Derivation Of Formula Research Articles

A Numerical Investigation of Flow Characteristics in a Cryogenic Perforated Plate Flowmeter for Vertical Pipe Applications

To address the research gap regarding the flow characteristics of cryogenic perforated plate flowmeters in vertical pipes and to enhance measurement reliability in challenging environments, this study investigates the flow characteristics of liquid hydrogen in a vertical pipe using a perforated plate flowmeter. Numerical simulations are performed based on an extended derivation of performance parameter formulas in the vertical direction. Various inlet Reynolds numbers, plate thicknesses, and equivalent diameter ratios are analyzed to assess their effects on key performance parameters, including the discharge coefficient, pressure loss coefficient, and stable region. The results indicate that the influence of flow direction on the performance parameters decreases with increasing Reynolds number. Downward flow is associated with smaller discharge coefficients, lower pressure loss coefficients, and reduced upper limits of Reynolds numbers in the stable region. Furthermore, the effects of gravity become more pronounced at larger thicknesses and greater equivalent diameter ratios.

Mechanical characteristics and design parameter analysis of spherical hinge structure for swivel bridge

As the core component of a swivel bridge, the spherical hinge structure exhibits complex mechanical behavior under high pressure, making it challenging to describe accurately through theoretical analysis or on-site monitoring. In this context, this paper systematically examined the mechanical characteristics and key design parameters of a simplified spherical hinge structure using numerical simulations and formula derivation. Additionally, an indoor scaled test was conducted based on a representative spherical hinge structure to further investigate its mechanical properties. The findings indicated that contact pressure along the spherical hinge interface increased radially outward and followed a cubic distribution pattern. The support radius was identified as the most critical factor influencing the maximum stress value within the spherical hinge structure. In the aperture ratio range of 0.042 to 0.083, the upper and lower spherical hinges are unable to achieve coordinated deformation under high pressure, resulting in negative pressure at the edge of the pin hole. Furthermore, for spherical hinge structures with sliders, the presence of the slider introduced additional complexity to the stress state, leading to a coexistence of tension and compression on the surface of the lower spherical hinge. The highest stress occurred between the inner ring slider and the pin hole, an area that requires careful monitoring. The research provides valuable insights for the design of spherical hinges in swivel bridges and can guide the development of spherical hinge structures tailored to different bridge types in the future.

Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine

Abstract In this article, the authors introduce Qi’s normalized remainder of the Maclaurin series expansion of Qi’s normalized remainder for the cosine function. By virtue of a monotonicity rule for the quotient of two series and with the aid of an increasing monotonicity of a sequence involving the quotient of two consecutive non-zero Bernoulli numbers, they prove the logarithmic convexity of Qi’s normalized remainder. In view of a higher order derivative formula for the quotient of two functions, they expand the logarithm of Qi’s normalized remainder into a Maclaurin series whose coefficients are expressed in terms of determinants of a class of specific Hessenberg matrices. In light of a monotonicity rule for the quotient of two series, they present the monotonicity of the ratio between two normalized remainders. Finally, the authors connect two of their main results with the generalized hypergeometric functions.

Implications of [formula omitted]-deformed statistics on stellar stability

Examination of Chandrasekhar’s equilibrium and stability condition for stars in the context of q-deformed Maxwell–Boltzmann statistics leads to the derivation of a new analytical formula that extends its classical case. This derivation assumes that the kinetic description of stellar matter conforms to a generalized Maxwell–Boltzmann distribution. It has been found that the maximum allowable radiation pressure at the center of a star for a given mass depends on the deformation parameter q. The classical Chandrasekhar condition is recovered in this framework when q equals 1.

κ-FRACTIONAL DERIVATIVE OPERATOR AND SOLUTION OF FRACTIONAL KINETIC EQUATIONS INVOLVING (κ,𝔍)-SECOND APPELL HYPERGEOMETRIC MATRIX FUNCTIONS WITH THE TWO VARIABLES

In this study, we present a new definition of [Formula: see text])-second Appell hypergeometric matrix functions ([Formula: see text])-SAHMFs). Then, we investigate analytical properties related to the novel matrix function such as derivative formulas and integral representations, and the [Formula: see text]-fractional derivative operators. Additionally, we obtain bilinear and linear generating relations for the [Formula: see text])-SAHMFs. This work presents novel results regarding [Formula: see text])-SAHMFs to solve the fractional kinetic equations.

