# Vibration Energy at Damage-Based Statistical Approach to Detect Multiple Damages in Roller Bearings
## 2 Vibration Energy at Damage-Based Statistical Approach  
One body has six degrees of freedom (DOF): three are translational, and three are rotational. The equation of motion for a shaft under torsional/rotational vibration is shown by Equation (1). The signal acquired from the machine in terms of displacement, velocity, or acceleration can be compared with a defected signature to perform fault diagnosis. \[I\ddot{\theta}+K_{d}\dot{\theta}+K_{t}\theta=T(t) \tag{1}\]  
where \(I\) is the mass moment of inertia, \(\ddot{\theta}\) is the angular acceleration, \(K_{d}\dot{\theta}\) is the damping coefficient in the torsional domain, \(K_{t}\theta\) is the torsional stiffness, and \(T(t)\) is the torque as a function of time. These torsional vibrations can be measured by a laser torsional vibrometer and optical encoders. In this study, accelerometers were used to measure rotational vibration, rather than measuring the torsional vibration. The data can be acquired in terms of displacement (\(x\)), velocity (\(\dot{x}\)) or acceleration (\(\ddot{x}\)). The followings are the basic parameters to measure vibration:  
\[|x|=X\quad,\ \left|\dot{x}\right|=\omega X\quad,\ \left|\ddot{x}\right|= \omega^{2}X \tag{2}\]  
where \(\omega\) is the rotational frequency in rad/s, which can also be expressed as \(2\pi f\), and \(f\) is the frequency in cycles per second, or hertz (Hz). Generally, the \(X\) is measured at \(0<f<10\), the \(\omega X\) is measured at \(10<f<1000\), and the \(\omega^{2}X\) is measured at \(f>1000\). On the right hand side of the equation of motion, there is a forcing function, and it depends upon \(\omega\), which is the forcing frequency. The natural characteristics of the machine can be obtained from Equation (3):  
\[\omega_{n}=\sqrt{\frac{k}{m}}\quad\quad f_{n}=\ \frac{1}{2\pi}\sqrt{\frac{k}{m}} \tag{3}\]  
where \(\omega_{n}\), \(k\), \(m\), and \(f_{n}\) are the natural rotational speed, stiffness, mass, and natural frequency of the machine, respectively. The resonance will occur if the forcing frequency is equal to the natural frequency, and it will result in larger displacement. Operators avoid the natural frequency during the operation, but this parameter is calculated by designers for safety. The following Equations (4) and (5) explain the frequency response functions, where \(F\) is the forcing function:  
\[Stiffness=\ \frac{F}{x}\quad\quad Damping=\ \frac{F}{\ddot{x}}\quad\quad Mass= \ \frac{F}{\ddot{x}} \tag{4}\]\[Compliance=\frac{x}{F}\quad Mobility=\frac{\dot{x}}{F}\quad Impedance=\frac{\ddot{x}}{F} \tag{5}\]  
In this study, a statistical approach is proposed to detect roller bearing damages. As the roller bearings are mainly used in industry, it is important to detect damages at the early stages. The roller bearings have to bear great loading at high rotating speeds. Therefore, it is necessary to use a proactive technique for inspection. This technique not only provide a safe environment the proper function of the machines, but it also decreases the maintenance costs. Basically, the roller bearings bear loading with their rolling elements and reduce friction. Roller bearings have three main components: the balls, the inner race, and the outer race. The disorder or sudden peak in the vibration-based frequency or time domain data are often considered faults. Because, as the ball passes through a discontinuity or defect, it can be subjected to an impulsive force as an effect of damages, peaks in courses of physical magnitudes can be noticed. Thus, most of the researchers have analyzed characteristic features of signals at frequency. In this research, the characteristic fault frequency multipliers for ball pass frequency outer race \((BPFO)\), ball pass frequency inner race \((BPFI)\), ball spin frequency \((BSF)\), and fundamental train frequency \((FTF)\) were calculated using Equations (6)-(9). However, our main objectives are to plot histograms and calculate kurtosis values so that the damage energy for an intact bearing and damaged bearing can be compared.