Identification of the Asymmetric Transmission Error and Gear Mesh Dynamic Parameters using Full-Spectrum Responses in a Geared-Rotor System

A dominant source of vibration in geared-rotor systems are the gear mesh fault parameters. They include the asymmetric transmission error, phases of transmission error, the gear mesh stiffness, the gear mesh damping and the gear runouts. The present work deals with the experimental identification of aforementioned parameters. A mathematical model of geared-rotor system has been developed using Lagrangian dynamics. Equations of motion are transformed into frequency domain using the full-spectrum response analysis. These transformed equations are used to develop an identification algorithm based on least-squares fit to estimate the transmission error and gear mesh dynamic parameters. The system identification algorithm is initially verified using numerical simulations. The robustness of the algorithm is checked by introducing white Gaussian noise in the simulated responses. A geared-rotor experimental rig was developed and used to measure responses at gear locations in two orthogonal directions. Measured responses are transformed in frequency domain using the full-spectrum analysis and used in the present novel identification algorithm to identify the gear parameters. The identified parameters are validated by comparing the numerically generated full-spectrum response using experimentally estimated parameters and that from the experimental rig.

Gear mesh damping ratio

Introduction and literature review
Complex nature of rotating machinery fascinated lot of researchers to study their dynamic behaviour.These studies resulted in the design of lighter, high-speed and exceptionally reliable automotive, aerospace and other industrial machinery.Geared-rotor systems are the most commonly used torque carrying members in most of the mechanical and electrical equipment.Gears are considered as dominant source of vibration due to its deviation from perfect conjugate profile and the dynamic nature of gear contact between flanks of mating gears.The parameters that cause this profile deviation are considered as the transmission error, gear mesh stiffness and gear runout errors, individually or in combination.These dynamic effects may sometimes cause catastrophic failure of mechanical systems and subassemblies attached to the supporting structure.Estimation of these gear parameters is having paramount importance [1].
Several researchers studied identification of rotating machinery faults, such as the gear-faults, misalignments, residual unbalances, shaft bow, shaft cracks, motor-faults and bearing faults using system mathematical models [2][3][4].In these attempts, authors used the analytical, transfer matrix and finite element methods.Wherein, the fault models with unknown parameter are considered.Through least-squares or other fitting techniques the unknown fault parameters are estimated with the help of system responses.Some researchers used feature extraction techniques, conventional neural network, discrete wavelet transform and thermal images [5][6][7][8][9].
Plenty of researchers studied geared-rotor vibration through numerous mathematical models.Most of the researchers considered transmission error (TE) as excitation source for the gear vibration problem.TE is specified at the pitch point of the gear mesh, and it is mathematically expressed as change in driven gear position with respect to the position of a perfect gear drive.The perfect gear drive is free from any geometrical deviations from its conjugate profile.Several researchers presented fundamental concepts of TE and its measurement techniques [10][11][12].Özgüven and Houser [13] reviewed works on mathematical models used in gear dynamics.They gave insights into sources of the mesh excitation and its contribution to the system excitation, pertaining to the gear noise.Houser [14] published the state of the art by summarizing research works on contribution of various sources of gear mesh excitations to gear noise.Kahraman et al. [15] presented finite element (FE) modelling of geared-rotor mounted on flexible bearings with a combination of TE excitation and constant gear mesh stiffness.
Wadkar and Kajale [16] studied geared-rotor natural frequencies and modes shapes.Also, the reliability of geared-rotor using the time-varying mesh stiffness was investigated.Mohammed et al. [17] proposed an improved mesh stiffness calculation for the purpose of fault detection in gearedrotors using vibration signals.Li et al. [18] studied coupled lateral-torsional-axial vibration problem of a helical gears mounted on flexible bearings using system model approach.Zhou et al. [19] analyzed a coupled vibration of a spur-gear pair considering the TE and presented vibration response using 3-dimensional frequency spectrum.They studied the influence of eccentricity, speed and bearing clearance on the nonlinear response.Temis et al. [20] studied gear vibration problem with proportional viscous damping by considering the time-varying mesh stiffness and the tooth separation.Mohamed et al. [21] presented the dynamic behavior of geared systems with multiple cracks using model-based approach.Rao and Ganti [22] presented a case study to mitigate gear whine in 6-speed automotive transmission using 3-dimensional multi-body dynamics simulation by exciting the loaded gear pair with TE.They also suggested procedures for mitigation of the gear whine.
To analyse the geared-rotor response in frequency domain, the signal processing techniques are very important.It separates response due to different fault frequencies and helps in the identification of TE and other gear mesh faults.Traditional FFT fails to provide the direction of precession of the pinon and gear harmonics with respect to the direction of drive shaft.Southwick [23] presented the full-spectrum analysis, which unwraps the traditional fast Fourier transform (FFT) spectrum into the forward and backward whirl frequencies.Full-spectrum is suitable for analysing response of geared-rotor system as it is an augmented version of the Campbell diagram.Qu et al. [24] studied vibration measurement of large machinery by short period Fourier Transform and Wigner distribution to nullify the shortfalls in conventional FFT methods.Southwick [25] studied response of a rotor using full-spectrum for obtaining the asynchronous and synchronous vibrations.He investigated the ellipticity of whirling orbits under various conditions that paved motivation for the progress to full-spectrum.Goldman and Muszynska [26] presented a method for the detection of various rotating machinery faults with full-spectrum.They also presented phase correlation of rotor orbits in the horizontal and vertical spectrum components.Bachschmid et al. [27] analysed rotating machinery vibrations using full-spectrum with shape and directivity index method for analysing the ellipticity of filtered orbit.Tuma and Bilos [28] studied rotor mounted on fluid film bearing to identify fluid-induced instability and whirl frequency components with full-spectrum response plots.Patel and Darpe [29] investigated cracked rotors using full-spectrum response for identification of the crack rub with directional nature of higher harmonics.Shravankumar and Tiwari [30] studied the model-based crack identification in cracked rotors with full-spectrum response plots.
Hong and Dhupia [31] demonstrated time domain approach to identify the gearbox faults using measured response from an experimental rig.Sawalhi and Randall [32] presented a method on the identification of teeth count of two parallel stages of gears without speed reference signal under variable speed condition.Feng et al. [33] proposed an order spectrum analysis method based on the iterative generalized demodulation for the characteristic frequency identification in a faulty planetary gearbox.Dogan and Karpat [34] presented a dynamic transmission error (DTE) based numerical fault detection model using finite element analysis for crack detection in spur gears with asymmetric teeth.Benatar et al. [35] presented a set of motion transmission error data for a family of helical gears having different profile and lead modifications operated under both low-speed (quasi-static) and dynamic conditions.Celikay [36] studied a subharmonic resonance observed in spur gear pairs as a parametrically excited system.The stiffness at the gear contact interface that couples the gear bodies is a periodically time-varying due the fluctuation of number of tooth pairs in contact.Flek et al. [37] presented time-varying stiffness of spur gears in the dynamic model of transmission systems as an internal excitation of the dynamic system.
Chin et al. [38] proposed a TE-based method for the estimation of gear root crack depth as an indicator of severity.Xue et al. [39] presented a gear system dynamic model for the non-stationary operating condition using the iterative convergence of the tooth mesh stiffness and demonstrated the effect of gear tooth crack on the resultant dynamic response for the non-stationary condition.Poletto et al. [40] presented identification of gear wear damage using topography analysis.Dai et al. [41] developed a simulation model to investigate the impact of defective bearings on the mesh characteristics of gear pairs.Talakesh et al. [42] presented experimental and analytical methods to calculate the time-varying mesh stiffness for healthy and cracked straight bevel gear systems.Dong et al. [43] presented a mediator algorithm between simulation and experiment to identify the error excitation in gear systems.The authors also proposed a signal processing procedure to eliminate the phase difference and improved the signal-to-noise ratio.Koutsoupakis [44] presented a damage identification and condition monitoring (CM) method based on CNNs and validated on an experimental two-stage gearbox.The CM-CNN was trained on numerical data (PSDs) generated by repetitive simulations of an optimized MBD model of the actual structure with randomly sampled parameters.
From the literature survey, a very limited literature is found on the identification the transmission error of a gear mesh using vibration signal.This research work focused on a novel model-based approach for quantitative identification of the transmission error along with other gear dynamic parameters.In this direction, the authors recently presented their work [45], where the response of geared-rotor is analysed using the full-spectrum to detect qualitatively the novel asymmetric transmission error of variable components of TE.However, authors have not attempted to have quantitative estimates of the transmission error.In this work, the authors have developed an identification procedure based on geared-rotor model of [45] to estimate the novel asymmetric transmission error along with the mean transmission error, phases of variable TE, gear mesh stiffness, gear mesh damping and gear runouts using the full-spectrum response.
The content provided in the following sections are as follows.Section 2 gives a brief on the modelling of geared-rotor, gear mesh fault parameters and the derived equations of motion using the Lagrangian dynamics.Section 3 depicts the formulation to transform equations of motion from time domain to frequency domain.Section 4 details the development of identification algorithm for quantitative identification of gear mesh parameters.Section 5 presents the numerically simulated fullspectrum responses of geared-rotor system.Section 6 describes the numerical testing of identification algorithm.Section 7 has information about experimental rig.Section 8 describes full-spectrum response analyses from the experimental rig response measurements.Section 9 describes the identification of gear mesh DTE parameters using experimental rig full-spectrum responses.Section 10 presents the validation of geared-rotor system model and the identification algorithm.Section 11 summarizes the conclusions of the present work.Appendices A and B gives frequency domain formulation transformation of equations of motion

