Nonlinear Vibration Study on the Gear System based on the Non-dimensional Differential Equation of Dynamic Model 1

Corresponding Author: Fan-Ping Qing, Automotive Engineering College, Shanghai University of Engineering Science, Shanghai, Tel.: 13524976219 This work is licensed under a Creative Commons Attribution 4.0 International License (URL: http://creativecommons.org/licenses/by/4.0/). 1125 Research Article Nonlinear Vibration Study on the Gear System based on the Non-dimensional Differential Equation of Dynamic Model


INTRODUCTION
The analysis of tooth mesh stiffness is based on the classic material mechanics early, which includes Equivalent toothed law, Cantilever method (Li and Jun, 1997).And then the elastic mechanics method is used to study the deformation of gear meshing, but this kind of method are very different compared with actual working condition in shape, load, boundary conditions, etc.With the development of computer technology, people begin using the finite element method to calculate the elastic deformation of the gear teeth and tooth root stress (Wang and Liu, 2003).However, the contact force (engaging force) is a distributed force rather than concentrated force.To solve this problem, the introduction of a numerical contact method is chosen.
There are two kinds of nonlinear dynamics model of gear system which are rigid impact model (Shaw, 1985) and elastic impact model.The first model hypothesizes that impact object is rigid and uses a compensation coefficient to describe the energy loss (Smith and Liu, 1992).Although this method can not be directly used for the analysis of the gear system, but some of the viewpoints and conclusions for the gear system is of great value.The elastic impact model (Dubowsky and Freudenstein, 1971;Veluswami and Crossley, 1975) can reveal some important characteristics of nonlinear clearance vibration.Two kinds of damping are considered in the model: viscous damping and impact damping (Dubowsky and Freudenstein, 1971).The analysis is based on the elastic impact model in study.

THE KINETICS MODEL OF GEAR
Nonlinear dynamic model: Assuming that the system is composed with the only having elastic spring and the only having inertial quality block, this study uses the lumped mass method (Li and Jun, 1997;Tamminana et al., 2006) to build the nonlinear dynamic model, ignoring the elastic deformation of the transmission shaft and support system and considering about the nonlinear factors of the time-varying stiffness k(t), mesh error e(t), gear back lash 2b.The model is showed as Fig. 1.
According to the Newton Law and the nonlinear dynamic model of the gear, the motion differential Eq. ( 1) is established as follow: where, t = Time The dynamic transmission error of gear: Deduce by the Eq. ( 1) and the Eq. ( 2): where, The defined transmission error is the difference between dynamic transmission error and the static transmission error: So the Eq. ( 3) can be transformed into as follows: where, Equation parameters: • Gear mesh stiffness ) (t k : Using the finite element contact method, the mesh stiffness curve is obtained as shown in Fig. 2. Time-varying stiffness changes periodically, so the gear mesh stiffness can be developed into Eq.( 7) in form of the Fourier series (Wang and Howard, 2005 = Amplitude of the j-th harmonic • General teeth errors ) (t e : In the gear transmission process, the mesh error takes the gear meshing frequency e ω (Munro, 1992; Raclot and   Velex, 1999) as the basic frequency.Assuming all the engaging position occurring on the theoretical meshing line, the mesh error is converted into the form of the Fourier series based on the gear mesh frequency, as displayed in Eq. ( 8): Every harmonic component amplitude of error is defined as the Munro experimental values (Munro, 1992) in analysis the affection meshing error on gear vibration, which are shown in Table 1 and Fig. 3.
• Meshing damping: The equation (Li and Jun, 1997) of meshing damping is as follows: where, ς is the damping ratio • Nonlinear function of gear back lash: The backlash (Wang and Liu, 2003;Theodossiades and Natsiavas, 2000) is circumferential wobbles of one gear calculated on the pitch circle when another tooth is fixed for the assembled gear pair.Backlash referred in this study is the measured value in the engagement line.Defining the gear backlash as 2b, the nonlinear function of the gear backlash is obtained Eq. ( 10) assuming the backlash is symmetric:

DIMENSIONLESS ON THE DIFFERENTIAL EQUATION OF GEAR SYSTEM
In the study, the differential equation of gear is converted into non-dimensional form (Tamminana et al., 2006;Raclot and Velex, 1999;Theodossiades and Natsiavas, 2000).This dimensionless equation doesn't depend on the physical quantity any longer, so that it can avoid excessive difference of magnitude between parameters in the numerical analysis and it can offer facilities for controlling error and defining step.
From the Eq. ( 6), the natural frequency ω 0 of SDOF gear system: q& & can be deduced to Eq. ( 11) and Eq. ( 12): Taking the meshing stiffness Eq. ( 7) and transmission error Eq. ( 8) into the Eq. ( 6), the analysis model of dimensionless is obtained as follows: ( ) is the dimensionless nonlinear function of clearance.

