GENERAL CASE
2.12 COMPLETE AND INCOMPLETE MODELS .1 Some Definitions
Most of the preceding theory has been concerned with complete models;
that is, the analysis has been presented for a n N degree-of-freedom system with the implicit assumption that all the mass, stiffness and damping properties are known and that all the elements in the eigenmatrices and the FRF matrix are available. While this is a valid approach for a theoretical study, it is less generally applicable for experimentally-based investigations where it is not usually possible to measure all the or to examine all the modes possessed by a structure. Because of this limitation, it is necessary to extend our analysis to examine the implications of having access to something less than a complete set of data, or model, and this leads us to the concept of a 'reduced' or 'incomplete' type of model.
I t is appropriate here to introduce a few additional definitions which will be used throughout this book when dealing with the various types of incomplete model. A complete model is one which is fully defined by its description. This can be achieved in any of the three types of model if all the individual mass stiffness and damping elements are included (spatial model), or if all the modes (natural frequencies and mode shapes) are included or if all the FRF data are known over a frequency range which includes all the modes (response model). This means that the full N x N matrices are available for the different mathematical descriptions. Models become incomplete when less than the above
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information is available, or displayed.
There are different types of incomplete model. There is the model which is reduced in size (from N to n) by simply deleting information about certain degrees-of-freedom. This process leads to a reduced model which retains full accuracy for the DOFs which are retained, but which loses access to those which have been deleted. The process can be applied only to the modal and response models and results in a modal model described by a n N x N eigenvalue matrix but by a n eigenvector matrix which is only n N. The corresponding response model is a n incomplete FRF matrix of size n n, although all the elements of that reduced matrix are themselves fully accurate. Another type of reduced model is one in which the number of modes natural frequencies) is again eliminated from the complete description but a n attempt is made to include the effects of the masses and stiffnesses which are thereby eliminated in the retained DOFs. This is the condensation process which is applied in the and other reduction techniques used to contain the size of otherwise very large finite element models. In such a condensed model, the spatial, modal and response models are all reduced to n x n matrices, and it must be noted that the properties of each are approximate in every respect.
Lastly, it should be mentioned that it is sometimes required to seek to recover a full-sized model of a structure's dynamics from the basis of a n incomplete model. This can be attempted by one of various processes of interpolation and leads to a n expanded model, usually of full size (N x but great care should be exercised in making any use of such a model.
2.12.2 Incomplete R e s p o n s e Models
As intimated, there are two ways in which a model can be incomplete - by the omission of some modes, by the omission of some
of-freedom
-
and we shall examine these individually, paying particular attention to the implications for the response model (in the form of the FRF matrix). Consider first the complete FRF matrix, whichand then suppose that we decide to limit our description of the system
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to include certain DOFs only (and thus to ignore what happens at the others, which is not the same a s supposing they do not exist). Our elements of interest and removing or deleting those to be ignored.
At this point, it is appropriate to mention the consequences of this type of reduction on the impedance type of FRF data. The impedance matrix which corresponds to the reduced model defined by will be denoted as and it is clear that
I t is also clear that the elements in the reduced impedance matrix such a s are not the same quantities as the corresponding elements in the full impedance matrix and, indeed, a completely different impedance matrix applies to each specific reduction. Thus:
(m) = but (m)
We can also consider the implications of this form of reduction on the other types of model, namely the modal model and the spatial model. For the modal model, elimination of the data pertaining to some of the DOFs results in a smaller eigenvector matrix, which then becomes rectangular or order n x N. This matrix still retains N columns, and the corresponding eigenvalue matrix is still N x N because we still have all N modes included.
For the spatial model it is more difficult to effect a reduction of this type. It is clearly not realistic simply to remove the rows and columns corresponding to the eliminated DOFs from the mass and stiffness matrices a s this would represent a drastic change to the system. I t is possible, however, to reduce these spatial matrices by a number of methods which have the effect of redistributing the mass and stiffness (and damping) properties which relate to the redundant DOFs amongst those which are retained. In this way, the total mass of the structure,
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and its correct-stiffness properties can be largely retained. The
reduction procedure is perhaps the best known of this type although there are several modelling techniques (see later, Chapter 5). Such reduced spatial properties will be denoted as:
Next, we shall consider the other form of reduction in which only of the N modes of the system are included. Frequently, this is a necessary approach in t h a t many of the high-frequency modes be of little interest and almost certainly very difficult to measure. Consider first the FRF matrix and include initially all the but suppose that each element i n the matrix is computed using not only m of the N terms in the summation,
I n full, we can write the FRF matrix as:
Of course, both types of reduction can be combined when the resulting matrix would be denoted:
It can be seen from (2.160) that the FRF matrix thus formed, , will, i n general, be rank deficient, and thus it will not be possible to obtain the impedance matrix by numerical inversion. This remains the case as long a s n m (and can even be found i n cases where n and so the numerical condition of matrices of incomplete models is frequently found to be a cause for concern. I n order to overcome these problems, it is often convenient to attempt to provide a n approximate correction to the FRF data to compensate for the errors introduced by leaving out some of the terms. This is usually effected by adding a constant or 'residual' term to each FRF, a s shown in the following equation:
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The consequence of neglecting some of the modes on the modal model is evident i n that the eigenvalue matrix becomes of order m x m and the eigenvector matrix is again rectangular, although in the other sense, and we have:
2.12.3 I n c o m p l e t e Modal and S p a t i a l Models
I t h a s been shown earlier that the orthogonality properties of the modal model provide a direct Link between the modal model and the spatial model:
which can be inverted to yield:
If the modal model is incomplete, then we can note the implications for the orthogonality properties. First, if we have a modal incompleteness (m N modes included), then we can write:
However, if we have a spatial incompleteness (only N
included), then we cannot express any orthogonality properties a t all because the eigenvector matrix is not commutable with the system mass and stiffness matrices. In both reduced-model cases, it becomes impossible to use (2.162) to re-construct the system mass and stiffness matrices from a n incomplete modal model. Even i n the special case where m in which case we have square reduced eigenvector and eigenvalue matrices and can i n theory - compute the inverses required by the equation these matrices are generally singular and thus not invertible. Even if they are numerically non-singular, there is no theoretical basis for applying (2.162) in case and any mass and stiffness matrices produced by such application have no physical significance and should not be used.
Fig. 2.43 shows the relationship between different forms of complete and incomplete models.
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they help t o locate errors in models in updating applications;
they are useful in guiding design optimisation procedures, and
they are used in the course of curve-fitting for the purposes of testing the reliability of the modal analysis processes.
It will be helpful to include here a short summary of the main sensitivity parameters and to show how they may be deduced from both theoretically- and experimentally-derived models.
2.13.2 Modal Sensitivities
The most commonly-used sensitivities are those which describe the rates of change of the modal parameters with the individual mass and stiffness elements in the spatial model. These quantities are defined in general a s follows:
where p represents any variable of interest.
(a) SDOF system
It is useful to approach the general expressions for these parameters via a very simple example based on a n undamped SDOF system. We can introduce the concept of sensitivity through the basic SDOF system comprising mass, and spring, k, and we can define the basic sensitivities of the system's natural frequency, , with respect to these two design parameters as:
respectively. We can readily show that these two sensitivities can be expressed as follows: