# Simplification of Complex Structural Dynamic Models:
## 2 Model Descriptions
### Analytical Theory
Even within the bounds of experimental uncertainty, which are largely acceptable for the second and third bending mode, the first bending mode was not correctly identified by this reduced numerical model FEfree-clamped. The arithmetically averaged relative deviations for the eigenfrequencies of the first three bending modes were \(-28.2\%\), \(-1.8\%\), and \(1.1\%\), respectively. In Figure 4, an envelope identifying the upper and lower limits around the mean values of the eigenfrequencies indicates the deviation due to the material and geometrical uncertainties. These limits were determined by applying the Interval Method, the details of which can be found in c.f. [23; 24; 25]. It is clear that the uncertainty range of modelling approach **2** (blue) did not overlap the uncertainty range of modelling approach **1** (red) for the first bending eigenfrequency. Hence, there is little likelihood of predicting the first \"true\" bending mode from a simplified free-clamped model. However, the uncertainty ranges for the two modelling approaches do overlap for eigenfrequencies of the second and third bending mode. The consequence is that, especially for low frequencies, it is not recommended to use a free-clamped boundary condition to approximate the effect of an attached, large mass. To understand the high deviation for the first eigenfrequency of bending modes, we take a closer look at the associated mode shapes. Figure 5 shows the mode shapes for the first three bending modes of the cantilever beam and the simple beam with the tip mass. In the second and third bending modes, the mode shapes are nearly identical. This means, that the location of the two end-nodes for the second mode and the three nodes for the third mode are situated at nearly the same point. This is not the case for the first bending mode. Although both mode shapes have exactly one node, as expected, the position of the nodes is not the same. Although it is difficult to assess directly from Figure 5a, for the cantilever beam, the node is located at the ideal rigid constraint, whereas, for the beam with the tip mass, the node is located slightly away from the boundary, i.e., closer to the centre of the beam. Having identified modelling approach **2** as insufficient, the focus is set on the approaches **3**, **4** and **5**, c.f. Figure 1. In this case, only one specimen is taken into account. Figure 6 shows that modelling approach **3**, i.e., replacing the tip-mass at the end of the beam by a coupling of a point mass, yielded the highest deviation over the whole frequency range. For the first natural frequency, the deviation even exceeded 100%. Thus, taking only the tip mass into account and neglecting the rotary inertia is not a valid modelling approach. However, when using modelling approach **4**, i.e., adding the rotary inertia of the tip-mass, far better results were achieved. The relative deviation between the numerical results and the experimental results was below 2% for all considered eigenfrequencies. In the case of modelling approach **5**, i.e., the analytic model Analyticfreefreefree assuming free-free boundary conditions, the highest deviation was found at the first eigenfrequency with 2.9%. For the second and third eigenfrequency, the deviations to the experiment were less than 1%. Thus, both the finite element model with the coupling of the point mass and rotary inertia as well as the analytical model yielded sufficiently accurate results. As a last step, we studied the effect of the uncertainties of the geometrical and material parameters for the tip mass. In this case the large attachment. Using the finite element model FEfree-tip mass & inertia, these uncertainties were included into the model by changing the point mass and rotary inertia values accordingly. Based on the values given in Table 1, a relative deviation from the mean value, or expected value, of the determined eigenfrequencies of less than \(\pm 0.8\%\) was found.