3) Review of Theory and Modal Constraints Crowley, et al. presented the original structural modification using FRFs (sometimes called SMURF) [3], and it is repeated here as the basis for this method. The FRF matrix equations can be partitioned as (1) where a is acceleration, H is a frequency response function and f is force. The subscripts associated with partitioning are f for free dof and c for dof which will be constrained mathematically at a later time. Only a few (perhaps just one) columns of equation (1) are measured, therefore the other columns must be reproduced from the modal parameters extracted from the measured columns. This method is limited by the effects of modal truncation. By constraining ac to zero, equation (1) can be rearranged (2) However, Hcc is usually poorly conditioned on inversion and gives uncertain results. Poor results are probably the reason why this constraint theory has not been more popular. By reducing the physical c dof in equation (2) to the modal coordinates of the table, the conditioning is much improved. Equation (3), which utilizes modal connection dofs, can be much more robust and is given by (3) where Φc is the mode shape matrix of the flexible table without a test article, and the superscript + indicates the pseudoinverse. This procedure can also be performed using the modes of the sytem raterh than measured FRFs. An equivalent modal approach begins with the equations of motion based on modes extracted from the full system modal test on the shaker table as [ ]{ } { } { }f I q q T n − =Ψ [ ] 2 2 \ \ ω ω (4) where ωn represents the circular natural frequencies from the system modal test, q represents the modal coordinates, f is the vector of forces and Ψ provides the mass normalized mode shapes. A constraint for the table degrees of freedom can be written as { } { }0 = cx (5) or using the modal substitution, then { } { }0 Ψ = q c . (6) Using modal constraints (which are weaker than physical dof constraints) produces { } { }0 Φ Ψ = + q c c (7) where cΦ comes from the previously characterized mode shapes of the bare table. This projects the constraint onto the vector space of the bare table shapes, which has the advantage of producing a least squares fit through any experimental modal analysis errors. After the constraint is applied, the eigenvalue problem is solved again to estimate the fixed base modes. The number of final modes will be reduced from the original number of modes in eqn (4) by the number of constraints. As a practical consideration, any significant dynamics of the table which are active in the full system test need to be constrained. Modes of the table that are not active are ignored. In the authors' experience, it can actually be detrimental to the results to include them in the constraint mode shapes, cΦ , particularly if there are no test article modes that exercise these inactive table modes. ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ c f cc cf fc ff c f f f H H H H a a { } [ ]{ }f cf fc cc ff f f a H H H H1− = − { } [ ] [ ]{ }f cf c T c cc c T fc c ff f f H H a H H + − + + + = − Φ Φ Φ Φ1 39
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