Impact Vibration Absorber of Pendulum Type

7th International DAAAM Baltic Conference"INDUSTRIAL ENGINEERING22-24 April 2010, Tallinn, EstoniaIMPACT VIBRATION ABSORBER OF PENDULUM TYPEPolukoshko, S.; Boyko A.; Kononova, O,; Sokolova, S. & Jevstignejev, V. Abstract: In this work the impactvibration absorber of pendulum type isexamined. It consists of pendulum withmotion limiting stops attached to thevibrating system. Pendulum vibrationabsorbers are widely used in practice. Theinfluence of pendulum parameters on thepossibility of suppression of vibrations ofthe basic system under harmonic excitationis discussed in this study.Key words: vibration, absorber, amplitude,pendulum, frequency.1. INTRODUCTIONVibration is a repetitive, periodic oroscillatory response of mechanical system.Since most of machines and structuresundergo some degree of vibrations,engineers have to consider the results ofvibrations in the designing process [ 4 ], [ 8 ].It is usually required to control thevibrations because it causes fatigue andfailure of the vibrating elements anddiscomfort for the people. One of the mosteffective passive control methods is addingan impact vibration absorber (IVA) to thesystem under excitation [ 1 ], [ 2 ], [ 5 ]. IVAconsists of impact mass which is placed onbasic vibrating mass so, that periodicallycollides with it. The transfer of momentumto the impact mass from the main mass anddissipation of energy in every impactprovide reduction in amplitude response ofthe main mass. IVA are fulfilled with one,two and more degrees of freedom;noncontrollable and regulated; withunilateral or with bilateral constraints. Inaccordance with structural type impactvibration absorbers may be spring (Fig.1),floating (Fig.2) and pendular (Fig.3).a) outer b) innerFig.1.a-b. Spring impact absorbersa) single –unit b) multi –unitFig.2.a-b. Floating impact absorbersFig.3. Pendulum impact absorbersIn this work the impact absorber ofpendulum type is examined. It consists ofpendulum with motion limiting stopsattached to the vibrating system.Pendulum vibration absorbers are used inpractice for decreasing of vibration level ofdifferent engineering structures: flue pipes,television towers, bridges, high-risebuildings, aerial masts, for shaftautobalancing and others [ 3 ], [ 6 ], [ 8 ].The purpose of this research is to study theinfluence of parameters of the pendulum

on possibility of vibration suppression ofthe basic system under harmonicexcitation, and the effect of the systemparameters on system dynamics. Thisinvolved determination the effect of massratio, excitation amplitude, and clearancebetween impact stop walls. A pendulumwith one and two impacts during the periodis considered. Dependence of suppressionability of absorber on pendulum length,coefficient of restitution at impact, massratio of the basic system and pendulum,and gap size are found.2. ANALITICAL MODEL OFABSORBER2.1 Mathematical modelIn Fig.4 the models of single and doubleimpact pendulum absorbers are presented.a) single-impact absorber modelb) double impacts absorber modelFig.4. Model of the pendulum impact absorberParameters of system:m 1 – mass of the main body;m 2 - mass of the damper;μ= m 2 /m 1 - mass ratio;d – inherent damping coefficient of themain system;k – stiffness coefficient of main system;r - coefficient of restitution of the velocityafter impact;λ - natural frequency of the main system;ω – the frequency of pendulum;l - length of pendulum;c – size of gap;α - max angle for two-impacts absorber,tan α = c / l ;The system is considered under harmonicexcitation:P( t)= P0 sin Ωt,P 0 - amplitude of excitation force;Ω – frequency of excitation.2.2 The equations of motion of thesystemUsing Lagrange’s equations the equation ofmotion of the examined system is derived:⎧(m1+ m2) x + m2l ϕ cosϕ= −kx− dx+⎪∞⎪ 2+ m2l ϕ sinϕ+ P0sin Ωt− S∑δ( t − RT );⎪R=0⎨ 2⎪m2l ϕ + m 2xl cosϕ= −m2gl sinϕ+⎪ ∞⎪+S −∑δ( t RT )(1)⎩ R=0where:S - impact impulse,δ ( t − RT ) − delta function,T – period of collisions.The stereomechanical theory of impact isused for impact impulse definition [ 7 ]:m1m2S = ( 1+r)( v01− v02) (2)m1+ m2wherev 01 and v 02 - velocity of main body andvelocity of impactor just before impact.The velocity of impactor consists oftranslational velocity and relative velocity:v01 − v02= v01− ( v01+ v2r ) = −lϕ01,(3)here the angular velocity is pendulumvelocity just before impact, ϕ01= ϕ(T ) .Taking into account (3) impact impulsemay be represented as:m2 S = (1 + r)( −l ϕ). (4)1+µAfter rearrangement of equation of system(1) and taking into account (4) system (1)may be written:

