An approach to the response Analysis of Shafts – Numerical Examples

18 02 2010

3Numerical examples

In the following examples, finite element simulations are used to demonstrate this concept. These simulations concern shaft loading with torsional moment and axial compression. This is a common loading condition for shafts, for example in rotating machinery used in power plants. For performing the simulations, the commercial finite element code Abaqus/Explicit 6 has been employed, using double precision arithmetic. Further, the POD processing is performed with custom software based on the LAPACK library 6. In all cases, POD analysis is performed over at least 300 snapshots (taking care to always observe the lower bound of the Nyquist–Shannon sampling theorem) including all the degrees of freedom of all nodes in the models. The results presented hereafter include the extracted mode shapes and plots of variable field distributions, amplitude vs. time fluctuation for each mode and Singular value percentage for each mode. It is noted here that in all cases multi-field POD is performed. This means that the input snapshot vectors include three displacement variables for each node. The resulting POMs include fields that correspond to each of the input variables. The amplitude vs. time curves for each mode represents the variation with time of the mode participation in the time-space domain of the simulation. In vibration problems this is usually an oscillating curve and a Fourier analysis is performed on this time variation so as to calculate the excited frequencies for each mode 6. Finally, the singular values relative percentage presents an overall estimated participation factor for the mode. Considering the fact that in the results there are three fields to each POM, it is interesting to calculate the norm of each field so as to determine its participation to the POM. In the results presented, the sum of the squared field norms equals unity and the field norms are presented next to the singular value percentage so as to easily determine which field is dominating the POM.

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An approach to the response Analysis of Shafts – Discussion

16 02 2010

4 Discussion

The study of shaft behaviour under dynamic loading and rotation is of profound importance in predicting resonance, in system control and system monitoring. The purpose of the work presented in this paper is to introduce the method of Proper Orthogonal Decomposition (POD) as a tool that can be effectively used in characterising the dynamics involved in the above tasks and extracting useful information for the real-time behaviour of the structure. The method is used in the form of meta-processing of finite element simulation results. It has been shown that the POD method is used to reproduce modes and corresponding frequencies that are systematically correlated under variations of initial conditions in dynamical problems of free vibration. Even when combined loading is applied, the POD method correctly discriminates and classifies these modes and frequencies. It has also been shown that the frequencies are affected by dynamical effects and pre-strain, a behaviour that is expected but is often difficult to calculate.The Proper Orthogonal Decomposition presents a considerable advantage: it is indifferent to the system that generates the input to the method. Nevertheless, it succeeds in extracting dominant modes and classifying them from the time-space response of the structure. Throughout this text, POMs have not been considered to coincide necessarily with natural modes of vibration. Rather, POMs are appropriate combinations of modes and therefore feasible configurations of a body. POMs, being orthogonal, they form a basis of the space where the configurations of the body in the particular process lie. Moreover, they are classified in an eigenvalue sense. These two properties are very important. In combination, they identify dominant POD modes in the response. This information can be interpreted in two ways, especially  in cases where a unilateral behaviour is desired. The first interpretation is that a strong dominant mode depicts a process that is consistent and “robust” to that mode. The second interpretation is that singular value percentage dispersion over more than one mode signifies a process that includes strong interference with the dominant mode.

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An approach to the response Analysis of Shafts – Introduction

8 06 2009

1 Introduction

There are several situations where investigating the dynamical behaviour of shafts and attachments is important: resonance prediction is very important in the dimensioning and design of shafts but equally important is the prediction of shaft response in system control and system monitoring. The finite element method is frequently employed in numerical investigations aiming in analysing the influence of parameters such as section properties, materials and loading conditions. A few theories are presented in the literature that cope with particular problems. Such applications are presented hereinafter but certainly there should exist much more. Gubran studies the dynamic performance and cross-section deformation of shafts made of steel and aluminium, composites (CFRP and GFRP) and hybrids of metals and composites. A layered degenerated shell finite element with transverse shear deformation is employed. Results obtained show that improvement in dynamic performance and reduction of cross-section deformation of hybrid shafts over metallic and composite shafts is possible. Khulief et al identify drill-string vibration as one of the root causes of a deteriorated drilling performance. In order to demonstrate the complex vibrational mechanisms in a drilling system, as well as control its operation and improve its performance, they employ a dynamic model of the drill-string including both drill-pipe and drill-collars. The model accounts for gyroscopic effects, torsional/bending inertia coupling and the effect of the gravity. Modal transformations are applied to obtain a reduced order modal form of the dynamic equations.

