Amplification of useful vibration using on-off damping

Nội dung text: Amplification of useful vibration using on-off damping

1. Vietnam Journal of Mechanics, VAST, Vol.42, No. 1 (2020), pp. 87 – 94 DOI: AMPLIFICATION OF USEFUL VIBRATION USING ON-OFF DAMPING La Duc Viet1,2,∗ 1Institute of Mechanics, VAST, Hanoi, Vietnam 2Graduate University of Science and Technology, VAST, Hanoi, Vietnam E-mail: ldviet@imech.vast.vn Received: 22 December 2019 / Published online: 17 March 2020 Abstract. This paper points out how much useful vibration can be extracted from a base- excited oscillator, which is controlled by the on-off electrical damping. We studies the class of on-off electrical damping controller, which switches the damping level from high to low and back at fixed times every quarter of period. The problem reduces to the maximization of a single-variable function. This result can open the new direction to amplify the useful vibration using controllable dampings. Keywords: vibration energy harvesting, frequency response, analytical optimization, on-off damping, optimal bound. 1. INTRODUCTION A base-excited oscillator is a well-known model to extract useful vibration [1]. The optimization of electrical damping to extract maximum useful vibration has been exten- sively investigated in [1] and many references therein. If the damping is too large, the useful vibration is suppressed. Conversely, the too small electrical damping can make large vibration due to the resonance effect but this large vibration is useless. Those are the reason why the damping should be optimized to maximize the useful vibration. To suppress the useless vibration, it is well-known that the performance of constant passive damping can be enhanced by the semi-active damping [2,3]. However, to am- plify the useful vibration, semi-active damping is still not considered much. The switch- ing technique applied to a piezoelectric harvester has been studied in some papers [4–6]. Some theoretical analyses of semi-active damping in energy harvesting have been pre- sented in [7,8]. In the field of vibration control, some papers pointed out how much useless vibration can be suppressed by a general controller of on-off damping [9–12]. In this paper, in the converse way, we derive clearly how much useful vibration can be extracted using on-off electrical damping. In literature, there is still no optimal solution (of constant damping) for the transient response. Therefore, for comparison, in this paper, only the c 2020 Vietnam Academy of Science and Technology
2. 88 La Duc Viet steady responses are considered. Energy extraction from transient response by on-off damping is indeed an interesting topic for the future studies. The found solution also gives a good direction to design the future practical on-off electrical damping controllers. 2. PROBLEM STATEMENT Let us consider a base-excited SDOF oscillator with electromagnetic transduction as shown in Fig.1. x m k r cm fe FigureFig. 1 .1. SDOF SDOF base-excited base-excited oscillator oscillator TheThe motion motion equation equation of the ofsystem the systemhas form has[1]: form [1] mx!!++ c cx ! - r ! + k x- r =0 (1) mxă +((cmem +)c(e)(x˙)− r˙)( + k)(x − r) = 0, (1) in whichin whichm ,mk, andk andc mcmrespectively respectively as as the the mass, mass, the the stiffness stiffness and and the mechanical the mechanical damping damping of spring-mass-damper system, c is the electrical damping provided by the electromotive force f . It is of spring-mass-dampere system, ce is the electrical damping provided by thee electromo- notedtive that force thef useful. It is vibration noted that mostly the dependsuseful vibration on the electrical mostly damping. depends Denote on the r and electrical x as the damp- absolute displacemente s of the foundation and the oscillator's mass, respectively. The non-dimensional forming. of Denote(1) is: r and x as the absolute displacements of the foundation and the oscillator’s mass, respectively. The