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Статті в журналах з теми "Elliptic curve determined in three dimensions"
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Huang, Zhi Dong, An Min Hui, Peng Chen, and Yu Wang. "Computer Aided Design and Three-Dimensional Modeling of High-Order Deformed Elliptical Gear." Applied Mechanics and Materials 391 (September 2013): 178–81. http://dx.doi.org/10.4028/www.scientific.net/amm.391.178.
Повний текст джерелаАнотація:
The characteristics of high-order deformed elliptical gear is analyzed. The parameters of high-order deformed elliptical gear are chosen and calculated. The modeling method of high-order deformed elliptical gear is presented. The shape of pitch curve is determined. The position and orientation of gear teeth are clarified. The three-dimensional solid model of high-order deformed elliptical gear is achieved. The method and the result facilitate finite element analysis and numerical control machining simulation of high-order deformed elliptical gear.
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Sun, Yi-Sui. "On the Measure-Preserving Mappings with Three-Dimensions." International Astronomical Union Colloquium 132 (1993): 73–89. http://dx.doi.org/10.1017/s0252921100065957.
Повний текст джерелаАнотація:
AbstractWe have systematically made the numerical exploration about the perturbation extension of area-preserving mappings to three-dimensional ones, in which the fixed points of area preserving are elliptic, parabolic or hyperbolic respectively. It has been observed that: (i) the invariant manifolds in the vicinity of the fixed point generally don’t exist (ii) when the invariant curve of original two-dimensional mapping exists the invariant tubes do also in the neighbourhood of the invariant curve (iii) for the perturbation extension of area-preserving mapping the invariant manifolds can only be generated in the subset of the invariant manifolds of original two-dimensional mapping, (iv) for the perturbation extension of area preserving mappings with hyperbolic or parabolic fixed point the ordered region near and far from the invariant curve will be destroyed by perturbation more easily than the other one, This is a result different from the case with the elliptic fixed point. In the latter the ordered region near invariant curve is solid. Some of the results have been demonstrated exactly.Finally we have discussed the Kolmogorov Entropy of the mappings and studied some applications.
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Journal, Baghdad Science. "Classification of Elliptic Cubic Curves Over The Finite Field of Order Nineteen." Baghdad Science Journal 13, no. 4 (December 4, 2016): 846–52. http://dx.doi.org/10.21123/bsj.13.4.846-852.
Повний текст джерелаАнотація:
Plane cubics curves may be classified up to isomorphism or projective equivalence. In this paper, the inequivalent elliptic cubic curves which are non-singular plane cubic curves have been classified projectively over the finite field of order nineteen, and determined if they are complete or incomplete as arcs of degree three. Also, the maximum size of a complete elliptic curve that can be constructed from each incomplete elliptic curve are given.
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Singer, David A. "The Trefoil Soliton." Mathematics 10, no. 9 (May 1, 2022): 1512. http://dx.doi.org/10.3390/math10091512.
Повний текст джерелаАнотація:
The Kiepert trefoil is an algebraic curve with remarkable geometric and number theoretic properties. Ludwig Kiepert, generalizing ideas due to Serret and Liouville, determined that it could be parametrized by arc length in terms of elliptic functions. In this note, we observe some other properties of the curve. In particular, the curve is a special example of a buckled ring, and thus a solitary wave solution to the planar filament equation, evolving by rotation. It is also a solitary wave solution to a flow in the (three-dimensional) filament hierarchy, evolving by translation.
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Bayrakdar, T., and A. A. Ergin. "Motion of an integral curve of a Hamiltonian dynamical system and the evolution equations in 3D." International Journal of Geometric Methods in Modern Physics 14, no. 12 (November 24, 2017): 1750172. http://dx.doi.org/10.1142/s0219887817501729.
