A NONLINEAR ELECTROMECHANICAL PENDULUM ARM WITH A NONLINEAR ENERGY SINK CONTROL(NES) APPROACH

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1 JOURNAL OF THEORETICAL AND APPLIED MECHANICS 54, 3, pp , Warsaw 2016 DOI: /jtam-pl A NONLINEAR ELECTROMECHANICAL PENDULUM ARM WITH A NONLINEAR ENERGY SINK CONTROL(NES) APPROACH G. Füsun Alışverişçi, Hüseyin Bayıroğlu Yıldız Technical University, Istanbul, Turkey; afusun@yildiz.edu.tr; hbayir@yildiz.edu.tr Jorge Luis Palacios Felix UFFS Federal University of Fronteira Sul, Cerro Largo, RS, Brazil; jorge.felix@uffs.edu.br José M. Balthazar ITA Aeronautics Technological Institute, São José dos Campos, SP, Brazil and UNESP-Bauru, SP, Brazil jmbaltha@gmail.com Reyolando Manoel Lopes Rebello da Fonseca Brasil Federal University of ABC, Santo André, SP, Brazil; reyolando.brasil@ufabc.edu.br This paper considers the nonlinear dynamics of an electromechanical device with a pendulum armandanonlinearenergysink(nes)putonthepointofthependulumsuspension.itis shown that the(nes) is capable of absorbing energy from the system. The numerical results are shown in a bifurcation diagram, phase plane, Poincaré map and Lyapunov exponents. Keywords: nonlinear electromechanical system, bifurcations, nonlinear energy sink 1. Introduction Mechanical oscillatory systems(e.g., a pendulum system which is a basic subsystem of any robot) are of special interest for research and applications as examples of simple systems that may exhibit complex nonlinear behavior. That is why mechanical pendulum-like models of robot arms and mechanical manipulators have recently drawn attention of researchers(mogo and Woafo, 2007). An interesting example of a driven pendulum device coupled with an electric circuit through a magnetic field. This enters the class of nonlinear electromechanical devices with a pendulum arm. It has been found that the device displays different nonlinear behavior, including chaos(mogo and Woafo, 2007). SDRE control and sensibility analysis of a chaotic double pendulum arm excited by a RLC circuit based nonlinear shaker was presented by Tusset et al. (2014, 2015). A non-ideal electromechanical damping vibration absorber, the Sommerfeld effect and energy transfer were studied by Felix and Balthazar(2009). The energy pumping, synchronization and beat phenomenon in a non-ideal structure coupled with an essentially nonlinear oscillator were discussed by Felix et al.(2009). Thispaperdealswithathinrodmountedtoaplatetowhichelectricalwindingsareapplied. Connected to an electric circuit(tusset et al., 2014, 2015), its oscillations are due to the electromagnetic force resulting from two identical and repulsive permanent magnets(mogo and Woafo,2007)aswellasaNonlinearEnergySink.NEShasrecentlydrawnattentionofmany researchers. The NES method represents a new and unique application of strong nonlinearity. The nonlinear energy sink is a local, simple, lightweight subsystem capable of completely alteringtheglobalbehavioroftheprimarysystemtowhichitisattached(vakakisetal.,2008). Elimination of chaotic behavior in a non-ideal portal frame structural system using both passive and active controls were described by Tusset et al.(2013). Steady-state dynamics of a linear

