Mathematical Problems in Engineering Volume, Article ID 77543, pages doi:.55//77543 Research Article Nonlinear Response of Vibrational Conveyers with Nonideal Vibration Exciter: Superharmonic and Subharmonic Resonance H. Bayıroğlu, G. F. Alışverişçi, and G. Ünal Yıldız Technical University, 34349 Istanbul, Turkey Yeditepe University, 34755 Istanbul, Turkey Correspondence should be addressed to H. Bayıroğlu, hbayir@yildiz.edu.tr Received 7 October ; Accepted 4 November Academic Editor: Swee Cheng Lim Copyright q H. Bayıroğlu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Vibrational conveyers with a centrifugal vibration exciter transmit their load based on the jumping method. Common unbalanced-mass driver oscillates the trough. The motion is strictly related to the vibrational parameters. The transition over resonance of a vibratory system, excited by rotating unbalances, is important in terms of the maximum vibrational amplitude produced and the power demand on the drive for the crossover. The mechanical system is driven by the DC motor. In this study, the working ranges of oscillating shaking conveyers with nonideal vibration exciter have been analyzed analytically for superharmonic and subharmonic resonances by the method of multiple scales and numerically. The analytical results obtained in this study agree well with the numerical results.. Introduction The load-carrying element of a horizontal shaking conveyer performs, as a rule, linear or sometimes circular or elliptical symmetrical harmonic oscillations with a sinusoidal variation of exciting force. In vertical shaking conveyers, the load-carrying element performs double harmonic oscillations: linear along the vertical axis and rotational around that axis i.e., longitudinal and torsional oscillations. Conveyer drives with centrifugal vibration exciters may have a single unbalanced-mass, two equal unbalancing masses, 3 a pendulum-type unbalanced-mass, 4 four unbalanced-masses in two shafts, 5 four rotating unbalanced-masses for three principal modes of oscillation, that is, linear, elliptical, and circular. To induce strictly oriented linear oscillations of the load-carrying element, the conveyer drive should be arranged so that the line of excitation force passes through the
Mathematical Problems in Engineering inertial centre of the entire oscillating system. Nonideal drives find application in suspended and supported vibrational conveyers and feeders. By the characteristics and adjustment of the elastic support elements oscillating system, we can distinguish between shaking conveyers with a resonant, subresonant, and superresonant system. A practical difficulty with unbalanced-mass exciters, observed as early as 94 by A Sommerfeld, is that local instabilities may occur in operating speed of such devices. Rocard, Mazert 3, Panovko, and I. I. Gubanova 4 have studied the problem of the stability of the unbalanced-mass exciter. The first detailed study on the nonideal vibrating systems is presented by Kononenko. He obtained satisfactory results by the comparison of the experimental analysis and the approximated method 5. After this publication, the nonideal problem was presented by Evan-Ivanowski 6 or Nayfeh and Mook 7. These authors showed that sometimes dynamical coupling between energy sources and structural response that had not to be ignored in reel engineering problems. Theorical studies and computations of Ganapathy and Parameswaran have indicated the beneficial effect of the material load during the starting and transition phase of an unbalanced-mass-driven vibrating conveyor 8. Bolla et al. analyzed through the multiple scales method a response of a simplified nonideal and nonlinear vibrating system 9. Götzendorfer in presented a macromechanical model for the transport of granular matter on linear and horizontal conveyors subject to linear, circular or elliptic oscillations and compared it to experimental results.. The Governing Equations of the Motion The equations of motion for the modified rocard system may be obtained by using Lagrange s equation d T D T V, dt q i q i q i q i. where T, the kinetic energy, is T Mẏ m [ ẏ θe cos θ θe sin θ ] I mot θ,. where e is the eccentricity of the mass m, m is the unbalanced-mass, M is the mass of the trough and the conveyed material on the trough of the conveyor, θ is the angle of rotation of the shafts carrying unbalanced-masses, I mot is the moment of inertia of the rotating parts in the motor, V, the potential energy, is V k y 4 k y 4 mge sin θ,.3
Mathematical Problems in Engineering 3 where the constants k and k are the linear and nonlinear elastic coefficients, respectively, D, the Rayleigh dissipation function, is D cẏ K ω s θ,.4 and q i is the generalized coordinate. Applying Lagrange s equation for the two coordinates q i y and q i θ gives the differential equations of motion Mÿ mÿ cẏ k y k y 3 me θ sin θ θ cos θ, I mot me }{{} θ me ÿ cos θ g cos θ [ ] K θ ω s K. θ I sys.5 Equation.5 can be rewritten as follows: ] ÿ ω [ μẏ y ε αy 3 e θ cos θ κ θ sin θ,.6 θ ε [ I cos θ ÿ g E θ ],.7 where I m M e me, κ I sys m M, μ c m, α k m, k m M ω, m ε, m M E θ m M L θ, mi sys L θ [ ] K θ ω s K, θ.8 where c is the damping coefficient of the vibrating conveyor, g is acceleration due to gravity, I sys I m I M I mot is the total moment of inertia of all the rotating parts in the system, and ω s is the synchronous angular speed of the induction motor 9. K is the instantaneous drive torque available at the shafts. Note that E contains L θ that is the active torque generated by the electric circuit of the DC motor, shown in Figure. 3. Analytical Solution Ideal system: if there is no coupling between motion of the rotor and vibrating system and θ constant θ ωt, θ,.6 becomes ÿ ω y ε [ μẏ αy 3] κω sin ωt. 3. On the right side of the equation, a function of time is present.
