called the modal mass and modal stiffness. There is a variety of terminology in this area which is worth mentioning so that a t least the different quantities can be identified, even if uniformity of terminology cannot be assured. Three terms are encountered in the literature: modal mass (and stiffness); generalised mass and effective or equivalent mass.
In this section we shall seek to explain what these different quantities are and how they may be interpreted or used.
We start with the modal mass, already defined above based on the mode shape vector for mode and the system mass matrix. As mentioned there, there is no unique value for the modal mass a s it is directly related to the scaling method which has been used to define the mode shape eigenvector, This scaling is completely arbitrary and so the modal mass could be any value, a s also can be the corresponding modal stiffness. However, a s already observed, the ratio between any modal stiffness and its associated modal mass is unique and is equal to the corresponding eigenvalue. The modal mass is generally used to convert the original mode shape vector, to the more useful
normalised mode shape vector, I t should be noted that the original vector is dimensionless, while the mass-normalised vector has dimensions of
Using the mass-normalised mode shape vectors, we can see how to derive quantities which provide us with information about the effective mass (or stiffness) a t any point on the structure, such a s DOF j. I t is helpful to a detail of the point FRF, that might be computed and plotted just in the immediate vicinity of the corresponding natural frequency: this would look something similar to the detailed plot shown earlier in Fig 2.15 and would be characterised by a skeleton which is based on a n effective mass line, , and a n effective stiffness line, These quantities can be related to the eigenvector elements by the simple formulae:
Effective mass a t DOF, for mode
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1 , which has units of mass,
and effective stiffness a t DOF; for mode r,
It can be seen t h a t since the mass-normalised eigenvectors are unique, and not subject to any arbitrary scaling factors, these effective mass and stiffness properties are also unique and represent a useful description of the underlying behaviour of the structure point by point, and mode by mode.
The other quantities which are sometimes referred to a s unique properties of each mode are the g e n e r a l i s e d mass and generalised stiffness. Although there s no universal agreement of the definitions of these properties, t h a t which is adopted in this work is to define the generalised mass (or stiffness) of the mode a s the effective mass (or stiffness) a t the DOF with the largest amplitude of response. This quantity serves to provide a comparison of the relative strength of each mode of the structure.
2.4.4 Repeated R o o t s o r Multiple Modes
There are situations where two (or more) different modes will have the same natural frequency. This is one of the exclusions made above a t equations (2.32) and (2.33) but occurs frequently in structures which exhibit a degree of symmetry, especially axisymmetry, a s found i n most discs, cylinders, rings, etc. I n these cases, there is no guarantee that the corresponding two (or more) eigenvectors, and will be orthogonal to each other a s required in those equations. However, it can be asserted t h a t two such orthogonal vectors do exist and that if this property is not already exhibited by the two vectors available, then two other linear combinations of these two vectors can always be found such that orthogonality is observed between the mode shapes used to describe the motion i n each of the two modes which have the same frequency. It should be noted, however, that free vibration a t that frequency is possible not only in each of the two vectors thus defined, but also in a deformation pattern which is given by any linear combination of these two vectors. This can be easily demonstrated using the example of a circular-section bar, clamped a t one end, a s shown in Fig. There clearly be. two modes which correspond to each bending deflection pattern along the shaft: one in the vertical plane and
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the other in the horizontal plane, see Figs. and (c). If the bar itself and the end supports are completely axisymmetric, these two modes will have identical natural frequencies. As a result, any combination of 'the vertical' mode and 'the horizontal' mode will also be a valid mode of vibration a t that natural frequency, such a s the example shown i n Fig.
As the existence of double modes is commonplace in many structures, we shall return to these features from time to time.
Fig. 2.17 Repeated modes of symmetric structures.
(a) Symmetric structure; Vertical mode; (c) Horizontal mode;
(d) Oblique mode
2.4.5 F o r c e d R e s p o n s e S o l u t i o n
-
T h e F R F C h a r a c t e r i s t i c s Turning now to a response analysis, we shall consider the case where the structure is excited sinusoidally by a set of forces all a t the same frequency, but with individual amplitudes and phases. Thenand, as before, we shall assume a solution exists of the form
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where {F) and {X} are N vectors of time-independent complex amplitudes.
The equation of motion then becomes
or, rearranging to solve for the unknown responses,
which may be written
where is the N N receptance FRF matrix for the system and constitutes its R e s p o n s e Model. The general element in the receptance FRF matrix, is defined a s
and a s such represents a n individual receptance FRF expression very similar to that defined earlier for the SDOF system.
It is clearly possible for us to determine values for the elements of a t any frequency of interest simply by substituting the appropriate values into (2.36). However, this involves the inversion of a system matrix a t each frequency and this has several disadvantages, namely:
it becomes costly for large-order systems (large
it is inefficient if only a few of the individual FRF expressions are required;
it provides no insight into the form of the various FRF properties.
For these, and other, reasons an alternative means of deriving the various FRF parameters is used makes use of the modal properties for the system instead of the spatial properties.
Returning to we can write
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Premultiply both sides by and postmultiply both sides by obtain
which leads to
It is clear from this equation that the receptance matrix is symmetric and this will be recognised a s the principle of reciprocity which applies to many structural characteristics. Its implications i n this situation are that:
Equation (2.40) permits us to compute any individual FRF parameter, using the formula (noting that the resulting expression is delivered by multiplying the row of by the diagonal frequency matrix by the kth column of
.
which is very much simpler and more informative than by means of the direct inverse, equation Here we introduce a new parameter, , which we shall refer to a s a Modal Constant*: i n this case, t h a t
* Note that other presentations of the theory sometimes refer to the modal constant as a 'Residue' together with the use of 'Pole' instead of our natural frequency.
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for mode for this specific receptance coordinates and k. The above is a most important result and is in fact the central relationship upon which the whole subject is based. From the general equation
the typical individual FRF element defined in would be expected to have the form of a ratio of two polynomials:
and in such a format it would be difficult to the nature of the function,
.
However, it is clear that a n expression such a s (2.42) can also be rewritten a sand by inspection of the form of it is also clear that the factors in the denominator, , etc. are indeed the natural frequencies of the system, (this is because the denominator is necessarily formed by the det [K] -
All means that a forbidding rational fraction expression such as (2.42) can be expected to be reducible to a partial fraction series form, such a s
Thus, the solution we obtain through equations (2.38) to is not unexpected, but its lies in the very simple and convenient formula it provides for the coefficients, in the series form.
We can observe some of the above relationships through our example. The forced vibration equations of motion give
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which, in turn, give (for example):
or, numerically,
Now. if we use the modal summation formula together the results obtained earlier, we can write
or, numerically,
which is equal to x - above.
The above characteristics of both the modal and response models of a n undamped MDOF system form the basis of the corresponding data for the more general, damped, cases.
The following sections will examine the effects on these models of adding various types of damping, while a discussion of the presentation MDOF frequency response data is given in Section
2.5 MDOF SYSTEMS WITH PROPORTIONAL DAMPING