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1 Coun Nonlinear Sci Nuer Siulat 17 (2012) Contents lists available at SciVerse ScienceDirect Coun Nonlinear Sci Nuer Siulat journal hoepage: Using delayed daping to iniize transitted vibrations Lahcen Mokni, Mohaed Belhaq Laboratory of Mechanics, University Hassan II-Casablanca, Morocco article info abstract Article history: Received 4 May 2011 Received in revised for 13 July 2011 Accepted 19 August 2011 Available online 30 August 2011 Keywords: Paraetric daping Delayed daping Vibration isolation Fast excitation Perturbation analysis In Mokni et al. [Mokni L, Belhaq M, Lakrad F. Effect of fast paraetric viscous daping excitation on vibration isolation in sdof systes. Coun Nonlinear Sci Nuer Siulat 2011;16: ], it was shown that in a single degree of freedo syste a fast nonlinear paraetric daping enhances vibration isolation with respect to the case where the nonlinear daping is tie-independent. The present work proposes additional enhanceent of vibration isolation using delayed nonlinear daping. Attention is focused on assessing the contribution of a delayed nonlinear daping over a fast paraetric daping in ters of iniizing transissibility. The results show that a nonlinear daping with delay greatly iproves vibration isolation. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction To iprove dynaic perforance of support structures subjected to vibrations, significant efforts have been ade toward investigating new strategies for reducing transitted vibrations to such equipent; see for instance [1 4] and references therein. A standard technique uses viscous daping in the vibration isolation device to enhance isolation perforance. However, when the viscous daping is linear, the transissibility is reduced near the resonance, but increased elsewhere. To overcoe this issue and enhance vibration isolation in the whole frequency range, cubic nonlinear viscous daping has been introduced [3]. It was revealed in the case of a single degree of freedo (sdof) spring daper syste that cubic nonlinear daping can produce an ideal vibration isolation such that the non-resonant regions reain unaffected [4]. Recently, a new technique was proposed to iprove transitted vibrations to a support structure [5]. This strategy, based on adding a nonlinear paraetric viscous daping to the basic cubic nonlinear daper, significantly enhanced vibration isolation coparing to the case where the nonlinear daping is tie-independent. Specifically, it was concluded that increasing the aplitude of the paraetric daping enhances substantially the vibration isolation over the whole frequency range. In the present paper, additional efforts are directed toward bringing further iproveent in enhancing isolation perforance with respect to the strategy proposed in [5]. In this effort, we introduce a delayed nonlinear daping in the syste considered in [5] and we investigate its influence on vibration isolation when acting alone or in the presence of a paraetric daping. The introduction of tie delay is inspired by Shin and Ki [6] in which a tie delay control is applied to a pneuatic isolator to enhance the isolation perforance by controlling the pressure in chaber. The effectiveness of this active control technique was shown experientally in a single pneuatic chaber and using nuerical siulation [6]. Here we use a hysteretic nonlinear suspension with tie delay. Note that a delayed feedback was used to quench undesirable vibrations in a van der Pol type syste [7]. Corresponding author. E-ail address: belhaq@yahoo.fr (M. Belhaq) /$ - see front atter Ó 2011 Elsevier B.V. All rights reserved. doi: /j.cnsns