Geared-rotor system model and gear mesh fault parameters
In this section, descriptions are presented related to formulation of equations of motion [45], which are used for development of the identification of transmission error and other uncertain system fault parameters.The shaft supports are assumed to be rigid in transverse directions, and shafts and gears are also considered to be rigid in torsion mode of vibration that gives the scope of ignoring the torsional vibration of geared system.This assumption also eliminated the non-linearities associated with resulting geared-rotor equations of motion.This approach provides the opportunity to study more about gear transmission dynamics problem with matured linear mathematical techniques of transverse vibration.A four-DOF system model is developed for modelling gear transverse vibration problem with the transmission error as shown in Fig. 1(a) and 1(b).The system model has two shafts of lengths l1 and l2, and a motor drive connecting to the input shaft with a rigid coupling.The input shaft carries a pinion, and the output shaft carried a gear.Both the pinion and the gear are mounted at the mid-span of the uniform flexible shafts to have a gear mesh.The output shaft is loaded with required torque to load the gear pair so that they remain in contact during operation without separation.
To consider the fluctuation in gear mesh stiffness due to deviation from the conjugate gear profile, the static transmission error (STE) is incorporated.This paper uses the orientation of the line of action (LOA) at an angle  with y-axis [45], as shown in Fig. 1(b), to model meshing of gears between two parallel shafts in a realistic power transmission.This work also assumes two different transmission errors in two lateral orthogonal directions to consider the different excitation in two orthogonal directions ( ( ) x e t and ( ) y e t ).It helps in studying the forward and backward whirls features of gear mesh when it is loaded.The angular displacements of the drive, pinion, gear, pinion and load are given, respectively, as Herein, p and g are the spin speed of the pinion and the gear, respectively,  is the initial phase, subscripts: d denotes the drive, p denotes the pinion, g denotes the gear and l denotes the load.Also, the gyroscopic effect is not significant in this formulation as the pinion and the gear are fixed at mid-span of the shafts.Geometric relations are given as where (x1, y1) are the displacements of the centre of pinion, (x2, y2) are the displacements of the centre of gear, (xp, yp) are the position of the centre of gravity of the pinion and (xg, yg) are the position of the centre of gravity of the gear, and ep and eg are runouts of the pinion and the gear, respectively.On substituting equation (1) into equation (2), we get x x e t y y e t x x e t y y e t The gear runouts generate unbalance force in the pinion and the gear, in addition, they give rise to relative deformation of shafts during the torque transfer.The STE along with the runout errors which contributes to no load transmission error (NLTE) constitute the dynamic transmission error (DTE).The DTE excites the gear drive at the pitch point, which is assumed as sum of mean and variable components with their initial phases.The difference of STE from tooth-to-tooth is not considered in this model for avoiding complex equations for derivation.As the LOA is slanted by an angle  with y-direction, the DTE is modelled to excite the gear mesh in x and y directions on the exterior of the contact region created at the pitch point due to gear mesh profile deviation in complete mesh cycle (start of engagement of teeth in a gear mesh to end of engagement of teeth).
It is found in the literature [15] that while modelling DTE, sinusoidal excitation along the pressure line is given at the gear mesh frequency by neglecting the initial phase.In this work, similar to the work of the authors [45], the initial phase is included in the excitation.Apart from a constant term, the sine and cosine variation with an initial phase, and higher harmonics of the gear mesh frequency are also included.As proposed by the authors in their work [45] and supported by literature [15], in the present work the variable gear mesh stiffness force has been modelled with a constant mesh stiffness multiplied by displacement in the form of loaded STE, which gives external forcing as in Kahraman et al. [15].Also, it is assumed that the relative deformation of gears is completely transformed into elastic deformation on surface of teeth.The dynamics of pair of gears is mathematically modelled as discs.These discs at its pitch circle radius joined using spring and damper having gear mesh stiffness ( m k ) and gear mesh damping ( m c ), respectively, along the LOA.The LOA is tangent to base circle of gears to ensure the contact of teeth surface during power transmission.So, the displacement (  ) between the pinion and gear along the LOA is given as, where m k and m c denotes the average mesh stiffness and damping along the LOA.The DTE modelled in the x and y directions depict a practical state of power transmission as shown in Fig. 1(b).The components of  in the x and y directions, from Eqn. ( 3) and ( 4), are given as, The dynamic gear mesh force given in Eqn. ( 5) is resolved in the x and y components.The elastic deformation in the contact region at the pitch point is given by Eqns.( 6) and (7).The components of dynamic transmission error ( ) x e t and ( ) y e t in the x and y directions are modelled as a mean value and a fluctuating part [45], and can be given as, ( The displacement vector of the centers of shafts in geared-rotors is given as, x y x y  (10) where subscripts 1 and 2 refer to the pinion and gear shafts, respectively.Using the Lagrangian dynamics, the equations of motion are derived and presented in the matrix form [45], which contain forces from the TE, the unbalances and runouts, are given as, ( ) with     where, f(t) is the force vector; M, C and K are the mass, damping and stiffness symmetric matrices.Also, m1 and m2 are the mass of the pinion and the gear, respectively; ks1 and ks2 are shaft stiffness for the pinion and the gear, respectively; and cs1 and cs2 are shaft stiffness for the pinion and the gear, respectively.This novel approach makes use of existing simple measurement techniques available for the lateral vibration in rotor dynamic systems to validate the system model and identification of asymmetric TE instead of going for expensive torsional vibration measuring equipment, which is if not available in hand.Also, it is practically easy to access gear transmission shafts with displacements probes for lateral vibration measurements rather than mounting high quality encoders to measure the TE of gear mesh by phase demodulation of the pulse signals of encoders as followed in transitional approach Munro [12] Figure 1(a): Overall geared-rotor system

