variable friction coefficient. In contrast, the

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A. Saha, P. Wahi / Journal of Sound and Vibration ] (]]]]) ]]]–]]]2

instability due to the drooping characteristics is caused purely due to a decrease in the friction coefficient (variable frictioncoefficient) with increasing relative velocities in the low velocity regime resulting in an effective negative damping. Thedrooping friction characteristic produces indefinite damping matrix in the linearized system which is a source ofsubcritical flutter and squeal [13,14]. In this paper, we will only consider the instability due to the drooping characteristicsof the friction force and analyse the effectiveness of the control force in quenching the resulting vibrations. For simplicityof analysis, we consider a single degree-of-freedom (SDOF) spring–mass–damper system on a moving belt.

Several empirical and semi-empirical models with varying levels of complexities have been proposed in the literatureto account for the drooping characteristics [5–7,24–33]. In this study, we have used the exponential friction modeloriginally proposed by Tustin [5] but with an alternate form suggested by Hinrichs et al. [6]. However, we briefly considerthe LuGre friction model proposed by Canudas de Wit et al. [29] which is a widely used dynamic friction model todemonstrate the robustness of our results with changes in the friction model. Previous studies have shown that the natureof the bifurcation associated with these friction models is subcritical [4,34–36] which is not desirable since the equilibriumis only locally stable near the stability boundary. Moreover, a slight intrusion into the unstable regime results in largeamplitude vibrations making it difficult to retract back to stable operation. In contrast, a supercritical bifurcationguarantees global stability in the linearly stable regime and small amplitude vibrations when the stability boundaries arecrossed preventing detrimental damage to the structure. Hence, supercritical bifurcations are more acceptable thansubcritical bifurcations. In this paper, we explore the possibility of changing the nature of the bifurcation using anappropriate control strategy.

Both passive and active means have been employed in the literature for the control of friction-induced vibrations[7,34–43]. Passive control achieves only partial quenching and reduces the amplitudes of the vibrations. On the other handactive control can achieve complete quenching and hence has become popular among researchers. A traditional controlstrategy based on PID controller for friction-induced vibrations due to the drooping characteristics essentially requires aderivative component. However, a time-delayed control strategy with only a proportional controller can stabilize thesesystems making it an attractive option. Previous work on control of friction-induced vibrations used a linear time-delayedcontroller and focussed only on increasing the stability domains [34,35,42,43]. We however consider both linear andnonlinear time-delayed controllers in this work and study both the increase of the stable regions as well as changing thenature of the associated bifurcations. An interesting result from our study is that a linear controller applied in the normaldirection couples with the inherent nonlinearity of the friction force to produce new nonlinear terms which can cause thenature of the bifurcation to change from subcritical to supercritical.

The rest of the paper is organized as follows. We introduce the mathematical models of the system with theexponential friction function in Section 2 with control forces applied parallel to the friction force (tangent control force) aswell as normal to the friction force (normal control force). In Section 3, the system with the tangent control force with anonlinear controller is analysed while the analysis of the system with a linear normal control force is performed in Section4. In Section 5, we consider the system with the LuGre friction model and a linear normal control force to validate therobustness of the results with respect to changes in the friction model. Finally, some conclusions are drawn in Section 6.

2. The mathematical model with exponential friction function

A very popular and simple model for the study of friction-induced vibrations is a spring–mass–damper system with themass resting on a friction belt moving with a constant velocity. We consider this model for the analysis in the presentwork. A schematic representation of the corresponding physical model is shown in Fig. 1. In Fig. 1(a) the control force isapplied parallel to the friction force, while in Fig. 1(b) the control force is applied normal to the friction force. Forsimplicity, we call these control forces as the ‘tangent control force’ and the ‘normal control force’, respectively.

2.1. Governing equation with tangent control force

The equation of motion for the system shown in Fig. 1(a) is given by

Md2X

dt2þC

dX

dtþKX ¼ FtðVrÞþFn

c , (1)

X(t) *0N

K

C

bV

M *cF

X(t) * *c0N +F

K

C

bV

M

Fig. 1. Damped harmonic oscillator on a moving belt as a model for friction-driven vibrations. (a) Tangent control force, (b) normal control force.

Please cite this article as: A. Saha, & P. Wahi, Delayed feedback for controlling the nature of bifurcations infriction-induced vibrations, Journal of Sound and Vibration (2011), doi:10.1016/j.jsv.2011.07.032

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A. Saha, P. Wahi / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 3

where FtðVrÞ ¼Nn

0mðVrÞ is the friction force, Nn

0 ¼MgþFnn is the total normal load acting on the slider with M as the mass of

the slider and Fnn as the externally applied normal load, mðVrÞ is the friction function and Fn

c is the control force.Vr ¼ Vb�dX=dt is the slip velocity where Vb is the belt velocity. We will consider the following nonlinear control force (as in[44]) for the analysis:

Fn

c ¼ Kn

c ðXðt�TnÞ�XÞþKn

cnðXðt�TnÞ�XÞ3, (2)

where Knc is the linear control gain, Kn

cn is the nonlinear control gain and Tn is the time-delay. More advanced time-delayedcontrol strategies (e.g., act-and-wait control [45,46]) are gaining popularity in the recent literature. However, we have notincorporated them in our study.

The nondimensional form of the equations of motion (using the characteristic time scale fixed by o0 ¼ffiffiffiffiffiffiffiffiffiffiffiK=M

pand the

characteristic length scale fixed by x0 ¼ g=o20 ¼Mg=K) is

x00 þ2xx0 þx¼N0mðvrÞþKcðxðt�TÞ�xðtÞÞþKcnðxðt�TÞ�xðtÞÞ3, (3)

where primes denote derivative with respect to the nondimensional time t¼o0t. The other nondimensional quantitiesare

T ¼o0Tn, x¼C

2Mo0, x¼

X

x0, N0 ¼

Nn

0

Mo20x0¼ 1þFn, Fn ¼

Fnn

Mo20x0

,

vb ¼Vb

o0x0, Kc ¼

Knc

Mo20

, Kcn ¼Kn

cnx20

Mo20

, vr ¼ vb�x0:

We will use the exponential friction model [6] given by

mðvrÞ ¼ ðmkþDme�a9vr9ÞsgnðvrÞ, (4)

where Dm¼ ms�mk, ms is the coefficient of static friction, mk is the minimum coefficient of kinetic friction and a is the slopeparameter. For the exponential friction model, the instability is through a subcritical bifurcation [4,34,35]. Note that forlarger values of the slope parameter a, the instability due to the drooping characteristics is more severe, i.e., the effectivenegative damping for a given relative velocity is higher in magnitude.

The steady-state equilibrium ðx00 ¼ x0 ¼ 0Þ for the equation of motion (3) is xeq ¼N0mðvbÞ. Shifting the origin to theequilibrium by introducing the coordinate transformation z¼ x�xeq we get

z00 þz¼�2xz0 þN0ðmðvrÞ�mðvbÞÞþKcðzT�zÞþKcnðzT�zÞ3, (5)

which for pure slipping motions (vr 40) reduces to

z00 þz¼�2xz0 þgN0ðeaz0�1ÞþKcðzT�zÞþKcnðzT�zÞ3, (6)

where g¼Dme�avb and zT ¼ zðt�TÞ.The analysis of the system with the linear tangent control force (with Kcn ¼ 0) has been performed in our previous work

[35] and it was shown that they enhance the stability margins with no effect on the nature of the bifurcation. Here we willonly analyse the case Kcna0 for which the control force can meet our objective of controlling the instability as well aschanging the nature of the bifurcation from subcritical to supercritical.