2 L. Mokni, M. Belhaq / Coun Nonlinear Sci Nuer Siulat 17 (2012) To achieve our analysis, we perfor a direct partition of otion (DPM) followed by averaging ethod [8] to obtain the slow dynaic and then, we ipleent the ultiple scales ethod [9] on the slow dynaic to derive its slow flow. Exaination of steady state solutions of this slow flow yields indications on transissibility (TR) versus the syste paraeters. More precisely, the contribution of delayed nonlinear daping over the fast paraetric viscous daping [5] is evaluated in ter of vibration isolation. 2. Equation of otion and slow dynaic Following [5], we consider a sdof odel of a suspension syste with nonlinear stiffness, nonlinear paraetric viscous daping and delayed nonlinear daping in the for X þ x 2 X þ B 1 X 3 þðb 2 X _ þ B3 X _ 3 Þ 1 þ b 2 cosðtþ ¼ gþyx 2 cosxt þ k 1 Xðt _ sþþk2 ð Xðt _ sþþ 3 ; ð1þ where x 2 ¼ k 1 ; B 1 ¼ k 2 ; B 2 ¼ c 1 and B 3 ¼ c 2. Here is the body ass, k 1 and k 2 are the linear and nonlinear stiffness coefficients, c 1 and c 2 are the linear and nonlinear daping coponents, g is the acceleration gravity and X is the relative vertical displaceent of the ass. The paraeters Y and X denote, respectively, the aplitude and the frequency of the external excitation, b 2 and are the acceleration aplitude and the frequency of the fast paraetric excitation, respectively, while k 1 and k 2 denote the gains of the delayed states, and s is the tie delay. In the new odel given by Eq. (1), the procedure of applying a tie delay control technique to the pneuatic isolator is otivated by the experiental and nuerical work [6], in which the effectiveness of this active control technique in enhanceent of transissibility perforance was deonstrated by controlling the pressure in chaber. In the odel (1), the control strategy is perfored using a hysteretic nonlinear suspension with tie delay. The particular case of linear stiffness (B 1 = 0), tie-independent daping (b = 0) and undelayed state feedback (k 1 = k 2 = 0) was studied in [3], while the case B 1 0, b 0 and k 1 = k 2 = 0 was treated in [5]. It was deonstrated that adding nonlinear paraetric daping (b 0) to the basic nonlinear daping enhances significantly the vibration isolation. The purpose here is to study the contribution of delayed nonlinear daping over the paraetric daping on vibration isolation of syste (1). It is worthy to notice that a siilar delayed nonlinear syste to Eq. (1) was investigated near priary resonances [10] in the absence of paraetric daping (b = 0). Attention was focused on perforing an approach to analyse the dynaic of the syste with arbitrarily large gains. Eq. (1) contains a slow dynaic due to the external excitation and a fast dynaic produced by the fast excitation. Following [5], we use the ethod of DPM [8,11 17] which consists in introducing two different tie scales, a fast tie T 0 = t and a slow tie T 1 = t, and splitting up X(t) into a slow part z(t 1 ) and a fast part /(T 0,T 1 )as XðtÞ ¼zðT 1 Þþ/ðT 0 ; T 1 Þ: ð2þ Thus Xðt sþ ¼zðT 1 sþþ/ðt 0 s; T 1 sþ; ð3þ where z describes the slow ain otions at tie-scale of oscillations and / stands for an overlay of the fast otions. The fast part / and its derivatives are assued to be 2p-periodic functions of fast tie T 0 with zero ean value with respect to this R tie, so that <X(t)>=z(T 1 ) and <X(t s)>=z(t 1 s) where <> 2p 1 2p ðþ; dt 0 0 defines tie-averaging operator over one period of the fast excitation with the slow tie T 1 fixed. Introducing D i yields d ¼ D j T i dt 0 þ D 1 ; 2 ¼ 2 D 2 dt 2 0 þ 2D 0D 1 þ D 2 1 and substituting Eqs. (2) and (3) into Eq. (1) gives z þ / þ x 2 z þ x 