# Estimation of Resonance Frequency for Systems with Contact Using Linear Dynamics Methods
## 3 Vibrations Research with Numerical Simulations  
For a detailed look at how systems behave during vibration with contact, the numerical simulation method was selected as the main tool of the preliminary investigation. To accurately include the contact phenomenon, it was mandatory to use full dynamic simulation. Because accuracy was a priority in the simulations, the implicit algorithm was selected over the explicit algorithm. For the preliminary tests, a flat rectangle specimen with a straight edge of contact was selected.  
The model built for analysis was made out of continuous shell elements fixed to the rigid clamp with an introduced area of contact. Additionally, a mass point was included to represent the accelerometer mass and allow for future repeating of the simulation in experimental tests. The model is presented in Figure 2.  
The simulation was following the tip of the specimen, where the accelerometer mass point was attached, and had an enforced displacement of 20 mm in the first step. The second step of the simulation was to release the enforced displacement. The specimen then started to vibrate at natural frequencies. The displacement of the accelerometer point in time was recorded. For the described simulation, Abaqus 2018 was used, with a nonlinear implicit dynamics algorithm, taking into account the contact and geometrical nonlinearity. The CFRP specimen was modelled with Continuum Shell-layered elements, and the accelerometer was modelled as single node mass, connected to the specimen with rigid elements. In addition, the fixation was modelled with discrete rigid elements. For contact, the definition penalization algorithm was used with a friction coefficient of value f = 0.2. To analyze the resonance frequency of the system, a Python script was created to apply a fast Fourier transform on the displacement- time characteristic.  
For research purposes, a flat rectangle specimen geometry was selected as an object of investigation. The dimensions of the specimen were 220 \(\times\) 20 mm with 2 mm thickness. The specimen material was 10 plies of carbon fiber twill weave composite with 0/90 layup. The specimen was fixed at 40 mm length.  
Additionally, for the experimental tests a hole in the free end of the specimen was introduced to attach additional mass as a parameter. The two masses that were included to change the resonance frequency of the system were 7.6 g and 9.7 g.  
Figure 2: View of the discrete model for numerical testing.  
The mechanical properties of the material were evaluated in the static tensile strength test for three manufactured pieces. The values of the properties are collected in Table 1.  
The exact placement of the holes in the samples is described in Table 2 as parameters hx--longitude distance from end of the sample, hy--side distance from end of the specimen.  
The described setup is presented in Figure 3.  
Dynamic FE simulation was performed with 10 different lengths of contact X in the range 0-100 mm, each 10 mm. The characteristics of the natural resonance frequency by contact length are shown in Figure 4. Additionally, the resonance frequencies calculated with the classical modal analysis with fixed contact edge and opened contact were included to visualize the difference.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Specimen** & **1** & **b** & **t** & **\(\alpha\)s** & **hx** & **hy** \\
**Name** & **(mm)** & **(mm)** & **(mm)** & **(\({}^{\circ}\))** & **(mm)** & **(mm)** \\ \hline TW090\_1 & 219.86 & 19.98 & 2.02 & 0.39 & 11.5 & 9.6 \\ TW090\_2 & 219.70 & 20.1 & 2.04 & 0.19 & 7.7 & 11.0 \\ TW090\_3 & 219.60 & 20.1 & 2.05 & 0.13 & 7.0 & 9.5 \\ \hline \end{tabular}
\end{table}
Table 2: Measured dimensions of manufactured test specimens.  
Figure 3: Scheme of the specimen mounted in the grip.