Tatsuhito Aihara , Kensho Sakamoto

Department of Mechanical Engineering, Faculty of Science and Engineering, Hosei University, 3-7-2 Kajinocho, Koganei-shi, Tokyo, Japan

Keywords:Nonlinear Vibration Gear Chaos Shaft Tooth separation

ABSTRACT The circumferential vibration of a gear pair is a parametric excitation caused by nonlinear tooth stiffness, which fluctuates with meshing.In addition, the vibration characteristics of the gear pair become com- plicated owing to the tooth profile error and backlash.It is considered that the circumferential vibration of the gear pair is affected by the torsional vibration of the shafts.It is important to understand quan- titatively the vibration characteristics of the gear system considering the shafts.Therefore, the purpose of this research was to clarify the nonlinear vibration characteristics of a gear pair considering the influ- ence of the shafts using theoretical methods.To achieve this objective, calculations were performed using equations of motion in which the circumferential vibration of the gear pair and the torsional vibration of the shafts were coupled.The nonlinear tooth stiffness was represented by a sine wave.The influence of tooth separation was considered by defining a nonlinear function using backlash and the tooth profile error.For the numerical calculations, both stable and unstable periodic solutions were obtained by using the shooting method.The effect of the shafts on the gear system vibration were clarified by comparing the results in the cases in which the shaft was not considered, one shaft was considered, and both shafts were considered.

Gears are used as power transmission mechanisms in many machines, but vibration and noise may be problematic [1] .In re- cent years, gears have been increasingly used in high-speed rota- tion ranges in high-speed machines such as electric vehicles [2-4] .The circumferential vibration of a gear system is a paramet- ric excitation caused by the nonlinear tooth pair stiffness, which changes with meshing [5] .In a gear pair rotating at high speed, tooth flank separation and gear rattle vibration occur due to tooth profile errors and backlash [ 1 , 6-8 ].Since tooth profile errors and backlash causes non-linear vibration, many studies on this non- linearity have been conducted experimentally [ 6 , 8-11 ] and theo- retically [ 1 , 5-8 , 11-13 ].Quantitative understanding of the complex vibration characteristics of gear systems is important for reducing vibration and increasing the safety and machine lives of gear de- vices.

Gear systems with backlash have been modeled and analyzed as piecewise linear systems [14] .Since the piecewise linear sys- tem is a strong nonlinear system, subharmonic and higher har- monic resonance occur.It has been also clarified from experiments and theoretical analysis that subharmonic resonance occurs in the gear system [14-16] .Celikay et al.[15] analyzed dynamic behav- ior observed in spur gear pairs as a parametrically excited sys- tem and studied an extensive experimental investigation to show such subharmonic resonances.Eritenel and Parker [16] investigates the three-dimensional nonlinear vibration of gear pairs where the nonlinearity is due to portions of gear teeth contact lines los- ing contact.And both primary and subharmonic resonances of the twist mode are excited is clarified.In addition, research on the chaotic vibrations generated in gear pairs has been published [ 14 , 17-21 ].Kahraman [14] conducted analytical study of the non- linear dynamics of a spur gear pair with backlash as excited by the static transmission error by the harmonic balance method and the chaotic and subharmonic resonances are observed if the mean load is too small for a lightly damped system.Raghothama and Narayanan [18] analyzed the periodic motions of a non-linear geared rotor-bearing system are investigated by the incremental harmonic balance method and the chaotic motions are investi- gated.However, these studies on chaotic vibrations in gear systems have not considered the torsional stiffness of the shafts, and the occurrence of subharmonic and chaotic vibrations in a gear system with shafts has not been clarified.It is presumed that the influence of the torsional stiffness of the shafts has a significant influence on the generated chaotic vibration.

Therefore, the purpose of this study was to clarify the vibra tion characteristics of a gear system with a pair of helical gears,considering the shafts.In order to consider the influence of the shafts, we performed analysis using equations of motion in which the torsional vibration of the shafts was coupled with the circum ferential vibration of the gear pair.Moreover, the differences in the vibration characteristics between cases in which neither shaft was considered and one shaft was considered were evaluated by com paring the corresponding calculation results.Furthermore,we clar ified how the shaft stiffness affects the resonance curve when both shafts are considered.

