Impact Vibration Absorber of Pendulum Type

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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
Page 1: 7th International DAAAM Baltic Conf
Page 5 and 6: The absorber with parameters ω=1.0

Impact Vibration Absorber of Pendulum Type

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