Analytical Formulas For Calculation Research Articles

A 3D slice–based analytical calculation formula for the reservoir landslide velocity

A 3D slice–based analytical calculation formula for the reservoir landslide velocity

The Attenuation calculation method for the underwater low-frequency magnetic field of finite-length straight wire

Usually, when solving the magnetic field distribution of a finite-length straight wire in a conducting medium, numerical calculation is required. However, this method has some disadvantages, such as complex calculations and not being conducive to engineering implementation. To solve this problem, this paper studied the attenuation calculation method for the underwater low-frequency magnetic field of the finite-length straight wire and then obtained the analytical calculation formula of magnetic field spatial distribution. The simulation results show that, within a certain distance, the analytical method is in good agreement with the numerical calculation results. The method presented in this paper is convenient for calculation and has good engineering practicability.

Initial impact force analysis and simulation of planetary helical gear transmission

The planetary gear transmission is widely used because of its advantages such as large transmission ratio and high transmission efficiency. But at the moment of starting, there is a speed difference between driving gear and driven wheel due to the backlash between the teeth, this will easily cause the initial collision and impact force, which seriously affect the life of gear transmission, even lead to tooth surface pitting and root fracture failure, therefore, it is necessary to conduct a quantitative analysis of the initial collision. Based on the gear meshing principle and the system dynamics theory, the analytical calculation formula of initial impact force for the internal and external meshing gear pair was derived. The impact simulation analysis model of planetary helical gear transmission was established based on the finite element LS-DYNA, and the simulation analysis results were compared with the analytical calculation results, which verified the correctness of the analytical calculation model. This paper reveals the impact collision mechanism and studies the influence law of the speed and temperature on the impact force, thus laying a good theoretical foundation for the dynamic modification design of the planetary gear transmission and preventing the failure of the gear teeth from impact collision.

Experimental investigation on the seismic performance of phosphogypsum-filled cold-formed thin-walled steel composite walls

Cold-formed steel (CFS) is generally used in the construction industry for low-rise and multi-story buildings. In such structures, CFS walls are the main load-carrying elements. The traditional CFS wall has a low load-carrying capacity and poor seismic performance, which is the main reason for structural damage under natural disasters and also restricts the application of CFS structures in medium- and high-rise buildings. In this study, to improve the shear capacity and seismic performance of CFS walls, an innovative CFS wall filled with phosphogypsum (PG), called phosphogypsum-filled cold-formed thin-walled steel (PFCFS) composite wall, is presented. Taking the axial compression ratio, PG strength, and steel tube’s wall thickness as the study parameters, seven full-scale specimens have been fabricated and tested under cyclic lateral loads. The failure processes, failure modes, load–displacement response, strength and stiffness degradations, and energy absorption capacities of all the specimens are comprehensively investigated. The test results indicate that the specimen with a low axial compression ratio exhibits a connection failure between the CFS frame and hold-downs. For other specimens, the damage to the component materials leads to the final specimen failure. The axial compression ratio and the wall thickness of steel tubes have a significant influence on the seismic performance of the composite walls, while the PG strength has a much lower impact on the seismic performance. In addition, an analytical model and calculation formulas are proposed for predicting the shear capacities of the PFCFS walls. The effectiveness and accuracy of the proposed formulas are verified by comparing the calculated and test values.

Analytical Algorithm for Capacities of Classical and Classical-Quantum Channels

We derive an analytical calculation formula for the channel capacity of a classical channel without any iteration while its existing algorithms require iterations and the number of iteration depends on the required precision level. Hence, our formula is its first analytical formula without any iteration. We apply the obtained formula to examples and see how the obtained formula works in these examples. Then, we extend it to the channel capacity of a classical-quantum (cq-) channel. Many existing studies proposed algorithms for a cq-channel and all of them require iterations. Our extended analytical algorithm have also no iteration and output the exactly optimum values.

Optimal Control of the Angular Momentum for a Solid (Spacecraft) Performing a Spatial Turn

A quaternion solution of the dynamic problem on the optimal turn of a solid (for example, a spacecraft) from a known initial to a given final angular position is presented. Optimization of the control program is carried out by using a combined indicator that combines a quadratic performance criterion and turn time; the minimized functional combines energy costs and maneuver duration in a given proportion. Based on the maximum principle, quaternion models, and methods for studying the controlled motion of a solid (spacecraft), a solution of the problem has been obtained. The construction of the optimal rotation is based on a differential equation relating the angular momentum and the orientation quaternion of a solid. The conditions of optimality are written in analytical form and the properties of the optimal motion are studied. Analytical equations and calculation formulas for finding the optimal control are presented. The control law is formulated as an explicit dependence of the control variables on the phase coordinates. Key relations that determine the optimal values of the parameters of the angular momentum control algorithm are given. In the case of a dynamically symmetric body, a complete solution of the turn problem in a closed form is obtained: analytical dependences as explicit functions of time for control variables and relations for calculating the parameters of the control law are given. A numerical example and the results of numeric simulation of the rotation of a spacecraft as a solid under optimal control that demonstrate the practical feasibility of the proposed control method are given.

