A Statistical Index for Monitoring Tooth Cracks in a Gearbox

1995 ◽  
Author(s):  
Fathy Ismail ◽  
Hugh R. Martin ◽  
Farag Omar
Keyword(s):  
Beta Distribution ◽  
General Equation ◽  
Beta Function ◽  
Statistical Moments ◽  
Tooth Root ◽  
Statistical Index ◽  
Damage Indicator ◽  
Small Cracks ◽  
Root Crack

Abstract This paper is concerned with detecting and monitoring the growth of a tooth root crack in a gearbox. It uses the reciprocal of the Kurtosis of the beta distribution 1/k for each tooth period, as the damage indicator. A general equation to generate the statistical moments strictly in terms of the α and β parameters of the beta function is developed. To monitor the growth of the crack, a new index, called the “Kurtosis Index”, is formulated from 1/k’s of all the teeth. This index correlates well with the size of the crack, and it shows that small cracks can be detected.

3-D elastic coupling vibration and acoustical radiation characteristics of cracked gear under elastic support condition

2015 ◽  
Vol 23 (9) ◽  
pp. 1548-1568 ◽  
Author(s):  
Shao Renping ◽  
Purong Jia ◽  
Xiankun Qi
Keyword(s):  
Dynamic Response ◽  
Support System ◽  
Three Dimensional ◽  
Elastic Support ◽  
Dynamic Responses ◽  
Tooth Root ◽  
Support Condition ◽  
Elastic Boundary ◽  
Root Crack ◽  
Elastic Disc

According to the actual working condition of the gear, the supporting gear shaft is treated as an elastic support. Its impact on the gear body vibration is considered and investigated and the dynamic response of elastic teeth and gear body is analyzed. On this basis, the gear body is considered as a three-dimensional elastic disc and the gear teeth are treated as an elastic cantilever beam. Under the conditions of the elastic boundary (support shaft), combining to the elastic disk and elastic teeth, the influence of three-dimensional elastic discs on the meshing tooth response under an elastic boundary condition is also included. A dynamic model of the gear support system and calculated model of the gear tooth response are then established. The inherent characteristics of the gear support system and dynamics response of the meshing tooth are presented and simulated. It was shown by the results that it is correct to use the elastic support condition to analyze the gear support system. Based on the above three-dimensional elastic dynamics analysis, this paper set up a dynamics coupling model of a cracked gear structure support system that considered the influence of a three-dimensional elastic disc on a cracked meshing tooth under elastic conditions. It discusses the dynamic characteristic of the cracked gear structure system and coupling dynamic response of the meshing tooth, offering a three-dimensional elastic body model of the tooth root crack and pitch circle crack with different sizes, conducting the three-dimensional elastic dynamic analysis to the faulty crack. ANSYS was employed to carry out dynamic responses, as well as to simulate the acoustic field radiation orientation of a three-dimensional elastic crack body at the tooth root crack and pitch circle with different sizes.

Logarithmic concavity of the inverse incomplete beta function with respect to the first parameter

2021 ◽  
Vol 127 (1) ◽  
pp. 111-130
Author(s):  
Dimitris Askitis
Keyword(s):  
Distribution Function ◽  
Beta Distribution ◽  
Probability Distributions ◽  
Beta Function ◽  
Incomplete Beta Function ◽  
Univariate Function ◽  
Convexity Properties ◽  
Logarithmic Concavity ◽  
Two Parameter ◽  
Monotonicity Results

The beta distribution is a two-parameter family of probability distributions whose distribution function is the (regularised) incomplete beta function. In this paper, the inverse incomplete beta function is studied analytically as a univariate function of the first parameter. Monotonicity, limit results and convexity properties are provided. In particular, logarithmic concavity of the inverse incomplete beta function is established. In addition, we provide monotonicity results on inverses of a larger class of parametrised distributions that may be of independent interest.

scholarly journals Frequency and Vibration Characteristics of High-Speed Gear-Rotor-Bearing System with Tooth Root Crack considering Compound Dynamic Backlash