4 Mathematical Problems in Engineering Figure : Vibrating model of the system. Nonideal system: θ ε [ I cos θ ÿ g E θ ], [ ] ÿ ωny ε μẏ αy 3 e θ cos θ κ θ sin θ, 3. where E θ M m θ H θ is the difference between the torque generated by the motor and the resistance torque. Function E θ u u θ is approximated by a straight line, where u is a control parameter that can be changed according to the voltage, and u is a constant parameter, characteristic for the model of the motor. We will obtain an approximate analytical solution to 3. by using the multiple scales method: θ t, ε θ T,T εθ T,T ε θ T,T, y t, ε y T,T εy T,T ε y T,T, 3.3 T n ε n t, n,,..., T t, T εt where the fast scale T t and slow scale T εt. The time derivatives transform according to d dt dt dt dt T dt T D εd d dt D εd D, 3.4 where D n / T n, n,,... ; then, θ d θ dt θ dθ dt D θ ε D θ D θ, D θ ε D D θ D θ,
Mathematical Problems in Engineering 5 ÿ d y dt ẏ dy dt D y ε D y D y, D y ε D D y D y. 3.5 Substituting 3.5 into 3., we will obtain [ D θ ε D D θ D θ I cos θ D y g E θ ], D y ε D D y D y ωn y εy [ ε μd y α ] ] 3 y ed θ cos θ κ D θ ε D θ D θ sin θ, 3.6 and equating coefficients of a like powers ε,weobtain i for ε D θ, D y ω y κ D θ sin θ, 3.7 ii for ε D θ D D θ I cos θ D y g E θ, D y ω y D D y μd y α y 3 ed θ cos θ κ D θ D θ D θ sin θ. 3.8 3.9 The solution of 3.7 can be written as θ BT σt, y Λsin T Ω A T e iω T A T e iω T, 3. 3. where cos θ εθ cos θ O ε, Λ κω Ω ω. sin θ εθ sin θ O ε, 3.
6 Mathematical Problems in Engineering Table : Vibrational conveyer parameters in SI units. ε α μ c I e m E E g ω k k m M.5...9..5.6 9.8 5 3.. Subharmonic Resonances Ω 3ω Table Near resonance: D θ Ω, Ω 3ω εσ. 3.3 3.4 The solution of 3.3 can be written as θ ΩT 3ω T σt. 3.5 Taking A T,T a T e iβ T, 3.6 where a and β are real, and substituting it into 3., we will obtain y e it ω iβ T a T eit ω iβ T a T Λsin T Ω, θ 8 Ω 3 Ωω e i T ω β T I 4Ω ω a T e i T ω β T Ω ω cos T Ω i e i T ω β T Ωω sin T Ω e i T ω β T Ω ω cos T Ω 4g κω Λω sin T Ω, y e i 3T Ω 4T ω 3β T 8e 3iβ T Ω Ω 3 3Ω ω Ωω 3ω3 ie it 4 Ω 3ω ω 3αΛ 3 8iΛμΩ Iκ Ω IκΛω 36Ω 4 3Ω ω ω4 ie it Ω 3ω ω 3αΛ 3 8iΛμΩ Iκ Ω IκΛω 36Ω 4 3Ω ω ω4 4e it Ω 3ω giκω 9Ω 4 Ω ω ω4 4e it 5 Ω 3ω giκω
Mathematical Problems in Engineering 7 9Ω 4 Ω ω ω4 8e 3iT Ω ω giκ 36Ω 6 49Ω 4 ω 4Ω ω 4 ω6 ie 3iT Ω ω ω 4Ω 4 5Ω ω ω4 αλ 3 Iκ κω Λω ie 3iT ω ω 4Ω 4 5Ω ω ω4 αλ 3 Iκ κω Λω e iβ T ω 36Ω 5 8Ω 4 ω 39Ω ω 3 Ωω4 3ω5 e it 5 Ω ω Ω ω IκΩω 3αΛ Ω ω e i T Ω 4ω β T Ω ω IκΩω 3αΛ Ω ω e i T Ω ω Ω ω IκΩω 3αΛ Ω ω e i 5T Ω 4T ω β T Ω ω IκΩω 3αΛ Ω ω a T 4ie i T Ω ω β T α ΛΩω e i T ω β T e i T Ω ω β T e i T Ω ω β T Ω 6e i T ω β T 6e i T Ω ω β T e i T Ω ω β T ω 36Ω 6 49Ω 4 ω 4Ω ω 4 ω6 a T e 3iT Ω e 6i T ω β T αω Ω ω 36Ω 5 8Ω 4 ω 3Ω 3 ω 39Ω ω Ωω4 3ω5 a T 3 cos T ω i sin T ω / 64 Ω 6Ω 5 Ωω ω Ω ω ω 3 6Ω 3 3Ω ω 6 Ωω 3ω3. 