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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2011, Article ID 546280, 5 pages doi:10.1155/2011/546280 Research Article Reducing Transmitted Vibration Using Delayed Hysteretic Suspension Lahcen Mokni and Mohamed Belhaq Laboratory of Mechanics, University Hassan II, Casablanca, Morocco Correspondence should be addressed to Mohamed Belhaq, firstname.lastname@example.org 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 Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Previous numerical and experimental works show that time delay technique is eﬃcient to reduce transmissibility of vibration in a single pneumatic chamber by controlling the pressure in the chamber. The present work develops an analytical study to demonstrate the eﬀectiveness of such a technique in reducing transmitted vibrations. A quarter-car model is considered and delayed hysteretic suspension is introduced in the system. Analytical predictions based on perturbation analysis show that a delayed hysteretic suspension enhances vibration isolation comparing to the case where the nonlinear damping is delay-independent. 1. Introduction performance by controlling the pressure in chamber in the low frequency range . Eﬀectiveness of this active control Various strategies for reducing transmitted vibrations have technique in enhancement of transmissibility performance been proposed; see, for instance, [1–5] and references was demonstrated using simulation as well as experiments therein. A standard technique using linear viscous damping testing in the case of a single pneumatic chamber. in the vibration isolation device reduces the transmissibility In the present paper, we use a delayed hysteretic suspen- near the resonance but increases it elsewhere. To enhance sion in the system and we examine its inﬂuence on vibration vibration isolation in the whole frequency range, cubic isolation. In this TDC technique, the values of all the state nonlinear viscous damping has been successfully introduced variables and their ﬁrst derivatives have to be provided by . It was shown, in the case of a single degree of freedom some means. (sdof) spring damper system, that cubic nonlinear damping can eﬀectively produce an ideal vibration isolation in the To achieve our analysis, we implement the multiple scales whole frequency range . Recently, a strategy based on method  on the equation of motion to derive the corre- adding a nonlinear parametric time-dependent viscous sponding modulation equations and we examine the steady damping to the basic cubic nonlinear damper has been state solutions of this modulation equation to obtain indica- proposed . This method signiﬁcantly enhances vibration tions on transmissibility (TR) versus the system parameters. isolation comparing to the case where the nonlinear damping is time-independent [3, 4]. Speciﬁcally, it was reported that increasing the amplitude of the parametric 2. Equation of Motion damping enhances substantially the vibration isolation over A representative model of a suspension system with non- the whole frequency range. On the other hand, time delay control (TDC) [6, 7]was linear stiﬀness, nonlinear viscous damping, and delayed applied to a pneumatic isolator to enhance the isolation hysteretic suspension under external excitation is proposed. 2 Advances in Acoustics and Vibration x(t) (λ , λ ) 1 2 (c , c ) 1 2 (k , k ) 1 2 y(t) = Y cos(Ωt) Figure 1: Schematic diagram of pneumatic vibration isolator under ground excitation. It consists of a sdof model, as shown in Figure 1,described 3. Frequency Response and Transmissibility by To obtain the frequency-response equation and TR, we 2 3 3 ¨ ˙ ˙ X + ω X + B X + B X + B X perform a perturbation method. Introducing a bookkeeping 1 2 3 parameter and scaling Y = Y, B = B , B = B , B = 1 1 2 2 3 ( ) = −g + YΩ cosΩt + λ X t − τ 1 (1) B , λ = λ ,and λ = λ ,(1)reads 3 1 1 2 2 + λ X(t − τ) , 2 2 3 3 z¨ + ω z = −g + YΩ cosΩt − B z − B z˙ − B z˙ where ω = (k /m), B = (k /m), B = (c /m), and 1 2 3 1 1 2 2 1 (2) B = (c /m). In the case of a single pneumatic chamber, as 3 2 +λ z˙(t − τ) + λ (z˙(t − τ)) . shown in Figure 1, m is the body mass, k and k are the 1 2 1 2 linear and the nonlinear