INVESTIGATION OF DYNAMIC PROCESSES IN A GEARBOX WITH DEFECTS

The article presents theoretical research of a transmission system that allows examination of the dynamic processes evaluating different sizes of defects in gear tooth (0%, 20%, 50%, 80% of teeth width) and gaps between the connection of the gear teeth. The dynamics of the transmission is described using the finite element method and discreet elements. This paper presents and analyzes the results of mathematical experiments on this system.


INTRODUCTION
A gearbox is widely used as one of the main transmission elements in industrial and military applications to transmit power. Gear or gear teeth surfaces in particular excessively wear because of the difficult shape of the geometry caused by intense loads in transmission systems. The contact area of two connecting gear teeth is variable, causing defects in gear teeth. Defects, such as gear tooth cracks, fractures, and chipping, increase vibration levels and generate noise simultaneously as the gearbox continues to work. Early fault detection might be crucial to stop incoming systems failure. At present, increasingly more investments are being made in the field of diagnostics. Many scientists perform mathematical modeling of defects to understand their impact on system dynamic characteristics. For example, Mohammed et al. [1] studied the gear system using three different dynamic models from 6 to 12 degrees of freedom (DOF) evaluating different crack sizes and pinion's displacements. The results of the first model show visible differences from other models that were made more realistic, including simulation of motor and load. From the second model, the influence of the friction on inter tooth could be seen. Jia et al. [2] for the object took 26 DOF transmission system which includes three shafts and two connected spur gears. This model is developed by calculating torsional stiffness and tooth load adding spalling and crack damage. For simulation, Matlab Simulink software tool was used. Patrick et al [3] focused on the identification of cracks in a planetary gear system using vibration sensors. Mathematical model results are compared with experiments from the results obtained from a helicopter. Using the same vibration signal modeling, Liang et al. [4] identified the exact gear tooth crack location. The presented results can help identify defects and it can be used as a tool for planetary gear system diagnostics. Group of researchers, Ma et al. [5] conducted a dynamic investigation of cracked gear systems. The whole research focused on 3 major important factors: crack propagation prediction, mesh stiffness calculation, and vibration response calculation. For the object they took three different gears. Tian et al. [6] carried out a model-based gear dynamic analysis and simulation research evaluating cracks' propagation level. The results suggested that the root mean square is a better statistical indicator than the Kurtosis indicator to reflect the crack propagation in the early stages.
Finite element modeling (FEM) is one of the methods to simulate gears with defects dynamic behavior; however, these models have large amount of DOF. Therefore, it becomes quite difficult to proceed with the modeling. Nevertheless, Parker et al. [7] proposed a nonlinear tooth contact and mesh 6 P. Bogdevičius, M. Bogdevičius, O. Prentkovskis stiffness variation model using FEM. Pruerter et al. [8], with the same gear teeth contact model, indicated that uneven loading on the teeth was because of differing constraints on the two sides of the teeth compared to the experimental results. Lin et al. [9] studied 3D dynamic contact problems. According to the results, mesh stiffness results were the same as those obtained from the conventional methods. Secondly, the case of initial speed impact is considered; the contact time is independent of the initial speed. Third, the impact time depends on the geometric parameters of the gear drive and the total contact force is proportional to the initial speed and transmission load. To insure lifetime of the gear, stiffness calculation is one of the key parameters that is being studied by the scientists. Xue et al. [10] presented work on torsional stiffness evaluation in involute spur planetary gears in mesh using FEM. The theoretical gear contact position was determined using an ANSYS APDL software. Walha et al. [11] investigated the dynamics of a two-stage gear system involving backlash and time-dependent mesh stiffness. To satisfy some conditions, nonlinear dynamic systems were linearized to a linear dynamic system and each system was solved using Newmark iterative algorithms. A hypoid gear pair was taken as an object in the research carried out by Wang et al. [12]. Nonlinear mesh stiffness results showed that the increase in the mean mesh stiffness ratio tends to worsen the dynamic response amplitude and stiffness parameters have an impact on dynamic responses. Saxena et al. [13], in the mesh stiffness model, included sliding friction and spalling defects.
In this paper, dynamic teeth coupling investigation with mesh and contact stiffness defects in a gearbox system is presented. Teeth of each gear are self-contained and may have defects. The deformation of each tooth is evaluated. The size of gear tooth's stiffness defect is presented and expressed as a percentage (0, 20, 50, and 80 %). Transmission with an idle with no external excitation reveals the presence of gaps between contacting two gear teeth 50 . The torsional vibrations of shafts and bending vibrations of gear teeth are simulated using finite and discrete elements methods.

