reducingtransmittedvibrationusingdelayed...

Hindawi Publishing CorporationAdvances in Acoustics and VibrationVolume 2011, Article ID 546280, 5 pagesdoi:10.1155/2011/546280

Research Article

Reducing Transmitted Vibration Using DelayedHysteretic Suspension

Lahcen Mokni and Mohamed Belhaq

Laboratory of Mechanics, University Hassan II, Casablanca, Morocco

Correspondence should be addressed to Mohamed Belhaq, [email protected]

Received 19 May 2011; Accepted 19 September 2011

Academic Editor: Marc Asselineau

Copyright © 2011 L. Mokni and M. Belhaq. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Previous numerical and experimental works show that time delay technique is efficient to reduce transmissibility of vibrationin a single pneumatic chamber by controlling the pressure in the chamber. The present work develops an analytical study todemonstrate the effectiveness of such a technique in reducing transmitted vibrations. A quarter-car model is considered anddelayed hysteretic suspension is introduced in the system. Analytical predictions based on perturbation analysis show that a delayedhysteretic suspension enhances vibration isolation comparing to the case where the nonlinear damping is delay-independent.

1. Introduction

Various strategies for reducing transmitted vibrations havebeen proposed; see, for instance, [1–5] and referencestherein. A standard technique using linear viscous dampingin the vibration isolation device reduces the transmissibilitynear the resonance but increases it elsewhere. To enhancevibration isolation in the whole frequency range, cubicnonlinear viscous damping has been successfully introduced[3]. It was shown, in the case of a single degree of freedom(sdof) spring damper system, that cubic nonlinear dampingcan effectively produce an ideal vibration isolation in thewhole frequency range [4]. Recently, a strategy based onadding a nonlinear parametric time-dependent viscousdamping to the basic cubic nonlinear damper has beenproposed [5]. This method significantly enhances vibrationisolation comparing to the case where the nonlineardamping is time-independent [3, 4]. Specifically, it wasreported that increasing the amplitude of the parametricdamping enhances substantially the vibration isolation overthe whole frequency range.

On the other hand, time delay control (TDC) [6, 7] wasapplied to a pneumatic isolator to enhance the isolation

performance by controlling the pressure in chamber in thelow frequency range [8]. Effectiveness of this active controltechnique in enhancement of transmissibility performancewas demonstrated using simulation as well as experimentstesting in the case of a single pneumatic chamber.

In the present paper, we use a delayed hysteretic suspen-sion in the system and we examine its influence on vibrationisolation. In this TDC technique, the values of all the statevariables and their first derivatives have to be provided bysome means.

To achieve our analysis, we implement the multiple scalesmethod [9] on the equation of motion to derive the corre-sponding modulation equations and we examine the steadystate solutions of this modulation equation to obtain indica-tions on transmissibility (TR) versus the system parameters.

2. Equation of Motion

A representative model of a suspension system with non-linear stiffness, nonlinear viscous damping, and delayedhysteretic suspension under external excitation is proposed.

2 Advances in Acoustics and Vibration

y(t) = Y cos(Ωt)

x(t)

m

(k1, k2)(c1, c2)

(λ1, λ2)

Figure 1: Schematic diagram of pneumatic vibration isolator under ground excitation.

It consists of a sdof model, as shown in Figure 1, describedby

X + ω2X + B1X3 +(B2X + B3X

3)

= −g + YΩ2 cosΩt + λ1X(t − τ)

+ λ2(X(t − τ)

)3,

(1)

where ω2 = (k1/m), B1 = (k2/m), B2 = (c1/m), andB3 = (c2/m). In the case of a single pneumatic chamber, asshown in Figure 1, m is the body mass, k1 and k2 are thelinear and the nonlinear stiffness of vehicle suspension, c1

and c2 are the linear and the nonlinear equivalent viscousdamping, g is the acceleration gravity, and X is the relativevertical displacement of the mass. The parameters Y andΩ denote, respectively, the amplitude and the frequency ofthe external excitation, while λ1 and λ2 denote the gains ofthe linear- and the nonlinear-delayed viscous damping, andτ is the time delay. This delayed viscous damping can bepractically implemented using delayed nonlinear dampersbased on magnetorheological fluid.

It is worthy to notice that the particular case of linearstiffness (B1 = 0) and undelayed state feedback (λ1 = λ2 = 0)was studied in [3], while the case of B1 /= 0, λ1 = λ2 =0, and time-dependent (parametric) damping was treatedin [5]. It was demonstrated that adding nonlinear para-metric damping to the basic nonlinear damping enhancessignificantly the vibration isolation [5]. The purpose of thepresent work is to study the effect of delayed nonlineardamping on vibration isolation of system (1). Note also thata similar delayed nonlinear system to (1) was investigatednear primary resonances [10]. Attention was focused onperforming an approach to analyze the dynamic of thesystem with arbitrarily large gains. Note also that a delayedfeedback was used to quench undesirable vibrations in a vander Pol type system [11].

