Nonlinear Dynamics of Rotors due to Large Deformations and Shear Effects

Shad, Muhammad Rizwan; Michon, Guilhem; Berlioz, Alain

doi:10.4028/www.scientific.net/amm.110-116.3593

Cited by 3 publications

(5 citation statements)

References 11 publications

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“…where 𝑇 and 𝑉 are the total kinetic and potential energy of the rotor system, respectively, and 𝛿 represents the variation. The kinetic energy of the disc and the shaft are given by [13][14][15]43]. (…”

Section: Derivation Of Governing Equations Using Hamilton's Principlementioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Malgol,

et al. 2023

Preprint

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In this paper, a continuous rotor system is modelled by considering some critical factors, like the gyroscopic and rotary inertia effects of disc and shaft cross-sections, large shaft deformation, and restriction to shaft axial motion at the bearings. The bearings are replaced by springs along horizontal and vertical directions. Governing partial differential equations (PDE’s) for the vibrations of the disc along the horizontal and vertical directions are derived by employing the Hamiltonian principle. The governing equation is then transformed into a set of ordinary differential equations (ODEs) using method of modal projection. The large deformation and restriction to axial motion of the shaft yields nonlinearities in the system governing equations. Only the linear system is analysed in this first part. The parameters in the dimensionless form of the governing equations are functions of some independent variables which are associated to the material and geometrical properties of the rotor system. Understanding the effect of the system properties on the response characteristics will lead to an appropriate design. Exact analytical solutions for the motion along horizontal and vertical directions are obtained. The effect of the independent parameters on the system dynamics are analysed wherein the variation of the dependent parameters are also monitored. The system dynamics are explored using time-displacement responses, phase-plane plots, frequency response diagrams, phase angle plots and Campbell diagrams.

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Section: Derivation Of Governing Equations Using Hamilton's Principlementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Malgol,

et al. 2023

Preprint

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show abstract

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Malgol

Jayaprakash

Vineesh

et al. 2023

Preprint

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The nonlinear analysis of a continuous rotor system (CRS) is performed in this paper. The CRS model is derived with the inclusion of some important factors, like the gyroscopic effect, the rotary inertia of disc and shaft, large shaft deformation, and constraint to the axial motion of the shaft at the bearing ends. The entire derivation is explained in the first part (Part I) of this two-part article. The nonlinear system is analysed in this second part. Method of multiple scales is applied to obtain the autonomous amplitude and phase equations for simultaneous resonance conditions. Comparison of the analytical and numerical results yield a close match. Localized and nonlocalized oscillations are examined. Linear stability analysis is performed to assess the stability of steady-state solutions. Expressions for critical value of a parameter along both directions are derived to determine the emergence of limit points (LPs). A comprehensive parametric analysis is conducted to investigate the impact of various system parameters on the system dynamics. To examine the nature of vibration of CRS, time-response, phase-plane, amplitude-frequency, and phase-frequency diagrams are plotted. Various intriguing non-linear phenomena such as multivalued solutions, multiple loops, and multiple jump phenomena are observed.

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“…Shad et al [14] studied the nonlinear vibration of the rotating shaft. The shaft was considered a beam with a circular cross section.…”

Section: Introductionmentioning

confidence: 99%

Analytical investigation on nonlinear vibration behavior of an unbalanced asymmetric rotor using the method of multiple scales

Jamshidi

Jafari

2019

J Braz. Soc. Mech. Sci. Eng.

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In this paper, the vibration behavior of a rotor with an asymmetric shaft subjected to unbalanced forces is analyzed theoretically. The model is a rotor composed of a rigid disk and a flexible shaft. The shaft is considered to be a beam with a rectangular cross section. The general equations of motion were first derived by considering the effect of high order large deformation in bending. In this process, a continuous shaft, gyroscopic effects, and rotor mass unbalance are taken into account to study the rotor's nonlinear vibratory behavior near the main resonances. The equations are discretized using the Rayleigh-Ritz method. The obtained equations are nonlinear coupled differential equations which are solved using the multiple-scales method. It can be concluded from the results that nonlinearity due to an asymmetric shaft severely affects the resonance behavior of the rotor. Keywords Rotor dynamics Á Nonlinear vibration Á Asymmetric shaft Á High order deformations List of symbols a 1 Amplitude at the equilibrium position (m) in x direction a 2 Amplitude at the equilibrium position (m) in z direction X Angular Velocity of the rotor (rad sec-1) x 1 , x 2 First and second critical speed of the rotor (rad sec-1) I Average area moment of inertia of shaft (m 4) c Coefficient of damping (N s m-1) R 1 Cross-sectional radius of shaft/internal radius of disk (m 2) A Cross-sectional area of shaft (m 2) q Density of material (kg m-3) r Detuning parameter (rad sec-1) u(y, t) Displacement along x-axis of rotor (m) w(y, t) Displacement along z-axis of rotor (m)

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Nonlinear Dynamics of Rotors due to Large Deformations and Shear Effects

Cited by 3 publications

References 11 publications

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

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

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

Analytical investigation on nonlinear vibration behavior of an unbalanced asymmetric rotor using the method of multiple scales

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