Double Angle Research Articles

Effect of stiffness on load-sharing characteristics of twin rotors concentric face gear power-split transmission system

A twin rotors concentric face-gear power-split transmission system (TRCFGPSTS) applied to the core transmission mechanism of advanced aviation twin rotors helicopter was studied. The effects of torsional stiffness and support stiffness on the load-sharing characteristics of system were mainly studied. By comprehensively considering the factors such as manufacturing error and installation error, meshing stiffness, torsional stiffness, support stiffness and elastic deformation, and according to the power flow double closed-loop mechanical characteristics of this system, the static double closed-loop torsional angle elastic deformation coordination conditions are derived. The static load-sharing model was established by combining the moment balance equation of system. The load-sharing coefficient (LSC) and sensitivity parameters characterizing the load distribution of system were obtained. The effects of the torsional stiffness of the split-torsion shaft and the double-torsion shaft and the support stiffness of each gear on the load-sharing characteristics of the system were analyzed. The results show that the torsional stiffness of the split-torsion shaft and the double-torsion shaft has a great influence on LSC and sensitivity of the system, when its value is in the range of (1–1.6) × 105 N·m /rad and (2.4–4.3) ×105 N·m /rad, the system can meet the requirements of ± 5% of load-sharing characteristics under service conditions. Reducing the torsional stiffness of the split-torsion shaft can improve the load-sharing characteristics of the system, and the torsional stiffness of the right-branching double-shaft has little effect on the load-sharing characteristics of the left-branching of the system. The support stiffness of I-stage small cone angle face gear and Ⅲ-stage face gear is the most sensitive to LSC. With the increase of the support stiffness, the LSC first decreases and then gradually flattens, and the support stiffness of Ⅲ-stage face-gear has little effect on the I-stage LSC. The research results can provide a theoretical basis for the optimal design of load-sharing structure.

SYSTEMATIZATION OF THE FORMULAS OF THE RESONANT FERRITE ISOLATOR LOSS

Context. The problem is to systematize and improve the models of a resonance ferrite isolator in the rectangular waveguide for the antenna-feeder devices, generating, receiving, measuring microwave equipment containing ferrite decoupling devices: ferrite isolators and circulators. Objective. The goal of the work is to verify the formula for the losses of the resonant ferrite isolator in the direct and reverse directions, as well as the isolator ratio. Method. The research method of the work is a critical analysis of literary sources, which was carried out, but did not bring the desired results, since it did not allow to verify the correctness of the derivation of the formula [17]. Therefore, a number of hypotheses were put forward, what the formula might mean. The difficulty lay in the presence in the formula of the product of trigonometric functions that can be attributed to frequency properties, which was taken as an initial hypothesis, which was not subsequently confirmed. The check included transformation of formulas using mathematical physics in terms of microwave electrodynamics, trigonometry and algebra. The beginning was the formula of the classics [16], similar to the formula of [18], accepted without proof. As it is known, for the main type of wave in a rectangular waveguide, the components of the magnetic field strength, obtained as a solution to the wave equation under the boundary conditions inherent in a rectangular waveguide. One component of the magnetic field strength is along the direction of wave propagation, and the second one is in the transverse direction in the section of the waveguide are proportional to the trigonometric functions cosine and sine with the same arguments. The equality of the two components of the strengths is traditionally uses to find the plane of circular polarization where to place the ferrite isolator, and so the authors use this proportionality to trigonometric functions in their derivation, namely the formulas of trigonometric functions of a double angle, the basic trigonometric identity sine squared plus cosine squared is equal to one for replacing the propagation constants with trigonometric functions, this allows to get rid of radicals in the formulas, these radicals in the formula are due to the phenomenon of dispersion in a rectangular waveguide. The rest of the manipulations with the formula are the reduction of similar terms. Results. There was obtained analytical expressions for the losses of the resonant ferrite isolator in the forward and reverse directions, as well as the isolator ratio by strict mathematical transformations. There was performed such transformations. The ratios of the longitudinal propagation constant to the transverse propagation constant are replaced by the ratios of the trigonometric functions sine and cosine, since they are continuous as opposed to tangents and cotangents. Such a transformation allows to avoid square roots in the formula for the losses of the ferrite isolator in the forward and reverse directions, which are associated with the presence of dispersion in the waveguide, as in the formula for wavelength in the waveguide. The conversion is based on microwave electrodynamics. The formulas are used for the distribution of fields in a rectangular waveguide for the main type of wave. Further transformations consist in taking the common factor out of brackets and other arithmetic transformations. Conclusions. Тhrer was obtained results partially coincide with the well-known [17], the derivation of the formula [17] was obtained for the first time, the studies carried out allowed us to reject the hypothesis that the product of cosines and sines in the loss formula of a ferrite isolator is a frequency characteristic, it appears as a result of arithmetic transformations. To take into account the frequency range, it is used that there is circular polarization at the middle frequency, there will also be circular polarization at the extreme frequency of the range, but the plane of circular polarization will shift in comparison with the position of the plane of circular polarization at the middle frequency. That is, a peculiar system of two equations is obtained with respect to two positions of the polarization plane relative to the wide side of the rectangular waveguide section. The scientific novelty consists in systematization and generalization of the formulas of the loss of the resonance ferrite isolator, the connection between the formulas from different literature sources, both foreign and domestic, is proved, which saves time for researchers of ferrite isolators for the verification of the formula. The practical significance. It may be useful for teaching purposes and in optimization of the ferrite isolator design.