Figure 1 shows the analytical model.In this study, in order to evaluate the influence of the shaft quantitatively, the analysis was performed by using the motion equations corresponding to cases in which neither shaft, one shaft, and both shafts were considered.The origin of the system is the state whereθandψis 0.

Neither-shaft case

The dimensionless equations of motion without considering ei ther shaft can be expressed as the following simultaneous differ ential equations:

In addition, the following nonlinear function can be defined using the dimensionless tooth profile errore(τ)and dimensionless backlashη:

The definitions of the symbols used in these equations are as follows:

kav: Average nonlinear meshing tooth stiffness;

C: Damping coefficient of the gear pair;

f(x1): Nonlinear function with backlash;

ε: Half of the backlash;

ψ1,2: Rotation angles of the gears;

TD: Driving torque;

TDav: Half of the driving torque;

ω: Ratio of the meshing angular frequency to the natural angu- lar vibration;

i: Tooth ratio.

One-shaft case

In the gear system shown in Fig.1, assuming that one of the shafts, here the drive-side shaft, is sufficiently soft with respect to the meshing tooth stiffness, the equations of motion can be expressed as the following simultaneous differential equations:

Fig. 1. Basic configuration of gear system analysis model with shafts.θ D.Lis the shaft rotation angle; ψ 1 and ψ2 are the gear rotation angles; I 1 and I2 are the moments of inertia of the gears; I Dand I Lare the moments of inertia of the drive and driven sides, respectively; Ks 1 and Ks 2 are the shaft stiffnesses; and C s 1 and Cs 2 are the damping coefficients of the shafts.

Fig. 2. Resonance curves of harmonic and 1/2-order subharmonic vibrations when neither shaft was considered.The vertical axis shows the dimensionless displace- ment amplitude, and the horizontal axis represents the ratio of the meshing an- gular frequency to the natural angular frequency.The black and red lines indicate stable and unstable solutions, respectively.

Fig. 3. Bifurcation diagrams obtained when neither shaft was considered.The horizontal axis shows the frequency ratio, and the vertical axis indicates the dimensionless displacementx1 at the moment at which the phase of tooth stiffness fluctuation is zero.(a) and (b) correspond to the cases of sweeping up and down, respectively.

Fig. 4. Poincare map obtained when neither shaft was considered, withω= 1.3 .ρis the Lyapunov exponent.The vertical and horizontal axes represent the dimen- sionless velocity and displacement, respectively, when the phase of tooth stiffness fluctuation is zero.

Fig. 5. Resonance curves of harmonic and 1/2-order subharmonic vibrations in the case in which one shaft was considered.The vertical axis represents the dimension- less displacement amplitude, and the horizontal axis shows the ratio of the meshing angular frequency to the natural angular frequency.The black and red lines indicate stable and unstable solutions, respectively.

The definitions of the symbols used in these equations are as follows:

Ks2: Driven shaft stiffness;

Cs2: Damping coefficient of the driven shaft;

TD: Drive torque;

TL: Load torque;

JL: Moment of inertia on the load side;

I1: Moment of inertia of the drive gear;

I2: Moment of inertia of the driven gear;

Θ2: Dimensionless relative angular displacement between the driven and driven shafts.

Both-shafts case

The equations of motion when considering both the drive and driven shafts can expressed as the following simultaneous differential equations:

The definitions of the symbols used in these equations are as follows:

Cs1: Damping coefficient of the drive shaft;

JD: Moment of inertia on the drive side;

Θ1: Dimensionless relative angular displacement between driven gear and drive shaft;

Ks2: Drive shaft stiffness.

The meshing tooth stiffness and tooth profile error can be de- fined as follows in all cases:

where the symbol definitions are as follows:

ka:Amplitude of the dimensionless meshing tooth stiffness;

ea: Amplitude of the dimensionless tooth profile error;

In this study, a shooting method was used that could efficiently produce a periodic solution by combining numerical integration and the Newton method when creating a resonance curve.