2 976 G.F. Alışverişçi et al. structure weakly coupled to an essentially nonlinear oscillator was studied by Malatkar and Nayfeh(2007). Steady state passive nonlinear energy pumping in coupled oscillators was studied by Jiang et al.(2003). Introduction of passive nonlinear energy sinks to linear systems was discussed by Vakakis(2001). Energy pumping in nonlinear mechanical oscillators for resonance capture was examined by Vakakis and Gendelman(2001). The energy transfer between linear and non-linear oscillators was investigated by Dantas and Balthazar(2008). Fig.1.IdealizationofaNESappliedtoapendulumarm,u r andu θ arethepolarunitvectors Inthiswork,aNESdeviceisappliedtothefreeendofthependulum(Fig.1).Thisset-up isasystemwiththreedegreesoffreedom:(i)chargeqofthenonlinearcondenser,(ii)angular displacementθofthependulum,(iii)displacementζofthem nes Equation of the electric drive 2. Equations of motion The electric oscillator used to drive the pendulum is an RLC series circuit with a sinusoidal excitatione(t)=v 0 cosωt(v 0 andωbeing,respectively,theamplitudeandfrequency,and ttime).denotingtheforcedmeshcurrentiintherlccircuit,asshowninfig.1,applying Kirchhoff s rules, and taking into account the contribution of Lenz s electromotive voltage for Nturnsbyintegratingoverθfromzeroto2πN(e= 0.5NBσ 2 l 2 dθ/dt)oneobtains L di dt +Ri+V c(q) e=e(t) (2.1) whereldi/dt,ri,v C (q)arethevoltagesacrosstheselenoidoftheinductancel,theresistorr, and the nonlinear capacitor C, respectively. In this electromechanical model, the electrical nonlinear term is introduced by considering that the voltage of the capacitor is a nonlinear function of the instantaneous electrical charge q of the following form V c (q)= 1 C 0 q+a 3 q 3 (2.2) wherec 0 isthelinearvalueofcanda 3 isthenonlinearcoefficientdependingonthetypeof the capacitor used. Inserting Eq.(2.2) in Eq.(2.1), the electric part of the model is described by the following nonlinear differential equation d 2 q dt 2+R L dq dt +ω2 e q+a 3 L q3 + NBσ2 l 2 2L dθ dt =v 0 L cos(ωt) (2.3) whereω 2 e=1/(lc 0 )istheresonancefrequencyoftheelectricoscillator.

3 A nonlinear electromechanical pendulum arm Equation of the pendulum arm and the Nonlinear Energy Sink InFig.1,theforcesactingonthependulumarerepresented.Themechanismconsistsofa uniformthinrodoaofmassmandlengthl,havingaplatetowhichnconductingelectric windingsoflengthσleachareapplied,withσ=1/2.thependulumishingedatoabouta horizontalaxis,withonlytheportionσloftherodinthemagneticfield.thetotalmassofthe conducting wire and the plate bathing in the magnetic field have been neglected compared to thependulummass.themomentofinertiaoftheoutputisthenreducedto I O = 1 3 ml2 +M nes l 2 (2.4) wherem,m nes aremassofthependulumandnes,respectively.whenthecurrentiflows through the conducting wire in the magnetic field, there appear, according to the directions of the current(upward or downward), two identical Laplace forces(direction and intensity) whose resultant f sets the pendulum into motion in a viscous medium with the frictional coefficient β. Accordingtotheequationforkineticmoment,themomentofinertiaI O timestheangular acceleration equals the sum of torques due to forces applied to the pendulum. The Laplace force f=nbσldq/dtisappliedtothecenteroftheplategravityandfrictionforces.nesspringand frictionforcesareappliedtothefreeendofthependulum. Thus, the pendulum motion is described by d 2 θ l 2 dq I O dt 2=NBσ2 2 dt mgl dθ 2 sinθ βl2 4 dt C nesl (l dθ ) dt dζ K nes l(lθ ζ) n (2.5) dt wheren=1andn=3.itcanbedividedbythemomentofinertiatoobtainthisequationin the standard form d 2 θ NBσ 2 l 2 dq dt 2= ) 2( m 3 +M nes l 2 dt mgl βl 2 dθ sinθ ) 2( m 3 +M nes )l 4( 2 m 3 +M nes l 2 dt C nes l ( m 3 +M nes) l 2 (l dθ ) dt dζ dt K nes l ( m 3 +M nes) l 2 (lθ ζ) n d 2 θ dt 2+ω2 m sinθ+ β ) 4( dθ m 3 +M dt NBσ 2 ) nes 2( dq m 3 +M dt + C nes ( (l m nes 3 nes) dθ ) +M l dt dζ dt (2.6) + K nes ( (lθ ζ) m 3 nes) n =0 +M l whereω 2 m=mg/[2( m 3 +M nes)l]istheresonancefrequencyofthependulum,c nes isthedamping coefficientsofthenes,k nes isthespringcoefficientofthenes. According to Newton s second law, the NES motion is described by d 2 ζ ( dζ ) M nes dt 2= C nes dt ldθ K nes (ζ lθ) n (2.7) dt ItcanbedividedbymassoftheNEStoobtainthisequationinthestandardform d 2 ζ ( dt 2+C nes dζ ) M nes dt ldθ + K nes (ζ lθ) n =0 (2.8) dt M nes