PENERAPAN KURVA TRIGONOMETRI UNTUK MOTIF KAIN BATIK - MEDIA PEMBELAJARAN PERSAMAAN TRIGONOMETRI

The subject of trigonometric equations is a learning material given to high school students majoring in Mathematics and Natural Sciences and also deepened in student lectures in the fields of science such as mathematics, engineering, informatics and others. The problem discussed in this study is that learning media is still needed to support learning trigonometric equations. The purpose of this research is to create a media for visual simulation of the results of trigonometric equation functions, namely the sine function with various equation models and by modifying the coefficient values in the equation. The results of this simulation are various curve images and are used as a choice of batik motifs. The research methodology uses mathematical modeling with trigonometric equations of sine functions. By changing some of the coefficients in the modification of the sine function, we get an interesting selection of images for batik patterns. The results of the study were in the form of a batik cloth motif design. This research can be used as an interesting learning medium on the subject of trigonometric equations and to addto the richness of various types of batik cloth motifs as an option for lovers of batik cloth art.

Old Iron and Steel Bridges. Historical and technical notes Static and strength of lattice truss girders. Part 4 (Conclusion)

A model was composed only of linear finite elements to simulate 5 consecutive spans (64m+3x66m+64m) of the D. Luiz bridge deck. It integrates the two main girders of the deck, the two horizontal bracings that exist between their upper and lower chords and the transverse vertical bracings located on the same planes of the girders’ verticals. It is already known (see previous parts of this article) that the diagonals of the trusses are single angles, the verticals are double angles and the chords are T-sections, formed by plates and angles, as it happens in many of the lattice truss girders built at that time (second half of the XIX century, first decades of the XX century).

Development of STEAM-based Teaching Materials on Trigonometric Equations to Improve Students’ Mathematical Creative Thinking Ability

Development of STEAM-based Teaching Materials on Trigonometric Equations to Improve Students’ Mathematical Creative Thinking Ability

Arduino Mega Based System Design for Sequence and Phase Difference Detection of Three-Phase Systems

In a three-phase system, the difference in the angle between the phases and the phase sequence of the system is very important to ensure the system functions normally and does not cause damage to the three-phase equipment connected to the system. Trigonometric formulas in multiphase systems are used to obtain the angle difference between the phases and the sequence of phases in the system. The trigonometric formula was tested in a simulation using MATLAB software, then applied to an Arduino Mega-based system. In the simulation, the data are two voltages vs. time with a certain phase angle difference, then using the trigonometric formula in the MATLAB program, the data is recovered from the phase angle difference and the direction of rotation of the two voltages. Based on the valid MATLAB simulation test results, the program algorithm is embedded in an Arduino Mega-based system equipped with 2 voltage sensors and a 2.4-inch TFT LCD. The Arduino Mega-based system has succeeded in detecting and visualizing in the form of a graph the angle difference between the phases and the direction of rotation of the three-phase system.

PENGEMBANGAN LKPD INTERAKTIF MENGGUUNAKAN LIVEWORKSHEET BERBASIS PENDEKATAN REALISTIC MATHEMATICS EDUCATION (RME) PADA MATERI PERSAMAAN TRIGONOMETRI UNTUK KELAS XI SMA

The learning process can run effectively and efficient if using teaching materials that are in accordance with the needs of students, support the competencies to be achieved and have a sustematic description of the material, and can facilitate students to actively from their own understanding. But in reality students still tend to be passive in learning and are not accustomed to constructing their own knowledge. The teaching materials used are printed books from erlangga and LKPD from publishers. The absence of LKPD development from educators is due to the limitations of educators in utilizing technology and time constraints in designing attractive LKPD. This resulted in the absence of a valid and practical LKPD. This study aims to create an LKPD based on a valid an Realistic Mathematics Education (RME) teaching methide, for grade XI hight school students. This type of study is a design study using the plomp development model, which includes needs analysis, program analysis, concept analysis, and student analysis. The prototyping phase consists of steps taken in the design of the LKPD. In step 1 of the prototype, a process of self-assessment is performed (self-evaluation). In prototype 2, an expert review was performed, which was then assessed by the experts to obtain a valid prototype. In phase 3 of the prototype, an individual assessment was performed by testing LKPD on 3 students with heterogeneous qualifications. During prototype 4 phase, a small group evaluation was conducted on six students with heterogeneous ability levels, to determine the practicality of the LKPD. From the studies that have been done, it os known that the LKPD based on the Realistic Mathematics Education (RME) approach on trigonometric equation material has a validity level of 86,5% with very valid criteria (based on three experts, namely teo mathematics lectures and one educator). LKPD also has a level of practicality. Using the very practical category as much as 88,38% by students in the one-to-one evaluation stage and 91,86% of the students in the small group assessment phase. Keywords – LKPD, Realistic Mathematics Education, valid, practical

Efficiency Comparison of Quaternions and Spherical Trigonometry Formulas for 3D Rotations

Efficiency Comparison of Quaternions and Spherical Trigonometry Formulas for 3D Rotations

A New Mechanical Analysis of a Crankshaft-connecting Rod Dynamics Using Lagrange’s Trigonometric Identities

The main objective of this study was to conduct a new and simple but accurate analysis of the dynamics of a crankshaft-connecting rod system based on Lagrange’s trigonometric identities. Actual and equivalent connecting rod mass approximations of single- and multi-cylinder reciprocating engines were studied. Several examples were studied to demonstrate the dynamics of the system. Lagrange’s trigonometric identities were used to simplify the model, while MATLAB was used to obtain the results. For both the proposed reduced model and the full model, the resultant forces and torques of an actual and an equivalent connecting rod mass were compared. The results showed that the proposed reduced model gives force and torque results that match the results of the full model very well. It was shown that the largest torque imbalance resultant on the crankshaft was exerted by the two-cylinder engine. In addition, it was shown that the largest external forces resultant acting in the x-direction was exerted by the one-cylinder engine. The results also revealed that the resultant of external forces acting in the y-direction was zero for multi-cylinder engines. The relative error, which mainly occurred at the points of maximum force and torque, ranged from about 1% to about 15%.

Elastic compressive buckling resistance for back-to-back double angle assemblies

The present study investigates the elastic buckling resistance of compression members consisting of non-slender hot-rolled double angles connected back-to-back with discrete interconnectors. Towards this goal, the study (a) formulates a thin-walled beam finite element buckling solution that treats both angles as independent members away from the intermediate connector locations while constraining the displacements for both angles at interconnector locations, and (b) develops an extension of the Vlasov thin-walled beam theory, originally intended for monolithic members, to determine the effective torsional/warping sectional properties for back-to-back double angle members. The validity of the thin-walled beam finite element model is then verified against experimental results and shell finite element models and its predictions are shown to asymptotically converge to the Vlasov theory extension developed in the present study as the number of interconnectors is increased. A systematic parametric study consisting of 1250 finite element runs is then conducted and the database of results generated is used to propose a simple equation to predict the elastic compressive buckling resistance of double angle compression members.

