Authors:
Sanaullah Dehraj,Rajab A. Malookani,Sajad H. Sandilo,DOI NO:
https://doi.org/10.26782/jmcms.2020.08.00027Keywords:
axially moving string,viscous damping,mode response,internal resonance,Laplace transform method,Abstract
This paper examines an (in) stability of an axially moving string system under the effect of external (viscous) damping. The string is taken to be fixed at both ends and general initial conditions are taken into consideration. The belt (string) speed is assumed to be non-constant harmonically varying about a relatively large means speed. The external damping is also considered to be small. Mathematically, the transverse vibrations of damped axially moving string system are modeled as second order linear homogeneous partial differential equation with variable coefficients. The approximate-analytic solution of the given initial-boundary value problem has been obtained by the application of two timescales perturbation method in conjunction of with Laplace transform method. It is found out that there are infinitely many values of resonant frequency parameter that gives rise to internal resonance in the system. However, in this study only non-resonant and the fundamental resonant cases has been studied. It turned out that the mode-response and the energy of system exhibits stability under certain values of damping parameter and mode-truncation for those parametric values is not problematic.Refference:
I. A. Maitlo, S. H. Sandilo, A. H. Sheikh, R. A. Malookani, and S. Qureshi, “On aspects of viscous damping for an axially transporting string”. Sci. int. (Lahore)., Vol. 28, No. 4, pp.3721–3727 (2016).
II. A. H. Nayfeh, “Introducation to Perturbation techniques”. John Wiley and Sons, Inc, 1981.
III. Darmawijoyo, W. T. Van Horssen and P. H. Clément, “On a Rayleigh wave equation with boundary damping”. Nonlinear Dyn., Vol. 33, pp. 399–429 (2003).
IV. Darmawijoyo and W. T. Van Horssen, “On boundary damping for a weakly nonlinear wave equation”. Nonlinear Dyn.,Vol.30, No. 2, pp.179–191(2002).
V. G. Suweken and W. T. Van Horssen, “On the transversal vibrations of a conveyor belt with a low and time-varying velocity. Part I: the string-like case”. J. Sound Vib.,Vol. 264, No. 1, pp.117–133 (2003).
VI. I. V. Andrianov and J. Awrejcewicz, “Dynamics of a string moving with time-varying speed”. J. Sound Vib.,Vol. 292, No. 3–5, pp.935–940 (2006).
VII. J. Kevorkian and J. D. Cole, “Multiple scale and singular perturbation methods”. Springer-Verlag, New Yark Inc.Vol. 114 (1996).
VIII. K. Marynowski and T. Kapitaniak, “Zener internal damping in modelling of axially moving viscoelastic beam with time-dependent tension”. Int. J. Non. Linear. Mech., Vol. 42, No. 1, pp. 118–131 (2007).
IX. L. Debnath and D. Bhatta, “Integral Transforms and their applications”. Chapman and Hall/CRC, second edit. (2007).
X. M. A. Zarubinskaya and W. T. van Horssen, “On aspects of boundary damping for a rectangular plate”. J. Sound Vib.,Vol. 292,No. 3–5, pp. 844–853 (2006).
XI. N. Jakšić and M. Boltežar, “Viscously damped transverse vibrations of an axially-moving string”. J. of Mech. Eng., Vol. 51, No. 9, pp. 560–569 (2005).
XII. N. V. Gaiko and W. T. Van Horssen, “On the transverse, low frequency vibrations of a traveling string with boundary damping”. J. Vib. Acoust. Trans. ASME.,Vol. 137, No. 4, pp. 10–12 (2015).
XIII. P. Zhang, J. H. Bao, and C. M. Zhu, “Dynamic analysis of hoisting viscous damping string with time-varying length”. J. Phys. Conf. Ser., Vol. 448, pp. 1-9 (2013).
XIV. Rajab A. Malookani, S. H. Sandilo, and A. H. Sheikh, “On (non) applicability of a mode-truncation of a damped traveling string”. Mehran Univ. research J. of Eng. & Tech., Vol. 38, No. 2, pp. 471–478 (2019).
XV. Rajab A. Malookani and W. T. Van Horssen, “On the asymptotic approximation of the solution of an equation for a non-constant axially moving string”. J. Sound Vib., Vol. 367, pp. 203–218 (2016).
XVI. Rajab A. Malookani, S. Dehraj, and S. H. Sandilo, “Asymptotic approximations of the solution for a traveling string under boundary damping”. J. of appl. and compt. Mech.,Vol. 5, No. 5, pp. 918–925 (2019).
XVII. R. A. Malookani and W. T. Van Horssen, “On resonances and the applicability of Galerkin’s truncation method for an axially moving string with time-varying velocity”. J. Sound Vib.,Vol.344, pp. 1–17 (2015).
XVIII. S. Krenk, “Vibrations of a taut cable with an external damper”. J. Appl. Mech., Vol. 67, No. 4, pp. 772–776 (2000).
XIX. S. H. Sandilo, “On boundary damping for an axially moving beam and on the variable length induced vibrations of an elevator cable”. ENOC (Conference paper) (2011).
XX. S. V. Ponomareva and W. T. Van Horssen, “On transversal vibrations of an axially moving string with a time-varying velocity”. Nonlinear Dyn.,Vol. 50, No. 1–2, pp. 315–323 (2007).
XXI. S. H. Sandilo and W. T. Van Horssen, “On boundary damping for an axially moving tensioned beam”. J. Vib. Acoust. Trans. ASME.,Vol.134, pp. 011005-1-011005-8 (2012).
XXII. S. H. Sandilo, Rajab A. Malookani, and A. H. Sheikh, “On vibrations of an axially moving beam under material damping”. IOSR J. Mech. Civ. Eng., Vol. 13, No. 05, pp. 56–61(2016).
XXIII. S. H. Sandilo, A. H. Sheikh, M. A. Soomro, and Rajab A. Malookani, “On oscillations of an axially translating tensioned string-like equation under internal damping”. Sci. int. (Lahore).,Vol. 28, No. 4, pp.3897–3901(2016).
XXIV. S. Dehraj, S.H. Sandilo, Rajab A. Malookani, “On applicabitlity of Galerkin’s truncation method for damped axailly moving string”. J. of Vibroeng., Vol. 22, No. 2, pp. 337-352 (2020).
XXV. T. Akkaya and W. T. van Horssen, “On constructing a Green’s function for a semi-infinite beam with boundary damping”.Meccanica., Vol. 52, No. 10, pp. 2251–2263 (2017).
XXVI. T. Akkaya and W. T. van Horssen, “On boundary damping to reduce the rain–wind oscillations of an inclined cable with small bending stiffness” Nonlinear Dyn.,Vol.95, No. 1, pp.783–808 (2019).
XXVII. V. S. Sorokin, “On the effects of damping on the dynamics of axially moving spatially periodic strings”. Wave motion, Vol. 85, pp. 165–175 (2019).
XXVIII. W. T. Van Horssen, “On the weakly damped vibrations of a string attached to a spring mass dashpot system”.Journal Vib. Control, Vol.9, No. 11, pp. 1231–1248 (2003).
XXIX. W. T. Van Horssen and S. V. Ponomareva, “On the construction of the solution of an equation describing an axially moving string”.J. Sound Vib., Vol. 287, No. 1–2, pp. 359–366 (2005).
XXX. Y. Li and Y. Tang, “Analytical analysis on nonlinear parametric vibration of an axially moving string with fractional viscoelastic damping”. Math. Probl. Eng., Vol. 2017,pp.1-9 (2017).
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