Magnetic stiffness calculation for the corresponding force between two current-carrying circular filaments arbitrarily oriented in the space

In this article, sets of analytical formulas for calculation of nine components of magnetic stiffness of corresponding force arising between two current-carrying circular filaments arbitrarily oriented in the space are derived by using Babic’s method and the method of mutual inductance (Kalantarov–Zeitlin’s method). Formulas are presented through integral expressions, whose kernel function is expressed in terms of the elliptic integrals of the first and second kinds. Also, we obtained an additional set of expressions for calculation of components of magnetic stiffness by means of differentiation of Grover’s formula of the mutual inductance between two circular filaments with respect to appropriate coordinates. The derived sets of formulas were mutually validated and results of calculation of components of magnetic stiffness agree well to each other.

A Simplified LPTN Model for a Fault-Tolerant Permanent Magnet Motor under Inter-Turn Short-Circuit Faults

This paper proposes a simplified lumped parameter thermal network (LPTN) model for thermal analysis under the inter-turn short-circuit (SC) faults of a five-phase fault-tolerant permanent magnet (PM) motor for electric vehicles. The proposed model can consider circumferential heat transfer, variable copper loss, and uneven loss distribution. Firstly, an analytical calculation formula of inter-turn SC current is proposed to separate the copper loss of SC turns from electromagnetic simulation, and the accuracy of the formula is verified by finite element analysis (FEA). Secondly, a simplified LPTN model is proposed, the calculation formulas of the thermal resistance are given, and the simplified principle is explained. By comparing the results of different simplifications, it is found that the error between the simplified model and the original model is small. Finally, the proposed model is verified by experiments. The results show that the simplified model can achieve not only a high calculation speed but also a high accuracy. Moreover, the proposed model is applicable to all cases of asymmetrical temperature distribution.

Calculation of surface settlement and pile foundation deformation caused by shield machine tunnelling along curve section

The shield machine tunnelling along the curve section causes more disturbance to the surrounding environment than the straight section. Pile foundation is the most commonly used foundation form in high-rise buildings, and high-rise buildings have extremely high requirements for controlling non-uniform deformation. Therefore, it is necessary to study the surface settlement and the deformation law of pile foundations caused by shield machine tunnelling along the curve section. Considering the shield machine tunnelling factors along the curve section and the coupling effect of piles and soil, the analytical calculation formula of the ground settlement and the pile foundation deformation caused by the shield machine tunnelling along the curve section is deduced. According to the actual project situation, a finite difference model (FDM) is constructed, and the correctness of the FDM and analytical prediction formula is verified by comparing the on-site monitoring data. The research shows that the error among the FDM results, the analytical prediction results, and the on-site monitoring data are small, and the surface settlement and pile foundation deformation meet the construction control standards. The friction resistance of the shield shell, the integrative gap at the shield tail (IGST), and the over-cutting gap (OG) are the main factors leading to the surface settlement. However, the shield shell friction and OG are the main reasons leading to the inner settlement of the curve section being more prominent than the outer side. The difference in the pile foundation settlement on both sides of the curve section is slight, the maximum settlement difference rate is 1.8%, and the maximum horizontal deformation rate of the pile foundation on the inner and outer sides is 9.2%, which shows that the horizontal deformation of the pile foundation is more sensitive to the asymmetrically distributed construction factors.

Structural deformation mechanism of the hat-stiffened plate used in marine structures under impact load

As the basic structure of the bottom or side of the ship structure or marine structure, the stiffened plate plays the main energy absorption role in the collision accident. In the design of ship crashworthiness structure, the design improvement of the stiffened plate structure can improve the crashworthiness of the ship. In this paper, the damage deformation mode of the hat-stiffened plate is determined by designing and conducting the falling weight impact test, and the analytical formulas for the instantaneous resistance of the structure is derived based on the theory of plastic mechanics. The accuracy of the analytical calculation method is verified by comparing with the test results, and the analytical calculation formula for calculating the instantaneous structural resistance of the hat-stiffened plate proposed in this paper can quickly evaluate the response of the structure under accident loads, which is of guiding significance for the structural design of marine structures.