4 978 G.F. Alışverişçi et al Nondimensional equations and values of the parameter Using the transformation x= q y= θ z= ζ Q 0 θ 0 lθ 0 dq dt =Q dx dθ 0ω e dτ dt =θ dy dζ 0ω e dτ dt =lθ dz 0ω e dτ d 2 q dt 2=Q 0ωe 2 d 2 x d 2 θ dτ 2 dt 2=θ 0ωe 2 d 2 y d 2 ζ dτ 2 dt 2=lθ 0ωe 2 d 2 z dτ 2 τ=ω e t (2.9) wherex,y,τaredimensionlessvariables,q 0 isthereferencechargeofthecondenser,andθ 0 is the reference pendulum angular displacement. The equations of motion of the complete system can be written as follows and d 2 x dτ 2+ R Lω e dx x 3 + NBσ2 l 2 θ 0 dy 2LQ 0 ω e dτ = v ( 0 Ω ) Q 0 ωel 2 cos τ ω e dτ +x+a 3Q 2 0 Lωe 2 d 2 y NBσ 2 dx dτ 2= ) Q 0 2( m 3 +M nes ω e θ 0 dτ mg β sin(θ 0 y) 2( m 3 +M nes )θ 0 ωe 4( 2l m 3 +M nes C ( nes dy ( ) m 3 nes) +M ω e dτ dz K nes(lθ 0 ) n 1 ( (y z) dτ m 3 nes) n +M ωe 2 d 2 z C ( dτ 2+ nes dz )+ M nes ω e dτ dy K nes(lθ 0 ) n 1 dτ M nes ωe 2 (z y) n =0 dy ) ω e dτ (2.10) x +µ 1 x +x+αx 3 +γ 1 y =Ecos(ωτ) y +µ 2 y +ω 2 2 sin(θ 0 y) γ 2x +c 2nes (y z )+k 2nes (y z) n =0 z +c 1nes (z y )+k 1nes (z y) n =0 (2.11) wheretheprimedenotesaderivativewithrespecttoτand µ 1 = R Lω e α= a 3Q 2 0 Lω 2 e ω= Ω ω e γ 1 = NBσ2 l 2 θ 0 2LQ 0 ω e µ 2 = E= v 0 LQ 0 ω 2 e β ) 4( m 3 +m nes ω e ω 2 2 = ω2 m ω 2 e θ 0 c 2nes = C nes ( m 3 +M nes) ω e NBσ 2 Q 0 γ 2 = ) 2( m 3 +m nes ω e θ 0 k 1nes = K nes(lθ 0 )n 1 M nes ω 2 e c 1nes = C nes m nes ω e k 2nes = ( Knes(lθ 0) n 1 m )ω 3 +Mnes e 2 Thephysicalparametersusedarethefollowing:C 0 =0.11F,a 3 =158VC 3,Q 0 =0.24C, R=0.97Ω,L=1.15H,B=0.02T,N =685,θ 0 =πrad,l=0.465m,g=9.81m/s 2, m = 1kg,M nes = 0.1kg,β = 0.49Ns/m,Ω = 5.6rad/s,σ = 0.5,K nes = 1.5N/m, C nes =0.3Ns/m. Thisgivesthefollowingvaluesfornon-dimensionalconstants:ω=2,ω 2 =1,µ 1 =0.30, µ 2 =0.1,α=1,γ 1 =1.5,γ 2 =0.1,c 1nes =1,c 2nes =5,k 1nes =2,k 2nes =8.

5 A nonlinear electromechanical pendulum arm Bifurcation structures and basin of chaoticity Theaimofthissubsectionistofindhowchaosarisesinourelectromechanicalmodelasthe parameters of the system evolve. For this purpose, we numerically solve equations of motion (2.11) and plot the resulting bifurcation diagrams, Lyapunov exponents, phase planes and PoincaremapsasE,ω,ω 2,α,µ 1,µ 2,γ 1,γ 2,c 1nes,c 2nes,k 1nes,k 2nes varies.figures2and3show the non-dimensional amplitude diagram for the pendulum arm as the other non-dimensional Fig. 2. Amplitude of the pendulum arm versus(a) excitation amplitude E,(b) excitation frequency ω; bleulinesn=1,blacklinesn=3,greylineswithoutnes Fig.3.Amplitudeofthependulumarmversus(a)functionofα,(b)functionofω 2,(c)functionofµ 1, (d)functionofµ 2,(e)functionofγ 1,(f)functionofγ 2 ;bleulinesn=1,blacklinesn=3, grey lines without NES