The equations of motion can be expressed by the following si- multaneous differential equations.

where T represents transposition.The variational equation of Eq.(16) is as follows:

The periodic solution condition is as follows:

Considering thatx(T) is determined based on the initial valuex(0) in Eq.(18) ,x(T) can be regarded as a function ofx(0), so the Newton method was applied to Eq.(18) .The following formula was obtained:

Here,Bis the fundamental matrix solution, which can be pro- duced by setting the solutions acquired by integrating the varia- tional equations for one period with the columns of thenth unit matrixEnas the initial values in the column direction, andis the corrected initial value.By solving Eq.(19) and usingx(0)+as the new initial value, the above calculation was repeated until |x(T)?x(0)| became very small and a periodic solution could be obtained.

In the shooting method, not only stable periodic solutions, but also unstable periodic solutions can be obtained in the same way, so it is necessary to determine the stability of the resulting solu- tion.Stability analysis can be performed by determining the eigen- value (characteristic multiplier) of the fundamental matrix solu- tionBof the variation after one period obtained as described in previous paragraph.The solution is considered to be asymptoti- cally stable if allneigenvalues are less than 1 and unstable if any of the eigenvalues is greater than 1.The Lyapunov exponent is used to identify chaos.There is a method [22] in which the vari- ational equations are integrated and Schmitt orthogonalization is performed at an appropriate time to calculate the Lyapunov expo- nent.However, in this study, we applied the direct numerical in- tegration method shown above, which enabled us to calculate the Lyapunov exponent.

A nonlinear function such as Eq.(3) can be generally expressed as

whereu(?) is the unit step function.In Eq.(20) , the discontinuous functionfis represented by the sum of the continuous functionf0and the discontinuous component.fhas a discontinuity ofrjat the pointgj(x,t) =cj= (constant).The variation process when such a function is included is as follows.If the time at which the discontinuous function reachescjist0 and the time immediately before it ist0, the variation including the discontinuous function is as follows:

Becauset0must be obtained with high precision, the numer- ical integration in the actual calculation was performed with the usual step size until the discontinuity was reached and the restor- ing force switching point (discontinuity) was exceeded.The step size was made small from immediately before to immediately af- ter the switching point, and the calculation was performed again.

In this study, the fourth-order Runge-Kutta method was em- ployed for numerical integration, using 1024 equal intervals as steps.At the switching point of the restoring force, the step was multiplied times 1/10, and the solution was obtained with high accuracy.The convergence condition of the shooting method was |x(T)?x(0)|＜10?8.

Neither-shaft case

The calculation was performed using the calculation method shown in Eqs.(1) and (2) .In the calculation, the torque transmit- ted by the gear pair was constant (T= 1), and the following values were utilized:ζ= 0.1,η= 7,ka= 0.1,ea= 1,= 1 , andi= 1 .

Figure 2 shows the main and 1/2 subharmonic resonance curves obtained from the calculations performed under the above condi- tions.The vertical axis represents the dimensionless displacement amplitudeA, and the horizontal axis indicates the ratioωbetween the meshing angular frequencyωgand natural angular frequencyωn.The dimensionless displacement amplitude was calculated asA= (x1max-x1min)/2.The black and red lines in the figure represent stable and unstable solutions, respectively.As shown in Fig.2 , the resonance curve is inclined to the left, indicating softened spring characteristics.This feature appears when tooth flank separation occurs, because the stiffness of the entire system decreases due to tooth flank separation.Unstable solutions appear in the main reso- nance in the rangeω1.06 ?1.415 .Such unstable solutions are not generally seen in a coefficient excitation system in which the am- plitude increases only owing to the tooth meshing stiffness vari- ation, and is considered to be unstable owing to the influence of the tooth profile error.Subharmonic resonance due to the period- doubling bifurcation occurs near this unstable solution.The max- imum amplitude of the 1/2 subharmonic resonance exceeds that of the main resonance by approximately 1.43 times.In the rangeω1.106 ?1.38 , an unstable solution appears in the 1/2 subhar- monic resonance.

Fig. 6. Bifurcation diagrams obtained when one shaft was considered.The horizontal axis shows the frequency ratio, and the vertical axis represents the dimensionless displacementx 1at the moment at which the phase of tooth stiffness fluctuation is zero.(a) and (b) correspond to the cases of sweeping up and down, respectively.

