| Gear transmission systems are widely used in many fields such as automobiles,ships,aviation,etc.The vibration signal of the faulty gear has the characteristics of amplitude modulation(AM)and frequency modulation(FM).It is of important research significance and engineering practical value to use the variation law of the modulation sideband of the vibration signal to diagnose gear faults.Based on the dynamic model of the single-stage fixed-axis gear train,considering the system’s harmonic speed fluctuations,the spectral characteristics of the vibration response signal of the gear under stationary fault and impact-type fault are theoretically analyzed.A new stationary fault vibration signal amplitude modulation and frequency modulation(AM-FM)mathematical model is proposed.The proposed mathematical model has frequency modulation in both the AM part and the meshing frequency carrier part.The frequency modulation intensity of the AM part is related to the speed fluctuation rate,and the frequency modulation intensity of the carrier part is related to the product of the speed fluctuation rate and the number of teeth of the faulty gear.The characteristics of the modulation sideband of the vibration signal in the numerical simulation of the lumped parameter model and the experiment of the single-stage fixed-axis gearbox are consistent with the theoretical analysis results.Based on the AM-FM mathematical model of gear fault,a new universal accurate amplitude demodulation method for vibration signals near the meshing frequency is proposed.The Hilbert transform is employed to construct the optimization objective function.The square demodulation is applied to construct the constraint equation.Besides,the least square optimization algorithm is used to solve the amplitude modulation parameters.Then,the real AM signal of undermodulation and overmodulation signals can be reconstructed.Simulation results show that the proposed amplitude demodulation method has high precision,good anti-noise performance,and no redundant frequency components.The demodulation effect for overmodulated signals is much better than those of the Hilbert transform demodulation method and the new amplitude demodulation method based on square demodulation and least square optimization algorithm.The experimental results show that the vibration signal near the meshing frequency is undermodulated in the case of slight stationary fault,while the vibration signal is overmodulated in the case of severe stationary fault.The fault degree of the gearbox can be judged according to the difference of demodulation results.For the compound fault signal of misalignment and broken teeth,the closer the meshing frequency is to the resonance region,the more likely the vibration signal is to be overmodulated.Based on the amplitude demodulation method of gear fault vibration signal,a frequency demodulation method is proposed.The iterative Hilbert transform is used to process the residual signal after amplitude demodulation.The Bessel function identities of the first kind are used to construct the constraint equation.Besides,the least squares optimization algorithm is used to solve the FM parameters.Finally,frequency demodulation and FM signal reconstruction are realized.Simulation results show that the Hilbert transform demodulation method fails when noise is added.The new frequency demodulation method based on Bezier function and least square optimization algorithm fails when the overmodulated signal is processed.The proposed method has high precision.The experimental results show that the more serious the stationary fault is,the higher order and modulation intensity of the FM signal are.For the compound fault of misalignment and broken teeth,the closer the meshing frequency is to the resonance zone,the larger the reconstruction error of the proposed AM-FM demodulation method is.However,the impact-type fault can be diagnosed based on the residual signal,which provides a new way for gear fault diagnosis. |
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