A method is described for deriving constrained modes and frequencies from a reduced model based on a subset of the free-free modes plus the residual effects of neglected modes. The method involves a simple modification of the MacNeal and Rubin component mode representation to allow development of a verified constrained (fixed-base) structural model. Results for two spaceflight structures having translational boundary degrees of freedom show quick convergence of constrained modes using a measurable number of free-free modes plus the boundary partition of the residual flexibility matrix. This paper presents the free-free residual flexibility approach as an alternative test/analysis method when fixed-base testing proves impractical. NomenclatureA -transformation matrix for flexibility F = external force vector G = flexibility matrix G r = residual flexibility matrix H = mass matrix associated with residual flexibility K = stiffness matrix K = reduced stiffness matrix M = mass matrix M = reduced mass matrix m = number of retained modes plus boundary coordinates TV = number of coordinates in unreduced structure n = number of retained or measured modes n b = number of boundary coordinates q = generalized displacement vector T = transformation matrix to reduce structure coordinates u = physical structure displacement vector $ = mode shapes w,12 = natural frequencies Subscripts b = boundary coordinates c = constrained structure / = flexible structure / = interior coordinates n = retained modes R = rigid body r = residual Introduction D EVELOPMENT of a dynamic model of a constrained structure having acceptable fidelity requires a modal survey test of the physical structure and subsequent improvement of the model using test data. Constrained-boundary or fixedbase testing has historically been the most common approach for verifying constrained mathematical models, since the boundary conditions of the test article are designed to match the actual constraints in service.Unfortunately, there are a number of difficulties involved with fixed-base testing, making the approach impractical in some cases. As stated in Refs. 1 and 2, it is not possible to conduct a truly fixed-base test due to coupling between the test article and the fixture. In addition, it is often difficult to accurately simulate the actual boundary constraints, and the cost of designing and constructing the fixture may be prohibitive.Alternate free-boundary test methods for deriving constrained modes have been investigated for use when measurement of fixed-base modes proves impractical or undesirable. The mass-additive technique 1 ' 2 is an interesting approach that shows promise, though large sets of free-free modes are required for some structures. Blair and Vadlamudi 3 presented a useful method for model improvement where the minima of the interface response functions were used to assess model adequacy in the interface regions. This data in combination with the free-free mode set was used to obtain a Shuttle-constrained space telescope model for loads analysi...
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