Using variable modulus to modify a rack cutter and generate a gear pair

In this paper, two normal imaginary helical rack cutters were first established. One of these cutters is a skewed-rack cutter with an asymmetrical straight edge. The other is a rack cutter with an asymmetric parabolic profile. Second, the gear’s tooth surface of the asymmetric parabolic rack cutter is modified to be barrel-shaped based on a variable modulus. The tooth thickness of the gear is gradually reduced along the face width of the tooth from the middle of the tooth surface. Then the coordinate relationship between the gears’ blanks and the imaginary helical rack cutters was established. Through the differential geometry, crowned and uncrowned helical gear pairs were generated. Because of human factors, when the gear pair is installed, it is easy to cause the gear pair edge contact. It is necessary to add artificial assembly error settings through the tooth contact analysis to investigate the kinematic errors and contact conditions of the crowned and uncrowned helical gear pair. The mathematical models and analysis methods proposed for the crowned imaginary rack cutter using variable modulus should be useful for the design and production of double crowned helical gears with asymmetric parabolic teeth.

The Association of Fatigue With Decreasing Regularity of Locomotion During an Incremental Test in Trained and Untrained Healthy Adults

Fatigue is a key factor that affects human motion and modulates physiology, biochemistry, and performance. Prolonged cyclic human movements (locomotion primarily) are characterized by a regular pattern, and this extended activity can induce fatigue. However, the relationship between fatigue and regularity has not yet been extensively studied. Wearable sensor methodologies can be used to monitor regularity during standardized treadmill tests (e.g., the widely used Bruce test) and to verify the effects of fatigue on locomotion regularity. Our study on 50 healthy adults [27 males and 23 females; <40 years; five dropouts; and 22 trained (T) and 23 untrained (U) subjects] showed how locomotion regularity follows a parabolic profile during the incremental test, without exception. At the beginning of the trial, increased walking speed in the absence of fatigue is associated with increased regularity (regularity index, RI, a. u., null/unity value for aperiodic/periodic patterns) up until a peak value (RI = 0.909 after 13.8 min for T and RI = 0.915 after 13.4 min for U subjects; median values, n. s.) and which is then generally followed (after 2.8 and 2.5 min, respectively, for T/U, n. s.) by the walk-to-run transition (at 12.1 min for both T and U, n. s.). Regularity then decreases with increased speed/slope/fatigue. The effect of being trained was associated with significantly higher initial regularity [0.845 (T) vs 0.810 (U), p < 0.05 corrected], longer test endurance [23.0 min (T) vs 18.6 min (U)], and prolonged decay of locomotor regularity [8.6 min (T) vs 6.5 min (U)]. In conclusion, the monitoring of locomotion regularity can be applied to the Bruce test, resulting in a consistent time profile. There is evidence of a progressive decrease in regularity following the walk-to-run transition, and these features unveil significant differences among healthy trained and untrained adult subjects.

Magnetic field diffusion in ferromagnetic materials: fractional calculus approaches

The paper addresses diffusion approximations of magnetic field penetration of ferromagnetic materials with emphasis on fractional calculus applications and relevant approximate solutions. Examples with applications of time-fractional semi-derivatives and singular kernel models (Caputo time fractional operator) in cases of field independent and field-dependent magnetic diffusivities have been developed: Dirichlet problems and time-dependent boundary condition (power-law ramp). Approximate solutions in all theses case have been developed by applications of the integral-balance method and assumed parabolic profile with unspecified exponents. Tow version of the integral method have been successfully implemented: SDIM (single integration applicable to time-fractional semi-derivative model) and DIM (double-integration model to fractionalized singular memory models). The fading memory approach in the sense of the causality concept and memory kernel effect on the model constructions have been discussed.

Analysis of parametric instability of a spring-attached pre-twisted beam with viscoelastic end support

This study investigated the parametric instability of a single elastic beam with spring attachment on the top and viscoelastic springs as end supports. The beam considered is pre-twisted with a pin connection at both ends that supports the beam. The analytical solution of the problem is expressed in the matrix form achieved from the implementation of Hamilton’s principle and General Galerkin’s method, from which both static and dynamic stability of the beam can be investigated. The results of various influential dimensionless parameters such as stiffness, mass, length, position of the spring attachment, and stiffness of the viscoelastic springs on both the stabilities are studied. This analysis concluded that the spring attachment on the system leads to substantial contribution in improving the stability. The viscoelastic springs also contribute in upsurging the beam’s stability. Three different profiles of the beam have been considered, and for each profile, three different types of springs have been examined. The results revealed that the beam with parabolic profile and stiffness of the spring attachment with parabolic variation is most effective towards strength-to-weight ratio.

