XI International Conference on the Theory of Machines and Mechanisms TMM 2012, September 4-6, 2012, Liberec, Czech Republic Nonlinear antiresonance vibrating screen Authors: Valeriy Belovodskiy <e-mail: belovodskiy@cs.dgtu.donetsk.ua>, Sergey Bukin <e-mail: s.bukin08@gmail.com>, Maksym Sukhorukov <e-mail: max.sukhorukov@gmail.com> Ukraine, Donetsk National Technical University, Computer Monitoring Systems department, Mineral Processing department 1
The issue of research Biharmonic vibrations are in demand in different technological processes: – transportation, – screening, – compacting and so on [1, 2]. Examples of such machines are shown in the figure. 02 2 XI International Conference on the Theory of Machines and Mechanisms TMM 2012, September 4-6, 2012, Liberec, Czech Republic
The issue of research 03 3 XI International Conference on the Theory of Machines and Mechanisms TMM 2012, September 4-6, 2012, Liberec, Czech Republic
The issue of research There are known vibrating machines which use the idea of antiresonance for decreasing dynamical forces on foundation [3]. Examples of such machines are shown in the figure. 04 4 XI International Conference on the Theory of Machines and Mechanisms TMM 2012, September 4-6, 2012, Liberec, Czech Republic
The issue of research 05 5 XI International Conference on the Theory of Machines and Mechanisms TMM 2012, September 4-6, 2012, Liberec, Czech Republic
The issue of research 06 6 XI International Conference on the Theory of Machines and Mechanisms TMM 2012, September 4-6, 2012, Liberec, Czech Republic
The issue of research One of the purposes of this work is the investigation of principal possibility of combining these properties in the nonlinear vibrating machine with harmonic excitation at the expense of realization of combination resonances. 07 7 XI International Conference on the Theory of Machines and Mechanisms TMM 2012, September 4-6, 2012, Liberec, Czech Republic
The model under consideration Here it is considered the vibrating two-masses screen with ideal harmonic inertial excitation and polynomial characteristic of the main elastic ties. 08 8 XI International Conference on the Theory of Machines and Mechanisms TMM 2012, September 4-6, 2012, Liberec, Czech Republic
The model under consideration The equations of its motion may be derived with use of Lagrange equations. In non-dimensional form they have a view: { 2 ξ 1 d ξ 1 d d ξ d τ + b 12 ξ d ξ 2 d ξ 2 + k 13 ξ 3 = P 1 cos η τ , 2 + b 10 d τ + b 11 d τ + b 13 ξ d τ + k 10 ξ 1 + k 11 ξ + k 12 ξ d τ d ξ 1 2 ξ d d τ + b 22 ξ d ξ d ξ 2 d ξ 2 + k 23 ξ 3 = P 2 cos η τ , 2 + b 20 + b 21 d τ + b 23 ξ d τ + k 20 ξ 1 + k 21 ξ + k 22 ξ d τ d τ ' Δ ' Δ ' + m 2 ) k 1 2 ' ' where b 10 = μ k 0 , b 11 =− μ k 1 , b 12 =− μ k 2 , b 13 =− μ k 3 , b 20 =− μ k 0 , b 21 = μ ( m 1 , ' ω 1 ' ω 1 ' ω 1 ' ω 1 ' ω 1 ' m 2 ω 1 m 1 m 1 m 1 m 1 m 1 m 1 ' Δ ' Δ ' + m 2 ) k 2 ' + m 2 ) k 3 2 2 b 22 = μ ( m 1 , b 23 = μ ( m 1 2 , k 12 =− k 2 Δ 2 , k 13 =− k 3 Δ (1) k 0 k 1 , k 10 = 2 , k 11 =− 2 , ' m 2 ω 1 ' m 2 ω 1 ' ω 1 ' ω 1 ' ω 1 ' ω 1 m 1 m 1 m 1 m 1 m 1 m 1 ' + m 2 ) ' + m 2 ) Δ ' + m 2 ) Δ 2 k 0 2 , k 21 = k 1 ( m 1 2 , k 22 = k 2 ( m 1 , k 23 = k 3 ( m 1 , P 1 = m 0 r 2 , P 2 =− m 0 r 2 , η η k 20 =− ' ω 1 ' Δ ' Δ ' m 2 ω 1 ' m 2 ω 1 ' m 2 ω 1 2 2 m 1 m 1 m 1 m 1 m 1 m 1 ξ 1 = x 1 / Δ , ξ = x / Δ , x = x 2 − x 1 , x 1 – displacement of a frame, x 2 – displacement of a working organ, ' = m 0 + m 1 , m 0 – unbalanced mass, m 1 – mass of a frame, m 2 – mass of a screen box, − 3 m, m 1 Δ = 10 ' , k 2 ' , k 3 ' , – of dissipation, k 0 – stiffness of isolators, k 1 , k 2 , k 3 – parameters of elastic ties and k 1 r – eccentricity of an exciter, μ – coefficient of dissipation, η = ω / ω 1 , ω – frequency of an vibroexciter, ω 1 – the first natural frequency of a vibromachine, τ = ω 1 t . 09 9 XI International Conference on the Theory of Machines and Mechanisms TMM 2012, September 4-6, 2012, Liberec, Czech Republic
The model under consideration The parameters of the experimental model are m 1 = 700 kg , m 2 = 550 kg , k 0 = 0.12·10 6 N/m , k 1 = 5.5·10 6 N/m , m 0 = 50 kg , r = 0.088 m , μ = 0.0008 s and the working frequency ω = 100 rad/s . The cases of linear ( k' 2 = 0, k' 3 = 0) and nonlinear ( k' 2 = k 2 , k' 3 = k 3 ) dissipation are considered. 10 10 XI International Conference on the Theory of Machines and Mechanisms TMM 2012, September 4-6, 2012, Liberec, Czech Republic
