# Simplification of Complex Structural Dynamic Models:
## 1 Introduction  
In structural mechanics, computational methods are often employed to simulate structural vibration by solving the underlying partial differential equations. With this method, an approximate function is computed for the exact solution for these differential equations: First, a continuous system is discretised into small _elements_. The response of the discrete system is then calculated with a _finite_ number of unknowns by solving a set of matrix equations. Within the past decade, as computer capability has increased, finite element methods (FEM) have become an important part in dynamic modelling for engineers. In particular, in industry, FEM is of importance for reducing the cost of experimental examinations. We assume that the reader is possibly unfamiliar with the fundamental formulations of the finite element method, but we refrain from including a detailed description here and refer the reader to the literature, such as [1, 2, 3, 4, 5]. To confront the majorissue, every finite element model possesses some degree of uncertainty due to a number of factors, such as topological and material uncertainties, initial and boundary conditions, forcing terms, etc. Therefore, experimental or analytical results are normally required to validate finite element models. [PERSON] [6] described the study of discrepancy between model-based simulations and real physical models as uncertainty quantification (UQ). General recommendations regarding model uncertainties are given in the article by [PERSON] [7], who showed the importance of modelling in the development process. This application of the method on various practical engineering problems, including uncertainty boundary conditions, has been studied widely, see [8; 9; 10; 11; 12; 13; 14], for example. In this work, although we consider the case where the details of a large complex attachment are simplified as modelled lumped-parameters. By considering the uncertainties of the structural dynamics, we show that poor modelling leads to poor results but that the models can be easily improved upon. This work primarily focusses on the free-vibration problem for a beam structure attached to a large mass, which could represent a vehicle engine block. Although it would be beneficial to extend our analysis to large (mass) complex geometries, which could include bolted or welded attachments, the simplified application presented here represents only the first few modes of vibration. Complex geometry in this frequency region plays a minor part; however, the connection between the two can rely strongly on the number of bolts, the torque applied, or damping at the interface. However, our analysis concerns a simplified case that can be referenced for more detailed studies. Moreover, if the vibration response on the mass, the beam, or at other locations--perhaps in the vehicle--then the vibrational amplitude or energy flow needs to be modelled accurately. Necessarily, this increases the modelling complexity significantly, which is beyond the scope of this work. In addition, a forced vibration model may need to include a (coarse) finite element mesh for the large mass if responses on the mass or physical connections to the vehicle were required. Indeed, in the free-vibration analysis considered here, coarse meshes were studied in this work; however, we decided not to include these, since these results would detract from the main focus of the work: simplified analytical solutions or finite element meshing of the structure connected to a large mass. [PERSON] [15] scrutinized the modelling process for structural dynamics and made the following descriptive factors responsible for hindering accuracy: simplifications, approximations, assumptions, and the sample selections made. He further divided up uncertainties into two distinct types, namely aleatoric and epistemic uncertainties, see also [16; 17; 18; 19]. Aleatory uncertainties are variations of physical properties and are unavoidable as is the case for the determination of material and geometry parameters.