Discontinuity Detection in the Vibration Signal of Turning Machines

Abstract

The chapter deals with the detection of discontinuities in the vibration signal created by a turning machine in soft and hard processes. This chapter presents an experiment with two embedded piezo-electrical sensors on the turning machine tool tip. AISI 4140 steel workpieces are used in the experiments to obtain the analyzed signal. The purpose of the experiment is to identify the nature and position of rapid changes as a measure of compliance with surface roughness. A new algorithm for wavelet selection is developed and the procedure proved the Daubechies wavelet of the 6th order to be the best choice. The proposed algorithm is based on a new technique of energy matching for wavelets. Due to the algorithm’s efficiency, it is well suited for real time monitoring. Furthermore, it is possible to build a real time monitoring system based on the digital signal system.

Appendices

Appendix 1: General Mathematical Conditions for the Selection of the Wavelet

There are four conditions, which have to be fulfilled by the scaling function, φ(t), in order to be considered as the solution:

• 1. ${\text{for}}\;{\text{t}} \in {\text{R:}}\;\left\| {\phi (t)} \right\| = 1$

• 2. $\sum\limits_{k} {\left| {\hat{\phi }\left( {\omega + 2k\pi } \right)} \right|^{2} } = 1$

• 3. $\mathop {\lim }\limits_{j \to \infty } \hat{\phi }\left( {2^{ - j} \omega } \right) = 1$

• 4. $\hat{\phi }(2\omega ) = \hat{H}(\omega ) \cdot \hat{\phi }(\omega )$

where:

• $\hat{\phi }$ is the Fourier transform of the function φ,

• $\hat{H}$ is the frequency response of the low-pass filter in form:

$$ \hat{H}(\omega ) = \sum\limits_{k} {h(k) \cdot e^{ - i\omega k} } . $$

(13)

Furthermore, φ(t) is defined for the orthonormal case in L ₂ (R) vector space.

The first condition is the general condition of the orthogonality. The second condition is the Poisson’s equation, which is equal to orthogonality property for the shifted scaling function, φ(t − k), where k ∈ Z. The third condition is the un-interruption property (the function is continuous) in ω = 0. The fourth condition is the so called dilatation equation.

It should be notice that the low-pass filter, H, and the high-pass filter, G, should be in the following relationship for all frequencies:

$$ \hat{G}(\omega ) = e^{i\omega } \cdot \bar{\hat{H}}(\omega + \pi ) . $$

(14)

Functions $\hat{G}$ and $\hat{H}$ are the periodic functions with period 2π.

Appendix 2: Conditions for the Amplitude and Phase of the Chosen Wavelet’s Spectrum

By the multiresolution theory, the spectrum of the frequency limited scaling function has the compact domain in the interval $\left\langle { - \omega_{\text{m}} ,\;\omega_{\text{m}} } \right\rangle$, where $\omega_{\text{m}} = \pi + \alpha$ and $0 \le \alpha \le \frac{\pi }{3}$.

The corresponding wavelet function, $\left| {\hat{\psi }(\omega )} \right|$, can be expressed with the scaling function with the substitution:

$$ g(\omega ) = \left| {\hat{\phi }(\omega )} \right| . $$

(15)

The obtained expression is more appropriate for further considerations:

$$ \left| {\hat{\psi }(\omega )} \right| = \left\{ {\begin{array}{ll} 0&0 \le \left| \omega \right| < \pi - \alpha \hfill \\ g(2\pi - \omega )&\pi - \alpha \le \left| \omega \right| < \pi + \alpha \hfill \\ 1&\pi + \alpha \le \left| \omega \right| < 2\pi - 2\alpha \hfill \\ g\left( {\frac{\omega }{2}} \right)&2\pi - 2\alpha \le \left| \omega \right| < 2\pi + 2\alpha \hfill \\ 0&2\pi + 2\alpha \le \left| \omega \right| \hfill \\ \end{array} } \right. $$

(16)

From the above equation, it can be concluded that g(ω) is a positive function. The orthonormal wavelet defined with Eq. (16) has the compact domain, which varies due to parameter α selection. For α = 0, frequency is the interval between $\pi$ and $2\pi$. For $\alpha = \frac{\pi }{3}$, the frequency range is: $\left[ {\frac{2\pi }{3},\frac{8\pi }{3}} \right]$.

The necessary condition for the chosen wavelet is the value of the parameter α, which must be equal to $\frac{\pi }{3}$ to satisfy the condition for the amplitude.

In order to derive the phase condition, we start with the relationship between the wavelet and the scaling function:

$$ \hat{\psi }(2\omega ) = e^{i\omega } \cdot \bar{\hat{H}}(\omega + \pi ) \cdot \hat{\phi }(\omega ) $$

(17)

Equation (17) can be written as Eqs. (18–20):

$$ \hat{H}(\omega ) = \frac{{\hat{\phi }(2\omega )}}{{\hat{\phi }(\omega )}} $$

(18)

$$ \hat{H}(\omega + \pi ) = \frac{{\hat{\phi }(2\omega + 2\pi )}}{{\hat{\phi }(\omega + \pi )}} $$

(19)

$$ \bar{\hat{H}}(\omega + \pi ) = \frac{{\bar{\hat{\phi }}(2\omega + 2\pi )}}{{\bar{\hat{\phi }}(\omega + \pi )}} $$

(20)

Combining Eqs. (17) and (20), it can be written:

$$ \hat{\psi }(2\omega ) = e^{{i\frac{\omega }{2}}} \cdot \frac{{\bar{\hat{\phi }}(2\omega + 2\pi )}}{{\bar{\hat{\phi }}(\omega + \pi )}} \cdot \hat{\phi }(\omega ) $$

(21)

or:

$$ \hat{\psi }(\omega ) = e^{{i\frac{\omega }{2}}} \cdot \frac{{\bar{\hat{\phi }}(\omega + 2\pi )}}{{\bar{\hat{\phi }}\left( {\frac{\omega }{2} + \pi } \right)}} \cdot \hat{\phi }\left( {\frac{\omega }{2}} \right) $$

(22)

From Eq. (22), the phase of the wavelet function can be written as:

$$ \theta_{\psi } (\omega ) = \frac{\omega }{2} - \theta_{\phi } (\omega + 2\pi ) + \theta_{\phi } \left( {\frac{\omega }{2} + \pi } \right) + \theta_{\phi } \left( {\frac{\omega }{2}} \right) $$

(23)

where:

• $\theta_{\psi }$ is the phase of the $\hat{\psi }$ function, and

• $\theta_{\phi }$ is the phase of the $\hat{\phi }$ function.

Due to periodicity of the $\hat{H}$ function, the phase of the $\hat{\phi }$ function satisfy the following equation:

$$ \theta_{\phi } (2\omega ) - \theta_{\phi } (\omega ) + \theta_{\phi } (4\pi - 2\omega ) - \theta_{\phi } (2\pi - \omega ) = 0 $$

(24)

Equation (24) is the phase condition for the wavelet choice.