non-dimensional form of (1) is 2 !!xxrxr++20(zzwmen) ( ! - ! ) + w n( -2) = (2) xă + 2 (ζm + ζe) ωn (x˙ − r˙) + ωn (x − r) = 0, (2) where w =q km is the natural frequency, zw= cm(2 ) and zw= cm(2 ) respectively where ω =n km is the natural frequency,mnζ = c(2mω )eeand ζ = c n(2mω ) respec- are the mechanicaln and electrical damping ratios. m n e e n tively are the mechanical and electrical damping ratios. TheThe actual actual useful useful vibration vibration is expressed is expressed by the average by the power average extracted power by the extracted electrical load: by the elec- TT trical load 1 2mw 2 P=T f x!!- r dt=n z x !!-T r dt (3) Zũũee( ) ( Z ) 1TT002mωn 2 P = fe (x˙ − r˙) dt = ζe(x˙ − r˙) dt, (3) where T is a certain simulationT time. Under harmonicT base excitation with flat acceleration 0 0 spwhereectrum, considerT is a certainthe harmonic simulation response time. at steady Under state, harmonicthe optimal electrical base excitation damping withratio z flate has accel- been derived as [1]: eration spectrum, consider the harmonic response at steady state, the optimal electrical opt damping ratio ζe has been derived aszz [em1]= (4) opt The existence of the optimal ze is explainedζ eas follows.= ζm. If ze is too large, the relative velocity is (4) suppressed that reduces the value of (3). Conversely, ze is too small, the vibration can be large but the useful vibrationThe existence in (3) once of again the optimalis reduced.ζ e is explained as follows. If ζe is too large, the rela- tive velocity is suppressed that reduces the value of (3). Conversely, ζe is too small, the vibrationIn this paper, can be we large consider but the the on useful-off electrical vibration damping in (in3) the once form: again is reduced. opt zgzee= ộg h certain condition (5) g = ờ ởg l otherwise in which gh and gl, respectively, are the on-value and off-value of gain of on-off electrical damping. If gh=gl=1, we return to the constant passive damping.
3. Amplification of useful vibration using on-off damping 89 In this paper, we consider the on-off electrical damping in the form opt ζe = γζe ,  γ certain condition (5) γ = h γl otherwise in which γh and γl, respectively, are the on-value and off-value of gain of on-off electrical damping. If γh = γl = 1, we return to the constant passive damping. 3. MAXIMIZATION OF USEFUL VIBRATION Let us consider a class of on-off damping controller, which switches the damping level from high to low and back at fixed times every half period of each given frequency [9]. This controller is quite3. general MAXIMIZATION that can beOF use USEFUL not only VIBRATION to suppress useless vibration but also to amplifyLet us consider useful a class vibration. of on-off damping Fig.2 controller,illustrates which the switches switching the damping law level of mentionedfrom high to low and back at fixed times every half period of each given frequency [9]. This controller is controller,quite where general tthat1 is can the be use switching not only to suppress time from useless thevibration high but to also low, to amplifyt2 is useful the vibration. back switching time andFig.Ω 2is illustrates the excitation the switching frequency. law of mentioned For all controller, permissible where t1 is switching the switching times, time from the the controller can be optimizedhigh to low, tot2 is maximize the back switching the useful time and vibration. Ω is the excitation frequency. For all permissible switching times, the controller can be optimized to maximize the useful vibration. Fig. 2.Figure Illustration 2: Illustration of switching of switching times times over over an anexcitation excitation period period To simplify the solution, the paper [9] introduced the normalized damping width t and the normalized centre of damping peak, t0. Both are illustrated in Fig.2 and nondimensional. All To simplifypermissible the switching solution, times can the be paper described [9 ] by introduced these two parameters. the normalized As mentioned