Infinitesimal Method Research Articles

The Performance Calculation Method of Fouled Axial Flow Compressor Based on the Stage Stacking Method

Fouling is the most important performance degradation factor, so it is necessary to accurately predict the effect of fouling on engine performance. This paper develops a performance calculation method of fouled multi-stage axial flow compressor based on experiment result and operating data. For multistage compressor, the whole compressor is decomposed into two sections. In this model, the performance of the first section is obtained by stage stacking method by combining scaling law method, linear progression model with traditional stage stacking method. On the other hand, the performance of the second section is calculated by averaged infinitesimal stage method. Finally, the model is successfully applied to predict the 8-stage axial flow compressor.

The new performance calculation method of fouled axial flow compressor.

Fouling is the most important performance degradation factor, so it is necessary to accurately predict the effect of fouling on engine performance. In the previous research, it is very difficult to accurately model the fouled axial flow compressor. This paper develops a new performance calculation method of fouled multistage axial flow compressor based on experiment result and operating data. For multistage compressor, the whole compressor is decomposed into two sections. The first section includes the first 50% stages which reflect the fouling level, and the second section includes the last 50% stages which are viewed as the clean stage because of less deposits. In this model, the performance of the first section is obtained by combining scaling law method and linear progression model with traditional stage stacking method; simultaneously ambient conditions and engine configurations are considered. On the other hand, the performance of the second section is calculated by averaged infinitesimal stage method which is based on Reynolds' law of similarity. Finally, the model is successfully applied to predict the 8-stage axial flow compressor and 16-stage LM2500-30 compressor. The change of thermodynamic parameters such as pressure ratio, efficiency with the operating time, and stage number is analyzed in detail.

Evolution and effect of the stress concentration and rock failure in the deep multi-seam coal mining

Supports crushing accident occasionally occurs in the protected seam exploitation of deep multi-seam coal mining structure and results in adverse effect to the production. To prevent its recurrence in a newly developed working field, a 3D numerical extraction model was built based on the geologic and mining conditions of Jining coal mine to evolve the changes, state and characteristics of the reconstructed vertical and lateral stress in rock interlayer after protective seam exploitation. Stress release and increase zones of this mining structure were separated. Mining-induced localized stress concentration and the interlayer rock failure behavior were explored. The action of concentrated stress on the hydraulic supports in protected seam was discussed upon the major stress redistribution. Using the infinitesimal strain method, a mechanical model was created to further explore, from the vertical and lateral directions, the cause and mechanism of localized stress concentration and rock failure behavior in rock interlayer. The field investigation was finally performed to verify the numerical and mechanical results, and the essential control measures were proposed to prevent this accident. Key findings of this study bring some new insights into the deep multi-seam coal extraction and help to promote a more reliable underground mining.

Differential invariants of a class of Lagrangian systems with two degrees of freedom

We consider systems of Euler–Lagrange equations with two degrees of freedom and with Lagrangian being quadratic in velocities. For this class of equations the generic case of the equivalence problem is solved with respect to point transformations. Using Lieʼs infinitesimal method we construct a basis of differential invariants and invariant differentiation operators for such systems. We describe certain types of Lagrangian systems in terms of their invariants. The results are illustrated by several examples.

Study on the Conversion Method of Three-Dimensional Shape Prediction Model of BGA Solder Joints

In the study of three-dimensional shape prediction of SMT solder joints, the software Surface Evolver has been widely applied as a quick and accurate effective tool for the prediction of solder joints shape. But the model it builds is not able to be directly imported into any finite element analysis software like ANSYS, and even after the import it still needs a lot of time to mend the import model. For this issue, to predict of the solder joints shape of ball grid array (BGA), the implement programs of three conversion methods of point-line-area method, axisymmetric method and infinitesimal method are given. By comparison, axisymmetric method and infinitesimal method are more suitable for the shape conversion of BGA solder joints.

Research of Baseball’s Best Batting Position Changes with the Selection of Material of Baseball

From the aspect of the fastest exit speed of the ball and through the collision model of mechanics, we analyzed the best hitting point which results in the fastest exit speed of the ball and the change of the best hitting point. Based on this model, we gave a very practical definition of the sweet spot, and we also found out the effect of the baseball bat’s material on the sweet spot. The distances from the best hitting point to the end of the baseball bat result from the three different materials, which are the bat’s original material, aluminum and the corked material. Through the sensitivity analysis, three parameters affecting the hitting position resulting in the ball’s fastest exit speed are the weight of the bat, the center of mass and the moment-of-inertia. In the process of the three parameters’ calculation, the infinitesimal method is used. As can be found out that the length of the sweet spot results from the corked and the aluminum are much longer than results from the original material. The longer the length of sweet spot, the possibility that the hitter hits the fast ball is higher. Also ball hit by the aluminum bat is faster than that hit by the bat made out of the original material. The above analysis can explain that why the aluminum bat and corked bat are prohibited by Major League Baseball

