# Diagnosing the Machine Set Rotor Using Its Relative Vibrations
## 4 Measurements of Relative Vibration  
Measurements of rotating element vibration make use of absolute vibration sensors with a slider or relative vibration sensors (eddy current sensor, inductive or capacitive sensor) [8]. In the case of vibration of rotating elements, diagnosis makes use of relative vibration, mainly radial vibration, in specific cases torsional vibration, and to a limited extent, the axial vibration of the rotating shaft. Relative radial vibration of rotating shafts is  
Figure 7: Model of the unilaterally fixed crankshaft. Measures of the crankshaft alignment symptom: a—amplitude, b—phase shift in relation to TDC of first crank, c—mean value. measured in each bearing plane by two sensors, placed at the right angle to each other [8]. Relative axial vibration is used mainly for monitoring the axial shaft position, which consists of monitoring the rotor position relative to a selected constant point, e.g., thrust bearing housing. The measurements are recorded by an axially fixed displacement sensor. For turbomachinery, measurements of absolute axial elongation of the machine body (measured by a sensor on a radial bearing that does not fix the shaft longitudinally) and relative elongation of the body (displacement sensor mounted on the body, monitoring the position of the face or flange of the shaft) are taken [49]. The displacement sensors are usually fixed in holes in non-rotating elements. It follows from points 2 and 3 of this article that both constant and variable components of the relative vibrations should be measured. Such relative vibration can be measure by common relative vibration sensors placed in the appropriate holder. The holder with the sensor is called here the sensor head. The vibration of points located on the surface of the selected rotor cross-section is measured relative to or in the plane perpendicular to the rotor axis, called the reference plane. Because the rotor performs all kinds of vibrations simultaneously, the points on the surface of the observed cross-section, apart from the measured motion, also perform other motions, and the reference plane of the measured vibration becomes non-perpendicular to the rotor axis. The problem can be solved by:  
* measurements of other types of vibration and correction of the results or;
* measurement of a given type of vibration by more than one sensor and mathematical operations on the measured signals or;
* fixing the reference plane with the rotor axis; the reference plane performs the same motions as the rotor, except for the measured motion. When vibrations of the same point are measured by a pair of sensors, there will be a phase shift \(\varphi\) between the measured signals depending on the location of the two sensors relative to each other. The measurement results can be recorded in the coordinate system \(x(t)\), \(y(t)\) or \(y(x)\). One of the methods of vibration tests in the \(y(x)\) system measured by two sensors shifted by the angle \(\varphi\) are tests using a dual channel oscilloscope working in the XY mode. In the case of vibration with a zero value of the variable (\(X_{\text{variable}}=0\), \(Y_{\text{variable}}=0\)), the result is a point, which for \(X=Y\
eq 0\) lies on the diagonal of the XY coordinate system. The interpretation of the obtained image in the XY coordinate system is facilitated by the Lissajous or Bowditch curve. If the vibrations measured by specific sensors were harmonic vibrations with zero average value (\(X_{\text{average}}=0\), \(Y_{\text{average}}=0\)), then their time-dependent variations would be described by the formulae:  
\[x(t)=X\sin(\omega_{x}\cdot t+\varphi),y(t)=Y\sin(\omega_{y}\cdot t) \tag{15}\]  
and in the XY coordinate system, they would be figures with a shape dependent on \(\varphi\), \(\omega_{x}/\omega_{y}\), \(X/Y\). Table 2 presents the Lissajous curves for \(\omega_{x}=\omega_{y}\), \(X=Y\) and the selected angles of phase shift \(\varphi\). The essential element of the time and harmonic analysis of vibration is information on the angular position of the rotor shaft.