# Simplification of Complex Structural Dynamic Models:
## 2 Model Descriptions
### Test Samples  
In the following, the influence of a significantly simplified boundary condition on finite element models is discussed. First, six reference structures consisting of a beam with a large mass attached to the free end are considered (see Figure 1a). The beam samples are made of steel with the material properties homogeneous and elastic were assumed. The large mass, also steel, is approximately 10-times the mass of the beam. Various mass ratios were considered that could provide a limit on the minimum beam to large mass ratio; however, no definitive conclusions could be drawn below the mass ratio of 10 preferred here. The nominal values of the material and geometrical parameters with averaged deviation \(\varepsilon\) over six samples are given in Table 1. The nominal value and uncertainty of the parameters related to the material properties is obtained by performing measurements with a parameter identification method described in [23]. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Parameters** & **1** & **2** & **3** & **4** & **5** & **6** & **Mean** &
\begin{tabular}{c} **Standard** \\ **Deviation** \(\varepsilon\) \\ \end{tabular} \\ \hline Length, \(l\) [m] & 0.2001 & 0.2001 & 0.1999 & 0.2002 & 0.2000 & 0.2002 & 0.2001 & \(\pm 0.02\times 10^{-3}\) \\ Width, \(w\) [m] & 0.0399 & 0.0399 & 0.0400 & 0.0400 & 0.0399 & 0.0400 & 0.0399 & \(\pm 0.02\times 10^{-3}\) \\ Thickness, \(h\) [m] & 0.0040 & 0.0040 & 0.0041 & 0.0040 & 0.0040 & 0.0040 & 0.0040 & \(\pm 0.02\times 10^{-3}\) \\ Density, \(\rho\) [kgm\({}^{-3}\)] & 7700 & 7700 & 7700 & 7700 & 7700 & 7700 & 7700 & \(\pm 0.8\) \\ Young’s modulus, \(E\) [GPa] & 203.1 & 202.6 & 203.1 & 203.3 & 205.2 & 204.1 & 203.6 & \(\pm 4.7\) \\ Poisson’s ratio, \(v\) [-] & 0.29 & 0.29 & 0.29 & 0.29 & 0.28 & 0.28 & 0.29 & \(\pm 0.01\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The nominal values of the geometrical and material parameters with the averaged deviation \(\varepsilon\) over six samples. The velocity of longitudinal \((c_{l})\) and transversal \((c_{t})\) waves in an elastic continuum are given by  
\[c_{l}=\sqrt{\frac{E\cdot(1-\
u)}{\varrho\cdot(1+\
u)\cdot(1-2\
u)}} \tag{1}\]  
and  
\[c_{t}=\sqrt{\frac{E}{2\varrho\cdot(1+\
u)}}\, \tag{2}\]  
respectively. Herein, \(\varrho\) denotes the structure's density. Transforming both equations yields two expressions for [PERSON]'s modulus \(E\), i.e.,  
\[E(c_{l})=\frac{c_{l}^{2}\cdot\varrho\cdot(1+\
u)\cdot(1-2\
u)}{(1-\
u)} \tag{3}\]  
and  
\[E(c_{t})=2\cdot{c_{t}}^{2}\cdot\varrho\cdot(1+\
u). \tag{4}\]  
Assuming that \(E(c_{l})=E(c_{t})\) results in  
\[\
u=\frac{2c_{t}^{2}-c_{l}^{2}}{2\cdot({c_{t}}^{2}-{c_{t}}^{2})}. \tag{5}\]  
Given the longitudinal and transversal wave velocities \(c_{l}\) and \(c_{t}\), [PERSON]'s ratio \(\
u\) can be easily obtained from Equation (5). For determining the [PERSON]'s modulus, Equation (3) and Equation (4) were evaluated, and their results were averaged, i.e., \(E=1/2\) (\(E(c_{l})+E(c_{t})\)). The ultrasonic measurements were performed using the 5-MHz ultrasonic module UT/Mate from Vogt ultrasonics and transducers. All measurements were taken in reflection mode, i.e., pulse-echo mode, on specimens with plane-parallel grinded surfaces. On each specimen, ten individual measurements were taken for both the longitudinal and transversal sound velocities. The density was measured employing hydrostatic weighing using Archimedes' principle. For this, a Mettler Toledo 204 AG of 0.1-mg scale is used. Each specimen was first weighed in air, yielding \(m_{\text{air}}\), and afterward weighed in water, yielding \(m_{\text{water}}\). Given the two masses and the density of water \(\varrho_{\text{water}}\) and air \(\varrho_{\text{air}}\), the density \(\varrho\) of the specimen can be calculated using Equation (6) without knowledge of the specimen's volume.