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The vibration behavior and the synchronization between some internal points of four coupled self-excited beams are numerically studied. Coupling through the root of the beams is considered. The transverse displacements of the internal points and the beam tips are monitored, and the power spectra of the resulting time series are employed to determine the oscillation frequencies. The synchronization between beams is analyzed using phase portraits and correlation coefficients. Numerical results show multiple frequencies in the vibration pattern, and complex patterns of synchronization between pairs of beams.

Vibration of elastic beams is an important issue given that failure by fatigue in mechanical systems may arise. The majority of the reports published on this topic are aimed to the analysis of vibration modes [

The collective behavior, particularly synchronization, of spatially extended coupled oscillators has been poorly studied in the past. Nowadays, the majority of the reports on the collective behavior of coupled oscillators are focused on oscillators described by ordinary differential equations, e.g. [

In this work, the vibration behavior along the length of a single beam and the synchronization among internal points of four coupled self-excited elastic beams are numerically studied. The transverse displacements of three internal points along the length plus the tip of each beam are monitored, and the power spectra of the resulting time series are employed to determine the oscillation frequencies. The synchronization between pairs of beams is studied by means of the corresponding phase portraits and correlation coefficients of the time series of each monitoring point. As in [

The complete derivation of the mathematical model can be found in [

where X is the distance from the root, Y(X, τ) is the transverse displacement, and τ is the time. As in [

where A is a constant which rules the self-excited oscillations. Assuming that there are four beams, Equation (2) is rewritten as

where i = 1, 2, 3, 4. As in [

The remaining boundary condition must consider the coupled nature of the beams. The displacements at the roots of beams are null, so the coupling through the slopes of the beams at the roots is considered [

where K is the coupling parameter. Equation (7) can form a linear system as follows:

The matrix of Equation (8) represents a coupling matrix, and is named here as K. It is verified that

phenomenon of beams may arise [

For the sake of simplicity, the finite differences method was selected to numerically solve the mathematical model of the coupled beams. The explicit finite difference schemes reported in [^{−}^{6}. To prevent undesired transient solutions, computer runs were carried out for long dimensionless integration times, e.g. τ = 10,000. The initial conditions of the four beams were as follows: X_{1} = 1, Y_{1} = 0, X_{2} = 0, Y_{2} = 1, X_{3} = 1, Y_{3} = 1, X_{4} = 0, Y_{4} = 0.

Fortran programming language was employed to create the computer program. The computer runs aim to analyze the influence of A, the self-oscillation constant, on the coupled behavior of the beams. Three values of A are considered: 0.1, 0.5 and 1.0, which cover from linear to nonlinear behavior of the self-excited beams. To test the internal synchronization, four monitoring points in the X coordinate of each beam were selected: X_{A} = 0.0965, X_{B} = 0.50, X_{C} = 0.8846 and X_{D} = 1.0. To simulate a weak coupling, during the computer runs the value of the coupling parameter was kept constant at 0.1, i.e. K = 0.1.

Given the large amount of data generated by the computer runs, and considering that the vibration behavior of the four beams are qualitatively similar, the time series and the power spectra presented here are just for the beam 1 of

Figures 3-5 show the time series of the transverse displacement for the monitors X_{A}, X_{B}, X_{C} and X_{D} of beam 1 for the considered values of the self-oscillating constant, i.e. A = 0.1, A = 0.5 and A = 1.0, respectively. _{A} and X_{C} monitoring points exhibit a similar oscillation pattern: a dynamic behavior in which a dominant high frequency wave (3615.78 Hz) is enveloped by a low frequency one (19.59 Hz). The values of these frequencies were obtained through the power spectra depicted in _{A} and X_{C} monitors are shown in _{B} and X_{D} monitoring points of the beam 1 share an analogous oscillation pattern for A = 0.1 and A = 0.5: a low frequency dominant component (19.59 Hz for A = 0.1, and 13.35 Hz for A = 0.5), and a high frequency component which was undetectable by the power spectra: see

The time series of the transverse displacements monitoring points for A = 1.0 are shown in _{A} and X_{B} monitors (see _{A} monitor wave has a value of 3, whereas the amplitude of the X_{B} monitor wave has a value of 2. Besides, the power of the low frequency component of the X_{A} monitor is larger than the power of the low frequency component of the X_{B} monitor, as is observed in the power spectra of _{C} monitoring point shows an inverse oscillation pattern related to the X_{A} and X_{B} monitors: a low frequency dominant pattern (67.0 Hz) and a high frequency (3667.5 Hz) weak component. For its part, the X_{D} monitor exhibits a single low frequency component of 67.06 Hz, as is shown in the time series of