Research methods Analysis of the model is performed with the help of original software worked out as a tool of program MATLAB. Searching of the bifurcation diagrams is based on the harmonic balance method. Stationary solutions of the system are found in the form of finite Fourier expansions N N ( 1 ) e in ητ , ξ ( τ )= ∑ c n e in ητ , ξ 1 ( τ )= ∑ c n n =− N n =− N where N is a number of harmonics taken into consideration. 11 11 XI International Conference on the Theory of Machines and Mechanisms TMM 2012, September 4-6, 2012, Liberec, Czech Republic
Research methods After substituting it into the differential equations and equating the coefficients of equal powers the polynomial system for determination of expansion coefficients is produced { N 2 n ( 1 ) + ( k 11 + b 11 i η n ) c n + ∑ 2 ) c n ( k 10 + b 10 i η n − η c j c n − j ( k 12 + b 12 i η ( n − j ))+ j =− N c j c m c n − j − m ( k 13 + b 13 i η ( n − j − m ))= { N N P 1 / 2, n =± 1 + ∑ ∑ , 0, n ≠± 1 j =− N m =− N N 2 n 2 ) c n + ∑ ( 1 ) + ( k 21 + b 21 i η n − η ( k 20 + b 20 i η n ) c n c j c n − j ( k 22 + b 22 i η ( n − j ))+ j =− N c j c m c n − j − m ( k 23 + b 23 i η ( n − j − m ))= { N N P 2 / 2, n =± 1 + ∑ ∑ , 0, n ≠± 1 j =− N m =− N where n , n − j ,n − j − m ∈[− N , N ] . 12 12 XI International Conference on the Theory of Machines and Mechanisms TMM 2012, September 4-6, 2012, Liberec, Czech Republic
Research methods Changing one of the parameters of the model and solving algebraic system of equations in step by step one can get the bifurcation curves. The construction of basins of attraction of periodic regimes is based on the scanning of the domain of initial conditions and implementation of Demidenko-Matveeva method [4]. 13 13 XI International Conference on the Theory of Machines and Mechanisms TMM 2012, September 4-6, 2012, Liberec, Czech Republic
Results Here are some amplitude- and phase frequency characteristics (AFC and PFC) for the different values of non-linearity of the system k 13 / k 11 = 0, 0.0001, 0.0002 and N = 5 in the finite Fourier expansions. One may mention that introduction of non-linearity causes usual nonlinear phenomena, – slope of AFC and appearance of zone of ambiguity (Fig. 1). 14 14 XI International Conference on the Theory of Machines and Mechanisms TMM 2012, September 4-6, 2012, Liberec, Czech Republic
Results Fig. 1 A k – are the amplitude of k -th harmonic 15 15 XI International Conference on the Theory of Machines and Mechanisms TMM 2012, September 4-6, 2012, Liberec, Czech Republic
Results In the zone between natural frequencies ω 1 and ω 2 the different combination resonances are possible ( p ω ≈ | p 1 | ω 1 + | p 2 | ω 2 [5]), where p, p 1 , p 2 . We ∈ ℤ consider only pure resonances of lower orders ( ω ≈ p 1 ω 1 , where p 1 = 2, 3 and p ω ≈ p 2 ω 2 , where p = 2, 3 , p 2 = 1 and p = 1 , p 2 = 2, 3 ), namely, the super- and subharmonic resonances of the orders 2:1, 3:1, 1:2, 1:3. Varying the non-linearity k 13 / k 11 of elastic ties and scanning the domain of initial conditions we succeed to discovery some of these resonances (Fig. 2 – 2:1; Fig. 3 – 3:1; Fig. 4 – 1:3). Here the case of linear dissipation is given. 16 16 XI International Conference on the Theory of Machines and Mechanisms TMM 2012, September 4-6, 2012, Liberec, Czech Republic
Results Fig. 2 – resonance 2:1, k 13 /k 11 = 1 A k , φ k – are the amplitude and initial phase of k -th harmonic 17 17 XI International Conference on the Theory of Machines and Mechanisms TMM 2012, September 4-6, 2012, Liberec, Czech Republic
Results Fig. 3 – resonance 3:1, k 13 /k 11 = 1 A k , φ k – are the amplitude and initial phase of k -th harmonic 18 18 XI International Conference on the Theory of Machines and Mechanisms TMM 2012, September 4-6, 2012, Liberec, Czech Republic
Results Fig. 4 – resonance 1:3, k 13 /k 11 = 1 A k , φ k – are the amplitude and initial phase of k -th harmonic 19 19 XI International Conference on the Theory of Machines and Mechanisms TMM 2012, September 4-6, 2012, Liberec, Czech Republic
Results The resonance of the order 2:1 is seemed to be one of the most perspectives. It gives an opportunity to carry out practically significant biharmonic vibrations: A 2 / A 1 ≈ 0.125 .. 0.250 , φ 2 – 2φ 1 ≈ 0 .. π /3 [1, 2] and takes place inside rather broad frequency range. It should be noted one of its peculiarities, – the existence of opposite regimes for symmetric characteristic of elastic ties. 20 20 XI International Conference on the Theory of Machines and Mechanisms TMM 2012, September 4-6, 2012, Liberec, Czech Republic
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