A Receiver Position Estimation Method Based on LSTM for Multi-Transmitter Single-Receiver Wireless Power Transfer Systems

The multi-transmitter single-receiver wireless power transfer (MTSR-WPT) system has good tolerance for coil misalignment because the magnetic fields generated by multiple transmitters can be shaped to adapt to position changes in the receiver coil. In order to achieve magnetic field shaping of the MTSR-WPT system and increase power transfer efficiency, accurately estimating the position of the receiver coil is a key issue that needs to be addressed. In this article, a receiver position estimation method based on long short-term memory (LSTM) is proposed, which utilizes a data-driven approach to establish a neural network model. By learning the relationship between the measured time-series voltage data of the transmitter coils and the position of the receiver coil, the proposed model can achieve accurate position estimation of the receiver. Compared with previous works, the proposed method does not require communication between the transmitter and receiver, which is conducive to simplifying the system structure and reducing costs. In addition, the proposed LSTM-based method requires less derivation of complex formulas and the internal mechanism analysis of the system. Finally, a MTSR-WPT prototype is built to verify the proposed method. The experimental results show that the proposed LSTM-based method can achieve high-accuracy position estimation of the receiver. When the receiver moves within a range of 160 mm × 160 mm, the average error between the estimated receiver coil position using the proposed method and the actual receiver coil position is less than 2.40 mm.

A method of protecting an articulated electric bus from side overturning

Justification. The current trend towards the use of electric buses has been growing in the last few years. Articulated electric buses also enter the routes. These vehicles, due to the presence of heavy traction batteries mainly on the roof, have a tendency to an increased roll angle and a tendency to roll over. Therefore, there is a need to apply anti-rollover measures for such vehicles. The purpose of the work – development of a law and a control algorithm that allows reducing the torque to reduce the tendency of an articulated electric bus to overturn Materials and methods. During the development and research of the algorithm, the MATLABSimulink simulation environment is used with the developed mathematical model of spatial motion of an articulated electric bus. Results. The derivation of formulas for calculating the critical turning speed of sections for articulated vehicles is presented, an algorithm and a traction control law are formulated depending on the turning parameters, graphs are presented that substantiate the operability and effectiveness of the algorithms. Conclusion. The practical value of the developed algorithm lies in its practical application on an articulated vehicle in order to reduce the tendency to increased roll angles and rollover protection.

A [formula omitted]-Weyl fractional operator of the extended [formula omitted]-type function in a complex domain

This paper proposes a new extension of the well-known S-special function, which is called SM-function. We introduce this function by drawing inspiration from exponential function. This new special function is studied from a variety of analytical perspectives, including differential and integral operators. Furthermore, the ϱ-Weyl fractional integral operator involving the SM-function is studied. These classes are defined by utilizing a new q-differential operator.•The SM- function is provided.•The derivative and integral formulas of the SM-function are studied.•An application of the ϱ-Weyl fractional integral operator associated with the SM-function is investigated.

Dependence of eigenvalues for higher odd-order differential operator with general transmission conditions

In this paper, we consider the dependence of eigenvalues on the parameter for higher odd-order differential operator with general boundary conditions and general transmission conditions. By defining an appropriate Hilbert space and its inner product, we prove that the self-adjointness of the operator associated with the problem. In addition, we show that the eigenvalues are not only continuously but also smoothly dependent on the parameters of the problem, and give the corresponding differential expressions. In particular, giving the general formula for derivative formulas of the eigenvalues with respect to the left and right sides of the inner discontinuity point.

Experimental Test and Analytical Calculation on Residual Strength of Prestressed Concrete T-Beams After Fire

High temperatures during a fire can lead to the evaporation of moisture and the degradation of hydration products within concrete, consequently compromising its mechanical properties. This paper thoroughly investigates the effect of fire-induced high temperatures on the residual load-bearing capacity of concrete structures, with a focus on prestressed concrete T-beams. By conducting constant temperature tests and residual load-bearing capacity tests, complemented by finite element modeling, this study examines the degradation of mechanical properties in prestressed concrete T-beams due to fire exposure and its impact on post-fire residual load-bearing capacity. Additionally, an equivalent concrete compressive strength method was employed to propose a calculation method for concrete material degradation under high temperatures and a corresponding concrete strength reduction factor. Simplified calculations were also performed for the high-temperature damage to reinforcement and prestressed tendons, leading to the derivation of a simplified formula for the residual load-bearing capacity of post-fire prestressed concrete T-beams. The results indicate that in prestressed concrete T-beams exposed to fire, an increase in holding time results in more severe damage modes, accelerated crack propagation, and wider crack widths during bending failure. Under the same load, a longer holding time corresponds to a more pronounced reduction in deflection. At holding times of 60 min, 120 min, and 180 min, the prestress losses were 48.17%, 85.16%, and 93.26%, respectively. The cracking load decreased by 15%, 27%, and 42%, while the residual load-bearing capacity decreased by 11%, 21%, and 28%. Comparison with experimental data demonstrates that both the finite element model and the simplified calculation formula exhibit high accuracy, offering a reliable reference for the performance evaluation of post-fire prestressed concrete T-beams.

