This book equips the reader to understand every important aspect of the dynamics of rotating machines. Will the vibration be large? What influences machine stability? How can the vibration be reduced? Which sorts of rotor vibration are the worst? The book develops this understanding initially using extremely simple models for each phenomenon, in which (at most) four equations capture the behavior. More detailed models are then developed based on finite element analysis, to enable the accurate simulation of the relevant phenomena for real machines. Analysis software (in MATLAB) is associated with this book, and novices to rotordynamics can expect to make good predictions of critical speeds and rotating mode shapes within days. The book is structured more as a learning guide than as a reference tome and provides readers with more than 100 worked examples and more than 100 problems and solutions.

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Dynamics of rotating machines

Mathematical models of elastic structures have become very sophisticated: given the crucial material properties (mass density and the several elastic moduli), computer-based techniques can be used to construct exotic finite element models. By contrast, the modeling of damping is usually very primitive, often consisting of no more than mere guesses at “modal damping factors.” The aim of this paper is to raise the modeling of viscoelastic structures to a level consistent with the modeling of elastic structures. Appropriate material properties are identified which permit the standard finite element formulations used for undamped structures to be extended to viscoelastic structures. Through the use of “dissipation” coordinates, the canonical “M , K ” form of the undamped motion equations is expanded to encompass viscoelastic damping. With this formulation finite element analysis can be used to model viscoelastic damping accurately.

Dynamics of viscoelastic structures: a time-domain finite element formulation

Mechanical model development of rolling bearing-rotor systems: A review

The presence of cracks in a structure is usually detected by adopting a linear approach through the monitoring of changes in its dynamic response features, such as natural frequencies and mode shapes. But these linear vibration procedures do not always come up to practical results because of their inherently low sensitivity to defects. Since a crack introduces non-linearities in the system, their use in damage detection merits to be investigated. With this aim the present paper is devoted to analysing the peculiar features of the non-linear response of a cracked beam. The problem of a cantilever beam with an asymmetric edge crack subjected to a harmonic forcing at the tip is considered as a plane problem and is solved by using two-dimensional finite elements; the behaviour of the breathing crack is simulated as a frictionless contact problem. The modification of the response with respect to the linear one is outlined: in particular, excitation of sub- and super-harmonics, period doubling, and quasi-impulsive behaviour at crack interfaces are the main achievements. These response characteristics, strictly due to the presence of a crack, can be used in non-linear techniques of crack identification.

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Non-linear dynamics of a cracked cantilever beam under harmonic excitation

Cracked beam identification by numerically analysing the nonlinear behaviour of the harmonically forced response

The breathing behaviour of closing cracks has been adequately simulated as a small-displacement, frictionless contact problem. The problem of a beam with an edge crack subjected to harmonic loading has been considered as a plane problem and an attempt is made to solve it by using finite elements employing eight-node plane isoparametric elements. The proposed model allows the crack size and position to be varied. Another physically important problem of a cantilever beam held between two heavy jaws at the top and bottom, which are not equally flushed, is considered. This beam is also excited by a harmonic load at the tip. The contact model and a simple single degree of freedom model are used to solve the problem. Both the above problems (cracked beam and beam in offset jaws) show presence of integral multiples of the forcing frequency in their frequency spectra.An important observation regarding cracked beams and beams with imperfect support is made. If the forcing frequency is such that it coincides or is ...

Modelling of a Beam with a Breathing Edge Crack and Some Observations for Crack Detection

A shaft is modelled using three-dimensional solid finite elements. The shear-deformation and rotary inertia effects are automatically included through the three-dimensional elasticity formulation. The formulation allows warping of plane cross-sections and takes care of gyroscopic effect. Unlike a beam element model, the present model allows the actual rotor geometry to be modelled. Shafts with complicated geometry can be modelled provided that the shaft cross-section has two axes of symmetry with equal or unequal second moment of areas. The acceleration of a point on the shaft is determined in inertial and rotating frames. It is found that the finite element formulation becomes much simpler in a rotating frame of reference that rotates about the centre-line of the bearings with an angular velocity equal to the shafts spin speed. The finite element formulation in the above frame is ideally suited to non-circular shafts with solid or hollow, prismatic or tapered sections and continuous or abrupt change in c...

Modelling of rotors with three-dimensional solid finite elements:

An efficient scheme for stability analysis of finite element asymmetric rotor models in a rotating frame

Reduction of finite element equations for a rotor model on non-isotropic spring support in a rotating frame

The present work deals with a two-step nonlinear finite element analysis for misaligned multi-disk rotors on short oil-film bearings of various types (cylindrical, pocket, symmetrical three-lobed, unsymmetrical three-lobed). As a first step, the conventional parallel, angular and combined parallel and angular misalignments are modelled using Lagrange multipliers. The static equilibrium position of the journal within the bearing is determined using an iterative nonlinear static finite element analysis. The present work proposes a method for computing the displacement-dependent stiffness terms from the experimental static load-displacement data. Finally, the orbit of the rotor around the static equilibrium is determined using a time-integration scheme.

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A. Nandi

Papers

Modelling of a Beam with a Breathing Edge Crack and Some Observations for Crack Detection

Modelling of rotors with three-dimensional solid finite elements:

An efficient scheme for stability analysis of finite element asymmetric rotor models in a rotating frame

Reduction of finite element equations for a rotor model on non-isotropic spring support in a rotating frame

Finite element analysis of misaligned rotors on oil-film bearings