Повний текст джерелаАнотація:
We show that all of the nonstretching curve motions specified in the Frenet–Serret frame in the literature can be described by the time evolution of an integral curve of a Hamiltonian dynamical system such that the underlying curve is a geodesic curve on a leaf of the foliation determined by the Poisson structure in three dimensions. As an illustrative example, we show that the focusing version of the nonlinear Schrödinger equation and the complex modified Korteweg–de Vries (mKdV) equation are obtained in this way.
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Mishra, Amaresh Chandra. "Micromagnetic simulation of hysteresis loop of elliptic permalloy nanorings." International Journal of Modern Physics B 30, no. 26 (October 12, 2016): 1650192. http://dx.doi.org/10.1142/s0217979216501927.
Повний текст джерелаАнотація:
Magnetic hysteresis behavior of isotropic permalloy elliptic nanorings of outer semi-major axis length [Formula: see text] 100 nm and thickness [Formula: see text] 20 nm were studied with respect to the variation of two parameters: outer semiminor axis length [Formula: see text] and the difference between outer and inner dimensions [Formula: see text]. The outer semiminor axis length [Formula: see text] varied from 90 nm to 20 nm which covers from nearly circular nanoring to elliptic nanoring of high aspect ratio. The value of [Formula: see text] varied in steps of 10 nm. Micromagnetic simulation of in-plane hysteresis curve of these nanorings revealed that the remanent state of all of these elliptic rings are onion states if the magnetic field is applied along the longer side of the elliptic rings. If the magnetic field is applied along the shorter side, then the remanent states turn out to be vortex state. The hysteresis loss indicated by area of the hysteresis loop was found to be decreasing gradually with the increment of either [Formula: see text] or [Formula: see text]. On the other hand, the remanent magnetization increased with increment of [Formula: see text] but decreased with the increment of [Formula: see text]. The changes were attributed to three parameters mainly: inner curvature, exchange energy and demagnetization energy. The changes in loop area were discussed in light of variation of these three parameters.
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Güner, Alper, Alexander Brosius, and A. Erman Tekkaya. "Analysis of the Hydraulic Bulge Test with FEA Concerning the Accuracy of the Determined Flow Curves." Key Engineering Materials 410-411 (March 2009): 439–47. http://dx.doi.org/10.4028/www.scientific.net/kem.410-411.439.
Повний текст джерелаАнотація:
This work covers the finite element analysis of geometric and process parameters in hydraulic bulge tests in terms of the accuracy of the evaluated flow curve. The important parameters are identified and varied to cover the whole range of possible uses. The effects of these parameters are analyzed for three representative materials: aluminium, mid-strength steel, and high-strength steel. The flow curves of the materials for each set of parameters are calculated by using the results of the simulations and the membrane theory. It is seen that even with simulation results, it is not always possible to obtain the input flow curve, especially towards the end of the test. The dimensions of the sheet and the tooling affect the plastic strain development and geometry of the bulge, leading to errors in computed flow curves. In order to observe the effect of the material flow from the flange on the determined yield stresses, the function and position of the drawbeads are also examined. These parameters, together with the method used to calculate the radius of the bulge, determine the accuracy of the calculated flow curve. Guidelines for an accurate flow curve determination regarding the test set-up and calculation methods are given.
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Kohn, Kathlén, and Kristian Ranestad. "Projective Geometry of Wachspress Coordinates." Foundations of Computational Mathematics 20, no. 5 (November 11, 2019): 1135–73. http://dx.doi.org/10.1007/s10208-019-09441-z.