Transformation of equations of motion to frequency domain
The force vector, given in Eqn.(14), is simplified after substituting Eqns.( 8) and ( 9).Using complex mathematics as described in Appendix A with Euler's equation, we have equation ( 15) in frequency domain with various frequency excitation components as, 2 2 j j j j j j j j j j j p r p j p r p j g r g j p p g p r p j p r p j g r g j g r g j e r e j g g r g j As described in Appendix B, the equations of motion in frequency domain are grouped into the components of forward and backward whirls.The frequency domain transformation helps in clarifying the whirling features of the gear mesh using its excitation frequency.By combining the static force, the gear mesh force, and the pinion and gear runout/unbalance forces, the equations of motion in frequency domain using Eqns.(A-63), (A-89) and (A-118) can be written as, Herein, i depicts the index of harmonic of gear order; n depicts number of harmonics which are assumed in TE.With presumed value of TE parameters, Eqn. ( 16) can be used to numerically simulate transverse vibration response of a geared-rotor system.Here, we can observe that the equations of motion are linear due to neglect of the torsional vibration coupling.The holistic solutions with assumed parameters of each harmonic should be summed up for i =1, 2, 3, …, n .The full responses can be written in the matrix form by combining the equations for all value of i and is written as, This can be put in a simple form for the required number of harmonics as,  Ap s (18) The solution of the above equation can be written as, Eqn. (19) calculates the response function of the forward and backward whirl gear mesh frequencies with obligatory number of harmonics based on the nature of excitation for known values of all dynamic parameters of system model.This will be used to test the identification algorithm described in the next section.

