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Internal Resonance of Axially Moving Beams with Masses

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Iranian Journal of Science and Technology, Transactions of Mechanical Engineering Aims and scope Submit manuscript

Abstract

Transverse vibrations of axially moving beams with multiple concentrated masses have been investigated. It is assumed that the beam is of Euler–Bernoulli type, and both ends have simply supports. Concentrated masses are equally distributed on the beam. This system is formulated mathematically and then sought to find out approximate solutions. In case of three-to-one internal resonance, analytical solutions are derived by means of method of multiple scales (a perturbation method). It is assumed that axial velocity of the beam is harmonically varying around a mean-constant velocity. Steady-state vibration characteristics are investigated from the amplitude-phase modulation equations. Then, the effects of both magnitude and number of the concentrated masses on nonlinear vibrations are investigated numerically in detail.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Institute of Natural Sciences, University of Manisa Celal Bayar, Yunusemre, Manisa, Turkey

    Murat Sarıgül

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  1. Murat Sarıgül
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Appendix

Appendix

$$\begin{aligned} f & = \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {\bar{Y}_{{\left( {r + 1} \right)s}} \cdot F_{{\left( {r + 1} \right)1}} \cdot {\text{d}}x} } \\ K_{1} & = - \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {\bar{Y}_{{\left( {r + 1} \right)k}} *} \left\langle {2 \cdot u_{0} \cdot Y_{{\left( {r + 1} \right)k}}^{\prime } + 2 \cdot i \cdot \omega_{k} \cdot Y_{{\left( {r + 1} \right)k}} } \right\rangle \cdot {\text{d}}x} - \left. {\sum\limits_{r = 1}^{n} {\alpha \cdot \left\langle {2 \cdot i \cdot \omega_{k} \cdot Y_{rk} \cdot \bar{Y}_{rk} + 2 \cdot u_{0} \cdot Y_{rk}^{\prime } \cdot \bar{Y}_{rk} - u_{0} \cdot Y_{rk} \cdot \bar{Y}_{rk}^{\prime } } \right\rangle } } \right|_{{x = x_{r} }} \\ K_{2} & = \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {\bar{Y}_{{\left( {r + 1} \right)k}} *} \left\langle {u_{1} \cdot \left( {i \cdot \left( {\omega_{k} + \omega_{s} } \right) - \frac{{\omega_{s} }}{2}} \right) \cdot \phi_{{\left( {r + 1} \right)1}}^{\prime } + u_{1} \cdot \left( {i \cdot \left( { - \omega_{k} + \omega_{s} } \right) - \frac{{\omega_{s} }}{2}} \right) \cdot \phi_{{\left( {r + 1} \right)2}}^{\prime } + u_{0} \cdot u_{1} \cdot \left( {\phi_{{\left( {r + 1} \right)1}}^{{\prime \prime }} - \phi_{{\left( {r + 1} \right)2}}^{{\prime \prime }} } \right)} \right.} \left. { + \frac{{u_{1}^{2} }}{2} \cdot Y_{{\left( {r + 1} \right)k}}^{{\prime \prime }} } \right\rangle \cdot {\text{d}}x \\ & \quad - \sum\limits_{r = 1}^{n} {\alpha \cdot \left\langle {u_{1} \cdot \left( {\left( { - i \cdot \left( {\omega_{k} + \omega_{s} } \right) + \frac{{\omega_{s} }}{2}} \right) \cdot \phi_{r1}^{\prime } + \left( {i \cdot \left( {\omega_{k} - \omega_{s} } \right) + \frac{{\omega_{s} }}{2}} \right) \cdot \phi_{r2}^{\prime } } \right) \cdot \bar{Y}_{rk} } \right.