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**Natural frequencies of nonlinear vibration of axially moving beams.**
*(English)*
Zbl 1215.74032

Summary: Axially moving beam-typed structures are of technical importance and present in a wide class of engineering problem. In the present paper, natural frequencies of nonlinear planar vibration of axially moving beams are numerically investigated via the fast Fourier transform (FFT). The FFT is a computational tool for efficiently calculating the discrete Fourier transform of a series of data samples by means of digital computers. The governing equations of coupled planar of an axially moving beam are reduced to two nonlinear models of transverse vibration. Numerical schemes are respectively presented for the governing equations via the finite difference method under the simple support boundary condition. In this paper, time series of the discrete Fourier transform is defined as numerically solutions of three nonlinear governing equations, respectively. The standard FFT scheme is used to investigate the natural frequencies of nonlinear free transverse vibration of axially moving beams. The numerical results are compared with the first two natural frequencies of linear free transverse vibration of an axially moving beam. And results indicate that the effect of the nonlinear coefficient on the first natural frequencies of nonlinear free transverse vibration of axially moving beams. The numerical results also illustrate the three models predict qualitatively the same tendencies of the natural frequencies with the changing parameters.

### MSC:

74H45 | Vibrations in dynamical problems in solid mechanics |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

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\textit{H. Ding} and \textit{L.-Q. Chen}, Nonlinear Dyn. 63, No. 1--2, 125--134 (2011; Zbl 1215.74032)

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### References:

[1] | Mote, C.D. Jr.: A study of band saw vibrations. J. Franklin Inst. 276, 430–444 (1965) |

[2] | Mote, C.D. Jr., Naguleswarn, S.: Theoretical and experimental band saw vibrations. ASME J. Eng. Ind. 88, 151–156 (1966) |

[3] | Wickert, J.A., Mote, C.D. Jr.: Classical vibration analysis of axially moving continua. J. Appl. Mech. 57, 738–744 (1990) · Zbl 0724.73125 |

[4] | Öz, H.R., Pakdemirli, M.: Vibrations of an axially moving beam with time dependent velocity. J. Sound Vib. 227, 239–257 (1999) |

[5] | Öz, H.R.: On the vibrations of an axially traveling beam on fixed supports with variable velocity. J. Sound Vib. 239, 556–564 (2001) |

[6] | Özkaya, E., Öz, H.R.: Determination of natural frequencies and stability regions of axially moving beams using artificial neural networks method. J. Sound Vib. 254, 782–789 (2002) |

[7] | Öz, H.R.: Natural frequencies of axially travelling tensioned beams in contact with astationary mass. J. Sound Vib. 259, 445–456 (2003) |

[8] | Kong, L., Parker, R.G.: Approximate eigensolutions of axially moving beams with small flexural stiffness. J. Sound Vib. 276, 459–469 (2004) |

[9] | Chen, L.Q., Yang, X.D.: Vibration and stability of an axially moving viscoelastic beam with hybrid supports. Eur. J. Mech. A-Solids 25, 996–1008 (2006) · Zbl 1104.74031 |

[10] | Ghayesh, M.H., Khadem, S.E.: Rotary inertia and temperature effects on non-linear vibration, steady-state response and stability of an axially moving beam with time-dependent velocity. Int. J. Mech. Sci. 50, 389–404 (2008) · Zbl 1264.74088 |

[11] | Wang, L., Ni, Q.: Vibration and stability of an axially moving beam immersed in fluid. Int. J. Solids Struct. 45, 1445–1457 (2008) · Zbl 1169.74427 |

[12] | Tang, Y.Q., Chen, L.Q., Yang, X.D.: Natural frequencies, modes and critical speeds of axially moving Timoshenko beams with different boundary conditions. Int. J. Mech. Sci. 50, 1448–1458 (2008) · Zbl 1264.74101 |

[13] | Özkaya, E., Sarigul, M., Boyaci, H.: Nonlinear transverse vibrations of a slightly curved beam carrying a concentrated mass. Acta Mech. Sin. 25(6), 871–882 (2009) · Zbl 1269.74101 |