3.7 Secular terms will be eliminated from the particular solution of 3.8 if we choose A to be a solution of εσ E θ. 3.8 In addition to the terms proportional to e ±iω T or proportional to e ±i Ω ω T there is another term that produces a secular term in 3.9. We express Ω ω T in terms of ω T according to Ω ω T ω T εσt ω T σt. 3.9
8 Mathematical Problems in Engineering Table : Vibrational conveyer parameters in SI units. ε α μ c I e m E E g ω k k m M.5...9..5.6 9.8 5 3.. Superharmonic Resonances Ω /3 ω Table We consider 3 Ω ω εσ. 3. In addition to the terms proportional to e ±iω T in 3.9, there is another term that produces a secular term in 3.9. Thisis αλ 3 e ±3iΩT. T eliminate the secular terms, we express 3 ΩT in terms of ω T according to 3 ΩT ω εσ T ω T σεt ω T σt. 3. Eliminating secular terms in equations, taking 3.6, and separating real and imaginary parts, the system of equations solved it is as follows : y e it ω iβ T a T eit ω iβ T a T Λsin T Ω, θ y 8 Ω 3 Ωω e i T ω β T I 4Ω ω a T e i T ω β T Ω ω cos T Ω i e i T ω β T Ωω sin T Ω e i T ω β T Ω ω cos T Ω 4g κω Λω sin T Ω, e i T Ω 5T ω 3β T 8e 3iβ T Ω Ω 4 Ω ω 9ω4 4e it 4 Ω 5ω giκω Ω ω 4e 5iT ω giκω Ω ω ie it 3 Ω 5ω ω 4Ω ω 3αΛ 3 8iΛμΩ Iκ Ω IκΛω e it Ω 5ω ω 4Ω ω 3iαΛ 3 8ΛμΩ iiκ Ω iiκλω 8e it Ω 5ω giκ 4Ω 4 5Ω 4 ω ω4 e iβ T ω 4Ω 4 37Ω ω 9ω
Mathematical Problems in Engineering 9 e 4iT Ω ω Ω ω IκΩω 3αΛ Ω ω e i 3T ω β T Ω ω IκΩω 3αΛ Ω ω e 4iT ω Ω ω IκΩω 3αΛ Ω ω e i T Ω 3T ω β T Ω ω IκΩω 3αΛ Ω ω a T 4ie iβ T αλωω 4Ω 4 5Ω ω ω4 e i 3T Ω 5T ω β T Ω 9ω e i T Ω 5ω β T Ω 9ω e it Ω 3ω Ω 4 Ωω 3ω e i 3T Ω 7T ω 4β T Ω 4 Ωω 3ω e 3iT Ω ω Ω 4 Ωω 3ω e i T Ω 7ω 4β T Ω 4 Ωω 3ω a T e it Ω ω e i T Ω 4ω 3β T αω Ω ω 4Ω 4 37Ω ω 9ω4 a T 3 / 64 Ω ω ω 3 4Ω 5 37Ω 3 ω 9 Ωω4. 3. 4. Stability Analysis We analyzed in this system the stability a, γ in the equilibrium point, using 4., where F is the Jacobian matrix of 4.,andγ sub 3β T T σ T, γ sup β T T σ T. Stability of the approximate solutions depends on the value of the eigenvalues of the Jacobian matrix F. The solutions are unstable if the real part of the eigenvalues is positives Figures a and a. Figure shows the frequency-response curves for the subharmonic and superharmonic resonance of the unbalanced vibratory conveyor: a 3 sub μa αλa cos [ γ ], 8ω γ sub 3 IκΩ ω 3αΛ Ω ω 4 Ω ω 9 αa 9 αλa sin [ γ ] σ, ω 8ω 8ω a 3 sub μa αλ 3 Iκ κω Λω cos [ γ ], 8ω IκΩ γ sub ω 3αΛ Ω ω 4 Ω ω 3αa αλ 3 Iκ κω Λω [ ] sin γ ω 8ω 8ω a σ, 4. F { a f, γ f }, { a f, γ f }, f a,f γ. 4.
