# Analysis of Dynamic Response of a Two Degrees of Freedom (2-DOF) Ball Bearing Nonlinear Model
## 3 Simulation Results
### Time Series of Deformation and Fast Fourier Transform (FFT)  
The bearing-rotor SYSTEM'S dynamic response is examined by the different values of eccentricity as specified in Table 1. The level of the eccentricity is a very common problem in rotational mechanical systems and its too big value, strongly affects the correct operation of the system. In the considered dimensionless model, the value of eccentricity is dependent on the constant clearance value. In Figure 4, the time series of deformation in the \(x\)-axis and obtained FFT spectra are presented. In the analyzed model, the ball bearing is radially loaded \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Angular velocity of the shaft & \(\omega_{s}\)\({}^{f}\) & 1.000 \\ Radial internal clearance & \(r_{c}\)\({}^{f}\) & 1.000 \\ Number of rolling elements & \(N\) & 13 \\ Rotor’s eccentricity & \(ec\)\({}^{f}\) & 0.05; 0.1; 0.2; 0.3; 0.35; 0.4 \\ The amplitude of inner raceway surface waviness & \(U_{inner}\)\({}^{f}\) & 0.020 \\ Number of waviness undulations per inner raceway circumference & \(N_{inner}\) & 28 \\ The amplitude of outer raceway surface waviness & \(U_{outer}\) & 0.016 \\ Number of waviness undulations per outer raceway circumference & \(N_{outer}\) & 28 \\ Ball stiffness & \(k_{b}\)\({}^{f}\) & 1.000 \\ Damping coefficient in \(x\)-axis & \(c_{x}\)\({}^{f}\) & 0.010 \\ Damping coefficient in \(y\)-axis & \(c_{y}\)\({}^{f}\) & 0.010 \\ External force subjected in the \(x\)-axis & \(F_{x}\)\({}^{f}\) & 1.200 \\ External force subjected in the \(y\)-axis & \(F_{y}\)\({}^{f}\) & 0.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the rolling element bearing taken for simulation in the dimensionless model, all the lengths are expressed in the clearance value. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Angular velocity of the shaft & \(\omega_{s}\) & 37.65 [Hz] \\ Ball diameter & \(D\) & 15.875 [mm] \\ Pitch diameter & \(d_{p}\) & 71.810 [mm] \\ Radial internal clearance & \(r_{c}\) & 20.0 [mm] \\ Rotor’s eccentricity & \(ec\) & 1; 2; 4; 6; 7; 8 [mm] \\ The amplitude of inner raceway surface waviness & \(U_{inner}\) & 0.4 [mm] \\ The amplitude of outer raceway surface waviness & \(U_{outer}\) & 0.32 [mm] \\ Mass of the bearing-rotor system & \(m\) & 6 [kg] \\ Ball stiffness & \(k_{b}\) & 189000 [N/m] \\ Damping coefficient in \(x\)-axis & \(c_{x}\) & 265.6 [Ns/m] \\ Damping coefficient in \(y\)-axis & \(c_{y}\) & 265.6 [Ns/m] \\ External force subjected in the \(x\)-axis & \(F_{x}\) & 7.05 [kg] \\ External force subjected in the \(y\)-axis & \(F_{y}\) & 0.0 [kg] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Physical parameters corresponding the dimensionless terms. only in one direction (\(x\)-axis) and there is no subjected load in the \(y\)-axis, so it is expected to obtain a periodic solution by all considered cases in unloaded direction. As the eccentricity level increases, the response with higher amplitudes is obtained and with stronger nonlinear effects (Figure 4). The effect of stronger nonlinearity in response is observed at FFT spectra, the magnitude of the main harmonic in 1 and its super-harmonics are increasing with the value of eccentricity value up to \(occ=0.35\). The main harmonic is marked with the red dashed line in Figure 4 and it corresponds to the frequency of the shaft. Moreover, the increasing value of the eccentricity shows the impact of the waviness on the response by appearing of numerous and small amplitude frequency peaks at FFT spectra. Figure 4: _Cont._  
The increase of eccentricity is affecting the system response with main and higher superharmonics. In the limit of small \(ecc\), we observe the domination of the second harmonic corresponding to parametric excitation with the shaft rotational velocity, \(\omega_{s}\). For medium \(ecc\), the peak corresponding to \(\omega_{s}\) is developing and dominates (see the case with \(ecc=0.3\)).