Analytical study of the natural bending oscillations of a concave beam with parabolic change in thickness

The synthesis of factorization and symmetry methods produced a general analytical solution to the fourth-order differential equation with variable coefficients. The form and structure of the variable coefficients correspond, in this case, to the problem of the oscillations of a concave beam of variable thickness. The solution to this equation makes it possible to study in detail the oscillations of such and similar, for example convex, beams at the different fixation of their ends' sections. A practical confirmation has been obtained that the beam whose thickness changes in line with the concave parabola law H=a2x2+1, where a is the concave factor, demonstrates an increase in the natural frequencies of its free oscillations with an increase in its rigidity. As an example, the object's maximum deflection dependence on the beam rigidity factor has been established. The nature of this dependence confirmed the obvious conclusion that the deflections had decreased while the rigidity had increased. The evidence from the calculation results can be a testament to the correctness of the reported procedure of problem-solving. The considered problem and the analytical solution to it could serve as a practical guide to the optimal design of beam structures. In this case, it is very important to take into consideration the place and nature of the distribution of cyclical extreme operating stresses. The resulting ratios to solve the problem make it possible to simulate the required normal stresses in both the fixation and central zones when the rigidity parameter is changed. Designers could predict such a parabolic profile of the beam, which would ensure the required reduction of maximum stresses in the place of fixing the beam. The considered example of solving the problem of the natural oscillations of the beam with rigid fixation of the ends illustrates the effectiveness of the factoring and symmetry methods used. The developed solution algorithm could be extended to study the natural bending oscillations of the beam at other fixing techniques, not excluding a variant of a completely free beam

The Effects of Quartic Term Mathematical Model on The Concentration Profile of Fixed Bed Gas Adsorber

The need for a reliable mathematical model depicting the process inside a column adsorber has become a requisite in designing an effective gas adsorber. Even though this task can be done by using commercial software, it is still important to get an understanding of how the entire process happens. In this paper, we discuss a new way to approximate the concentration profile inside the porous solids. It is an extension of the work of Liaw et al., who adopted a parabolic (i.e., quadratic) profile, which is a function of pellet radius while retaining the spherical symmetry. We extend their work by adding the quartic term. The inclusion of this new term still preserves the form of linear driving force approximation with some correction to Glueckauf’s parameter (i.e., the effective diffusivity coefficient). The addition of the correction will affect the breakthrough curve so that it affects the saturation time. In the binary system of hydrogen/methane discussed in this study, we found that a negative correction to the diffusivity coefficient will make the saturation happen earlier compared to that of the parabolic case, and vice versa. This study may help us design an efficient gas purifier, in particular when we set out for the regeneration of the adsorbent.Keywords: activated carbon; effective diffusivity coefficient; hydrogen; linear driving force; methane; parabolic profile A B S T R A KKebutuhan model matematis yang dapat menggambarkan proses penyerapan dalam kolom adsorpsi telah menjadi kebutuhan yang tak terelakkan dewasa ini. Walaupun kini telah tersedia berbagai perangkat lunak komersial, namun tidak dapat dipungkiri bahwa memahami bagaimana proses tersebut terjadi tetap menjadi suatu hal yang berguna. Paper ini bertujuan untuk menampilkan cara baru dalam pemodelan konsentrasi adsorbat di dalam adsorben padat berpori. Kami memperluas metode yang dikembangkan Liaw et al. dengan menambahkan suku pangkat empat (kuartik). Penambahan ini akan mengoreksi koefisien difusivitas efektif dari persamaan linear driving force (LDF). Koreksi yang bernilai negatif, misalnya, akan mengurangi nilai koefisien difusivitas tersebut sehingga akan menghambat kemampuan adsorpsi. Hasil perhitungan kami pada sistem biner hidrogen/ metana menunjukkan bahwa suatu koreksi bernilai negatif dapat menyebabkan saturasi berlangsung lebih cepat dari kasus profil parabolik. Begitu pula sebaliknya, koreksi positif akan menambah daya adsorpsi sehingga saturasi dapat diperlambat. Studi ini kami harapkan dapat diterapkan untuk mendesain suatu kolom adsorpsi yang efisien, terutama dalam perencanaan proses regenerasi adsorben.Kata kunci: hidrogen; karbon aktif; koefisien difusivitas efektif; linear driving force; metana; profil parabolik

Thermo-mechanical and metallurgical model of leaf spring industrial quenching process

This study is a numerical analysis of the industrial quenching process for leaf springs developed by the CAVEO company. The leaf chosen for this study is of a parabolic profile made of EN-51CrV4 steel (AISI 6150). The aim of this study is to set up a numerical model to predict thermal, metallurgical, and mechanical behavior of a leaf spring from exit of the heating furnace to exit of the quenching bath going through a cambering operation. This study would therefore allow the company to switch from a development scheme based on experiments using physical prototypes tested on the production line to a new scheme based on virtual prototypes using numerical simulation. The development of the numerical model using the finite element method is carried out using the ABAQUS/Implicit solver coupled with two user subroutines Phase and UMAT. The first one have been developed to compute microstructure evolution and the second one to define the constitutive law taking into account phase transformations. This model helps us to follow the spatio-temporal evolutions of temperature and microstructure in the leaf, as well as the variation of the leaf deflection during the process. The proposed numerical model is supported by an experimental protocol based on infrared thermographic images, Rockwell-C hardness measurements, metallographic observations, and deflection measurements. Indeed, the results of the proposed thermo-mechanical and metallurgical model are closed to experimental results.