2⎧ b λ µ 2⎪x + x+ x = l ϕ sinϕ−⎪1+µ 1+µ 1+µ⎪ µp0⎪−l ϕ cosϕ+ sin Ωt+⎪1+µ 1+µ⎪∞(1 + r)µ⎨+l ϕ(T ) ∑δ( t − RT );2⎪ (1 + µ )R=0⎪2 x⎪ϕ + ω sinϕ= − cosϕ−⎪l∞⎪ 1+r(5)⎪− ϕ(T ) ∑δ( t − RT )⎩1+µ R=0where:kλ = ,m 1db = ,m 1P0p0= ,m1gω = .l2.3 Analytical solution of the simplifiedequations of motionFor the simplified variant of equations ofmotion of the system – without taking intoaccount dissipation and inertias forces:∞⎧ k P0S⎪x + x = sin Ωt− ∑δ( t − RT)m m m R=0⎨(6)∞⎪ 2 x S ϕ + ω ϕ = − + −⎪∑δ( t RT ),⎩l m2lR=0where m=m 1 +m 2 , with help of method offitting an analytical solution is found [ 6 ].Purely forced vibrations of the system andabsorber for a time domain (0, Т) betweenimpacts under conditions of tuning:ΩT=2π; 2ω= Ωfor resonance condition λ/(1+μ)= Ω:~ p 3( )0 ⎛ λtx t = cos2 ⋅ ⎜ −+2λ⎝ 2 1+µ⎛ 3 (1 2 )(1 )⎞ ⎞⎜π + µ − r λt− + ⎟λtπ sin ⎟,4 (1 ) 11+⎝ µ + r+ µ ⎠ µ ⎠~ 2 p ⎛0 (3 2 )( ) ⎜π − µϕ t = ⋅ sin23lλ⎝ 2µ2⎛⎜ 3π(1 + 2µ)(1 − r)−π+⎜⎝4µ(1 + r)λt−1+µλt⎞⎟sin1+µ⎠where the notations are as agreed above.(7)λt⎞⎟,1+µ ⎟⎠2.4 Numerical solution of equations ofmotionIn this work the numerical solution ofsystem (5) was obtained with help of Eulermethod using the kinematics conditions –pre-impact and post- impact velocities ofmoving bodies if coefficient of restitutionis known. The velocity of the main body v1and velocity of impactor v 2 just afterimpact are:µ (1 + r)v1= v01+ l ϕ01. (8)1+µµ − rv2= v01+ l ϕ01. (9)1+µAlgorithm of Euler’s method for the sinleimpactdamper, taking into account (8),(9):tn+1 = tn+ ∆t,xnxnx+ 1=+n∆tϕn( ϕ + 1=n+ ϕn∆t)if ( ϕn≥ 0,0,1)µ (1 + r)xnxnxnt l + 1=+ ∆ + ϕnif ( ϕn≤ 0,1,0)1+µµ − r ϕ nϕ nϕnt + 1= + ∆ + ϕnif ( ϕn≤ 0,1,0)1+µ2b λ p0 xn1x+= −n− xn+ sin Ωtn1+µ 1+µ 1+µµ 2 µ+ l ϕnsinϕn− l ϕncosϕn1+µ1+µ2 xn ϕ= − ϕ sinϕ− cosϕn+1nlEuler method gives good results if timeinterval Δt is small. The equations ofmotion are solved numerically with help ofMatcad program. The received resultsenable to analyze all parameters of motionof the system.Examples of the solution of motion arepresented below for single and two-impactabsorbers.3. NUMERICAL EXAMPLEFor the numeral solution next value ofparameters are accepted λ=1.5, b = 0.1,p 0 = 0.5. Parameters values are chosen forcivil engineering conditions. The structureis modeled as single-degree of freedomn