Methodology for calculating the support reactions of a statically indeterminate tractor drive axle final drive shaft

BACKGROUND: Currently, in the design of vehicle drive axles, bevel final drives are used, forming both statically determinate structures and statically indeterminate ones. In the first case, the bearings on which the bevel final drive shafts rest form two races of rolling elements, in the second case - three or more. Despite the prevalence of bevel final drive designs that form statically indeterminate structures, in the process of teaching the design and calculation of machine elements, statically determinate structures are considered. The relevance of the work lies in the formalization of the analytical calculation of the reactions of the supports of the bevel final drive, forming a statically indeterminate system, with the derivation of formulas for determining the reactions in the extra support. AIMS: The article aim is the formalization the methodology for calculating the support reactions of the drive axle bevel final drive, forming a statically indeterminate system, for calculating bearings. METHODS: The article uses mathematical dependencies of Analytical Mechanics to calculate reactions in the main supports, a corollary of Castigliano theorem and the Force method to calculate reactions in extra supports. "Mathcad" was used for mathematical calculations. RESULTS: A formalized method for the analytical calculation of support reactions in statically indeterminate systems has been developed and presented, using Castigliano theorem to solve static indetermination. The methodology application for calculating the support reactions of the statically indeterminate the Kirovets K-7 series tractors drive axle bevel final drive is considered. CONCLUSIONS: The dependencies obtained during the calculations make it possible to calculate the bevel final drive support reactions, the design of which forms a statically indeterminate system. The bevel final drive mathematical model obtained during the calculations can be used as a basis for developing software that automates this calculation.

Several Derivative Formulas of Two Exponential Functions and Real Power of Hyperbolic Secant Function with a Generalization of a Formula for Specific Partial Bell Polynomials

In the paper, by virtue of some identities for the partial Bell polynomials and with the aid of the Faá di Bruno formula, the author presents several derivative formulas of two exponential functions and the real power of the hyperbolic secant function, and generalizes a formula for specific partial Bell polynomials.

Investigation of the approximate solution of one class of curvilinear integral equations by the projection method

UDC 517.9 We prove the existence theorem for the normal derivative of the double-layer potential and establish the formula for its evaluation. A new method for the construction of quadrature formulas for the normal derivatives of simple- and double-layer potentials is developed, and the error estimates are obtained for the constructed quadrature formulas. By using these quadrature formulas, the integral equation of the exterior Dirichlet boundary-value problem for the Helmholtz equation in two-dimensional space is replaced by a system of algebraic equations, and the existence and uniqueness of the solution to this system is proved. The convergence of the solution of the system of algebraic equations to the exact solution of the integral equation at the control points is proved and the convergence rate of the method is determined.

Finite‐time stability and numerical approximations of fractional neutral delay systems involving proportional Caputo derivative

This paper deals with a class of fractional neutral delay systems involving proportional Caputo derivative. Maintaining the finite‐time stability of fractional‐order systems is a major challenge, as their capacity to mimic complex dynamics draws more attention to them. Therefore, the paper presents a novel finite‐time stability criterion based on the Banach fixed‐point theorem. A decomposition formula for proportional Caputo derivative is provided. This formula allows us to derive a new numerical technique for efficiently solving the proposed problem. Finally, extensive numerical results are performed to illustrate and validate the proposed theoretical results.

Non-Inertial Dynamic Analysis of 3-SPS/U Parallel Platform by Screw Theory and Kane’s Method

This paper presents an improved method for the non-inertial dynamic analysis of the 3-SPS/U parallel platform (3-SPS/U PM), employing the screw theory and Kane’s method, where S, P, and U denote spherical, prismatic, and universal joints, respectively. The proposed method extends the traditional inertial dynamic analysis to non-inertial systems. First, the generalized screw method is introduced, followed by the derivation of a transformation formula that adapts the screw method to various co-ordinate systems. Subsequently, the velocities and accelerations of each rigid body within the platform under non-inertial conditions are examined by combining the extended screw method with the system’s inverse kinematics model. The extended screw method is not only conceptually simple, but also adaptable to other non-inertial systems. Finally, the standard non-inertial dynamic model of the 3-SPS/U PM is derived through the Kane’s method and validated by the co-simulations with RecurDyn (V9R5) and MATLAB/Simulation (2019b).

