Deflection Surface Research Articles

On cutter deflection surface errors in peripheral milling

Cutter deflections induce significant amount of surface error on machined components and it is one of the major obstacles towards achieving higher productivity in peripheral milling operation. These surface errors do not take one particular form and their shape and profile measured along axial direction, varies significantly with cutting conditions. The understanding and characterization of all possible surface error types is of immense value to process planners as it forms a basis for controlling and compensating them. This paper presents a methodology to classify surface error profiles and to relate the same with cutting conditions in terms of axial and radial engagement between cutter and workpiece. The proposed characterization scheme has been validated using computational studies and machining experiments. The importance of proposed characterization is further demonstrated in understanding peripheral milling of curved geometries where workpiece curvature influences radial engagement of the cutter that often changes surface error shape both qualitatively and quantitatively. Computational and experimental studies undertaken to study machining of curved geometries underline the importance of proposed characterization scheme in identifying correct cutting conditions for a given machining situation.

Study on Gradual Deformation and Prediction Model of the Mined Overburden

Based on the overburden three caving feature, the deformation of mining rock process was devided and the criterion of gradual transformation of each stages deformation were given. Then , combined the thin-plate theory, the differential models were derived for rock deformation in level and similar to level bured condition. The boundary condition of each models and exchange condition between different models were put forward and the gradual mechanics model was set up.The subsidence model before roof collapse was solved by Navier double trigonometric series and the deflection surface expression of rock deformation was put forword. At last, the reliability and practicality of the models was verified by engineering examples.

Effect of stiffener on nonlinear characteristic behavior of a rectangular plate: A single mode approach

In this paper, nonlinear dynamic of an excited square plate with a single stiffener is investigated using a single mode approach. In a first part, we present an exact solution of the buckling mode of the stiffened plate under uni-axial compression. The differential equation of the deflection surface of the rectangular plate with different boundary conditions is used and the effect of the stiffener upon natural frequency of the plate is examined. In a second part, we analyze the dynamic of the excited mode near the principal resonance. Specifically, we study the influence of stiffener on nonlinear characteristic behavior of the excited plate.

Behaviour evolution of automotive connectors

Automotive connector ageing has been studied in the literature, but none of these studies is completely satisfactory for finding the relation between the insertion force, the contact surface, and the spring deflection and wear after several insertion phases. The main objective of this work is to find and establish a correlation between insertion parameters and wear behaviour using numerical and experimental approaches. This study is concerned with automotive connectors made of copper with a tin coating of thickness 2 μm. The evolution of the insertion force, spring deflection, and contact surface versus the insertion—extraction operation number will provide information about the connector wear behaviour. Using a microbalance, the mass of the sample was determined before and after the experimental insertion test, to quantify the mass loss and the wear evolution. In addition, the wear shape was analysed. The connector includes four spring lamellae on each side and the pin has a flat shape. Owing to the friction effect and after several insertion—extraction tests, eight tracks are visible. Using a scanning profilometer, the length, the depth, and the width of the track can be measured. At the same time, a numerical analysis using the finite element code Ansys is used with an elastoplastic behaviour law to find the fatigue law, to measure the contact surface, and to predict the wearing surface for the materials employed, which can support a long connector use period. The selected assumptions introduced in the numerical modelling lead to a good correlation with the experimental results.

A Simple Spline Integral Equation Method for Circular Plates with Variable Thickness

A simple spline integral equation method is presented in this paper for the axisymmetrical bending of circular plates with variable thickness. Firstly, the fundamental solution of a second-order differential equation is derived. With the slope of the deflection surface taken as an unknown function, an integral equation is then established for circular plates with variable thickness. The integral equation is solved numerically by cubic spline interpolation and the deflection and bending moment at any point within the circular plate are obtained. Finally, the validity of the proposed method is verified with the analytical solution obtained from the literature.

