>@ >@ >@ >@ >@ 1 B AB A 1 AB cc cc cc cc cc H H I H H (11) Both approaches obtain the same solution when only connection DOF are used, as seen by comparing (9) to (11). Inverse Frequency Based Substructuring Approach The Inverse Frequency Based Substructuring Approach (Inverse FBS) reformulates the Frequency Based Substructuring (FBS) equations to solve for a single component [15]. For this technique, measurements are required at either side of the connection. The subscript c(a) represents a connection DOF on component A of the system, and the subscript c(b) represents a connection DOF on component B of the system. With this nomenclature, the drive point FRF on the unknown component can be written in terms of purely system FRFs as, > @ > @ > @ > @ > @ 1 B AB AB AB AB c(b)c(b) c(a)c(a) c(b)c(a) c(b)c(b) c(a)c(b) H H H H H (12) > @ > @ > @ 1 AB AB AB c(a)c(a) c(a)c(b) c(a)c(b) H H H x APPLICATION System Description – Analytical Models A basic six DOF mass-spring has been previously studied [17] to investigate the effects of noise and modal parameter estimation on the Mobility and Impedance decoupling approaches. Both techniques produced promising results, especially with the use of internal DOF. An additional technique, the Constraint Force Approach [17] was also developed to uncouple a system, and compared to the Mobility and Impedance approaches using the six DOF model. While this technique could successfully uncouple the model, only a single connection could be uncoupled at a time. This feature made the technique impractical on complex models and experimental structures where it may be impossible to couple a single DOF at a time. While the basic six DOF model was useful to study the effects of noise on each technique, a more complex model was needed to study issues inherent with experimental data. Issues such as truncation and lack of rotational DOF [6] have been shown to be detrimental to frequency based system modeling approaches. To address these issues, a finite element model (FEM) of a laboratory test structure was created in MATLAB [18]. The structure consists of two identical aluminum hollow rectangular beams, Beam A and Beam B, bolted together at two locations. Figure 1 displays the cross section of the beams, and Table 1 lists the physical properties used in the model. Figure 1. Beam Cross Section. Table 1. Physical Properties of Beam Models. The beams were considered planar and modeled using thirty elements each having two DOF per node (translational and rotational). Soft springs were connected to each end node to simulate a free-free boundary condition. Figure 2 displays a physical representation of each beam model. Table 2 lists the natural frequencies of each beam. As the models contained only mass and stiffness matrices, a one percent damping was assigned to each mode to limit the dynamic range. 175
RkJQdWJsaXNoZXIy MTMzNzEzMQ==