Повний текст джерелаАнотація:
Abstract We show that there is a unique hypersurface of minimal degree passing through the non-faces of a polytope which is defined by a simple hyperplane arrangement. This generalizes the construction of the adjoint curve of a polygon by Wachspress (A rational finite element basis, Academic Press, New York, 1975). The defining polynomial of our adjoint hypersurface is the adjoint polynomial introduced by Warren (Adv Comput Math 6:97–108, 1996). This is a key ingredient for the definition of Wachspress coordinates, which are barycentric coordinates on an arbitrary convex polytope. The adjoint polynomial also appears both in algebraic statistics, when studying the moments of uniform probability distributions on polytopes, and in intersection theory, when computing Segre classes of monomial schemes. We describe the Wachspress map, the rational map defined by the Wachspress coordinates, and the Wachspress variety, the image of this map. The inverse of the Wachspress map is the projection from the linear span of the image of the adjoint hypersurface. To relate adjoints of polytopes to classical adjoints of divisors in algebraic geometry, we study irreducible hypersurfaces that have the same degree and multiplicity along the non-faces of a polytope as its defining hyperplane arrangement. We list all finitely many combinatorial types of polytopes in dimensions two and three for which such irreducible hypersurfaces exist. In the case of polygons, the general such curves are elliptic. In the three-dimensional case, the general such surfaces are either K3 or elliptic.
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Bor, Gil, Mark Levi, Ron Perline, and Sergei Tabachnikov. "Tire Tracks and Integrable Curve Evolution." International Mathematics Research Notices 2020, no. 9 (May 17, 2018): 2698–768. http://dx.doi.org/10.1093/imrn/rny087.
Повний текст джерелаАнотація:
Abstract We study a simple model of bicycle motion: a segment of fixed length in multi-dimensional Euclidean space, moving so that the velocity of the rear end is always aligned with the segment. If the front track is prescribed, the trajectory of the rear wheel is uniquely determined via a certain first order differential equation—the bicycle equation. The same model, in dimension two, describes another mechanical device, the hatchet planimeter. Here is a sampler of our results. We express the linearized flow of the bicycle equation in terms of the geometry of the rear track; in dimension three, for closed front and rear tracks, this is a version of the Berry phase formula. We show that in all dimensions a sufficiently long bicycle also serves as a planimeter: it measures, approximately, the area bivector defined by the closed front track. We prove that the bicycle equation also describes rolling, without slipping and twisting, of hyperbolic space along Euclidean space. We relate the bicycle problem with two completely integrable systems: the Ablowitz, Kaup, Newell, and Segur (AKNS) system and the vortex filament equation. We show that “bicycle correspondence” of space curves (front tracks sharing a common back track) is a special case of a Darboux transformation associated with the AKNS system. We show that the filament hierarchy, encoded as a single generating equation, describes a three-dimensional bike of imaginary length. We show that a series of examples of “ambiguous” closed bicycle curves (front tracks admitting self bicycle correspondence), found recently F. Wegner, are buckled rings, or solitons of the planar filament equation. As a case study, we give a detailed analysis of such curves, arising from bicycle correspondence with multiply traversed circles.
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Jin, Rong, Bing Fei Gu, and Guo Lian Liu. "Research on Generation Rules of Chest Width Point Curve Based on 3D Female Mannequin." Advanced Materials Research 796 (September 2013): 513–18. http://dx.doi.org/10.4028/www.scientific.net/amr.796.513.
Повний текст джерелаАнотація:
In this paper 110 female undergraduates in Soochow University are measured by using 3D non-contact measurement system and manual measurement. 3D point cloud data of human body is taken as research objects by using anti-engineering software, and secondary development of point cloud data is done on the basis of optimizing point cloud data. In accordance with the definition of the human chest width points and other feature points, and in the operability of the three-dimensional point cloud data, the width, thickness, and length dimensions of the curve through the chest width point are measured. Classification of body type is done by choosing the ratio values as classification index which is the ratio between thickness and width of the curve. The generation rules of the chest curve are determined for each type by using linear regression method. Human arm model could be established by the computer automatically. Thereby the individual model of the female upper body mannequin modeling can be improved effectively.
Дисертації з теми "Elliptic curve determined in three dimensions"
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Альбрехт, Йосип Омелянович. "Система аутентифікації на базі еліптичних кривих з використанням векторних операцій". Master's thesis, Київ, 2018. https://ela.kpi.ua/handle/123456789/25522.
