• East Asian mathematical journal
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• v.33 no.1
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• pp.103-113
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• 2017

We provide quantitative two weight estimates for the commutator of the Hilbert transform under certain conditions on a pair of weights (u, v) and b in $Carl_{u,v}$. In [10] and [11], Bloom's inequality is shown in a modern setting, and the boundedness of the commutators is provided by assuming both weights u, v are $A_2$ and $b{\in}BMO_{\rho}$. In the present paper we show that the condition on b can be replaced by $Carl_{u,v}$ by using the joint $A^d_2$ condition.

• Interaction and multiscale mechanics
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• v.5 no.3
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• pp.169-185
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• 2012

The concrete bridge is likely to produce fatigue cracks during long period of service due to the moving vehicular loads and the degeneration of materials. This paper deals with the time-frequency analysis of a coupled bridge-vehicle system. The bridge is modeled as an Euler beam with breathing cracks. The vehicle is represented by a two-axle vehicle model. The equation of motion of the coupled bridge-vehicle system is established using the finite element method, and the Newmark direct integration method is adopted to calculate the dynamic responses of the system. The effect of breathing cracks on the dynamic responses of the bridge is investigated. The time-frequency characteristics of the responses are analyzed using both the Hilbert-Huang transform and wavelet transform. The results of time-frequency analysis indicate that complicated non-linear and non-stationary features will appear due to the breathing effect of the cracks.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2007.11a
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• pp.895-902
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• 2007

Rotating machinery has two typical faults with clearance, one is partial rub and the other is looseness. Due to these faults, non-linear and non-stationary signals are occurred. Therefore, time-frequency analysis is necessary for exact fault diagnosis of rotating machinery. In this paper newly developed time-frequency analysis method, HHT(Hilbert-Huang Transform) is applied to fault diagnosis and compared with other method of FFT, SFFT and CWT. The results show that HHT can represent better resolution than any other method. Consequently, the faults of rotating machinery are diagnosed efficiently by using HHT.

• Smart Structures and Systems
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• v.15 no.1
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• pp.1-13
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• 2015

This paper proposes an analytical mode decomposition (AMD) and Hilbert transform method for structural nonlinearity quantification and damage detection under earthquake loads. The measured structural response is first decomposed into several intrinsic mode functions (IMF) using the proposed AMD method. Each IMF is an amplitude modulated-frequency modulated signal with narrow frequency bandwidth. Then, the instantaneous frequencies of the decomposed IMF can be defined with Hilbert transform. However, for a nonlinear structure, the defined instantaneous frequencies from the decomposed IMF are not equal to the instantaneous frequencies of the structure itself. The theoretical derivation in this paper indicates that the instantaneous frequency of the decomposed measured response includes a slowly-varying part which represents the instantaneous frequency of the structure and rapidly-varying part for a nonlinear structure subjected to earthquake excitations. To eliminate the rapidly-varying part effects, the instantaneous frequency is integrated over time duration. Then the degree of nonlinearity index, which represents the damage severity of structure, is defined based on the integrated instantaneous frequency in this paper. A one-story hysteretic nonlinear structure with various earthquake excitations are simulated as numerical examples and the degree of nonlinearity index is obtained. Finally, the degree of nonlinearity index is estimated from the experimental data of a seven-story building under four earthquake excitations. The index values for the building subjected to a low intensity earthquake excitation, two medium intensity earthquake excitations, and a large intensity earthquake excitation are calculated as 12.8%, 23.0%, 23.2%, and 39.5%, respectively.

• Journal of the Chungcheong Mathematical Society
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• v.35 no.3
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• pp.269-275
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• 2022

In this note, we verify that the bounded shadowing property and quasi-hyperbolicity of bounded linear operators on Hilbert spaces are preserved under Aluthge transforms.

• Proceedings of the Korea Water Resources Association Conference
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• 2006.05a
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• pp.1450-1454
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• 2006

Empirical mode decomposition (EMD) is applied to analyze time series characterized with nonlinearity and nonstationarity. This decomposition could be utilized to construct finite and small number intrinsic mode functions (IMF) that describe complicated time series, while admitting the Hilbert transformation properties. EMD has the capability of being adaptive, capture local characteristics, and applicable to nonlinear and nonstationary processes. Unlike discrete wavelet transform (DWT), IMF eliminates spurious harmonics and retains meaningful instantaneous frequencies. Examples based on data representing natural phenomena are given to demonstrate highlight the power of this method in contrast and comparison of other ones. A presentation of the energy-frequency-time distribution of these signals found to be more informative and intuitive when based on Hilbert transformation.

