Nonlinear responses of externally excited rotor bearing system

Abstract

Abstract. 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.

Modelling and analysis of a continuous rotor system Part II: Analyzing the effect of nonlinearities on the system dynamics

“…Setting the discriminant of eqn. (25) to zero, the expression for determining the critical values of 𝑎 1 is obtained as…”

“…Only the literatures on the modelling and analyses of non-linear rotor system are discussed in this part. The rotor system models can be broadly categorized into: (i) discrete [1-16] and (ii) continuous [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. Only the stiffness of the shaft is considered in the discrete system model, not its inertia.…”

“…In many cases, these PDEs with boundary conditions are reduced to a set of ODEs using different discretization method for further analysis. Some influencing factors considered in the literature for analyzing the dynamics of a discrete rotor system are as follows: shear effects There is an extensive amount of literature on the analysis of continuous rotor systems [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. The Hamilton's [17,31,32,35,38] and Lagrange's [18,23,26,38] equations are mainly used to derive the governing partial differential equations and boundary conditions.…”

“…Different discretization methods used to transform the PDEs into ODEs are the Galerkin [17,32,35] method and the Rayleigh-ritz [23,25] method. To solve or discretize the difficulties associated with complex rotor and continuous models, techniques such as the finite element method (FEM) [10, 31, 34, 38-43], and variational asymptotic approach [32] are developed.…”

Modelling and analysis of a continuous rotor systemPart I: Derivation of governing equations and analysis of linear system

“…Several models are available in literature for the study of rotor system vibrations. They can be categorized into: (i) discrete or lumped parameter model There is a considerable number of literatures on the analysis of continuous rotor systems [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] wherein the inertia of the shaft is also considered along with the inertia of the disc. In the continuous model, the governing equations of the shaft-rotor system are first derived in the form of partial differential equations (PDE's) either by using Lagrange's principle Galerkin method [6, 18, 21, 24, 28], or Raleigh-ritz method [3, 9, 11, 16], or any other dimensionality reduction technique.…”