水下监测平台动态特性计算模型及影响因素分析

论文针对水下监测平台动力学特性问题，利用流场中结构振动模态的有限元计算方法及微元法的思想，考虑了流体对结构动态特性的影响，建立了水下监测平台浮体在流体的拖曳力、流体的垂直升力、浮力作用下的计算模型。通过数值分析方法，探讨了水下监测平台浮体固有频率的影响因素，通过对影响因素的调整，可以使浮体结构的固有频率远离水流漩涡释放频率，有效地减小共振给监测平台带来的危害，对水下监测平台的设计具有十分重要的意义。 The paper aimed at the dynamic characteristic of underwater monitoring platform, which endured drag force, normal lift along flow and buoyancy, and taken into account the influence of flow force to structures dynamic characteristic, the computational model was formulated with the finite computing method of structure vibration modal on flow field and infinitesimal method idea. The influencing factors were discussed for natural frequency of underwater monitoring platform by numerical analysis method. Natural frequency of buoy was far from the vortex release frequency by adjusting the influencing factors. This method can effectively decrease the vibration by resonance and it has important implications to design the underwater monitoring platform.

Preliminary group classification of the nonlinear differential–difference equations

Preliminary group classification of the nonlinear differential–difference equations of the form ün=Fn(t,un−1,un,un+1) is considered by using the classical infinitesimal Lie method and the theory of classification of abstract low-dimensional Lie algebras. Restrict to Lie point symmetries, and assume that the Lie algebra of the symmetry group is realized by a particular vector field. It is proved that there is only one equation admitting three-dimensional simple Lie algebras. What is more, all the inequivalent equations admitting semi-simple Lie algebras are nothing but the one. Also there exist inequivalent equations admitting low-dimensional solvable Lie algebras and semi-direct sums of semi-simple and solvable Lie algebras.

Studies of TLP dynamic response under wind, waves and current

Investigated is the coupled response of a tension leg platform (TLP) for random waves. Inferred are the mass matrix, coupling stiffness matrix, damping matrix in the vibration differential equation and external load of TLP in moving coordinating system. Infinitesimal method is applied to divide columns and pontoons into small parts. Time domain motion equation is solved by Runge-Kutta integration scheme. Jonswap spectrum is simulated in the random wave, current is simulated by linear interpolation, and NPD spectrum is applied as wind spectrum. The Monte Carlo method is used to simulate random waves and fluctuated wind. Coupling dynamic response, change of tendon tension and riser tension in different sea conditions are analyzed by power spectral density (PSD). The influence of approach angle on dynamic response of TLP and tendon tension is compared.

Based on the Target Track Cam Design Method and Numerical Solution Research

In this paper, a new design method of cam is developed .To illustrate this method,a car which adopts cam mechanism to achieve specific movements is designed. By establishing the relationship between the point coordinates of cam profile and the track through the relevant mathematical models, and using the infinitesimal elements method to solve the models, the discrete points coordinates of the cam profile are obtained. The displacement, velocity and acceleration of the direction arm are simulated by Pro/E. the accurate trace of cam profile can be obtained by the method. To make the CAD/CAE/CAM integrated design of cam is reasonable.

Factorisation properties of group scheme actions

Let H be an algebraic group scheme over a field k acting on a commutative k-algebra A which is a unique factorisation domain. We show that, under certain mild assumptions, the monoid of nonzero H-stable principal ideals in A is free commutative. From this we deduce, in certain special cases, results about the monoid of nonzero semi-invariants and the algebra of invariants. We use an infinitesimal method which allows us to work over an arbitrary base field.

Equivalence of Linear Systems of Two Second-Order Ordinary Differential Equations

An equivalence problem is solved completely for a linear system of two second-order ordinary differential equations. Using Lie's infinitesimal method we construct the basis of differential invariants for this class of equations and provide the operators of invariant differentiation. Certain types of linear systems are described in terms of their invariants. Some examples are given to illustrate our results. Key Words: Linear Equation; System of Second-Order Ordinary Differential Equations; Equivalence Problem; Differential Invariant

Smoothness of certain functions in two kinds of risk models with a barrier dividend strategy

In this paper, we study the smoothness of certain functions in two kinds of risk models with a barrier dividend strategy. Mainly using technique from the piecewise deterministic Markov processes theory, we prove that the function is continuously differentiable in the first risk model. Using the weak infinitesimal generator method of Markov processes, we prove that the function is twice continuously differentiable in the second risk model. Intego-differential equations satisfied by them are derived.