Identification of gear mesh parameters
Using the frequency domain formulation discussed in previous section, it is attempted to identify critical unknown dynamic parameters of gear pair in a three-step estimation process as described below.Fig. 2 gives schematic representation of the step-down identification algorithm (IA).
 In the first step, the gear mesh stiffness and the mean transmission error are estimated using static components of response taken from numerically simulated full-spectrum plots. In the second step, gear mesh damping is calculated with the help of estimated gear mesh stiffness and average gearing mass (details of this is given subsequently).Also, variable components of transmission errors and their initial phases are identified using the gear mesh stiffness estimated from the first step, gear mesh damping and the gear mesh frequency harmonic responses from full-spectrum plots. In the third step, the gear runouts and their initial phases are estimated using the estimates of first and second steps, and response at the pinion and gear mesh frequencies taken from fullspectrum plots.
Figure 2: Schematic represeantion of identification algorithm in a step down process The solution to the identification problem starts with initial identification of mean transmission error and gear mesh stiffness using static response components.For this purpose, static displacement coefficients from the frequency domain formulation, given in Appendix A, Eqn.(A-61) and (A-62) are combined in a matrix form as, by substituting Eqn.(A-41) into Eqn.(20) and unknown terms are rearranged as, ) The gear mesh stiffness and mean TE are separated, and written in matrix form as, In the first step of identification problem, the gear mesh stiffness ( m k ) and the mean transmission error ( m e ) are estimated using the static response (P) terms.On the left and right sides of the matrix in equation ( 22), the P terms are taken from full-spectrum plots.On pseudo-inverting the first (left) matrix in Eqn.(22) we get, m k and m e .
In the second step of identification process, the gear mesh frequency components are considered for the identification of the gear mesh damping, and variable transmission error with its magnitudes and initial phases.For this purpose, Eqn.
Now, the forcing functions due to gear mesh frequency on the input and output shafts are grouped for the input shaft by substituting these equations in Eqns.(A-42) and (A-43) to get the defined pinion and gear force real and imaginary components (R) in terms of critical gear mesh parameters on the input shaft as, Similarly, for the output shaft we can write as, These equation formulations are used for second step of identification by substituting them into Eqns.(A-77), (A-80), (A-83), (A-86), (A-106), (A-109), (A-112) and (A-115) to get the identification equations of the input and out shafts.
These equations are written in a matrix form by rearranging terms for the input shaft as, ( ) Similarly, for the output shaft, it can be arranged as, Herein, the gear mesh damping is estimated from gear mesh stiffness, km, and average pinion (mp) and gear (mg) masses.The average mass of pinion and gear is given by average gearing mass as, The gear mesh damping can be calculated using free vibration damping formular with average gearing mass and gear mesh stiffness by using assumed gear mesh damping ratio ( m Eqns. ( 26) and ( 27) are combined in matrix form as, Herein, subscripts 1 and 2 represent the input and output shafts.The harmonic number i is from 1 to 5 harmonics of gear mesh frequencies considered in modelling variable TE, the gear mesh damping from Eqn. (29), the gear mesh stiffness calculated from Step 1, the input and output shaft stiffness, the masses of gears and the full-spectrum responses (P terms) calculated numerically are substituted in Eqn.(30).On taking the pseudo-inverse of the regression Eqn.(30) to identify the four unknown quantities of fluctuating part of TE and their phases are written as, On noting Eqn.(31), the fluctuating part of the static TE in the x and y directions are estimated using the sine and cosine trigonometric relations by combining the first and third components; then by combining the second and fourth components as, From the first and second components of Eqn.(31), we get the initial phases of fluctuating components of the static TE in the x and y directions and are estimated as, In the third and last step, the gear runouts and their phases are identified with the pinion and gear mesh frequency components of frequency domain transformations.The defined pinion and gear force real and imaginary components (R) in Eqns.(A-30) through (A-33), (A-35), (A-36), (A-38) and (A-39) are substituted into Eqns.(A-107), (A-113), (A-110), (A-116), (A-79), (A-85), (A-81) and (A-88), respectively, to get the identification equations on the input and output shafts for the ith harmonic and is written in matrix form by rearranging pinion equations as, and gear equations as, g j g r g j g g j g r g r g r g j g r g r g r g j g j g j g r g j g r g r g r g By substituting the identified value of gear mesh stiffness ( m k ) and gear mesh damping ( m c ) from the first and second steps, chosen geared-rotor parameters given in Table 1 and full-spectrum amplitudes at respective harmonics (P terms) in Eqns.( 27) and ( 28) and resulting equations are simplified to get the combined form as, p j p r p j p j p j p r p r g r g j g r g r g r g j g j In the third step of identification problem, the matrix in Eqn. ( 36) is pseudo-inverted using the regression fit for identifying the pinion and gear runouts.The resulting form of the vector is written as, On noting Eqn.(37), the pinion and gear runouts are estimated using the sine and cosine trigonometric relations by combining the first and second components; then by combining the third and fourth components as, From the first and second components of Eqn.(37), we get the initial phases of the pinion and gear runout frequencies, and are estimated as, With this three-step process, the estimation of all eleven-gear mesh DTE parameters that influences the lateral responses of the spur geared-rotor with parallel shafts has been presented.Now through numerical simulation the displacement responses are generated and analysed for chosen system parameters.These responses will be used to evaluate the proposed identification algorithm to get estimation of the chosen system parameters.