} \\ & \left. {\left. {\quad +\, \left( {\frac{{u_{1}^{2} }}{2} \cdot Y^{\prime}_{rk} + \frac{{u_{1} }}{2} \cdot i \cdot \left( {\omega_{k} + \omega_{s} } \right) \cdot \phi_{r1} - \frac{{u_{1} }}{2} \cdot i \cdot \left( {\omega_{k} - \omega_{s} } \right) \cdot \phi_{r2} + u_{0} \cdot u_{1} \cdot \left( {\phi_{r1}^{\prime } - \phi_{r2}^{\prime } } \right)} \right) \cdot \bar{Y}_{rk}^{\prime } } \right\rangle } \right|_{{x = x_{r} }} \\ K_{3} & = \frac{1}{2} \cdot v_{k1}^{2} \cdot \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {\bar{Y}_{(r + 1)k} *} \left\langle {Y_{{\left( {r + 1} \right)k}}^{{\prime \prime }} \cdot \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {2 \cdot Y_{{\left( {r + 1} \right)k}}^{\prime } \cdot \bar{Y}_{{\left( {r + 1} \right)k}}^{\prime } \cdot {\text{d}}x} } + \bar{Y}_{{\left( {r + 1} \right)k}}^{{\prime \prime }} \cdot \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {Y_{{\left( {r + 1} \right)k}}^{\prime 2} \cdot {\text{d}}x} } } \right\rangle \cdot {\text{d}}x} \\ K_{4} & = \frac{1}{2} \cdot v_{k1}^{2} \cdot \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {\bar{Y}_{{\left( {r + 1} \right)k}} *} \left\langle {Y_{{\left( {r + 1} \right)k}}^{{\prime \prime }} \cdot \sum\limits_{r = 0}^{s} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {2 \cdot Y_{{\left( {r + 1} \right)s}}^{\prime } \cdot \bar{Y}_{{\left( {r + 1} \right)s}}^{\prime } \cdot {\text{d}}x} } + Y_{{\left( {r + 1} \right)s}}^{{\prime \prime }} \cdot \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {2 \cdot Y_{{\left( {r + 1} \right)k}}^{\prime } \cdot \bar{Y}_{{\left( {r + 1} \right)s}}^{\prime } \cdot {\text{d}}x} } } \right.} \left. { + \bar{Y}_{{\left( {r + 1} \right)s}}^{{\prime \prime }} \cdot \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {2 \cdot Y_{{\left( {r + 1} \right)k}}^{\prime } \cdot Y_{{\left( {r + 1} \right)s}}^{\prime } \cdot {\text{d}}x} } } \right\rangle \cdot {\text{d}}x \\ K_{5} & = - \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {\bar{Y}_{{\left( {r + 1} \right)k}} } *\left\langle {u_{1} \cdot \omega_{s} \cdot \left( {\frac{1}{2} + 2 \cdot i} \right) \cdot \phi_{(r + 1)3}^{\prime } + u_{0} \cdot u_{1} \cdot \phi_{(r + 1)3}^{{\prime \prime }} + \frac{{u_{1}^{2} }}{4} \cdot Y_{{\left( {r + 1} \right)s}}^{{\prime \prime }} } \right\rangle \cdot {\text{d}}x} + \sum\limits_{r = 1}^{n} {\alpha \cdot \left\langle { - u_{1} \cdot \omega_{s} \cdot \left( {\frac{1}{2} + 2 \cdot i} \right) \cdot \phi_{r3}^{\prime } \cdot \bar{Y}_{rk} } \right.} \\ & \left. {\left. {\quad +\, \left( {\frac{{u_{1}^{2} }}{4} \cdot Y_{rs}^{\prime } + u_{0} \cdot u_{1} \cdot \phi_{r3}^{\prime } + u_{1} \cdot i \cdot \omega_{s} \cdot \phi_{r3} } \right) \cdot \bar{Y}_{rk}^{\prime } } \right\rangle } \right|_{{x = x_{r} }} \\ K_{6} & = \frac{1}{2} \cdot v_{k1}^{2} \cdot \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {\bar{Y}_{{\left( {r + 1} \right)k}} *\left\langle {Y_{{\left( {r + 1} \right)s}}^{{\prime \prime }} \cdot \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {Y_{{\left( {r + 1} \right)s}}^{\prime 2} \cdot {\text{d}}x} } } \right\rangle } \cdot {\text{d}}x} \\ \end{aligned}$$
