Modeling and diagnostics of the gear power transmissions

V. Barzdaitis*, P. Mažeika**, J. Grigonienė**, R. Didžiokas**, V. Kartašovas** *Kaunas University of Technology, A. Mickevičiaus str. 37, 44244 Kaunas, Lithuania, E-mail: vytautas.barzdaitis@ktu.lt **Klaipeda University Mechatronis Science Institute, Bijūnų str. 17, 91225 Klaipėda, Lithuania, E-mail: pranasmazeika@centras.lt, jurgita.grigoniene@gmail.com, rimantas.didziokas@ku.lt, valentines.kartasovas@ku.lt


Introduction
Rotating systems are the main mechanical units of power generating and technological machines.The integral part of modern machines with rotating systems introduces IT based condition monitoring, protection, failure diagnostics and expert systems for prediction of unexpected failures and based on vibration and technological parameters measurement.To increase technical condition monitoring and vibration severity evaluation accuracy in practice with the identification of vibration sources the modeling and simulation of rotor systems are inevitable processes [1 -7].
Systematic machine vibration monitoring and failure diagnostics prevent from unexpected failures of machines.The most frequently occurring breakdown of rotor systems -failure of bearings and rotor-stator part rub [1].Authors evaluate diagnostics methods for high speed rotors with rolling bearings [2].The papers [8 -10] concern vibration monitoring and failure diagnostics of low rotation speed, but high torque rotors and gear transmission rotors, for the identification of rolling bearings defects.The gears teeth meshing malfunctions are estimated through measurement of rotors radial and axial vibrations displacement in situ and modeling and simulations of designed results.The authors present new research data on gear transmission dynamics and diagnostics.
The heavy duty gear power transmission have been modeled and tested in situ to protect unexpected failures [11,12].
This paper presents effectiveness of new designed diagnostic method for prediction of the unexpected failures of low rotation speed high torque gear transmission with antifriction bearings.

Objects of research
High mass horizontal axis cylinder machine kinematic scheme is presented in Fig. 1.The heavy loaded low speed gear transmission elements are pinion z 1 and gear z 2 , four low rotation speed cylinders 1K, 2K, 1D, 2D and eight roller bearings: four on the left cylinder side 1K1, 1K2, 2K1, 2K2 and four on the right cylinder side 1D1, 1D2, 2D1, 2D2.The impacts generated by pinion z 1 teeth meshing are the main sources of vibration.The teeth module is 16 mm, z 1 = 25 and z 2 = 176 teeth.The rotational speed of the pinion z 1 is low ~38 rpm, rotation torque is high 8250 Nm and the speed of asynchronous electric motor rotor is 1500 rpm.All rotors rotate in double row spherical roller bearings SKF 22228 CCK/W33 with a tapered bore H3128.
The operation data indicated that the main defect of low speed rotors is the failure of bearing elements: rolled bearings, a tapered bore wrenched on an inner ring ' and z 4 were seriously damaged (Fig. 3).The new diagnostics method was designed to identify the defects of rotors antifriction bearings and gears.

The high mass horizontal cylinder machine vibration testing in situ
During the machine operation defects of the gears' z 1 and z 2 teeth surface were identified.For the evaluation of rotor's with z 1 gear transmission technical condition, failure diagnosis and identification causality of damages, the experimental research and theoretical modeling was carried out.Absolute vibration velocity and aceleration, and rotors radial displacement were measured for http://dx.doi.org/10.5755/j01.mech.17Fig. 3 The damaged z 3 teeth and vibration displacement measurement sensor the 9th and the 10th bearings housings (Fig. 1).The measurement results were analyzed using Adash Compressed Time (ACMT) method [10].The measurement of shaft radial vibration displacement were based on shaft displacement measurements with contactless sensors following methods of discreet modeling designed at the Klaipeda University and Kaunas University of Technology.Condition monitoring of bearings continues systematically.Mechanical vibration measurement data showed that conventional vibration testing methods were ineffective in diagnostics of low rotation high power machines, because the vibration signal was stochastic.The comparison of 9th bearing housing vibration V RMS parameters of the rotor with defective bearing and new one is illustrated in Table 1.The FFT vibration velocity and aceleration spectra are shown in Figs. 4 -7 and the vibrations velocity and acceleration time in Figs. 8 -11.At the comparison of vibration data with a damaged bearing and with new bearing outstanding difference between vibrations parameters was not noticed, Table 1.
For the gears z 1 and z 2 vibration amplitudes of teeth meshing frequency ~15.6 Hz and harmonics (~31.1 Hz, ~47 Hz, ~63 Hz, ~78 Hz, ~94 Hz, ~109 Hz,   The vibration data FFT formats and V RMS values, presented, are uninformative in condition monitoring and failure diagnostics of defective 9th bearing.The defective 9th bearing (rolled bearing's elements and damaged tapered bore wrenches in the bearing and on the shaft cylindrical surface) generates low amplitude vibration, which were impossible to be identified with ordinary vibration data formats.The more advanced ACMT (Adash Compressed Time Method) method based on measurements of bearings housings vibration accelerations were used as shown in Figs. 12 and 13.The diagrams show vibration of defective and new bearing.Shocks generated due to the change of teeth meshingimpacts when rolling tracks of the bearing are rolled or bearings housings elements have heavy damages.However, the ACMT method with vibrations acceleration evaluates the severance of teeth meshing but did not allow identify the defects in 9th bearing.Diagrams of ACMT vibrations velocity time-base plots show limited information for diagnostics.
Diagrams of vertical vibrations displacements of a shaft to the 9th bearing support, before and after replace-

