A self-starting multistage, time-domain procedure is presented for the identification of nonlinear, multi-degree-of-freedom systems undergoing free oscillations or subjected to arbitrary direct force excitations and/or nonuniform support motions. Recursive least-squares parameter estimation methods combined with non-parametric identification techniques are used to represent, with sufficient accuracy, the identified system in a form that allows the convenient prediction of its transient response under excitations that differ from the test signals. The utility of this procedure is demonstrated in a companion paper.

It is worth noting from Fig. 8 that the spread of the results (i.e., dimensionless ordinate scales of the two plots) pertaining to the damping and stiffness influence coefficients differ by more than an order of magnitude (a factor of about 50). This behavior is consistent with the fact that, in the example under discussion, the relative contribution of damping-related forces and stiffens-related forces is nearly inversely proportional to the above-mentioned spread.4.4 Determination of Nonlinear Forces. Using the available measurements and the previously identified system matrices, the nonlinear system forces can now be computed from With that, the time history of the nonlinear force vector f, ( t ) components can be determined and are shown in Fig. 9. For convenience, identical scales are used for the three plots.At this stage of the identification procedure, the "best" (in least-squares sense) equivalent linear model has been determined in the form of the identified matrices. Thus, if for the purposes of a particular application the norm of the residual error, Ilf, (t) II, as computed from equation (6) is sufficiently small, then the identification task can be terminated. For more demanding situations, additional processing is required to more accurately identify the residual forces that have been determined.As pointed out earlier, if there is a need to augment the parametric identification results with additional results from the nonparametric phase of the data processing, one can proceed directly to develop approximating analytical representations, for as many of the components of fN(t) as warranted, in terms of a series expansion involving suitable generalized coordinates. However, when the order of the dynamic system is relatively large, dealing with a transformed set of nonlinear forces may lead to a faster rate of convergence of the applicable series.A convenient and natural transformation to use with realistic dynamic systems is the one expressed by equation (A33): where and @ is the modal matrix associated with MjjlKl,. Although the linear modal transformation of equation (8)

The computational problems associated with the estimation of damping, particularly in piping systems, is discussed in detail and appropriate procedures are recommended to minimize their impact. The theoretical background and the related basic assumptions in each of these methods is given first in order that the approximations introduced in practical problems is better understood. This understanding then leads to alternative procedures where more adequate and useful results are expected. The objective of this effort is to help homogenize the existing damping data by the use of systematic procedures and hence to reduce the variability of damping estimates in piping systems.

No abstract