Forced Response Solution - FRF Characteristics

We turn next to the analysis of forced vibration for the particular case of harmonic excitation and response, for which the governing equation of motion is:

As before, a direct solution to this problem may be obtained by using the equations of motion to give:

but again this is very inefficient for numerical application and we shall make use of the same procedure a s before by multiplying both sides of the equation by the eigenvectors. Starting with and following the same procedure as between equations (2.38) and we can write:

and from this matrix equation we can extract any one FRF element, such a s , and express it explicitly in a series form:

which may also be rewritten in various alternative ways, such as:

I n these expressions, the numerator (as well a s the denominator) is now complex a s a result of the complexity of the eigenvectors. I t is in this respect that the general damping case differs from that for proportional damping.

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2.6.3 Excitation by a General Force Vector 2.6.3.1 Operating deflection shape

Having derived a n expression for the general term in the frequency response function matrix, it is appropriate to consider next the analysis of a situation where the system is excited simultaneously a t several points (rather than at just one, a s is the case for the individual FRF expressions).

The general behaviour for this case is governed by equation (2.63) and the solution in (2.64). However, a more explicit (and perhaps useful) form of this solution may be derived from (2.64) - although not very easily! - as:

This equation permits the calculation of one or more individual responses to a n excitation of several simultaneous harmonic forces (all of which must have the same frequency but may vary in magnitude and phase) and it may be seen that the special case of one single response to a single force (a frequency response function) is clearly that quoted in The resulting vector of responses is sometimes referred to a s forced vibration mode or, more commonly, a s a n operating deflection shape When the excitation frequency is close to one of the system's natural frequencies, the will usually reflect the shape of the nearby mode because one term in the series of (2.67) will dominate, but will not be identical to it because of the contributions, albeit small, of all the other modes.

2.6.3.2 Pure mode excitation 1

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damped system normal modes There are a number of cases of multi-point harmonic excitation of special interest which are worth mentioning here. These are generally associated with the notion that by choosing or tuning the vector of individual forces it is possible to set up a response of the structure which is entirely controlled by a single normal mode of the structure. When we enter this domain, we sometimes r u n into difficulties of nomenclature, especially in respect of the meaning of the term 'normal mode'. As explained earlier in this work, the normal modes are the characteristic modes of the structure in its actual, damped, state. While it is possible to talk of the modes 'that the structure would have if the damping could, by some magic, be removed', these are not the 'normal' modes of the structure in any strict sense.

They may be referred to a s the 'normal modes of the associated undamped structure' and it is true that they are properties of some

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interest because in most cases of test-analysis comparison, the analytical model will be undamped and so there is a desire to be able to extract the test structures 'undamped' modes from the test data in order to effect a direct comparison between prediction and measurement.

So, we find ourselves seeking procedures which would enable us the normal modes of the structure directly, one by one, by generation of a suitable excitation force vector, {F) .

The first of these cases that we shall describe is the genuine normal mode excitation, in which a n excitation vector {F} is sought such that response in just mode s. Depending upon the exact damping conditions, this exclusive excitation vector may be more or less easy to define, and indeed, its elements may well be complex they will each have different phases) but it will always exist.

2.6.3.3 Pure m o d e excitation 2

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associated u n d a m p e d system n o r m a l modes

I t is also worth mentioning another special case of some interest:

namely, that where the harmonic excitation is described by a vector of mono-phased forces. Here, the complete generality admitted in the previous paragraph is restricted somewhat by insisting that all forces have the same frequency and phase, although their magnitudes may vary. What is of interest in this case is to see that there exist conditions under which it is possible to obtain a similarly mono-phased response (the whole system responding with a single phase angle). This is not generally the case in the solution to equation (2.67) above.

Thus, let the force and response vectors be represented by

where both and are vectors of real quantities, and substitute these into the equation of motion, (2.63). This leads to a complex equation which can be split into real and imaginary parts to give:

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The second of this pair of equations can be treated as a n eigenvalue problem which has 'roots' 8, and corresponding 'vectors' {K),

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These may be inserted back into the first of the pair of equations (2.69) in order to establish the form of the (mono-phased) force vector necessary to bring about the (mono-phased) response vector described by . Thus we find t h a t there exist a set of N mono-phased force vectors each of which, when applied a s excitation to the system, results in a

phased response characteristic.

I t must be noted that this analysis is even more complicated than it appears a t first, mainly because the equations used to obtain the above mentioned solution are functions of frequency. Thus, each solution obtained a s described above applies only at one specific frequency, However, it is particularly interesting to determine what frequencies must be considered in order that the characteristic phase lag (8) between (all) the forces and (all) the responses is exactly Inspection of equation (2.69) shows that if is to be then that equation reduces to:

which is clearly the equation to be solved to find the undamped system natural frequencies and mode shapes. Thus, we have the important result that it is always possible to find a set of mono-phased forces which will cause a mono-phased set of responses and, moreover, if these two sets of mono-phased parameters are separated by exactly then the frequency a t which the system is vibrating is identical to one of its undamped natural frequencies and the displacement 'shape' is the corresponding undamped mode shape.

This most important result is the basis for many of the multi-shaker test procedures used (particularly in the aircraft industry) to isolate the undamped modes of structures for comparison with theoretical predictions. I t is also noteworthy that this is one of the few methods for obtaining directly the undamped modes a s almost all other methods extract the actual damped modes of the system under test. The physics of the technique are quite simple: the force vector is chosen so that it exactly balances all the damping forces, whatever these may be, and so the principle applies equally to other types of damping.

2.6.4 Postscript

I t is often observed that the analysis for hysteretic damping is less than rigorous when applied to the free vibration situation, a s we have done above. However, it is a n admissible model of damping for describing harmonic forced vibration and this is the objective of most of our studies. Moreover, it is always possible to express each of the receptance

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(or other FRF) expressions either as a ratio of two polynomials (as explained in Section or as a series of simple terms such a s those we have used above. Each of the terms in the series may be identified with one of the 'modes' we have defined in the earlier free vibration analysis for the system. Thus, whether or not the solution is strictly valid for a free vibration analysis, we can and confidently consider each of the uncoupled terms or modes a s being a genuine characteristic of the system. As will be seen in the next section, the analysis required for the general case of viscous damping which is more rigorous

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is considerably more complicated than that used here which is, in effect, a very simple extension of the undamped case.

In document Modal Testing_Theory and Practice 2nd Edition (Page 83-87)