Impact Vibration Absorber of Pendulum Type

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