Comparison of long-term astigmatic changes following cataract surgery among types of corneal astigmatism

AimsTo examine whether long-term corneal astigmatic changes after stabilisation of surgically induced astigmatism (SIA) following cataract surgery differ among eyes having against-the-rule (ATR), with-the-rule (WTR), and oblique astigmatism.MethodsAnterior corneal astigmatism...

Evaluation of Lumbar Myofascial Release Effects on Lumbar Flexion Angle and Pelvic Inclination Angle in Patients with Non-Specific Low Back Pain

BackgroundMany studies have shown that changes in lumbar flexion angle and the pelvic inclination angle can be affected by the shortening of the lumbar muscles, which can cause low back pain. Decreased lumbar flexion angle and pelvic inclination angle can cause or exacerbate low back pain by disrupting the lumbo-pelvic rhythm.PurposeThis study aimed to use myofascial release techniques as a specialized treatment on muscle tissue to cause muscles to reach the optimal length and improve lumbar flexion angle and pelvic inclination angle, and thus improve low-back pain.SettingNon-specific low back pain patients, Tarbiat Modares University, Iran.Participants30 chronic non-specific low back pain participants were randomly assigned into two groups.Research DesignThis is a randomized control trial.InterventionsThe myofascial release group (n=15) underwent 4 sessions of myofascial release treatment based on Myer’s techniques, and the control group (n=15) underwent 10 sessions of routine electrotherapy for two weeks.Main Outcome MeasuresBefore starting the intervention and after the last treatment session, both groups were evaluated by the lumbar flexion angle with a flexible ruler, calculating the pelvic inclination angle by a trigonometric formula, and VAS measured the pain score of the participants.ResultsThe results of the paired t test showed that, after treatment in both groups, the severity of pain and lumbar flexion angle changed significantly (p ≤ .001). However, the pelvic inclination angle was changed considerably only in the myofascial release group, and we did not see significant changes in the control group (p = .082). Also, the independent sample t test results to examine the between-group changes showed that changes in the myofascial release group were significantly different from the control group (p ≤ .000). Also, the effect size shows the large effect of the myofascial release technique compared to the control group (effect size ≥ 1.85).ConclusionThe present study results showed that myofascial release techniques in patients with low back pain could help decrease pain intensity and increase lumbar flexion and pelvic inclination angle. Based on the present study results, myofascial release can be a treatment to correct posture in patients with chronic non-specific, low back pain. Due to the prevalence of the COVID-19 pandemic, it was not possible to evaluate the long-term effects of treatment.

On Gosper’s $\Pi_q$ and Lambert series identities

In an interesting article entitled “Experiments and discoveries in $q$-trigonometry”, R. W. Gosper conjectured few beautiful $\Pi_q$ and Lambert series identities. Many people have attempted confirming some of those identities in the Gosper’s list, mainly by using Gosper’s $q$-trigonometric identities. In this paper we either prove or disprove all the $\Pi_q$ and Lambert series identities in the Gosper’s list by mainly using S. Ramanujan’s theta function identities and W. N. Bailey’s summation formula. In the process, we obtain three new Gosper kind of identities.

Useful zero friction simulations for assessing MBS codes Pascal's formula giving wheelsets frequency for zero wheel-rail friction

Present study provides a simple analytical formula, the “Klingel-like formula” or “Pascal's Formula” that can be used as a reference to test some results of existing railway codes and specifically those using rigid contact. It develops properly the 3D Newton-Euler equations governing the 6 degrees of freedom (DoF) of unsuspended loaded wheelsets in case of zero wheel-rail friction and constant conicity. Thus, by solving numerically these equations, we got pendulum like harmonic oscillations of which the calculated angular frequency is used for assessing the accuracy of the proposed formula so that it can in turn be used as a fast practical target for testing multi-body system (MBS) railway codes. Due to the harmonic property of these pendulum-like oscillations, the square ω2 of their angular frequency can be made in the form of a ratio K/M where K depends on the wheelset geometry and load and M on its inertia. Information on K and M are useful to understand wheelsets behavior. The analytical formula is derived from the first order writing of full trigonometric Newton-Euler equations by setting zero elastic wheel-rail penetration and by assuming small displacements. Full trigonometric equations are numerically solved to assess that the formula provides ω2 inside a 1% accuracy for usual wheelsets dimensions. By decreasing the conicity down to 1 × 10−4 rad, the relative formula accuracy is under 3 × 10−5. In order to test the formula reliability for rigid contact formulations, the stiffness of elastic contacts can be increased up to practical rigidity (Hertz stiffness × 1000).