2.2. Governing equation with normal control force

The equation of motion for the system with a normal control force as shown in Fig. 1(b) is

Md2X

dt2þC

dX

dtþKX ¼ FnðVrÞ, (7)

where FnðVrÞ ¼ ðNn

0þFnc ÞmðVrÞ is the friction force and Nn

0 ¼MgþFnn is the total normal load acting on the slider in the

absence of the control force. The other parameters are as described in Section 2.1. We will use a linear time-delayedcontrol force as given by Eq. (2) with Kn

cn ¼ 0 to check the feasibility of the normal control force in controlling the vibrationas well the nature of the bifurcation. Using the same nondimensionalization as used in Section 2.1, the nondimensionalequation of motion for the system with the normal control force is

x00 þ2xx0 þx¼ ðN0þKcðxðt�TÞ�xðtÞÞÞmðvrÞ: (8)

The steady-state equilibrium ðx00 ¼ x0 ¼ 0Þ for the equation of motion (8) is xeq ¼N0mðvbÞ. The equation of motion with theshifted coordinate z¼ x�xeq becomes

z00 þz¼�2xz0 þN0ðmðvrÞ�mðvbÞÞþKcðzT�zÞmðvrÞ, (9)

which for pure slipping motions (vr 40) reduces to

z00 þz¼�2xz0 þgN0ðeaz0�1ÞþmkKcðzT�zÞþgKcðzT�zÞeaz0 , (10)

where g¼Dme�avb and zT ¼ zðt�TÞ.


dx.doi.org/10.1016/j.jsv.2011.07.032

−0.2 0 0.2−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Fig. 2. Phase portrait of a typical stick-slip limit cycle with and without smoothing. Parameters: ms ¼ 0:3, mk ¼ 0:1, a¼2.5, x¼ 0:05, N0 ¼ 1, Kc ¼ 0:6,

T ¼ 1:0, k¼ 500.


In deriving Eq. (10) from Eq. (9) and Eq. (6) from Eq. (5), we have considered only pure slipping motions (vr 40) andreplaced sgnðvrÞ by 1. Therefore the analysis in this work is restricted to pure slipping motions only. However, tocompletely describe the dynamics of the system through the bifurcation diagrams the stick-slip limit cycles are also foundnumerically along with the pure slipping motions using the software package DDE-BIFTOOL [47]. For this purpose, we havereplaced the non-smooth function sgnðvrÞ by its smooth approximation tanhðkvrÞ with k¼ 500. We note at this point thatthis smoothing operation can lead to errors in the solutions involving stick-slip motions. Fig. 2 shows a phase portrait of atypical stick-slip limit cycle with and without the smoothing and it clearly demonstrates that the stick-slip limit cycle iscaptured fairly well by the smooth approximation. The solutions presented in Fig. 2 have been obtained using the in-builtMATLAB command dde23 with event detection on and a tight tolerance of 1e�6. The time taken for integrating thenon-smooth version is around two orders of magnitude larger than the smoothened version. We emphasize that the mainobjective of this work is to study the change in the nature of the bifurcation leading to vibrations and the associated smallamplitude motions which involve only pure slipping regimes. The stick-slip motions have been presented here only for thesake of completeness of the results and hence the errors incurred by the smoothing operation is acceptable.

3. Analysis of the system with tangent control force

In this section, we analyse the system with tangent control force (Eq. (6)). We start with linear stability analysis. Thelinearized form of Eq. (6) is the same as for the linear controller (Kcn ¼ 0) which has been studied in [35]. However, webriefly present the main results for completeness. We next present the nonlinear analysis to understand the nature of thebifurcation and its dependence on the control parameters.

3.1. Linear stability analysis

Linearizing Eq. (6) about the equilibrium (z¼ 0) we get

z00 þð1þKcÞz�h31z0 ¼ KczT , (11)

where h31 ¼�2xþgaN0. The corresponding characteristic equation is obtained by substituting z¼ cest in Eq. (11). Toinvestigate the stability boundary corresponding to the Hopf bifurcation we first substitute s¼ 7 jo into the characteristicequation, where o determines the frequency of the ensuing periodic solution. Separating the real and imaginary parts ofthe resulting equation gives two consecutive equations which can be solved for the two unknowns (Kc and T) with theother unknown o as the parameter for a given vb (h31). The parametric dependence of the critical values of Kc and T on o isgiven by

Kco ¼ðo2�1Þ2þh2

31o2

2ðo2�1Þ,

To ¼2

otan�1 o2�1

h31o

� �þ2ip

� �, 8i¼ 0,1,2, . . . ,1: (12)

For a Hopf bifurcation to occur, the characteristic roots must cross the imaginary axis transversally. Therefore, thequantity (considering the control gain as the bifurcation parameter)

Rtðoc ,KcoÞ ¼ sgn Reds

dKc

��joc ,Kco

!" #¼ sgnððh31þKcoToÞð1�cosðocToÞÞ�2ocsinðocToÞÞ (13)


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must also be checked. If Rt o0, the equilibrium becomes stable for Kc 4Kco and if Rt 40, the equilibrium becomes unstablefor Kc 4Kco.

3.2. Nonlinear analysis by averaging method

It was demonstrated in [35] that the averaging method yields simple expressions to obtain an understanding of thequalitative behavior of the system but might give crude results. However, we still use it in our analysis to investigate thefeasibility of changing the nature of the bifurcation and numerically verify the qualitative results obtained from thismethod. We rewrite Eq. (6) as

z00 þz¼ ef ðzðtÞ,z0ðtÞ,zT Þ, 0oe51, (14)

where

ef ðzðtÞ,z0ðtÞ,zT Þ ¼�2xz0 þgN0ðeaz0�1ÞþKcðzT�zÞþKcnðzT�zÞ3: (15)

Following [34,35,48] the solution of Eq. (14) is assumed to be

zðtÞ ¼ AtðtÞ sinðtþftðtÞÞ,

z0ðtÞ ¼ AtðtÞ cosðtþftðtÞÞ, (16)

where the amplitude AtðtÞ and phase ftðtÞ are slowly varying functions of time t. Accordingly, it is assumed thatAtðt�TÞ � AtðtÞ, and ftðt�TÞ �ftðtÞ (see [48]) and following the standard averaging method, we obtain the amplitudeequation as

A0t ¼�12 ð2xþKc sinðTÞÞAtþgN0I1ðaAtÞþ

38Kcnðsinð2TÞ�2 sinðTÞÞA3

t : (17)

where I1ðaAtÞ is the modified Bessel’s function of order 1. Expanding I1ðaAtÞ in a Taylor series around At ¼ 0 and retainingterms till third order we get

A0t ¼ LtaAtþNLtaA3t , (18)

where

Lta ¼12ðh31�Kc sinðTÞÞ, (19a)

NLta ¼N0ga3

16þ

3

8Kcnðsinð2TÞ�2 sinðTÞÞ: (19b)

The coefficient of the linear term contains the bifurcation parameter and changes sign at some critical value of thebifurcation parameter (the stability boundary) which is given by Lta ¼ 0. The linearly stable and unstable regimes aredefined by Ltao0 and Lta40, respectively. On the other hand, the nature of the bifurcation is determined by the sign of thecoefficient of the cubic term (NLta). We get the amplitude of the limit cycle by equating A0t to zero as

At ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi�

Lta

NLta

s:

Therefore, for a limit cycle to exist the signs of Lta and NLta must differ. Hence, the bifurcation will be supercritical whenNLtao0 and subcritical when NLta40 at the stability boundary. Therefore, the nature of the bifurcation changes forparameter values which satisfy

Lta ¼ 0, NLta ¼ 0: (20)

Since NLta in Eq. (19b) depends on the control parameters Kcn and T along with the friction parameters, there is apossibility of changing the nature of the bifurcation by choosing these control parameters appropriately. For Kcn ¼ 0, wealways get NLta40 reaffirming that the bifurcation is subcritical in nature.