Sugar production diffusion machine vibration testing in situ
Experimental testing of gear driver in situ of II rotor shaft with gear z 3 , were carried out under variable load from 80% to 110% of rated load values.The aim of the research was to determine the displacement and vibration displacement of rotors II shaft located near bearing as was shown in Fig. 3. Experimental research was carried out with RB 6423 contactless sensor and signal analyzer module DMA4 (Epro, Germany).The experimental measurement results are presented in Figs.16 -18.The results of vibration displacement measurement for the driving rotor shaft II show that antifriction bearing support has low radial and axial stiffness and large mechanical looseness.
The vibration displacement amplitude s p of the driving gear z 3 rotor shaft measured in a horizontal (radial) direction is 150 -200 μm and in the vertical 120 -150 μm.The axial displacement of rotor II is high with low frequency 0.01217 Hz, Fig. 17.These results determined that during the operation of a gears z 3 , z 3 ' and z 4 , the teeth base pitch permanently changes its position that caused valuable changes in teeth meshing pitch point positions of the involutes teeth surfaces.The analysis of theoretical and experimental research materials showed that there are no finished and reliable diagnostics technologies for defects identification of low speed rotors with antifriction bearings and gear transmissions.Therefore the authors designed and validated new diagnostics method RSMMD as showed in Fig. 19.It has the main difference that vibration measuring is performed with contact less sensors instead accelerometers Fig. 19 The scheme of RSMMD method used for diagnostics of low speed rotor Fig. 20 The shaft radial displacement measurement scheme or vibration velocity seismic transducers.The vibration displacement measurements of rotor are performed near the bearing as showed in Fig. 3.The measured vibration displacement amplitude zero-to-peak s o-p is involved in mathematical model and using mathematical simulation it is desirable to analyze gears teeth meshing conditions as showed in Fig. 20.

Modeling and numerical example
Modeling of high mass horizontal cylinder machine and diffusion machine's gear transmissions and tooth contact is carried out by the FEM with ANSYS software.Experimentally measured radial displacements s o-p values of cooler's gear z 1 and diffusion machine's rotor gear z 3 were integrated into discreticised models (separates models).The models were made under least favorable conditions of teeth meshing, i.e. when only one pair of teeth were in contact and the both machines are fully loaded.The models were settled with a change of rotor's radial displacement from 0 to 500 µm in the case of cooler and from 0 to 250 µm in the case of diffusion machine.The models were solved when both rotors materials have linear stiffness characteristics.Geometric models of teeth are divided into lower order finite elements of hexahedron SOLID 45 type, having eight nodes and three degrees of freedom (displacements in respect of three coordinates) as showed in Fig. 21.For the reason that the modeled gear transmissions are of low speed, the solutions were worked out in static where [K] is the stiffness matrix of finite element; {U} is the nodal element displacement vector; {F} is the load vector.
Nodal displacement vector {U} is composed of Having solved the teeth contact discreticised models, it was determined, that in case of a rotor's radial displacements, located next to support of a bearing, are equal to 0, teeth contact stresses reach 341 MPa value in the case of cooler, and 378 MPa in the case of diffusion machine.When radial displacement of the rotor of high mass horizontal cylinder machine exceeds the value of 130 µm and 100 µm in the case of diffusion machine, permissible contact stresses value (σ adm =~500 MPa) is exceeded, therefore plastic deformations occur in the teeth, and the teeth are damaged.Radial displacement of the rotor is limited by values 130 µm of high mass horizontal cylinder machine and 100 µm of diffusion machine in continuous long term operation.A diagram, presented in Fig. 22, can be used in order to determine the residual resources of service for rotors only if bearing's rolling tracks and rolling elements are rolled or damaged the tapered bore-shaft outer ring-case mounting tolerance of the bearing housings.