2 / þ B 1 ðz þ /Þ 3 þ B 2 _z þ B 2 / _ þ B2 b 2 cosðtþ_z þ B 2 b 2 cosðtþ / _ þ B 3 ð_z þ /Þ _ 3 þ B 3 b 2 cosðtþð_z þ /Þ _ 3 ¼ gþyx 2 cosðxtþþk 1 _zðt 1 sþþ /ðt _ 0 s; T 1 sþ þ k 2 _zðt 1 sþþ /ðt _ 3: 0 s; T 1 sþ ð4þ Averaging (4) leads to z þ x 2 z þ B 1 z 3 þ 3B 1 z < / 2 > þb 1 < / 3 > þb 2 _z þ B 2 b 2 < cosðtþ / _ > þb 3 _z 3 þ 3B 3 _z < / _ 2 > þb 3 < / _ 3 > þ 3B 3 b 2 _z 2 < cosðtþ / _ > þ3b 3 b 2 _z < cosðtþ / _ 2 > þb 3 b 2 < cosðtþ / _ 3 > ¼ gþyx 2 cosðxtþþk 1 _zðt 1 sþþk 2 _z 3 ðt 1 sþþ3k 2 _zðt 1 sþ < / _ 2 ðt 0 s; T 1 sþ > þ k 2 < / _ 3 ðt 0 s; T 1 sþ >: ð5þ

3 1982 L. Mokni, M. Belhaq / Coun Nonlinear Sci Nuer Siulat 17 (2012) Subtracting (5) fro (4) yields / þ x 2 / þ 3B 1 z 2 / þ 3B 1 z/ 2 3B 1 z < / 2 > þb 1 / 3 B 1 < / 3 > þb 2 _ / þ B2 b 2 cosðtþ _ / B 2 b 2 < cosðtþ _ / > þ 3B 3 _z 2 / _ þ 3B3 _z / _ 2 3B 3 _z < / _ 2 > þb 3 / _ 3 B 3 < / _ 3 > þ 3B 3 b 2 _z 2 / _ cosðtþ 3B3 b 2 _z 2 < / _ cosðtþ > þ3b 3 b 2 _z / _ 2 cosðtþ 3B 3 b 2 _z < / _ 2 cosðtþ > þ B 3 b 2 / _ 3 cosðtþ B 3 b 2 < / _ 3 cosðtþ > k 1 /ðt0 _ s; T 1 sþ 3k 2 _z 2 ðt 1 sþ /ðt _ 0 s; T 1 sþ þ 3k 2 _z 2 ðt 1 sþ < /ðt _ 0 s; T 1 sþ > 3k 2 _zðt 1 sþ / _ 2 ðt 0 s; T 1 sþ þ 3k 2 _z 2 ðt 1 sþ < / _ 2 ðt 0 s; T 1 sþ > k 2 < / _ 3 ðt 0 s; T 1 sþ >¼ b 2 ðb 2 _z þ B 3 _z 3 Þ cosðtþ: ð6þ Using the so-called inertial approxiation [5,8], i.e. all ters in the left-hand side of Eq. (6), except the first, are ignored, one obtains / ¼ bðb 2 _z þ B 3 _z 3 Þ cosðtþ: ð7þ This siplification when solving Eq. (6) consists in finding / in the for of a su of a sall nuber of haronics of the fast tie T 0 taking into account that / is sall as copared to z and then it is possible to consider only the linear doinant ters in the left-hand side of Eq. (6). For ore details on this approxiation, the reader can refer to chapter 2 in [8]. Inserting / fro Eq. (7) into Eq. (5), using that <cos 2 T 0 > = 1/2, and neglecting ters of orders greater than three in z, give the equation governing the slow dynaic of the otion z þ x 2 z þ B 1 z 3 þ B 2 _z þ H 1 z_z 2 þ H 2 _z 3 ¼ gþyx 2 cosxt þ k 1 _zðt sþþh 3 _z 3 ðt sþ; ð8þ where H 1 ¼ 3 B 2 1B 2 2 b2 ; H 2 ¼ B 3 1 þ 3 2 B2 2 b2 2 and H 3 ¼ k 2 1 þ 3 2 B2 2 b Frequency response and transissibility To obtain the frequency response equation and TR, we perfor a perturbation ethod. Introducing a bookkeeping paraeter and scaling Y ¼ Y e ; B 1 ¼ B e 1 ; B 2 ¼ B e 2 ; H 1 ¼ H e 1 ; H 2 ¼ H e 2 ; H 3 ¼ H e 3 and k 1 ¼ ~ k 1, Eq. (8) reads h z þ x 2 z ¼ gþ Y e X 2 cosxt B e 1 z 3 B e 2 _z H e 1 z_z 2 H e 2 _z 3 þ ~ k 1 _zðt sþþh e i 3 ð_zðt sþþ 3 ð9þ Using the ultiple scales technique [9], we seek a two-scale expansion of the solution in the for XðtÞ ¼z 0 ðt 0 ; T 1 Þþz 1 ðt 0 ; T 1 ÞþOð 2 Þ; ð10þ where T i = i t. In ters of the variables T i, the tie derivatives becoe d ¼ D dt 0 þ D 1 þ Oð 2 Þ and d2 ¼ D 2 dt 2 0 þ 2D 0D 1 þ Oð 2 Þ, where D j i. Substituting Eq. (10) into Eq. (9), we obtain the following j T i h ðd 2 0 þ 2D 0D 1 Þðz 0 þ z 1 Þþx 2 ðz 0 þ z 1 Þ¼ gþ Y e X 2 cosðxtþ B e 2 ðd 0 þ D 1 Þðz 0 þ z 1 Þ B e 1 ðz 0 þ z 1 Þ 3 H e 1 ðz 0 þ z 1 ÞðD ð 0 þ D 1 Þðz 0 þ z 1 ÞÞ 2 H e 2 ððd 0 þ D 1 Þðz 0 þ z 1 ÞÞ 3 þ ~ k 1 ðd 0 þ D 1 Þðz 0 ðt sþ þ z 1 ðt sþþ þ H e 3 ððd 0 þ D 1 Þðz 0 ðt sþþz 1 ðt sþþþ 3i ð11þ and equating coefficients of the sae power of, we obtain at different orders D 2 0 z 0 þ x 2 z 0 ¼ g; ð12þ D 2 0 z 1 þ x 2 z 1 þ 2D 1 D 0 z 0 ¼ Y e X 2 cosðxtþ B e 1 z 3 0 B e 2 D 0 z 0 H e 1 z 0 ðd 0 z 0 Þ 2 H e 2 ðd 0 z 0 Þ 3 þ ~ k 1 D 0 z 0 ðt sþ þ H e 3 ðd 0 z 0 ðt sþþ 3 : ð13þ In the case of the principal resonance, i.e. X = x + r, where r is a detuning paraeter, standard calculations yield the firstorder solution zðtþ ¼ g þ a cosðxt cþþoðþ; x ð14þ 2 where the aplitude a and the phase c are given by the odulation equations 8 < _a ¼ e Y X 2 2x sinðcþ s 1a s 2 a 3 : a _c ¼ e Y X 2 2x cosðcþ s 3a s 4 a 3 Here s 1 ¼ e B 2 ~ k 1 cosðxsþ ; s ¼ 3 e H 2 x 2 3 e H3 x 2 cosðxsþ ; s ¼ 3 e B1 g 2 2 x r ~ k 1 sinðxsþ and s ¼ H 1x þ 3 e B1 8 8x 3 e H 3 x 2 sinðxsþ. Periodic solutions of 8 Eq. (9) corresponding to stationary regies ð _a ¼ _c ¼ 0Þ of the odulation Eq. (15) are given by the algebraic equation ð15þ