Повний текст джерелаАнотація:
Магістерська дисертація містить результати розробки системи аутентифікації на базі еліптичних кривих з використанням векторних операцій. Актуальність даної роботи полягає у суттєвому зменшенні необхідної довжини ключа для реалізації електронного цифрового підпису за рахунок використання еліптичних кривих, визначених у трьох вимірах, при збереженні рівня криптостійкості, і збереженні швидкодії за рахунок використання векторних операцій.The master's dissertation contains the results of the development of an authentication system based on elliptic curves using vector operations. The relevance of this work is in significant reducing the required key length for the implementation of the digital signature through the use of elliptical curves determined in three dimensions, while maintaining the level of cryptostability, and maintaining the performance through the use of vector operations.
Магистерская диссертация содержит результаты разработки системы аутентификации на базе эллиптических кривых с использованием векторных операций. Актуальность данной работы заключается в существенном уменьшении необходимой длины ключа для реализации электронной цифровой подписи за счет использования эллиптических кривых, определенных в трех измерениях, при сохранении уровня криптостойкости, и сохранении быстродействия за счет использования векторных операций.
Тези доповідей конференцій з теми "Elliptic curve determined in three dimensions"
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Aluko, O., and H. A. Whitworth. "Bearing Strength Analysis of Pin-Loaded Elliptical Holes in Laminated Composite Joints." In ASME 2011 International Mechanical Engineering Congress and Exposition. ASMEDC, 2011. http://dx.doi.org/10.1115/imece2011-62507.
Повний текст джерелаАнотація:
An analysis was performed to predict bearing strength and mode of failure of pin loaded orthotropic plates with elliptic holes of varying sizes using two dimensional stress analyses and a characteristic curve model. The stresses required to analyze joint failure were obtained by utilizing complex stress functions that were determined from assumed displacement expressions that satisfy the boundary conditions around the hole. Three different joint geometries with major-to-minor diameter ratios ranging from 1 to 5 were evaluated and the analysis revealed that the joint strength was found to vary with increasing major-to-minor diameter ratios. The material properties of graphite/epoxy laminates were used in this investigation.
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Bhimanadam, V. R., and F. J. Blom. "Probabilistic PTS Analysis." In ASME 2016 Pressure Vessels and Piping Conference. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/pvp2016-63112.
Повний текст джерелаАнотація:
This paper describes the effect of variability of fracture toughness of nuclear pressure vessels during a PTS event. The model used in this paper is based on the NESC-1 experiment. To determine the behavior of the surface breaking defect NRG performed three dimensional finite element calculations and subsequently extended these calculations to the probabilistic calculations. Three-dimensional finite-element model of the cladded cylinder was generated using ANSYS with semi-elliptical surface crack having a crack depth of 75 mm and a crack length of 205 mm. The cylinder specimen was subjected to thermal-shock and centrifugal loading conditions and analyzed with a themo-elastic-plastic material model and subsequently determined the fracture mechanics parameters (J and K) along the elliptical crack front as a function of time and temperature. The determined stress intensity factor K has been compared with the available cleavage fracture toughness (KJC) data with 50% fracture probability which has been obtained from the Master Curve according to BS7910. The comparison has been performed for the locations in the base metal as well as the locations in the heat affected zone. Deterministic analysis has been extended to probabilistic analysis to calculate the failure probability for the crack initiation at the locations in the base metal as well as the locations in the heat affected zone along the crack front by considering probabilistic distributions from Master Curve and FAVOR. Master Curve analysis through the ASME code case N-629 has been applied to the material. Results obtained from these two methods have been compared and also the results are used to compare the inherent safety factors in the deterministic analysis using RTNDT and Master Curve.