Analytical Research on the Bearing Characteristics of Oil Film Supplied with Constant Oil Flow Hydrostatic Turntables under Fixed Eccentric Load Condition

This study was initiated in view of the phenomenon that hydrostatic turntable rails are prone to scrape damage when the liquid hydrostatic turntable is running under fixed partial load. The existing bias-load hydrostatic turntable oil film bearing characteristics are mainly calculated by using the calculus integration method and CFD fluid simulation method. The calculating formula obtained by the calculus integration method is complex and inefficient. The CFD fluid simulation calculation method requires 3D modeling and meshing of the oil film, which is a tedious and time-consuming process and may not yield convergent calculation results due to improper meshing methods or boundary condition settings. In order to solve the shortage of the above calculation methods, this paper simplifies and equates the uneven thickness of the oil film of each sealing edge of the oil pad of the bias-loaded hydrostatic rotary table to the equivalent uniform thickness of the oil film, and based on this idea, the analytical calculation formula of the oil film bearing capacity, bending moment and stiffness under constant bias-load conditions of the constant-flow liquid hydrostatic rotary table is derived. In this paper, Fluent software was used to numerically simulate the oil film under this working condition, and a hydrostatic turntable test bench was established to conduct an experimental study on the biased load hydrostatic turntable; the experimental data and simulation results were compared with the results obtained from this simplified method of calculating the oil film loading characteristics. The results show that the error of oil film bearing capacity is less than 6% and the error of overturning moment is less than 7%, which has verified the validity of the calculation method. The simplified analytical calculation method proposed in this paper is used to study the influence of tilt displacement rate, lubricant flow rate and turntable speed on the basic performance parameters of oil film, which provides a theoretical basis for the study of oil film load-bearing characteristics under constant bias-load conditions of the constant-flow liquid hydrostatic turntable.

Analytic Design of on-Chip Spiral Inductor with Variable Line Width

On-chip spiral inductors with variable line width layouts are known for their high quality factor (Q-factor). In this paper, we present an analytical approach to facilitate the design of such inductors. Based on an analysis of ohmic and eddy-current losses, we first derive an analytical formula for the metal resistance calculation of a spiral inductor. By minimizing the metal resistance, a simple design equation for finding the proper line width of each coil is then presented. Several 0.18 μm CMOS spiral inductors are investigated, via electromagnetic simulations and experimental studies, to test the proposed resistance calculation, as well as the variable line width design method. It is found that the developed resistance calculation can effectively model the metal-line resistance of a spiral inductor. Moreover, the inductor with a variable line width obtained using the proposed method can significantly improve the Q-factor with little compromise to inductance, which validates the capacity of the developed variable line width design technique. Since the proposed approach can be carried out using analytical calculations, it may be a more efficient design method than those previously reported in the literature.

Rotor Strength Analysis of FeCo-Based Permanent Magnet High Speed Motor

To solve the problem of tension stress caused by centrifugal force and caused by high-speed operation of permanent magnet (PM) rotor, a FeCo-based PM rotor structure model is proposed. Based on the thick-walled cylinder theory, the uniform analytical calculation formulas of strain field, displacement field, and stress field of high-speed permanent magnet rotor are derived, and the stresses of FeCo-based PM and sleeve are obtained. The correctness of analytical calculation method (ACM) is verified by finite element method (FEM). Based on the derived analytical formula, the influence of static interference, sleeve thickness, rotating shaft, and PM thermal expansion coefficient on the strength of high-speed permanent magnet rotor is analyzed, and the mechanical design law of high-speed permanent magnet rotor is summarized. Alloy sleeves are usually fitted outside the high-speed permanent magnet rotor to protect the PM from damage. In order to ensure safe operation of PM rotor under hot rotating conditions, strength calculation and check of sleeve and PM must be carried out. Based on the characteristics of slender structure of high-speed rotor, a high-speed permanent magnet motor (HSPMM) with a rated power of 100 kW and a rated speed of 18,000 r/min was produced, and a continuous operation test was carried out to verify the validity of the above theoretical analysis.

Internal impedance analysis of rectangular conductors at high frequencies

This paper presents the internal impedance analysis of rectangular conductors using analytical calculations. A novel 2D analytical formula is presented using a variational method. And its impedance calculation is compared with other analytical formulas with the format of sum of series for internal impedances calculations of rectangular conductors at high frequencies. It is shown that the novel analytical formula for internal impedance calculation has excellent accuracy despite its simple and closed form function. Dirichlet boundary condition is used by applying zero magnetic vector potential on the edges of rectangular conductor for the analytical calculations of internal impedance. The internal inductance and resistance are calculated at different frequencies between 1 MHz and 1000 MHz. The accuracy and simplicity of different analytical formulas for the internal impedance of rectangular conductors and equivalent circular conductors are compared.