The analytical curvature distribution model of columns and mathematical solution for pushover analysis

AbstractThe acquisition of a flexural backbone curve for columns primarily relies on finite element analysis and experiments, of which the complexity and high cost are significant challenges. Hence, this study proposes an analytical curvature distribution model (CDM) along the full height of the column, which is the key point of formula derivation and theoretical solution of the column backbone curve. The proposed CDM is a piecewise function model composed of linear and quadratic functions, with the characteristics of continuity and differentiability at the intersection point. The deformation compatibility equations, equilibrium equations, and material constitutive equations based on the variables of CDM are constructed to form an equation system, the solution of which can be theoretically determined by an iterative algorithm. Furthermore, 155 test specimen columns of different materials, for example, reinforced concrete, high‐strength reinforced concrete, and shape memory alloy are used to validate the proposed CDM and mathematical solutions of pushover analysis through three crucial indicators, that is, goodness of fit of backbone curve, ultimate displacement, and peak force, indicating strong practicability, high accuracy, and wider applicability. This theoretical method can be applied to deal with the key issues of interest during pushover analysis, that is, predicting the flexural backbone curves with different materials, determining the curvature distribution throughout the entire process of pushover analysis, and characterizing the evolution process of the plastic region.

A Calculation Method of Bearing Balls Rotational Vectors Based on Binocular Vision Three-Dimensional Coordinates Measurement

The rotational speed vectors of the bearing balls affect their service life and running performance. Observing the actual rotational speed of the ball is a prerequisite for revealing its true motion law and conducting sliding behavior simulation analysis. To address the need for accuracy and real-time measurement of spin angular velocity, which is also under high-frequency and high-speed ball motion conditions, a new measurement method of ball rotation vectors based on a binocular vision system is proposed. Firstly, marker points are laid on the balls, and their three-dimensional (3D) coordinates in the camera coordinate system are calculated in real time using the triangulation principle. Secondly, based on the 3D coordinates before and after the movement of the marker point and the trajectory of the ball, the mathematical model of the spin motion of the ball was established. Finally, based on the ball spin motion model, the three-dimensional vision measurement technology was first applied to the measurement of the bearing ball rotation vector through formula derivation, achieving the analysis of bearing ball rolling and sliding characteristics. Experimental results demonstrate that the visual measurement system with the frame rate of 100 FPS (frames per second) yields a measurement error within ±0.2% over a speed range from 5 to 50 RPM (revolutions per minute), and the maximum measurement errors of spin angular velocity and linear velocity are 0.25°/s and 0.028 mm/s, respectively. The experimental results show that this method has good accuracy and stability in measuring the rotation vector of the ball, providing a reference for bearing balls' rotational speed monitoring and the analysis of the sliding behavior of bearing balls.

Framework of acoustic analysis and shape optimization for three-dimensional doubly periodic multilayered structures

In this research, a framework of acoustic analysis and shape optimization, based on isogeometric boundary element method (IGA-BEM), is proposed for three-dimensional doubly periodic multilayered structures. The study addresses a gap in the literature by focusing on the shape optimization of such structures, which has not been extensively explored previously. The interface between different acoustic media is an infinite doubly periodic surface, which can be constructed by an open non-uniform rational B-splines. A periodic IGA-BEM is developed for the sound field analysis of the doubly periodic multilayered structure, in which the Ewald method is used to accelerate the calculation of periodic Green function. Furthermore, the shape derivative of the doubly periodic multiple boundaries is derived by imposing boundary perturbation and using the adjoint variable method. The control points of the NURBS surfaces are defined as the shape design variables, and all shape sensitivities can be quickly calculated by discretizing the shape derivative formula. Finally, in according with shape sensitivities, the corresponding shape optimization problem is solved by the method of moving asymptotes, so that the optimized shape design can be obtained. A series of numerical examples validates the accuracy and applicability of the proposed approaches.

De Bruijn’s Short Route to Rényi’s Parking Constant

Rényi’s parking constant describes the mean density of randomly parked cars in a long street. We present a short elementary derivation of the exact formula for Rényi’s constant, which is inspired by de Bruijn’s research in number theory and differential equations.

Extension of Pochhammer symbol, generalized hypergeometric function and τ-Gauss hypergeometric function

Abstract We introduce new extension of the extended Pochhammer symbol and gamma function by using the extended Mittag-Leffler function. We also present extension of the generalized hypergeometric function as well as some of their special cases by using this extended Pochhammer symbol. Further, we define the extension of the τ-Gauss hypergeometric function. Integral and derivative formulas involving the Mellin transform and fractional calculus techniques associated with this extended τ-Gauss hypergeometric function are also given. Also, new extended τ-Gauss hypergeometric function also provides a few more interesting and well-known results. This enriches the theory of special functions. The obtained results are believed to be newly presented.