A New Numerical Method for Axisymmetrical Bending of Circular Plates with Large Deflection

A simple and efficient spline integral equation method is presented in this paper for the axisymmetrical bending of circular plates with large deflection. Based on two second-order differential equations in terms of the slope of the deflection surface and the radial displacement of the circular plate, two integral equations are derived. The circular plate is then equidistantly divided into a circular plate element and a series of annular plate elements along its radial direction and the slope of the deflection surface and the radial displacement are both approximated by cubic spline interpolation. The two integral equations are solved numerically and the displacements and internal forces at any point within the circular plate can be obtained. Finally, some numerical results are presented for illustrating the validity of the proposed method. It can be concluded that the proposed numerical method can be used to analyze circular plates with large deflection with reasonable accuracy.

A kind of bivariate spline space over rectangular partition and pure bending of thin plate

The mechanical background of the bivariate spline space of degree 2 and smoothness 1 on rectangular partition is presented constructively. Making use of mechanical analysis method, by acting couples along the interior edges with suitable evaluations, the deflection surface is divided into piecewise form, therefore, the relation between a class of bivariate splines on rectangular partition and the pure bending of thin plate is established. In addition, the interpretation of smoothing cofactor and conformality condition from the mechanical point of view is given. Furthermore, by introducing twisting moments, the mechanical background of any spline belong to the above space is set up.

The solution of certain loss of contact between a plate and unilateral supports

This paper examines the loss of contact between a square plate and the unilateral supports under uniformly distributed load. Since the plate is rested on the unilateral supports, it will have the regions of lost contact between a plate and the supports due to the absence of restraining corner force at the plate corners. This leads to the mixed boundary conditions and these conditions are then written in the form of dual-series equations which can further be reduced to a Fredholm integral equation by taking advantage of finite Hankel transform technique. Numerical results are given for the deflections of free edge and deflections along the middle line of the plate with deferent values of the Poisson?s ratio. In addition, the deflection surface is also presented. From the investigation, it can be indicated that the loss of contact is decreased upon the increasing Poisson?s ratio.

Geometrically nonlinear analysis of elastic membranes with embedded rigid inclusions

In this paper the deformation of membranes containing rigid inclusions is analyzed. These rigid inclusions can significantly change the entire stress distribution in the membrane and therefore create major difficulties for the design. The initially flat membrane, which may be prestretched by boundary in-plane tractions or displacements, is subjected to externally applied loads and to the weight of the rigid inclusions. The composite system is examined in cases where its deformation reaches a state for which the undeformed and deformed shapes are substantially different. In such cases large deflections of membranes are considered, which result from nonlinear kinematic relations. The three coupled nonlinear equations in terms of the displacements governing the response of the membrane are solved using the analog equation method, which reduces the problem to the solution of three uncoupled Poisson's equations with fictitious domain source densities. The problem is strongly nonlinear [Katsikadelis JT, Nerantzaki MS. The boundary element method for nonlinear problems. Eng Anal Boundary Elements 1999;23:365–73]. In addition to the geometrical nonlinearity, the problem is itself nonlinear, since the membrane's reactions on the boundary of the rigid inclusions are not a priori known as they depend on the produced deflection surface. Iterative schemes are developed for calculation of deformed membrane's configuration, which converge to the final equilibrium state of the membrane with the given external applied loads. Several example problems are presented, which illustrate the method and demonstrate its accuracy and efficiency. The method employed for the solution is boundary only with all the advantages of the pure BEM.

Analytical Solution for Piles Supporting Combined Lateral Loads

Analytical solutions of normalized maximum deflection and normalized maximum moment for laterally loaded long piles in homogeneous elastoplastic soil under combined loads are presented in this paper. Both the normalized deflection surface and normalized moment surface are continuous and increasing constantly with normalized applied force and moment with various slopes. It shows that the normalized applied force and moment have different contributions to the deflection and moment. In general, the variation of normalized moment surface is relatively moderate compared to normalized maximum deflection. Due to the nonlinear effect, the deflections and moments using superposition approach will be on the unsafe side. The analytical solutions can be used for any elastic materials, any type of soil, and any shapes of pile cross section. Above all, the analytical solutions may be easily applied to calculate the maximum deflection or moment of lateral long piles subject to combined loads accurately using a calculator.

