Nonlinear Dynamics for Gear Fault Level

Since there are backlash friction and time-varying stiffness, the characteristics of gear pair has strong nonlinearity. When a gear has faults such as pitting, spalling and tooth breakage, there are often some nonlinear characteristics as super-harmonics or sub-harmonics along with vibration signal. These characteristics will cause some difficulty for fault diagnostics, so it is necessary to study the mechanism of nonlinear vibration. Fourier expression was used to describe time-varying stiffness, a factor was used to depict gear fault level and nonlinear dynamics model for gear fault level was established. Simulation results show that, with certain friction and rotating speed, the nonlinear vibration has direct relationship with rotating speed and gear friction and has no direct relationship with fault types.


INTRODUCTION
Much work had been done since the first paper on gear vibration was presented in 1920 [1].The gear vibration has strong nonlinearity because of teeth friction, backlash and time-varying stiffness when the gear pair works normally [2][3][4][5][6].As analytic solutions of the nonlinear vibration equation cannot be obtained, Japanese researcher got the linear approximate equation by doing some linear work on the equation and verified it by experiments [7].Obviously, when a gear has faults such as pitting, spalling or a broken tooth and so on, the fault source will lead the time-varying stiffness to vary severely, and that is the most important reason leading to aggravate nonlinear vibration of gear pair.When a gear has faults, there are often some nonlinear phenomena such as super-harmonics and sub-harmonics.However, up till now, the research on mechanism of nonlinear vibration of gear fault is very few, the fault diagnostics of a gear are more concentrated on methods of signal processing, such as time domain analysis, frequency domain analysis, time-frequency joint domain analysis, etc. [8,9].In [10], the authors investigated the phenomena of cycle vibration and chaos excited by the piecewise continuous stiffness and depicted fault gear stiffness using the piecewise continuous function.However that article does not consider the influences of friction and backlash which is more.In addition, there is no shock phenomenon in the simulation result and they have not been verified by experiments.
Since the characteristics of gear pair and meshing stiffness have vast changes and the nonlinear characteristics of the gear fault signal perform more significantly when a gear has fault.It is necessary to study the nonlinearity of a fault gear, especially the steady-state response of the faulty gear's nonlinear vibration to help the gear's fault diagnostics.The nonlinear dynamics integrated model of normal gears and fault gears is established considering friction, backlash and time-varying stiffness.Fourier series is used to depict varying stiffness, a gear fault factor is induced to depict gear fault level and simulation results are also presented.Analysis on characteristics of the gear box vibration shows that simulation results are consistent with experimental results.

NONLINEAR DYNAMICS MODEL FOR A NORMAL GEAR PAIR CONSIDERING FRICTION AND BACKLASH
The nonlinear dynamics model of a gear pair is shown in Fig. (1) considering friction and backlash.
The nonlinear dynamics, according to works [2,11] with friction and backlash of the gear pair are as follows. Here, Deal with eq(3) and eq (4) with non-dimension, let, ( ) Then, from eq. ( 3)., we obtain ) ) where, = ω e r p ω p + r p  θ p ( ) Let g 1 τ ( ) equal eq.( 5).As known, Then, eq(3) is simplified at last as below, Let Substituting above equation to eq.( 4)., and the necessary simplification, we obtain, Now the non-dimensional equations of the gear pair nonlinear dynamics are obtained as below,  y τ

Time-Varying Meshing Stiffness of a Fault Gear
When a gear has faults such as pitting or broken tooth, the characteristics of meshing stiffness will change [12], in particular, when a tooth is broken, the variation of the meshing stiffness is very profound.Fig. (2) depicts timevarying stiffness of a normal gear, supposing each tooth has the same variance [6,9].
When a tooth of a gear has fault, there must be reduction of meshing stiffness of the corresponding tooth meshing.Suppose a gear has teeth number equal to Z , and there is one tooth broken, the time-varying meshing stiffness coincides with Fig. (2).Suppose the broken tooth has the same peakto-peak value as that of normal teeth.
The Fourier series of the signal F t , n = 1, 3, 5,... (10) where, The below is to obtain the parameters of the Fourier series k m t ( ) of the signal F t ( ) in Fig. (2).Let.Since, Then, the Fourier translation of Fig. ( 4) is obtained as below, Then the expression of the meshing stiffness of a gear pair with a broken tooth fault, or the Fourier transmission of Fig. (3) is equal to, The curves in Figs.(5)(6)(7) are the simulation results of eq (10) to eq (12), respectively using Matlab.The meshing period is T = 20 1024 Sec ; teeth of the gear is Z = 6 ; mean of meshing stiffness is k av = 20 ; peak-to-peak of meshing stiffness is k a = 0.5 ;with the maximum Fourier order of