Geared-rotor response numerical simulation
The equations of motion of geared-rotor system are solved numerically using Runge-Kutta technique in MATLAB with function ode45 for time domain solution.Traditionally, transformation of the calculated solution into frequency domain FFT gives information about various frequency components existing in the vibration spectrum.It contains both magnitude and phase information of response to analyse the system dynamics.However, the response spectrum does not indicate about relative phase direction among different vibration signals.In addition, it fails to provide the direction of pinon and gear harmonics with respect to the direction of drive shaft.
The geared-rotor vibration signal in the x and y directions are plotted for two transverse directions to get orbit.It may include multiple components of the forward and backward whirls.Both magnitude and phase of various frequency components are required to get actual shape of the rotor orbit.Full-spectrum plot is a convenient tool to identify whether the orbit at a frequency component is of forward or backward whirl with respect to rotor spin direction [26].Rao and Tiwari [45] analysed geared-rotor response to detect qualitatively the asymmetric transmission error using full-spectrum.Relative phase correlation of two vibration signals was used in full-spectrum to splits the geared-rotor frequency into the forward (positive) whirl frequency component and backward (negative) whirl frequency component, which constitute the orbit.For gears due to asymmetric transmission error the forward and backward whirls come in to existence and with conventional FFT methods this asymmetry cannot be found out.In conventional FFT methods both forward and backward whirl at same frequency overlaps.Displacement responses are calculated in time domain using linear differential equations of motion using geared-rotor data given in Table 1.Fig. 3 shows the orbit plot using numerically calculated time domain response.Different orbit shapes for the input and output shafts are attributed to different parameters chosen for the pinion and the gear.Fig. 4 shows the full-spectrum in the form of Bode plots calculated using matlab fftshift function of MATLAB applied on the numerically calculated time domain response.The response displays the features of grouping of multiple harmonics.The five harmonics 176, 352, 528, 704 and 880 Hz are manifested in the full-spectrum in the forward and backward whirl frequencies.Of which intial three harmonics are dominant and rest of the two are minor due to the assumed lower TE at higher harmonics.The phase shift is also observed at each of the harmonic in Fig. 4 for both input and output shaft responses.The utility of full-spectrum is to analyse the whirl amplitudes of different harmonics present in the gear drive due to the TE.
The impact of measurment noise is analysed using numerically generated resposne by mixing 5% Gaussian white noise into signal in time scale.Full-spectrum plot in Fig. 5 with measurement noise shows the noise has hardly any effect on the response in the frequency range of analysis.

Numerical testing of identification algorithm
To test the three-step identification algorithm (IA) described in Section 4, estimation of TE parameters is performed using numerically generated full-spectrum responses (P terms, which are real and imaginary components of full-spectrum amplitudes) as shown in Fig. 4 (a) and 4 (b) where magnitudes of the real and imaginary components are shown for the first harmonic of gear mesh frequency.Since the present identification algorithm is based on linear least-squares fit so the algorithm does not dependent upon initial guest and no iteration is involved.For improving the conditioning of matrix suitable scaling has been done.
The estimates are compared with parameters chosen for simulating numerical responses as given in Table 2.The percentage deviation of the estimates with assumed parameters for numerical simulation are also presented.It is observed that the proposed IA estimates are perfectly matching with the assumed values without any deviation.Also, the estimations are checked by changing the harmonic number from 1 to 5. For all five harmonics the estimates are perfectly matching with assumed variables.
To test the robustness of proposed identification algorithm against noise present in measurement data, 1 % and 5 % white Gaussian noise is introduced in the numerically simulated data.The full-spectrum response is plotted with 5 % Gaussian noise, as shown in Fig. 6, in the amplitude and phase form (Bode plot) and is used for the identification purpose.The comparison of phase between these two plots, Figs. 4 and 6, shows the effect of adding 5% Gaussian measurement noise with quickly varying phase in the entire spectrum in Fig. 6.
With the same identification procedure used for without noise case, the estimates are calculated with measurement noise in the full-spectrum by increasing from 1% up to 5% serially.It is observed that, with noise the estimate of gear mesh stiffness and mean transmission error are deviating.Table 2 shows the percent deviations of estimates with noise and without noise for varied presumed gear mesh parameters by changing the shaft speed, gear mesh stiffness, mean TE, variable TE, gear runouts and damping.It is observed that without considering measurement noise, the identified estimates are perfectly matching with presumed parameters used in numerical simulations, as shown in Table 2, for the first harmonic.
Table 2 shows the percent deviation in the estimates with the increase in measurement noise mainly due to deviation in gear mesh stiffness and slight deviation in mesh TE in all cases, which is sensitive to the static deflection.Also, it is observed that without passing the deviated gear mesh stiffness and the mean TE due to measurement noise in the second and third steps of IA, the estimates of TE and runout parameter estimate are robust to the measurement noise.To avoid these deviations, one may calculate gear mesh stiffness using Hertzian contact formulation presented in Flek et al. [37] using analytically or by the FE model.3 show the deviation in parameter estimates for higher harmonics from 1 to 5.These deviations are helpful in correcting the parameters of the actual gear drive for practical purposes.Also, the deviations in estimates are checked by feeding data from two speeds, which are 600 rpm apart to the IA to improve the least-squares fit estimates.Last part of Table 3 shows that with two speeds data the deviation in phase of TE has reduced from 103 % to 40 % but the same has increased the deviation in the corresponding fluctuating TE from -4.1 to -10%.

Geared rotor experimental parameter rig
An experimental geared-rotor system was designed and fabricated for model validation through measured vibration responses.It consists of two shafts supported on bearings and they were connected by a gear pair as shown in Fig. 7.The test rig can accommodate different gear sets having shaft centre distance (CD) varies from 45 mm to 100 mm.Threaded holes with 10 mm pitch are provided in the mounting plate to insert the shaft with bearing heads.Gear hub has 11 mm bore and 20 mm width (10 mm width is reserved for face of gear) are designed and fabricated to resist up to 50 Nm torque during the power transmission.The experimental rig consists of two parallel shafts, the drive shaft couples the motor shaft and the driven shaft, which is loaded with a torque brake.The pinion and the gear are mounted at the mid-span of these parallel shafts, which are coupled due to gear tooth meshing.The gear tooth is loaded in torsion to avoid separation of gears during experiment by a magnetic torquer with 0.11 Nm capacity.This causes vibration at the gear mesh due to TE introduced on gear teeth, which transfers to the bearing blocks through the shafts.
The horizontal and vertical shaft displacements at gears are captured using proximity probes, which are mounted on the base plate.The eddy current type proximity transducers have sensitivity of 7870 mv/mm.The displacement probe placement, motor and magnetic torquer of experimental setup are depicted in Fig. 7.The reference signal is measured on the input shaft, close to coupling using another proximity probe.This reference signal is used for the output shaft by expanding the time scale using gear ratio 2.18.
The power was given to the motor through a variable frequency drive, as shown in Fig. 7, to adjust the input shaft speed.The magnetic torquer was set close to maximum capacity 0.1 Nm of resisting torque with the adjustment key provided with it.The experimental probes were connected to oscilloscope to adjust the sensor with a screw to maintain a specified gap between the probe and the shaft for getting the right measurement.Each probe was connected to an eight-channel dataacquisition system to digitize the signal further processing.
The gear set has gear ratio 2.18 with the driven shaft is mounted with a bigger gear.When viewed from the hub of gears, the torque transfer happens through left flank.The smaller gear is  Fig. 8: Crowned spur gear tooth A variable frequency drive (VFD) VFD-M power source was used as a regulator to control the speed of motor as per requirement of the shaft spin speed.The motor was set to run at 970 rpm (16.15 Hz) for taking measurements.The geared-rotor experimental rig was set to rotate few minutes to attain steady state condition before taking the measurement.The dSPACE DAQ system was utilized to store the measurement signal at a sampling frequency of 5,000 samples per second.A reference signal was utilized for acquiring displacement signals of the shaft for complete multiple shaft rotational cycles, i.e., for ωt =2πn, where n is the number of complete cycles during post processing of acquired signals.Using complete cycles of signals avoids leakage error [46] and that gives consistent estimates while using the identification algorithm.The horizontal and vertical measurements taken on time scale on both input and output shafts at the pinion and the gear are combined for each shaft to plot their orbits as shown in Fig. 9.

Full-spectrum response analyses from the experimental rig
Gear mesh frequency of 258.4 Hz is calculated from the number of teeth in pinion ( 16) multiplied with its rotational frequency (16.15 Hz).When the plots of orbits are analyzed, it is found that they are different for both input and output shafts.Also, many cycles are manifested in the orbit plot specifies presence of several frequencies.Phase compensation of the measured signal is done by subtracting the phase of input and output shafts with the reference phases on the input and output shafts (output shaft phase is adjusted with the help of gear ratio) with the help of measured reference signal at the input shaft.Phase compensation helps to avoid leakage error by synchronizing the measurements related to different harmonics present in the system.
One block of measured time domain responses is chosen for transforming them into frequency domain.The phase compensated full-spectrum plot of experimental rig responses is shown in Fig. 10, which shows predominant peaks at five gear mesh forward and backward whirl frequencies.These harmonics are at 258.4, 516.8, 775.2, 1033.6, and 1292 Hz.The asymmetric amplitudes of the forward and backward whirls of all five harmonics, which excites simultaneously at all harmonics is seen in the full-spectrum.The asymmetric full-spectrum amplitudes prove the asymmetric static TE hypothesis proposed in this research with the help of system model and with the numerical simulation in the research work.
As there is relatively small TE of the fourth and fifth harmonics in the present gear, relatively smaller amplitude peaks are reflected in the respective frequencies in the full-spectrum.Also, there is relative difference in the peak amplitudes in the forward and backward whirls, which demonstrates asymmetricity in TE.Peak amplitudes at the driver and driven gear rotational frequencies are also seen at 16.15 Hz and 7.38 Hz, respectively.

. Identification of gear mesh DTE parameters using experimental rig full-spectrum responses
After getting full-spectrum responses (both magnitude and phase) of test measured signal, the identification of the experimental rig gear mesh dynamic TE parameters is attempted in this section.Phase compensated full-spectrum responses (real and imaginary components of full-spectrum amplitudes) from experimental rig al all the five harmonics of gear mesh frequencies along with responses at the gear and pinion shaft rotational frequency, as shown in Fig. 10, are used in the identification algorithm.
Using the numerically tested identification algorithm as discussed in Setion4, which has three steps in multiple parameter estimation procedure.The schematic of the three-step identification algorithm is shown in Fig. 11.These estimates are presented in Table 4. Table 5 shows percentage deviation of identified parameter using first harmonic by capturing measurement under same operating conditions.It shows only minor deviation.4.These estimates are fed to the mathematical model for numerically generating the full-spectrum responses at both input and output shafts.These numerically generated full-spectrum responses are now compared with the measured full-spectrum responses from the experimental rig for validating both the system model and the identification algorithm for correctness of estimates.
For this purpose, numerically generated full-spectrum responses of the input shaft generated with the dynamic TE parameters estimated through experimental rig is compared with the measured full-spectrum responses from experimental rig as shown in Fig. 12.The comparison shows perfect matching of the gear rotation frequency, pinion rotation frequency and gear mesh frequencies up to five harmonics.Also, we can observe the perfect matching of amplitudes at the pinion and gear shaft frequencies, and at higher harmonics of the gear mesh frequencies of the input shaft in Fig. 12.Here we can also observe the asymmetry in amplitude in both forward and backward whirls at the same harmonic due to the asymmetric TE, which is present in the system model and same is manifested in measured response from experimental rig.A novel asymmetric TE based geared-rotor system model has been developed.The equations of motion of the system model have been obtained using the Lagrangian dynamics.The response generated using time domain numerical solution is transformed into frequency domain to obtain in full-spectrum form.It depicts variable amplitudes in the forward and backward whirl frequencies.
A novel three-step identification algorithm has been developed to quantitatively identify the ten gear mesh parameters.The algorithm is initially tested against numerically simulated fullspectrum response.This identification summary shows stable solution even when the Gaussian measurement noise considered in the numerical identification.Also, it is observed that the estimates improve on considering responses for more speeds.
The geared-rotor experimental rig is designed and fabricated to authenticate the system model and identification algorithm.The full-spectrum are generated using experimentally measured responses using displacement probes.Using the same three-step identification algorithm, the gearedrotor experimental rig parameters are identified.
Finally, in the validation part, the full-spectrum response has been plotted using experimentally estimated parameters with numerical model and by comparing this full-spectrum with that of original experimental rig full-spectrum plot.Excellent correlation between these two fullspectrum plots is observed at input shaft.Overall, the objective of getting reasonably good estimates of all dynamic parameters of a geared-rotors are achieved both using numerical and using experimental data.The attempt of identifying gear mesh dynamic parameters by linear model of the geared-rotor system using test measured data will surely give new direction for solving gear real world problem.
The proposed gear parameter identification algorithm facilitates researchers for identifying all the critical gear parameters using lateral vibration measurements using simple displacement probes (by simple displacement probes (it is practically easy to access gear transmission shafts with displacements probes) rather than mounting high quality encoders to measure the TE of gear mesh by phase demodulation of the pulse signals of encoders as followed in transitional approach.The identified parameters using proposed IA is very much helpful for researchers in wide range of industrial application in automotive and aerospace applications in fault prediction, design optimization and preventive maintenance to avoid catastrophic failures.

Appendix A. Time domain to frequency domain transformation using Euler's complex identity
From Euler's complex identity, we can convert the exponential terms into the sine and cosine terms as, (A-6) On substituting above equations (A-4) to (A-6) in the force vector, equation (14) ) ) The force given in Eqn. ( 14) is acting at the gear mesh pitch point, which is resolved in the x and y components.Forces are now split into various components as follows, The equations of motion Eqn. ( 11) can be written, in terms of the complex displacements as given in equation (A-9) as, Defining a force vector in complex form as, Replacing for force component terms form Eqn. (A-7), to get ) ) On collecting similar exponential terms, to get

e c e e e c e e e k e e e k e e e k e e e k e e e k e e e k e e e k e e e k e e e
Excitations at various frequency components are divided into the forward and backward whirls, by regrouping the coefficients as per its whirl components, to get The static, and the forward and backward whirl components are present in Eqn.(A-16).For the simplicity, the static components of force vector are expressed, as, For input shaft, we get On separating the real and imaginary components, we can write Similarly, for output shaft, we get On separating the real and imaginary components, we can write where subscripts: f signifies the forward whirl, r signifies the real part and j signifies the imaginary part.
In the same way, for the backward whirl of the gear mesh frequency, we have For the input shaft, we define On separating the real and imaginary components, we can write For the output shaft, we define On separating the real and imaginary components, we can write where subscript b signifies the backward whirl, r signifies the real part and j signifies the imaginary part.
As there are no forward whirl components of pinion runout frequency component, we can write from Eqn. (A-16) Similarly, the pinion backward whirl runout frequency component is written as, The real and imaginary terms, for the input shaft, are written as The gear forward whirl runout frequency component is written as, The real and imaginary terms, for input shaft, are written as, Similarly, for the output shaft the forward whirl gear runout frequency components is written as,

e k e k e e c e c e e
On separating the real and imaginary parts, we get

R c e c e m e k e k e
As there is no backward whirl of the gear runout frequency, so we can write 1 0;

Appendix B. Grouping of the forward and backward whirl components
Similar frequency components are collected and expressed them in frequency domain for the static, forward and backward whirl components, as follows On substituting Eqns.(A-41) to (A-43) in Eqn.(15), we get the force vector as, On substituting forces from Eqn. (A-10) to Eqn. (A-11) and Eqn.(A-44) into Eqn.( 11), equations of motion in a matrix form is written as, Here , , , and S S S S S contains the TE amplitude and phase of the gear mesh, i signifies harmonics of gear order and n represents number of harmonics considered in the TE.For presumed value of TE and its variable components, Eqn.(A-45) can be solved to get the response of the gearedrotor system.Since equations of motion are linear, assumed solution for each harmonic can be added up using the principle of superposition, as For i th harmonic, Eqn.(A-45) can be written as, p g e e r e j p r p j g r g j p g e e r e j p r p j g r g j For the backward whirl force, we have Assuming a solution due to the static force for Eqn.(A-50), as where 1, 2 , , i n   .These are organized into the real and imaginary components, as 1 2 1 j j j j j j j j g r g j g r g j g r g j g r g j g g j g g j On combining Eqns.(A-77) through (A-88), we get These are organized into the real and imaginary components, as 2 2 2 2 2 j j j j j j j j g r g j g r g j g r g j g r g j g r g j g r The real and imaginary components are separated on both sides of Eqns.(A-100) through (A-105), to get: The real component of the backward whirl The imaginary component of the backward whirl

A
(A-122) The displacement and force vectors are given in the matrix form as,      Table 1: Gear mesh vibration problem assumptions Table 2: Identification summary based on numerical simulation of first harmonic Table 3: Identification summary with noise and higher harmonics Table 4: Estimated experimental rig gear mesh DTE parameters

Dynamic TE in y directions 1 Gof gravity of pinion 2 G
Center Center of gravity of gear i Number of harmonics of TE Angle of LOA orientation with y-axis  Deformation of the meshing teeth along the LOA x The component of  in x-direction y The component of  in y-direction d Initial Angular speed of the pinion m 

and 2 lr
are elastic deformations of the input and output shafts; lp e and lg e are gear runouts of pinion and gear along line of action.For a constant gear ratio, the fifth and sixth terms of above equation remains same and together they become zero.Along the LOA, the dynamic gear mesh force is given in Eqn.(5) as,

Figure 3 :
Figure 3: (a) Orbit plot of input shaft; (b) Orbit plot of output shaft

Figure 7 :
Figure 7: Experimental rig and probe positioning at Vibration and Acoustics lab of IIT Guwahati Magnetic torquer

Figure 11 :
Figure 11: Identification algorithm for estimation of geared-rotor parameters.

10 .
Validation of geared-rotor system model and the identification algorithmThis section is dedicated for validation of proposed of geared-rotor system model discussed in Section2 and for further validation of identification algorithm (IA) developed in Setion4.Using the full-spectrum responses measured through the experimental rig the estimated dynamic TE parameters are obtained and presented in Table

Figure 12 :
Figure 12: (a) Numerically generated full-spectrum responses of the input shaft by feeding estimated dynamic TE parameters of the experimental rig (b) Full-spectrum responses of the input shaft from the experimental rig 11.ConclusionsA novel asymmetric TE based geared-rotor system model has been developed.The equations of motion of the system model have been obtained using the Lagrangian dynamics.The response generated using time domain numerical solution is transformed into frequency domain to obtain in full-spectrum form.It depicts variable amplitudes in the forward and backward whirl frequencies.A novel three-step identification algorithm has been developed to quantitatively identify the ten gear mesh parameters.The algorithm is initially tested against numerically simulated fullspectrum response.This identification summary shows stable solution even when the Gaussian measurement noise considered in the numerical identification.Also, it is observed that the estimates improve on considering responses for more speeds.The geared-rotor experimental rig is designed and fabricated to authenticate the system model and identification algorithm.The full-spectrum are generated using experimentally measured responses using displacement probes.Using the same three-step identification algorithm, the gearedrotor experimental rig parameters are identified.Finally, in the validation part, the full-spectrum response has been plotted using experimentally estimated parameters with numerical model and by comparing this full-spectrum with that of original experimental rig full-spectrum plot.Excellent correlation between these two fullspectrum plots is observed at input shaft.Overall, the objective of getting reasonably good estimates of all dynamic parameters of a geared-rotors are achieved both using numerical and using experimental data.The attempt of identifying gear mesh dynamic parameters by linear model of the geared-rotor system using test measured data will surely give new direction for solving gear real world problem.The proposed gear parameter identification algorithm facilitates researchers for identifying all the critical gear parameters using lateral vibration measurements using simple displacement probes (by simple displacement probes (it is practically easy to access gear transmission shafts with displacements probes) rather than mounting high quality encoders to measure the TE of gear mesh by phase demodulation of the pulse signals of encoders as followed in transitional approach.The identified parameters using proposed IA is very much helpful for researchers in wide range of industrial application in automotive and aerospace applications in fault prediction, design optimization and preventive maintenance to avoid catastrophic failures.
real and imaginary parts, for the output shaft, are written as fluctuating components with harmonics of gear mesh frequency.Herein, f and b subscripts signify the forward and backward whirls.From Eqn. (A-46) and (A-47), we can write Only the static force, it will give For the forward whirl force, we have 93)On substituting back Eqn.(A-57) in Eqns.(A-53) and (A-54), for the backward whirl force, we get

Figure 1 :
System model for identification problem of gear parameters Figure 2: Pictorial representation of identification algorithm step down process Figure 3: (a) Orbit plot of drive shaft; (b) Orbit plot of driven shaft Figure 4: (a) Full-spectrum bode plot of drive shaft; (b) Full-spectrum bode plot of driven shaft Figure 5:(a) Orbit response plot of input shaft with noise; (b) Orbit response plot of output shaft with noise Figure 6: (a) Full-spectrum bode plot of drive shaft when noise is introduced; (b) Full-spectrum bode plot of driven shaft when noise is introduced Figure 7: Experimental rig and probe positioning at Acoustics and Vibration lab of IIT Guwahati Figure 8: Crowned spur gear tooth Figure 9: (a) Amplitudes of full-spectrum of input shaft lateral displacements using identified variables; (b) Phase compensation of input shaft with respect to reference signal using identified parameters Figure 10: (a) Full-spectrum responses of the input shaft from the experimental rig; (b) Full-spectrum responses of the output shaft from the experimental rig Figure 11: Identification algorithm for estimation of geared-rotor parameters.

Figure 12 :
(a) Numerically generated full-spectrum responses of the input shaft by feeding estimated dynamic TE parameters of the experimental rig (b) Full-spectrum responses of the input shaft from the experimental rig List of Tables

Figure 12 :
Figure 12: (a) Numerically generated full-spectrum responses of the input shaft by feeding estimated dynamic TE parameters of the experimental rig (b) Full-spectrum responses of the input shaft from the experimental

Table 1 :
Geared-rotor data used for numerical simulation

Table 4 :
Estimated experimental rig gear mesh DTE parameters

Table 5 :
Deviation of identified parameters with repeated measurement

Table 1 :
Gear mesh vibration problem assumptions

Table 4 :
Estimated experimental rig gear mesh DTE parameters

Table 5 :
Deviation of identified parameters with repeated measurement