$$\begin{aligned} S_{1} & = - \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {\bar{Y}_{{\left( {r + 1} \right)s}} } *\left\langle {2 \cdot u_{0} \cdot Y_{{\left( {r + 1} \right)s}}^{\prime } + 2 \cdot i \cdot \omega_{s} \cdot Y_{{\left( {r + 1} \right)s}} } \right\rangle \cdot {\text{d}}x} - \sum\limits_{r = 1}^{n} {\alpha \cdot \left. {\left\langle {2 \cdot i \cdot \omega_{s} \cdot Y_{rs} \cdot \bar{Y}_{rs} + 2 \cdot u_{0} \cdot Y_{rs}^{\prime } \cdot \bar{Y}_{rs} - u_{0} \cdot Y_{rs} \cdot \bar{Y}_{rs}^{\prime } } \right\rangle } \right|_{{x = x_{r} }} } \\ S_{2} & = \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {\bar{Y}_{{\left( {r + 1} \right)s}} *} \left\langle { - u_{1} \cdot \omega_{s} \cdot \left( {\frac{1}{2} - 2 \cdot i} \right) \cdot \phi_{(r + 1)3}^{\prime } - u_{1} \cdot \frac{{\omega_{s} }}{2} \cdot \phi_{(r + 1)4}^{\prime } u_{0} \cdot u_{1} \cdot \left( {\phi_{(r + 1)3}^{{\prime \prime }} - \phi_{(r + 1)4}^{{\prime \prime }} } \right) + \frac{{u_{1}^{2} }}{2} \cdot Y_{{\left( {r + 1} \right)s}}^{{\prime \prime }} } \right\rangle {\text{d}}x} \\ & \quad - \sum\limits_{r = 1}^{n} {\alpha \cdot \left. {\left\langle {\left( {u_{1} \cdot \omega_{s} \cdot \left( {\frac{1}{2} - 2 \cdot i} \right) \cdot \phi_{r3}^{\prime } + u_{1} \cdot \frac{{\omega_{s} }}{2} \cdot \phi_{r4}^{\prime } } \right) \cdot \bar{Y}_{rs} + \left( {\frac{{u_{1}^{2} }}{2} \cdot Y_{rs}^{\prime } + u_{1} \cdot i \cdot \omega_{s} \cdot \phi_{r3} + u_{0} \cdot u_{1} \cdot \left( {\phi_{r3}^{\prime } - \phi_{r4}^{\prime } } \right)} \right) \cdot \bar{Y}_{rs}^{\prime } } \right\rangle } \right|_{{x = x_{r} }} } \\ S_{3} & = \frac{1}{2} \cdot v_{k1}^{2} \cdot \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {\bar{Y}_{{\left( {r + 1} \right)s}} } *\left\langle {Y_{{\left( {r + 1} \right)s}}^{{\prime \prime }} \cdot \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {2 \cdot Y_{{\left( {r + 1} \right)s}}^{\prime } \cdot \bar{Y}_{{\left( {r + 1} \right)s}}^{\prime } \cdot {\text{d}}x} } + \bar{Y}_{{\left( {r + 1} \right)s}}^{{\prime \prime }} \cdot \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {\bar{Y}_{{\left( {r + 1} \right)s}}^{\prime 2} \cdot {\text{d}}x} } } \right\rangle \cdot {\text{d}}x} \\ S_{4} & = \frac{1}{2} \cdot v_{k1}^{2} \cdot \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {\bar{Y}_{{\left( {r + 1} \right)s}} } *} \left\langle {Y_{{\left( {r + 1} \right)k}}^{{\prime \prime }} \cdot \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {2 \cdot \bar{Y}_{{\left( {r + 1} \right)k}}^{\prime } \cdot Y_{{\left( {r + 1} \right)s}}^{\prime } \cdot {\text{d}}x} } + \bar{Y}_{{\left( {r + 1} \right)k}}^{{\prime \prime }} \cdot \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {2 \cdot Y_{{\left( {r + 1} \right)k}}^{\prime } \cdot Y_{{\left( {r + 1} \right)s}}^{\prime } \cdot {\text{d}}x} } } \right.