# Vibration Diagnosis of Sand Units in a Stone Crusher Plant: An On-Site Field Test
## 4 Vibration Reduction  
As observed from the measured vibration signals shown in Figure 8a, the beating occurred due to close driving (or exciting) frequencies of actuating motors during multiple operation of the artificial sand manufacturing plant. The beating phenomenon at this plant occurred within a short period of time with the pattern of the periodic propagation. It is well known that the beating phenomenon is frequently observed when two harmonic motions with frequencies close to each other are combined. To briefly review the beating, consider a displacement \(x(t)\) that is produced by two harmonic forces \(f_{1}\) and \(f_{2}\)[22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 36; 37; 38; 39; 40]:  
\[f_{1}\left(t\right)\,=\,F_{1}\,\cos\,\omega_{1}\,t\,\text{and}\,f_{2}\left(t \right)\,=\,F_{2}\,\cos\,\omega_{2}\,t \tag{7}\]  
Assume that the steady state responses to \(f_{1}\) and \(f_{2}\) are given by  
\[x_{1}\left(t\right)\,=\,X_{1}\,\cos\,(\omega_{1}\,t+\phi_{1})\,\text{and}\,x_{ 2}\left(t\right)\,=\,X_{2}\,\cos(\,\omega_{2}t+\phi_{2}) \tag{8}\]  
Then, because of the linear properties of the system, the following equation is obtained:  
\[ \begin{array}{c}x\left(t\right)\,=\,\left(X_{1}+X_{2}\right)\cos\left(\Omega _{1}\,t-\Phi_{1}\cos(\Omega_{2}\,t-\Phi_{2})\right.\\ \left.-\left(X_{1}-X_{2}\right)\sin\left(\Omega_{1}\,t-\Phi_{1}\right)\sin( \Omega_{2}\,t-\Phi_{2})\right)\end{array}  \tag{9}\]  
where,  
\[\Omega_{1}\,=\,\frac{\omega_{1}+\omega_{2}}{2},\quad\Omega_{2}\,=\,\frac{ \omega_{1}-\omega_{2}}{2},\Phi_{1}\,=\,\frac{\phi_{1}+\phi_{2}}{2},\quad\Phi_{ 2}\,=\,\frac{\phi_{1}-\phi_{2}}{2} \tag{10}\]  
Thus, the beat frequency and beat mean value frequency (\(f_{bm}\)) are given by  
\[f_{b}=\left|f_{1}-f_{2}\right|\left[\text{Hz}\right],\text{ for }f_{1}>f_{2} \tag{11}\]  
\[f_{bm}=\frac{f_{1}+f_{2}}{2} \tag{12}\]  
Therefore, the beating magnitude and cycle can be determined by the difference of two frequencies and phase angles. On the other hand, the vibration level due to the beating can be effectively reduced by adjusting the two frequencies. In this work, to avoid the beating behavior, the operating speed (or driving frequency) of the sand unit is adjusted. The original operating speeds of three motors for three sand units are fixed at 1320 rpm, but this changed a little as time went on. Therefore, three operating speeds for three sand units become close, resulting in the beating behavior. In this work, after measuring the operating speed of each sand unit, it is adjusted to 1050 rpm, 1320 rpm and 1180 rpm for the sand unit #1, sand unit #2 and sand unit #3, as shown in Table 1. Figure 9 presents vibration signals measured at the bottom point of the sand unit during full operation of entire experimental facilities for artificial sand production. It is clearly observed that the vibration level is significantly reduced after adjusting the operating speed (or driving frequency for motors to activate the sand units). More specifically, the power spectrum is remarkably reduced; \(\Delta VAL\) is 28 dB, and the transmissibility is 0.04. Thus, the percent isolation is calculated by 96%, which is very high. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Before Adjusting Frequency**} & \multicolumn{3}{c}{**After Adjusting Frequency**} \\ \cline{2-7}
**Devices No.** & **Power Efficiency** & **Operating Speed (rpm)** & **Fundamental Frequency** & **Power Efficiency** & **Operating Speed (rpm)** & **Fundamental Frequency** \\ \hline Sand Unit \#1 & 100\% & 1320 rpm & 22.0 Hz & 80\% & 1050 rpm & 17.5 Hz \\ Sand Unit \#2 & 100\% & 1320 rpm & 22.0 Hz & 100\% & 1320 rpm & 22.0 Hz \\ Sand Unit \#3 & 100\% & 1320 rpm & 22.0 Hz & 90\% & 1180 rpm & 19.6 Hz \\ Screen Unit \#1 & 100\% & 1320 rpm & 22.0 Hz & 73\% & 960 rpm & 16 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of driving (excitation) frequencies of the sand unit and screen unit. However, the vibration signals measured at the sand unit do not clearly show the beating phenomenon.