Sine Law Research Articles

Constitutive Description of 7075 Aluminum Alloy During Hot Deformation by Apparent and Physically-Based Approaches

Hot flow stress of 7075 aluminum alloy during compressive hot deformation was correlated to the Zener-Hollomon parameter through constitutive analyses based on the apparent approach and the proposed physically-based approach which accounts for the dependence of the Young’s modulus and the self-diffusion coefficient of aluminum on temperature. It was shown that the latter approach not only results in a more reliable constitutive equation, but also significantly simplifies the constitutive analysis, which in turn makes it possible to conduct comparative hot working studies. It was also demonstrated that the theoretical exponent of 5 and the lattice self-diffusion activation energy of aluminum (142 kJ/mol) can be set in the hyperbolic sine law to describe the peak flow stresses and the resulting constitutive equation was found to be consistent with that resulted from the proposed physically-based approach.

Hot compression behavior of GZ31 magnesium alloy

Hot deformation behavior of Mg–3Gd–1Zn (GZ31) magnesium alloy was studied by hot compression tests over the temperature range of 300–500°C under strain rates of 0.0001–0.1s−1. This material exhibited typical broad single-peak dynamic recrystallization behavior followed by a gradual drop towards the steady state stress. The constitutive behavior of the tested alloy was studied by the power, exponential, and hyperbolic sine laws. The stress multiplier and the hyperbolic sine exponent were calculated as 0.024MPa−1 and 3.42, respectively. The deformation activation energy was found to be about 173.2kJ/mol, which is higher than the lattice self-diffusion activation energy of magnesium (135kJ/mol). The latter can be ascribed to the presence of gadolinium, which shows the importance of rare earth elements in increasing the deformation resistance at high temperatures.

Linear Theory for Self-Localization: Convexity, Barycentric Coordinates, and Cayley–Menger Determinants

Localization, finding the coordinates of an object with respect to other objects with known coordinates—hereinafter, referred to as anchors, is a nonlinear problem, as it involves solving circle equations when relating distances to Cartesian coordinates, or, computing Cartesian coordinates from angles using the law of sines. This nonlinear problem has been a focus of significant attention over the past two centuries and the progress follows closely with the advances in instrumentation as well as applied mathematics, geometry, statistics, and signal processing. The Internet-of-Things (IoT), with massive deployment of wireless tagged things, has renewed the interest and activity in finding novel, expert, and accurate indoor self-localization methods, where a particular emphasis is on distributed approaches. This paper is dedicated to reviewing a notable alternative to the nonlinear localization problem, i.e., a linear-convex method, based on Khan et al. ’s work. This linear solution utilizes relatively unknown geometric concepts in the context of localization problems, i.e., the barycentric coordinates and the Cayley–Menger determinants. Specifically, in an $m$ -dimensional Euclidean space, a set of $m+1$ anchors, objects with known locations, is sufficient (and necessary) to localize an arbitrary collection of objects with unknown locations—hereinafter, referred to as sensors, with a linear-iterative algorithm. To ease the presentation, we discuss the solution under a structural convexity condition, namely, the sensors lie inside the convex hull of at least $m+1$ anchors. Although rigorous results are included, several remarks and discussion throughout this paper provide the intuition behind the solution and are primarily aimed toward researchers and practitioners interested in learning about this challenging field of research. Additional figures and demos have been added as auxiliary material to support this aim.

DESARROLLO DE UNA GEOMETRIA EXPERIMENTAL MEDIANTE EL PROCESO DE CONFORMADO INCREMENTAL DE LAMINA EN DOS PUNTOS

Industrialized countries with modern manufacturing technologies forming sheet, have studied various manufacturing processes of sheet metal parts (stamping, hydroforming, super forming). Recently have worked with pieces that use the principle of forming a sheet whit a tool that builds up a particular geometry without supporting cast with a single point of contact for the forming process (Dieless-SPIF) [1] [2]. This paper presents and explains the experiment on a pyramidal geometry with a supporting cast on the same principle of incremental deformation having two contact points, which give creation to a new method derived from Dieless, that allows to extend the radius of application various geometries, complex and unique shapes, Dieless - DPIF. The design of the geometry, the respective CAM and G-code generation, for the tool path to be processed by a machine center and CNC computer and as the ratio of the thicknesses measured in different areas of it, depending on the angle ? of formability, through the principles of the sine law, to determinate the applicability of it in this innovative manufacturing process.

Constitutive behaviors of magnesium and Mg–Zn–Zr alloy during hot deformation

The flow stress of pure magnesium and ZK60 (Mg–6Zn-0.6Zr) magnesium alloy during high-temperature deformation were correlated to the Zener-Hollomon parameter through analyses based on the apparent and physically-based approaches. It was demonstrated that the theoretical exponent of 5 and the lattice self-diffusion activation energy of magnesium (135 kJ/mol) can be set in the hyperbolic sine law to describe the peak flow stress of either the pure Mg or highly alloyed one with zinc and zirconium. As a result, the influence of alloying elements upon the hot flow stress of the ZK60 alloy was characterized by the proposed approach based on the simple material's constants. One of the main prospects of the proposed approach, which is not possible by the conventional approach, is its ability to be utilized in the comparative hot working and alloy development studies.

An efficient method for thickness prediction in multi-pass incremental sheet forming

Incremental sheet forming (ISF) is a highly versatile and flexible process for rapid manufacturing of complex sheet metal parts. In the ISF process, efficient and accurate prediction of part thickness variation is still a challenging task, which is especially true for the multi-pass ISF process. The Sine law equation and the finite element method (FEM) are the two commonly used conventional prediction methods. However, these approaches are either with limited accuracy or very time consuming. For the multi-pass ISF process, the thickness prediction is even more challenging since two or more forming steps are involved. Focusing on the thickness prediction of multi-stage ISF process, this work proposes a thickness prediction model based on the geometrical calculation of intermediate shapes of the formed part and backward tracing of nodal points of the forming tool. By developing this method, the thickness distribution can be calculated through the predicted nodal displacement in the ISF process. To verify the proposed model, four different geometrical shapes, i.e., conic, parabolic conic, non-axisymmetric, and hemispherical parts, are physically formed by using a NC ISF machine. The geometric shapes and the detailed thickness distributions of the formed parts are carefully measured and compared with the prediction model developed. Good agreements between the analytical predictions, and the experimental results are obtained. This indicates the effectiveness and robustness of the developed thickness prediction approach.

Constitutive analysis to predict the hot deformation behavior of 34CrMo4 steel with an optimum solution method for stress multiplier

The hot deformation behaviors of steel 34CrMo4 is investigated by hot compression test with the temperature range of 1073–1373 K and the strain rate range of 0.01–10 s−1. The flow behaviors of 34CrMo4 steel were characterized based on the true stress–true strain curves. The hyperbolic sine law in Arrhenius type is adopted in the constitutive modeling for 34CrMo4. Solving algorithm of the stress multiplier α in hyperbolic sine law is a key factor to guarantee the constitutive model accuracy. How to solve the stress multiplier α is investigated and an optimum solution method for α is proposed. Meanwhile, the influence of strain is incorporated in constitutive analysis by considering the effect of strain on material constants α, n, Q and A. With the optimum solution method for stress multiplier α proposed, the stress prediction is satisfactory with the higher correlation coefficient, R = 0.988 and the lower average absolute relative error, AARE = 3.44% for the entire strain rate-temperature domain. The optimum solution method for stress multiplier α can also be applied for other materials to predict the flow behavior more accurately.

Constitutive analysis of Mg–Al–Zn magnesium alloys during hot deformation

The constitutive behaviors of Mg–Al–Zn magnesium alloys during hot deformation were studied over a wide range of Zener–Hollomon parameters by consideration of physically-based material’s parameters. It was demonstrated that the theoretical exponent of 5 and the lattice self-diffusion activation energy of magnesium (135kJ/mol) can be used in the hyperbolic sine law to describe the flow stress of AZ31, AZ61, AZ80, and AZ91 alloys. The apparent hyperbolic sine exponents of 5.18, 5.06, 5.17, and 5.12, respectively for the AZ31, AZ61, AZ80, and AZ91 alloys by consideration of deformation activation energy of 135kJ/mol were consistent with the considered theoretical exponent of 5. The influence of Al upon the hot flow stress of Mg–Al–Zn alloys was characterized by the proposed approach, which can be considered as a versatile tool in comparative hot working and alloy development studies. It was also shown that while the consideration of the apparent material’s parameters may result in a better fit to experimental data, but the possibility of elucidating the effects of alloying elements on the hot working behavior based on the constitutive equations will be lost.

Hot deformation behaviour of Mg–Al–Zn alloys

The hot working behaviour of Mg–Al–Zn alloy was studied in the temperature range of 270–370°C and strain rate range of 0·001–0·1 s−1. In order to eliminate the influence of Mg17AL12 phase, the magnesium alloy was solutionised at 420°C for 24 h and then quenched in water before hot compression. Results show that the microstructures are mainly characterised by twinning and dynamic recrystallised grains during the hot compression process. The twinning forms at lower temperature, while the dynamic recrystallised grains are usually found at higher temperature and lower strain rate. Moreover, constitutive analysis was carried out by the hyperbolic sine law function, which illustrated that the deformation mechanism of the present alloy is controlled by dislocation climb. The critical conditions associated with the onset of dynamic recrystallisation determine the changes of the strain hardening rate (θ), and the amount of dynamic recrystallised grains increases with the increasing strain in the form of sigmoid scheme.

Experimental and numerical studies on formability of extra-deep drawing steel in incremental sheet metal forming

This paper focuses on the formability and thickness distribution in incremental sheet forming (ISF) of extra-deep drawing steel (EDD). In ISF, the formability of the material is primarily measured by the maximum formable wall angle and maximum allowable thinning. The maximum wall angle is generally obtained by forming frustum of cones and square pyramids having different wall angles till fracture, which requires a large number of experiments. Therefore in the present study, a continuously varying wall angle conical frustum (VWACF) was used to predict the maximum wall angle to minimize the number of experiments. VWACF is generated using circular, parabolic, elliptical and exponential generatrices. In order to get the maximum allowable thinning, the thickness of the formed geometry has been measured at various points along the depth. In addition, the thickness distribution has been computed theoretically based on the sine law and also using finite element code LS-DYNA. Theoretical and simulated thickness values have been compared with measured thickness values. It was found from the results that the finite element model was more accurate than theoretical model in predicting thickness distribution.

Dynamic Optimization Design of Sand Milling Collection Machinery Suspensions

Sand milling collection machinery is a kind of large machinery to collect the surface soil containing ores. The suspension system of milling machines directly affects the ride comfort and driving safety for the operator; for this end, the design of suspension system has been optimized so as to satisfy the requirements of reliability and comfort by making the console and operator set at the maximum station and minimum absolute acceleration and the amplitude within the range. This paper first established a suspension system model of milling collection machines and set up the state equation of suspension system by applying Lagrange equation. On this basis, this paper conducted the optimized calculation encouraged bt the pavement with sine rule change and obtained an optimal suspension system parameters ki and ci.

Study on Robot's Obstacle Avoidance Walking

The paper researches on robots obstacle avoidance walking in a certain range, searching the shortest path between two points and the shortest walking time under some limitation of speed conditions. The paper first uses AutoCAD software to carry out field simulation, drawing out various robot walking route options. Then applying the distance between two points, sine and cosine theorems and related knowledge of plane geometry such as arc length formula to carry out modeling, and then uses Mathematica software to calculate various length of the path, finally take the shortest after analysis and comparison. On this basis, and after summed up various walking routes, we can see that from the start point to destination in this paper, the less the wheel number is, the narrower the distance between obstacles is, the shorter the path is. Finally, the paper carries out systematic evaluation and improvement on the disadvantages and advantages of the model, and gives the model promotes ideas.

Influence of Geometrical Parameters, Wall Angle and Part Shape on Thickness Reduction of Single Point Incremental Forming

The aim of this paper is to study the thickness reduction during the whole single point incremental forming process. The experimental setup is realized in such a way that the blank sheet is in the vertical position and forming is done using an anthropomorphic robot with six degrees of freedom. The vertical positioning of the blank sheet allows the on-line visualization using an optical measuring system of the major and minor strains and of the thickness reduction. The wall angle varied between values of 450, 550 and 650 respectively. The logarithmic major strain and minor strains and logarithmic thickness reduction were analyzed in order to allow a comparison with values obtained based on the sine law. The results show that indeed, the maximal values of the thickness reduction observe the sine law for the parts but only after a “critical value” of the part's height is reached, the variation of the thickness reduction on the lateral faces of the pyramidal frustum are large, the thickness reduction is significant in some areas as compared to the maximal values.

Accurately Measuring the Height of (Real) Forest Trees

Q uick and accurate tree height measurement has always been a goal of foresters. The techniques and technology to measure height were developed long ago—even the earliest textbooks on mensuration showcased hypsometers (e.g., Schlich 1895, Mlodziansky 1898, Schenck 1905, Graves 1906), and approaches to refine these sometimes remarkable tools appeared in the first issues of Forestry Quarterly, Proceedings of the Society of American Foresters, and the Journal of Forestry. For example, one such hypsometer based on the geometric principle of similar triangles (top of Figure 1) employed rotary mirrors to allow the user to simultaneously see the top and bottom of the tree in “proper parallax” (Tieman 1904). Other early hypsometers applied different approaches that used angles and distance (e.g., Graves 1906, Detwiler 1915, Noyes 1916, Krauch 1918). Of these trigonometric hypsometers, those that calculated total tree height (HT) as a function of the tangent of the angles to the top (B2) and bottom (B1) of the tree and a baseline horizontal distance (b) to the stem were most common (Figure 1). Because they are easy to apply and required only simple technology, these approaches (hereafter, the similar triangles and tangent methods) have dominated tree height measurement. That is not to say the challenge of accurate tree height measurement was solved— many early foresters reported problems with getting consistent data in uneven terrain or dense understories or with the use of different types of hypsometers. Good measurement practices usually mitigated these issues and became standard components of forester training programs. Others addressed these challenges by designing new techniques based on different trigonometric relationships. As an example, Haig (1925) proposed a slide rule solution that calculated tree height using the sine law and slope distance from the observer to the tree base (s1)

Cutoff wavenumbers of eccentric circular metallic waveguides

Cutoff wavenumbers knm are determined analytically for an eccentric circular metallic waveguide. Separation of variables technique is used for the solution. For small eccentricities kd, where d is the distance between the axes of the cylinders, cosine and sine laws are used instead of the translational addition theorem, in order to satisfy the boundary conditions at the surface of the outer cylinder. Keeping terms up to the order (kd)2 exact, analytical expressions of the form knm(d) = knm(0)[1 + gnm(knmd)2 + O(knmd)4] are obtained for the cutoff wavenumbers of the waveguides, where knm(0) corresponds to the coaxial geometry, with d = 0. Both transverse magnetic and transverse electric modes are considered. Numerical results are given for all types of modes and for various values of the parameters. The method used here is an alternative of the one using the translational addition theorem, in the case of small eccentricities.

The sporangiophore of Phycomyces blakesleeanus: a tool to investigate fungal gravireception and graviresponses

The giant sporangiophore of the single-celled fungus, Phycomyces blakesleeanus, utilises light, gravity and gases (water and ethylene) as environmental cues for spatial orientation. Even though gravitropism is ubiquitous in fungi (Naturwissenschaftliche Rundschau, 1996, 49, 174), the underlying mechanisms of gravireception are far less understood than those operating in plants. The amenability of Phycomyces to classical genetics and the availability of its genome sequence makes it essential to fill this knowledge gap and serve as a paradigm for fungal gravireception. The physiological phenomena describing the gravitropism of plants, foremost adherence to the so-called sine law, hold even for Phycomyces. Additional phenomena pertaining to gravireception, specifically adherence to the novel exponential law and non-adherence to the classical resultant law of gravitropism, were for the first time investigated for Phycomyces. Sporangiophores possess a novel type of gravisusceptor, i.e. lipid globules that act by buoyancy rather than sedimentation and that are associated with a network of actin cables (Plant Biology, 2013). Gravitropic bending is associated with ion currents generated by directed Ca(2+) and H(+) transport in the growing zone (Annals of the New York Academy of Sciences, 2005, 1048, 487; Planta, 2012, 236, 1817). A set of behavioural mutants with specific defects in gravi- and/or photoreception allowed dissection of the respective transduction chains. The complex phenotypes of these mutants led to abandoning the concept of simple linear transduction chains in favour of interacting networks with molecular modules of physically interacting proteins.

Influence of Thermomechanical Parameters on the Hot Deformation Behavior of AA1070

The experimental stress–strain data from isothermal hot compression tests, in a wide range of temperatures (350–500 °C) and strain rates (0.005–0.5 s−1), were employed to develop constitutive equations in a commercially pure aluminum (AA1070). The effects of temperature and strain rate on the hot deformation behavior were represented by Zener–Hollomon parameter including Arrhenius term. The results show that the hardening rate and flow stress are evidently affected by both deformation temperature and strain rate. The power law, exponential, and hyperbolic sinusoidal types of Zener–Hollomon equations were used to determine the hot deformation behavior of AA1070. The results suggested that the highest correlation coefficient was achieved for the hyperbolic sine law for the studied material. So the proposed deformation constitutive equations can give an accurate and precise estimate of the flow stress for AA1070, which means it can be used for numerical simulation of hot forming processes and for choosing proper forming parameters in engineering practice accurately.

A Phase Locked Loop Method for Low Voltage Ride-Through Control in Three-Phase Three-Wire Photovoltaic Grid System

Low voltage ride through (LVRT) is regarded as one of the biggest challenges in photovoltaic grid equipments designing, manufacturing and control technology. The fast and accurate automatic phase locked control is the premise to achieve photovoltaic grid low voltage ride through. Considering that the traditional phase locked loop method is difficult to adapt the performance requirements of low voltage ride through (LVRT) in three-phase three-wire photovoltaic grid system, this paper presents a new three-phase unbalanced phase locked loop method. Improved second order generalized integral orthogonal signal generator (SOGI-OSG) is used for line voltage detection; Synchronous reference frame for phase locked loop (SRF-PLL) is used for line voltage phase locked and phase angle detection; Heron formula and Sine theorem is adopted to calculate the three-phase phase voltage amplitude and phase angle. Research results show that when grid voltage dips, this phase locked loop method can detect the fault signal accurately. It provides a technical support for photovoltaic grid low voltage ride through system.

A brightness exceeding simulated Langmuir limit

When an excitation of the first lens determines a beam is parallel beam, a brightness that is 100 times higher than Langmuir limit is measured experimentally, where Langmuir limits are estimated using a simulated axial cathode current density which is simulated based on a measured emission current. The measured brightness is comparable to Langmuir limit, when the lens excitation is such that an image position is slightly shorter than a lens position. Previously measured values of brightness for cathode apical radii of curvature 20, 60, 120, 240, and 480 μm were 8.7, 5.3, 3.3, 2.4, and 3.9 times higher than their corresponding Langmuir limits, respectively, in this experiment, the lens excitation was such that the lens and the image positions were 180 mm and 400 mm, respectively. From these measured brightness for three different lens excitation conditions, it is concluded that the brightness depends on the first lens excitation. For the electron gun operated in a space charge limited condition, some of the electrons emitted from the cathode are returned to the cathode without having crossed a virtual cathode. Therefore, method that assumes a Langmuir limit defining method using a Maxwellian distribution of electron velocities may need to be revised. For the condition in which the values of the exceeding the Langmuir limit are measured, the simulated trajectories of electrons that are emitted from the cathode do not cross the optical axis at the crossover, thus the law of sines may not be valid for high brightness electron beam systems.

Thomas Harriot’s optics, between experiment and imagination: the case of Mr Bulkeley’s glass

Some time in the late 1590s, the Welsh amateur mathematician John Bulkeley wrote to Thomas Harriot asking his opinion about the properties of a truly gargantuan (but totally imaginary) plano-spherical convex lens, 48 feet in diameter. While Bulkeley’s original letter is lost, Harriot devoted several pages to the optical properties of “Mr Bulkeley his Glasse” in his optical papers (now in British Library MS Add. 6789), paying particular attention to the place of its burning point. Harriot’s calculational methods in these papers are almost unique in Harriot’s optical remains, in that he uses both the sine law of refraction and interpolation from Witelo’s refraction tables in order to analyze the passage of light through the glass. For this and other reasons, it is very likely that Harriot wrote his papers on Bulkeley’s glass very shortly after his discovery of the law and while still working closely with Witelo’s great Optics; the papers represent, perhaps, his very first application of the law. His and Bulkeley’s interest in this giant glass conform to a long English tradition of curiosity about the optical and burning properties of large glasses, which grew more intense in late sixteenth-century England. In particular, Thomas Digges’s bold and widely known assertions about his father’s glasses that could see things several miles distant and could burn objects a half-mile or further away may have attracted Harriot and Bulkeley’s skeptical attention; for Harriot’s analysis of the burning distance and the intensity of Bulkeley’s fantastic lens, it shows that Digges’s claims could never have been true about any real lens (and this, I propose, was what Bulkeley had asked about in his original letter to Harriot). There was also a deeper, mathematical relevance to the problem that may have caught Harriot’s attention. His most recent source on refraction—Giambattista della Porta’s De refractione of 1593—identified a mathematical flaw in Witelo’s cursory suggestion about the optics of a lens (the only place that lenses appear, however fleetingly, in the writings of the thirteenth-century Perspectivist authors). In his early notes on optics, in a copy of Witelo’s optics, Harriot highlighted Witelo’s remarks on the lens and della Porta’s criticism (which he found unsatisfactory). The most significant problem with Witelo’s theorem would disappear as the radius of curvature of the lens approached infinity. Bulkeley’s gigantic glass, then, may have provided Harriot an opportunity to test out Witelo’s claims about a plano-spherical glass, at a time when he was still intensely concerned with the problems and methods of the Perspectivist school.