Dynamic stability and bifurcation phenomena of an axially loaded flexible shaft-disk system supported by flexible bearing:

Abstract: The parametric instability of a rotor system with electromechanically coupled boundary conditions under periodic axial loads is studied Based on the current flowing piezoelectric shunt damping technique, the detailed rotor model is established by the finite element (FE) method In the matrix assembly procedure, a novel simple process is proposed to make the equations of shunt circuits more conveniently to be introduced into the global FE equations The discrete state transition matrix method which is used for determining the influence of circuit parameters on instability regions in this paper has also been presented The numerical simulation shows that only the combination instability regions exist when the shaft is rotating The mechanical damping has different effect on the simple and combined instability regions These two points are consistent with the previous references, which verifies the obtained FE model In addition, the simulated results also reveal that the introduction of shunt circuits has little influence on the rotor’s original whirling frequencies It gives rise to the appearance of new synchronous whirl modes The new whirling frequencies are combined with the original ones to form the new combination instability regions Furthermore, the resistance of shunt circuits has the same performance as the mechanical damping has That is, moving up the start points of instability regions and expanding its width

Abstract: The parametric instability of an electromechanically coupled single-span rotor-bearing system subjected to periodic axial loads is studied. Here, the rotor system is equipped with two piezoelectric dampers, which has been developed in our previous work. The so-called electromechanically coupled characteristic is namely derived from that damper. By using assumed mode method and Lagrange equation, the equations of motion are derived. The multiple scales method is utilized to obtain the analytical instability boundaries. Numerical simulations based on the discrete state transition matrix method (DSTM) are conducted to verify the analytical results. With the comparison between analytical results and simulated results, we find that the additional combination instability regions are created due to the usage of piezoelectric dampers.

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.

Abstract: The present work deals with the parametric instability regions of a cantilever beam with tip mass subjected to time-varying magnetic field and axial force. The nonlinear temporal differential equation of motion having two frequency parametric excitations is solved using second-order method of multiple scales. The closed-form expressions for the parametric instability regions for three different resonance conditions are determined. The influence of magnetic filed, axial load, damping constant and mass ratio on the parametric instability regions are investigated. These results obtained from perturbation analysis are verified by solving the temporal equation of motion using fourth-order Runge–Kutta method. The instability regions obtained using this method is found to be in good agreement with the experimental result.

Abstract: In this study, the interactive behaviors among transverse magnetic fields, axial loads and external force of a magneto-elastic beam with general boundary conditions are investigated. The axial force and transverse magnetic force are assumed to be periodic with respect to time and two specified frequencies, one for axial force and the other for oscillating transverse magnetic field, are applied to the whole system. The equation of motion for the physical model is derived by using the Hamilton's principle and the vibration analysis is performed by employing the characteristic orthogonal polynomials as well as the Galerkin's method. The displacement of the beam with the effect of the magnetic force, axial force and spring force are determined from the modal equations by using the Runge–Kutta method. Based on the present study, we can conclude that the effect of the magnetic field not only reduces the deflection but also decreases the natural frequencies of the system, also it should be noted that the specified beam model can be adopted to simulate several structures in mechanical, civil and electronic engineering.

Abstract: A mathematical model incorporating the higher order deformations in bending is devel- oped and analyzed to investigate the nonlinear dynamics of rotors. The rotor system con- sidered for the present work consists of a flexible shaft and a rigid disk. The shaft is modeled as a beam with a circular cross section and the Euler Bernoulli beam theory is applied with added effects such as rotary inertia, gyroscopic effect, higher order large deformations, rotor mass unbalance and dynamic axial force. The kinetic and strain (defor- mation) energies of the rotor system are derived and the Rayleigh–Ritz method is used to discretize these energy expressions. Hamilton’s principle is then applied to obtain the mathematical model consisting of second order coupled nonlinear differential equations of motion. In order to solve these equations and hence obtain the nonlinear dynamic response of the rotor system, the method of multiple scales is applied. Furthermore, this response is examined for different possible resonant conditions and resonant curves are plotted and discussed. It is concluded that nonlinearity due to higher order deformations significantly affects the dynamic behavior of the rotor system leading to resonant hard spring type curves. It is also observed that variations in the values of different parameters like mass unbalance and shaft diameter greatly influence dynamic response. These influences are also presented graphically and discussed.