• Journal of Korean Society of Industrial and Systems Engineering
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• v.35 no.2
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• pp.106-112
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• 2012

Real-life time series characteristic data has significant amount of non-stationary components, especially periodic components in nature. Extracting such components has required many ad-hoc techniques with external parameters set by users in case-by-case manner. In our study, we evaluate whether Hilbert-Huang Transform, a new tool of time-series analysis can be used for effective analysis of such data. It is divided into two points : 1) how effective it is in finding periodic components, 2) whether we can use its results directly in detecting values outside control limits, for which a traditional method such as ARIMA had been used. We use glass furnace temperature data to illustrate the method.

• Structural Engineering and Mechanics
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• v.25 no.6
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• pp.653-674
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• 2007

This paper discusses the application of piezoelectric sensors used for evaluation of damping ratio of PVC plastics. The development of the mathematical formulation based on the Empirical Mode Decomposition for calculating the damping coefficient and natural frequency of the system is presented. A systematic experimental and analytical investigation was also carried out to demonstrate the integrity of several methods commonly used to evaluate the damping of materials based on a single degree freedom formulation. The influence of the sensors' location was also investigated. Besides the commonly used methods, a newly emerging time-frequency method, namely the Empirical Mode decomposition, is also employed. Mathematical formulations based on the Hilbert-Huang formulation, and a frequency spacing technique were also developed for establishing the natural frequency and damping ratio based on the output voltage of a single piezoelectric sensor. An experimental investigation was also conducted and the results were compared and verified with Finite Element Analysis (FEA), revealing good agreement.

• Earthquakes and Structures
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• v.16 no.2
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• pp.197-208
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• 2019

The objective of this study is to propose new seismic intensity parameters based on the Hilbert spectrum and to associate them with the seismic damage potential. In recent years the assessment of even more seismic features derived from the seismic acceleration time-histories was associated with the structural damage. For a better insight into the complex seismic acceleration time-history, Hilbert-Huang Transform (HHT) analysis is utilized for its processing, and the Hilbert spectrum is obtained. New proposed seismic intensity parameters based on the Hilbert spectrum are derived. The aim is to achieve a significant estimation of the seismic damage potential on structures from the proposed new intensity parameters confirmed by statistical methods. Park-Ang overall structural damage index is used to describe the postseismic damage status of structures. Thus, a set of recorded seismic accelerograms from all over the word is applied on a reinforced concrete frame structure, and the Park-Ang indices through nonlinear dynamic analysis are provided and considered subsequently as reference numerical values. Conventional seismic parameters, with well-known seismic structural damage interrelation, are evaluated for the same set of excitations. Statistical procedures, namely correlation study and multilinear regression analysis, are applied on the set of the conventional parameters and the set of proposed new parameters separately, to confirm their interrelation with the seismic structural damage. The regression models are used for the evaluation of the structural damage indices for every set of parameters, respectively. The predicted numerical values of the structural damage indices evaluated from the two sets of seismic intensity parameters are inter-compared with the reference values. The numerical results confirm the ability of the proposed Hilbert spectrum based new seismic intensity parameters to approximate the postseismic structural damage with a smaller Standard Error of Estimation than this accomplished of the conventional ones.

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
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• 2007.06a
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• pp.739-742
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• 2007

In the step processing multimedia information signals transmitted by a variety of mediums, noises are generated by the internal or exterior causes of systems and these noises degrade the perception about information signals. Therefore, in order to remove noises and restore signals a great number of researches have been progressed and recently, many noise removal methods using time-frequency localization of wavelet have been applied in wide applications. Moreover, when two wavelet bases are designed to accomplish the Hilbert transform pair, wavelet can be efficiently applied to detect characteristics of signals. Therefore, in this paper, in order to restore the corrupted signal by noises, a noise removal algorithm using the Hilbert transform pair of wavelet was proposed.