One final word by way of introduction to this important aspect of the work: this subject many others but perhaps more than most) has generated a wealth of jargon, not all of it consistent! We have adopted a particular notation and terminology throughout this work but i n order to facilitate 'translation' for compatibility with other references, and i n particular the manuals of various analysis equipment and software in widespread use, the alternative names will be indicated a s the various parameters a r e introduced.
In the past decade (the 1990s) some attempts have been made to the notation and the terminology used i n our subject, While not succeeding completely, there has been some notable progress, not least by a number of journals, and significant conferences, recommending to their prospective authors a basic notation which was proposed i n 1990, and refined over a period of three to four years before being used a s the basis of the documentation published by the Dynamic Testing Agency, This notation is presented here in a n Appendix and will be strictly adhered to throughout this text.
It is not only in respect of notation that care is needed in order to ensure clarity i n the explanations, and the necessary lack of ambiguity.
The terminology that we use should also be carefully considered and there are certain words that are used rather imprecisely, and others that are used by different authors to mean different things. Many of these are 'ordinary' words, and not technical jargon, and so it is worth mentioning a few of these a t the outset of our text. We shall be dealing extensively with the imprecision and uncertainty which accompanies all experimental work. Thus we must be clear about the meaning of words such a s 'errors', 'uncertainty', inaccuracy', 'imprecision'; and 'repeatability' and 'reproducibility'.
An Error is the difference between an obtained value and the true or correct value
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U n c e r t a i n t y refers to the range of values within which we can define a quantity (that we have measured, or otherwise obtained) I n a c c u r a c y relates to the size of the error which may be associated with a quoted value
I m p r e c i s i o n generally refers to the degree of resolution or detail with which a calculation is performed, and often accompanies the introduction of a truncated or otherwise unrefined element in the computation of a given quantity
Repeatability refers to the extent to which a quantity will be found to have exactly the same (measured) value in a second or subsequent acquisition, using exactly the same
Reproducibility refers to the extent to which a given result (experiment, measurement) can be reproduced a t a later date using similar procedures or equipment
We shall need to consider the difference between directly measured data and indirect measurements. Usually, we shall make direct measurements of excitation forces and the resulting responses: any response functions which are then derived are indirectly measured, as are the ensuing estimates of natural frequencies and' mode shapes. We must always be aware that we do not m e a s u r e these modal properties, but we derive them by some inexact analysis of the quantities that we actually do measure.
Later, we shall be concerned with the reconciliation between predicted properties and measured ones. Indeed, a s already mentioned, process is one of the major reasons for modal tests to be performed.
In such applications, there is much talk of 'errors' in the theoretical model and this can lead to unrealistic expectations of the whole analysis comparison process. We shall talk then of verification and validation and although these terms will be defined in the relevant chapter, it is perhaps appropriate to introduce the subtle difference intended to be represented by these two words. 'Verification' refers to the process of determining whether something (an algorithm, a calculation, a model) is correct or not. I t is black or white: the object is either correct or it is not: the matrix inversion routine is either coded correctly or it is not: the cables are either correctly labelled or they are not. 'Validation', on the other hand, is less black and white and is more concerned with whether the object being described is fit for its intended purpose. 'Valid' is taken here to mean that the object a measurement, or a mathematical model) is capable of representing the quantity or behaviour of interest sufficiently well to s e r v e t h e n e e d s of that object. Thus, we have a degree of judgement to exercise in deciding whether something we have obtained or created is valid
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good enough) and this satisfies many of the real life situations in which we are obliged to apply our skills.
These are not the only examples where precise use of the language is necessary, but they serve to illustrate the concern and to justify the comments t h a t w i l l be made from time to time where a similar confusion is to be avoided by careful choice of the words we use.
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'SPATIAL MODEL'
'MODAL MODEL'
Response Levels Descrlptlon
of structure
'RESPONSE MODEL'
Mass. Natural Frequency Responses
Mode Shapes Impulse Responses
Fig. 2.1 Theoretical route to vibration analysis
Experimental Modal
Vlbratlon Modes
Fig. 2.2 Experimental route to vibration analysis
1
STRUCTURAL MODEL MODES
RESPONSE PROPERTIES
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f a c t o r s and vibration m o d e shapes. It is important to remember that this solution always describes the various ways in which the structure is capable of vibrating naturally, without any external forcing or excitation, and so these are called the 'natural' or 'normal' modes of the structure.
The third stage is generally that in which we have the greatest interest; namely, the analysis of exactly how the structure will respond under given excitation conditions and, especially, with what amplitudes.
Clearly, this will depend not only upon the structure's inherent upon this characteristic that the principles of modal testing are founded.
As indicated in Fig. it is possible to proceed from the spatial model through to a response analysis. I t is also possible to undertake a n analysis in the reverse direction - from a description of the response properties (such a s measured frequency response functions) we can deduce modal properties and, in the limit, the spatial properties.
This is the 'experimental route' to vibration analysis which is shown in Fig. 2.2 and which will be discussed in detail in Chapter 5.
As a parting comment before we embark on a moderately lengthy voyage through the underlying theory upon which our subject is based, it must be noted that it can seem to be a n extreme irony that a n experimentally-based technology such a s we are describing here demands a richness of theoretical methods that significantly outstrips the corresponding material that would be found in a theoretically-based study of the same general subject. This is simply because in the experimental field we must be prepared to explain and to interpret the most general of circumstances (uncertain damping type; almost inevitable arbitrariness of damping distribution; the distinct possibility of non-linear behaviour, and so on). The luxury of being able to dictate the conditions (or assumptions) a t the outset of a study
-
as we are wont to do in theoretical analyses - is not one that can
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generally be extended to the experimentalist, and so he or she must be armed with the most general of models.
2.2 SINGLE-DEGREE-OF-FREEDOM (SDOF) SYSTEM