system, after adding the pendulumabsorber it becomes two freedom degrees,the exiting force is harmonic.Parameters of motion of single-impactabsorber are presented in Fig. 5 a-e, multiimpactabsorbers - in Fig. 6 a-e, 7a-e.Plots in Fig.5-7: a) x=x(t) - displacement ofmass m 1 , b) φ=φ(t) - rotation angle ofpendulum, c) v 2 r=v 2 r (t) - relative velocityof mass m 2 , d) v=v(t) - velocity of massm 1 , as functions of time t, e) v=v(x) -velocity of mass m 1 as function of massdisplacement x.a) x=x(t)a) x=x(t)b) φ=φ(t)b) φ=φ(t)c) v 2 r=v 2 r (t)c) v 2 =v 2 r (t)d) v=v(t)d) v=v(t)e) v=v(x)Fig.5.a-e. Plots of dependence of motionparameters on time and phase map for oneimpactabsorber in case of: Ω=1.5, λ=1.5,ω=0,75, μ=0.04, e=0.6e) v=v(x)Fig.6.a-e. Plots of dependence of motionparameters on time and phase map fordouble-impact absorber in case of: Ω=1.5,λ=1.5, ω=0.75, α=0.1, μ=0.04, e=0.6

The absorber with parameters ω=1.0α=0.1 appears multi-impacts – it showsfour impacts during period (Fig.7).a) x=x(t)b) φ = φ(t)Fig.8. Maximal amplitude Amax in relation toexiting force frequency Ω for different multiimpactspendulum frequencies and α, μ=0.04,r =0.6c) v 2 r = v 2 r (t)d) v=v(t)Fig.9. Maximal amplitude Amax in relation toexiting force frequencies Ω for single impactpendulum absorber (μ=0.04, r=0.6)e) v=v(x)Fig.7.a-e. Plots of dependence of motionparameters on time and phase map for thecase of Ω=1.5, λ=1.5, ω=1.0, μ=0.04,α=0.1 e=0.6Plots of maximal amplitude A max of mainbody in relation to exiting force frequencyΩ for different pendulum frequences ω arepresented in Fig.8-9, plots of A maxdeprnding on mass ratio μ – in Fig.10,depending on angle α–in Fig.11.Fig.10. Maximal amplitude Amaxdepending on μ –ratio for multi-impactabsorber (ω=0.75, r=0.6, α=0.05)

5. REFERENCESFig.11. Maximal amplitude Amaxdepending on pendulum clearance angle α(ω=0.75, μ=0.04, r=0.6)4. CONCLUSIONThe differential equations of motion of thevibrating system are derived on the basis ofLagrange’s equation of the second type.The impacts in the system are described asimpacts of perfectly rigid bodies takinginto account the coefficient of restitution.The equations of motion are solvednumerically with help of Matcad program,using Euler’s method. Numerical solutionallows calculating not only the parametersof motion in the steady-state mode, butalso in a transitional process. Allparameters of transient motion and steadystatemotion were defined, results wereanalyzed. Dependences of amplitude ofvibrations are shown graphically oncorrelation of the masses, maximal ofpendulum Amplitude in the graphs isshown maximal, instead of amplitude ofthe set motion.For a one-impact absorber, adjusted onresonance frequency, attenuation ability isgreater, but velocity of collisions is great,that can result in the damage of material.In future it is necessary to take into accountresilient properties of impact contacts usingthe dynamics conditions – to add thecontact forces in impact contact point.1.Cheng Jianlian, Hui Xu. Inner massimpact damper for attenuating structurevibration. International Journal of Solidsand Structures. 2006, 43, Issue 17, 5355-5369, www.sciencedirect.com2.Cheng C.C., Wang J.Y. Free vibrationanalysis of a resilient impact damper.International Journal of MechanicalSciences. 2003, 45,589– 604.3.Chua Sy G., Pacheco B.M., Fujuino Y.and Ito M. Classical impact damper andpendulum impact damper for potentialcivil engineering application. StructuralEng./Earthquake Eng.1990,7, No1,110-112.4.Clarence W. de Silva. Vibration.Fundamental and Practice. CRC PressLLC, New-York, 2000.5. Ema S., Marui E. A fundamental studyon impact dampers, InternationalJournal of Machine Tools andManufacturers . 1996, 36 , 93–306.6. Korenev B.G., Reznikov L.M. Dynamicvibration absorber. Nauka, Moscow,1988. (in Russian)7. Viba J. Optimisation and syntesis ofvibro-impact mashines. Zinatne, Riga1988. (in Russian)8. Vibrations in engineering: V.6.Protection from vibrations and impacts.Frolov, K.F., editor, Mashinostroenie,Moscow, 1995. (in Russian)6. CORRESPONDING AUTHORSvetlana Polukoshko, Dr. sc .ing., leadingresearcher, Engineering Research CenterVentspils University CollegeAddress: 101 Inzenieru street, Ventspils,LV-3601, Latvia,Phone: +371-29294968E-mail: pol.svet@inbox.lv