Parametric and External Resonances in Kinematically and Externally Excited Nonlinear Spring Pendulum

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

doi:10.1142/s0218127411030313

Cited by 39 publications

(31 citation statements)

References 14 publications

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“…= 0.01 and 1 = −0.01, respectively, while parts (e) and (f) display the projections of the phase planes at the same values of 1 . Therefore, the present results are in a good agreement with the known previous results as in [17] (in the absence of the damping force ) and [18,19] (with the consideration of ).…”

Section: Examplessupporting

confidence: 93%

“…The main task of this section is to determine the resonances cases. For this purpose, we note that solutions (17)- (18) have singularities when any of their denominators equals zero. So the secular terms appearing on the right sides of (14)-(15) make system (17)- (18) insignificant when some frequency conditions are hold.…”

Section: Modulation Equations Near Resonancesmentioning

confidence: 99%

“…For this purpose, we note that solutions (17)- (18) have singularities when any of their denominators equals zero. So the secular terms appearing on the right sides of (14)-(15) make system (17)- (18) insignificant when some frequency conditions are hold. According to this discussion and to the previous solutions, the resonance cases can be classified as follows:…”

Section: Modulation Equations Near Resonancesmentioning

confidence: 99%

“…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. However, in [17] the authors investigated the motion of a spring pendulum which is moving in circular path.…”

Section: Introductionmentioning

confidence: 99%

“…However, in [17] the authors investigated the motion of a spring pendulum which is moving in circular path. In [18,19] they generalized their previous work by considering the effect of a damper to the motion.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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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Section: Examplessupporting

confidence: 93%

Section: Modulation Equations Near Resonancesmentioning

confidence: 99%

Section: Modulation Equations Near Resonancesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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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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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On the solutions and stability for an auto-parametric dynamical system

2022

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The main goal of this study is to look at the motion of a damped two degrees-of-freedom (DOF) auto-parametric dynamical system. Lagrange’s equations are used to derive the governing equations of motion (EOM). Up to a good desired order, the approximate solutions are achieved utilizing the method of multiple scales (MMS). Two cases of resonance, namely; internal and primary external one are examined simultaneously to explore the solvability conditions of the motion and the corresponding modulation equations (ME). These equations are reduced to two algebraic equations, through the elimination of the modified phases, in terms of the detuning parameters and the amplitudes. The kind of stable or unstable fixed point is estimated. In certain plots, the time histories graphs of the achieved solutions, as well as the adjusted phases and amplitudes are used to depict the motion of the system at any instant. The conditions of Routh–Hurwitz are used to study the various stability zones and their analysis. The achieved outcomes are considered to be novel and original, in which the used strategy is applied on a particular dynamical system. The significance of the studied system can be observed in its applications in a number of disciplines, such as swaying structures and rotor dynamics.

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Parametric and External Resonances in Kinematically and Externally Excited Nonlinear Spring Pendulum

Cited by 39 publications

References 14 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

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

On the solutions and stability for an auto-parametric dynamical system

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