Paraxial and Aberration Analysis of Off-Axial Optical Systems

Off-axial optical systems are an extended concept of co-axial optical systems, where deflection surfaces are arranged along the folded reference axis. A newly proposed aberration expression with two kinds of 4-element vectors is simple and comprehensive in its order relation for off-axial optical systems. Using these vectors and tensor algebra, it is shown that aberration expansion coefficients for an arbitrary azimuth can be calculated by transforming the coefficients of 4-element fundamental ray vectors, and the aberration contributions of each surface are in general represented not in simple sum form, but in converted sum form. The aberration contribution of cross terms that exist over the third order, can also be clearly represented using tensor coefficient representation. This paraxial and aberration analysis of off-axial optical systems is very useful in the actual design of optical systems.

Ponding on floating membranes

Membranes subjected to ponding loads and floating on a liquid are analyzed. The initially flat membrane, which may be prestressed by edge in-plane tractions or displacements, is subjected to the weight of a liquid (e.g. rain water) filling the space created by the deflection of the membrane. Large deflections of membranes are considered, which result from nonlinear kinematic relations. The three coupled nonlinear equations in terms of the displacements governing the response of the membrane are solved using the analog equation method, which reduces the problem to the solution of three uncoupled Poisson's equations with fictitious sources. The problem is strongly nonlinear. In addition to the geometrical nonlinearity, the ponding problem is itself nonlinear, because the ponding load and the liquid reaction are not a priori known as they depend on the produced deflection surface. Iterative schemes are developed which converge to the equilibrium state of the membrane. Example problems are presented, which illustrate the method and demonstrate its efficiency. The method has all the advantages of the pure BEM.

The ponding problem on elastic membranes: an analog equation solution

In this paper, the nonlinear response of elastic membranes with arbitrary shape under partial and full ponding loads has been analyzed. Large deflections are considered, which result from nonlinear kinematic relations. The problem is formulated in terms of the displacements components and the three coupled nonlinear governing equations are solved using the analog equation method (AEM). The membrane may be prestressed either by prescribed boundary displacements or tractions. Using the concept of the analog equation the three coupled nonlinear equations are replaced by three uncoupled Poisson's equations with fictitious sources under the same boundary conditions. Subsequently, the fictitious sources are established using a procedure based on the BEM and the displacement components as well as the stress resultants at any point of the membrane are evaluated from their integral representations. In addition to the geometrical nonlinearity, the ponding problem is itself nonlinear, because the ponding load depends on the deflection surface that it produces. Iterative schemes are developed which converge to the equilibrium state of the membrane under the ponding loads. Several membranes are analyzed which illustrate the method and demonstrate its efficiency and accuracy. The method has all the advantages of the pure BEM, since the discretization and integration is limited only to the boundary.

Buckling mode localization in rib-stiffened plates with misplaced stiffeners – Kantorovich approach

Buckling mode localization in rib-stiffened plates with randomly misplaced stiffeners is studied in this paper. The method of Kantorovich on reducing a partial differential equation to a system of ordinary differential equations is employed to obtain the deflection surface of the rib-stiffened plates under axial compressive load. The edges of the plates normal to the stiffeners can be either simply supported or clamped. The solutions of the deflection surface are then expressed in the form of transfer matrices. The expressions of the solutions obtained for the case of one edge simply supported and one edge clamped and the case of two edges clamped are similar to those for the case of two edges simply supported. When the two edges are simply supported, the method of Kantorovich yields the exact results. Localization factors, which characterize the average exponential rates of growth or decay of amplitudes of deflection, are determined using the method of transfer matrix. The method of Kantorovich is a general approximate method, which is applicable for various support conditions.

Studies on High-precision Slicing with ID-blade. Compliance of ID-blade.

This paper elucidates the size effect of ID-blade on its lateral compliance at the inside edge. The obtained results are summarized as follows. (1) The deflection surface of ID-blade by a single-point lateral load acting at the inside edge has a large slope around the loading point especially in the radial direction. (2) The deflection of the larger-sized ID-blade by a single point unit load usually extends wider in the angle having its vertex at the center of ID blade. (3) The compliance of ID blade decreases exponentially with an increase of the tangential stress at the inside edge σIaθ. (4) While the larger-sized ID-blade has the larger compliance, the difference becomes small with an increase of the stress σIaθ. (5) Under the condition that the outside diameter of ID blade is constant, the compliance of ID-blade increases with an increase of the inside diameter. (6) When there being enough difference in size between the inside and outside diameters of ID blade, the effect of the outside diameter on the compliance is small. (7) The compliance of ID-blade is almost inversely proportional to the thickness of D-blade. (8) The compliance of ID-blade increases with a decrease of the radial width of the electroplated diamond layer.