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Stratil, Ludek, Filip Siska, Ivo Dlouhy, and Marta Serrano. "The Application of Miniaturized Three-Point-Bend Specimens for Determination of the Reference Temperature of JRQ Steel." In ASME 2015 Pressure Vessels and Piping Conference. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/pvp2015-45706.
Повний текст джерелаАнотація:
This contribution deals with determination of the reference temperature of JRQ steel using miniaturized specimens. The dimensions of used miniaturized specimens were 3 × 4 × 27 mm (thickness × width × length). This specimen type offers the utilization of limited amount of test material or broken halves of precracked Charpy and larger specimens. The test material comes from the broken halves of 0.5T SEB specimens previously tested for purposes of the reference temperature determination in Ciemat, Madrid. The fracture toughness tests of specimens were performed in the transition region of the steel according to the recommendations of standard ASTM E1921 and according Wallin’s recommended temperature range for miniaturized specimens. The determined reference temperature of the Master Curve was very similar to the determined ones from three-point-bend specimen of sizes 0.2T, 0.4T and 0.5T. The obtained results confirm a necessity of conduct of tests at low temperatures and testing sufficient number of specimens in order to generate enough valid data for determination of the reference temperature.
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Figliolini, Giorgio, and Jorge Angeles. "Kinematic Analysis of Spherical Four-Bar Linkages via the Inflection Cubic and Thales Ellipse." In ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/detc2015-47571.
Повний текст джерелаАнотація:
The subject of this paper is the formulation of a specific algorithm for the kinematic analysis of spherical four-bar linkages via the inflection spherical cubic and spherical Thales ellipse by devoting particular attention to the crossed four-bar linkage (anti-parallelogram). Moreover, both the inflection and the elliptic cones, which represent the equivalent of the Bresse cylinders of the planar case in three-dimensions, are obtained by showing the particular properties of the spherical motion in terms of the curvature of a coupler curve and both the velocity and acceleration vector fields. Of special interest are also the cases in which the three acceleration poles coincide at one unique point or in two plus one, which depends on the intersections of two spherical curves of third and second degree.
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Haque, Mohammad Shafinul, and Calvin M. Stewart. "Exploiting Functional Relationships Between MPC Omega, Theta, and Sinh-Hyperbolic Continuum Damage Mechanics Model." In ASME 2016 Pressure Vessels and Piping Conference. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/pvp2016-63089.
Повний текст джерелаАнотація:
The MPC Omega and Theta models for creep deformation and life prediction have become popular in recent years. Both models offer better prediction than classical constitutive models such as Norton Power law, Bailey-Norton law, and Norton-Soderberg law to name a few. The Omega model uses a strain hardening approach and requires two material constants for creep deformation and life prediction. An analytical solution to the constants are available and it is easy to manipulate and implement numerically. However, the analytical damage of the Omega model predicts an unrealistic linear damage evolution. The Theta model uses a time-hardening approach, and requires four constant that are a function of stress and temperature. For materials under isothermal conditions, with tertiary creep dominant deformation, the Theta model constants can be determined using only two constants. Life prediction using the Theta and Omega models depends on the final creep strain. The final creep strain observed in an experiment is stochastic; dependent on the material, testing conditions, and operator. The statistics of final creep strain must be investigated before the Theta or Omega models can be applied. In literature, some authors add a nonlinear damage variable to the Theta model; however, critical damage at rupture is not unity violating the assumptions of continuum damage mechanics. There is a superior Sin-hyperbolic continuum damage model available in the literature that can be used to overcome these problems. It is hypothesized that a functional relationships exist between the three models and these relationships can be exploited to achieve more accurate and easy to implement creep deformation and life predictions. In this study, the relationships between the constants of MPC Omega, Theta, and a Sin-hyperbolic CDM models are determined analytically. The sin-hyperbolic model incorporates a continuum damage variable in the creep strain rate equation. The damage function exhibits a more realistic elliptical path and is constructed such that damage is always unity at rupture. This function facilitates conversion of one models’ constants to the constants of the other two. The relationships between the constants are identified, while maintaining dimensional homogeneity. Using the derived relationships, the three models can be easily compared and the disadvantages of each respective model can be avoided. Experimental data at four different configurations of stress (6.3 to 36.5 ksi) and temperature (1200 to 1800°F) (sixteen data sets) for Hastelloy X is used to compare the models. Creep rupture data at seven temperature levels (600 to 1000°C) and a wide stress range (5 to 500 MPa) is used to analyze life prediction. The constants for each model are determined. Using one models’ constants and the derived relationships; the predictions of the other two models are generated. It is observed that the relationship generated curves agree with experimental data. Finally, it is demonstrated that using the derived relationships, the most useful aspects of each model can combined. An elliptical damage evolution curve is obtained for the Omega model. The final creep strain rate dependency problem of the Theta model can be avoided. It is observed that the Sinh model becomes more flexible and easy to implement.