damping in [1], the width t¯ and the normalizedacceleration spectrum centre of vibration of damping sources peak,commonlyt0 . available Both are in daily illustrated lives is relatively in Fig. flat2 and with nondi- mensional.frequency. All permissibleTherefore, we consider switching the foundation times displacement can be describedin the form: by these two parameters. As mentioned in [1], the acceleration spectrumD of vibration sources commonly available rt=2 sin W (6) in daily lives is relatively flat with frequency.W Therefore, we consider the foundation displacementIn in which the D form is the acceleration constant amplitude. Let the relative displacement be written in the approximated form: ∆ xra-ằr = cos Wsin tb+Ω sint,W t (7) (6) 11Ω2 in which ∆ isThe the solution acceleration in [9] gives: constant amplitude. Let the relative displacement be writ- -DW2wzz+ cos2t ten in the approximated form nts( 0 ) a1 = (8) 22 2 2 2 22 4wzzwntsnW-( ) +( -W) x − r ≈ a1 cos Ωt + b1 sin Ωt. (7) 22 D-W-W(wwznns2 sin 2t0 ) b1 = (9) 22 2 2 2 22 4wzzwntsnW-( ) +( -W) in which, we denote:
4. 90 La Duc Viet The solution in [9] gives −2∆ωnΩ (ζt + ζs cos 2t0) a1 = , (8) 2 2 2 2
   2 2 2 4ωnΩ ζt − ζs + (ωn − Ω ) 2 2
   ∆ ωn − Ω − 2ωnΩζs sin 2t0 b1 = , (9) 2 2 2 2
   2 2 2 4ωnΩ ζt − ζs + (ωn − Ω ) in which, we denote   γ − γ opt γ − γ opt ζ = h l t¯ + γ ζ + ζ , ζ = h l ζ sin t¯. (10) t π l e m s π e Over one vibration period, substitute T = 2π/Ω into (3) give the useful vibration in form 2π/Ω mω Ω Z P = n ζ (x˙ − r˙)2dt (11) π e 0 From the motion equation (2), we have 2 − ωnζe (x˙ − r˙) = xă + 2ζmωn (x˙ − r˙) + ωn (x − r) (12) which changes (11) to 2π/Ω −mΩ Z   P = xă (x˙ − r˙) + ω2 (x − r)(x˙ − r˙) + 2ζ ω (x˙ − r˙)2 dt (13) 2π n m n 0 Substituting the harmonic forms (6) and (7) into (13) and simplifying give −mΩ P = ∆a + 2Ωζ ω a2 + b2
   
   (14) 2 1 m n 1 1 Substituting (8) and (9) into (14), some manipulations yields m∆2α2 P =   ì 2 2 2
   2 2 ωn 4α ζt − ζs + (1 − α )   2 2 2
   2
   2 2  2
   2  (15) cos 2t0 4α ζt − ζs + 1 − α − 8α ζtζm + 2ζmα 1 − α sin 2t0 − 8ζmζsα ζt − ζm + ζs  , 2 2 2
   2 2 4α ζt − ζs + (1 − α ) in which, the normalized frequency is introduced Ω α = . (16) ωn It is important to noted that, in the passive case, we have γh = γl = 1, ζs = 0, (17)
5. Amplification of useful vibration using on-off damping 91 which turns the solution (15) to the one presented in [1]. Otherwise, in the on-off electrical damping case, the problem now is to maximize P, which is a function of two variables t¯ and t0. By using the Cauchy–Schwarz inequality, it is not difficult to check that m∆2α2 P (t¯, t0) ≤ P0 (t¯) =   ì 2 2 2
   2 2 ωn 4α ζt − ζs + (1 − α ) r   2  (18) 8ζ ζ α2 − 4α2 ζ2 − ζ2
   + (1 − α2)2 − 8α2ζ ζ + 4ζ2 α2(1 − α2)2  m s t s t m m  ζt − ζm − ζs  .  2 2 2
   2 2  4α ζt − ζs + (1 − α ) The equality holds when 2 2 2
   2
   2 2 4α ζt − ζs + 1 − α − 8α ζtζm cos 2t0 opt = r ,  2 2 2 2
   2 2 2 2 2 2 2 4α ζt − ζs + (1 − α ) − 8α ζtζm + 4ζmα (1 − α ) (19) 2
   2ζmα 1 − α sin 2t0 opt = r ,  2 2 2 2
   2 2 2 2 2 2 2 4α ζt − ζs + (1 − α ) − 8α ζtζm + 4ζmα (1 − α ) where t0 opt is the optimal value of t0. The problem now is to find the maximum of a single-variable function P0 in a fixed interval (0 ≤ t¯ ≤ π) PM = max P0 (t¯) (20) 0≤t¯≤π where PM is the maximum useful vibration can be extracted by on-off electrical damping. It is noted that maximizing a single-variable function in a bound interval can be done by many efficient algorithms. 4. COMPARISONS BETWEEN ON-OFF DAMPING AND PASSIVE DAMPING 2 Fig.3 shows some plots of the normalized useful vibration Pωn/m/∆ in the fre- quency domain for varying values of higher gain γh and lower gain γl. It is seen that the useful vibration can be amplified remarkable by the on-off electrical damping in com- parison with the optimal passive electrical damping. The peak of useful vibration can not be heightened. This conclusion can be drawn from (18) as follows. The peak useful vibration of the passive case is obtained when α = 1 [1]. Substitute α = 1 into (18) we have 2 2 2 ! m∆ 2ζmζs − ζ − ζ − 2ζtζm P (t¯)| = ζ − ζ − ζ t s . (21) 0 α=1 2 2
