Finding the Limits of Beam Elements: Modeling Bolted Joints as an Effective Joint Region 131 Fig. 2 Experimental instrumentation setup used for impact hammer modal testing. Bolted joints, specifically the BRB, have been demonstrated to exhibit non-repeatability and sensitivity to bolt tightening order [8]. To account for this, six bolt tightening orders were tested, ten impacts were recorded for each order and the average frequency response function (FRF) was computed for each order in units of accelerance aas A(n)(f)=˜a(n)(f)/˜F(n)(f) (1) where ˜a is the frequency domain acceleration, ˜F is the frequency domain force, f is the frequency, and nis the hit number index. The average FRF is thus Aave(f)= 1 10 10X n=1 A(n)(f). (2) Finite element model updating This work uses the Euler-Bernoulli beam element as a test case for the EJR method. Despite its lower performance compared to other finite element formulations for beams, the Euler-Bernoulli element is computationally inexpensive and widely accessible. The properties chosen for updating are the damping loss factor η, the Young’s modulus E, and the density ρ. These properties were selected because joints have been observed to influence stiffness and damping, and the addition of bolts, nuts, washers and holes necessarily changes the density of an otherwise homogeneous region. Using this formulation, the dynamic response of an element at a given frequencyωas a function of these properties is⃗ U(η,E,ρ)= (1+ηi)K(E) −ω2M(ρ) −1⃗ F (3) where⃗ F are the nodal applied forces,⃗ U are the nodal displacements, Kis the stiffness matrix, andMis the mass matrix. In order to compare to modal test data, the response is converted to accelerance⃗ A.⃗ A=(iω2)⃗U⊘⃗F (4) A unit force is applied at the instrumented hammer impact location by finding it’s equivalent nodal location and substituting the corresponding element of⃗ F in equation (3). The frequency response function for comparison to modal data is calculated by finding the equivalent nodal location of the accelerometer and extracting the corresponding element from the accelerance vector in equation (4). All models evaluated in this work use free-free boundary conditions and neglect the added mass of the accelerometer. Mesh convergence studies were performed and an element length of 1 cm was chosen for model updating. The EJR method separately defines properties for the joint region as ηj, Ej, and ρj, allowing them to differ from the known properties for the monolithic sections of the structure as shown in Figure 3. In the non-joint region, these material properties are assigned according to test specimen material data. EJR properties are assigned by finding the equivalent nodal locations of the joint boundaries and replacingη, E, andρwithηj, Ej, andρj for the elements within these boundaries. The resulting model is that of a solid beam with properties η, E, andρvarying as a function of beam length. The known material properties and bounds used for optimization of EJR properties are summarized in Table 1. The finite element model used in this work was developed in MATLAB and benchmarked against Abaqus. Firstly, a set of frequencies is extracted from modal test data when the experimental frequency response function is calculated. Next, the finite element model is evaluated at each frequency in this set to construct a frequency response function that can be compared directly with the experimental modal test data. The degree of similarity between the experimental FRF and the
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