Lee et al present a method for obtaining the out of balance response orbit of a gear-coupled two-shaft rotor-bearing system, based on finite element simulations. Bumps in the out of balance response are observed at the first torsional natural frequency because of the coupling between the lateral and torsional dynamics due to gear meshing. Neogy et al present work related to the stability analysis of finite element models of asymmetric rotors (non-axisymmetric shaft and non-isotropic bearings) in a rotating frame, which rotates synchronously about the undeformed centre-line of the rotor. Chatelet et al outline the necessity to predict at the design stage the dynamic behaviour of rotating parts of turbomachinery in order to be able to avoid resonant conditions at operating speeds. In their study, the global non-rotating mode shapes of flexible bladed disc-shaft assemblies are used in a modal analysis method for calculating the dynamic characteristics of the corresponding rotating system. The non-rotating mode shapes are computed using a finite element cyclic symmetry approach. Rotational effects, such as centrifugal stiffening and gyroscopic effects are taken into account. Turhan et al study the stability of parametrically excited torsional vibrations of shafts connected to mechanisms with position-dependent inertia. The shafts are considered to be torsionally elastic, distributed parameter systems and discretised through a finite element scheme. The mechanisms are modelled by a linearised Eksergian equation of motion. A general method of analysis is described and applied to examples with slider-crank and Scotch-yoke mechanisms. Li et al carry out finite element simulations using a speed-increase gearbox. Profile error curves are determined by the gear precision and the time-variable stiffness curves of the gears were acquired using 3D contact. After the stiffness excitation and error excitation are determined and the dynamic model of the speed-increase gearbox is established, the normal frequencies and vibration response of the whole gearbox and transmission shafts are analysed. Sekhar outlines the importance of dynamics and diagnostics of cracked rotors. In his study a model-based method is proposed for the on-line identification of two cracks in a rotor. The fault-induced change of the rotor system is taken into account by equivalent loads in the mathematical model. The rotor has been modelled using finite elements, while the cracks are considered as local flexibility changes. The cracks have been identified for their depths and locations on the shaft. The nature and symptoms of the crack are ascertained using FFT. Nandi presents a simple method of reduction for a finite element model of non-axisymmetric rotors on non-isotropic spring support in a rotating frame. In this frame the stiffness matrix, mass matrix and Coriolis matrix for the non-axisymmetric rotor (rotor with rectangular cross-section, cracked rotor, etc.) is independent of time but the support forces become periodic. Therefore, a large set of linear ordinary differential equations with periodic coefficients at support degrees of freedom is formed, which requires substantial computational effort to solve. To effectively handle this large system, a reduction method is introduced that keeps the essential information almost intact. Jun introduces Timoshenko beam theory for modelling shaft behaviour. Complex variables are used to represent the displacement, slope, moment and shear force. The complex transfer matrix between the variables at both ends of the shaft element is derived. The influence coefficients are analytically derived for the general flexible rotor having two resilient bearings at both ends. Modelling and derivation of the influence coefficients is based on the transfer matrix method. Simulated influence coefficients are compared to the results using the finite element method based on Timoshenko beam theory. Hu et al present a finite element-based formulation for modelling the dynamic behaviour of a rotating flexible shaft supported by a flexible structure. The coupling effect between the rigid-body rotation and the flexible deformation of the shaft is considered and represented by non-linear coupling terms in the mass matrix and forcing vectors in the global system of equations. The rigid-body rotation is treated as one of the degrees of freedom (d.o.f.) of the entire system. The interaction between the rotating shaft and the flexible support is modelled by either linear or non-linear springs distributed around the circumference of the shaft. The coupling between the flexibility of the shaft and the flexibility of the support structure are considered. The equations of motion are solved in the time domain using a modified Newmark scheme. Analyses are performed to validate the new development for different combinations of load condition, spring type, and rigid-body rotation. Mohiuddin et al present the finite element formulation of the dynamic model of a rotor-bearing system. The model accounts for the gyroscopic effects as well as the inertia coupling between bending and torsional deformations. A reduced order model is obtained using modal truncation. The modal transformation invokes the complex mode shapes of a general rotor system with gyroscopic effects and anisotropic bearings. The reduced modal form of the dynamic model is numerically simulated and the dynamic responses due to different excitations are obtained.

It is obvious that there is considerable research work devoted to the subject of rotating shafts and attachments and this is only part of it. Many of the afore-mentioned approaches involve considerable assumptions on the response of shafts and subsystems as well as for the involved physical phenomena. In this article I try to establish a way of dealing with such problems, either at the design stage or the monitoring stage by use of the Proper Orthogonal Decomposition (POD) as a tool that can provide insight to the behaviour of systems by appropriately processing data, either from simulations, experiments or measurements. Towards this, the ideal configuration of the system is established and deviations from this either in simulations or reality are quantified. The advantage is that no special algorithms or measurement devices are required by the method itself. Rather the mothod extracts such deviations from the available data. The quality and completeness of the result depends on the ohysical content of the data.

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