NUMERICAL CALCULATION AND RESULTS ANALYSIS
Numerical calculation: This study uses the variable step size fourth order Runge-Kutta to solve the nonlinear differential equation.First, the second order differential Eq. ( 13) is turned into two first differential equations (He et al., 2008) the equation directly by the fourth order Runge In order to obtain two first-order differential equations from Eq. ( 13), it is necessary to define a new variable So the Eq. ( 13) can be deduced into the form of state space (Eq.16):  is the dimensionless nonlinear function of

NUMERICAL CALCULATION AND RESULTS ANALYSIS
This study uses the variable Kutta to solve the nonlinear differential equation.First, the second order is turned into two first-order ., 2008) and then solve the equation directly by the fourth order Runge-Kutta.
order differential equations 13), it is necessary to define a new variable u: The static deformation solved by tooth stiffness is considered as initial values.Successive iterating won't stop until results are close to the desired solution.Numerical solution of Eq. ( 15) is finished by matlab software.In addition, considering the impact of the initial value, it is necessary to delete the response of hundreds of cycle at the beginning, so that the ideal phase diagram is gained.

Analysis of results:
In analysis, the parameters of the gear is shown in Table 2 (Al-Shyyab and Kahraman 2005; Ambarisha and Parker, 2007;2005).
• The effect of time varying mesh stiffness for gear vibration: The above convert the mesh stiffness into the form of the Fourier series the high-order harmonic has the greater impact at low frequencies because the produces resonance only in Ω mesh stiffness is converted into the form of harmonic function, as shown in ( 18), so as to study the effect of time stiffness on system vibration: Addendum coefficient Tooth width 2.5 1 85 2.5 1 85 domain chart and spectrogram of the system with σ = 0 The static deformation solved by tooth stiffness is considered as initial values.Successive iterating won't stop until results are close to the desired solution.
is finished by matlab software.In addition, considering the impact of the initial value, it is necessary to delete the response of hundreds of cycle at the beginning, so that the ideal In analysis, the parameters of the Shyyab and Kahraman, Wang and Howard,