To obtain the critical values of the parameter Kcn corresponding to the change in the nature of the Hopf bifurcation forgiven values of other parameters, we substitute Eqs. (19a) and (19b) in Eq. (20) and solve them simultaneously to get

Kcn ¼a2ð2xþKcao sinðTaoÞÞ

6ð2 sinðTaoÞ�sinð2TaoÞÞ: (21)

Alternately, we first get the critical belt velocity from Lta ¼ 0 as

ðvbÞtao ¼1

aln

aN0DmKcao sinðTaoÞþ2x

� �: (22)

The critical belt velocity for a supercritical bifurcation should satisfy the condition (using NLtao0)

ðvbÞtao41

aln

a3N0

6Kcnð2 sinðTaoÞ�sinð2TaoÞÞ

� �¼ ðvbÞtacr: (23)


dx.doi.org/10.1016/j.jsv.2011.07.032


The values of the bifurcation parameters corresponding to the criticality curve where the nature of the bifurcation changesfrom subcritical to supercritical can now be obtained by equating ðvbÞtao and ðvbÞtacr. In all the expressions, the subscript ‘t’stands for tangent control force, ‘a’ is used to signify the averaging method, ‘o’ stands for the critical point corresponding tothe Hopf bifurcation (supercritical or subcritical) and ‘cr’ stands for the criticality point where the nature of the bifurcationchanges.

Note from Eqs. (22) and (23) that ðvbÞtao depends on the linear control gain Kc while ðvbÞtacr depends on Kcn. Therefore,the control gain Kc shifts the bifurcation point while Kcn changes the nature of the bifurcation as evident in Figs. 3 and 4. InFig. 3(a), ðvbÞtao and ðvbÞtacr are plotted as a function of Kcn for a fixed value of Kc ¼ 0:2 (i.e., a fixed critical belt velocity)while in Fig. 3(b), ðvbÞtao and ðvbÞtacr are plotted as a function of Kc for Kcn ¼ 3 with the other parameters as ms ¼ 0:3,mk ¼ 0:1, a¼2.5, x¼ 0:05, N0 ¼ 1 and T ¼ 0:4. The subcritical and supercritical zones are marked in these figures. Fig. 3(b)has another regime marked as ‘completely stable region’ which corresponds to the region for which the system is stable forany belt velocity. In Fig. 4, the bifurcation diagrams are plotted as a function of belt velocity vb for different Kcn values withother parameters as used in Fig. 3(a). It is observed from Fig. 4 that the nature of the bifurcation is subcritical for Kcn ¼ 0and 2 wherein an unstable pure slipping limit cycle coexists with a stable equilibrium. There also exist a large amplitudestable stick-slip limit cycle. The unstable limit cycle and the stable stick-slip limit cycle merge at the cyclic fold point afterwhich the equilibrium becomes globally stable. We note that the cyclic fold point is prone to errors due to our smoothingapproximation but this is acceptable since we are mainly interested in the nature of the primary (the Hopf) bifurcationwhich is unaffected by the approximation. The nature of the bifurcation is supercritical for Kcn ¼ 4 and 6 wherein a stablelimit cycle coexists with an unstable equilibrium. As a result the equilibrium is globally stable for belt velocities higherthan the critical belt velocity ðvbÞto. Due to the approximate nature of the averaging analysis, the estimates of theparameters corresponding to the criticality curve is not very accurate and hence, we have used values of Kcn in Fig. 4 whichare sufficiently inside the subcritical and the supercritical zones of Fig. 3(a). A refined analysis using the method ofmultiple scales (MMS) is not performed here since the nonlinear component of the control force (KcnA3

t ) required forchanging the nature of the bifurcation is found to be comparable to the linear component (KcAt).

2 4 6 80

0.5

1

1.5

BA

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

B

A

C

Fig. 3. (a) Variation of ðvbÞtao and ðvbÞtacr with Kcn, (b) variation of ðvbÞtao and ðvbÞtacr with Kc . Inside the figure: region A—subcritical region, region

B—supercritical region, region C—completely stable region. Parameters: ms ¼ 0:3, mk ¼ 0:1, a¼2.5, x¼ 0:05, N0 ¼ 1, T ¼ 0:4, (a) Kc ¼ 0:2, (b) Kcn ¼ 3.

0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5Stable stick−slip

limit cycle

Stable pureslipping

limit cycle

Unstablelimit cycle

Cyclic fold bifurcation

Fig. 4. Amplitude of oscillations for different Kcn values obtained using the software package DDE-BIFTOOL [47]. Parameters: ms ¼ 0:3, mk ¼ 0:1, a¼2.5,

x¼ 0:05, N0 ¼ 1, Kc ¼ 0:2, T¼0.4.


dx.doi.org/10.1016/j.jsv.2011.07.032


4. Analysis of the system with normal control force

In this section, we consider the system derived in Section 2.2 for a linear normal control force (i.e., Eq. (10)). Weperform a linear stability analysis to obtain the stability boundaries corresponding to the Hopf bifurcation. To determinethe effect of the linear control parameters on the nature of the bifurcation, we perform a nonlinear analysis. The nonlinearanalysis is presented using averaging method (for simplicity of understanding) and the method of multiple scales (foraccurate estimates of the criticality curve).


To perform stability analysis, we first linearize Eq. (10) about the equilibrium (z¼ 0) and get

z00 þð1þKcmðvbÞÞz�h21z0 ¼ KcmðvbÞzT , (24)

where h21 ¼�2xþgaN0 and mðvbÞ ¼ mkþDme�avb . A substitution of KcmðvbÞ by Kc in Eq. (24) reduces it to Eq. (11). Hence,the critical values of Kc and T can be directly derived from Eq. (12) as

Kco ¼ðo2�1Þ2þh2

21o2

2mðvbÞðo2�1Þ,

To ¼2

otan�1 o2�1

h21o

� �þ2ip

� �, 8i¼ 0,1,2, . . . ,1: (25)

The stability boundaries corresponding to the Hopf bifurcation in the plane of the control parameters (Kc vs. T) areplotted in Fig. 5. It is observed from Fig. 5 that the stable region increases with an increase in the value of vb and decreaseswith an increase in the value of N0 which is consistent with the friction force versus relative velocity relationship.

4.2. Nonlinear analysis by averaging

We first perform a nonlinear analysis using the method of averaging to get a qualitative understanding of thedependence of the nature of the bifurcation on the control parameters. More accurate results using the method of multiplescales [35,49] with a better quantitative correspondence with numerics will be presented in Section 4.3. To performaveraging we rewrite Eq. (10) as

z00 þz¼ ef ðzðtÞ,z0ðtÞ,zT Þ, 0oe51, (26)

where

ef ðzðtÞ,z0ðtÞ,zT Þ ¼�2xz0 þgN0ðeaz0�1ÞþmkKcðzT�zÞþgKcðzT�zÞeaz0 : (27)

Following the procedure outlined in Section 3.2 we obtain the amplitude equation as

A0n ¼�ð2xþmkKc sinðTÞÞ

2AnþgN0I1ðaAnÞ�

gKc

2sinðTÞAnðI0ðaAnÞþ I2ðaAnÞÞ, (28)

where I0ðaAnÞ, I1ðaAnÞ and I2ðaAnÞ are modified Bessel’s functions of order 0, 1 and 2, respectively. Expanding I0ðaAnÞ, I1ðaAnÞ

and I2ðaAnÞ in a Taylor series around An ¼ 0 and retaining up to third-order terms we get

A0n ¼ LnaAnþNLnaA3n , (29)

0 1 2 30

0.5

1

1.5

2

2.5

3

Unstableequilibrium

Stable equilibrium

0 1 2 3

0.5

1

1.5

2

2.5

3

Unstableequilibrium

Stable equilibrium

Fig. 5. Stability boundaries for normal control force. Parameters: ms ¼ 0:3, mk ¼ 0:1, a¼2.5, x¼ 0:05, (a) N0 ¼ 1 with various vb , (b) vb ¼ 0:4 for various N0.