Differential invariants for systems of linear hyperbolic equations

In this paper we consider a general class of systems of two linear hyperbolic equations. Motivated by the existence of the Laplace invariants for the single linear hyperbolic equation, we adopt the problem of finding differential invariants for the system. We derive the equivalence group of transformations for this class of systems. The infinitesimal method, which makes use of the equivalence group, is employed for determining the desired differential invariants. We show that there exist four differential invariants and five semi-invariants of first order. Applications of systems that can be transformed by local mappings to simple forms are provided.

A symmetry classification for a class of (2+1)-nonlinear wave equation

In this paper, a symmetry classification of a (2+1)-nonlinear wave equation u t t − f ( u ) ( u x x + u y y ) = 0 where f ( u ) is a smooth function on u , using Lie group method, is given. The basic infinitesimal method for calculating symmetry groups is presented, and used to determine the general symmetry group of this (2+1)-nonlinear wave equation.

Preliminary group classification of quasilinear third-order evolution equations

Group classification of quasilinear third-order evolution equations is given by using the classical infinitesimal Lie method, the technique of equivalence transformations, and the theory of classification of abstract low-dimensional Lie algebras. We show that there are three equations admitting simple Lie algebras of dimension three. All non-equivalent equations admitting simple Lie algebras are nothing but these three. Furthermore, we also show that there exist two, five, twenty-nine and twenty-six nonequivalent third-order nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvable Lie algebras, respectively.

Invariants of a family of third-order ordinary differential equations

We consider a family of third-order ordinary differential equations (ODEs) with the cubic nonlinearity in second-order derivative, which is closed with respect to contact transformations. Using Lie's infinitesimal method we construct the basis of differential invariants for this family of equations and the operators of invariant differentiation. Invariants provide a simple way of finding the equations, which may be equivalent to a given nonlinear ODE and the transformation connecting two equivalent equations. Some examples are given to illustrate the results obtained in the paper.

Dynamic impedances of pile groups with embedded caps in homogeneous elastic soils using CIFECM

Significant research has been reported on the dynamic analysis of pile groups. However, in most of the cases, the effect of pile cap is neglected despite the fact that there may be additional interactions due to the presence of the cap. This paper presents the dynamic impedances for the pile groups with caps embedded in isotropic homogeneous elastic soils. A general three-dimensional finite element procedure is developed. The system is sub-structured into bounded near-field and an unbounded far-field. The pile-soil system of the near-field is modeled using solid finite elements, and the unbounded elastic soil system of the far-field is modeled using the consistent infinitesimal finite element cell method (CIFECM) in the frequency domain. The method has shown excellent agreement with the published results for dynamic response of pile groups with a free-standing pile-cap. The dynamic impedances of pile groups with embedded cap are presented. Each pile group is capped with a thick pile-cap, embedded in the near-field soil domain. The dynamic impedances of 2×2 pile groups are presented, for different spacing of piles and thickness of the pile-cap with different embedment ratios. It was found that even for a small embedment ratio, the dynamic interactions due to cap can be significant.

Nonparametric Estimation of Standard Errors in Covariance Analysis Using the Infinitesimal Jackknife

The infinitesimal jackknife provides a simple general method for estimating standard errors in covariance structure analysis. Beyond its simplicity and generality what makes the infinitesimal jackknife method attractive is that essentially no assumptions are required to produce consistent standard error estimates, not even the requirement that the population sampled has the covariance structure assumed. Commonly used covariance structure analysis software uses parametric methods for estimating parameters and standard errors. When the population sampled has the covariance structure assumed, but fails to have the distributional form assumed, the parameter estimates usually remain consistent, but the standard error estimates do not. This has motivated the introduction of a variety of nonparametric standard error estimates that are consistent when the population sampled fails to have the distributional form assumed. The only distributional assumption these require is that the covariance structure be correctly specified. As noted, even this assumption is not required for the infinitesimal jackknife. The relation between the infinitesimal jackknife and other nonparametric standard error estimators is discussed. An advantage of the infinitesimal jackknife over the jackknife and the bootstrap is that it requires only one analysis to produce standard error estimates rather than one for every jackknife or bootstrap sample.

Stability of a two-dimensional airfoil with time-delayed feedback control

This paper presents a systematic study on aeroelastic stability of a two-dimensional airfoil with a single or multiple time delays in the feedback control loops. Firstly, the delay-independent stability region of the aeroelastic system with a single time delay is determined on the basis of the generalized Sturm criterion for polynomials. Then, the stability switches with variations in time delay are analyzed when the system parameters fall out of the delay-independent stability region. Flutter boundaries of the controlled aeroelastic system as time delay varies are predicted in a continuous way by the predictor-corrector technique. Finally, two methods, the polynomial eigenvalue method and the infinitesimal generator method, are introduced to investigate the stability of the controlled aeroelastic system with multiple time delays. Numerical simulations are made to demonstrate the effectiveness of all the above approaches.