   t m s 2 2 4ωn ζt − ζs ζt − ζs
6. 92 La Duc Viet Figure 3. Normalized useful vibration versus normalized frequency for varying gains and for ζ =0.1 Fig. 3. Normalized useful vibration versus normalized frequency m It is seen that the useful vibrationfor varying can be gains amplified and remarkable for ζ = by 0.1 the on-off electrical damping in comparison with the optimal passive electrical damping. Them peak of useful vibration can not be heightened. This conclusion can be drawn from (18) as follows. The peak useful vibration of the Afterpassive some case manipulations, is obtained when a= we1 [1]. simplify Substitute a (21=1 )into as (18) we have  2 ổử22 m(Dζt − ζs) − ζm 22zzms2 z t2 z s zz tm Pt0 ( ) = ỗữzztms z ζ − ζ − 2ζtζm ≥ 0 (21) ¯ a =1  22ỗữ2 t zz22-s ωnP0 (t) 4wznt( -4 z( sζ) − ζ ) ts = tốứs (22) 2 ( + ) − m∆ = ζt ζs ζm 2 2 After some manipulations,α 1 we simplify (21) as: ζ − ζ − 2ζtζm < 0  ( + )2 t s ỡ(zz4 ζt ) ζ zs ts m zz22 20 zz ³ It is noted that for any u we haveù 2 ts tm wn Pt0 ( ) ù 4(zzts- ) 2 = ớ (22) mD (zzts+− ) - z m a =1 ù u ζm 1 zz22 20 zz< 2 ≤ .ts tm (23) ù 4 zz+2 ợ ( tsu ) 4ζm The equalityIt is noted that holds for any when u we have uu -=z 2ζ 1. (24) m Ê m (23) u2 4z In (22), if we substitute u by ζt − ζs or ζt + ζsm, we have The equality holds when ωnP0 (t¯) 1 ≤ , (25) 2 u = 2z m (24) m∆ α=1 16ζm In (22), if we substitute u by zz- or zz+ , we have Because the value 1/16/ζmtsis thets peak of normalized useful vibration in the passive case, the formula (24) implies that the on-off damping can not heighten the peak of useful vibration as shown in Fig.3. However, it is also seen in Fig.3, the width of the curve can be increased, which implies the useful vibration can be amplified in a wider frequency range. The effects of gains are simple: the on-value should be as large as possible while the off-value should be as small as possible. To see the effect of on-off damping in the limit case, let us consider the zero value of off-gain (γl = 0) and the very large value of on-gain, i.e γh increases to a very large value.
7. w Pt( ) 1 n 0 Ê (25) mD2 16z a =1 m Because the value 1/16/zm is the peak of normalized useful vibration in the passive case, the formula (25) implies that the on-off damping can not heighten the peak of useful vibration as shown in Fig.3. However, it is also seen in Fig.3, the width of the curve can be increased, which implies the useful vibration can be amplifiedAmplification in a wider offrequency useful vibration range. usingThe effects on-off dampingof gains are simple: the on-value 93 should be as large as possible while the off-value should be as small as possible. To see the effect of on-off damping in the limit case,2 let us consider the zero value of off-gain (gl=0) The normalized useful vibration Pωn/m/∆ is plotted versus the normalized frequency and the very large value of on-gain, i.e gh increases to a very large value. The normalized useful 2 α in Fig.4vibration. Pwn/m/D is plotted versus the normalized frequency a in Fig. 4. FigureFig. 4. Normalized 4 useful vibration versus normalized frequency for ζm=0.1, gl=0 and gh tends to . Normalized useful vibrationinfinity versus normalized frequency for ζm = 0.1, γl = 0 and γh tends to infinity It is seen that the useful vibration tends to the maximum one of the passive case in a wider range of frequency. The same proof can be done as above when we observe (10) and (18). From (10), when It isz seenl=0 and that zh tends the to usefulinfinity, two vibration terms zt and tends zs also totend the to infinity. maximum The expression one of (18 the) reduces passive to: case in a wider range of frequency. The same proof can be done as above when we observe (10) 2 ổử22 mD 22zzms z t z s zz tm lim Pt== ỗữzz z (26) and (18). From (10), when0 (ζl) 0 and22ζh tendstms to infinity, two22 terms ζt and ζs also tend to z h đƠ ỗữzz- z =0 4wznt( - z s) ts infinity. The expressionl (18) reduces to ốứ The expression (26) completely coincides2 with (21). From (25) we also have 2 2 ! m∆ 2ζmζs − ζt − ζs − 2ζtζm lim P (t¯) = ζw−Ptζ − ζ . (26) 0 2 2
