# Vibration Diagnosis of Sand Units in a Stone Crusher Plant: An On-Site Field Test
## 2 Artificial Sand Production Plant
### Vibration Isolation  
Vibration isolation is a procedure by which undesirable vibrations are eliminated or reduced. Basically, it involves the insertion of a resilient member (or isolator) between the vibrating mass (or equipment or payload) and the source of vibration so that the reduction in the dynamic response of the system can be achieved under specified vibration excitation conditions. Therefore, this can be achieved using not only passive but also controllable vibration isolators. It is well known that the isolation system can be classified into passive, active and semi-active. The passive isolator consists of a resilient material (stiffness), an energy dissipator (damper) and mass as a single degree of freedom of the mechanical system. The example of passive isolators in a mechanical system includes metal coil spring, cork, felt, pneumatic springs and viscoelastic material, such as elastomer or rubber springs. Among the passive isolators, the rubber springs have been popularly used in terms of the shear mode, or combinations of the shear and compression modes along with sophisticated viscoelastic elements. The active vibration isolator features actuators with a closed-loop feedback control system. Thus, the isolation performance of the active isolation is high, but it requires high cost and sophisticated  
Figure 1: Artificial sand manufacturing plant and process: (**a**) Photograph of the artificial sand manufacturing plant; (**b**) Overall process of manufacturing in artificial sand production. sensors and control algorithms. The semi-active vibration isolation is featured by addition of damping property in real time manner. This method is known to be simple, but very effective. Among three vibration isolation approaches, the passive method is mostly used in the production of the artificial sand or in the building of the civil engineering structures. The vibration isolation for the artificial sand production plant is mainly achieved by utilizing rubber mounts and/or coil springs. More specifically, many rubber mounts and coil springs are installed under the structures of screen and sand units to reduce the vibration caused from driving (or exciting) actuators such as large-sized motors. Therefore, this isolation system is weak to external disturbances and time-varying uncertainty of the driving frequencies. This isolation system is normally designed to protect against the lowest frequency of the system since it causes the highest vibration amplitude. To achieve the vibration transmissibility of the artificial sand production plant, a single degree of freedom (DOF) model shown in Figure 2 can be considered. The resilient material is assumed to have both elasticity and damping and modeled as a spring \(k\) and a dashpot \(c\), respectively. It is assumed that the operation of the sand manufacturing plant is undertaken by applying a harmonically varying force \(F(t)=F_{0}\cos\omega\,t\). Then, the governing equation of the machine sand unit (or screen unit) body is given by  
\[m\ddot{x}+c\dot{x}+kx=F_{0}\cos\omega\,t \tag{1}\]  
Using the steady state solution of the above equation, the magnitude of the total transmitted force is determined by [32; 33; 34; 35; 36; 37]  
\[F_{T}=\frac{F_{0}\sqrt{k^{2}+\omega^{2}\,c^{2}}}{\sqrt{\left(k-m\,\omega^{2} \right)^{2}+\omega^{2}c^{2}}} \tag{2}\]  
Thus, the transmissibility of the vibration isolation model is defined as the ratio of the magnitude of the force transmitted to the exciting force [32; 33; 34; 35; 36; 37]:  
\[T=\left|\frac{F_{T}}{F_{0}}\right|=\ \frac{\sqrt{k+\omega^{2}c^{2}}}{\sqrt{ \left(k-m\omega^{2}\right)^{2}+\omega^{2}c^{2}}}=\frac{\sqrt{1+\left(2\zeta r \right)^{2}}}{\sqrt{\left(1-r^{2}\right)^{2}+\left(2\zeta r\right)^{2}}} \tag{3}\]  
In the above, \(F_{T}\) is the amplitude of the force transmitted to the sand unit, \(F_{0}\) is amplitude of the excitation force from the sand unit body and \(r\) is the frequency ratio. It is defined by \(f\!/\!f_{n}\).