# Simplification of Complex Structural Dynamic Models:
## 2 Model Descriptions
### Analytical Theory  
The Euler-Bernoulli beam theory is the most commonly used theory for simple vibration problems because of its simplicity due to reasonable approximations for many real physical problems. The theory is accurate for slender beams and for natural frequencies of the lower modes, see [27]. For this reason, the developed analytical formalism is based on the Euler-Bernoulli beam theory. We assume that the reader is familiar with the fundamental formulations of the Euler-Bernoulli model. Therefore, the authors refrain from including a detailed description here but refer the reader to the ample literature available, such as [31; 32; 33; 34]. However, the basic assumptions of the Euler-Bernoulli theory should be mentioned as follows:  
1. One of the spatial dimensions is significantly larger than the other two. 2. The material behaves according to [PERSON]'s law. 3. The Poisson effect is neglected. 4. The angle of rotation is small; hence, the small angle assumption holds. The transverse displacements and cross-section rotations are small. Hence, the formulations are geometrically linear. 5. Cross-sections remain perpendicular to the neutral axes after deformation. 6. The rotational inertia of the cross-sections is neglected. Figure 3 shows the parameters of the one-dimensional model to develop the analytical formula to calculate eigenfrequencies for the bending modes. The equation of motion for free vibrations according to the Euler-Bernoulli theory is given as:  
\[\frac{\partial^{4}W(x,t)}{\partial x^{4}}+\frac{\rho A}{E_{\rm b}}\frac{ \partial^{2}W(x,t)}{\partial t^{2}}=0, \tag{8}\]  
where \(W(x,t)\) is the transverse displacement in \(z\) direction at the point \(x\) and the time \(t\). Considering the freely supported structure presented in Figure 3, _interface_ boundary conditions (at \(x=l\)) can be formulated for the simplified model in terms of relating the bending moment and shear forces at the beam end and equivalent terms for the tip-mass, see [PERSON] and Shillor [35],  
\[\left[m_{t}\frac{\partial^{2}W(x,t)}{\partial t^{2}}-E_{\rm b}\frac{\partial^ {3}W(x,t)}{\partial x^{3}}\right]_{x=l}=0,\left[\theta_{t}  
Using the standard separation of variables applied to \(W(x,t)\) into two functions:  
\[W(x,t)=\phi(x)T(t). \tag{11}\]  
and the following fourth-order differential-equation, which describes free flexural vibrations of a thin beam follows from Equation (8):  
\[EI_{\text{b}}\varDelta A\frac{d^{4}\phi(x)}{dx^{4}}+\left(\frac{\lambda}{T} \right)^{4}\phi(x)=0, \tag{12}\]  
where \(\lambda\) is a positive constant. It is well established that the spatial solutions of the homogeneous Equation (8) can be considered as flexural mode shapes:  
\[\phi(x)=A\cos\lambda_{n}\frac{x}{l}+B\sin\lambda_{n}\frac{x}{l}+C\cosh\lambda_ {n}\frac{x}{l}+D\sinh\lambda_{n}\frac{x}{l},\ \ n\geq 1 \tag{13}\]  
where \(\lambda_{n}\) and A, B, C, and D are the eigenvalues and mode shape parameters. In Equation (13)  
\[\lambda_{n}^{4}=\frac{\varrho A}{EI_{\text{b}}}\omega_{n}^{2}t^{4}. \tag{14}\]  
is the well-known relation between eigenvalues and the natural frequency of vibrations, \(\omega\). Employing the notation \(c(\lambda)=cos(\lambda)\), \(s(\lambda)=sin(\lambda)\), \(ch(\lambda)=cosh(\lambda)\), \(sh(\lambda)=sinh(\lambda)\), and after some standard algebraic manipulations, the characteristic equation follows:  
\[\begin{split}& C_{T}(\lambda)=\frac{\lambda^{4}m_{\text{t}}\theta_{ \text{t}}}{l^{2}m_{\text{b}  
Clearly, from Figure 4, modelling the specimen in full detail with three dimensional solid elements and applying free-free boundary conditions is a valid modelling approach for engineering purposes. The numerical results of this reference model FEfreefreefree share a high level of agreement with the experimental results. The deviation was lower than 1% over the whole frequency range as indicated in Table 2.  
In contrast, the modelling approach **2**, i.e., replacing the tip-mass at the end of the specimen with ideal clamped boundary conditions, yielded insufficient results.