Mathematical Problems in Engineering a 9 8 7 6 5 4 3 9 8 Subharmonic resonance.5.5.5 σ, α =.8,μ =.5 a a Superharmonic resonance 7 6 5 4 3.5.5.5 σ, α =.8,μ =.5 a 3.5 a 7 6 5 a.5.5 4.5.5.5 4 6 8 κσ =.5,σ = 3,σ = 8 κσ =.5,σ =,σ = 3 σ =.5 σ = 3 σ = 8 b σ =.5 σ = σ = 3 b a 8 6 4 a 3 4 5 3 σ, μ =.,μ= σ, μ =.,μ=.5 μ =. μ =.5.5 μ =. μ =.5 c c Figure : Subharmonic resonance and superharmonic resonance. a Frequency-response curves with stability, stable, unstable { a : a.78, σ.4358.69 and a : a.73 7., σ.678.986448}, b effect of detuning parameter, and c effect of damping parameter.
Mathematical Problems in Engineering Power spectrum.8.6.4. 3 4 5 6.4 a Velocity-displacement Power spectrum.8.6.4. 3 4 5 6.4 b Poincare map velocity-displacement.. v v...4.4.5.5 y.5.5 y c d Figure 3: Subharmonic resonance: a power spectrum and superharmonic resonance, b power spectrum, c phase portrait, and d Poincaré sections. 5. Numerical Results The numerical calculations of the vibrating system are performed with the help of the software Mathematica 3, 4. We analyze the subharmonic resonance Ω 3ω and superharmonic resonance 3 Ω ω. Figure 3 shows the power spectrum, phase portrait and the Poincaré map for superharmonic resonance and Figure 3 shows the power spectrum for superharmonic resonance. 6. Conclusions The vibrating system is analyzed, analytically, and numerically for superharmonic and subharmonic resonance by the method of multiple scales. Very often in the motion of the system near resonance the jump phenomenon occurs. The frequency-response curves of the subharmonic resonance consist of two branches; the left one is stable and the right one is unstable saddle node bifurcation. The frequency-response curves of the superharmonic resonance consist of three branches; the left one is stable, the middle one is unstable, and the right one is stable pitchfork bifurcation. The stable motions of the oscillator are shown
Mathematical Problems in Engineering with one peaks in the power spectrum for superharmonic resonance and with two peaks in the power spectrum for subharmonic resonance. Both analytical and numerical results that we have obtained are in good agreement. The system studied here exhibits chaotic behaviour in case of strong nonlinearity. This will be reported in the forthcoming paper. Furthermore, control methods in the passage through resonance await for future publication. Acknowledgment The authors greatly appreciate the comments of an anonymous referee. References J. M. Balthazar, D. T. Mook, H. I. Weber et al., An overview on non-ideal vibrations, Mechanica, vol. 38, no. 6, pp. 63 6, 3. Y. Rocard, General Dynamics of Vibrations, Ungar, New York, NY, USA, 3rd edition, 96. 3 R. Mazert, Mécanique Vibratoire,C.Béranger, editor, Paris, France, 955. 4 Y. G. Panovko and I. I. Gubanova, Stability and Oscillations of Elastic Systems, Consultans Bureau, New York, NY, USA, 965. 5 V. O. Kononenko, Vibrating Problems with a Limited Power Supply, Iliffe, London, UK, 969. 6 R. M. Evan-Ivanowski, Resonance Oscillations in Mechanical Systems, Elsevier, Amsterdam, The Netherlands, 976. 7 A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, John Wiley & Sons, New York, NY, USA, 979. 8 S. Ganapathy and M. A. Parameswaran, Transition over resonance and power requirements of an unbalanced mass-driven vibratory system, Mechanism and Machine Theory, vol., no., pp. 73 85, 986. 9 M. R. Bolla, J. M. Balthazar, J. L. P. Felix, and D. T. Mook, On an approximate analytical solution to a nonlinear vibrating problem, excited by a nonideal motor, Nonlinear Dynamics, vol.5,no.4,pp. 84 847, 7. A. Götzendorfer, Vibrated granular matter: transport, fluidization, and patterns, Ph.D. thesis, University Bayreuth, 7. V. Piccirillo, J. M. Balthazar, and B. R. Pontes Jr., Analytical study of the nonlinear behavior of a shape memory oscillator: part II-resonance secondary, Nonlinear Dynamics, vol. 6, no. 4, pp. 53 54,. J. Awrejcewicz and C. H. Lamarque, Bifurcation and Chaos in Nonsmooth Mechanical Systems, World Scientific, River Edge, NJ, USA, 3. 3 A. H. Nayfeh and C. M. Chin, Perturbation Methods with Mathematica, Dynamics Press, Virgina, va, USA, 999. 4 S. Lynch, Dynamical Systems with Applications Using Mathematica, Birkhäauser, Boston, Mass, USA, 7.
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