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where

Lna ¼12ðh21�KcmðvbÞ sinðTÞÞ, (30a)

NLna ¼ga2

16ðaN0�3Kc sinðTÞÞ: (30b)

It is observed from Eq. (30) that both the coefficients of the linear and the cubic part of Eq. (29) depend on the controlparameters (Kc and T). Hence, there is a possibility that these control forces can meet both our requirements, i.e.,quenching the vibrations as well as changing the nature of the bifurcation from subcritical to supercritical. To analyse theeffect of the normal control force, we first get a relationship between the critical parameters at the Hopf bifurcation point(stability boundary) corresponding to Lna ¼ 0. From (30a) we can write this relationship either as

Kcao ¼h21

mððvbÞnaoÞ sinðTaoÞ(31)

or

ðvbÞnao ¼1

aln

DmðaN0�Kcao sinðTaoÞÞ

mkKcao sinðTaoÞþ2x

� �, (32)

where ðvbÞnao is the belt velocity and Kcao and Tao are the control parameters corresponding to the Hopf point. For asupercritical Hopf bifurcation we require

NLna ¼ga2

16ðaN0�3Kcao sinðTaoÞÞo0: (33)

Inserting the expression for Kcao from Eq. (31) into Eq. (33) and after some straightforward manipulations, we get thecondition for a supercritical Hopf bifurcation as

ðvbÞnaoo1

aln

2aN0DmaN0mkþ6x

� �¼ ðvbÞnacr: (34)

The subscripts have the same meaning as in Section 3.2 except that ‘n’ here denotes the normal control force. Thecriticality curve is obtained by equating ðvbÞnao ¼ ðvbÞnacr to give

3Kcao sinðTaoÞ ¼ aN0: (35)

The above analysis using averaging gives us the feasibility of changing the nature of the bifurcation with a linear normalcontrol force. To get better estimates of the control parameters corresponding to this change, we employ the method ofmultiple scales (MMS) which is presented in the next subsection.

4.3. Nonlinear analysis using the method of multiple scales (MMS)

We briefly outline the procedure for the application of MMS [35,49] to the system with the normal control force.A prerequisite to apply the MMS is the smallness of nonlinear terms and the presence of a small parameter using which anasymptotic series solution can be developed. To this end, we substitute z¼ ey into Eq. (10), where 0oe51. Expanding theexponential term in a series and retaining terms up to third order we obtain

y00 þyþKcmðvbÞy�KcmðvbÞyT�h21y0 þegaKcyy0�egaKcyT y0

�e ga2

2N0y02�e2 ga3

6N0y03�e2 ga2

2KcyT y02þe2 ga2

2Kcyy02 ¼ 0, (36)

where yT ¼ yðt�TÞ.While considering the belt velocity vb as the bifurcation parameter, we perturb vb as vb ¼ ðvbÞnoþe2D. Accordingly, the

terms dependent on vb get modified to g¼ go�e2goaD, mðvbÞ ¼ mððvbÞnoÞ�e2goaD, h21 ¼ h21o�e2goa2N0D, where go, mððvbÞnoÞ

and h21o are the values of g, mðvbÞ and h21 at the critical belt velocity ðvbÞno. The above expressions are substituted inEq. (36) and only terms till Oðe2Þ are retained to result in

y00 þyþKcmððvbÞnoÞy�KcmððvbÞnoÞyT�h21oy0 þeðgoaKcyy0�goaKcyT y0

�12goa2N0y02Þþe2ð�KcgoaDyþKcgoaDyTþgoa2N0Dy0

�16 a3N0goy03�1

2 a2KcyT y02goþ12a2Kcyy02goÞ ¼ 0: (37)

For e¼ 0, Eq. (37) reduces exactly to the linearized Eq. (24) whose solution is known. The solution to Eq. (37) is assumedto be a function of several independent time scales as yðtÞ ¼ Yðt0,t1,t2Þ, where t0ðtÞ ¼ t is the original time scale andt1ðtÞ ¼ et, t2ðtÞ ¼ e2t are slow time scales. Next, we expand Yðt0,t1,t2Þ in a series in e as

yðtÞ � Yðt0,t1,t2Þ ¼ Y0ðt0,t1,t2ÞþeY1ðt0,t1,t2Þþe2Y2ðt0,t1,t2Þþ � � � : (38)


dx.doi.org/10.1016/j.jsv.2011.07.032


The delayed term yðt�TÞ is accordingly written as

yðt�TÞ � Yðt0�T ,t1�eT ,t2�e2TÞ ¼ Y0ðt0�T ,t1�eT,t2�e2TÞþeY1ðt0�T,t1�eT,t2�e2TÞþ � � � : (39)

We further expand Eq. (39) in a Taylor series about e¼ 0. The first-order term in this series is Y0ðt0�T ,t1,t2Þ. This series isfinally substituted into Eq. (37) along with Eq. (38) and various powers of e are collected. The resulting equation at the firstorder, i.e., Oðe0Þ is

q2Y0

qt20

�h21oqY0

qt0þY0þKcmððvbÞnoÞY0 ¼ KcmððvbÞnoÞY0ðt0�TÞ, (40)

where Y0 � Y0ðt0,t1,t2Þ and Y0ðt0�TÞ � Y0ðt0�T ,t1,t2Þ. The steady-state solution to Eq. (40) (from linear analysis) can bewritten as

Y0 ¼ C1 sinðot0ÞþC2 cosðot0Þ, (41)

where C1 ¼ C1ðt1,t2Þ and C2 ¼ C2ðt1,t2Þ. The equation at OðeÞ after substituting Eq. (41) for Y0 is

q2Y1

qt20

�h21oqY1

qt0þY1þKcmððvbÞnoÞY1�KcmððvbÞnoÞY1ðt0�TÞ

þP1 sinðot0ÞþP2 cosðot0ÞþP3 sinð2ot0ÞþP4 cosð2ot0ÞþP5 ¼ 0, (42)

where Y1 � Y1ðt0,t1,t2Þ, Y1ðt0�TÞ � Y1ðt0�T ,t1,t2Þ and the other terms are

P1 ¼ ð�h21oþKcmððvbÞnoÞT cosðoTÞÞqC1

qt1�ð2o�KcmððvbÞnoÞT sinðoTÞÞ

qC2

qt1,

P2 ¼ ð2o�KcmððvbÞnoÞT sinðoTÞÞqC1

qt1þð�h21oþKcmððvbÞnoÞT cosðoTÞÞ

qC2

qt1,

P3 ¼12 aKcgoðC

21�C2

2 Þo�12 aKcgoðC

21�C2

2 Þo cosðoTÞþ12a2N0goo2C1C2�aKcgoo sinðoTÞC1C2,

P4 ¼ aKcgooC1C2ð1�cosðoTÞÞþ12 aKcgoðC

21�C2

2 Þo sinðoTÞ�14a2N0gooðC2

1�C22 Þ,

P5 ¼12 aKcgoðC

21þC2

2 Þo sinðoTÞ�14a2N0goðC

21þC2

2 Þo2:

Setting the coefficients of sinðot0Þ and cosðot0Þ in Eq. (42) to zero to avoid secular terms, we get qC1=qt1 ¼ 0 andqC2=qt1 ¼ 0. The particular solution of Eq. (42) can then be assumed as

Y1 ¼ C3þC4 sinð2ot0ÞþC5 cosð2ot0Þ: (43)

Substituting the expression of Y1 from Eq. (43) into Eq. (42) along with qC1=qt1 ¼ 0 and qC2=qt1 ¼ 0, we obtain thecoefficients C3, C4 and C5 in terms of C1 and C2. The results obtained so far, including the expressions qC1=qt1 and qC2=qt1

are now substituted into the equation at Oðe2Þ. Again, we equate the coefficients of sinðot0Þ and cosðot0Þ to zero to avoidthe secular terms and obtain qC1=qt2 and qC2=qt2. Finally the rate of change of C1 and C2 in the original time t (up to secondorder) is given by

qC1

qt¼ e qC1

qt1þe2 qC1

qt2and

qC2

qt¼ e qC2

qt1þe2 qC2

qt2:

A change to polar coordinates through C1ðtÞ ¼ RnðtÞ cosðfnðtÞÞ and C2ðtÞ ¼ RnðtÞ sinðfnðtÞÞ is introduced next. The solutionin the original variable becomes zðtÞ ¼ eyðtÞ ¼ AnðtÞ sinðtþfnðtÞÞ, where AnðtÞ ¼ eRnðtÞ. The slow-flow equations governingthe evolution of the amplitude and the phase are

dAn

dt¼ ðB1D1ÞAnþB2A3

n, (44a)

dfn

dt ¼ ðB3D1ÞAnþB4A2n, (44b)

where D1 ¼ e2D and

B1 ¼H0þH1oþH2o2þH3o3þH4o4þH5o5þH6o6

D0þD1oþD2o2þD3o3þD4o4þD5o5þD6o6, (45a)

B2 ¼G2o2þG3o3þG4o4þG5o5þG6o6þG7o7þG8o8

D0þD1oþD2o2þD3o3þD4o4þD5o5þD6o6, (45b)

B3 ¼U0þU1oþU2o2þU3o3þU4o4þU5o5

D0þD1oþD2o2þD3o3þD4o4þD5o5þD6o6, (45c)


dx.doi.org/10.1016/j.jsv.2011.07.032


B4 ¼W2o2þW3o3þW4o4þW5o5þW6o6þW7o7

D0þD1oþD2o2þD3o3þD4o4þD5o5þD6o6: (45d)

The expressions for H0, H1, H2, etc. obtained using the symbolic algebra package MAPLE are long and are provided inAppendix A. The amplitude of the limit cycle from Eq. (44a) is

An ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�B1D1

B2

s: (46)

The criticality curve can be obtained by setting B2 ¼ 0. The results obtained using the approximate nonlinear analyses forthe normal control force are compared against numerical solutions in the next subsection.

4.4. Results and discussions for normal control force

Fig. 6(a) shows the variation of ðvbÞnao and ðvbÞnacr with the control gain Kcao for Tao ¼ 1. In this section, we will use thefollowing standard set of parameters: ms ¼ 0:3, mk ¼ 0:1, a¼ 2:5, x¼ 0:05 and N0 ¼ 1. It is observed from Fig. 6(a) that withan increase in the control gain Kcao the system crosses the criticality boundary where the nature of the Hopf bifurcationchanges from subcritical to supercritical. With a further increase in the control gain we have ðvbÞnao ¼ 0 which implies thatthe equilibrium is always stable for any vb and vibrations are completely quenched. This region has been termed as the‘completely stable region’. The stability surface along with the criticality curve both obtained using the averaging methodin the space of control parameters Kc and T and the belt velocity vb is shown in Fig. 6(b). As mentioned earlier, both thestability surface and the criticality curve obtained using the averaging method are only approximate and might differsignificantly from the actual values (see [35]). Better estimates for the criticality curve obtained using the method ofmultiple scales (MMS) along with the exact stability surface obtained from the linear stability analysis of Section 4.1 areshown in Fig. 6(c). It can be seen from Figs. 6(b) and (c) that the stability surface and the criticality curve match for smallervalues of Kc and T. However, they differ significantly for larger values of Kc which is the more interesting regime since itresults in smaller ðvbÞno.

0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

A B C

01

2

0

1

2

0

0.2

0.4

0.6

0.8

1

B

A

C

Criticalitycurve

01

2

0

1

2

0

0.2

0.4

0.6

0.8

1

A

B

C

Criticalitycurve

Fig. 6. (a) Variation of ðvbÞnacr and ðvbÞnao with Kcao. (b) Criticality curve on the stability surface obtained using the averaging method. (c) Criticality curve

obtained using MMS. Inside the figure: region A—subcritical region, region B—supercritical region, region C—completely stable region. Parameters:

ms ¼ 0:3, mk ¼ 0:1, a¼2.5, x¼ 0:05, N0 ¼ 1, (a) Tao ¼ 1:0.


dx.doi.org/10.1016/j.jsv.2011.07.032

0.38 0.4 0.420

0.1

0.2

0.3

0.4

Unstablelimit cycle

Stick−sliplimit cycle

0.15 0.155 0.16 0.165 0.17 0.1750

0.05

0.1

0.15

Stick−sliplimit cycle

Pure slippinglimit cycle

Fig. 7. Bifurcation diagram with vb as the bifurcation parameter. Parameters: ms ¼ 0:3, mk ¼ 0:1, a¼2.5, x¼ 0:05, N0 ¼ 1, Tao ¼ To ¼ 1:0, (a) Kcao ¼ Kco ¼ 0:6

(corresponding to a subcritical bifurcation), (b) Kcao ¼ Kco ¼ 1:2 (corresponding to a supercritical bifurcation).

12

34

5

01

23

0

0.2

0.4

0.6

0.8

B

C

A

Criticalitycurve

Fig. 8. Criticality curve on the stability surface obtained using MMS. Inside the figure: region A—subcritical region, region B—supercritical region, region

C—completely stable region. Parameters: ms ¼ 0:3, mk ¼ 0:1, x¼ 0:05, N0 ¼ 1, To ¼ 0:4.


To validate the results obtained using the approximate analytical methods of averaging and MMS, bifurcation diagramswith vb as the bifurcation parameter are plotted in Fig. 7(a) and (b) for values of the control gains in the subcritical and thesupercritical zone of Fig. 6(a), respectively. The dashed lines representing the amplitude ðAnÞmms are obtained using MMS,the dashed-dotted lines representing the amplitude ðAnÞavg are obtained using the averaging method and the solid lines areobtained using the software package DDE-BIFTOOL [47]. Co-existence of the unstable limit cycle in Fig. 7(a) with the stableequilibrium (vb4ðvbÞnao) signifies a subcritical Hopf bifurcation. Similarly, a stable limit cycle coexisting with an unstableequilibrium (vbo ðvbÞnao) in Fig. 7(b) implies that the bifurcation is supercritical in nature. It is observed from Fig. 7(a) thatwith an increase in the value of vb past ðvbÞno the amplitude An of the unstable limit cycle grows in magnitude until itmerges with the stable stick-slip limit cycle in a cyclic fold bifurcation leaving the static equilibrium globally stableafterwards. It is clear from the bifurcation diagrams shown in Fig. 7(a) and (b) that the amplitudes obtained from theaveraging method do not match well with the amplitudes obtained from the numerical simulation. Also, the bifurcationpoint (ðvbÞnao) deviates from the exact one (ðvbÞno) quite significantly. These observations substantiate our earlier claimabout the approximate nature of the results obtained using the averaging method.

The controllability of friction-induced vibrations depends largely on the friction parameters ms, mk and a. We do notpresent here the effect of each one of them on the stability properties. However, we analyse the effect of the change in theslope parameter a on the system dynamics as a representative study which is shown in Fig. 8. The critical belt velocitycorresponding to both the stability surface as well as the criticality curve increases with an increase in a as observed fromFig. 8 which is consistent with the fact that the instability increases with an increase in the slope parameter.

5. Linear normal control force with a dynamic (LuGre) friction model

In this section, we consider the LuGre friction model [29] which is one of the most widely used form of a dynamicfriction model and it includes more complicated physical phenomenon like hysteresis associated with friction. We performthis study to ascertain the robustness of the linear normal time-delayed control force in quenching vibration as well as


dx.doi.org/10.1016/j.jsv.2011.07.032


controlling the nature of the bifurcation with changes in the friction model. We present the linear stability analysis indetail and obtain the stability boundary corresponding to the Hopf bifurcation. The nature of the bifurcation at the criticalvalues obtained from the stability analysis is studied through bifurcation diagrams generated numerically using thesoftware package DDE-BIFTOOL.