The coupled vibration behavior of the beams is analyzed by means of phase portraits and correlation coefficients, which were obtained from the time series of the transverse displacements monitoring points. _{A} monitor. Adjacent pairs of beams 1 - 2 (

the above pair of beams have a value of −0.7670, as is shown in the first column of

The phase portraits and the correlation coefficients for A = 0.5 and the X_{A} monitor are shown in

10(d)) and 3 - 4 (

Pair of beams | X_{A } | X_{B } | X_{C } | X_{D } |
---|---|---|---|---|

1 - 2 | −0.7670 | 0.9712 | −0.7650 | 0.9970 |

1 - 3 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

1 - 4 | −0.7670 | 0.9712 | −0.7650 | 0.9970 |

2 - 3 | −0.7670 | 0.9712 | −0.7650 | 0.9970 |

2 - 4 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

3 - 4 | −0.7670 | 0.9712 | −0.7650 | 0.9970 |

Pair of beams | X_{A } | X_{B } | X_{C } | X_{D } |
---|---|---|---|---|

1 - 2 | −0.0667 | −0.9147 | 0.7439 | 0.9915 |

1 - 3 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

1 - 4 | −0.0667 | −0.9147 | 0.7439 | 0.9915 |

2 - 3 | −0.0667 | −0.9147 | 0.7439 | 0.9915 |

2 - 4 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

3 - 4 | −0.0667 | −0.9147 | 0.7439 | 0.9915 |

pairs of beams, i.e. 1 - 3 (

The phase portraits between the diverse pair of beams for A = 1.0 and the X_{A} monitor are displayed in

For the X_{B}, X_{C} and X_{D} transverse displacement monitors, instead of the phase portraits just the correlation coefficients are displayed at the columns 2 - 4 of Tables 1-3 for A = 0.1, A = 0.5 and A = 1.0, respectively. These data show that for A = 0.1 and A = 0.5 the X_{D} monitor, located at X = 1.0, i.e. the tip of the beams, exhibits complete in-phase synchronization for every pair of beams. However, for A = 1.0 the X_{D} monitor presents a behavior which goes from small synchronization to anti-phase and in-phase synchronization with different amplitudes of the transverse displacement. The most irregular coupled behavior is presented by all the monitors for A = 1.0, as is shown in

From the numerical results, the following conclusions arise:

1) A complex and different vibration behavior is exhibited along the distance from the root for a single coupled beam.

2) The power spectra show that for a single coupled beam with A = 0.1 and A = 0.5, the transverse displacements of the beams generally exhibit two vibration frequencies, and for A = 1.0 generally just a single frequency is present.

3) Generally speaking, not adjacent coupled beams exhibit complete in-phase or anti-phase synchronization for A = 0.1 and A = 0.5.

4) Irregular and diverse coupled behavior is presented along the distance from the root by all the monitors for A = 1.0.

5) Among the considered monitoring points, the location at the tip of the beams, i.e. X = X_{D}, presents complete

synchronization for A = 0.1 and A = 0.5.

Pair of beams | X_{A } | X_{B } | X_{C } | X_{D } |
---|---|---|---|---|

1 - 2 | −0.0732 | 0.6226 | 0.0178 | 0.6049 |

1 - 3 | −0.9927 | −0.2311 | −0.8638 | −0.0216 |

1 - 4 | 0.0804 | −0.8054 | −0.0171 | −0.6182 |

2 - 3 | 0.0804 | 0.6019 | 0.1497 | 0.5985 |

2 - 4 | −0.9927 | −0.0853 | −0.8606 | −0.0067 |

3 - 4 | −0.0730 | 0.7299 | 0.0203 | 0.6159 |

6) As the value of the self-oscillation constant of the beams is increased from 0.1 to 1.0, i.e. the beams become more nonlinear, the vibration pattern become more complex and disordered.

The author gratefully acknowledges the original inspiration and guidance of Professor Mihir Sen, from the University of Notre Dame, IN, during the initial conception of this research.

Miguel A. Barron, (2016) Internal Vibration and Synchronization of Four Coupled Self-Excited Elastic Beams. Open Journal of Applied Sciences,06,501-513. doi: 10.4236/ojapps.2016.68050