6 A Brief History of 30 Years of Model Updating in Structural Dynamics 57 a b Element 3 Node 1 Node 1 Node 2 Node 3 Node 2 Node 3 Element 1 Element 2 Element 1 Element 3 Element 4 Element 2 Fig. 6.2 Illustrations of the two fundamental rules of the stiffness FE method. (a) Assembly of three truss elements. (b) Assembly of four shell elements flexibility representation, as opposed to the conventional stiffness approach, would make it much easier to compare vibration measurements to FE predictions. Discussion presented in the remainder focuses on the stiffness method. The stiffness method is articulated around two fundamental rules: Rule 1: The summation of all forces contributed by finite elements that share a common node, or DOF, is equal to the externally applied force. Rule 2: Displacements contributed by different finite elements at the same node, or DOF, are equal. These rules are illustrated on the left side of Fig. 6.2 in the case of truss (or bar) elements. The right side of Fig. 6.2 suggests that similar rules apply to continuous elements such as shell (or membrane) finite elements. The aforementioned rules are briefly illustrated for the three-truss example of Fig. 6.2a, which is useful to highlight properties of FE matrices that are discussed later in the context of model updating. The first rule provides the principle to equilibrate the forces at the kth node, or DOF, of the FE discretization. For the three-truss example, it can be written as: Rule 1 ! F.1/ CF.2/ CF.3/ DFExt;k (6.7) Representations of the kinetic condition and material behavior, discussed further in Eqs. (6.11)–(6.13) below, lead to a force-displacement equation expressed within each element as: F.e/ Dk.e/ U.e/ (6.8) The second rule converts the generalized displacement unknowns contributed by each element, denoted by U(e), into a single unknown per DOF: Rule 2 ! U.1/ DUk; U.2/ DUk; U.3/ DUk (6.9) Inserting Eq. (6.8) in the summation [Eq. (6.7)] for each element, then, replacing element-specific displacements U(e) with the global DOF unknown Uk from Eq. (6.9), leads to the definition of the master stiffness matrix as: k.1/ U.1/ C k.2/ U.2/ C k.3/ U.3/ D FExt;k (6.10a) k.1/ Ck.2/ Ck.3/ Uk D FExt;k (6.10b) X eD1 NE k.e/ „ ƒ‚ … K UDFExt (6.10c) It is noted that, for simplicity, Eqs. (6.7)–(6.10) are derived symbolically in the local orientation frame of each element and the transformation to the global frame-of-reference is not included. For completeness, this omission is corrected in Eq. (6.14). What remains to complete this description of the stiffness method, is a brief discussion of three so-far-overlooked aspects. The first two are assumptions made to describe, first, the kinetic relation that connects the element-level displacements to
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