3. Frequency Response and Transmissibility

To obtain the frequency-response equation and TR, weperform a perturbation method. Introducing a bookkeepingparameter ε and scaling Y = εY , B1 = εB1, B2 = εB2,B3 =εB3, λ1 = ελ1, and λ2 = ελ2, (1) reads

z + ω2z = −g + ε[YΩ2 cosΩt − B1z

3 − B2z − B3z3

+λ1z(t − τ) + λ2(z(t − τ))3].

(2)

Using the multiple scales technique [9], a two-scale expan-sion of the solution is sought in the form

X(t) = z0(T0,T1) + εz1(T0,T1) + O(ε2), (3)

where Ti = εit. In terms of the variables Ti, the timederivatives become d/dt = D0 + εD1 + O(ε2) and d2/dt2 =D2

0 +2εD0D1 +O(ε2), where Dji = (∂j/∂jTi). Substituting (3)

into (2), we obtain

(D2

0 + 2εD0D1)(z0 + εz1) + ω2(z0 + εz1)

= −g + ε[YΩ2 cos(Ωt)− B2(D0 + εD1)(z0 + εz1)

− B1(z0 + εz1)3 − B3((D0 + εD1)(z0 + εz1))3

+ λ1(D0 + εD1)(z0(t − τ) + εz1(t − τ))

+λ2((D0 + εD1)(z0(t − τ) + εz1(t − τ)))3]

,

(4)

Advances in Acoustics and Vibration 3

and equating coefficients of the same power of ε, we obtainat different orders

D20z0 + ω2z0 = −g,

D20z1 + ω2z1 + 2D1D0z0

= YΩ2 cos(Ωt)

− B1z30 − B2D0z0 − B3(D0z0)3

+ λ1D0z0(t − τ) + λ2(D0z0(t − τ))3.

(5)

In the case of the principal resonance, that is, Ω = ω + εσ ,where σ is a detuning parameter, standard calculations yieldthe first-order solution:

z(t) = − g

ω2+ a cos

(Ωt − γ

)+ O(ε), (6)

where the amplitude a and the phase γ are given by themodulation equations:

a = YΩ2

2ωsin(γ)− s1a− s2a

3,

aγ = YΩ2

2ωcos(γ)− s3a− s4a

3.

(7)

Here s1 = (B2/2) − (λ1 cos(ωτ)/2), s2 = (3B3ω2/8) −(3λ2ω2 cos(ωτ)/8), s3 = (3B1/2)(g2/ω5)−σ−(λ1 sin(ωτ)/2),

and s4 = (3B1/8ω) − (3λ2ω2 sin(ωτ)/8). Periodic solutionsof (2) corresponding to stationary regimes (a = γ = 0)of the modulation equations (7) are given by the algebraicequation:

(s2

2 + s24

)a6 + (2s1s2 + 2s3s4)a4 +

(s2

1 + s23

)a2 −

(YΩ2

2ω

)2

= 0.

(8)

On the other hand, the relationship between displacementtransmissibility and the system parameters is defined by

TR = X

Y=√(

1 +a

Ycos(γ))2

+(a

Y

)2

sin2(γ). (9)

4. Influence of Delayed Damping

The model we consider consists in quarter-car modelwith softening spring in which m = 240 kg, k1 =160000 N/m, k2 = −30000 N/m3, c1 = 250 N · s/m, andc2 = 25 N · s3/m3. The amplitude of the excitation frequencyis fixed as Y = 0.11.

Figure 2 illustrates the relative amplitude of motion aversus the frequency Ω, as given by (8) (solid line), andfor validation we plot the result obtained by numericalintegration using a Rung-Kutta method (circles).

In Figure 3(a) is shown the TR versus r = Ω/ω for variousfeedback gain λ1 and for λ2 = 0. It can be seen in this figurethat as λ1 increases from 0.01 to 15, the TR reduces. Theeffect of the feedback gain λ2 (with λ1 = 0) on the TR is also

15 20 25 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

a

Ω

35

Figure 2: Amplitude a versusΩ, for λ1 = 0.01, λ2 = 0.01, and τ =0.1. Analytical prediction: solid line; numerical simulation: circles.

illustrated in Figure 3(b) showing also a decrease of TR asλ2 increases from 0.01 to 4. The plots in the figures indicatethat to ensure a significant decrease of TR, a small increase inthe nonlinear gain λ2 is sufficient while a larger value of thelinear gain λ1 is necessary to have an equivalent effect.

Figure 4 shows the variation of TR with respect to timedelay τ for different values of the gains λ1 and λ2. Theseplots indicate that increasing the gains causes TR to reducein repeated periodic intervals of τ. Also, a small increase ofλ2 (λ2 = 0.5) produces this reduction (Figure 4(b)), while alarger value of λ1 (λ1 = 10) should be introduced to obtaina comparable effect (Figure 4(a)). To validate the analyticalpredictions (solid lines), we show in Figures 5 and 6 compar-isons with the numerical simulations (circles).