stiﬀness of vehicle suspension, c and c are the linear and the nonlinear equivalent viscous damping, g is the acceleration gravity, and X is the relative Using the multiple scales technique , a two-scale expan- vertical displacement of the mass. The parameters Y and sion of the solution is sought in the form Ω denote, respectively, the amplitude and the frequency of the external excitation, while λ and λ denote the gains of 1 2 X(t) = z (T , T ) + z (T , T ) + O , the linear- and the nonlinear-delayed viscous damping, and (3) 0 0 1 1 0 1 τ is the time delay. This delayed viscous damping can be practically implemented using delayed nonlinear dampers where T = t. In terms of the variables T , the time based on magnetorheological ﬂuid. i i 2 2 2 derivatives become d/dt = D + D + O( )and d /dt = It is worthy to notice that the particular case of linear 0 1 2 2 j j stiﬀness (B = 0) and undelayed state feedback (λ = λ = 0) D +2D D +O( ), where D = (∂ /∂ T ). Substituting (3) 1 1 2 0 0 1 i was studied in , while the case of B = 0, λ = λ = into (2), we obtain 1 / 1 2 0, and time-dependent (parametric) damping was treated in . It was demonstrated that adding nonlinear para- 2 2 D +2D D (z + z ) + ω (z + z ) metric damping to the basic nonlinear damping enhances 0 1 0 1 0 1 signiﬁcantly the vibration isolation . The purpose of the = −g + YΩ cos(Ωt) − B (D + D )(z + z ) present work is to study the eﬀect of delayed nonlinear 2 0 1 0 1 damping on vibration isolation of system (1). Note also that 3 3 − B (z + z ) − B ((D + D )(z + z )) a similar delayed nonlinear system to (1) was investigated 1 0 1 3 0 1 0 1 near primary resonances . Attention was focused on ( )( ( ) ( )) + λ D + D z t − τ + z t − τ performing an approach to analyze the dynamic of the 1 0 1 0 1 system with arbitrarily large gains. Note also that a delayed +λ ((D + D )(z (t − τ) + z (t − τ))) , 2 0 1 0 1 feedback was used to quench undesirable vibrations in a van der Pol type system . (4) Advances in Acoustics and Vibration 3 and equating coeﬃcients of the same power of ,weobtain 0.35 at diﬀerent orders 0.3 2 2 D z + ω z = −g, 0 0 0.25 2 2 D z + ω z +2D D z 1 1 1 0 0 a 0.2 = YΩ cos(Ωt) (5) 0.15 − B z − B D z − B (D z ) 1 2 0 0 3 0 0 0.1 + λ D z (t − τ) + λ (D z (t − τ)) . 1 0 0 2 0 0 0.05 In the case of the principal resonance, that is, Ω = ω + σ, 15 20 25 30 35 where σ is a detuning parameter, standard calculations yield the ﬁrst-order solution: Figure 2: Amplitude a versusΩ,for λ = 0.01, λ = 0.01, and τ = 1 2 ( ) ( ) z t = − + a cos Ωt − γ + O ,(6) 0.1. Analytical prediction: solid line; numerical simulation: circles. where the amplitude a and the phase γ are given by the modulation equations: illustrated in Figure 3(b) showing also a decrease of TR as 2 λ increases from 0.01 to 4. The plots in the ﬁgures indicate YΩ a˙ = sin γ − s a − s a , 1 2 that to ensure a signiﬁcant decrease of TR, a small increase in 2ω the nonlinear gain λ is suﬃcient while a larger value of the (7) 2 linear gain λ is necessary to have an equivalent eﬀect. YΩ aγ˙ = cos γ − s a − s a . 3 4 Figure 4 shows the variation of TR with respect to time 2ω delay τ for diﬀerent values of the gains λ and λ . These 1 2 Here s = (B /2) − (λ cos(ωτ)/2), s = (3B ω /8) − plots indicate that increasing the gains causes TR to reduce 1 2 1 2 3 2 2 5 (3λ ω cos(ωτ)/8), s = (3B /2)(g /ω )−σ −(λ sin(ωτ)/2), in repeated periodic intervals of τ. Also, a small increase of 2 3 1 1 λ (λ = 0.5) produces this reduction (Figure 4(b)), while a 2 2 and s = (3B /8ω) − (3λ ω sin(ωτ)/8). Periodic solutions 4 1 2 larger value of λ (λ = 10) shouldbeintroducedtoobtain 1 1 of (2) corresponding to stationary regimes (a˙ = γ˙ = 0) acomparableeﬀect (Figure 4(a)). To validate the analytical of the modulation equations (7) are given by the algebraic predictions (solid lines), we show in Figures 5 and 6 compar- equation: isons with the numerical simulations (circles). YΩ 2 2 6 4 2 2 2 s + s a + (2s s +2s s )a + s + s a − = 0. 