Fig. 7. Poincare map obtained when one shaft was considered, withω= 1.32 .ρis the Lyapunov exponent.The vertical and horizontal axes represent the dimension- less and dimensionless displacement, respectively, when the phase of tooth stiffness fluctuation is zero.

Fig. 8. Resonance curves of harmonic and 1/2nd order subharmonic vibrations when both shafts were considered.The vertical axis represents the dimensionless displacement amplitude, and the horizontal axis shows the ratio of the meshing an- gular frequency to the natural angular frequency.The black and red lines indicate stable and unstable solutions, respectively.

Fig. 9. Bifurcation diagrams obtained when both shafts were considered.The horizontal axis represents the frequency ratio, and the vertical axis shows the dimensionless displacementx 1 at the moment at which the phase of the tooth stiffness fluctuation is zero.(a) and (b) correspond to the cases of sweeping up and down, respectively.

Fig. 10. Resonance curve showing the effect of the shaft stiffnessk s2 .The solid, dotted, and dotted-dashed lines correspond to the cases ofk s2 = 0.1 , 0.25, and 0.5, respectively.The black and red lines indicate stable and unstable solutions, respec- tively.

Figure 3 presents the bifurcation diagram obtained when the frequency ratio was changed.The horizontal axis represents the changed frequency ratio, and the vertical axis indicates the value of the dimensionless displacementx1when the phase of tooth mesh- ing stiffness fluctuation is zero.Figure 3 a and b depict the cases of sweeping from the low- and high-frequency sides, respectively.In the bifurcation diagram, 1/4 subharmonic resonance appears near the frequency ratio at which unstable solutions appear in the main and 1/2 subharmonic resonances, and period-doubling bifurcation can be confirmed.In addition, chaotic behavior is observed after the period-doubling bifurcation.

Figure 4 depicts the Poincaré map obtained withω= 1.3 .The vertical and horizontal axes represent the dimensionless velocity and displacement, respectively, at the instant at which the phase of tooth meshing stiffness fluctuation is zero.An attractor pecu- liar to chaotic oscillation appears in the Poincaré map.Because the maximum Lyapunov exponent at this time isρ= 0.0227, it can be seen that the oscillation is chaotic.

One-shaft case

Next, we performed analysis using Eqs.(4) -(7) for the case in which one shaft was considered.The analysis conditions were con- stant torque (T1=T2= 1),ks2= 0.1,ζs2 = 0.1, andIL= 20.The other parameters were the same as those in “neither-shaft case”.

Figure 5 shows the resonance curve obtained in the same way as that presented in Section 4.1.The amplitude is generally smaller than when neither shaft was considered, which is understood to be because the damping of the whole system became large due to the shaft damping.The maximum amplitude of the 1/2 sub- harmonic resonance is larger than that of the main resonance by approximately 1.28 times, as in the case without considering ei- ther shaft.Atω1.113 ?1.413 , an unstable solution appears in the main resonance, where the 1/2 subharmonic resonance occurs.In the case of 1/2 subharmonic resonance, an unstable solution ap- pears atω1.261 ?1.376 .These features are very similar to those in the case in which neither shaft was considered, but the unsta- ble solutions in the 1/2 subharmonic resonance are reduced.

Figure 6 presents the bifurcation diagrams obtained when the frequency ratio was changed.Figure 6 a and b correspond to the cases of sweeping from the low- and high-frequency sides, respec- tively.As in the case in which neither shaft was considered, 1/4 subharmonic resonance appears near the frequency ratio at which unstable solutions appear in the main and 1/2 subharmonic reso- nances and period-doubling bifurcation is observed.It can also be seen that chaotic oscillation occurs after the period-doubling bifur- cation.

Figure 7 shows the Poincaré map obtained withω= 1.32.The attractors are distributed in a narrow range, unlike when neither shaft was considered.The maximum Lyapunov exponent this time isρ= 0.0504, which can be determined to indicate chaotic oscil- lation.Comparing the case in which neither shaft was considered with that in which one shaft was considered, the resonance curves appear to have similar tendencies in both the main and 1/2 sub- harmonic resonances.Comparison of the bifurcation diagram and Poincaré map when chaos oscillation occurs reveals that the char- acteristics are different.When neither shaft is considered, the at- tractor spreads over a wide range.However, when one shaft is con- sidered, the attractors are distributed in a narrow range.