MATHEMATICAL MODEL OF TRANSMISSION SYSTEMS
A mathematical model of a transmission system is considered (Fig 1.). The transmission system includes an asynchronous electrical motor, a gearbox, and connection shafts. The main parameter of the electric motor is the moment of inertia , rotation angle , and torque . The gearbox includes two connected gear pairs, where is the evaluated moment of inertia , , , , rotational angles , , , , and gear Pitch radius , , , . For torsional rotation, only gear tooth mass is evaluated. The transmission system is unloaded (torque of resistance is equal zero, ). To evaluate the rotation of the transmission elements, a mathematical model suggested by Aladjev [14] may be used: ( where is the rotation moment of the asynchronous electric motor; , are the parameters of the electric motor; is an angular velocity of idle running; is an angular velocity of the electric motor, and -glide and torque, -angular velocity of the motor, -ideal angular velocity of the asynchronous motor.
Investigation of dynamic processes in a gearbox with defects 7.
where , , and are the mass, damping, and stiffness matrix, respectively; , , and are the vectors of acceleration, velocity, and displacement, respectively; and is the vector of torques.
Rotational angle in the finite element is equal to: where is the vector of angles; is a matrix of shape functions; and is a local coordinate. Mass, stiffness, and damping matrices of the finite element of the transmission system are equal to: , , where is the shaft material density; is the polar moment of inertia, is the shear module; and and are optional coefficients by Aladjev [14]. To evaluate the interaction of possible cracks on tooth's surface mesh stiffness, it is important to determine the size of the defect. In this model, index k indicates gear or index i -gear tooth. The bending stiffness with a defect of the k-th gear and the i-th tooth is described by the formula: , (8) where is the size of the defect ,  ; (9) where is the defect depth and is the tooth thickness, are the moduli of elasticity of the tooth without and with a defect is the stiffness coefficient of the i-th tooth without a defect.
where , is the stiffness and the dumping coefficient between contact; n is the nonlinear power exponent determined from material and geometric properties; and is penetration, , where: and are the displacement vectors, is the gap between two connected gear teeth, is the restitution coefficient, and is the Heaviside function.
The potential energy of k-th gear i-th tooth and l-th gear j-th tooth is equal to: ; (11) , where , and , are base radii and rotation angles of the k-th and l-th gears, respectively, and and are the stiffness coefficients of the k-th and l-th gears, respectively.
The dissipation functions of k-th gear i-th tooth and l-th gear j-th tooth are equal to: ; (13) , (14) where and are the damping coefficients.
The equations of rotation k-th gear and motion of i-th tooth area described below: , (16) and l-th gear and j-th tooth are equal to: where , are inertia moments; , are k-th and l-th gear torques; , are mases of k-th gear i-th tooth and l-th gear j-th tooth, respectively; nzk and nzl are the numbers of gear teeth; is the parameter that takes into account the coupling of the k-th gear i-th tooth and the l-th gear j-th tooth , and . In the annex A, the derivation of the stiffness coefficient of gear is presented.

Initial data of the research
The purpose of this research is to determine the dynamic characteristics (velocity, acceleration, torque) because of the effect of different sizes of the defect 20 %, 50 % and 80 % of whole width on tooth surface with 50 gap between gear teeths. Initial conditions of the variables: initially, the torque of the asynchronous electro motor, vectors of rotation angles, and angular velocity are equal ( , ). The number of first and third gears is equal, ; the number of second and fourth gears is equal, . The integration time step is and the simulation time is 5 s. The data of the transmission system are shown in Tab 1.