The nondimensional equation of motion for the system with a linear normal control force is given by Eq. (8) which isrewritten here as

x00 þ2xx0 þx¼ ðN0þKcðxðt�TÞ�xðtÞÞÞf ðvrÞ, (47)

where f ðvrÞ is the friction function (friction force per unit of normal load). The LuGre friction model incorporates themicroscopic degrees of freedom wherein the asperities in the contact surface are considered as elastic spring-like bristleswith damping. These bristles deflect when an external force is applied and the total friction force is a summation of theaverage force associated with the deflection of the bristles and a viscous term proportional to the relative velocity betweenthe surfaces in contact. The friction function f ðvrÞ for the LuGre friction model is accordingly given by

f ðvrÞ ¼ s0zþs1dz

dt þs2vr , (48)

where z is the average bristle deflection, s0 is the bristle stiffness, s1 is the bristle damping and s2 is the viscouscomponent of the frictional force. The equation governing the evolution of the average bristle deflection (see [29]) is

dz

dt ¼ vr 1�s0z

gðvrÞsgnðvrÞ

� �: (49)

The function gðvrÞ in the above equation determines the stribeck effect (or the drooping characteristic) and it isrepresented as

gðvrÞ ¼ mkþDme�ðvr=vsÞ2

, (50)

where vs is the stribeck velocity which is the relative velocity at the transition from microslip (stick) to macroslip.Eqs. (47)–(50) describe the complete equations of motion for our system with the normal control force and the LuGrefriction function. These equations can be written compactly in the state-space form as

x0 ¼ u, (51a)

u0 ¼ �2xu�xþðN0þKcððxÞT�xÞÞ s0zþs1dz

dt þs2vr

� �, (51b)

z0¼ vr 1�

s0z

gðvrÞsgnðvrÞ

� �, (51c)

where vr ¼ vb�x0.


We first find the fixed points of the system given by Eq. (51) by setting ðx0 ¼ u0 ¼ z0¼ 0Þ as xs ¼N0ðgðvbÞþs2vbÞ, us ¼ 0

and zs ¼�gðvbÞ=s0. Linearizing Eq. (51) about these fixed points, we get

~x 0 ¼ L ~xþRð ~xÞT , (52)

where ~x ¼ ðx�xs,u�us,z�zsÞT and the matrices L and R are

L¼

0 1 0

�1�KcðgðvbÞþs2vbÞ �2x�N0s1vbg0ðvbÞ

gðvbÞþs2

� �s0N0 1�

s1vb

gðvbÞ

� �

0 �vbg0ðvbÞ

gðvbÞ

s0vb

gðvbÞ

2666664

3777775 (53)

and

R¼

0 0 0

KcðgðvbÞþs2vbÞ 0 0

0 0 0

264

375: (54)

Substituting ~x ¼ ~cest in Eq. (52), we obtain the characteristic equation

detðLþRe�st�sIÞ ¼ 0, (55)

where I is the identity matrix. Substitution of L and R from Eqs. (53) and (54) in Eq. (55) results in

s3þa2s2þa1sþa0þb0Kcðsþa0Þð1�e�sT Þ ¼ 0, (56)


dx.doi.org/10.1016/j.jsv.2011.07.032


where

a2 ¼ 2xþa0þN0 s2þa0s1

s0g0ðvbÞ

� �, (57a)

a1 ¼ 1þ2xa0þa0N0ðs2þg0ðvbÞÞ, (57b)

a0 ¼s0vb

gðvbÞ, (57c)

b0 ¼ gðvbÞþs2vb: (57d)

At the stability boundary corresponding to the Hopf bifurcation, we have s¼ jo. Substituting this in Eq. (56) and separatingthe real and imaginary parts, we get

ob0Kc sinðoTÞþa0b0Kc cosðoTÞ ¼ a0þa0b0Kc�a2o2, (58a)

a0b0Kc sinðoTÞ�ob0Kc cosðoTÞ ¼o3�a1o: (58b)

The critical values of Kc and T are obtained by solving Eqs. (58a) and (58b) as

Kco ¼o6þða2

2�2a1Þo4þða21�2a0a2Þo2þa2

0

2b0ðo4�ða1�a0a2Þo2�a20Þ

, (59a)

To ¼2

o tan�1 o4þða0a2�a1Þo2�a20

ða0�a2Þo3�ða0a1þa2Þo

� �þ2ip

� �8i¼ 0,1,2, . . . ,1: (59b)

As before, the stable and the unstable regions in the control parameter plane are identified by checking the transversalitycondition of the critical s with respect to the control gain as the bifurcation parameter at the Hopf point. The stabilityboundary on the control parameter plane is plotted in Fig. 9 using Eqs. (59). The stable region increases with an increase inthe belt velocity vb and a decrease in the pre-normal load N0 as noted in Section 4 for the exponential friction model.

Having established the effectiveness of the linear normal control force in quenching vibrations for the LuGre frictionmodel, we need to study its effect on the nature of the bifurcation. The nature of the bifurcation cannot be ascertainedfrom a linear analysis and hence, a nonlinear analysis is required. However, we point out that details of the analyticalcalculations involved in the nonlinear analysis for the LuGre friction model are tedious and beyond the scope of the currentpaper. We instead present the bifurcation diagrams generated numerically using the software package DDE-BIFTOOL [47]for two typical set of control parameters, one belonging to the subcritical regime (Kc ¼ 1 with T¼1) and the other to thesupercritical regime (Kc ¼ 7 with T¼1) in Fig. 10. In the bifurcation diagrams, the variation of the amplitudes of the limitcycles are plotted against the belt velocity vb. It is observed from Fig. 10 that for lower values of the control gain Kc thebifurcation is subcritical in nature while when the control gain is increased above a certain critical value, the nature ofbifurcation becomes supercritical. The change in the nature of the bifurcation from subcritical to supercritical with anincrease in the control gain Kc is qualitatively similar to the scenario obtained previously for the exponential frictionmodel. Thus, our control strategy of using a linear normal control force can achieve the dual objective of quenching thevibrations as well as changing the nature of the associated bifurcations irrespective of the complexity of the function usedin modeling the friction force.

0 0.5 1 1.5 2 2.50

5

10

15 Stable

Equilibrium

UnstableEquilibrium

0 0.5 1 1.5 2 2.50

5

10

15

UnstableEquilibrium Stable

Equilibrium

Fig. 9. Stability boundaries for normal control force with LuGre friction model. Parameters: ms ¼ 0:3, mk ¼ 0:1, vs ¼ 0:1, x¼ 0:05, s0 ¼ 100, s1 ¼ 10,

s2 ¼ 0:05, (a) N0 ¼ 1, (b) vb ¼ 0:18.


dx.doi.org/10.1016/j.jsv.2011.07.032

0.1 0.15 0.2 0.25 0.30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Stablelimit cycle

SubcriticalHopf bifurcation

Cyclic fold bifurcation

Unstablelimit cycle

0.1 0.11 0.12 0.13 0.140

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Stablelimit cycle

SupercriticalHopf bifurcation

Fig. 10. Bifurcation diagrams with vb as the bifurcation parameter for the LuGre friction model with normal control force. Parameters: ms ¼ 0:3, mk ¼ 0:1,

vs ¼ 0:1, x¼ 0:05, N0 ¼ 1, s0 ¼ 100, s1 ¼ 10, s2 ¼ 0:05, To ¼ 1, (a) Kco ¼ 1, (b) Kco ¼ 7.


6. Conclusion

In this paper, we have used time-delayed position feedback to control the friction-induced vibrations in a single-degree-of-freedom spring–mass–damper system on a moving belt. We consider the instability caused by the droopingcharacteristics of the friction force which has been modeled using the exponential friction model and the LuGre frictionmodel. The bifurcation for the uncontrolled system is subcritical in nature [4,34–36] and the system is prone to vibrationseven in the linearly stable regime close to the stability boundary. We have explored the possibility of changing the natureof the bifurcation from subcritical to supercritical such that the equilibrium is globally stable in the linearly stable regime.We have considered control forces applied both in the direction parallel to the friction force as well as normal to it. Linearstability analysis has been performed to obtain the stability boundaries corresponding to the Hopf bifurcation delineatingstable and unstable regimes. Nonlinear analysis using both the averaging method and the method of multiple scales hasbeen performed to obtain the criticality curve separating the regions on the stability boundaries with subcritical andsupercritical bifurcations. The averaging method is utilized to ascertain the feasibility of the change in the nature of thebifurcation while the method of multiple scales is used to get a better estimate of the exact criticality curve. Theseanalytical estimates for the criticality curves are verified against bifurcation diagrams generated numerically using thesoftware package DDE-BIFTOOL. We have found that a nonlinear feedback controller is required to achieve our objective ofcontrolling the nature of the bifurcation when applied in the tangential direction. However, a linear time-delayeddisplacement feedback applied in the normal direction can serve both the requirements of quenching the vibrations aswell as changing the nature of the bifurcation. Hence, the direction of the control force plays a critical role in quenchingfriction-induced vibrations. We also observe that our qualitative conclusions about the nature of the bifurcation under ourcontrol strategy are robust with respect to changes in the friction model.

