Elsevier

Mechanism and Machine Theory

Volume 51, May 2012, Pages 131-144
Mechanism and Machine Theory

Primary and parametric resonances of asymmetrical rotating shafts with stretching nonlinearity

https://doi.org/10.1016/j.mechmachtheory.2011.12.012Get rights and content

Abstract

In this paper, primary and parametric resonances of a simply supported nonlinear rotating asymmetrical shaft with unequal mass moments of inertia and bending stiffness in the direction of principal axes are simultaneously considered. The nonlinearities are due to the stretching and large amplitude. The method of multiple scales is applied to the ordinary and partial differential equations of motion. The achieved results are in a very good agreement. The influences of inequality of mass moments of inertia and bending stiffness in the direction of principal axes, inequality between two eccentricities corresponding to the principal axes and external damping on the steady state response of the asymmetrical rotating shaft are investigated. The loci of bifurcation points are plotted as functions of damping coefficient. The numerical simulation is utilized to verify the multiple scales method results. The results of multiple scales method are in a good agreement with those of numerical simulation.

Highlights

► Simultaneous resonances of a nonlinear asymmetrical rotating shaft are considered. ► The stability of the asymmetrical and symmetrical shaft is investigated. ► Asymmetry due to mass moments of inertia would change the bifurcation. ► Difference between eccentricities would change the stability and bifurcation. ► In the asymmetrical shaft bifurcations occurred for any speed change.

Introduction

Fan et al. [1] considered the linear vibration control of a horizontal rotor with an asymmetrical mass moments of inertia. The shaft was rigid, and the bending stiffness was neglected. They investigated the effect of asymmetrical mass moments of inertia on the system performance. Badlani et al. [2] studied the stability of asymmetrical linear shafts with unequal bending stiffness in the direction of principal axes. They found new instability region by considering Timoshenko beam theory. Sabanucu et al. [3] investigated dynamic stability of pre-twisted Timoshenko beam having asymmetric aerofoil cross section subject to lateral parametric excitation. They investigated combination resonances by a finite element model, which the rotary inertia and shear deformation were included. They showed that pre-twist angle has an influence on coupling and shear coefficient.

Turhan et al. [4] considered dynamic stability of linear rotating beam with a periodically fluctuating speed. They showed that a speed fluctuation might have a stabilizing effect on statically unstable inward-oriented beams. Kamel and Bauomy [5] considered the nonlinear vibrations of a rotor with active magnetic bearing under multi-parametric excitations. They found response of the system under primary and internal resonance conditions, and showed that in the system much behavior such as chaotic, hardening nonlinearities, softening nonlinearities and jump phenomenon would occur. Al-Nassar et al. [6] investigated nonlinear vibration of a rotating blade on a torsionaly flexible shaft. They obtained instability regions of the blade vibrations when the torsional excitation frequency is lower than the blade bending natural frequency. Sheu [7] investigated parametric instability of a cantilever shaft-disk system subjected to axial and follower loads, respectively. They used Bolotin method to find the instability regions. They showed that instability due to periodic axial load was more than periodic follower force in the same static load factor.

Rajalingham et al. [8] investigated vibrations of a rotor with unequal stiffness in direction of two principal axes. They showed that instability of the rotor could be removed using suitable parameters for supports. Pei [9] studied stability boundaries of a spinning linear rotor with parametric excitation. It was shown that the instability region obtained by Bolotin method was larger than that of the Floquet method. Wang et al. [10] investigated the nonlinear-coupled dynamic of the blade-rotor-bearing systems, and studied influence of nonlinear vibrations of rotor on the blade vibrations. It was shown that the system has a various kind of phenomenon such as period-doubling bifurcation, the multi-period and quasi-period motions. Verichev et al. [11] considered damping vibration of a rotary system without change in the visco-elastic properties of the system. They used a harmonic addition to the constant speed of the system for reduction of vibration. They showed that damping vibration could be achieved by selection of the best parameters for harmonic addition.

Sinou [12] probed nonlinear vibrations of a rotor supported by ball bearings. The source of excitation was due to imbalances. The nonlinearities were due to radial clearance and Herztian contact. It was shown that amplitude of vibrations in the critical and sub-critical resonances depends on the radial clearance and imbalances. In addition, for small excitation, system would have softening type of nonlinearity, but for the large excitation, system would have hardening type of nonlinearity. Chang-Jian and Chen [13] investigated nonlinear vibrations of rotor-bearing system with rub impact effect. They showed that system has quasi-periodic, periodic and 2T-periodic responses for various kinds of speed ratio. In addition, the same authors investigated dynamic analysis of the rotor-bearing system supported by journal bearings. They used bifurcation diagrams, Lyapunov exponent, power spectrum and Poincare maps to analyze the nonlinear system. It was shown that for small speed ratio, system has 2T-periodic motion and for other values of speed ratio, system has chaotic, quasi-periodic and periodic motions [14].