5. Conclusions

In this work, a strategy based on adding hysteretic nonlinearsuspension with time delay to control transmitted vibrationis presented. The analytical prediction, based on perturba-tion method, shows clearly that increasing the amplitudegains of the delayed damping reduces transmitted vibrationsto a support structure isolation. This analytical predic-tion confirms previous numerical and experimental worksobtained in the case of a single pneumatic chamber [8].The results revealed that the case where only the nonlineargain is acting improves greatly vibration isolation comparingto the case where only the linear gain is applied. Vibrationisolation enhancement can be obtained for a small increaseof the nonlinear gain λ2, while a larger increase of the lineargain λ1 is required to obtain a comparable effect. It is alsoshown that for small values of the nonlinear gain, vibrationisolation can be reduced in repeated periodic intervals oftime delay, whereas larger values of the linear gain is neededto obtain a similar result. The method performed in thiswork provides an approximate expression of transmissibilityrelating the parameters of the system, thereby showing theexplicit dependence between transmissibility and controlparameters which is important from monitoring view point.

4 Advances in Acoustics and Vibration

0.6 0.8 1 1.2 1.4

1

1.5

2

2.5

3

3.5

TR

r = Ωω

λ1 = 0.01λ1 = 4λ1 = 15

(a)T

R

1

1.5

2

2.5

3

λ2 = 0.01λ2 = 1.5λ2 = 4

0.6 0.8 1 1.2 1.4

r = Ωω

(b)

Figure 3: Transmissibility versus r for τ = 0.1. (a) λ2 = 0, and (b) λ1 = 0.

0

0.05 0.

1

0.15 0.

2

0.25 0.

3

0.35 0.

4

0.45 0.

5

0.55 0.

6

012345678

λ1 = 1λ1 = 5λ1 = 10

TR

τ

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6

λ2 = 0.01λ2 = 0.1λ2 = 0.5

τ

0

1

2

3

4

5

6

7

8

TR

(b)

Figure 4: Transmissibility versus τ for r = 1. (a) λ2 = 0, and (b) λ1 = 0.

0 0.1 0.2 0.3 0.4 0.5 0.6

τ

0

1

2

3

4

5

6

7

8

TR

λ1 = 1

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6

τ

0

1

2

3

4

5

6

7

8

TR

λ1 = 5

(b)

Figure 5: Transmissibility versus τ for r = 1 and λ2 = 0.

Advances in Acoustics and Vibration 5

0 0.1 0.2 0.3 0.4 0.5 0.6

τ

0

1

2

3

4

5

6

7

8T

Rλ2 = 0.01

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6

τ

0

1

2

3

4

5

6

7

8

TR

λ2 = 0.1

(b)

Figure 6: Transmissibility versus τ for r = 1 and λ1 = 0.

References

[1] E. I. Rivin, “Vibration isolation of precision equipment,”Precision Engineering, vol. 17, no. 1, pp. 41–56, 1995.

[2] R. A. Ibrahim, “Recent advances in nonlinear passive vibrationisolators,” Journal of Sound and Vibration, vol. 314, no. 3–5, pp.371–452, 2008.

[3] Z. Q. Lang, S. A. Billings, G. R. Tomlinson, and R. Yue,“Analytical description of the effects of system nonlinearitieson output frequency responses: a case study,” Journal of Soundand Vibration, vol. 295, no. 3–5, pp. 584–601, 2006.

[4] Z. Q. Lang, X. J. Jing, S. A. Billings, G. R. Tomlinson, and Z.K. Peng, “Theoretical study of the effects of nonlinear viscousdamping on vibration isolation of sdof systems,” Journal ofSound and Vibration, vol. 323, no. 1-2, pp. 352–365, 2009.

[5] L. Mokni, M. Belhaq, and F. Lakrad, “Effect of fast parametricviscous damping excitation on vibration isolation in sdofsystems,” Communications in Nonlinear Science and NumericalSimulation, vol. 16, no. 4, pp. 1720–1724, 2011.

[6] K. Youcef-Toumi and O. Ito, “Time delay controller for sys-tems with unknown dynamics,” Journal of Dynamic Systems,Measurement and Control, vol. 112, no. 1, pp. 133–142, 1990.

[7] T. C. Hsia and L. S. Gao, “Robot manipulator control usingdecentralized linear time-invariant time-delayed joint con-trollers,” in Proceedings of the IEEE International Conference onRobotics and Automation, vol. 3, pp. 2070–2075, Cincinnati,Ohio, USA, May 1990.

[8] Y. H. Shin and K. J. Kim, “Performance enhancement ofpneumatic vibration isolation tables in low frequency range bytime delay control,” Journal of Sound and Vibration, vol. 321,no. 3–5, pp. 537–553, 2009.

[9] A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, Wiley,New York, NY, USA, 1979.

[10] M. F. Daqaq, K. A. Alhazza, and Y. Qaroush, “On primaryresonances of weakly nonlinear delay systems with cubicnonlinearities,” Nonlinear Dynamics, vol. 64, pp. 253–277,2011.

[11] M. K. Suchorsky, S. M. Sah, and R. H. Rand, “Using delay toquench undesirable vibrations,” Nonlinear Dynamics, vol. 62,pp. 407–416, 2010.

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