Results and discussion of the transmission system
The dynamical characteristics of the transmission system may alter due to defects in gear tooth. The developed model of transmission with teeth defects allows a detailed analysis of the dynamic characteristics of a transmission system. The gap between two connected gears is 50 .    Dependencies in time of asynchronous electric motor torque and angular velocity of gear1 and gear 2 are shown in Fig. 4 when the defect size is 80%. With such defects, clearance, and idle rotation, pulsations of the torque of the electric motor appear .   b) 20% ; c) 50% Fig. 4 shows that acceleration changes due to an increase in the size of the defect. In the graphic, when there are no defects in the gear tooth system, the maximum value of the acceleration reaches 0,95 . On changing the defect size to 20% and 50%, acceleration increases to 1,05 and 2,85 . When there are gaps between the teeth 50 and the various defects of the first tooth of the first gear wheel, torsional vibrations of the shafts and bending vibrations of the teeth are generated in the transmission. Fig. 4 shows that complex torsional vibrations occur in the transmission.
When simulating the operation of the transmission for 5s at idle rotation, the transmission does not have torsional vibrations of the elements. Therefore, it can be argued that when an alternating rotating moment of resistance is applied in the transmission, the torsional vibrations of the transmission of the elements will be more complex.
Angular acceleration of four gears dependency on frequency is shown in Fig. 5 when the size of the defect is 50%.
The four graphs in Fig. 5 show that when a defect occurs, the resonant frequency at the first gear is visible. First gear vibration by coupling is transferred to the second gear, but decreases at the third gear, which connected to the second one by the shaft and completely disappears with in fourth gear. This analysis makes it possible to determine which gear has the defect, but it does not allow us to determine which gear tooth is damaged. This is done by another spectrum analysis of the gear teeth with different defect sizes and this is shown in Fig 6. According to Fig. 6, because of the effect of different defect sizes, the rotational frequency decreases with the maximum amplitude value. Tooth without defect , maximum amplitude value is 220 at 20000Hz frequency, when 20% value is 190 at 19 000Hz frequency, at

CONCLUSION
A mathematical model of transmission system is established that allows the analysis of dynamic processes evaluating different sizes of defects on gear tooth (0%, 20%, 50%, and 80% of tooth) and gaps between connections of the gear teeth 50 . The resonance frequency changes to a different size of defect. The bigger the size of the defect, the smaller the stiffness, and the resonance frequency decreases. Within 50% of the defect of tooth, gear 1 angular velocity reaches 314 rad/s and gear 2 angular velocity reaches 376 rad/s.
In a system without defects of the first tooth on gear 1 , the maximum acceleration value of the first tooth reaches 0,95 m/s 2 , when the defect size is 20% , acceleration of 1 tooth increases to 1,05 m/s 2 , and when the defect size is 50% , acceleration reaches 2,85 m/s 2 . To study the torsional vibrations of the transmission of elements and bending vibrations of the teeth of gear wheels with defects and gaps, the developed software has been developed.

Annex A
The fundamental equation for structural analysis, based on the Euler beam theory, is presented where one point is fixed; the force-load behavior can be described by: , where is the stiffness matric; is the displacement vector , and is the aggregated force vector.
. When the first point of the beam is fixed the equation lightens: .
Then . Fig. 7 shows a simplified model of a localized gear tooth of irregular shape evaluating an involute curve. where is the modulus elasticity of defected gear tooth; and are shape functions of the beam element, is the y coordinate, is the height of the tooth; and A is the cross section area of the tooth that depends on the y coordinate. The coordinate of the cross section of the tooth is approximated by 3  q q é ù ì ü ì ü ê ú ï ï ï ï ï ï ï ï ê ú = í ý í ý ê ú ï ï ï ï ê ú ï ï ï ï î þ î þ ë û