Primary resonances of a nonlinear in-extensional rotating shaft

Khadem, S.E.; Shahgholi, Majid; Hosseini, S.A.A.

doi:10.1016/j.mechmachtheory.2010.03.012

Cited by 53 publications

(25 citation statements)

References 27 publications

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“…To analyze aforementioned resonances damping and base excitation are scaled; hence it can be considered = 2 ,̇=̇. Substituting Equation 30 into Equation 26, using Equation 29 and equating the coefficients of the same power of on both sides, the following equations are obtained:…”

Section: Solution Methodologymentioning

confidence: 99%

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Forced vibrations of nonlinear symmetrical and asymmetrical rotating shafts mounted on a moving base

Shahgholi

Payganeh

2018

Z Angew Math Mech

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This paper deals with the combinational resonances of symmetrical and asymmetrical rotating shafts with base motions. The nonlinearities are owing to the large amplitude of the hinged‐ hinged the shaft. Generally, the asymmetrical shaft has unequal principal mass moments of inertia and flexural stiffnesses. The system is exposed to three sources of excitation as a result of the asymmetry of the shaft, dynamic imbalances, and base excitations. Using the variational approach the nonlinear governing equations of motion and related boundary conditions are derived in the complex form. These equations of motion are analyzed by the use of the perturbation method. It is shown that three combinational resonances occur in the system. In the two cases of combinational resonances the asymmetry of shaft has no effect on the steady‐state responses. However, in one case of the combinational resonances, the asymmetry of the shaft has a significant effect. Also, in each case, the steady‐state solutions and their stability are investigated. The effect of various parameters such as damping coefficient, eccentricities, and asymmetry of the shaft on the periodic solutions, the stability of the solutions and bifurcations are probed. Additionally, the analytical results are verified by the numerical simulations that are obtained a good agreement between these results.

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Section: Solution Methodologymentioning

confidence: 99%

“…It ought to be noted that the angles ψ and θ can be obtained as []:

ψ = {prefixsin}^{- 1} \frac{v^{'}}{\sqrt{{((), 1 + u^{'})}^{2} + {v^{'}}^{2}}}, θ = {prefixsin}^{- 1} \frac{- w^{'}}{\sqrt{{((), 1 + u^{'})}^{2} + {v^{'}}^{2} + {w^{'}}^{2}}}

…”

Section: Equations Of Motionmentioning

confidence: 99%

Forced vibrations of nonlinear symmetrical and asymmetrical rotating shafts mounted on a moving base

Shahgholi

Payganeh

2018

Z Angew Math Mech

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“…The present paper tries to answer these questions. This article is a continuation of the author's previous papers [29][30][31][32], in which bifurcation and chaotic behavior of the rotor system is investigated. The shaft is modeled as a Rayleigh simply supported beam, spinning with 4 constant rotational speed.…”

Section: Introductionmentioning

confidence: 91%

“…In previous articles [29][30][31][32], dynamical behavior of nonlinear inextensional rotating shafts was studied by the author. In those papers, the periodic response of the shaft was considered and other types of shaft motion like quasi periodic and chaotic responses were not studied.…”

Section: Introductionmentioning

confidence: 99%

Chaos and Bifurcation in Nonlinear Inextensional Rotating Shafts

Hosseini

2018

Scientia Iranica

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In this paper, bifurcation and chaotic behavior of in-extensional rotating shafts are investigated. The shaft is modeled as a Rayleigh simply supported beam, spinning with constant rotational speed. Using two-mode Galerkin truncation, the partial differential equations of motion are discretized and then with the aid of numerical simulations, the dynamical behavior of the rotating shaft is studied. Using tools from nonlinear dynamics, such as time history, bifurcation diagram, Poincaré map, Lyapunov exponents, and amplitude spectra, a comprehensive analysis is made to characterize the complex behavior of the rotating shaft. Periodic (synchronous), quasiperiodic, chaotic and transient chaotic responses are observed in the neighborhood of the second critical speed. The effect of rotary inertia and damping on the dynamics of the rotating shaft is considered. It is shown that the chaotic response is possible for a shaft with weak nonlinearity without the existence of any internal resonance.

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“…(22), creates first-and second-order nonlinearity in the temporal equation of motion. If the gravity effects are neglected, nonlinearity terms are only cubic [11][12][13][14]. Due to this fact, to apply the multiple scales method, three time scales are applied, T 0 = t, T 1 = εt and T 2 = ε 2 t. q (t) and p (t) are expanded in the following forms…”

Section: Formulation Of Dynamic Displacementmentioning

confidence: 99%

Nonlinear forced vibrations analysis of overhung rotors with unbalanced disk

2015

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In this paper, primary resonances of a cantilever flexible shaft carrying a rigid disk at its free end (overhung rotor) are investigated. Unbalance forces due to eccentricity and disk skew are simultaneously considered, and the shaft has initially static deflection due to the rotor's weight which causes asymmetry in the equations of motion. The rotor has large amplitude vibrations, which lead to nonlinearities in curvature and inertia. In the model, rotary inertia and gyroscopic effects are included, but shear deformation is neglected. The method of multiple scales is applied to the discretize differential equations of motion. It is shown that the static deflection creates second-order nonlinear terms and near the primary resonances only the forward modes are excited. With considering the gravity, the inertial nonlinearities become stronger near the primary resonances. So, gravity decreases the hardening effect, and the nonlinear system tends to a linear system. It is concluded that the gravity effect has a softening effect. By using this property of gravity, a relation between weight and external forces is derived, in which by applying this relation the jumping phenomenon is eliminated. Numerical examples are presented, and the result is verified by numerical simulations.

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Primary resonances of a nonlinear in-extensional rotating shaft

Cited by 53 publications

References 27 publications

Forced vibrations of nonlinear symmetrical and asymmetrical rotating shafts mounted on a moving base

Forced vibrations of nonlinear symmetrical and asymmetrical rotating shafts mounted on a moving base

Chaos and Bifurcation in Nonlinear Inextensional Rotating Shafts

Nonlinear forced vibrations analysis of overhung rotors with unbalanced disk

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