Vibrations of Clamped Rectangular Plates on Elastic Foundations Subjected to Uniform Compressive Forces

This paper deals with a different solution technique for free vibrations of rectangular plates resting on elastic foundations with clamped boundaries and subjected to uniform and constant compressive, unidirectional forces in the midplane. The applied method is based on the use of a nonorthogonal series expansion consisting of some specially chosen trigonometric functions for the deflection surface W¯ of the plate. The orthogonalization of the series and other calculations are performed using Fourier expansion of Bernoulli polynomials under some realistic approximations for the limiting values of the boundary conditions. In this method one need not use the solution of the differential equation of the problem. The results obtained for the problem are consistent with the well-known solutions.

Efforts by North Carolina Department of Transportation to Develop Mechanistic Pavement Design System

A first generation mechanistic empirical pavement design procedure was developed using falling weight deflectometer deflections taken over a 3-year period at 16 test sections in Siler City, North Carolina. Information available for use in developing the procedure included deflection data, surface and air temperature, coring thicknesses at each test location, pavement performance records regarding rate of cracking, and traffic records. Jung's method, based on the curvature of the deflection bowl, was used to calculate strain at the bottom of the asphalt layer as a measure of fatigue. This calculated strain was used to obtain a calculated number of load repetitions to failure. Comparision of actual loads to failure with calculated loads to failure resulted in a table of shift factors by pavement type.

Efforts by North Carolina Department of Transportation to Develop Mechanistic Pavement Design System

A first generation mechanistic empirical pavement design procedure was developed using falling weight deflectometer deflections taken over a 3-year period at 16 test sections in Siler City, North Carolina. Information available for use in developing the procedure included deflection data, surface and air temperature, coring thicknesses at each test location, pavement performance records regarding rate of cracking, and traffic records. Jung's method, based on the curvature of the deflection bowl, was used to calculate strain at the bottom of the asphalt layer as a measure of fatigue. This calculated strain was used to obtain a calculated number of load repetitions to failure. Comparision of actual loads to failure with calculated loads to failure resulted in a table of shift factors by pavement type.

Flexural strength, fracture toughness and fatigue crack growth behaviour of chromium alloyed ?-base TiAl

Mechanical properties of γ-base titanium aluminides strongly depend on microstructural parameters. Flexural strength, fracture toughness and fatigue crack growth properties were estimated for the cast and heat-treated (HT-TiAlCr) and cast, heat treated and isothermal forged (ITF-TiAlCr) chromium alloyed γ-base titanium aluminides at room temperature. HT-TiAlCr possessed superior fracture properties compared to ITF-TiAlCr. Toughening due to microcracks, and crack bridging by uncracked ligaments were observed in the test materials. Presence of lamellar grains in HT-TiAlCr increased the crack growth resistance and contributed positively to fracture properties. The coarse grain size promoted large crack deflection and fracture surface mismatch and caused high levels of crack closure in HT-TiAlCr. Combined crack-tip blunting and bridging by ductile β-phase was significant in the case of ITF-TiAlCr. Fracture mechanisms of test materials were investigated and correlated to the fracture properties.

Linearity of deflections of long-term loaded slabs

The effects of different long-term loading levels including those when cracks occur on the deflections of one-way reinforced slabs were investigated. Three series of slabs with different types of reinforcement were subjected to a sustained load at five levels over a period of about 800 days. The results have confirmed that a linear relationship between the initial and time-dependent deflections can be assumed up to a high loading level (0.8). Moreover, the effects of the age of concrete at the application of long-term load were studied. A series of 20 two-way slabs reinforced with welded deformed wire fabrics, made of concrete using river and crushed aggregate, were tested. Sustained load at the level of about 0.5 (with regard to the failure limit) was applied to slabs at the age of 7, 14, 28, 60 and 90 days and maintained for about 400 days. The results showed that the deflection surfaces at the two arbitrary time intervals were similar in spite of intensive crack development. The dependence of the deflection on the age of concrete at the time of load application was proved.