# 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.

Next: the POD method