4 L. Mokni, M. Belhaq / Coun Nonlinear Sci Nuer Siulat 17 (2012) ðs 2 2 þ s2 4 Þa6 þð2s 1 s 2 þ 2s 3 s 4 Þa 4 þðs 2 1 þ s2 3 Þa2! 2 YX2 ¼ 0: ð16þ 2x On the other hand, the relationship between displaceent transissibility (TR) and the syste paraeters is defined by TR ¼ X rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Y ¼ 1 þ a 2 Y cosðcþ a 2 þ sin 2 ðcþ: ð17þ Y 4. Influence of delayed daping The odel we consider consists in pneuatic vibration isolation table in which the ass of payload is = 240 kg, k 1 = N/, k 2 = 30,000 N/ 3, c 1 = 250 N s/ and c 2 =25Ns 3 / 3. The paraeters and Y are fixed as in [5] ( = 400 and Y = 0.11). Fig. 1. Aplitude a versus X, for k 1 = 0.01, k 2 = 0.01, s = 0.1, b = 0. Analytical prediction: solid line, nuerical siulation: circles. (a) (b) Fig. 2. Transissibility versus r for s = 0.1. (a) k 2 = 0, (b) k 1 = 0. Undelayed case: heavy line (fro [5]).

5 1984 L. Mokni, M. Belhaq / Coun Nonlinear Sci Nuer Siulat 17 (2012) Fig. 3. Transissibility versus r for s = 0.1 and for different values of k 1 and k 2. Undelayed case: heavy line (fro [5]). Fig. 4. Transissibility versus r for s = 0.1, b = Undelayed case: heavy line (fro [5]). To begin, we illustrate in Fig. 1 the relative aplitude of otion a versus the frequency X, as given by Eq. (16) (solid line), and for validation we plot the result obtained by nuerical integration using a Runge Kutta ethod (circles). In Fig. 2a is shown the TR versus r ¼ X x for various feedback gain k 1 and for k 2 = 0. It can be seen in this figure that the TR reduces by increasing k 1 fro 0.01 to 15 (see the corresponding curves for b = 0). The effect of the feedback gain k 2 (with k 1 = 0) on the TR is also illustrated in Fig. 2b showing also a decrease of TR (see the corresponding curves for b = 0). The plots in the figures indicate that to ensure a significant decrease of TR, a sall increase in the nonlinear gain k 2 is sufficient while a larger value of the linear gain k 1 is necessary to have an equivalent effect. Fig. 3 depicts the effect of the feedback gains k 1 and k 2 when applied siultaneously showing the TR decrease in this cobined case. To show the superiority of the delayed daping over the paraetric daping in ter of vibration isolation, we plot in Figs. 2 and 3 the TR curve (heavy line) in the presence of fast paraetric daping (b 0) and without tie delay (k 1 =0,k 2 =0)[5]. It can be clearly seen that the delayed daping provides ore iproveent of vibration isolation copared to the case of paraetric daping without delay (heavy line). In Fig. 4, we plot the effect of the feedback gains k 1 and k 2 on TR in the presence of paraetric daping. This figure indicates that adding the gains siultaneously induces ore iproveent of vibration isolation coparing to the case where the gains are acting separately. Finally, Fig. 5 shows the variation of TR with respect to tie delay s for different values of the

