# Experimental Verification of a Compressor Drive Simulation Model to Minimize Dangerous Vibrations
## 3 Results and Discussion
### Determination of the Operating State of the System in Terms of Resonance
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Mass Moments of Inertia**} & **Element Torsional Stiffness** \\ \hline \(I_{1}=I_{AM}\) & \(I_{7}=\frac{I_{7}}{I_{3}}\) & \(k_{1}=k_{H1}\) & \(k_{7}=\frac{k_{H10}}{I_{43}}\) \\ \(I_{2}=I_{P1}\) & \(I_{8}=\frac{I_{8}}{I_{8}}\) & \(k_{2}=k_{H2}\) & \(k_{8}=\frac{k_{H11}}{I_{3}}\) \\ \(I_{3}=2\The analysis of torsional vibration of a mechanical system is based on a system of differential equations of motion. The equations describing a two-mass reduced mechanical system without damping (Figure 6) have the following form:  
\[I_{1}^{*}\cdot\bar{\varphi}_{1}+k^{*}\cdot(\varphi_{1}-\varphi_{2})=M_{AM}, \tag{2}\]  
\[-I_{2}^{*}\cdot\bar{\varphi}_{2}+k^{*}\cdot(\varphi_{1}-\varphi_{2})=M_{PK}. \tag{3}\]  
The variable load torque of a piston compressor is given by:  
\[M_{PK}=M_{NPK}+\sum_{i=1}^{\infty}M_{iPK}\cdot cos(i\cdot\omega_{PK}\cdot t+ \gamma_{iPK}). \tag{4}\]  
The mode shape of the oscillation at coupling pressure is shown in Figure 7.  
A Campbell diagram is an interference diagram that identifies possible matches between the expected excitation frequencies and the calculated natural torsional frequencies over the operating speed range. Each ascending line on the interference diagram represents a single mechanical or electrical excitation frequency or harmonic; when they intersect with the torsional natural frequency (horizontal line), torsional resonance occurs. The intersection of those lines indicates a possible torsional resonance condition. For some systems, an interference diagram may be all that is needed to show that torsional resonances will not be excited [45; 46]. Figure 8 shows the Campbell diagram for a system operating at coupling pressure 700 kPa with the following parameters marked in it: the operating rotational frequency \(n_{P}=600\) min\({}^{-1}\), the critical revolutions from the principal harmonic element \(n_{K3}=641.2\) min\({}^{-1}\) and the natural rotational frequency \(N_{0}=1923.526\) min\({}^{-1}\). The detuning coefficient for the \(i\)-th harmonic component is defined by the formula:  
\[\eta_{i}=\frac{n\cdot i}{N_{0}}. \tag{5}\]  
Table 4 gives the detuning coefficients of the harmonic components operating rotational frequency \(n_{P}=600\) min\({}^{-1}\). Since this is a three-cylinder piston compressor, the main harmonic components will be integer multiples of the third harmonic component. Figure 6: Two-mass reduced mechanical system. Figure 7: Mode shape of the strain curve for \(\Omega_{0}=201.431\) rad\(\cdot\)s\({}^{-1}\). These will always appear during the load torque; the other harmonic components are the so-called secondary harmonic components, which will only appear if the compressor cylinders do not operate evenly [45; 46; 47; 48]. According to the literature [27], the system operates in the resonance region if:  
\[0.8<\eta_{i}<1.2. \tag{6}\]  
From the results shown in Table 4, we can say that the system at coupling pressure 700 kPa operates in the resonance region.