[14] | Wickert, J.A.: Non-linear vibration of a traveling tensioned beam. Int. J. Non-linear Mech. 27, 503–517 (1992) · Zbl 0779.73025 |

[15] | Thurman, A.L., Mote, C.D. Jr.: Free, periodic, nonlinear oscillation of an axially moving strip. J. Appl. Mech. 36, 83–91 (1969) · Zbl 0175.23001 |

[16] | Tabarrok, B., Leech, C.M., Kim, Y.I.: On the dynamics of an axially moving beam. J. Franklin Inst. 297, 201–220 (1974) · Zbl 0306.73044 |

[17] | Wang, K.W., Mote, C.D. Jr.: Vibration coupling analysis of Band/wheel mechanical systems. J. Sound Vib. 109, 237–258 (1986) |

[18] | Hwang, S.J., Perkins, N.C.: Supercritical stability of an axially moving beam Part I: Model and equilibrium analysis. J. Sound Vib. 154, 381–396 (1992) · Zbl 0924.73105 |

[19] | Hwang, S.J., Perkins, N.C.: Supercritical stability of an axially moving beam Part II: Vibration and stability analysis. J. Sound Vib. 154, 397–409 (1992) · Zbl 0924.73105 |

[20] | Hwang, S.J., Perkins, N.C.: High speed stability of coupled band/wheel systems: Theory and experiment. J. Sound Vib. 169, 459–483 (1994) |

[21] | Riedel, C.H., Tan, C.A.: Coupled, forced response of an axially moving strip with internal resonance. Int. J. Non-linear Mech. 37, 101–116 (2002) · Zbl 1117.74305 |

[22] | Kong, L., Parker, R.G.: Coupled belt-pulley vibration in serpentine drives with belt bending stiffness. J. Appl. Mech. 71, 109–119 (2004) · Zbl 1111.74487 |

[23] | Sze, K.Y., Chen, S.H., Huang, J.L.: The incremental harmonic balance method for nonlinear vibration of axially moving beams. J. Sound Vib. 281, 611–626 (2005) |

[24] | Chen, L.Q., Ding, H.: Steady-state transverse response in coupled planar vibration of axially moving viscoelastic beams. J. Vib. Acoust. 132(1), 011009 (2010) |

[25] | Ding, H., Chen, L.Q.: On two transverse nonlinear models of axially moving beams. Sci. China Ser. E 52, 743–751 (2009) · Zbl 1422.74058 |

[26] | Ding, H., Chen, L.Q.: Equilibria of axially moving beams in the supercritical regime. Arch. Appl. Mech. DOI: 10.1007/s00419-009-0394-y (2009) · Zbl 1271.74223 |

[27] | Nayfeh, A.H., Pai, P.F.: Linear and Nonlinear Structural Mechanics. Wiley, New York (2004) · Zbl 1092.74001 |

[28] | Chen, L.Q., Ding, H.: Steady-state responses of axially accelerating viscoelastic beams: approximate analysis and numerical confirmation. Sci. China Ser. G 51, 1707–1721 (2008) |

[29] | Chen, L.Q., Yang, X.D.: Steady-state response of axially moving viscoelastic beams with pulsating speed: Comparison of two nonlinear models. Int. J. Solids Struct. 42, 37–50 (2005) · Zbl 1093.74526 |

[30] | Chen, L.Q., Yang, X.D.: Nonlinear free vibration of an axially moving beam: Comparison of two models. J. Sound Vib. 299, 348–354 (2007) · Zbl 1241.74021 |

[31] | Boresi, A.P., Chong, K.P., Saigal, S.: Approximate Solution Methods in Engineering Mechanics, 2nd edn. Wiley, New York (2003) |

[32] | Duhamel, P., Vetterli, M.: Fast Fourier transforms: A tutorial review and a state of the art. Signal Process. 19, 259–299 (1990) · Zbl 0704.65106 |

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