   t n 0 ( m) 1s 2 2 ζh→∞ 4ω ζ − ζ lim Ê ζ − ζ (27) n t s z đƠ 2 t s = h mD 16z m ζl 0 z l =0 The expression (26) completely coincides with (21). From (24) we also have ω P (t¯) 1 n 0 ≤ lim 2 . (27) ζh→∞ m∆ 16ζm ζl =0 The value 1/16/ζm is indeed the peak of useful vibration in the passive case in seen in Figs.3 and4. The difference between (25) and (27) is that (27) holds for all frequencies. 5. CONCLUSIONS This paper considers the problem of amplifying useful vibration from a base-excited oscillator with on-off electrical damping. We derive the theoretical solution of the max- imum available useful vibration can be extracted. The on-off damping can not heighten the peak of useful vibration in the frequency domain in comparison with optimal pas- sive damping. However the larger on-damping and smaller off-damping can widen the useful vibration curve in the frequency domain. Moreover, in the theoretical limit case,
8. 94 La Duc Viet if the on-damping tends to infinity and off-damping is zero, the simple solution shows that, the peak of useful vibration in the passive case can be extracted in all frequency by the on-off damping. This conclusion opens the opportunities to find the future practical on-off damping controller to extract remarkable useful vibration. ACKNOWLEDGMENT This paper is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number “08/2018/TN”. REFERENCES [1] W.-C. Tai and L. Zuo. On optimization of energy harvesting from base-excited vibration. Journal of Sound and Vibration, 411, (2017), pp. 47–59. [2] F. Casciati, G. Magonette, and F. Marazzi. Technology of semiactive devices and applications in vibration mitigation. John Wiley & Sons, (2006). [3] S. M. Savaresi, C. Poussot-Vassal, C. Spelta, O. Sename, and L. Dugard. Semi-active suspension control design for vehicles. Butterworth-Heinemann (UK), (2010). [4] E. Lefeuvre, A. Badel, C. Richard, and D. Guyomar. Piezoelectric energy harvesting device optimization by synchronous electric charge extraction. Journal of Intelligent Material Systems and Structures, 16, (10), (2005), pp. 865–876. [5] D. Guyomar, A. Badel, E. Lefeuvre, and C. Richard. Toward energy harvesting us- ing active materials and conversion improvement by nonlinear processing. IEEE Trans- actions on Ultrasonics, Ferroelectrics, and Frequency Control, 52, (4), (2005), pp. 584–595. [6] H. Ji, J. Qiu, P. Xia, and D. Inman. Analysis of energy conversion in switched-voltage control with arbitrary switching frequency. Sensors and Actuators A: Physical, 174, (2012), pp. 162–172. [7] F. Di Monaco, M. G. Tehrani, S. J. Elliott, E. Bonisoli, and S. Tornincasa. Energy harvest- ing using semi-active control. Journal of Sound and Vibration, 332, (23), (2013), pp. 6033–6043. [8] G. Caruso, S. Galeani, and L. Menini. Semi-active damping and energy harvesting using an electromagnetic transducer. Journal of Vibration and Control, 24, (12), (2018), pp. 2542–2561. [9] J. N. Potter, S. A. Neild, and D. J. Wagg. Generalisation and optimisation of semi-active, on– off switching controllers for single degree-of-freedom systems. Journal of Sound and Vibration, 329, (13), (2010), pp. 2450–2462. [10] L. D. Viet and P. T. T. My. Theoretical upper and lower bounds of the performance of an on-off damping dynamic vibration absorber attached to a multi-degree-of-freedom system. Journal of Vibration and Acoustics, 141, (3), (2019). [11] L. D. Viet and P. T. T. My. Lower bound of performance index of an on–off damper in a state-space system. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 233, (12), (2019), pp. 4288–4298. [12] L. D. Viet and N. T. Kien. Lower bound of performance index of anti-sway control of a pendu- lum using on-off damping radial spring-damper. Vietnam Journal of Mechanics, 41, (2), (2019), pp. 193–201.