The effect of time varying mesh stiffness for
The above convert the mesh the form of the Fourier series Eq. ( 7), order harmonic has the greater impact at low frequencies because the n-th harmonic n / 1 = Ω near.The mesh stiffness is converted into the form of as shown in Eq. ( 17) and Eq.18), so as to study the effect of time-varying Vibration responses are calculated as defining σ = 0, σ = 0.4, σ = 0.6 and the results are shown in to c, observing the vibration situation with the stiffness amplitude.
When σ is equal to zero, as shown in mesh stiffness is a constant and the time history of the system is harmonic steady response.When σ = 0.4, as shown in Fig. 4b, the mesh stiffness is quasi response.
domain chart and spectrogram of the system with σ = 0.4 pectrogram and phase plane portrait of the system with σ = 0.6 (18) The alternating component of the mesh stiffness The average component of the mesh stiffness fluctuation coefficient of the mesh stiffness which shows the fluctuation degree of mesh stiffness.It means that the greater σ is, the more intensely the mesh Vibration responses are calculated as defining σ = 0, σ = 0.4, σ = 0.6 and the results are shown in Fig. 4a c, observing the vibration situation with the stiffness When σ is equal to zero, as shown in Fig. 4a, the mesh stiffness is a constant and the time history of the system is harmonic steady response.When σ = 0.4, as 4b, the mesh stiffness is quasi-periodic When σ = 0.6, as shown in F produces a chaotic response.Evidently, with σ increasing which is the fluctuation coefficient of the time-varying mesh stiffness, the vibration experienced harmonic response response and chaotic response.In addition, we can also see that the vibration amplitude increases accordingly with the ascent of the fluctuation coefficient σ from the spectrogram Тτ.
• Gear backlash: The gear system nonlinear vibration characteristics with the gear backlash.The effect of the gear back lash on the vibration of the gear system is various.It is easy to get the influence of gear backlash on vibration cycle from the phase diagram of the response.In order to highlight the clearance of nonlinear effect, assuming the meshing stiffness is a constant.Take the frequency ratio Ω = 0.85 and the response results are shown in Fig. 5 by changing the size of Fig. 4a, the system e.Evidently, with σ increasing which is the fluctuation coefficient of the varying mesh stiffness, the vibration state experienced harmonic response, quasi-periodic chaotic response.In addition, we can also ude increases accordingly with the ascent of the fluctuation coefficient σ from the The gear system shows a strong nonlinear vibration characteristics with the gear The effect of the gear back lash on the vibration of the gear system is various.It is easy to get the influence of gear backlash on vibration cycle from the phase diagram of the response.In order to highlight the clearance of nonlinear effect, he meshing stiffness is a constant.Take the frequency ratio Ω = 0.85 and the response 5 by changing the size of As Fig. 5 shown, the gear clearance has slight influence on the aperiodic vibration in a certain range.But the system response quickly converts to chaos response from single frequency response with the increase of gap and the mesh impact is more serious.
• Meshing error: From the differential Eq. ( 13), the external load of the gear is constant and meshing error is alternating incentive.If only change the size of the amplitude of the first step error in the Eq. ( 8) to study the effect of the error on the system response which is shown in Table 3.
The Fig. 6 is frequency response curve of the gear system under the two different error amplitudes (e 1 = 3.45, e 1 = 6.5), meanwhile, the frequency ratio Ω is equal to 0.85 and other parameters remain unchanged.From the Fig. 6, it is obviously to see that the amplitude of the error excitation has significant impact on the dynamic response amplitude of the system.When the error amplitude increases, the vibration will be greatly enhanced.Thus it can be seen that, in the gear transmission process, the size of the alternating component of the internal incentive also directly affects the non-linear dynamic response in the meshing process.Under the constant average component of internal incentive, the greater alternating error motivation is, the more serious the nonlinear response of the system is and so the vibration is fiercer.
• Exciting frequency: Changing the frequency to study the affect of the frequency on the nonlinear dynamical response with the other parameters unchanged.Obviously, it has practical significance to study the vibration response in the near the resonance frequency and high-speed working state, so let the variable Ω change between 0.8 and 3.5 and other parameters remain unchanged.The results are as follows: As the Fig. 7 shown, when the frequency ratio Ω = 0.8, the Fig. 7a is a periodic solution with the cycle T = 2π/Ω, which is known as a single period attractor in the nonlinear vibration theory; when Ω = 1.15, the Fig. 7b is a neither repeated nor closed track filled with a part of the phase space and the system generates a chaotic response; When Ω = 1.85, the system is also harmonic response and its cycle is twice the original.While Ω is equal to 2.35, the system is a curve band with a certain width and it's a quasi cycle response.When Ω = 3.0, the system occurred chaotic response; when Ω = 3.5, the response of the system is the harmonic response and this shows, when the frequency is far from the natural frequency, the response of the system tends to be steady again.It's visible that the exciting frequency has a very significantly effect on the gear vibration system.

CONCLUSION
This chapter used lumper parameter approximation to establish the gear system dynamics model, consider the effection of the time-varying mesh's stiffness, error and gear back lash.Using variable space 4th-order Runge-kutta numerical method solves the gear system's nonlinear kinetics differential equations, to obtain the domain diagram, spectrogram and phase plane of the response of the system by the larger number of numerical calculation analyzed the impact of various parameters on the gear dynamic characteristics, thus to predict the dynamic performance at the design stage, make the system as much as possible to avoid the resonance frequency.

Fig. 1 :=
Fig. 1: The nonlinear dynamic model of the gear c = The meshing damping bi r = The radius of base circle of the gear i (i = 1, 2) i θ = The torsional displacement of the gear i (i = 1, 2)

=
The gear's internal excitation (the error stimulation) Fig. 2: Gear mesh stiffness curve The mean of mesh stiffness m e = Equivalent mass Defining dimensionless time t

Fig. 5 :
Fig. 5: The response of the system phase diagram with different running clearances the gear backlash with other parameters unchanged.

Fig. 7 :
Fig. 7: The phase diagram under the different frequency ratio

Table 2 :
The parameters of the gear system

Table 3 :
Error excitation Fig. 6: Frequency response curve under the two different error amplitudes