Acknowledgments

We thank Department of Science and Technology (DST), India (Project Number: IITK/DST-ME-20090007) for financialsupport.

Appendix A. Expressions involved in the slow-flow equations using MMS

The expressions involved in the slow-flow (44) are

D0 ¼ 16ðK4com

4oT2

o�h21oK3com

3oTocT�h21oK2

com2oTocTþh21oK2

com2oToc3Tþ0:5h2

21o

�h221oKcomoc2T�h21oKcomoTocTþh21oK3

com3oToc3T�K4

com4oT2

o c2Tþh221oKcomo

�K3com

3oT2

o c2T�h221oK2

com2oc2Tþ0:5K2

com2oT2

o þK3com

3oT2

o þh221oK2

com2oÞ,

D1 ¼ 32ð�3K3com

3oTosTþK3

com3oTos3TþK2

com2oTos3T�h3

21oKcomos2T�KcomoTosT

þh221oK2

com2oTos3T�K3

com3oT2

o h21os2Tþh221oK2

com2oTosT�3K2

com2oTosTÞ,

D2 ¼ 32ð1þh421oþ2K2

com2o�4h21oK2

com2oToc3Tþ2h2

2oKcomoc2Tþ4h21oKcomoTocT

�2K3com

3oT2

o�2K2com

2oT2

o�2KcomoToh321ocT�2h2

21o�2Kc1omoc2T�2K2com

2oc2T

þK2com

2oT2

o h221o�2h2

21oKcomoþ2K3com

3oT2

o c2Tþ4h21oK2com

2oTocTþ2KcomoÞ,


dx.doi.org/10.1016/j.jsv.2011.07.032


D3 ¼ 128Kcomoð�h221oTosT�h21os2Tþ3KcomoTosT�KcomoTos3Tþ2TosTÞ,

D4 ¼ 256ð�1þh221o�h21oKcomoTocTþ0:5K2

com2oT2

o�KcomoþKcomoc2TÞ,

D5 ¼�512KcomoTosT ,

D6 ¼ 512,

H0 ¼ 8agoð�K3com

2oh21oc3T�2K3

com2oh21o�K3

com2oToc3TþK2

comoTocT�K2comoTo

þ2K2comoh21oc2TþK3

com2oTocTþK2

comocTh21o�2K2comoh21oþK4

com3oTocT

þK3com

2ocTh21oþKcocTh21o�K4

com3oToc3T�K2

comoh21oc3T�2K3com

2oTo

þ2K3com

2oToc2Tþ2K3

com2oh21oc2T�2K4

com3oToþ2K4

com3oToc2T�Kcoh21oÞ,

H1 ¼ 16goaKcoð2K2com

2oToh21os2Tþ3KcomosTþ1:5aN0K2

com3oTosT�Kcomos3T

�K2coh21om2

oTosT�Kcomoh221os3Tþ2Kcomoh2

21os2T�0:5aN0K2com

3oTos3T

þ1:5aN0Kcom2oTosTþ3K2

com2osT�0:5aN0Kcom2

oTos3T�KcomosTh221o

�K2com

2os3T�K2

com2oToh21os3Tþ0:5aN0moTosTþsTÞ,

H2 ¼ 32goaKcoð�2KcomoTocT�aN0moþ2K2com

2oTo�2cTh21o�KcomoToh2

21oþ2h21o

þh321ocTþ0:5aN0Kcom2

oh21oToc3T�2Kcomoh21oc2T�2KcomocTh21o

þaN0moc2T�aN0Kcom2oþ2Kcomoh21oc3TþaN0Kcom2

oc2TþK2com

2oToc3T�h3

21o

þ2Kcomoh21o�0:5aN0Kcom2ocTh21oTo�2K2

com2oToc2T�K2

com2oTocT

þKcomoTocTh221oþ2KcomoToÞ�16goa2N0,

H3 ¼ 32goaKcoð�4sTþ2Kcomos3TþaN0Kcom2oTos3Tþ2sTh2

21o�6KcomosT

�2aN0moTosTþaN0moTosTh221oþ2aN0moh21os2T�3aN0Kcom2

oTosTÞ,

H4 ¼ 128goaKcoðcTh21oþKcomoTocTþaN0�h21o�KcomoToþaN0mo�aN0moc2TÞ�64goa2N0h221o,

H5 ¼ 128goaKcoðaN0moTosTþ2sTÞ,

H6 ¼�256goa2N0,

G2 ¼ goa2K2coð�moToc2T�moTocT�8goKcomoTocTþKcom2

oToc3Tþ2goh21oc2T

�2goh21oc3T�6gocTh21o�6Kcom2oToc2T�h21omocTþKcom2

oToc4Tþ2moTo

þh21omoc3T�2h21omoc2T�Kcom2oTocTþ6goKcomoToc2Tþ6goK2

com2oToc3T

þ6goh21oþ4goK2com

2oToc2T�4goK2

com2oToc4T�2Kcom2

oh21oc2Tþ2Kcom2oh21o

�6goK2com

2oTocTþK2

com3oToc3TþKcom2

oh21oc3T�KcocTh21om2o�K2

com3oTocT

þ2goKcomoh21oc3T�4goKcomoh21oc2T�2goKcomocTh21o�6K2com

3oToc2T

þK2com

3oToc4Tþ5K2

com3oToþ2h21omoþ2goKcomoToþ5Kcom2

oToþ4goKcomoh21oÞ

þgoa2Kcoh21oð1�cTÞ,

G3 ¼ goa2KcoðaN0Kcom2oTos3T�goaN0h21os2Tþ2goaN0KcomoTos2T

�6goaN0sTh21o�aN0moTosTþ4goKcoh221os2Tþ4goKcos2Tþ4goa2KcosT

�3aN0Kcom2oTosTþ6Kcomos3Tþ4goKcosTh2

21o�4goKcoh221os3T�4goKcos3T

�4Kcoh221omos2T�4goK2

coh21omoTos2T�18K2cosTm2

oþ2goaN0Kcomoh21os2T

þ2KcomosTh221oþ8goK2

cosTh21omoToþ6K2com

2os3Tþ6goaN0K2

com2oTos2T�6sT

�2goaN0K2com

2oTos4Tþ2Kcomoh2

21os3Tþ2goaN0Kcomoh21os3Tþ4goK2comosT

þ16goK2comos2Tþ2h21oK2

com2oTos3T�10goaN0KcomosTh21o�3aN0K2

com3oTosT

�12goK2comos3T�8h21oK2

com2oTos2Tþ2h21oK2

com2oTos4Tþ5goaN0K2

com2oTosT

�3goaN0K2com

2oTos3TþaN0K2

com3oTos3T�18KcosTmo

þ2sTKcomoToðh21oKcomoþ2goaN0ÞÞ,

G4 ¼ goa2ð4K2comoToð2�h2

21oþgoaN0h21oÞðc2TþcT�2Þ�4Kcoh321oðcT�1Þ

þ2goaN0Kcoðc2Tþ2cT�3Þð1þh221oÞþ8goK3

comoToð4cT�1Þþ2aN0

þ2K2com

2oToðgoa2N2

0�aN0h21o�2KcoÞðc3T�cTÞþ16goK2coh21oðc3T�c2TÞ


dx.doi.org/10.1016/j.jsv.2011.07.032


�2goa2N20Kcomoh21oc2Tþ48goK2

cocTh21oþ24K3com

2oToc2Tþ16K2

coh21omocT

�4goaN0K2comocTþ4goaN0K2

comoc3T�4K3com

