Amplification of useful vibration using on - off damping

 Vietnam Journal of Mechanics, VAST, Vol.42, No. 1 (2020), pp. 87 – 94
 DOI: https://doi.org/10.15625/0866-7136/14851
 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
 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) 
 mx¨ +((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) 
 x¨ + 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) 
 Zòòee( ) ( 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  ... wise
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-»r = 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: 
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˙) = x¨ + 2ζmωn (x˙ − r˙) + ωn (x − r) (12)
which changes (11) to
 2π/Ω
 −mΩ Z  
 P = x¨ (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 =   ×
 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)
 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¯) =   ×
 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
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 æö22
 m(Dζt − ζs) − ζm 22zzms2--- z t2 z s zz tm
 Pt0 ( ) = ç÷zztms-- z ζ − ζ − 2ζtζm ≥ 0 (21) 
 ¯  a =1  22ç÷2 t zz22-s
 ωnP0 (t)  4wznt( -4 z( sζ) − ζ ) ts
 = tèøs (22)
 2  ( + ) −
 m∆  = ζt ζs ζm 2 2
 After some manipulations,α 1 we simplify (21) as: ζ − ζ − 2ζtζm < 0
  ( + )2 t s
 ì(zz4 --ζt ) ζ zs
 ts m zz22--20 zz ³
 It is noted that for any u we haveï 2 ts tm
 wn Pt0 ( ) ï 4(zzts- )
 2 = í (22) 
 mD (zzts+− ) - z m
 a =1 ï u ζm 1 zz22--20 zz<
 2 ≤ .ts tm (23)
 ï 4 zz+2
 î ( tsu ) 4ζm
 The equalityIt is noted that holds for any when u we have 
 uu -=z 2ζ 1. (24)
 m £ 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.
 w Pt( ) 1
 n 0 £ (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 æö22
 mD 22zzms--- z t z s zz tm
 lim Pt== ç÷zz-- z (26) 
and (18). From (10), when0 (ζl) 0 and22ζh tendstms to infinity, two22 terms ζt and ζs also tend to
 z h ®¥ ç÷zz-
 z =0 4wznt( - z s) ts
infinity. The expressionl (18) reduces to èø
 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 £ ζ − ζ (27) 
 n t s z ®¥ 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,
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”.
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