Determination of the thrust of an ion thruster by the aerodynamic method of double angle (AMαβ-method)

This paper presents the results of an experimental study of the jet thrust of a mixed jet of neutral and ionized nitrogen flowing from the working chamber of an experimental sample of an accelerator ion thruster into a vacuum space. The main method of the study was the aerodynamic double angle method (AMαβ-method). The method of determining low thrust and the design of the stand for the implementation of the AMαβ-method are proposed. The object of research is a prototype of an accelerator two-phase ion thruster with a solid-state microwave plasma generator and a toroidal resonator. The study showed that a 2.5 W microwave generator together with a 2.5 W accelerating potential difference voltage source with a nitrogen consumption of 0.05 -1 mg/s can increase the speed of the working gas by 2.3 times. The jet thrust, according to the test results, was 1-3.6 μN. It is shown that the thrust components (the gas-dynamic component, the ion component of a high-frequency discharge, the ion component of a glow discharge) have different levels depending on the mass flow rate of the gas.

Three algebraic number systems based on the q-addition with applications

In the spirit of our earlier articles on \(q\)-\(\omega\) special functions, the purpose of this article is to present many new \(q\)-number systems, which are based on the \(q\)-addition, which was introduced in our previous articles and books. First, we repeat the concept biring, in order to prepare for the introduction of the \(q\)-integers, which extend the \(q\)-natural numbers from our previous book. We formally introduce a \(q\)-logarithm for the \(q\)-exponential function for later use. In order to find \(q\)-analogues of the corresponding formulas for the generating functions and \(q\)-trigonometric functions, we also introduce \(q\)-rational numbers. Then the so-called \(q\)-real numbers \(\mathbb{R}_{\oplus_{q}}\), with a norm, a \(q\)-deformed real line, and with three inequalities, are defined. The purpose of the more general \(q\)-real numbers \(\mathbb{R}_{q}\) is to allow the other \(q\)-addition too. The closely related JHC \(q\)-real numbers \(\mathbb{R}_{\boxplus_{q}}\) have applications to several \(q\)-Euler integrals. This brings us to a vector version of the \(q\)-binomial theorem from a previous paper, which is associated with a special case of the \(q\)-Lauricella function. New \(q\)-trigonometric function formulas are given to show the application of this umbral calculus. Then, some equalities between \(q\)-trigonometric zeros and extreme values are proved. Finally, formulas and graphs for \(q\)-hyperbolic functions are shown.

Identifikasi Hambatan Belajar Siswa Kelas X Madrasah Aliyah Negeri 1 Pekanbaru Pada Topik Trigonometri

The purpose of study was to identifying student’s learning obstacle of Trigonometry by using student’s worksheet.This research is descriptive analysis research. This research was conducted in class X Madrasah Aliyah Negeri 1 Pekanbaru with a total of 30 students. The students worksheet (LKS) provided contains two learning activities. In first activity, students in groups were asked to find the rules of trigonometric formulas and use them in solving problems. Futhermore, in second activity students were asked to work on practice questions based on the rules of the Trigonometry formulas they discussed in the previous activity. Data collection were carried out based on student’s answerson the LKS. Furthermore, the student’s answer were analyzed to indentifying student’s learning obstacle on the topic of Trigonometry. Based on results of the analysis of student answers, it can be seen that most students are wrong in using the concept of the cosine rule because the are wrong determining the front side and the side flanking the angle and are unable to know the formula that must be used to solve the problem. Based on the result of this study, it can be seen that the use of student’s worksheets (LKS) is effevtive in identifying student learning obstacle, especially on topic of Trigonomentry so that they can be followed up in the next learning process.