6 980 G.F. Alışverişçi et al. parameters(e,ω,ω 2,α,µ 1,µ 2,γ 1,γ 2 )vary.thedarkgreylinesareshownforn=1,theblack linesforn=3andthegreyonesforwithoutnes.theinvestigationofthesefiguresshowsthat thependulumarmwithnesandn=3diminishesthechaoticeffectandamplitude. Figure 4 shows the non-dimensional amplitude diagram for the pandulum arm as the other non-dimensionalparameters(c 1nes,c 2nes,k 1nes,k 2nes )vary.thedarkgreylinesareforn=1,the blacklinesforn=3.itisclearthatforn=3thechaoticeffectandamplitudearediminished. Fig.4.Amplitudeofthependulumarmversus(a)functionofc 1nes,(b)functionofc 2nes, (c)functionofk 1nes,(d)functionofk 2nes ;darkgreylinesn=1,blacklinesn=3 Table1showsthestabilityconditionasafunctionofE,ω,ω 2,α,γ 2,k 1nes,k 2nes,c 1nes,c 2nes. It has been constructed with Figs. 5-8 and Figs According to this Table, the pendulum armwithnesandn=3diminishesthechaoticeffectandamplitude.figure9showsthatonly forµ1<0.01thesystemexhibitschaoticeffectwithoutnes. Fig.5.BifurcationdiagramsandLyapunovexponentsofthependulumarmasfunctionsofEinthe ydirection,(a)nes,n=1,(b)nesn=3,(c)withoutnes.theotherparametersaree=30,ω=2, µ 1 =0.3,µ 2 =0.1,θ 0 =π,γ 1 =1.5,γ 2 =0.1,c 1nes =1,c 2nes =5,k 1nes =2,k 2nes =8; dark grey line bifurcation diagram, grey line Lyapunov exponent

7 A nonlinear electromechanical pendulum arm Fig.6.BifurcationdiagramsandLyapunovexponentsofthependulumarmasfunctionsofωinthe ydirection,(a)nes,n=1,(b)nesn=3,(c)withoutnes.theotherparametersaree=30, ω 2 =1,µ 1 =0.3,µ 2 =0.1,θ 0 =π,γ 1 =1.5,γ 2 =0.1,c 1nes =1,c 2nes =5,k 1nes =2,k 2nes =8; dark grey line bifurcation diagram, grey line Lyapunov exponent Fig.7.BifurcationdiagramsandLyapunovexponentsofthependulumarmasfunctionsofω 2 inthe ydirection,(a)nes,n=1,(b)nesn=3,(c)withoutnes.theotherparametersaree=30,ω=2, µ 1 =0.3,µ 2 =0.1,θ 0 =π,γ 1 =1.5,γ 2 =0.1,c 1nes =1,c 2nes =5,k 1nes =2,k 2nes =8; dark grey line bifurcation diagram, grey line Lyapunov exponent Fig.8.BifurcationdiagramsandLyapunovexponentsofthependulumarmasfunctionsofαinthe ydirection,(a)nes,n=1,(b)nesn=3,(c)withoutnes.theotherparametersaree=30,ω=2, µ 1 =0.3,µ 2 =0.1,θ 0 =π,γ 1 =1.5,γ 2 =0.1,c 1nes =1,c 2nes =5,k 1nes =2,k 2nes =8; dark grey line bifurcation diagram, grey line Lyapunov exponent

8 982 G.F. Alışverişçi et al. Fig.9.BifurcationdiagramsandLyapunovexponentsofthependulumarmasfunctionsofµ 1 inthe ydirection,(a)nes,n=1,(b)nesn=3,(c)withoutnes.theotherparametersaree=30,ω=2, α=1,µ 2 =0.1,θ 0 =π,γ 1 =1.5,γ 2 =0.1,c 1nes =1,c 2nes =5,k 1nes =2,k 2nes =8; dark grey line bifurcation diagram, grey line Lyapunov exponent Fig.10.BifurcationdiagramsandLyapunovexponentsofthependulumarmasfunctionsofγ 2 inthe ydirection,(a)nes,n=1,(b)nesn=3,(c)withoutnes.theotherparametersaree=30,ω=2, µ 1 =0.3,µ 2 =0.1,θ 0 =π,α=1,γ 1 =1.5,c 1nes =1,c 2nes =5,k 1nes =2,k 2nes =8; dark grey line bifurcation diagram, grey line Lyapunov exponent Fig.11.BifurcationdiagramsandLyapunovexponentsofthependulumarmasfunctionsofk 1nes with NES,(a)n=1,(b)n=3.TheotherparametersareE=30,ω=2,µ 1 =0.3,µ 2 =0.1,θ 0 =π,α=1, γ 2 =0.1,c 1nes =1,c 2nes =5,k 2nes =8;darkgreyline bifurcationdiagram, grey line Lyapunov exponent

9 A nonlinear electromechanical pendulum arm Fig.12.BifurcationdiagramsandLyapunovexponentsofthependulumarmasfunctionsofk 2nes with NES,(a)n=1,(b)n=3..TheotherparametersareE=30,ω=2,µ 1 =0.3,µ 2 =0.1,θ 0 =π,α=1, γ 2 =0.1,c 1nes =1,c 2nes =5,k 1nes =2;darkgreyline bifurcationdiagram, grey line Lyapunov exponent Fig.13.BifurcationdiagramsandLyapunovexponentsofthependulumarmasfunctionsofc 1nes with NES,(a)n=1,(b)n=3.TheotherparametersareE=30,ω=2,µ 1 =0.3,µ 2 =0.1,θ 0 =π,α=1, γ 2 =0.1,c 1nes =1,c 2nes =5,k 1nes =2,k 2nes =8;darkgreyline bifurcationdiagram, grey line Lyapunov exponent Fig.14.BifurcationdiagramsandLyapunovexponentsofthependulumarmasfunctionsofc 2nes with NES,(a)n=1,(b)n=3.TheotherparametersareE=30,ω=2,µ 1 =0.3,µ 2 =0.1,θ 0 =π,α=1, γ 2 =0.1,c 1nes =1,k 1nes =2,k 2nes =8;darkgreyline bifurcationdiagram, grey line Lyapunov exponent

10 984 G.F. Alışverişçi et al. Table1.StabilityconditionsasafunctionofE,ω,ω 2,α,γ 2,k 1nes,k 2nes,c 1nes,c 2nes Periodic Quasi periodic Chaotic Fig.5a,n=1, E<45, 45<E<57, 57<E<73, variouse E>93 73<E<87 87<E<93 Fig.5b,n=3, E<20, 20<E<28, 58<E<72, variouse 93<E, 45<E<58, 86<E<93 28<E<45 72<E<86 Fig.5c,without 5<E<20, E=32, E<5, NES,variousE E>95 48<E<60 60<E<73, 85<E<95 Fig.6a,n=1, ω<0.7, 1.5<ω<1.8, 0.7<ω<0.8, variousω ω> <ω< <ω<1.45 Fig.6b,n=3, ω<1.45, 1.5<ω< <ω<1.5, variousω ω>2.3 ω=1.7 Fig.6c,without ω<0.4, 0.5<ω<0.6, 1.2<ω<1.5, NES,variousω ω> <ω< <ω<1.7 Fig.7a,n=1, 0>ω 2 <5 variousω 2 Fig.7b,n=3, 0>ω 2 <5 variousω 2 Fig.7c,without ω 2 <1.3, 1.3<ω 2 <1.6, NES,variousω 2 ω 2 >2.55, 2.5<ω 2 < <ω 2 <2.5 Fig.8a,n=1, α< <α<3.7 α>3.7 various α Fig.8b,n=3, α< <α<3.7, 3.7<α<4.4, variousα 4.4<α<4.6 α>4.6 Fig.8c,without 0.2<α<0.4, 1.2<α<1.4, 0<α<0.2, NES,variousα 0.5<α<1, 2.2<α< <α<0.5, 1.4<α< <α< <α<3.6, 3.8<α<4.4, 4.6<α<5 Fig.10a,n=1, 0<γ 2 <2 variousγ 2 Fig.10b,n=3, 0<γ 2 <0.8, 0.8<γ 2 <1.3 variousγ <γ 2 <2 Fig.10c,without 0<γ 2 <0.15, 1.9<γ 2 < <γ 2 <1.1, NES,variousγ <γ 2 < <γ 2 <1.9 Fig.11a,n=1, 0<k 1nes <2.1 3<k 1nes <3.2, 2.1<k 1nes <3, variousk 1nes 4.8<k 1nes <5, 3.1<k 1nes <5, 6.4<k 1nes <10 5<k 1nes <6.4 Fig.11b,n=3, 6.4<k 1nes <10 0<k 1nes <6.4 variousk 1nes Fig.12a,n=1, 7<k 2nes <15 0<k 2nes <7 variousk 2nes Fig.12b,n=3, 0<k 2nes <15 variousk 2nes