2019 ◽  
Vol 2019 ◽  
pp. 1-19 ◽  
Author(s):  
Jie Liu ◽  
Weiqiang Zhao ◽  
Weiwei Liu
Keyword(s):  
Time Domain ◽  
Crack Depth ◽  
Transmission System ◽  
Gear System ◽  
Time Varying ◽  
Tooth Root ◽  
Meshing Stiffness ◽  
The Time Domain ◽  
Center Distance ◽  
Root Crack

Considering the microstructure of tooth surface and the dynamic characteristics of the vibration responses, a compound dynamic backlash model is employed for the gear transmission system. Based on the fractal theory and dynamic center distance, respectively, the dynamic backlash is presented, and the potential energy method is applied to compute the time-varying meshing stiffness, including the healthy gear system and the crack fault gear system. Then, a 16-DOF coupled lateral-torsional gear-rotor-bearing transmission system with the crack fault is established. The fault characteristics in the time-domain waveform and frequency response and statistics data are described. The effect of crack on the time-varying meshing stiffness is analyzed. The vibration response of three backlash models is compared. The dynamic response of the system is explored with the increase in crack depth in detail. The results show that the fault features of countershaft are more obvious. Obvious fluctuations are presented in the time-domain waveform, and sidebands can be found in the frequency domain responses when the tooth root crack appears. The effect of compound dynamic backlash on the system is more obvious than fixed backlash and backlash with changing center distance. The vibration displacement along meshing direction and dynamic meshing force increases with the increase in crack depth. Backlash and variation of center distance show different tendencies with increasing crack depth under different rotational speeds. Amplitude of the sidebands increases with crack depth increasing. The amplitude of multiplication frequency of rotational frequency has an obvious variation with growing crack depth. The sidebands of the multiplication frequency of meshing frequency show more details on the system with complex backlash and crack fault.

scholarly journals Modeling the Liberation of Comminuted Scheelite Using Mineralogical Properties

Minerals ◽  
2019 ◽  
Vol 9 (9) ◽  
pp. 536 ◽  
Author(s):  
Sarbast Ahmad Hamid ◽  
Pura Alfonso ◽  
Josep Oliva ◽  
Hernan Anticoi ◽  
Eduard Guasch ◽  
...  
Keyword(s):  
Beta Distribution ◽  
Test Performance ◽  
Distribution Density ◽  
Beta Function ◽  
Size Fraction ◽  
Distribution Functions ◽  
Plant Design ◽  
Mineral Liberation ◽  
Size Grade ◽  
Liberation Analysis

In this paper, the modeling of the liberation of scheelite is presented. A pattern of concentration experiments was performed to investigate the scheelite liberation and distribution density calculation procedure. In this work, one sample from a Mittersill tungsten ore was studied. This work describes a method for determining the downstream milling energy requirements for rod mill products based on a Bond mill test performance. The grade distribution of particles at a given size fraction was calculated using a predictive liberation model. The concentration behavior of these particles in size fractions was evaluated using batch concentrate tests. The recovery of particles in size/grade classes, image analysis using mineral liberation analysis (MLA), and function calculations were implemented for the modeling of the liberation. By describing the size, grade, and recovery data of particles in size/grade classes, a technique for the measurement of distribution functions was developed that relates beta distribution, a model for the function based on the incomplete beta function, and a solution to produce liberation modeling. It was shown that the predicted results agreed well with the observed results. With a procedure for measuring the liberation, it was possible to carry out the first experimental measurement of the beta distribution. This liberation/concentrate model has wide potential applications for metallurgy and plant design, where the liberation modeling is to be determined with the distribution density solution to the predictive mineral liberation function equation, which includes the liberation of ore samples and their liberation characteristics.

ESTIMATING SIZE OF GEAR TOOTH ROOT CRACK USING EMBEDDED MODELLING

2002 ◽  
Vol 16 (5) ◽  
pp. 841-852 ◽  
Author(s):  
C. JAMES LI ◽  
HYUNGDAE LEE ◽  
SUK HWAN CHOI
Keyword(s):  
Gear Tooth ◽  
Tooth Root ◽  
Root Crack
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