2oToc4T�16K2

coh21omoc3T

þ8K2coh21omoc2T�4aN0K2

com2oc2T�24goK3

comoToc2Tþ8goaN0K2comoc2T

�48goK2coh21o�8Kcoh21oþ2goa2N2

0h21oþ2goa2N20Kcomoh21o�8goaN0K2

como

þ8KcocTh21o�8K2coh21omoþ4aN0K2

com2o�20K3

com2oToþ4aN0Kcomo

�2goa2N20KcomoTocT�4aN0Kcomoc2TÞ,

G5 ¼ 4goa2KcoðKcomoð13sT�6s3TÞþgoa2N20h21omoTosT�2goaN0KcomoTos2T

�4goKcosTþ2aN0moTosT�4goKcos2Tþ12goaN0sTh21oþ2goaN0h21os2T

�amoTosTN0h221o�aN0Kcom2

oTos3Tþgoa2N20mos2T�6a2KcosTh2

21oþ12sT

þ4goKcos3T�2aN0moh21os2Tþ12a3N0Kcom2oTosT�4goaN0KcomoTosTÞ,

G6 ¼ 8goa2Kcoð2h21o�2goaN0cT�2cTh21oþ24goaN0�goaN0c2Tþ2aN0moc2T

�2KcomoTocTþgoa2N20moTocTþ4KcomoTo�2KcomoToc2T�2aN0moÞ

þ8goa3ð�2goaN20h21o�2N0þN0h2

21oÞ,

G7 ¼�16goa2KcosTðaN0moToþ6Þ,

G8 ¼ 32goa3N0,

U0 ¼ 8goaKcoðKcomoTosTþK2com

2oh21os3T�sTh21oþKcomoh21os3T�K3

com3oTos3T

þ3K3com

3oTosTþ3K2

com2oTosT�3KcomosTh21o�K2

com2oTos3T�3K2

com2osTh21oÞ,

U1 ¼ 8goaKcoð2aN0moh21oþ4Kcomoc2TþaN0K2com

3oToc3T�2aN0Kcom2

oh21oc2T

�2Kcomoh221oc3Tþ2K2

com2ocTþ2K2

com2oToh21oc3Tþ2aN0Kcom2

oh21oþ2cT

þ2KcomocTh221o�2aN0moh21oc2T�16Kcomoc3T�4K2

com2o�2Kco�4Kcomo

�2K2com

2oc3T�aN0moTocT�aN0K2

com3oTocTþ4K2

com2oc2TþaN0Kcom2

oToc3T

�aN0Kcom2oTocT�2K2

com2oTocTh21oþ2KcomocTÞþ8goa2N0h21o,

U2 ¼ 16goaKcoð�2sTh321oþaN0h21oKcom2

oTosTþ4KcomosTh21o�2aN0moh221os2T

�4KcomoTosTþ4KcosTh21oþaN0h21oKcom2oTos3T�6K2

com2oTosT

þ2K2com

2oTos3Tþ4Kcomoh21os2Tþ2KcomoTosTh2

21o�4Kcomoh21os3TÞ,

U3 ¼ 32goaKcoð2aN0moh21oc2T�2h221oþaN0Kcom2

oTocTþ2Kcomoc3T�4Kcomoc2T

�aN0Kcom2oToc3Tþ4þ128Kcomo�2KcomocTþ2h2

21ocT�aN0h221omoTocT

�4cTþ2aN0moTocT�2aN0moh21oÞþ32goa2N0h21oðh221o�2Þ,

U4 ¼�128goaKcosTh21oþ128goaK2comoTosT ,

U5 ¼ 128goða2N0h21oþ256goaKcocT�256goaKco�128goa2N0KcomoTocTÞ,

W2 ¼ goa2Kcoð�KcomoToðsTþs2TÞþ3sTh21oþK2com

2oToðs3Tþs4T�3sT�2s2TÞ

þ2goK2comoToðs2T�2sTÞ�3Kcomoh21oðs3T�3sTÞ�3K2

coh21om2oðs3T�3sTÞ

�2goKcoh21oðsTþs2T�s3TÞþ2goK2comoh21oð3s3T�4s2T�sTÞ

þK3com

3oToðs4Tþs3T�2s2T�3sTÞ�2goK3

com2oToð2s4T�3s3TþsTÞÞ,

W3 ¼�goa3N0h21oþgoa2Kcoð�4goKcoðc3T�c2Tþ3cTÞ�6Kcomoh221oðcT�c3TÞ

þ2goaN0Kcoh21omoð2þcT�c3T�2c2TÞþ2aN0h21oKcom2oðc2T�1Þ

þgoaN0h21oð3�2cT�c2TÞþaN0moTocTþ2K2com

2oToh21oð1þcT�c3T�c4TÞ

þ2goaN0K2com

2oToð2þc4Tþ1:5c3T�3c2T�1:5cTÞþ2aN0h21omoðc2T�1Þ

þ4goK2coh21omoToð1�4cTþ3c2TÞþaN0Kcom2

oToðKcomoþ1ÞðcT�c3TÞ

þ2Kcomoð1þ2goKcoþKcomoÞð2�cT�2c2Tþc3TÞþ2ð1�cTÞþ12goKco

�2goaN0KcomoToðcTþc2T�2Þþ4goKcoh221oð3�3cTþc2T�c3TÞÞ,


dx.doi.org/10.1016/j.jsv.2011.07.032


W4 ¼ goa2Kcoð�2goaN0ð6sTþs2TÞþð8KcomoTo�4KcomoToh221oÞðsTþs2TÞ

�24sTh21o�8Kcomoh21oð4sTþs2T�2s3TÞþ16goKcoh21oðsTþs2T�s3TÞ

þ4K2com

2oToð3sTþ2s2T�s3T�s4TÞþ4goaN0Kcomoðs2Tþs3T�5sTÞ

þ8goK2comoToð2sT�s2TÞþ4aN0h2

21omos2Tþ4goaN0Kcoh21omoToð2sTþs2TÞ

�2aN0Kcom2oh21oToðsTþs3TÞþ12sTh3

21o�2goaN0h221oð6sTþs2TÞ

�2goa2N20moh21os2T�2goa2N2

0moTosT�2goa2N20Kcom2

oToðsT�s3TÞÞ,

W5 ¼ 4goa2Kcoðgoa2N20mo�4Kcomoþ4cTþ4goKcoðc3T�c2TÞ�4Kcoþ2KcocTmo

þ12goKcocT�6goaN0h21oþ2aN0h21omo�2aN0moTocTþ4goaN0h21ocT

þ2goaN0KcomoTocTþaN0Kcom2oToc3T�2Kcomoc3T�aN0Kcom2

oTocT

�2aN0h21omoc2Tþ2goaN0KcomoToc2Tþ2goaN0h21oc2T�12goKcoþ2h221oKco

þ4Kcomoc2T�2cTh221o�goa2N2

0moc2T�goa2N20h21omoTocTþaN0h2

21omoTocT

�4goaN0KcomoToÞþ4goa3N0ð2h21oþgoaN0ð1þh221oÞ�h3

21oÞ,

W6 ¼ 8goa2Kcoðgoa2N20moTosT�2KcomoToðsTþs2TÞþgoaN0ðs2Tþ6sTÞþ6sTh21oÞ,

W7 ¼ 16goa2ð2Kcoð1�cTÞ�aN0ðh21oþgoaN0�KcomoTocTÞÞ,

where the following notations have been used to write the expressions in a compact form:

mo ¼ mðvboÞ, sT ¼ sinðoToÞ, cT ¼ cosðoToÞ, s2T ¼ sinð2oToÞ, c2T ¼ cosð2oToÞ,

s3T ¼ sinð3oToÞ, c3T ¼ cosð3oToÞ, s4T ¼ sinð4oToÞ, c4T ¼ cosð4oToÞ:

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dx.doi.org/10.1016/j.jsv.2011.07.032

Delayed feedback for controlling the nature of bifurcations in friction-induced vibrations

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