Nonlinear responses of externally excited rotor bearing system

A mathematical model incorporating higher order deformation in bending is developed to investigate the nonlinear behavior of rotor. Transverse harmonic base excitation is imparted to rotor system and Euler-Bernoulli beam theorem is applied with effects such as rotary inertia, gyroscopic effect, higher order large deformations, rotor mass unbalance and dynamic axial force. Discretization of the kinetic and strain (deformation) energies of the rotor system is done using the Rayleigh–Ritz method. Second order coupled nonlinear differential equations of motion are obtained using Hamilton’s principle. Nonlinear dynamic response of the rotor system is obtained by solving above equations using the method of multiple scales. This response is examined for resonant condition. It is concluded that nonlinearity due to higher order deformations and variations in the values of different parameters like mass unbalance and shaft diameter significantly affects the dynamic behavior of the rotor system. It is also observed that the external harmonic excitation greatly affects the dynamic response.


Introduction
Rotating machinery [1][2][3] such as steam turbines, gas turbines, internal combustion engines, and electric motors, are the most widely used elements in mechanical systems.The prediction and analysis of the dynamic behavior of rotor system are crucial because their rotating components possess unlimited amounts of energy that can be transformed into vibrations.Therefore, studying the dynamic response, both theoretically and experimentally, of the flexible rotor system under various loading conditions would help in understanding and explaining the behaviour of rotor system.To simplify the analysis, researchers often try to use the linear analysis.But, application of nonlinear analysis is sometimes inevitable.Many phenomena should be described with nonlinear equations which are not explainable with linear analysis.The importance of considering the nonlinear effects in the dynamic analysis of rotating equipment has increased so as to obtain more accurate and optimized performance.Nonlinearities in rotor systems can be due to higher order large deformations, rotor-base excitations [3], geometric nonlinearities, oil film in journal bearings, magnetic bearings.The analysis of the nonlinear effects in rotorbearing systems is extremely difficult and there are a few analytical procedures that will generate valid results over a wide range of parameters.A geometrically nonlinear model of a rotating shaft was introduced by Luczko [4].The model included Von-Karman nonlinearity, nonlinear curvature effects, large displacements and rotations as well as gyroscopic and shear effects.Vibration problems involving nonlinearities do not generally lend themselves to closed form solutions obtained by using conventional analytical techniques.The perturbation methods [5] are a collection of techniques that can be used to simplify, and to solve, a wide variety of mathematical problems, involving small or large parameters.The solutions may often be constructed in explicit analytical form or, when it is impossible, the original equation may be reduced to a simpler one that is much easier to solve numerically.In this paper, analytical and numerical investigation of rotor system having transverse harmonic excitation is done considering the nonlinearity due to higher order deformation in bending.The restricted axial motion of rotor results in axial dynamic force, causing large deformation in bending.A nonlinear mathematical model has been developed; which includes various secondary effects like rotary inertia effects, gyroscopic effects and rotor mass unbalance.The method of multiple scales [5] is applied is used to solve this model including nonlinear terms.This method is applied directly to the partial differential equation of motion and to the discretized equations.The results of perturbation method are validated with numerical simulations.
Here, P and Q are generalized indepe and f ( y) is the displacement functi considered as the normalized first mode with a constant cross section in b ሺߨ‫ݕ‬ ‫ܮ‬ Τ ሻ, ݂ ᇱ ሺ‫ݕ‬ሻ ൌ ݃ሺ‫ݕ‬ሻ , and Following is the expressions for the c equations (1-2).
shaft and a rigid etry as used by gure 1.The shaft ection of radius R 1 inal axis y with a external radius R 2 shaft at a distance e denoted by m u is shaft is flexible, so energies, whereas etic energy only.
ndent coordinates on and has been e shape of a beam bending.݂ሺ‫ݕ‬ሻ ൌ ݂ ᇱᇱ ሺ‫ݕ‬ሻ ൌ ݄ሺ‫ݕ‬ሻǤ coefficients in the

