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=U∑V* (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 u_{1},…,u_{m} of U yield an orthonormal basis of Ω^{ m} and the columns v_{1},…,v_{n} 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 t_{i} 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 a_{i} 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 a_{i} 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 ω_{i} its eigenvectors,
Ω=span(ω_{i})(2)
On the other hand, using (1), POMs can be expressed as:
v_{i}= (3)
(3) shows that v_{i} can be expressed as linear combinations of a_{i}. Since a_{i} Ω, also v_{i} Ω, this observation means that POMs are linear combinations of the eigenvectors ω_{i}. 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…