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The translational axis is one of the most important subsystems in modern machine tools, as its degradation may result in the loss of the product qualification and lower the control precision. Condition-based maintenance (CBM) has been considered as one of the advanced maintenance schemes to achieve effective, reliable and cost-effective operation of machine systems, however, current vibration-based maintenance schemes cannot be employed directly in the translational axis system, due to its complex structure and the inefficiency of commonly used condition monitoring features. In this paper, a wavelet bicoherence-based quadratic nonlinearity feature is proposed for translational axis condition monitoring by using the torque signature of the drive servomotor. Firstly, the quadratic nonlinearity of the servomotor torque signature is discussed, and then, a biphase randomization wavelet bicoherence is introduced for its quadratic nonlinear detection. On this basis, a quadratic nonlinearity feature is proposed for condition monitoring of the translational axis. The properties of the proposed quadratic nonlinearity feature are investigated by simulations. Subsequently, this feature is applied to the real-world servomotor torque data collected from the X-axis on a high precision vertical machining centre. All the results show that the performance of the proposed feature is much better than that of original condition monitoring features.

The translational axis is one of the most important subsystems in modern machine tools, which has been widely used for transforming rotational motion into linear motion, due to its advantages of low sensitivity to inertia variations, good positional accuracy and high driving speeds [

As shown in _{u}_{v}_{w}_{p}_{q}_{d}

During the cutting process, the measured disturbance torque is composed of frictional torque and the torque caused by the cutting force (as shown in _{q}_{T}_{q}_{T}_{f}_{c}

When the table is fed steadily without cutting, the measured disturbance torque only represents the frictional torque of guideways, ball screw and other rotating components (such as support bearing and reducer). Then, _{c}

During the assembly of machine tools, a preload must be exerted to minimize the clearance of the ball screw and support bearing to improve the dynamical operational behavior [

The degradation of the part in the translational axis system will lead to an additional friction. Suppose that the additional friction

To avoid the additional friction, an additional torque should be provided by the servomotor. Substituting

In the frequency domain, the continuous wavelet transform (CWT) spectrum of _{ψ}

As shown in _{1} and 2 _{2} are denoted as the harmonics, and those at _{2} − _{1} and _{2}+_{1} are the nonlinear interaction components. In addition, the phase of the signals caused by the nonlinearity satisfies the relationship, _{1} + _{2} = _{3}, where _{1} and _{2} are the phase of the two coupling components, _{3} is the phase of the new interaction component. This is the quadratic phase coupling (QPC) due to system nonlinear behavior, which provides a way to distinguish signals at a particular frequency. Because only the components contributed to the QPC are of interest, all other terms are neglected, and

The simplified QPC model is employed widely for condition monitoring [

The WB is firstly introduced by van Milligen [

In an analogous definition to the usual Fourier bispectrum, the wavelet bispectrum is defined as follows [_{1}_{2}_{3}_{3}_{1}_{2}_{1}_{2}_{1}_{2}_{1}_{2}_{1}, _{2}) = _{1}(_{1}) + _{2}(_{2}) − _{3}(_{3}). Here, _{1}(_{1}), _{2}(_{2}) and _{3}(_{3}) are phases uniformly distributed within (−

Conventional wavelet spectrum based WB is incapable of eliminating the spurious peaks coming from long coherence time waves and non-QPC waves [^{iRφ}^{(}^{f}^{1,}^{f}^{2)} is the biphase randomization term, _{1}_{2}

After normalization, the biphase randomization wavelet bicoherence (BRWB) is given as [_{1}_{2}

The summed feature of BRWB (SBRWB feature) is used commonly to measure the distribution of phase coupling, which is defined as [

At the same time, a BRWB-based quadratic nonlinearity feature, namely the maximum eigenvalue of the BRWB (MEBRWB feature) [_{k}_{k}_{k}

The estimation of the BRWB, the SBRWB and MEBRWB features is detailed in the following steps as illustrated in

Step 1: Divide the servomotor torque signal into

Generally, in computing of BRWB, the signal with long data record length is firstly divided into a series of epochs, and then, the reliable bicoherence estimation of the signal is obtained by averaging the BRWB of all epochs. In this study, the BRWB of torque signal is calculated based on smallest meaningful signal segment, due to its non-stationary property. Because that the signal segment of 1/_{PFL}_{PFL}_{PFL}

Step 2: Compute the wavelet transform _{ψ}

For the torque signals, impulse components often correspond to the degradation of the translational axis system, thus the basic wavelet used for feature extraction should be similar to an impulse [_{0} to 6 to satisfy the admissibility condition, and takes

Step 3: Calculate the biphase _{1}, _{2}) of each epoch by using

Step 4: Generate a random variable _{1}, _{2}) and _{1}, _{2}).

