Vertical Asymptote Research Articles

Impact of printing orientation on inter and intra-layer bonds in 3D printed thermoplastic elastomers: A study using acoustic emission and tensile tests

The use of the Material Extrusion technique with Thermoplastic Elastomers is currently growing because of the large number of benefits of this family of materials. They are processable materials with high flexibility, which makes them very useful, for example, in biomedical applications that require flexible objects with complex geometries. This study aims to characterize a specific polymer, namely polyether-block-amide-based polymer (PEBA), by analyzing its anisotropic behavior in printed samples and investigating the mechanical properties based on different printing orientations. Three orientations (X, Y, and Z) were used to relate the printing orientation to the type of bonds formed in the samples: intra-layer bonds, inter-layer bonds, and the deposited filament. Tensile tests following ASTM D638 were conducted to measure sample rigidity, while Acoustic Emission, an advanced Non-Destructive Technique, was employed to examine the trend of the failure process. The presence of voids was also observed to assess printing quality, which is influenced by the printing orientation and alters the quality of bonds. The results revealed that samples printed horizontally exhibited higher Young’s Modulus values and fewer voids in the inner parts. Vertically printed samples displayed inferior mechanical properties and a greater number of voids. Consequently, the intra-layer yielded better bond formation and minimized voids. Acoustic Emission analysis corroborated these findings by demonstrating distinct energy distribution patterns among the different printing orientations. Hits were concentrated at maximum stresses, primarily observed in the vertically printed samples, which experienced macroscopic failure. Furthermore, this particular specimen exhibited a vertical asymptote near the maximum stress level. The analysis of the energy of Acoustic Emission hits demonstrated a reasonably good fit with the Gutenberg-Richter (GBR) law based on the printing direction.

Mechanism of formic acid oxidation on Bi modified Pt(111): Implication from the concentration effect of formic acid and different coverages of Bi

The concentration dependence of the activity for the formic acid oxidation at pH 1.2 and 4 has been studied on Pt(111) with different coverages of bismuth. The results clearly show: (1) The FAOR on bismuth-modified Pt(111) has no relation with HCOOb adsorbed on platinum but depends on the bismuth coverage. (2) The Tafel slope is ca. 90 - 110 mV in lower potential and increases to nearly the vertical asymptote when the potential increases. (3) Free Pt sites are required for FAOR on bismuth-modified Pt(111). (4) Solution formate, rather than formic acid, is the active species. Combined with previous reports, a kinetic model in which monodentate formate, adsorbed on a Pt-Bi ensemble, is the active intermediate and solution formate is the active species has been proposed. The model is able to reproduce the experimental behavior and agrees with the reaction order measured experimentally. For low potentials, the reaction order for formate is 1, whereas a reaction order of 0.5 is obtained for high potentials.

A Simplified Algorithm for Setting the Observer Parameters for Second-Order Systems with Persistent Disturbances Using a Robust Observer.

The properties of the convergence region of the estimation error of a robust observer for second-order systems are determined, and a new algorithm is proposed for setting the observer parameters, considering persistent but bounded disturbances in the two observation error dynamics. The main contributions over closely related studies of the stability of state observers are: (i) the width of the convergence region of the observer error for the unknown state is expressed in terms of the interaction between the observer parameters and the disturbance terms of the observer error dynamics; (ii) it was found that this width has a minimum point and a vertical asymptote with respect to one of the observer parameters, and their coordinates were determined. In addition, the main advantages of the proposed algorithm over closely related algorithms are: (i) the definition of observer parameters is significantly simpler, as the fulfillment of Riccati equation conditions, solution of LMI constraints, and fulfillment of eigenvalue conditions are not required; (ii) unknown bounded terms are considered in the dynamics of the observer error for the known state. Finally, the algorithm is applied to a model of microalgae culture in a photobioreactor for the estimation of biomass growth rate and substrate uptake rate based on known concentrations of biomass and substrate.

A special self-similar solution and existence of global solutions for a reaction-diffusion equation with Hardy potential

Existence and uniqueness of a specific self-similar solution is established for the following reaction-diffusion equation with Hardy singular potential∂tu=Δum+|x|−2up,(x,t)∈RN×(0,∞), in the range of exponents 1≤p<m and dimension N≥3. The self-similar solution is unbounded at x=0 and has a logarithmic vertical asymptote, but it remains bounded at any x≠0 and t∈(0,∞) and it is a weak solution in L1 sense, which moreover satisfies u(t)∈Lp(RN) for any t>0 and p∈[1,∞). As an application of this self-similar solution, it is shown that there exists at least a weak solution to the Cauchy problem associated to the previous equation for any bounded, nonnegative and compactly supported initial condition u0, contrasting with previous results in literature for the critical limit p=m.

Tracking Thermal Decomposition Chemistry in Secondary Solid Explosives with X-ray Diffraction

We present an approach for measuring thermal decomposition kinetics in crystalline solids using X-ray diffraction to track the loss of crystallinity that accompanies condensed phase decomposition chemistry. We apply this method to systems for which extracting thermodynamic parameters has been historically difficult: organic molecular crystals that thermally decompose below their melting points, such as solid explosives. To demonstrate this method, we measured the rate of solid, thermal decomposition versus temperature in three different secondary solid explosives and the sugar fructose. In all cases, we observed an acceleration in the thermal decomposition rate with increasing temperature, which forms a vertical asymptote, a phenomenon known as melt acceleration. We show that observing the vertical asymptote in the thermal decomposition rate allows for identifying the thermodynamic melting point, which is not trivial to determine when melting and thermal decomposition happen simultaneously. We expect this method to be useful for studying thermal decomposition and for extracting thermodynamic data for secondary solid explosives, data that are needed for modeling and understanding faster phenomena, such as detonation. We also expect this method to be relevant to other organic molecular crystals in which thermal decomposition and melting overlap, such as sugars or pharmaceuticals.

Una nota sobre transitividad de transformaciones crecientes a trozos sobre R

In this work a sufficient condition is shown to obtain transitivity in families of piecewise increassing maps with an inevitable discontinuity in x=0. Specifically, it is shown that the characteristics of a large class of transformations of the real line with a discontinuity in x=0 to be transitive (exhibits a dense orbit), they are the following: f has no fixed points, f has a vertical asymptote at x=0 and the preimage of zero is different from empty. In particular, the famous Boole transformation together with some of its parameterizations they exhibit these characteristics. As a particular case, for the family to a parameter of hyperbolas its dynamic behavior is explicitly determined according to the values of the parameter p &gt; 0.

On the Dynamics of Sexually Transmitted Diseases Under Awareness and Treatment

In this paper, we develop and extend the work of Jia and Qin on sexually transmitted disease models with a novel class of non-linear incidence. Awareness plays a central role both in the susceptible and the infectious classes. The Existence, uniqueness, boundedness, and positivity of solutions are systematically established. Concavity arguments and the occurrence of a vertical asymptote are essential in the proof of the existence of a unique endemic equilibrium. Conditions for the stability of all steady states are investigated. In particular, numerical simulations are performed in order to capture the asymptotic behavior of solutions.AMS Classification: 92D30, 34D23.

A predictive model for fracture in human ribs based on in vitro acoustic emission data.

The aim of this paper is to propose a fracture model for human ribs based on acoustic emission (AE) data. The accumulation of microcracking until a macroscopic crack is produced can be monitored by AE. The macrocrack propagation causes the loss of the structural integrity of the rib. The AE technique was used in in vitro bending tests of human ribs. The AE data obtained were used to construct a quantitative model that allows an estimation of the failure stress from the signals detected. The model predicts the ultimate stress with an error of less than 3.5% (even at stresses 15% lower than failure stress), which makes it possible to safely anticipate the failure of the rib. The percolation theory was used to model crack propagation. Moreover, a quantitative probability-based model for the expected number of AE signals has been constructed, incorporating some ideas of percolation theory. The model predicts that AE signals associated with micro-failures should exhibit a vertical asymptote when stress increases. The occurrence of this vertical asymptote was attested in our experimental observations. The total number of microfailures detected prior to the failure is and the ultimate stress is MPa. A significant correlation (p<0.0001) between and the predicted value is found, using only the first N=30micro-failures (correlation improves for N higher). The measurements and the shape of the curves predicted by the model fit well. In addition, the model parameters seem to explain quantitatively and qualitatively the distribution of the AE signals as the material approaches the macroscopic fracture. Moreover, some of these parameters correlate with anthropometric variables, such as age or Body Mass Index. The proposed model could be used to predict the structural failure of ribs subjected to bending.

Analytical construction of the projectile motion trajectory in midair

A classic problem of the motion of a projectile thrown at an angle to the horizon is studied. Air resistance force is taken into account with the use of the quadratic resistance law. An analytic approach is mainly applied for the investigation. Equations of the projectile motion are solved analytically for an arbitrarily large period of time. The constructed analytical solutions are universal, that is, they can be used for any initial conditions of throwing. As a limit case of motion, the vertical asymptote formula is obtained. The value of the vertical asymptote is calculated directly from the initial conditions of motion. There is no need to study the problem numerically. The found analytical solutions are highly accurate over a wide range of parameters. The motion of a baseball, a tennis ball, and a shuttlecock of badminton are presented as examples.

Intermittent Chaos in the CSTR Bray-Liebhafsky Oscillator-Specific Flow Rate Dependence.

Dynamic states with intermittent oscillations consist of a chaotic mixture of large amplitude relaxation oscillations grouped in bursts, and between them, small-amplitude sinusoidal oscillations, or even the quiescent parts, known as gaps. In this study, intermittent dynamic states were generated in Bray–Liebhafsky (BL) oscillatory reaction in an isothermal continuously-fed, well-stirred tank reactor (CSTR) controled by changes of specific flow rate. The intermittent states were found between two regular periodic states and obtained for specific flow rate values from 0.020 to 0.082 min−1. Phenomenological analysis based on the quantitative characteristics of intermittent oscillations, as well as, the largest Lyapunov exponents calculated from experimentally obtained time series, both indicated the same type of behavior. Namely, fully developed chaos arises when approaching to the vertical asymptote which is somewhere between two bifurcations. Hence, this study proposes described route to fully developed chaos in the Bray-Liebhafsky oscillatory reaction as an explanation for experimentally observed intermittent dynamics. This is in correlation with our previously obtained results where the most chaotic intermittent chaos was achieved between the periodic oscillatory dynamic state and stable steady state, generated in BL under CSTR conditions by varying temperature and inflow potassium iodate concentration. Moreover, it was shown that, besides the largest Lyapunov exponent, analysis of chaos in experimentally obtained intermittent states can be achieved by a simpler approach which involves using the quantitative characteristics of the BL reaction evolution, that is, the number and length of gaps and bursts obtained for the various values of specific flow rates.

Maximum Sensitivity-Constrained Data-Driven Active Disturbance Rejection Control with Application to Airflow Control in Power Plant

The increasing energy demand and the changing of energy structure have imposed higher requirements on the conventional large-scale power plants control. Complexity of the power plant processes and the frequent change of operation condition make the accurate physical models hard to obtain for control design. To this end, a data-driven control strategy, the active disturbance rejection control (ADRC) has received much attention for the estimation and mitigation of uncertain dynamics beyond the canonical form of cascaded integrators. However, the robustness of ADRC is seldom discussed in a quantitative manner. In this study, the maximum sensitivity is used to evaluate and then constrain the robustness of ADRC applied to high-order processes. Firstly, by using the new idea of the vertical asymptote of the Nyquist curve, a preliminary one-parameter-tuning method is developed. Secondly, a quantitative relationship between the maximum sensitivity and the tuning parameter is established using optimization methods. Then, the feasibility and effectiveness of the proposed method is initially verified in the total air flow control of a power plant simulator. Finally, field tests on the secondary airflow control in a 330 MWe circulating fluidized bed confirm the merit of the proposed maximum sensitivity-constrained ADRC tuning.

On the Variation of the Polytropic Exponent in a High Pressure Fan Impeller

Fan impellers are usually designed considering that the pumped air is incompressible and homogeneous, i.e. its density remains constant. When the incompressibility hypothesis can lead to significant errors, as in the case of high pressure fans, the analysis of the air flow can be made by considering that the air undergoes a polytropic process of constant polytropic exponent. In this paper, the concept of polytropic process of variable exponent depending on impeller radius is introduced, in order to better approximate the phenomena that take place inside blade passages. Numerical results obtained for an impeller of a high pressure fan without spiral casing suggest that the pumped air undergoes two different processes: an expansion in the first part of the impeller and the usual compression in the second part. The two processes are reflected in the strong variation of the polytropic exponent, which shows a vertical asymptote where the change of the process takes place. The results also suggest that high pressure fan impellers could consist of two stages, each stage being designed according to the process that takes place inside it: expansion or compression.

First-Year Non-STEM Majors’ Use of Definitions to Solve Calculus Tasks: Benefits of Using Concept Image over Concept Definition?

Six US first-year university students in humanities or social science degree programmes were interviewed while solving 4 tasks on continuity and asymptotes in a required mathematics course. The focus was on how the students referred to the definitions or to the concept images when solving the tasks and if partial understandings appeared. Partial understanding denotes when the definition or the concept image has correct parts but essential parts are missing or incoherencies occur. Some students confused continuity with differentiability or perceived it as a graph not having holes, which is also reported by other studies. Another misconception that emerged is to perceive points on the x-axis as non-continuous. The partial understandings of asymptotes were related to the vertical asymptote as it was confused with a function property. Some students referred to the definitions when correctly solving some of the tasks and some students with coherent concept images were successful solving some of the tasks even when they did not refer to the definitions.

Border Collision and Smooth Bifurcations in a Family of Linear-Power Maps

In this work we describe some properties and bifurcations which occur in a family of linear-power maps typical in Nordmark’ systems. The continuous case has been investigated by many authors since a few years, while the discontinuous case has been considered only recently. In particular, having a vertical asymptote, it gives rise to new kinds of bifurcations. Organizing centers related to codimension-two bifurcation points, due to the intersection of a border collision bifurcation and a smooth fold bifurcation of cycles having a different symbolic sequence are evidenced. It is shown the relevant role played by a codimension-two point existing on any border collision bifurcation curve, and related to the smooth fold bifurcation of cycles with the same symbolic sequence. We recall some of the properties proved up to now, evidencing the rich structure which is still to be understood.

Border collision and fold bifurcations in a family of one-dimensional discontinuous piecewise smooth maps: unbounded chaotic sets

In this work we consider a class of generalized piecewise smooth maps, proposed in the study of engineering models. It is a class of one-dimensional discontinuous maps, with a linear branch and a nonlinear one, characterized by a power function with a term and a vertical asymptote. The bifurcation structures occurring in the family of maps are classified according to the invertibility or non-invertibility of the map, depending on the parameters characterizing the two branches. When the map is non-invertible we prove the persistence of chaos. In particular, the existence of robust unbounded chaotic attractors. The parameter space is characterized by intermingled regions of attracting cycles born by smooth fold bifurcations, issuing from codimension-two bifurcation points. The main result is related to the description of the relationship between two types of bifurcations, smooth fold bifurcations and border collision bifurcations (BCBs). We describe the particular role of codimension-two bifurcation points associated with these bifurcations related to cycles with the same symbolic sequences. We show that they exist related to the border collision of any admissible cycle. We prove that each BCB, each fold bifurcation and each homoclinic bifurcation is a limit set of infinite families of other BCBs. We prove that in the considered range all the unstable cycles are always homoclinic, and that an unbounded chaotic set always exists, either in an invariant set of zero measure or of full measure.

Linear Stability of a Developing Thermal Front Induced by a Constant Heat Flux

A developing thermal front is set up by suddenly imposing a constant heat flux on the lower horizontal boundary of a semi-infinite fluid-saturated porous domain. The critical time for the onset of convection is determined using two main forms of analysis. The first of these is an approximate method which is effectively a frozen-time model while the second implements a set of parabolic simulations of monochromatic disturbances placed in the boundary layer at an early time. Results from the two approaches are compared and it is found that instability only occurs when the nondimensional disturbance wavenumber, $$k$$ , is less than unity. The neutral curve for the primary mode possesses a vertical asymptote at $$k=1$$ in wavenumber/time parameter space which is in contrast to the more usual teardrop shape which occurs when the surface is subject to a constant temperature. Asymptotic analyses are performed for the frozen-time model which yield excellent predictions for both branches of the neutral curve and the locus of the maximum growth rate curve at late times.

Examining mathematical discourse to understand in-service teachers’ mathematical activities

In this article I use Sfard’s theory of commognition to examine the surprising activities of a pair of in-service mathematics teachers in South Africa as they engaged in a particular mathematical task which allowed for, but did not prescribe, the use of GeoGebra. The (pre-calculus) task required students to examine a function at an undefined point and to decide whether a vertical asymptote is associated with this point or not. Using the different characteristics of mathematical discourse, I argue that the words that students use really matter and show how a change in one participant’s use of the term ‘vertical asymptote’ constituted and reflected her learning. I also show how the other participant used imitation in a ritualised routine to get through the task. Furthermore I demonstrate how digital immigrants may resist the use of technology as the generator of legitimate mathematical objects.

Shuttlecock dynamics

We study experimentally the dynamics of shuttlecocks. We show that their trajectory is completely different from classical parabola: for a same launch, the flight of the shuttlecock quickly curves downwards and almost reaches a vertical asymptote. We solve the equation of motion with gravity and drag at high Reynolds number and find an analytical expression for the range. At high velocity, this reach does not depend on the velocity anymore. This phenomenon, that we call the “aerodynamic wall”, is highly observable in badminton. Then we study how the shuttlecock shape influences the badminton game. The shuttlecock always flies the nose forehead, which means after the impact it has to flip. Actually it returns, oscillates and then stabilizes. We understand these damping oscillations by distinguishing the mass and aerodynamic center of a shuttlecock.

Approximate formula for the vertical asymptote of projectile motion in midair

The classic problem of the motion of a point mass (projectile) thrown at an angle to the horizon is reviewed. The air drag force is taken into account with the drag factor assumed to be constant. An analytical approach is used for the investigation. An approximate formula is obtained for one of the characteristics of the motion – the vertical asymptote. The value of an asymptote is determined directly by the initial conditions of throwing. Analytically derived values of asymptotes in comparison with numerical values obtained by integrating the equations of motion are given. The motion of a baseball is presented as an example.

Solutions of ordinary differential equations with vertical asymptote

Conditions ensuring that solutions of ordinary differential equations have vertical asymptotes are obtained.Bibliography: 8 titles.