Zamanla Değişken Kütleye Sahip Kütle Yay Sönüm Sisteminin Periyodik Olmayan Titreşimleri

Year 2020, , 2110 - 2121, 31.07.2020

Gözde Sarı , Yasemin Nur Aydın

https://doi.org/10.29130/dubited.670658

Abstract

Bu çalışmada zamana bağlı üstel fonksiyon ile değişken kütleli, sönümlü, tek serbestlik dereceli sistemin titreşimleri araştırılmıştır. Literatürde sunulan birçok çalışma değişken kütleli sistemlerde reaktif kuvvetin dikkate alınması gerektiğini belirtmektedir. Bu nedenle sistemin matematiksel modelinde sistemden ayrılan/eklenen kütle hızı ile sistem hızı arasındaki farkın etkisinden dolayı meydana gelen reaktif kuvvet dikkate alınmıştır. İkinci mertebe adi diferansiyel denklemin çözümü çok ölçekli metot ile elde edilmiştir. Sistemin çözümü periyodik titreşime göre farklılık göstermektedir. Kütlenin zamanla değişmesi, sistemin periyodik olmayan titreşim yapmasına neden olmaktadır. Sistemin genliği ve frekansı zamana bağlı olarak değişmektedir. Sistemin genliği sönüm ve reaktif kuvvet parametrelerine bağlı olarak değişmektedir. Reaktif kuvvet dikkate alınmadığında sistemin kütlesi arttığında veya azaldığında sistemin genliği, sönüm etkisi ile zamanla azalmaktadır. Reaktif kuvvet dikkate alındığında ise, kütle zamanla arttığında reaktif kuvvet ile sönüm kuvveti etkisi ile sistemin genliği azalmaktadır. Kütlenin azaldığı durumda reaktif kuvvet sistemin genliğini arttırırken, sönüm kuvveti sistemin genliğini azaltmaktadır.

Keywords

Değişken Kütleli Sistem, Serbest Titreşim, Çok Ölçekli Metot, Periyodik Olmayan Titreşimler, Sönümlü Titreşim

Nonstationary Vibrations of Time-varying Mass Oscillators

Abstract

In this study, free vibration of a single-degree-of-freedom system which had a time-varying mass, a damper was investigated. Many studies in the literature state that reactive force should be taken into account in variable mass systems. Therefore the reactive force due to the effect of the difference between the mass speed separated / added from the system and the system speed was taken into account in the mathematical model of the system. The solution of the second-order ordinary differential equation was obtained by the multi-scale method. The solution of the system changed according to periodic vibration. The change of mass with respect to time caused the system to vibrate non-periodically. The amplitude and frequency of the system varied with respect to time. The amplitude of the system varied depending on the damping and reactive force parameters. When the reactive force was not taken into account, whether the mass of the system increases or decreases, the amplitude of the system decreases with respect to time with the damping effect. When the reactive force is taken into account, when the mass increases with respect to time, the amplitude of the system decreases with the effect of reactive force and damping force. When the mass decreases, the reactive force increases the amplitudes of the system while the damping force decreases the amplitudes.

Keywords

Variable Mass System, Free Vibration, Multiple Scales Method, Nonstationary Vibrations, Damped Vibration

References

• [1] L. Cveticanin, Dynamics of Bodies with Time-Variable Mass. Mathematical and Analytical Techniques with Applications to Engineering, Switzerland: Springer, 2016.
• [2] H. Saruhan, M. Kam, F. Kara, “Dynamic behavior analysis of rotor supported by damped rolling element bearing housing,” Journal of Polytechnic, vol. 20, no.1, pp. 159-164, 2017.
• [3] N. Alçelik, M. Kam, “Dönen makinelerde eksenel kaçıklık ve dengesizliğin titreşim analizi - vibration analysis of axis misalignment and unbalance in rotating machineries,” BSEU Journal of Science, vol. 7, pp. 256-269, 2020.
• [4] A. K. Abramian, W. T. Van Horssen, S. A. Vakulenko, “On oscillations of a beam with a small rigidity and a time-varying mass,” Nonlinear Dynamics, vol. 78, pp. 449-459, 2014.
• [5] L. Cveticanin “Approximate solution of a time-dependent differential equation,” Meccanica, vol. 30, pp. 665-671, 1995.
• [6] Y. Terumichi, M. Ohtsyka, M. Yoshizawa, Y. Fukawa, Y. Tsujioka “Nonstationary vibrations of a string with time-varying length and a mass-spring system attached at the lower end,” Nonlineer Dynamics, vol. 12, pp.3 9–55, 1997.
• [7] L. Cveticanin “Self-excited vibrations of the variable mass rotor/fluid system,” Journal of Sound and Vibration, vol. 212, no. 4, pp. 685-702,1998.
• [8] H.J. Holl, A.K. Belyaev, H. Irschik “Simulation of the duffing-oscillator with time-varying mass by a BEM in time,” Computers and Structures, vol. 73, pp. 177-186, 1999.
• [9] J. Flores, G. Solovey, S. Gil “Variable mass oscillator,” American Association of Physics Teachers, vol. 71, no. 7, pp. 721-725, 2003.
• [10] A.H.P. Van der Burgh, Hartono, A.K.Abramian, “A new model for the study of the rain-wind-ınduced vibrations of a simple oscillator,” International Journal of Non-Linear Mechanics, vol. 41, pp. 345–358, 2016.
• [11] W.T. Horssen Van, O.V. Pischanskyy, “On the stability properties of a damped oscillator with a periodically time-varying mass,” Journal of Sound and Vibration, vol. 330, pp. 3257-3269, 2011.
• [12] Y. Zhu, S. Wang, “Analyzing the vibration system with time-varying mass,” Applied Mechanics and Materials, vol. 50-51, pp. 160-165, 2011.
• [13] M. Zukovic, I. Kovacic “An insight into the behaviour of the oscillators with a periodically piecewise-defined time-varying mass,” Communications in Nonlinear Science and Numerical Simulation, vol. 42, pp. 187-203, 2017.
• [14] W.T. Van Horssen, O.V. Pischanskyy, J.L.A. Dubbeldam, “On the forced vibrations of an oscillator with a periodically time-varying mass”, Journal of Sound and Vibration, vol. 329, pp. 721-732, 2010.
• [15] B. Ji-hu, Z. Peng, Z. Chang-ming, “Modeling of rope longitudinal vibration on flexible hoisting system with time-varying length,” Applied Mechanics and Materials, vol. 130-134, pp. 2783-2788, 2012.
• [16] W.T. Van Horssen, A.K. Abramian, Hartono, “On the free vibrations of an oscillator with a periodically time-varying mass,” Journal of Sound and Vibration, vol. 298, pp. 1166–1172, 2006.
• [17] O.V. Pischanskyy, W.T. Van Horssen, “On the nonlinear dynamics of a single degree of freedom oscillator with a time-varying mass,” Journal of Sound and Vibration, vol. 331, pp. 1887-1897, 2012.
• [18] H. Irschik, H. J. Holl, “Mechanics of variable-mass systems-part 1: Balance of mass and linear momentum,” American Society of Mechanical Engineers, vol. 57, no. 2, pp. 145-160.
• [19] A.H. Nayfeh, Introduction to Perturbation Techniques, New York, USA: John Wiley & Sons, 1981.
• [20] A.H. Nayfeh, J.F. Nayfeh, D.T. Mook, Nonlinear Oscillations, New York, USA: John Wiley & Sons, 1995.
• [21] A.H. Nayfeh, Perturbation Methods, A. Wiley Interscience, New York, USA: John Wiley & Sons, 1973.