Karpenko et al. [15] considered nonlinear dynamics of rotor system with bearing clearance effect. Using construction of basin attraction, they showed periodic, quasi-periodic and chaotic motions that occurred. Zhang and Meng [16] studied stability, chaotic dynamics and bifurcation in the micro rotor system. Their model was a Jeffcott rotor with rub-impact. They obtained periodic, chaotic and quasi-periodic motion in the micro rotor system. It was shown that damping coefficient could restrict chaotic motion and reduce amplitude of vibration. Khadem et al. [17] investigated the primary resonances of nonlinear in-extensional rotating shaft. They used multiple scales method to analyze of the rotor system. They showed effect of damping and eccentricity on the system response, and loci of bifurcation points. Luczko [18] developed a geometrically non-linear model of a rotating shaft. Their model included the Von-Karman non-linearity, non-linear curvature, large displacements and rotations as well as gyroscopic and shear deformation effects. He investigated internal resonances. Hosseini et al. [19] investigated free vibration analysis of a rotating shaft with nonlinearities in curvature and inertia. They used multiple scales method and numerical method to analyze the rotor with in-extensional nonlinearities.

In this study, primary and parametric resonances of a simply supported stretching asymmetrical rotating shaft with large amplitudes are simultaneously investigated. Rotary inertia and gyroscopic effects are included, however, shear deformation is ignored. The inequality between both mass moments of inertia and bending stiffness in the direction of principal axes is considered. The over mentioned effects are not considered simultaneously, before. In addition, in this paper the parametric excitation is due to inequality between mass moments of inertia and bending stiffness in the direction of principal axes. However, mostly in the literature, the parametric excitation is due to lateral force, speed fluctuations or stiffness anisotropy.

The equations of the motion are derived by the Hamiltonian approach with the stretching assumption in the complex plane. The system is analyzed by the multiple scales method. It is shown that in the primary and parametric resonances of asymmetrical rotating shaft with gyroscopic effect, only forward whirl mode is excited which is similar to primary resonances in a symmetrical rotating shaft [17]. The frequency response curves are plotted for the first two modes of the symmetrical and asymmetrical shafts. It is shown that these resonance curves are of hardening type. The effects of inequality between diametrical mass moments of inertia in the direction of principal axes, inequality between eccentricities in the direction of principal axes and external damping on the steady state response of the rotating asymmetrical shaft are investigated. The characteristics of the steady state solutions in the symmetrical and asymmetrical shafts are probed and the imaginary part of the eigenvalues corresponding to the steady state solution is plotted versus damping coefficient. It is shown that the characteristics of the steady state stable solution for asymmetrical shaft are a stable node or a stable spiral, but in the symmetrical shaft for the same detuning parameter, it is only a stable spiral. The loci of bifurcation points for symmetrical and asymmetrical shafts are plotted as functions of damping coefficient. To validate the perturbation results, a numerical simulation is utilized and a good agreement is achieved.

Section snippets

Theoretical formulation and equations of motion

Fig. 1 represents a rotating shaft with undeformed length l. In this figure, X, Y, and Z are used as inertial coordinate system. Also, the set of x, y, z are defined as principal axis originated from the centroid of the cross section of the shaft. The components of the deformation in the X, Y, and Z directions in arbitrary location x are denoted as u(x, t), v(x, t), and w(x, t) respectively. The shaft is asymmetrical and slender for which the gravity effect is neglected.

The kinetic energy for a

Solution of equations of motion

In the following sections multiple scales method is applied to both ordinary and partial differential equations of motion. The partial differential equations are converted to ordinary differential equations by the Galerkin method.

Numerical study

For numerical study of primary and parametric resonances of nonlinear rotating asymmetrical shaft, the asymmetrical shaft with the following dimensionless parameters is used:I3=0.003275,I2=0.002482,α= 0.0005,c=0.01.

In addition, in some cases the other values have been considered.

Figs. 2 and 3 show the frequency response curves of the symmetrical and asymmetrical shafts for the first and the second modes. These figures are plotted for e1 = e2 = 0.05. It is seen that the curves near the pick are bent

Conclusion

Simultaneous primary and parametric resonances due to inhomogeneous part of equations of motion and asymmetry of shaft for a rotating shaft with stretching nonlinearity are investigated. Rotary inertia and gyroscopic effects are considered, but shear deformation is ignored. Multiple scales method is applied to both the ordinary and partial differential equations of the motion. The effect of difference between two eccentricities on the asymmetrical shaft is considerable. The difference between

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