\left. { + Y_{{\left( {r + 1} \right)s}}^{{\prime \prime }} \cdot \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {2 \cdot Y_{{\left( {r + 1} \right)k}}^{\prime } \cdot \bar{Y}_{{\left( {r + 1} \right)k}}^{\prime } \cdot {\text{d}}x} } } \right\rangle \cdot {\text{d}}x \\ S_{5} & = \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {\bar{Y}_{{\left( {r + 1} \right)s}} } *\left\langle {u_{1} \cdot \left( {i \cdot \left( {\omega_{k} - \omega_{s} } \right) - \frac{{\omega_{s} }}{2}} \right) \cdot \phi_{(r + 1)2}^{\prime } u_{0} \cdot u_{1} \cdot \phi_{(r + 1)2}^{{\prime \prime }} - \frac{{u_{1}^{2} }}{4} \cdot Y_{{\left( {r + 1} \right)k}}^{{\prime \prime }} } \right\rangle \cdot {\text{d}}x} \\ & \quad + \sum\limits_{r = 1}^{n} {\alpha \cdot \left\langle {u_{1} \cdot \left( {i \cdot \left( {\omega_{k} - \omega_{s} } \right) - \frac{{\omega_{s} }}{2}} \right) \cdot \phi_{r2}^{\prime } \cdot \bar{Y}_{rs} } \right.} \left. {\left. { + \left( {\frac{{u_{1}^{2} }}{4} \cdot Y_{rk}^{\prime } - \frac{{u_{1} }}{2} \cdot i \cdot \left( {\omega_{k} - \omega_{s} } \right) \cdot \phi_{r2} - u_{0} \cdot u_{1} \cdot \phi_{r2}^{\prime } } \right) \cdot \bar{Y}_{rs}^{\prime } } \right\rangle } \right|_{{x = x_{r} }} \\ S_{6} & = \frac{1}{2} \cdot v_{k1}^{2} \cdot \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {\bar{Y}_{{\left( {r + 1} \right)s}} } *\left\langle {Y_{{\left( {r + 1} \right)k}}^{{\prime \prime }} \cdot \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {\bar{Y}_{{\left( {r + 1} \right)s}}^{\prime 2} \cdot {\text{d}}x} } + \bar{Y}_{{\left( {r + 1} \right)s}}^{{\prime \prime }} \cdot \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {2 \cdot Y_{{\left( {r + 1} \right)k}}^{\prime } \cdot \bar{Y}_{(r + 1)s}^{\prime } \cdot {\text{d}}x} } } \right\rangle \cdot {\text{d}}x} \\ S_{7} & = - \sum\limits_{r = 0}^{n} {\int\limits_{{x_{r} }}^{{x_{r + 1} }} {\bar{Y}_{{\left( {r + 1} \right)s}} } *\left\langle {u_{1} \cdot \frac{{\omega_{s} }}{2} \cdot \bar{\phi }_{(r + 1)4}^{\prime } + u_{0} \cdot u_{1} \cdot \bar{\phi }_{(r + 1)4}^{{\prime \prime }} + \frac{{u_{1}^{2} }}{4} \cdot \bar{Y}_{{\left( {r + 1} \right)s}}^{{\prime \prime }} } \right\rangle \cdot {\text{d}}x} + \sum\limits_{r = 1}^{n} {\alpha \cdot \left. {\left\langle { - u_{1} \cdot \frac{{\omega_{s} }}{2} \cdot \bar{\phi }_{r4}^{\prime } \cdot \bar{Y}_{rs} + \left( {\frac{{u_{1}^{2} }}{4} \cdot \bar{Y}^{\prime}_{rs} + u_{0} \cdot u_{1} \cdot \bar{\phi }_{r4}^{\prime } } \right) \cdot \bar{Y}_{rs}^{\prime } } \right\rangle } \right|}_{{x = x_{r} }} . \\ \end{aligned}$$

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Sarıgül, M. Internal Resonance of Axially Moving Beams with Masses. Iran J Sci Technol Trans Mech Eng 43, 1–16 (2019). https://doi.org/10.1007/s40997-017-0109-x

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