6926475979

t

by an analytical expression. The two common meth-ods of curve fitting are single-degree-of-freedom (d.o.f.) curve fit and multi-degree-of-freedom curve fit.

In the single d.o.f. curve fit each resonance of the measured transfer function is approximated to a simple oscillator. This approximation can be done in the time domain in terms of an impulse response function or a differential equation for the simple oscillator or in the frequency domain by using the transfer function

one resonance at a time. The Argand, or Nyqui$t, curve fit method is a graphical single d.o.f. curve fitting method in which each resonance is fitted into a circle in the real-imaginary (complex) piane. The Nyquist plot for a simple oscillator with velocity as the output - i.e., mobility function - is a circle. In a multi d.o.f. curve fit the entire transfer function is fitted into a higher order model (i.e., many reso-nances simultaneously) either in the time domain or in the frequency domain using an expression of the form

1    _rw_x(j>)

In this case all resonances are considered simultaneously. It follows that the multi d.o.f. curve fit is morę accurate but, at the same time, computationally morę demanding.

It is not necessary to measure the complete set of n^a transfer functions in the H matrix in order to determine the modal matrix. The symmetry suggests that we do not need morę than 1/2 n(n+1) transfer functions. It is easy to show, however, that just n transfer functions are adequate for this purpose. A convenient choice woufd be to measure any one row or any one column of the H matrix. For exam-ple, if the k th column i$ measured (H_ik with i * 1,2,.., n), by applying a forcing excitation at the k th degree of freedom and measuring the corre-sponding response signals at all degrees of freedom, the procedurę for constructing the modal matrix would be as follows:

i.    Examir>e the (diagonal) transfer function H_kk, curve fit it to equation (7). and determine the k th element of the n modę shape vectors; (^h. (^k)j. • •• <lMn* ^This consti-tutes the k th row of the modal matrix.

ii.    Examine the transfer function    and

from equation (7) determine    ^fc)|,

(^k+i ^k>a* • •* Wk+i *k>n- ^Th'S 9'ves the k+i th row of the modal matrix; Wk+j)i.

Wk+Pa* • •• (^k+iW

iii.    Repeat step (ii) for i = 1,2,.., n-k, and for i * -1, -2,.., -k+1.

The above procedurę for determining the modal matrix reveals that it is in fact not essential to measure a complete row or a complete column of the transfer matrix. As long as a diagonal element - a point transfer function or an auto-transfer function -is measured, the remaining n-1 transfer functions can be chosen arbitrarily, provided all n degrees of freedom are covered as either an excitation point, a measurement location, or both. Notę, however, that it is advisable to measure redundant transfer functions (those providing Information that isalready determined by previous measurements) as well. Such redundant data are usefuJ for checking the accuracy of the modal estimates. Figurę 1 shows three sets of transfer functions on the transfer function matrix. The first set is an example for the case in which redundant transfer functions are present. In the second set no redundant transfer functions are present. but the set is adequate to extract the modal matrix. The third set is clearly not adequate to determine the entire modal matrix.

In practice, the frequency response data for higher resonances are relatively less accurate. It is therefore customary, in most modal software, to extract modal parameters only for the first several modes. Then it is not possible to recover the mass. stiffness, and damping matrices. Even if these matrices were determined, their accuracy would be questionable unless great care is exercised to improve the accuracy at higher frequencies. due to their increased sensitivity to noise.

HARDWARE AND SOFTWARE SELECTION

The hardware structure of a typical modal analysis system is shown in Figurę 2. Either the response

£

V:

. •.

*.v.

i

v

$

%

:-:s