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Razelos, P., and S. Das. "Optimization of Arrays of Rectangular Longitudinal Fins and Circular or Elliptical Cross-Section Spines." In ASME 2010 International Mechanical Engineering Congress and Exposition. ASMEDC, 2010. http://dx.doi.org/10.1115/imece2010-40057.
Повний текст джерелаАнотація:
The purpose of this study is to illustrate a method for obtaining the thermal performance and optimum dimensions of arrays consisting of rectangular longitudinal fins, cylindrical and elliptical cross-section spines. Since the majority of fin problems in an array have constant boundary temperatures, our endeavor is focused on the analysis and optimization of these types of fins. The temperature distribution, heat released to the environment, and the fins’ optimum dimensions in the array have been determined. Our analysis is based on a new set of dimensionless parameters, which are more suitable than those frequently used in fin analyses in the available literature. The effects of the boundary temperature ratio on the temperature variation in the fin have been examined and the results are shown graphically. Three salient results are derived regarding the thermal characteristics of these types of fins. They are: (1) the fin’s heat dissipation is expressed exactly with the same expression which describes the optimum heat dissipated by a fin with insulated tip, having half the fin’s dimensionless height. (2) The optimum fins’ dimensionless height and thickness are identical with those obtained for a fin with adiabatic tip of twice the dimensionless height (3) The optimum semi-axes ratio of elliptical-cross section spines is uniquely defined. In addition, it is shown that this approach can be successfully applied to solve problems in several practical applications, including pin fins having elliptic-cross sections. Two examples serve to illustrate the usefulness of our method.
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Szwedowicz, J., C. Gibert, T. P. Sommer, and R. Kellerer. "Numerical and Experimental Damping Assessment of a Thin-Walled Friction Damper in the Rotating Set-Up With High Pressure Turbine Blades." In ASME Turbo Expo 2006: Power for Land, Sea, and Air. ASMEDC, 2006. http://dx.doi.org/10.1115/gt2006-90951.
Повний текст джерелаАнотація:
Under-platform friction dampers are preferably solutions for minimizing vibrations of rotating turbine blades. Solid dampers, characterized by their compact dimensions, are frequently used in real applications and often appear in patents in different forms. A different type of the friction damper is a thin-walled structure, which has larger dimensions and smaller contact stresses on a wider contact area in relation to the solid damper. The damping performance of a thin-walled damper, mounted under the platforms of two rotating, freestanding high pressure turbine blades is investigated numerically and experimentally in this paper. The tangential and normal contact stiffness, that are crucial parameters in optimal design of each friction damper, are determined from three-dimensional finite element (FE) computations of the contact behaviour of the thin-walled damper on the platform including friction and centrifugal effects. The computed contact stiffness values are applied to non-linear dynamic simulations of the analysed blades with the friction damper of a specified mass. These numerical analyses are performed in the modal frequency domain with a code, which is based on the Harmonic Balance Method (HBM) for the complex linearisation of friction forces. The blade vibrations are characterised by a set of the lowest FE mode shapes of one freestanding blade without damper. The dynamic results of the calculated blades with the damper are in good agreement with the measured data of the real mistuned system. In the analysed excitation range, the numerical performance curve of the thin-walled damper is obtained within the scatter band of the experimental results. For the known friction coefficients and available FE and HBM tools, the described numerical process confirms its usability in the design of under-platform dampers.
