Variable scale-convex-peak method for weak signal detection

Tian, Ruilan; Zhao, ZhiJie; Xu, Yong

doi:10.1007/s11431-019-1530-4

Cited by 31 publications

(27 citation statements)

References 28 publications

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“…According to Ref. 8, in the mean‐square sense, the stochastic subharmonic Melnikov function can be detected as

- {\overset{falseˆ}{M}}^{2} + {\bar{M}}_{1}^{2} false(t_{0} false) + {\bar{M}}_{2}^{2} false(t_{0} false) + E false[{\tilde{M}}^{2} (t_{0}) false] = 0,

where

\begin{matrix} left \\ \overset{falseˆ}{M} = - μ \frac{4 (2 - k^{2}) E (k) - 8 k^{' 2} F (k)}{3 {()(, 2 - k^{2})}^{3 / 2}}, \\ {\bar{M}}_{1} = f \sqrt{2} π ω sec h (\sqrt{2 - k^{2}} F^{'} (k) ω) \sin ω t_{0}, \\ {\bar{M}}_{2} = r \sqrt{2} π ω_{1} sec h (\sqrt{2 - k^{2}} F^{'} (k) ω_{1}) \sin ω_{1} t_{0}, \end{matrix}

\begin{matrix} left \\ E [{\tilde{M}}^{2} (t_{0})] = \int_{- \frac{T (k)}{2}}^{\frac{T false(k false)}{2}} {∣C (ω)∣}^{2} K normald ω \\ = - \frac{σ^{4} π}{2 Z^{3}} (\begin{matrix}  \end{matrix}) \end{matrix}

…”

Section: Periodic‐phase‐diagram Similarity Methodsmentioning

confidence: 99%

“…Consider Duffing system 8

({, \begin{array}{l} \dot{x} = y, \\ \dot{y} = x - x^{3} + f \cos ω t - μ y, \end{array})

where

x = x (t)

is the displacement at time

t

y

is the first derivative of

x

with respect to time

t

, and

f, ω, μ

represent the parameters of Duffing system.…”

Section: Phase Diagram Similaritymentioning

confidence: 99%

“…White Gaussian noise is added into system (), 8 which leads to

({, \begin{array}{l} \overset{x}{true} = y, \\ \overset{y}{true} = x - x^{3} + f \cos ω t - μ y + σ Γ, \end{array})

where Γ is the standard Gaussian white noise,

σ

is the noise intensity and the power spectral density of white Gaussian noise is

K = σ^{2} / 2 π

.…”

Section: Phase Diagram Similaritymentioning

confidence: 99%

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Periodic‐phase‐diagram similarity method for weak signal detection

Tian

Zhang

Yang

et al. 2021

Int Journal of Mech Sys Dyn

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The periodic‐phase‐diagram similarity method is proposed to identify the frequency of weak harmonic signals. The key technology is to find a set of optimal coefficients for Duffing system, which leads to the periodic motion under the influence of weak signal and strong noise. Introducing the phase diagram similarity, the influences of strong noise on the similarity of periodic phase diagram are discussed. The principle of highest similarity of periodic phase diagram with the same frequency is detected by discussing the persistence of similarity of periodic motion phase diagram under the strong noise and the periodic‐phase‐diagram similarity method is constructed. The weak signals of early fault and strong noise are input into Duffing system to obtain the identified system. The stochastic subharmonic Melnikov method is extended to obtain the conditions of the optimal coefficients for the identified system. Based on the results, the constructed frequency conversion harmonic weak signals are considered to form a datum periodic system. With the change of frequency in the datum periodic system, the phase diagram similarity of the two constructed systems can be calculated. Based on the periodic‐phase‐diagram similarity method, the frequency of weak harmonic signals can be identified by the principle of highest similarity of periodic phase diagram with the same frequency. The results of numerical simulation and the early fault diagnosis results of actual bearings verify the feasibility of the periodic‐phase‐diagram similarity method. The accuracy of the detection effect is 97%, and the minimum signal‐to‐noise ratio is −80.71 dB.

