Correcting indefinite mass matrices due to substructure uncoupling
Introduction
Experimental–analytical substructuring has been a topic of interest since modal testing was first introduced several decades ago. It is appealing because it has the potential to allow one to replace complicated subcomponents with experimental models that may be much less expensive to obtain. It also allows the experimentalist to re-use the experimental model, predicting its response in a multitude of other configurations without repeating the test. One can also think of structural modification [1] as a special case of substructuring, where the modification is a special substructure that one wishes to determine in order to reduce the response of the assembly to an acceptable level (although the terms “substructuring” and “structural modification” are often used interchangeably [2]). The basic theory underlying substructuring is described in several textbooks [3], [4], and an excellent review of analytical substructuring methods was recently presented by de Klerk, Rixen and Voormeeren [5]. Several additional challenges emerge when one of the substructures of interest exists as hardware and so a model must be created from test, and the authors are not aware of a comprehensive review paper of experimental–analytical substructuring, although several excellent Ph.D. theses have been published on this topic [6], [7], [8] and the authors recently presented a brief review paper exploring sensitivity in experimental/analytical substructuring [9]. Some of the challenges encountered include: need for rotational measurements, which are difficult to measure experimentally, inadequacy of the experimental modal basis since modes can only be measured over a limited frequency range, and sensitivity to experimental errors.
In [10], [11] the authors recently presented a new substructuring methodology that addresses some of these challenges. The substructure of interest is tested with a specially designed fixture, called a transmission simulator, attached. The transmission simulator mass-loads the interface, enriching the modal basis of the substructure and it is designed to be easy to model analytically or numerically so all of its relevant modes can be readily extracted, allowing modal filtering or sensor set expansion techniques to be used to obtain the unmeasured rotations.
One key component of the transmission simulator approach is the method whereby the transmission simulator is analytically removed from the tested assembly in order to obtain the substructure model. This substructure uncoupling approach can be very sensitive to experimental errors, depending on the sensor set used to perform the decoupling [12], [13]. This work employs modal constraints (or the Modal Constraints for Fixture and Subsystem (MCFS) approach) according to the method outlined in [10] to avoid these difficulties. In the MCFS approach, the transmission simulator is removed from the assembly by creating a model of the subcomponent that is to be removed, making its mass, stiffness, and damping negative and then coupling the negative subcomponent to the assembly. Whereas, in conventional substructuring one enforces constraints between the points where the substructures are joined, the MCFS method estimates a set of modal coordinates on the substructure and enforces constraints on those coordinates. This reduces the sensitivity of the method to experimental errors and assures that an appropriate number of constraints is enforced.
Unfortunately, removing a subcomponent (the transmission simulator) requires one to introduce a system with negative mass, stiffness and damping, and as a result the substructure model obtained is sometimes found to no longer have a positive definite mass matrix. Similar problems have been encountered by other researchers when removing rigid masses from a structure [14]. While good results have been found in a few previous studies even though the substructure mass matrix was not positive definite, a model such as this cannot be imported into most finite element packages and even when it can be it may interfere with the convergence of the eigenvalue solver. (For information regarding how to import an experimental model obtained using the transmission simulator method into an FEA package, see [15].) Furthermore, some Monte Carlo simulations have revealed that substructuring can be more sensitive to experimental errors when negative mass is present [16], [17]. This paper presents two methods that can be used to assure that the mass matrix of a subcomponent is positive definite. The theory underlying the MCFS method is reviewed in Section 2.1, and the two proposed methods are presented in 2.2 Modal scale factor method, 2.3 Added mass method. The methods are evaluated in Section 3 by applying them to two sample systems that have been studied previously [10] that were found exhibit negative mass.
Section snippets
Review of subtraction of modal substructures
Suppose that one has an assembly, denoted system C, that consists of the substructure of interest connected to a transmission simulator. Modal testing methods can be used to measure the natural frequencies, , damping ratios, , and matrix of mass-normalized mode shapes, , of the assembly. The modal parameters of the transmission simulator are also known. (Here we shall refer to the substructure that is being removed as the transmission simulator, but in a general problem it could be any
T-beam system
The first system considered is a 294 mm (12-in.) long steel beam with a 18.4 mm (0.75 in.)×24.5 mm (1.0 in.) cross section, as pictured in Fig. 1. This is the same system that was considered in [18] and the same finite element models were used with 21 and 30 nodes for A and B respectively. A 147 mm (6.0-in.) long transmission simulator with the same cross section is attached to the beam and the modes of the assembly (the C system) are computed; in the usual practice these modes would be found
Conclusions
This work presented two methods that can be used to obtain a positive definite mass matrix from the indefinite result that is often obtained when removing one structure from another using modal substructuring. The first method, called the mode scale factor method, used a nonlinear optimization algorithm to vary the modal masses (or equivalently mode scale factors) of the modes of the transmission simulator. This approach was found to work well in cases where the required transmission simulator
Acknowledgments
This work was supported by, and some of this work was performed at Sandia National Laboratories. Sandia is a multi-program laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy's National Nuclear Security Administration under Contract DE-AC04-94AL85000.
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Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U.S. Department of Energy under Contract DE-AC04-94AL85000.