# Diagnosing the Machine Set Rotor Using Its Relative Vibrations
## 2 Machine Set Rotor, Vibration and Mechanical Efficiency of the Machine Set Rotor  
In this paper, it was assumed that:  
* machine diagnosing is part of machine maintenance; and
* the relative vibrations of the machine rotor contain diagnostically useful information. Maintenance-oriented diagnosis can be improved by dividing the machine set into functional units and replacing the term of condition with the term of wear margin. The description of diagnostic relations in real machines is difficult; a sufficiently detailed description is possible on models of machines. In diagnosis, the relationship between the wear margin measure and the symptom measure must be reversible. The algorithm of diagnosing the machine set rotor using its relative vibrations is shown in Figure 1.  
A machine set, also referred to as a machine train, is a set consisting of at least one driving machine and one driven machine, while the power transmission to the driven machine is performed by a rotating shaft (Figure 2). Each machine of the machine set has a supported bearing-fixed rotor; each rotor can by modeled as consisting of a disk and a shaft. In the case of sufficiently rigid shafts, the connected rotors form the rotor of a machine set with a specific mass moment of inertia \(\theta\) (kg\(\cdot\)m\({}^{2}\)), rotary (angular) speed \(\omega\) (rad/s) and kinetic energy \(E_{\text{kin}}\) = \(\theta\cdot\omega^{2}/2\). The driving machine rotor disk is affected by a driving moment, while the rotor disk of the driven machine is affected by the driven, i.e., effective, moment. In machine set rotor supports the existing motion resistance, generates a moment of friction. The functional unit specified in this article, called a \"machine set rotor\", has a shaft made in one piece (Figure 2a) or a shaft consisting of two shafts connected by a coupling (Figure 2b). Where two machines are connected through a transmission (Figure 2c), two rotors of a machine set are distinguished: one is a rotor of the driving machine and a rotor of the first transmission shaft; the other is the last rotor of the transmission and the rotor of the driven machine. Other elements can be placed between the coupling discs of the machines: intermediate shaft, flexible coupling, a clutch or others. Both single machines of a machine set may have more than one rotor. The machine set rotor can be modeled as at least two masses connected by a massless shaft with specific rigidity and damping. The two-mass rotor has one node in the case of torsional and longitudinal vibration and three nodes in the case of the second mode radial vibrations (Figure 2). In the nodes, the values of vibration of a given type are equal to zero and increase along with the distance from the node. All vibrations of the machine set rotor are damped. Damping takes place in the shaft material and in the rotor bearings (viscosity  
Figure 1: The algorithm of diagnosing the machine set rotor using its relative vibrations. Figure 2: Rotors of the driving machine-driven machine set (on the left) and vibrations of the two-mass machine set rotor (on the right): (**a**) one rotor with one shaft, (**b**) one rotor with shafts connected by a coupling, (**c**) two rotors of a machine set with a transmission, \(E_{\text{in}}\)—energy powering the driving machine, \(E_{\text{1 out}}\)—energy carried away from the driving machine, \(E_{\text{2 in}}\)—energy delivered to the driven machine, \(E_{\text{2 out}}\)—energy at driven machine output, \(R_{1}\)—rotor disc of the driving machine, \(R_{2}\)—rotor disc of the driven machine, \(R_{1+}\), \(R_{2}\)—rotor discs of the transmission, \(\varphi\)—angular displacement of torsional vibration, \(\Delta\)—displacement of axial vibrations, \(y\)—displacement of radial vibration. damping). If the forces or moments acting on the rotor are periodic forces, then resonance vibrations may occur. The periodic forces can be decomposed into harmonic components. One of the characteristics of the driving-driven machine set common in use is the energy efficiency of the set.