# Diagnosing the Machine Set Rotor Using Its Relative Vibrations
## 2 Machine Set Rotor, Vibration and Mechanical Efficiency of the Machine Set Rotor
In terms of energy, the measure of the wear margin of the machine set is its mechanical energy efficiency or the moment of friction. In the case of rotating machines, the mechanical energy efficiency is equal to the mechanical energy efficiency of the machine set rotor. The mechanical efficiency of the machine set rotor \(\eta_{m.MSR}\) can be written as  
\[\eta_{m.MSR}=\Delta E_{2}/\Delta E_{1}=(E_{\text{2 in}}-E_{\text{2 out}})/(E_{ \text{1 out}}-E_{\text{1 in}})=(M_{U}\cdot\alpha)/(M_{N}\cdot\alpha) \tag{1}\]  
where: \(MN\)--driving moment, \(MU\)--effective moment, \(\alpha\)--angle of shaft rotation [rad]. Moments of friction \(M_{R}\) created in the bearings depend on radial and axial forces acting on the shaft and friction coefficients. The driving moment is equal to the sum of the effective moment and the moment of friction. \[M_{N}=M_{U}+M_{R} \tag{2}\]  
\[\eta_{m.MSR}=M_{U}/M_{N}=(M_{N}-M_{R})/M_{N}=1-M_{R}/M_{N} \tag{3}\]  
If the value of one of the moments is dependent on the rotation angle \(\alpha\) of the rotor, then the rotors in the torsional vibration node will experience an instantaneous change of rotor speed \(\omega=da/dt=\dot{\alpha}\) and, consequently, the kinetic energy of the rotor will change. The power balance and change of kinetic energy can then be written [12; 13] as  
\[M_{N}\cdot\dot{\alpha}-Mu\cdot\dot{\alpha}-M_{R}\cdot\dot{\alpha}=dE_{\text{ kin}}/dt \tag{4}\]  
\[dE_{\text{kin}}/dt=d\left(0.5\dot{\theta}\cdot\dot{\alpha}^{2}\right)/dt=0.5 \left(\dot{\alpha}^{2}\cdot d\theta/dt+2\theta\cdot d\dot{\alpha}/dt\right)=\]  
\[=0.5\cdot(d\theta/dt)\cdot\dot{\alpha}^{2}+\theta\cdot\dot{\alpha}\cdot\ddot{ \alpha}=0.5\cdot(d\theta/d\alpha)\cdot(da/dt)\cdot\dot{\alpha}^{2}+\theta \cdot\dot{\alpha}\cdot\ddot{\alpha} \tag{5}\]  
After substitution and dividing by \(\dot{\alpha}=da/dt\), we obtain the balance of moments, i.e., the equation of the moments of the machine set rotor  
\[M_{N}-M_{u}-M_{R}=0.5\cdot(d\theta/da)\cdot\dot{\alpha}^{2}+\theta\cdot\ddot{\alpha} \tag{6}\]  
For \(\dot{\alpha}_{\text{average}}=\text{const}\), \(\ddot{\alpha}_{\text{average}}=0\), the moments of forces and the moment of inertia can have instantaneous values dependent on the rotor rotation angle. For a given machine set rotor, variations of \(d\theta/da\) are known. \(d\theta/da\
eq 0\) mainly refers to piston machines and is determined at the design stage, and the period of changes is one revolution of the crankshaft. For the rotor of the rotating machine set \(d\theta/da=0\), the value of the product \(\theta\cdot\ddot{\alpha}\) with respect to the rotation angle is the sum values of the other components of the equation of moments. The values of the driving and effective moments as a function of the rotation angle are specific to the given type of machine. Equation (6) can be seen as one of the relationships between the wear margin measure given as the friction moment and the symptom measure of the machine set rotor given as the angular acceleration of torsional vibrations. If direct and precise measurements of the angular acceleration in the torsional vibration node of a machine set rotor were feasible, two specific forms of diagnosis would be available:  
1. In these working states, the driven machine can be in the working mode \(M_{U}>0\) or run idle, without a load, \(M_{U}=0\). For a constant average angular speed, the value of the components of Equation (6) with respect to time can be considered as periodic signals, and each of them can be presented as the sum of harmonic signals with specific values of the harmonic order, amplitude and phase angle. It is possible to determine the reference spectrum of the driving moment and effective moment. Additionally, the reference spectrum of the friction moment can be determined. It can be assumed that for a given machine set rotor and for a specific average rotary speed, there exists a model of value as a function of the rotation angle and a spectrum model of the angular acceleration \(\check{\alpha}\).