Evaluation of heat supply with maintaining a safe mine water level during operation of open geothermal systems in post-coalmining areas

Purpose. Evaluation of the efficiency of open geothermal systems in flooded and drained mines of the Donetsk basin for heat supply of buildings with maintaining a safe mine water level. Methods. Both circulation and non-return geothermal systems for the mine water heat recovery are analyzed. We proposed the energy and cost criteria for evaluating the effectiveness of open geothermal systems based on a comparison of the produced thermal energy with the energy costs for its production. The criteria use the relationships of thermodynamics, hydraulics, analytical formulas for calculation of ground water flow and methods to calculate the heat demand of indoor spaces. Findings. The estimated ranges of thermal capacity from a few tens kW to a few MW and a coefficient of performance (COP) conversion factor of 3.5-6.8 achievable by geothermal systems for the studied closed mines of Donbas correlate well with the values of these indicators at open geothermal systems operated in different countries, which shows the technical and economic feasibility of the installation to cover local heat demands. The possibility to fully cover the needs for thermal energy is shown on the example of buildings with office spaces for staying of a few hundred people. We demonstrate how to preliminary calculate the parameters of mine water circulation with maintaining the safe level in terms of keeping the ground water quality in the areas adjacent to the mine. Originality. The developed criteria and calculation methodology allow to realistically evaluate the parameters of the efficiency of operation for open non-return and circulation geothermal systems, taking into account mining, geological and technology conditions, to prioritize the exploration of geothermal resources in mines and to evaluate the parameters of mine water circulation with maintaining the safe mine water level. Practical implications. The study showed the feasibility of installation and effective operation of open geothermal systems at the mines of the Donetsk basin for heat supply of buildings located in the adjacent areas with maintaining a safe mine water level.

Fast calculation method of directly buried cables ampacity

Accurate calculation of the ampacity of underground cables is an important prerequisite for ensuring the safe and economical operation of cables, the realization of this goal mainly depends on the accuracy of the calculation of the external thermal resistance of the cable. The accuracy of numerical method is high but slow calculation speed, the analytical algorithm (IEC standard supply) is fast, but the calculation error is large when dealing with the non-uniform soil problems with poor heat conduction performance backfill soil. In this paper, the external thermal resistance of cable is calculated by finite volume method and equivalent thermal circuit method, substitutes the calculation result into the analytical calculation formula, which can quickly and accurately calculate the ampacity of the cable group. This method is faster than the pure numerical method, and which is more accurate than the analytical algorithm. The superiority of this method is proved by comparing with analytical algorithm, coordinate combination method and field-circuit combination method in uniform and non-uniform soils.

VPA: Computer program for the computation of the phase shift in atom–atom potential scattering using the Variable Phase Approach

A computer code for the computation of the phase shift in atom–atom and electron–atom potential scattering is presented. The phase shift is the central quantity required for the calculation of all types of scattering cross sections. The program uses the Variable Phase Approach (VPA). This is the only exact method for the direct calculation of the scattering phase shift. All other methods are based on examining the large distance behavior of the exact solution of the Schrödinger equation. Such methods yield the phase shift only modulo π. The absolute value of the phase shift and its variation with scattering energy is, however, needed for a full understanding of the scattering process, such as for instance in the study of shape resonances and Glory oscillations. The VPA has been sparingly used owing to the instability of the underlying equations and the consequent difficulty of writing computer code to solve them. We present a computer code for the efficient implementation of the VPA method for atom–atom scattering problems over a wide range of scattering energies. The code works for potentials which are singular and for those that are non-singular at the origin. An example of the implementation of the code is given for both an interaction potential with an attractive well and for a purely repulsive potential. Program summaryProgram title: VPA:Variable Phase ApproachCPC Library link to program files: https://dx.doi.org/10.17632/zh248d2362.1Code Ocean capsule: https://dx.doi.org/10.24433/CO.8146850.v1Licensing provisions: GNU General Public License 3 (GPL)Programming language: Fortran 90Nature of problem: To compute the quantum mechanical phase shift in an atom–atom or an electron–atom potential scattering problem. The interaction potential, dependent on a single radial coordinate, is defined by the user.Solution method: The Variable Phase Approach is used. This is the only available method for the direct calculation of the phase shift. The initial internuclear separation for the integration is chosen with the help of the analytical solution for a hard sphere core for all orbital angular momentum quantum numbers L of interest. The first-order nonlinear equation is integrated numerically with a modified LSODA19,20 program. The analytic formula for calculation of the Jacobian of the equation is also provided.Additional comments including restrictions and unusual features: Computer codes for the atom–atom interaction potential, POTENT1, and for its derivative with respect to the inter nuclear separation, DPOTENT1, must be supplied by a user. The calculation of the spherical Bessel and Neumann functions required is performed explicitly for large l angular momentum quantum numbers where the standard upward iterative generation of the functions becomes unstable.