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Cronin, Duane S. "Finite Element Analysis of Complex Corrosion Defects." In ASME 2002 Pressure Vessels and Piping Conference. ASMEDC, 2002. http://dx.doi.org/10.1115/pvp2002-1288.
Повний текст джерелаАнотація:
Aging gas and oil transmission pipeline infrastructure has led to the need for improved integrity assessment. Presently, external and internal corrosion defects are the leading cause of pipeline failure in Canada, and in many other countries around the world. The currently accepted defect assessment procedures have been shown to be conservative, with the degree of conservatism varying with the defect dimensions. To address this issue, a multi-level corrosion defect assessment procedure has been proposed. The assessment levels are organized in terms of increasing complexity; with three-dimensional elastic-plastic Finite Element Analysis (FEA) proposed as the highest level of assessment. This method requires the true stress-strain curve of the material, as determined from uniaxial tensile tests, and the corrosion defect geometry to assess the burst pressure of corrosion defects. The use of non-linear FEA to predict the failure pressure of real corrosion defects has been investigated using the results from 25 burst tests on pipe sections removed from service due to the presence of corrosion defects. It has been found that elastic-plastic FEA provides an accurate prediction of the burst pressure and failure location of complex-shaped corrosion defects. Although this approach requires detailed information regarding the corrosion geometry, it is appropriate for cases where an accurate burst pressure prediction is necessary.
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Fan, Muwei, Yang Wu, Wenhui Kong, and Jing Gong. "The Reliability Estimation of Simplified Natural Gas Pipeline Compressor Stations Based on Statistics Principles." In 2016 11th International Pipeline Conference. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/ipc2016-64084.
Повний текст джерелаАнотація:
With the rapid economic and rigorous requirement of environmental governance, natural gas serves as important energy source in industry consumption. In 2015, the total consumption of natural gas in China is approximately 142 billion m3. As the primary mode of natural gas transport, early pipeline operated for more than 20 years in China, and its reliability also attracts abundant concerns by the increasing potential risks. However, the process of pipeline reliability evaluating and estimation in China is still located under development. To solve the problem, a simplified reliability estimation method is introduced in this article. To begin with the main equipment of a compressing station, the filter, air cooler and compressors are three research objectives. Because of redundant design for enhancing reliability, the major equipment usually operates with a same spare unit. Thus, the simplified station is consisted by 3 main sections with multiple units as filter section, compressing section and cooling section. By assuming their reliability following normal distribution, the multivariate normal distribution model is available. Each unit is characterized by one dimension of the multivariate normal distribution. This article considers both relevant and irrelevant processes while the equipment is operating simultaneously and illustrates the results via two and three dimensions normal distribution calculation. Due to the model being singular function because of be established based on multivariate normal distribution and parameter estimation principles, the analytical solution is not available. Therefore, numerical computation is the compulsory solution. However, the reliability analysis of pipe is different from equipment and it has been studied for years in details from theory to experiments. Consequently, the reliability of pipe is analyzed by statistics method from collected operating data. A pipeline with 25-years operating data in China is applied in the case study chapter for reliability estimation and analysis. According to the operating data collected by China National Petroleum Corporation, the failure rate of major equipment is estimated by determining the parameters of each distribution and fitted a curve like bathtub curve. Similarly, the failure rate of pipe is indicated as failures per 103 km • year. For comparing, the operating data is also fitted a curve to validate the result of the model with the purpose of reasonably estimating the whole period reliability. The result demonstrates that the model is available in practice if the parameters are determined properly.