Automatic Detection of Sfe: A Step Forward

Solar flare effects (Sfe) are magnetic variations caused by solar flare events. They only show up in the illuminated hemisphere. Their detection is a difficult task because they do not have a definite pattern and, additionally, they must be separated from other magnetic perturbations. However, we attempted to automatize these detections by using two different strategies. The first strategy takes advantage of one of the Sfe characteristics, as they usually have a rapid rise, followed by a smooth decay, which typically produces a crochet-like shape in the magnetograms. Thus, we created several morphological models for each magnetic component. Then, we identified a definite Sfe time interval by setting the conditions for various parameters, such as the correlations of the measured data with the models, or the model similarities among the different components. In the second stage of this strategy, we observed clusters of time intervals. Each of these clusters were attributed to a timespan of event possibility. We found the statistical optimal value of the correlation parameters by using the ROC curve method and Youden index. The second strategy was based on some of the properties of Sfe ionospheric electric currents, such as their spherical symmetry around the vortex. Here, the algorithm calculated the derivative of the data in order to avoid contamination of the daily variation Sq, and, by means of trigonometric formulas, computed the magnetic radial component relative to the Sfe current vortex (the focus). It then created an Sfe index with this data. A prior assumption of the focus position in a preceding work is no longer needed since we made a wide patrol of the space area to find it. Through a progressive thresholding process, we found its statistical optimal value (0.4 nT min−1) again by using the ROC curve method and Youden index. For both of the strategies, we have made a large calculation of Sfe detection (for the period of 2000–2020), which included 33 Sfe. Finally, we combined the results of both methods—which in fact are complementary—and obtained a unified list that gave a higher hit ratio than those that were obtained separately. This unified method gave promising results towards the possibility of Sfe automatic detection.

Permanent Magnet Tracking Method Resistant to Background Magnetic Field for Assessing Jaw Movement in Wearable Devices.

There is a large gap between primitive bruxism detectors and sophisticated clinical machines for jaw kinematics evaluation. Large, expensive clinical appliances can precisely record jaw motion, but completely restrain the patient for the duration of the test. Wearable bruxism detectors allow continuously counting and recording bites, but provide no information about jaw movement trajectories. Previously, we developed a permanent magnet and three-axis magnetometer-based method for wearable, intra-oral continuous jaw position registration. In this work, we present an effective solution of the two main drawbacks of the method. Firstly, a two-adjacent-magnetometer approach is able to compensate for background magnetic fields with no reference sensor outside of the system’s magnetic field. Secondly, jaw rotational angles were included in the position calculations, by applying trigonometric equations that link the translation of the jaw to its rotation. This way, we were able to use a three-degree-of-freedom (3-DOF) magnetic position determination method to track the positions of the 5-DOF human masticatory system. To validate the method, finite element modeling and a 6-DOF robotic arm (0.01 mm, 0.01°) were used, which showed a 37% decrease in error in the average RMSE = 0.17 mm. The method’s potentially can be utilized in small-scale, low-power, wearable intra-oral devices for continuous jaw motion recording.

Natural Frequencies Optimization of Thin-Walled Circular Cylindrical Shells Using Axially Functionally Graded Materials.

One method to avoid vibration resonance is shifting natural frequencies far away from excitation frequencies. This study investigates optimizing the natural frequencies of circular cylindrical shells using axially functionally graded materials. The constituents of functionally graded materials (FGMs) vary continuously in the longitudinal direction based on a trigonometric law or using interpolation of volume fractions at control points. The spatial change of material properties alters structural stiffness and mass, which then affects the structure’s natural frequencies. The local material properties at any place in the structure are obtained using Voigt model. First-order shear deformation theory and finite element method are used for estimating natural frequencies, and a genetic algorithm is used for optimizing material volume fractions. To demonstrate the proposed method, two optimization problems are presented. The goal of the first one is to maximize the fundamental frequency of an FGM cylindrical shell by optimizing the material volume fractions. In the second problem, we attempt to find the optimal material distribution that maximizes the distance between two adjoining natural frequencies. The optimization examples show that building cylindrical shells using axially FGM is a useful technique for optimizing their natural frequencies.