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“…According to Ref. 8, in the mean‐square sense, the stochastic subharmonic Melnikov function can be detected as

- {\overset{falseˆ}{M}}^{2} + {\bar{M}}_{1}^{2} false(t_{0} false) + {\bar{M}}_{2}^{2} false(t_{0} false) + E false[{\tilde{M}}^{2} (t_{0}) false] = 0,

where

\begin{matrix} left \\ \overset{falseˆ}{M} = - μ \frac{4 (2 - k^{2}) E (k) - 8 k^{' 2} F (k)}{3 {()(, 2 - k^{2})}^{3 / 2}}, \\ {\bar{M}}_{1} = f \sqrt{2} π ω sec h (\sqrt{2 - k^{2}} F^{'} (k) ω) \sin ω t_{0}, \\ {\bar{M}}_{2} = r \sqrt{2} π ω_{1} sec h (\sqrt{2 - k^{2}} F^{'} (k) ω_{1}) \sin ω_{1} t_{0}, \end{matrix}

\begin{matrix} left \\ E [{\tilde{M}}^{2} (t_{0})] = \int_{- \frac{T (k)}{2}}^{\frac{T false(k false)}{2}} {∣C (ω)∣}^{2} K normald ω \\ = - \frac{σ^{4} π}{2 Z^{3}} (\begin{matrix}  \end{matrix}) \end{matrix}

…”

Section: Periodic‐phase‐diagram Similarity Methodsmentioning

confidence: 99%

“…Consider Duffing system 8

({, \begin{array}{l} \dot{x} = y, \\ \dot{y} = x - x^{3} + f \cos ω t - μ y, \end{array})

where

x = x (t)

is the displacement at time

t

y

is the first derivative of

x

with respect to time

t

, and

f, ω, μ

represent the parameters of Duffing system.…”

Section: Phase Diagram Similaritymentioning

confidence: 99%

See 1 more Smart Citation

Periodic‐phase‐diagram similarity method for weak signal detection

Tian

Zhang

Yang

et al. 2021

Int Journal of Mech Sys Dyn

Self Cite

View full text Add to dashboard Cite

show abstract

“…Traditionally, infinite-dimensional systems are described as PDEs (partial differential equations), and then the PDEs are truncated into ODEs (ordinary differential equations) [16][17][18][19][20][21][22]. eoretically, ODEs are topologically equivalent to PDEs, but the dynamical phenomena cannot be perfectly exhibited.…”

Section: Introductionmentioning

confidence: 99%

Analytical and Numerical Results on Global Dynamics of the Generalized Boussinesq Equation with Cubic Nonlinearity and External Excitation

Zhang

2021

Mathematical Problems in Engineering

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This paper analytically and numerically presents global dynamics of the generalized Boussinesq equation (GBE) with cubic nonlinearity and harmonic excitation. The effect of the damping coefficient on the dynamical responses of the generalized Boussinesq equation is clearly revealed. Using the reductive perturbation method, an equivalent wave equation is then derived from the complex nonlinear equation of the GBE. The persistent homoclinic orbit for the perturbed equation is located through the first and second measurements, and the breaking of the homoclinic structure will generate chaos in a Smale horseshoe sense for the GBE. Numerical examples are used to test the validity of the theoretical prediction. Both theoretical prediction and numerical simulations demonstrate the homoclinic chaos for the GBE.

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“…It is necessary to solve the modal function and natural frequency for studying the vibration characteristics of the beam [5][6][7][8]. For the uniform beam, section area and stiffness of the microsection in the discrete body model are constant and equal.…”

Section: Introductionmentioning

confidence: 99%

An Improved Analytical Method for Vibration Analysis of Variable Section Beam

Feng

Chen

Hao

et al. 2020

Mathematical Problems in Engineering

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The variable section structure could be the physical model of many vibration problems, and its analysis becomes more complicated either. It is very important to know how to obtain the exact solution of the modal function and the natural frequency effectively. In this paper, a general analytical method, based on segmentation view and iteration calculation, is proposed to obtain the modal function and natural frequency of the beam with an arbitrary variable section. In the calculation, the section function of the beam is considered as an arbitrary function directly, and then the result is obtained by the proposed method that could have high precision. In addition, the total amount of calculation caused by high-order Taylor expansion is reduced greatly by comparing with the original Adomian decomposition method (ADM). Several examples of the typical beam with different variable sections are calculated to show the excellent calculation accuracy and convergence of the proposed method. The correctness and effectiveness of the proposed method are verified also by comparing the results of the several kinds of the theoretical method, finite element simulation, and experimental method.

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Variable scale-convex-peak method for weak signal detection

Cited by 31 publications

References 28 publications

Periodic‐phase‐diagram similarity method for weak signal detection

Periodic‐phase‐diagram similarity method for weak signal detection

Analytical and Numerical Results on Global Dynamics of the Generalized Boussinesq Equation with Cubic Nonlinearity and External Excitation

An Improved Analytical Method for Vibration Analysis of Variable Section Beam

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