1 2 3 4 2 4 1 3 2ω 5. Conclusions (8) In this work, a strategy based on adding hysteretic nonlinear On the other hand, the relationship between displacement suspension with time delay to control transmitted vibration transmissibility and the system parameters is deﬁned by is presented. The analytical prediction, based on perturba- tion method, shows clearly that increasing the amplitude 2 2 X a a gains of the delayed damping reduces transmitted vibrations (9) TR = = 1+ cos γ + sin γ . to a support structure isolation. This analytical predic- Y Y Y tion conﬁrms previous numerical and experimental works obtained in the case of a single pneumatic chamber . 4. Inﬂuence of Delayed Damping The results revealed that the case where only the nonlinear The model we consider consists in quarter-car model gain is acting improves greatly vibration isolation comparing with softening spring in which m = 240 kg, k = to the case where only the linear gain is applied. Vibration 160000 N/m, k = −30000 N/m , c = 250 N · s/m, and isolation enhancement can be obtained for a small increase 2 1 3 3 c = 25 N · s /m . The amplitude of the excitation frequency of the nonlinear gain λ , while a larger increase of the linear 2 2 is ﬁxed as Y = 0.11. gain λ is required to obtain a comparable eﬀect. It is also Figure 2 illustrates the relative amplitude of motion a shown that for small values of the nonlinear gain, vibration versus the frequency Ω,asgiven by (8) (solid line), and isolation can be reduced in repeated periodic intervals of for validation we plot the result obtained by numerical time delay, whereas larger values of the linear gain is needed integration using a Rung-Kutta method (circles). to obtain a similar result. The method performed in this In Figure 3(a) is shown the TR versus r = Ω/ω for various work provides an approximate expression of transmissibility feedback gain λ and for λ = 0. It canbeseeninthisﬁgure relating the parameters of the system, thereby showing the 1 2 that as λ increases from 0.01 to 15, the TR reduces. The explicit dependence between transmissibility and control eﬀect of the feedback gain λ (with λ = 0) on the TR is also parameters which is important from monitoring view point. 2 1 4 Advances in Acoustics and Vibration 3.5 2.5 2.5 1.5 1.5 0.6 0.8 1 1.2 1.4 0.6 0.8 1 1.2 1.4 r = r = ω ω λ = 0.01 λ = 0.01 2 λ = 4 λ = 1.5 1 2 λ = 15 λ = 4 1 2 (a) (b) Figure 3: Transmissibility versus r for τ = 0.1. (a) λ = 0, and (b) λ = 0. 2 1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 λ = 0.01 λ = 1 2 λ = 0.1 λ = 5 2 λ = 10 λ = 0.5 1 2 (a) (b) Figure 4: Transmissibility versus τ for r = 1. (a) λ = 0, and (b) λ = 0. 2 1 λ = 1 8 1 8 λ = 5 7 7 6 6 5 5 4 4 2 2 1 1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 τ τ (a) (b) Figure 5: Transmissibility versus τ for r = 1and λ = 0. TR TR TR 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 TR TR TR Advances in Acoustics and Vibration 5 λ = 0.01 λ = 0.1 8 8 2 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 τ τ (a) (b) Figure 6: Transmissibility versus τ for r = 1and λ = 0. References  E. I. Rivin, “Vibration isolation of precision equipment,” Precision Engineering, vol. 17, no. 1, pp. 41–56, 1995.  R. A. Ibrahim, “Recent advances in nonlinear passive vibration isolators,” Journal of Sound and Vibration, vol. 314, no. 3–5, pp. 371–452, 2008.  Z. Q. Lang, S. A. Billings, G. R. Tomlinson, and R. Yue, “Analytical description of the eﬀects of system nonlinearities on output frequency responses: a case study,” Journal of Sound and Vibration, vol. 295, no. 3–5, pp. 584–601, 2006.  Z. Q. Lang, X. J. Jing, S. A. Billings, G. R. Tomlinson, and Z. K. Peng, “Theoretical study of the eﬀects of nonlinear viscous damping on vibration isolation of sdof systems,” Journal of Sound and Vibration, vol. 323, no. 1-2, pp. 352–365, 2009.  L. Mokni, M. Belhaq, and F. Lakrad, “Eﬀect of fast parametric viscous damping excitation on vibration isolation in sdof systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 4, pp. 1720–1724, 2011.  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.  T. C. Hsia and L. S. Gao, “Robot manipulator control using decentralized linear time-invariant time-delayed joint con- trollers,” in Proceedings of the IEEE International Conference on Robotics and Automation, vol. 3, pp. 2070–2075, Cincinnati, Ohio, USA, May 1990.  Y. H. Shin and K. J. 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Published: Dec 13, 2011
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