11 A nonlinear electromechanical pendulum arm Fig.13a,n=1, 0.2<c 1nes <0.4, 0<c 1nes < <c 1nes <0.7 variousc 1nes 0.7<c 1nes <2 Fig.13b,n=3, 0<c 1nes <2 variousc 1nes Fig.14a,n=1, 0<c 2nes <2 2<c 2nes <20 variousc 2nes Fig.14b,n=3, 0<c 2nes <2 2<c 2nes <20 variousc 2nes 3. Conclusion WehavestudiedtheeffectoftheNESonaelectro-mechanicaldevicewithapendulum. The system exhibits complex dynamical behavior such as multi-periodic, quasi-periodic and chaotic responses, and these are strongly dependent on non-dimensional control parameters E, ω andanonlinearitycoefficientαforrlccircuit,andnesparametersc 1nes,c 2nes,k 1nes,k 2nes. Moreover,thesystemwithoutNESexhibitsachaoticresponsedependingonγ 2.TheNES parameters(c 1nes,c 2nes,k 1nes,k 2nes )forn=1leadtochaoticresponses,butforn=3,they induce periodic and quasi-periodic responses. It is shown that the NES is capable of absorbing energy from the system and decreases the amplitude as well as diminishes the chaotic effect. References 1. Dantas M.J.H., Balthazar J.M., 2008, On energy transfer between linear and non-linear oscillator, Journal of Sound and Vibration, 315, 4/5, Felix J.L.P., Balthazar J.M., 2009, Comments on a nonlinear and a non-ideal electromechanical damping vibration absorber, sommerfeld effect and energy transfer, Nonlinear Dynamics, 55, 1, Felix J.L.P., Balthazar J.M., Dantas M.J.H., 2009, On energy pumping, synchronization and beat phenomenon in a non-ideal structure coupled to an essentially nonlinear oscillator, Nonlinear Dynamics, 56, 1, Jiang X., McFarland D.M., Bergman L.A., Vakakis A.F., 2003, Steady state passive nonlinear energy pumping in coupled oscillators: theoretical and experimental results, Nonlinear Dynamics, 33, Malatkar P., Nayfeh A.H., 2007, Steady-state dynamics of a linear structure weakly coupled to an essentially nonlinear oscillator, Nonlinear Dynamics, 47, Mogo J.B., Woafo P., 2007, Dynamics of a nonlinear electromechanical device with a pendulum arm, Journal of Computational and Nonlinear Dynamics, 2, 4, , DOI: / Tusset A.M., Balthazar J.M., Felix J.L.P., 2013, On elimination of chaotic behavior in a non-ideal portal frame structural system, using both passive and active controls, Journal of Vibration and Control, 19, 6, Tusset A.M., Bueno A.M., Sado D., Balthazar J.M., Colón D., 2014, SDRE control and sensibility analysis of a chaotic double pendulum arm excited by a RLC circuit based nonlinear shaker, Conference on Dynamical Systems Theory and Applications, Łódź, Poland, Tusset A.M., Picirillo V., Bueno A.M., Sado D., Balthazar J.M., Felix J.L.P., 2015, Chaos control and sensitivity analysis of a double pendulum arm excited by an RLC circuit based nonlinear shaker, Journal of Vibration and Control,

12 986 G.F. Alışverişçi et al. 10. Vakakis A.F., 2001, Inducing passive nonlinear energy sinks in linear vibrating systems, Journal of Vibration and Acoustics, 123, 3, Vakakis F., Gendelman O., 2001, Energy pumping in nonlinear mechanical oscillators. II: Resonance capture, Journal of Applied Mechanics, 68, 1, Vakakis A.F., Gendelman O.V., Bergman L.A., McFarland D.M., Kerschen G., Lee Y.S., 2008, Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems(Solid Mechanics and Its Applications), Springer Manuscript received January 13, 2015; accepted for print January 8, 2016