Conclusions
Experimental research confirms that ordinary vibration research methods with standard vibration data formats (root mean square values, FFT spectrum, time-base plots, etc.) are not suitable for the evaluation of technical condition of low speed rotors with gears and antifriction bearings.It is also impossible to identify defective bearing of a low speed rotation rotor with gears using ACMT method.Diagrams of ACMT vibrations acceleration timebase plots allowed only the identification of shocks generated by the impacts of gears teeth meshing.
Diagnostics and evaluation residual performance of low speed rotor is possible using absolute (or relative) measurements values of radial displacements s o-p or s p-p provided by contact less sensors.
Theoretical modeling together with experimental research of dynamics of a gear driver and roller bearings in low rotation speed machines is realized as a new Rotors Supervising, Mathematical Modeling and Diagnostics method.
Experimental and simulation results approved that low frequency vibration displacement values are the main parameters that describe technical condition of involute teeth meshing quality.

Fig. 1
Fig. 1 Kinematic scheme of high mass horizontal cylinder machine: EM -asynchronous electric motor, GB -gear box, C -elastic coupling, z 1 and z 2 -driving and driven spur gears; 1D, 1K, 2D and 2Kcylindrical wheels; D1, 1D2, 1K1, 1K2, 2D1, 2D2, 2K1 2K2, 9 and 10-double row spherical roller bearings SKF 22228 CCK/W33 with a tapered bore H3128 and shaft and mechanical looseness in the outer ring-case housing.The more sophisticated low rotation speed gear power transmission is operating in sugar production diffusion machine, as shown in Fig. 2: driving spur gears z 3 = z 3 ' = 17 with 15.45 rpm and driven z 4 = 115, contact ratio 1.6 and module m = 20 mm.The radial double row spherical roller bearings with cylindrical bore SKF 22230 CC/W33 are used.During continuous long term operation teeth surfaces of the gears z 3 , z 3' and z 4 were seriously damaged (Fig.3).The new diagnostics method was designed to identify the defects of rotors antifriction bearings and gears.

Fig. 10
Fig. 10 Horizontal vibration acceleration time-base plot of defective 9th bearing

Fig. 13
Fig. 13 Horizontal vibration acceleration time-base plot of new 9th bearing interpreted by the ACMT method

Fig. 14
Fig. 14 Radial displacements of the rotor with gear z 1 relative to bearing housing time-base plots with 9th defective bearing Identification of the defects in the 9th bearing was successful only with the measurements of rotor radial vibration displacements peak-to-peak s p-p or zero-to-peak s o-p values with contactless sensors.Sensors were fixed near the bearing of low speed rotor with a gear z 1 and upon carrying out modeling and theoretical calculations.Measurement results of rotor radial vibration displacements s o-p in horizontal and vertical directions are presented in Figs. 14 and 15.Radial vibration displacements of the rotor's measured before and after replacement of the 9th defective bearing by a new, are presented in Table2.

Fig. 18
Fig. 18 Displacement (eccentricity) and vibration displacement time plot of rotor III (with gear z 4 ) shaft during two rotations: in the vertical axis -shaft displacement measured in volts, in the horizontal axis -time measured in seconds 5. Rotors supervising, mathematical modeling and diagnostics method (RSMMD)

Fig. 21
Fig. 21 Main view of gears teeth meshing of high mass horizontal cylinder machine

Table 1
Vibrations velocity V RMS values of 9th and 10th bearings housing Vibration velocity (10 -1000 Hz) V RMS , mm/s (V-vertical, H-horizontal vibrations)RMS -root mean square

Table 2
Low speed rotor with a gear z 1 radial vibration displacements zero-to-peak values of the defective 9th bearing, are presented in Figs.14 and 15.In case of a defective bearing, the highest measured radial vibrations displacement (relative to 9 th bearing support) value s o-p reached 132 µm and with new bearing, it reduced ~2 times and reached 67 µm.Increased radial displacements of the shaft changed the position of teeth mesh pole of a gear z 1 and z 2 . ment