Perturbation tech
By applying the perturb putting the expre ߝߤ ଵ ǡ ߤ ଶ ൌ ߝߤ ଶ ǡ ݉ ଵ ൌ ߝ݉ have the following expres coefficients of the like po resulting equations.
System of order 0 equation The solution of above equa Here.
[cc] denotes the com equations 8-9 into equa ߱ ൌ ߱ ଵ ߝߪ ଵ where ı 1 controlling the nearness of solvability conditions.The solvability conditions is to equate the coefficients of both sides of the resultin following similar procedu real and imaginary terms, of four first order different ations is given as mplex conjugate.Substitution of ations 6-7 and then putting, is a detuning parameter for f Ȧ e to Ȧ 1 , one may obtain the e first step in determining these o substitute ߗ ൌ ߱ ଵ ߝߪ ଵ , and f exp(iȦ 1 T 0 ) and exp(iȦ 2 T 0 ) on ng particular solution and by ures in [5,6] with separation of one may obtain, following a set tial equations

Result
In all the simulations, Euler-Bernoulli beam element with length L = 0.4 m, mass density ρ =7800 kg/m 3 , Young's modulus = 2 × 10 11 N m 2 , mass moment of inertia in xdirection = 0.11 kgm 2 , mass moment of inertia in y direction = 0.19 kg m 2 , cross-sectional area = 3.1×10 -4 m 2 , area moment of inertia = 7.854×10 -9 , thickness of the disk = 0.03 m, and inner and outer radius of the disk are equal to 0.01 and 0.15 m, respectively has been considered.The solid line represents responses of shaft in the direction z while the dotted line represents the time history of the shaft in the direction x in Fig. 2. Time response has been obtained for P 0 equal to 0.001m and m u equal to 0.01 kg when other parameters are keeping constant.From the Fig. 2, it has been observed that amplitude for the response (Q) in transverse direction is higher than the response (P) amplitude generally occurred in x direction.Hence, rotor bearing system is more stiff in xy plane than in the plane perpendicular to the xy i.e., in z direction.One more observation has been noticed that while time response in the transverse direction shows a quasiperiodic in nature, a beating phenomenon is observed in the time history representing the response in x direction as forcing frequency is closed to first natural frequency of the shaft in x direction.

Conclusion
Nonlinear dynamic response of the rotor system is obtained by solving above equations using the method of multiple scales.This response is examined for resonant condition when the forcing frequency is nearly equal to the first natural frequency of shaft in x direction.It is concluded that nonlinearity due to higher order deformations and variations in the values of different parameters like mass unbalance and shaft diameter significantly affects the dynamic behavior of the rotor system.It is also observed that the external harmonic excitation greatly affects the dynamic response DOI: 10.1051/ C Owned by the authors, published by EDP Sciences, The rotor system consists of a flexible disk and has the same rotor geome Duchemin et al. [1] as shown in fig considered, is a beam of circular cross se and length L, and spins about longitudi constant speed .The disk of mass M d , and internal radius R 1 , is positioned on L d along the y axis.The mass unbalance also situated at a distance, y = L d .The s it is modeled by its kinetic and strain the rigid disk is modeled by its kine Transverse harmonic base excitation Q 2 applied to rotor system as shown in Fig.

Fig. 1 .
Fig.1.Rotor with shaft and diskUsing extended Hamilton's principle, structural dynamics of rotor bearing system u motion is expressed as[1]

Fig. 2 :
Fig. 2: Times history for P and Q for P 0 equal to 0.001.

Figure 3 Fig. 3 :Fig. 4 :Fig. 5 :
Figure 3 and 4 illustrate the effect of amplitude of external forcing stimulated due to support motion for P 0 equal to 0.001 and 0.01 on the dynamics performance of the rotor bearing system.It has been observed that as expected with increase in amplitude of base motion, the amplitude gets increased for the responses of both Figure5and 6 illustrate the influence of mass imbalance m u equal to 0.01 and 0.1 on the dynamics behaviour of the rotor bearing system.While with increase in mass imbalance, the amplitude of responses in transverse direction remains constant although amplitude gets increased for the response in x direction.Figure7depicts a comparison study between the response obtained using temporal equation of motion and the response obtained using first order method of multiple scale method.It has been observed that for both responses, the maximum