6 L. Mokni, M. Belhaq / Coun Nonlinear Sci Nuer Siulat 17 (2012) (a) (b) Fig. 5. Transissibility versus r for r = 1. (a) k 2 = 0, (b) k 1 =0. gains k 1 and k 2. It can be seen that increasing the gains causes TR to reduce drastically in repeated periodic intervals of s. In addition, only a sall increase of k 2 (k 2 = 0.3) produces this reduction (Fig. 5b), while a large value of k 1 (k 1 = 10) should be introduced to obtain a coparable effect (Fig. 5a). 5. Conclusions Nonlinear fast paraetric daping has been successfully used to reduce transitted vibrations to a support structure [5]. In the present work, a strategy based on adding hysteretic nonlinear suspension with tie delay to paraetric daping was explored. The analytical predictions, using the averaging ethod and the ultiple scale technique, shown clearly that increasing the aplitude gains of the delay induces ore vibration isolation over the whole frequency range deonstrating its superiority over the nonlinear fast paraetric daping. The results revealed that the case where the gains are acting siultaneously iproves greatly vibration isolation coparing to the case where the gains are applied separately. When the gains are acting separately, this effect can be obtained for a sall increase of the nonlinear gain k 2, while a large increase of the linear gain k 1 is required to obtain a coparable effect. It is also shown that for sall values of the nonlinear delay gain k 2, vibration isolation can be reduced to its iniu in repeated periodic intervals of tie delay, whereas a large value of the linear delay gain k 1 is needed to obtain a siilar result. References [1] Rivin RI. Vibration isolation of precision equipent. Precision Eng 1995;17: [2] Ibrahi RI. Recent advances in nonlinear passive vibration isolators. J Sound Vib 2008;314: [3] Lang ZQ, Billings SA, Tolinson GR, Yue Y. Analytical description of the effects of syste nonlinearities on output frequency responses: a case study. J Sound Vib 2006;295: [4] Lang ZQ, Jing XJ, Billings SA, Tolinson GR, Peng ZK. Theoretical study of the effects of nonlinear viscous daping on vibration isolation of sdof systes. J Sound Vib 2009;323: [5] Mokni L, Belhaq M, Lakrad F. Effect of fast paraetric viscous daping excitation on vibration isolation in sdof systes. Coun Nonlinear Sci Nuer Siulat 2011;16: [6] Shin YH, Ki KJ. Perforance enhanceent of pneuatic vibration isolation tables in low frequency range by tie delay control. J Sound Vib 2009;321: [7] Suchorsky MK, Sah MS, Rand RH. Using delay to quench undesirable vibrations. Nonlinear Dyn 2010;62: [8] Blekhan II. Vibrational echanics-nonlinear dynaic effects, general approach, application. Singapore: World Scientific; [9] Nayfeh AH, Mook DT. Nonlinear oscillations. New York: Wiley; [10] Daqaq MF, Alhazza KA, Qaroush Y. On priary resonances of weakly nonlinear delay systes with cubic nonlinearities. Nonlinear Dyn 2011;64: [11] Thosen JJ. Vibrations and stability: advanced theory, analysis, and tools. Berlin-Heidelberg: Springer-Verlag; [12] Belhaq M, Fahsi A. 2:1 and 1:1 frequency-locking in fast excited van der Pol Mathieu Duffing oscillator. Nonlinear Dyn 2008;53: [13] Belhaq M, Fahsi A. Hysteresis suppression for priary and subharonic 3:1 resonances using fast excitation. Nonlinear Dyn 2009;57: [14] Lakrad F, Belhaq M. Suppression of pull-in instability in MEMS using a high-frequency actuation. Coun Nonlinear Sci Nuer Siulat 2010;15: [15] Lakrad F, Belhaq M. Suppression of pull-in in a icrostructure actuated by echanical shocks and electrostatic forces. Int J Non-Linear Mech 2011;46: [16] Belhaq M, Sah SM. Horizontal fast excitation in delayed van der Pol oscillator. Coun Nonlinear Sci Nuer Siul 2008;13: [17] Sah SM, Belhaq M. Effect of vertical high-frequency paraetric excitation on self-excited otion in a delayed van der Pol oscillator. Chaos Soliton Fract 2008;37: