Sawtooth Functions Research Articles

Electromagnetohydrodynamic (EMHD) Flow in a Microchannel with Random Surface Roughness.

This study investigates the effect of small random transverse wall roughness on electromagnetohydrodynamic (EMHD) flow is in a microchannel, employing the perturbation method based upon stationary random function theory. An exact solution of a random corrugation function ξ, which is a measure of the flow rate deviated from the case without the roughness of two plates, is obtained by integrating the spectral density. After the sinusoidal, triangular, rectangular, and sawtooth functions that satisfy the Dirichlet condition are expanded into the Fourier sine series, the spectral density of the sine function is used to represent the corrugation function. Interestingly, for sinusoidal roughness, the final expression of the corrugation function is in good agreement with our previous work. Results show that no matter the shape of the wall roughness, the flow rate always decreases due to the existence of wall corrugation. Variations of the corrugation function and the flow rate strongly depend on fluid wavenumber λ and Hartmann number Ha. Finally, the flow resistance is found to become small, and the flow rate increases with roughness that is in phase (θ = 0) compared with the one that is out of phase (θ = π).

Active control for system with multiple microchannel heat exchangers using disturbance observer

To prevent flow maldistribution and imbalanced temperature, a disturbance observer-based control strategy is presented for microchannel cooling systems with multiple heat exchangers. A disturbance observer is designed using the extended Kalman filter for estimating the states and disturbance. The performance of the observer is validated using experimental data from a pumped liquid cycle testbed. Disturbance observer-based control is applied to systems with two and three heat exchangers. The findings demonstrate that the wall temperatures and the temperature differences for the heat exchangers could be maintained at the desired values. When the heat exchangers are subjected to heat loads in the form of square, sinusoidal, and sawtooth functions, the temperature fluctuation could be suppressed by 60%-80% and the temperatures could be maintained at the same values. Our research demonstrates that the proposed control strategy has acceptable accuracy and robustness for the system with multiple heat exchangers undergoing various operating situations.

Oscillatory valve effect on pressure drop oscillation in microchannel cooling system

Pressure drop oscillation (PDO) is one of the most common flow instabilities occurred in microchannel cooling systems, which could result in partial dry-out, heat transfer deterioration, and temperature fluctuation. In this study, the effects of three oscillatory electronic valve settings (square, sinusoidal, and sawtooth) on mass flow rate and PDO characteristics are examined numerically and experimentally. A one-dimensional lumped model is developed for the microchannel cooling system, which indicates that a specific range of valve oscillation amplitude and frequency can reduce PDO amplitude significantly. The PDO amplitude can be reduced by 85 % when the valve amplitude is 30 % and the frequency is 0.1 Hz. A smaller valve oscillation amplitude shows a weaker effect on the suppression of PDO. The square function has a better PDO suppression than the sinusoidal and sawtooth functions under the same pulsating valve amplitude and frequency. Experimental results demonstrate a similar effect of valve setting on PDO, for instance, a valve setting oscillates from 20 % to 80 % and a fixed frequency of 0.05 Hz can reduce pressured drop amplitude from 3 kPa to 0.5 kPa, and reduce the mass flow rate oscillation amplitude from 60 ml/min to 10 ml/min. The overly large oscillation amplitude of the valve setting could increase the oscillation amplitude of the mass flow rate and the overall pressure drop significantly. Thus, both the pressure drop through the microchannel evaporator and the valve should be considered in the selection of the oscillatory valve parameters. Based on the analysis of this study, an active control strategy could be developed to suppress flow oscillation and optimize the overall cooling performance using the oscillatory valve setting.

Thermal performance analysis of artificially roughened solar air heater under turbulent pulsating flow with various wave shapes

This paper presents a numerical investigation of the operation of a solar air heater (SAH) with cylindrical shape ribs under turbulent pulsating flow based on thermal and frictional considerations. Turbulence and thermal natures of flow physics are modeled by taking the relevant governing equations into the account using RNG k−ε turbulence model and transient state Reynolds-Averaged Navier-Stokes (RANS) equations. The pulsating flow is modeled with four different wave shapes including rectangular, sinusoidal, triangular, and sawtooth functions. The influence of various factors such as wave shape, Reynolds number (Re), frequency, and amplitude of pulsating flow on the thermal and hydraulic performance of SAH has been evaluated. Results indicated that the SAH with ribbed absorber section under pulsed flow has a larger Nu number compared to the air heater with smooth duct under steady flow. Moreover, the pulsating flow with a sawtooth velocity profile leads to the greatest value of the thermo-hydraulic performance parameter (THPP). The growth of pulsation frequency up to 200 enhances the heat transfer coefficient of the air heater while for frequencies larger than 200, the thermal performance declines. In addition, the increase in pulsation amplitude has a negative effect on the THPP. It was proved that the increment of the Re number enhances the THPP. Up to 53.3% improvement of THPP can be achieved by applying pulsed flow with sawtooth wave shape and ribbed absorber section compared to air heat with smooth duct under steady flow.

Quantifying periodic, multi-periodic, hidden and unstable regimes of a magnetic pendulum via semi-analytical, numerical and experimental methods

The dynamics of a magnetic pendulum mounted on a shaft with an elastic element is considered. The impulsive excitation of the pendulum is performed through the interaction of a strong neodymium magnet and an electric coil. The mathematical model of the pendulum is represented by the non-autonomous ordinary nonlinear differential equation. The model is verified in a wide range of motion regimes by the laboratory experiment. Periodic motions are considered in more detail by the semi-analytic approximate method utilizing the averaging ideas. The amplitude and phase shift of system solutions possess highly nonlinear features which we propose to approximate by the saw-tooth functions. This new approach allows one to improve the results of the application of the conventional averaging method and is promising for the application and analysis of nonlinear systems with a strong nonlinearity.

Generation of grid multi-wing chaotic attractors and its application in video secure communication system

In order to study the application of chaos in video image encryption, a real-time video secure communication system based on a new grid multi-wing chaotic system is proposed in this paper. First, by introducing sawtooth wave functions to the Lorenz system, a new grid multi-wing butterfly chaotic system with complicated dynamical behaviors is obtained. Compared with the existing multi-scroll and multi-wing chaotic systems, The system structure is simple and more easier to be implemented in a digital system. Then, a chaos-based pseudorandom random number generator is developed by implementing post-processing procedure. The present video secure communication system is designed with the closed-loop feedback scheme. The corresponding hardware implementation is developed by FPGA platforms, and the experimental results are given to verify its feasibility. Furthermore, a series of widely used secure analyses are applied to prove that the system has good security performance.

Effect of different frequency functions on ferrofluid FHD flow

Abstract A finite volume method is used to investigate the unsteady thermomagnetic convection of ferrofluid flow in a miniature channel under the effect of constant and oscillating magnetic fields. A magnetic line dipole as a magnetic source is placed under the lower wall of the two-dimensional channel whose lower and upper walls are subjected to a constant heat flux. To generate the oscillating magnetic field, four different frequency functions including rectangle, sine, triangle and sawtooth functions are implemented in the constant magnetic field. The study parameters include four frequency functions, magnetic field intensity as a magnetic number (Mn = 3.833 × 108, 8.624 × 108, 1.533 × 109), frequencies (f = 0.5–5 Hz) and Reynolds numbers (Re = 20–40). Results revealed that applying the constant magnetic field enhances heat transfer and as the Reynolds number increases, the effectiveness of magnetic field on heat transfer rate diminishes. The oscillating magnetic field, regardless of the type of the applied frequency function, improves heat transfer rate due to thermal boundary layer suppression and increase in velocity field. For all applied functions, results show that there is an optimum frequency where maximum heat transfer happens and this frequency increases for higher Reynolds numbers. For rectangle and sawtooth functions with instant change in the magnetic field, two simultaneous vortexes emerges which leads to better heat transfer than sine and triangle functions. The alteration of the pressure drop in one period is proportionate with the shape of the applied frequency even though the range of the alterations reduces with the increase of the Reynolds number.

Numerical study on the thermal and electrical performance of an annular thermoelectric generator under pulsed heat power with different types of input functions

The study of annular thermoelectric generators (ATEG) has attracted more attention in recent years due to their ability to cover round shaped surfaces. Moreover, transient and pulsed inputs have been introduced as some of the effective methods to enhance the performance of thermoelectric devices. Hence, in the present study, a three-dimensional transient simulation of ATEG including the influence of Thomson effect is carried out based on finite element method. The effect of pulsed heat inputs on the performance of ATEG is studied considering rectangular, triangular, sinusoidal and two types of the sawtooth input functions for modeling the transient heat load. Electric current, voltage, output power and conversion efficiency of ATEG under pulsed and steady-state heating are compared. The effects of two critical parameters, the ratio of maximum heat flux to minimum heat flux (b/a) and duty cycle (t0/τ) values on the performance of ATEG are studied. Results indicate that transient pulsed heating enhances the efficiency of ATEG for all types of heat input functions. In addition, it is found that rectangular input function leads to better results compared to other functions. Also, it is shown that for a constant amount of duty cycle, the increment of (b/a) increases the performance of ATEG unit. For duty cycle value of 0.1 and (b/a=48) a maximum efficiency enhancement of 249.36% is achieved when the rectangular input function is employed. It is found that in the range of the studied values of (b/a), the duty cycle of 0.4 leads to better results. A comparison of results with and without Thomson effect proves that the Thomson effect decreases the performance of ATEG. Additionally, it is observed that for sawtooth input functions, the implementation of heat load gradient plays an important role determining the performance.

Discontinuous modulation functions and their application for analysis of modulated structures with the computing system JANA2006

Abstract Discontinuous modulation functions called crenel and saw-tooth have been developed for description and refinement of strongly modulated crystal structures with abrupt changes of modulation parameters. Although used for refinement of many modulated structures and mentioned in books on aperiodic crystals, technical details of such refinements have never been published and remained hidden in the source code of the refinement program Jana2006. In this article we explain how to recognize discontinuous modulations in a Fourier map and how to refine structures where crenel or saw-tooth functions are combined with additional secondary modulation. Three sets of functions suitable for such combination are presented: the traditional ortho-harmonics, and newly developed sets of Legendre polynomials and x-harmonics. Tiny differences between refinements based on particular function sets are demonstrated using simulated as well as existing modulated structures.

Function Approximation Technique based Sliding Mode Controller Adaptive Control of Robotic Arm with Time-Varying Uncertainties

The controlling of robotic arm is really challenging due to the involvement of various uncertainties such as- time varying payload, friction and disturbances. These challenges attract many researchers to develop advanced control strategies for robot arm. However, most of the developed controllers focus on time invariant uncertainties. This paper presents the formulation of a new Sliding Mode Control- Function Approximation Technique (SMC-FAT) based adaptive controller for a robot arm carrying unknown time-varying payload with the presence of time-varying disturbance and friction. The limitation of previous controllers to cope up with wide range time-varying uncertainty is solved using FAT expression. The stability of the controller can be proven by selecting a proper Lyapunov function and the update law can be derived easily. Three different time-varying uncertainties in sinusoidal, sawtooth and random functions have been considered as the payload and disturbance in the computer simulation to evaluate the controller's performance. The results with error less than 0.02 percentages proved the effectiveness of the proposed controller.

Simulation on Force Tracking Control of a Magnetorheological Damper under Impact Loading

This paper focuses on the design of the control structure which consists of inner loop controller employed for MR damper under impact loading by using computer simulation. The simulation is done by using MATLAB 7.0. The structure of the inner loop control for the proposed MR damper model uses a simple PI control to achieve the desired force. In this simulation, the MR damper model that has been validated with the experimental result is used to simulate the actual force that produced by MR damper. The performance of inner loop controller to track the actual force produced by MR damper by obtaining the several input functions which are half wave of sinusoidal, saw-tooth, square and random functions of desired force with the variation in pendulum mass of 15 kg and 20 kg are investigated. It can be seen clearly that under several input functions, the proposed polynomial model with PI controller has the good ability to track the desired damping force under impact loading.

Understanding the PWM Nonlinearity: Single-Sided Modulation

This paper provides a new look at the mechanisms underlying pulse width modulation (PWM). A simple approach to analyze the behavior of a single-sided pulse width modulator is presented. By using elementary methods, the pulse-width-modulated waveform is written as the sum of two sawtooth functions and the original modulating waveform. Simply applying the well-known Fourier series expansion of the sawtooth function, an equivalent model of the pulse width modulator, which shows that it is in essence a sequence of phase modulators, is derived. This model provides a clear understanding of the nonlinearities involved in the PWM process. It is shown how the superposition of modulating waveforms in the time-domain translates into the convolution of the sidebands in the frequency domain. Finally, the interaction of the pulse width modulator and a sample-and-hold register is studied and a general expression for the Fourier transform of a regular-sampled PWM waveform is derived. The analysis applies to periodic as well as aperiodic modulating waveforms.

Striped Periodic Minimizers of a Two-Dimensional Model for Martensitic Phase Transitions

In this paper we consider a simplified two-dimensional scalar model for the formation of mesoscopic domain patterns in martensitic shape-memory alloys at the interface between a region occupied by the parent (austenite) phase and a region occupied by the product (martensite) phase, which can occur in two variants (twins). The model, first proposed by Kohn and Mueller, is defined by the following functional: $${\cal E}(u)=\beta||u(0,\cdot)||^2_{H^{1/2}([0,h])}+ \int_{0}^{L} dx \int_0^h dy \big(|u_x|^2 + \epsilon |u_{yy}| \big)$$ where $u:[0,L]\times[0,h]\to R$ is periodic in $y$ and $u_y=\pm 1$ almost everywhere. Conti proved that if $\beta\gtrsim\epsilon L/h^2$ then the minimal specific energy scales like $\sim \min\{(\epsilon\beta/L)^{1/2}, (\epsilon/L)^{2/3}\}$, as $(\epsilon/L)\to 0$. In the regime $(\epsilon\beta/L)^{1/2}\ll (\epsilon/L)^{2/3}$, we improve Conti's results, by computing exactly the minimal energy and by proving that minimizers are periodic one-dimensional sawtooth functions.

Experimental investigations of the scanning functions of galvanometer-based scanners with applications in OCT

We analyze the three most common profiles of scanning functions for galvanometer-based scanners (GSs): the sawtooth, triangular and sinusoidal functions. They are determined experimentally with regard to the scan parameters of the input signal (i.e., frequency and amplitude). We study the differences of the output function of the GS measured as the positional error of the oscillatory mirror from the ideal function given by the input signal of the device. The limits in achieving the different types of scanning functions in terms of duty cycle and linearity are determined experimentally for the possible range of scan parameters. Of particular importance are the preservation of an imposed duty cycle and profile for the sawtooth function, the quantification of the linearity for the sinusoidal function, and the effective duty cycle for the triangular, as well as for the other functions. The range of scan amplitudes for which the stability of the oscillatory regime of the galvo mirror is stable for different frequencies is also highlighted. While the use of the device in certain scanning regimes is studied, certain rules of thumb are deduced to make the best out of the galvoscanner. Finally, the three types of scanning functions are tested with a Fourier domain optical coherence tomography (FD OCT) setup and the conclusions of the study are demonstrated in an imaging application by correlating the determined limits of the scanning regimes with the requirements of OCT.

Multi-scroll hyperchaotic system based on colpitts model and its circuit implementation

Based on the framework of Colpitts oscillator, a four-dimensional multi-scroll hyperchaotic system is proposed, which generates (2M+1)×(2N+1)-scroll chaotic and hyperchaotic attractors. The key strategy is to increase the number of index-2 equilibrium points by introducing two unit saw-tooth functions to extend and modify the Colpitts oscillator model. By using bifurcation diagram and phase portrait, the dynamical characteristics of the multi-scroll hyperchaotic system are briefly studied. Moreover, micro-controller based circuit realization is introduced and the experimental results demonstrate that 7×5-scroll chaotic and hyperchaotic attractors can be obtained in the digital circuit.

CHAOTIFICATION FOR A CLASS OF FIRST-ORDER PARTIAL DIFFERENCE EQUATIONS

This paper is concerned with chaotification for a class of first-order partial difference equations, in which the system size is finite or infinite. Nine new chaotification schemes for the class of first-order partial difference equations with general controllers, mod-operation, and sawtooth functions are established, respectively. All the controlled systems are proved to be chaotic in the sense of both Devaney and Li–Yorke. In addition, five new chaotification schemes for general discrete dynamical systems in finite-dimensional real spaces and l∞are established. Two illustrative examples are provided with computer simulations.

The dynamics of odd sawtooth functions

This note considers a family of piece-wise linear functions that can be used in the classroom to illustrate various concepts involved in iteration theory such as periodicity. These functions require minimal background on the part of the student.

Near-Periodic Local Minimizers of Singularly Perturbed Functionals with Nonlocal Term

We consider singular perturbed functionals with a nonlocal term which are a one-dimensional model for an austenite/twinned martensite interface. The parameter range is chosen in a way that a limit functional can be derived. By examining the limit problem, we show the existence of highly oscillating, near-periodic minimizers of the original functionals which are close to periodic sawtooth functions.

Modulated one-dimensional structure of [Cd(NH3)3Ni(CN)4

The crystal structure of [Cd(NH(3))(3)Ni(CN)(4)] has been solved and refined as an incommensurate structure in four-dimensional superspace. The Xcalibur point-detector diffractometer was used for data collection up to the third-order satellites. The structure was refined from 3496 observed reflections to a final R value of 0.0371. The modulation strongly affects all the atoms of the structure. The dominating feature of the modulation is the incommensurate alternation between octahedrally and tetrahedrally coordinated Cd atoms in the electroneutral chains [-Cd(NH(3))(n)-NC-Ni(CN)(2)-CN-](infinity) (n alternates between 2 and 4). The chain is propagated along the a axis, in which the [Cd(NH(3))(n)](2+) cations are linked by square [Ni(CN)(4)](2-) anions. The atomic positional modulations, having mainly a transversal character in the c-direction, exhibit a switching character that can be described by the crenel and sawtooth functions. The effectively pentagonal coordinated cadmium positions were detected in the intermediate regions.

Intergrowth polytypoids as modulated structures: the example of the cation deficient oxides LaTi1-xO3

A new, essentially composition-independent structural model is proposed for the perovskite-related compound series LanTin-δO3n, with δ ≤ n/4, based on superspace formalism. In the standard 3-dimensional crystallographic approach, one must specify space group and cell parameters separately for each composition. In the superspace approach, the superspace group is invariant while the cell parameters and primary modulation wave-vector vary slowly and continuously with composition. To a first approximation, the structure at any particular composition can be described in terms of mixed cubic and hexagonal stacking of close-packed LaO3 (and Ti) layers along the z-axis. The actual layer stacking sequence varies abruptly with composition, and the titanium vacancies occur as entire layers between neighbouring hexagonal close-packed LaO3 layers. The continuously variable, composition-dependent character of reciprocal space indicates that the various layer stacking sequences must be intimately correlated and shows that all the compounds can be interpreted in terms of a structural modulation over a common average structure. It is shown that a very simple model in superspace is sufficient to describe the layer stacking sequences for any particular composition. The atomic surfaces representing the atoms in superspace are shown to be well described by means of crenel functions. The cation displacive modulations are approximately described by sawtooth functions. The only variable parameters are the composition-dependent (but smoothly varying) primary modulation wave-vector and associated Ti occupation domain width. Conventional space groups can be easily determined from the superspace group and the composition. From a practical point of view, this means that we have a very good single starting point to refine the actual atomic positions for any composition, including incommensurate or long period structures. Even more importantly, the superspace approach has the flexibility to represent in a simple manner the symmetry-allowed deviations from the ideal model. A comparison with the scarce number of refined structures in this system and in the homologous (Ba,La)nTin-1O3n system, demonstrates the suitability of the formalism for the analysis of these compound series and for the analysis of compositionally flexible systems in general.