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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Response Analysis of Shafts – The POD method

8 06 2009

2 The POD method

The POD is a powerful method for system identification aiming at obtaining low-dimensional approximate descriptions for multidimensional systems. It provides a basis for the modal decomposition of a system of functions, usually data obtained from experiments, measurements or numerical simulations. The basis functions retrieved are called Proper Orthogonal Modes (POMs). From a different point of view, the method provides an efficient way of capturing the dominant components of a multidimensional system and represents it to the desired precision by using the relevant set of modes, effectively reducing the order of the system. The POD is interpreted in quite a few ways and appears in the literature in three equivalent forms: the Karhunen-Loeve decomposition (KLD), the Principal Component Analysis (PCA) and the Singular Value Decomposition (SVD). The latter is preferred in this text for its simplicity. The method is well known and its mathematical background and numerical implementation are explained in detail in many classic textbooks and papers. In practical applications, the input to the method is the distribution of a variable in time and space in matrix form, either measured or calculated from a simulation. For example, in structural response analyses, the distribution of nodal displacements over a body at consecutive time steps is used. The POD method provides then a decomposition of this time-space variation into Proper Orthogonal Modes (POMs) in space, fluctuation in time and participation factors. Moreover, since the method operates on the actual response, any complex dynamical effects are included and represented in the results as soon as the space and time discretisation is adequate. The advantage of the method is therefore, that it is indifferent to the system itself: it does not introduce any restrictions or assumptions on the mechanical or material response. My attempt here is to illustrating these unique properties with simple examples of continuum bodies, such as solid cylinders, which are nevertheless quite common and critical components in rotating machinery assemblies. The simplicity of the structures does not necessarily lead to equally simple responses.

In order to formulate the decomposition from the SVD aspect, it is assumed that the A is an m-by-n matrix defined over the field Ω, which is generally either the field of real or complex numbers. Then there exists a factorization of the form:

A=UV* (1)

where U is an m-by-m unitary matrix over Ω, the matrix is a m-by-n diagonal matrix of the positive singular values sorted in descending order and V* is the conjugate transpose of V, an n-by-n unitary matrix over Ω. U and V are orthogonal matrices and the columns u1,…,um of U yield an orthonormal basis of Ω m and the columns v1,…,vn of V yield an orthonormal basis of Ω n. The existence and uniqueness theorems of this decomposition have been extensively analyzed and can be found in the literature.

In the finite element context, the SVD can be used to reduce finite element matrices by expressing them using a much smaller set of approximation basis functions. In this way the solution of nonlinear finite element equations can be significantly accelerated.  But using the SVD purely for facilitating numerical computations is not its only application and indeed not the application of interest here. The SVD is a powerful tool that can be used to process finite element results just as any other measurement. From this aspect,  given a discretized deformable body under a set of external forces, its distinct configurations at times ti can be arranged in a matrix form A and analyzed using the SVD. The physical meaning of the three matrices produced by the decomposition of such a matrix has been extensively discussed. V contains the “empirical eigenvectors” or POMs of A, U contains the time coefficients of the corresponding POM and the singular values in can be thought of as the intensities or magnification factors of the participation of each mode in the whole process. For this latter case, the correlation of POMs with normal modes is largely argued. It is considered that only the first or dominant mode is approximated correctly while the deviation of the other modes depends on several parameters, most dominant being the sampling points and the structure of the mass matrix.

I will try to take another view on the POMs through the following simple observation: let matrix A be a row ordered matrix, where each row ai for i[1,m] is an n-sized vector containing the displacement values calculated at each node or sampling point of a structure. By definition, vector ai Ω, where Ω is the space of all possible configurations of a body. If K is the generalized stiffness matrix of a deformable body and ωi its eigenvectors,


On the other hand, using (1), POMs can be expressed as:

vi= (3)

(3) shows that vi can be expressed as linear combinations of ai. Since ai Ω, also vi Ω, this observation means that POMs are linear combinations of the eigenvectors ωi. A first conclusion from this observation is that POMs are feasible configurations of the deformable body. The space generated by the POMs, the latter being an orthogonal basis, is a subspace of Ω. The orientation of the basis is such that the POMs are those combinations of eigenvectors whose participation in the process described by A can be classified on the basis of the singular values. This observation is important in cases where the response of the body is so complex (due to loading, material and structural arrangement) that it is difficult to calculate normal modes or even impossible. POMs guarantee a decomposition of the response to modal forms, albeit linear. It remains to investigate the usefulness of this decomposition: “The attractiveness of the POD lies in the fact that it is a linear procedure. The mathematical theory behind it is the spectral theory of compact, self-adjoint operators. This robustness makes it a safe haven in the intimidating world of non-linearity; although this may not do the physical violence of linearization methods[14]”.

In most mechanical systems, each component that performs a particular function and is under a particular loading condition, there are only a few – possibly one or two – modes of ideal motion, deformation or both. These can be represented in a convenient coordinate system. This functional behaviour can be desirable or undesirable. A simulation of this process will provide a set of configurations that, when analysed with the POD method will possibly return a dominant mode as well as other modes. The subset of dominant modes corresponds to the functional behaviour of the component or structure. The singular values demonstrate the relative participation of each mode in the process. These features together, illustrate the consistency or “robustness” of the process with respect to the particular mode or modes. If singular values are comparable or in other words there is dissipation of the contribution between the modes, then the process is not robust with respect to the desired modes. Therefore the issue here is not whether POMs approximate normal modes, but rather whether the particular process is consistent in the sense that the simulated response approximates the desired response. Ideally, for example in transmission systems, one would expect a single mode that represents rigid body rotation or equally in a rotating coordinate system, no deformations. An analysis of such a simulation shows that the POMs and singular values quantify the departure from the ideal condition. This notion can be extended to situations where normal modes are difficult or impossible to calculate. Such a case may be a rotating shaft driving a ship propeller, where the effective mass of the water on the blades varies with the depth, and the propeller is prestrained under gravity, inertia, pressure and turbulence effects. Even in such complex situations, the desired behaviour of the system is known by requirements and comparing the departure from this behaviour under parameter changes can be very helpful in designing and controlling such systems. On the other hand, it may be desirable to mitigate the dominant mode and minimize the difference between the largest and smallest singular values. In either case, parametric studies can be used to investigate the convergence or divergence from the state in question and can be used as an effective tool in design and optimisation. Having seen that POMs are feasible configurations of a body, numerical computations are used to demonstrate that the different aspects in the loading of rotating machinery are actually represented and quantified by the modes.

To be continued…

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.

Next: the POD method