# Simplification of Complex Structural Dynamic Models:
## 4 Conclusions  
In this paper, we presented a study of five different approaches for reducing the computational effort for a full finite element modal analysis when facing structures where flexible subcomponents are attached to other heavier subcomponents. The approaches include a reference model without any simplifications and finite element models where the heavy subcomponent is replaced by ideal rigid clamping, by a free-tip mass model or by a free-tip mass model with an additional rotary inertia. Furthermore, an analytic approach based on the Euler-Bernoulli beam theory was chosen. The accuracy of the presented modelling approaches was assessed by comparing the corresponding numerical results with experimental results based on the eigenfrequencies of the first three bending modes. Seeking to give generic results, a monolithic beam-like structure with ideal free-free boundary conditions and a tip mass attached to one end of the beam were considered for comparison in this work. This latter configuration permits an in-depth comparison and uncertainty quantification due to the low cost of calculations and allows for a simplified model validation process. Although a rather simple model was chosen, the conclusions of the presented studies can be extended to complex structures. The investigations showed that, beyond the first bending mode, the ideal rigid boundary condition provides a good approximation. An error analysis between numerical and experimental results showed that, for the second and third bending mode, the deviations were only 2% and 1%, respectively, while this value was as high as 28% for the first bending mode. This indicates that, for this mode, the model was very sensitive to the boundary conditions, and applying a small change will affect the associated mode shapes significantly. In summary, at low natural frequencies, the mode shapes were sensitive to the ideal clamping boundary conditions, and these conditions must be chosen carefully to achieve satisfactory results. However, this does not hold for all of the other modelling approaches. Especially when replacing the tip-mass by coupling a corresponding point mass and rotary inertia to the free end, sufficiently accurate results can be achieved for all eigenfrequencies of the three bending modes. Based on this approach, relative errors of less than 2% were achieved, proving it as the most favourable modelling approach if an analytical model is not feasible. Given that the mass-ratio of the attached mass to the beam, was rather low in our study, 12.5, higher mass ratio masses should lead to similar results at the first three and higher natural frequencies. General but important remarks regarding simulation models that include large lumped-masses as secondary components, are:  
* A full finite element model for all the components, including large heavy masses, using ideal-free conditions for the model is better suited where comparisons to experimental results are necessary, given that sufficient computer memory resources are available. * For the simplification of a large attached mass by an ideal clamped condition in a finite element model, a very high mass ratio is necessary, although the end results can still be quite poor even for the primary first bending mode. * It is commonly known, and was demonstrated here, that an analytical model can yield satisfactory results for a beam with an attached tip-mass, when the structure is beam-like. * Reduced size modelling by a finite element approach using only an offset point-mass term for simplification can be easily improved upon by adding a rotary inertia term. \begin{table}
\begin{tabular}{c c c c c} \hline
**Mode** & **Experiment [Hz]** & **FEfree-tip mass (3)** & **FEfree-tip mass \& inertia (4)** & **Analyticfree-free (5)** \\ \hline
1 st & 118 & 112.7\% & 0.8\% & 2.9\% \\
2 nd & 540 & 63.1\% & 1.5\% & 0.6\% \\
3 rd & 1462 & 33.7\% & 1.7\% & 0.7\% \\ \hline \end{tabular}
\end{table}