In practice, signals often have very long coherence time, which makes the biphase component dependent over each epoch. The biphase component dependent over each epoch may cause spurious bicoherence. In this step, a biphase randomization term ^{iRφ}^{(}^{f}^{1,}^{f}^{2)} is generated to damage the phase component dependent condition of each epoch by using the new biphase _{1}, _{2}). At the same time, the biphase randomization term is also used to eliminate the spurious bicoherence coming from non-QPC waves.

Step 5: Calculate the wavelet bispectrum by

In this step, the biphase randomization term ^{iRφ}^{(}^{f}^{1,}^{f}^{2)} generated by Step 4 is introduced for the wavelet bispectrum calculation, and the result is retained as the sample for the reliable bispectrum estimation. Actually, ^{iRφ}^{(}^{f}^{1,}^{f}^{2)}.

Step 6: Estimate the biphase randomization wavelet bispectrum of each signal epoch.

In this step, the biphase randomization wavelet bispectrum of this signal epoch is estimated by ensemble average operation of the 100 data samples obtained in Step 5. The spurious bispectrum coming from non-QPC waves can be eliminated by the ensemble averaging operation. After normalization, the BRWB of this signal epoch can be obtained, and the result is retained as the samples for reliable BRWB estimation. From this step,

Step 7: Estimate the BRWB by

Since

Step 8: parameter estimation by

The SBRWB feature is calculated by

According to the QPC model introduced in _{1}_{2}_{1} and _{2} are the initial phase distributed within (−_{1}_{2}

In this signal, the global QPC energy of the simulation signal is adjusted by increasing the coupling coefficient

It can be seen from

The proposed method is applied to monitor the translational axis of a high precision vertical machining centre. Experiments are conducted on the X- axis of this machining centre in an in-service environment, and the experimental time is the whole maintenance period of the X-axis. The experiment system is shown in

As shown in

To describe the degradation progress, the proposed BRWB is used to detect the quadratic nonlinear phase coupling frequencies band of the torque signature. Here, the bandwidth from 0–500 Hz is selected for the bicoherence calculation.

To further investigate the interactions, the phase coupling distribution of each data sample is estimated by using the SBRWB feature, as shown in

For convenience, the single-value MEBRWB feature of the eight torque data samples is calculated to monitor the condition of X-axis. The MEBRWB feature is compared with other commonly used feature during the whole maintenance period of the X-axis, as shown in

Results show that the commonly used features don't perform well in describing the progress of X-axis degradation. By contrast, as shown in

A BRWB based quadratic nonlinearity feature is established for translational axis condition monitoring of the machine tools, which shows that it is possible to perform condition monitoring by using servomotor torque signature. Some conclusions are drawn as follows:

A BRWB is established to overcome the problem of current WB, which can eliminate the spurious peaks coming from long coherence time waves and non-QPC waves efficiently. Based on the proposed BRWB, a quadratic nonlinearity MEBRWB feature is proposed for translational axis condition monitoring. Numerical example results show that the global QPC of the simulation signal can be tracked approximately linearly by the proposed feature, especially when the SNR is greater than 3 dB.

The proposed quadratic nonlinearity MEBRWB feature is also used to study the torque data collected from a high precision vertical machining centre. Experimental results illustrate the robust experimental performance of the proposed feature, compared with commonly used features. The advantage of MEBRWB feature is that it can exploit the true global QPC of the signal at different frequencies, which contains useful additional information for detection of quadratic nonlinear phenomena induced by mechanical faults. Potentially, this single-valued feature can be used in prognostic models for remanding useful life estimation of mechanical components.

The work is supported by the National Key Science & Technology Specific Projects (2010ZX04014-016), which is highly appreciated by the authors.

The authors declare no conflict of interest.

Illustration of the translational axis system.

Flow chart of the proposed BRWB and its feature estimation.

Simulation signals (

The effect of global QPC on MEWB feature (

Experimental system.

Illustration of the X-axis.

Servomotor torque of X-axis at different number of the days before maintenance.

BRWB of X-axis servomotor torque data: (

Features of different number of the days before maintenance: (

Trend of torque MEBRWB feature compared with the common used features: (

The characteristic frequencies of X-Axis system.

Feed rate of X-Axis | 550 (mm/min) |

Rotary frequency of servomotor | 5.35 (Hz) |

Rotary frequency of idler shaft | 1.34 (Hz) |

Rotary frequency of output shaft | 1.03 (Hz) |

Meshing frequency of G#1 and G#2 | 32.1 (Hz) |

Meshing frequency of G#3 and G#4 | 32.1 (Hz) |

Passing frequency of ball screw lead | 0.57 (Hz) |