Asymptotic Analysis of Resonances in Nonlinear Vibrations of the 3-dof Pendulum

Awrejcewicz, Jan; Starosta, R.; Sypniewska–Kamińska, Grażyna

doi:10.1007/s12591-012-0129-3

Cited by 50 publications

(34 citation statements)

References 33 publications

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“…As we developed previously, substitute (16)- (17) into (15) and then remove secular terms to obtain the third-order approximations in the form…”

Section: ̃3mentioning

confidence: 99%

“…Substituting (19) into 14- (15) leads to appearance of secular terms. Their elimination produces the following solvability conditions.…”

Section: Modulation Equations Near Resonancesmentioning

confidence: 99%

“…The authors concentrated on predicting the resonance conditions and they studied these cases. This problem was generalized in [15] to study the vibrations of a rigid body as a pendulum model. Many interested examples for the motion of nonlinear systems can be found in [16][17][18][19][20] and the references herein.…”

Section: Introductionmentioning

confidence: 99%

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On the Motion of Harmonically Excited Spring Pendulum in Elliptic Path Near Resonances

Amer

Bek

Hamada

2016

Advances in Mathematical Physics

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The response of a nonlinear multidegrees of freedom (M-DOF) for a nature dynamical system represented by a spring pendulum which moves in an elliptic path is investigated. Lagrange’s equations are used in order to derive the governing equations of motion. One of the important perturbation techniques MS (multiple scales) is utilized to achieve the approximate analytical solutions of these equations and to identify the resonances of the system. Besides, the amplitude and the phase variables are renowned to study the steady-state solutions and to recognize their stability conditions. The time history for the attained solutions and the projections of the phase plane are presented to interpret the behavior of the dynamical system. The mentioned model is considered one of the important scientific applications like in instrumentation, addressing the oscillations occurring in sawing buildings and the most of various applications of pendulum dampers.

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“…As we developed previously, substitute (16)- (17) into (15) and then remove secular terms to obtain the third-order approximations in the form…”

Section: ̃3mentioning

confidence: 99%

“…Substituting (19) into 14- (15) leads to appearance of secular terms. Their elimination produces the following solvability conditions.…”

Section: Modulation Equations Near Resonancesmentioning

confidence: 99%

See 1 more Smart Citation

On the Motion of Harmonically Excited Spring Pendulum in Elliptic Path Near Resonances

Amer

Bek

Hamada

2016

Advances in Mathematical Physics

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“…However, they are a valuable complement to the topic of rigid body motion, which includes the load motion. The motion of rigid bodies as a motion of the mathematical and physical pendulum is shown in works [21][22][23][24][25][26][27]. The deformability of the rope system was included in the paper [21].…”

Section: Introductionmentioning

confidence: 99%

“…Transport of energy between modes of vibration is also presented. The problem of spring pendulum nonlinear vibrations and resonances has also been extensively described in articles [25,26]. In both of these works, the asymptotic method was used for analytical calculations.…”

Section: Introductionmentioning

confidence: 99%

The dynamic analysis of load motion during the interaction of wind pressure

2020

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This work presents the analysis of the load motion during the interaction of wind pressure. The load was treated as a rigid body, and the rope system model as a non-deformed. The influence of effective area of wind pressure on load motion was considered. The theoretical model of load motion was presented, which may be an universal approach for transporting machines equipped with a rope-lifting system. To define the orientation of the movable Cartesian coordinate system related to the load, Bryant angles were used. An algorithm and computational program were developed to allow for analysis of dynamic phenomena. The initial problem was solved with the use of the ode45 calculation procedure in the Matlab software on the basis of the Runge-Kutta 4th Order Method. The obtained results were verified with the experimental ones achieved in the wind tunnel and Tracker program. Numerical calculations using commercial software SolidWorks were also presented. In the experiment, the spatial motion of the load was analysed. Experimental tests were carried out for gust of wind and constant temperature and humidity. In addition, the paper presents the application of the proposed method for a load carried by a rotary crane. After taking into account the control functions resulting from the nature of the work of any machine, the formulated model can be a full description of the carried load motion taking into account external forces.

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Short comments on chaotic behavior of a double pendulum with two subharmonic frequencies and in the main resonance zone

Avanço

Balthazar

Tusset³

et al. 2021

Z Angew Math Mech

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The parametric pendulum and others different pendular mechanisms were extensively analyzed in current literature because of the chaotic motions.Recently, practical reasons, like energy harvesting, motivated the study of some pendular systems. The resonance and the chaos may occur in multiple and submultiple frequencies based on the natural frequency of the pendulum. In the present paper, a double pendulum is analyzed with two subharmonic frequencies and in the main resonance zone. The results found demonstrate high similarities of the double pendulum with the parametric pendulum when compared the bifurcation diagrams based on varying the amplitude. The conditions in the numerical simulation include a variation in the viscous friction in the joints and different amplitudes and frequency of excitation. The differential equations used dimensionless parameters representing time, frequency, amplitude, friction, position and velocities. Concerning the results, all tests performed have pointed that the same type of motion takes place in the two pendulums after the transient time have already elapsed.

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Asymptotic Analysis of Resonances in Nonlinear Vibrations of the 3-dof Pendulum

Cited by 50 publications

References 33 publications

On the Motion of Harmonically Excited Spring Pendulum in Elliptic Path Near Resonances

On the Motion of Harmonically Excited Spring Pendulum in Elliptic Path Near Resonances

The dynamic analysis of load motion during the interaction of wind pressure

Short comments on chaotic behavior of a double pendulum with two subharmonic frequencies and in the main resonance zone

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