Nonlinear Dynamics Integrated Model
Since there will be varying when a gear has faults such as pitting and spalling and so on, and the variation of the meshing stiffness is less than that of tooth broken fault, transmit eq.( 12).as below, where, [ ]is gear fault factor, when a gear is in normal status, α = 0 ; when a gear has serious fault of tooth broken, α = 1 ; otherwise when a gear has light faults such as crack, pitting, and spalling 0 < α < 1 .
Dividing eq (13) by ω e 2 m e , then simplify it into its non- dimensional form, we have, Therefore, from eq(1) and eq (2) the nonlinear dynamics integrated model of a normal gear and a fault gear, considering backlash and friction, is satisfied, The Non-dimensional form is equal to,  y τ

4． SOLUTION
Eq. ( 18) is a second order differential equation which can be solved by 4-5 order Runge-Kutta method.In order to supply theoretical reference of gear faults diagnosis, the steady-state response of a normal gear and a fault gear vibration is focused as well on the frequency domain characteristics.For simplicity the teeth of the driving gear is z p = 6 , and the other parameters are as shown in Table 1.
Parts of them are according to reference [2].
Given µ , ω and α , the other parameters can be solved using parameters in Table 1.Non-dimensional frequency, (1) Let µ = 0.05 , ω = 0.3 , then, non-dimensional frequency is f nde = 0.048 ; non-dimensional period is T nde = 20.9 ; non-dimensional rotational period is T nd = 125 ; non-dimensional rotational frequency is f nd = 0.008 .Steady-state response vibrations are shown in Figs.(8)(9)(10).with the frequency resolution of Δf = 0.0019 .α = 0 , this means the gear is in normal status.And Fig. (8) depicts the steady state vibration response of a normal gear.α = 0.2 , this means the gear has some slight faults such as initial crack, pitting or spalling.Fig. (2).depicts the steady-state vibration response of a gear with fault of initial crack, pitting or spalling.α = 1 , this means the gear has serious faults as tooth broken.Fig. (10) depicts the steady-state vibration response of a gear with fault of broken tooth.
From the time domain signal of Fig. (10)., it can be seen that the shock phenomenon is more and more clear along with the increase of the fault factor α , with shock period about 125.Moreover, in the frequency domain signal of Fig. (10), the sidebands of 1x, 2x and high frequency multiplication of meshing frequency with an interval about 0.008 are more and more clear.
(2) µ = 0 ,ω = 0.7 , with non-dimensional meshing frequency α = 0 , means the gear is in normal status.Fig. (11)  depicts the steady state vibration response of a normal gear.α = 1 , means the gear has serious faults as broken tooth.Fig. (12) depicts the steady-state vibration response of a gear with fault of tooth broken.
From Fig. (11), it can be seen that the normal gear vibration signal also has sidebands, however, when α = 0 , there is no modulation frequency in the dynamics eq(18), so the sidebands should be superharmonics.In Fig. (11), the signal is about a gear with serious fault such as broken tooth, it is obvious that there are sidebands at the harmonics of meshing frequency with interval of rotational frequencies.Then we conclude that when a gear is in nonlinear vibration, the super-harmonics are related with meshing frequency ω and have nothing to do with gear faults.From Fig. (11), it reveals that the amplitudes of super-harmonics are much lower than that of sidebands of meshing frequency with gear faults.

CONCLUSION
The nonlinear dynamics integrated model of normal gears and fault gears is established considering friction, backlash and time-varying meshing stiffness.Fourier series is used to depict varying stiffness that is more suitable to solve the dynamics equation.The gear fault factor provides convenience for the integrated model to depict normal gear vibration and fault gear vibration.Simulations show that,

CONFLICT OF INTEREST
We declare that this article content has no conflict of interest with other people or organizations that can inappropriately influence our work.

T
Fig. (2).Meshing stiffness curve of a normal gear pair. b

Fig. ( 4 )
Fig. (4) depicts the peak-to-peak of the meshing stiffness of a broken tooth meshing.It can be seen that the Fig. (4) is the difference of Figs.(2, 3).

n max = 10 .
From Fig.(7), it can be seen that the curve of Fig.(7) can express the time-varying stiffness of a fault gear in Fig.(3).approximately.

=
Amplitude of static error ω h = Meshing frequency m e = Equivalent inertial mass of gear pair
). Time-varying meshing stiffness of a gear with fault of a broken tooth.