Improved coining force calculations through incorporation of key process parameters

Among the sheet forming processes, coining is a specific operation which makes it possible to correct shape defects or to perform thickness reductions in the parts. This operation often requires a very high force which is likely to have an impact on the functioning of the tool or the press. It is therefore important that the coining forces be evaluated accurately, but this is not allowed by the existing analytical calculation formulas. The objective of this study is to improve the accuracy of the calculation of the coining forces via the adaptation of an existing formula. Such adaptation was carried out based on a study of the influence of the flat coining parameters. To that end, a series of experimental measurements was performed. An instrumented force measuring setup made it possible to measure the maximum coining force. Several parameters were analysed, including the coining ratio and the sheet thickness. Numerical simulations were carried out at the same time, in order to understand the influence of certain parameters on the coining force. 2D numerical models were developed using the Forge NXT2 software. The simulations and the experimental tests were analysed and the results revealed the influential phenomena which have to be taken into account in the analytical formula, in particular the coining surface and the friction. In order to study the friction more thoroughly, ring compression tests were performed so as to determine the friction coefficients, based on the Coulomb’s law limited to Tresca. This type of test is representative of the stresses undergone by the metal during the coining operation. Finally, a new calculation formula is proposed, in order to integrate the coining surface and friction more accurately.

Calculation of Stretch Stress of Double-Layer Plate Composites

The stretch stress of each layer of laminate composite undergoes complex changes after being stretched, which significantly affects the stretch strength. In order to determine the magnitude of the change, the paper directly solved the stretch stress using the analytical method through simple assumptions, and obtained the analytical calculation formula. Studies on double-layer plates show that the magnitude of stretch stress is closely related to the elastic modulus, length and thickness dimensions of the plate layer, the shear modulus of the adhesive, the thickness of the adhesive layer, and the external stress, and the maximum stress occurs at the middle section. The calculation of the formula provides a convenient way to check the stretch strength.

A Disturbed Grid Voltage Interharmonic Analysis with Fourier Series of Several Variables

Increasing the range of electronic devices, especially nonlinear impulse loads worsen the parameters of electricity quality. One of the most negative factors for the electricity quality are low-frequency interharmonics, which have a negative impact on electric motors, transformers and other AC equipment. The interharmonics generated by disturbances forming by grid loads which operates with frequency that is not multiple with the electric grid one. The interharmonics detection is a complex problem because of their permanent frequency fluctuation. The common numerical methods implementation for interharmonic analysis leads to large number of the mathematical operation, therefore analytical methods are more attractable. Usually, interharmonics forming is researched in wide range of disturbance parameters for further analysis and their suppression. Often, for math operation reducing, interharmonic analysis is performed by probabilistic approach in several points of parameters range with subsequent expanding its result on all parameter values that cause to inaccurate results. In the paper is proposed a fully analytical method based on a Fourier series of several variables, which allows: describe the voltage model immediately in the frequency domain; calculate the interharmonics of the voltage in all disturbance parameter range directly with a minimum number of mathematical operation due to representing the parameters in final analytical formula for interharmonics calculation; minimize error. In considered case the electric grid model consists of active-inductive internal resistance and a batch of nonlinear loads: AC system which operates on different frequency comparing with the grid frequency; pulse loads that switches with different frequency than grid frequency; parametric load with non-multiple period to the grid frequency. The properties of Fourier series of several variable are analyzed and advantages of describing signals with several components are shown. Principle of projection of multidimensional spectrum to one variable is demonstrated. The grid voltage disturbance for each load are analyzed and voltage model in frequency domain of several variable are obtained. The shape of each disturbance for batch of the input parameters are constructed and total grid voltage shape and its spectrum are built. The nature of interharmonic forming as combinational harmonics are shown. The mathematical operation elimination is analyzed for general case and shown the total advantage of the proposed method in comparing with numerical Fourier series of one variable.