Both-shafts case

Next, we performed analysis using Eqs.(8) -(13) for the case in which both shafts were considered.The parameters wereks1 = 0.1,ζs1 = 0.1, andID= 20 in the analysis, and the other parameters were as described above.

Figure 8 depicts the resonance curve obtained in the same man- ner as before.As in the case in which one shaft was considered, the maximum amplitude of the resonance curve is smaller than that in the case in which neither shaft was considered because the damping in the entire system is large.The maximum amplitude of the 1/2 subharmonic resonance exceeds that of the main resonance even when both shafts are considered and is approximately 1.191 times that of the main resonance.The maximum amplitudes of the main and 1/2 subharmonic resonances are approximately 0.593 and 0.492 times those in the case in which neither shaft was con- sidered.In addition, the small peak atω.=·0.263 is caused by the changes in the eigenvalues of the system due to the considera- tion of both shafts.These results demonstrate that the existence of shafts affects the resonance curve and that it is necessary to con- sider the shafts.Even when both shafts are considered, unstable solutions appear in the main resonance curve atω1.161 ?1.407 , where 1/2 subharmonic resonance occurs.However, the unstable solution that appeared in 1/2 subharmonic resonance curve in the other two cases is no longer present.

Figure 9 presents the bifurcation diagrams obtained when both shafts were considered.It can be seen that 1/2 subharmonic res- onance occurs where the unstable solution appears in the main resonance curve.However, there are no more period-doubling branches, and chaotic oscillations cannot be confirmed.

Comparison of the calculation results obtained using the above three models revealed that the amplitude of the resonance curve decreases owing to the increase in the damping of the entire sys- tem caused by the existence of the shaft.Moreover, chaotic vibra- tion did not occur with the parameter values used in this study, even if the calculations were performed using the same tooth pro- file error, backlash, and nonlinear tooth meshing stiffness when considering both axes.

Next, we examined how the shaft stiffness affects the reso- nance curve.Specifically, we investigated the changes in the res- onance curve when one of the stiffness values of the shaft,ks2 , was changed to 0.1, 0.25, and 0.5.Figure 10 presents the resulting resonance curve.The solid, dotted, and dashed-dotted lines cor- respond to the cases ofks2= 0.1, 0.25, and 0.5, respectively.The black and red lines indicate stable and unstable solutions, respec- tively.It can be observed that the maximum amplitude tends to decrease as the shaft stiffness increases.With the parameters used here, 1/2 subharmonic resonance is evident in all cases, but the range in which unstable solutions appear in the main resonance curve becomes smaller as the shaft stiffness increases.When both shafts were considered, chaotic vibration could not be confirmed when the shaft stiffness was 0.1-0.5 times the average tooth mesh- ing stiffness.

Numerical calculations of the circumferential vibration of a gear pair were performed for cases in which neither shaft, one shaft, and both shafts were considered, and the results were compared.Furthermore, the effect of the shaft stiffness on the resonance curve shape was investigated when both shafts were considered.The main findings are as follows:

All three models yielded 1/2 subharmonic resonance, and 1/4 subharmonic resonance was also observed in the bifurcation dia- grams when neither shaft or one shaft was considered.

Chaotic oscillation occurred when neither shaft or one shaft was considered.The resonance curves in these cases exhibited sim- ilar shapes, but the characteristics of the chaotic oscillations such as the Poincare maps and bifurcation diagram shapes were differ- ent.Further, when both shafts were considered with the same pa- rameters, chaotic vibration did not occur when the shaft stiffness was 0.1-0.5 times the average tooth meshing stiffness.

When both shafts were considered, the maximum amplitude of the resonance curve tended to decrease as the stiffness of the driven side shaft increased, and 1/2 subharmonic resonance oc- curred in each case.

Declaration of competing interest

The authors declare that they have no known competing finan- cial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.