Unified Formulae Research Articles

A unified approach to inversion formulae for vector and tensor ray and radon transforms and the Natterer inequality

Most derivations of inversion formulae for x-ray or Radon transform are based on the projection theorem, where for fixed direction the Fourier transform of x-ray or Radon transform is calculated and compared with the Fourier transform of the searched-for function. In contrast to this we start here off from the searched-for field, calculate its Fourier transform for fixed direction, which is now a vector or tensor field, that we then expand in a suitable direction dependent basis. The expansion coefficients are recognized as the Fourier transform of longitudinal, transversal or mixed ray transforms or vectorial Radon transform respectively. The inverse Fourier transform of the searched-for field then directly leads to inversion formulae for those transforms applying problem adapted backprojections. When considering the Helmholtz decomposition of the field we immediately find inversion formulae for those transversal or longitudinal transforms. First inversion formulae for the longitudinal ray transform, similar to those given by Natterer (1986 The Mathematics of Computerized Tomography (Teubner and Wiley)) for x-ray tomography, were given by Natterer-Wübbeling in 2001, Natterer and Wübbeling (2001 Mathematical Methods in Image Reconstruction (SIAM)), but then not pursued by other authors. In this paper, we present the above described method and derive in a unified way inversion formulae for the ray transforms treated in Louis (2022 Inverse Problems 38 065008) containing the results from Louis (2022 Inverse Problems 38 065008) as special cases. Additionally we present new inversion formulae for the vectorial Radon transform. As a consequence the inversion formulae directly give Plancherel’s formulae for the vectorial or tensorial transforms. Together with the Natterer inequality, which is independent of the ray or Radon transforms, we present the Natterer stability of those vectorial and tensorial transforms.

Velocity-pressure admittance for structural analysis of large cooling towers against strong winds

To facilitate dynamic structural analyses of large cooling towers subjected to strong winds in structural design and safety assessments, the auto-spectra of the local wind pressures on the tower shell are required. To avoid expensive and laborious physical model tests, Pirner provided empirical normalized pressure spectra for direct use, which imply that the concept of velocity-pressure admittance can be applied in the windward region on the tower shell. However, Pirner's empirical results are subjected to the fact that the dependence of pressure spectra on height or turbulence intensity of the oncoming flow is wrongfully neglected. To this end, velocity-pressure admittances are measured at various heights on a 1:200 scaled tower model in an atmospheric boundary layer wind tunnel, and unified formulae for velocity-pressure admittance on windward region of large cooling towers are fitted out for use considering the turbulence intensity effects. The unified formulae are further validated using data measured on a 167 m high full-scale cooling tower in China. The full-scale/empirical formulae comparison indicates that the unified formulae for velocity-pressure admittance proposed are applicable on the windward side [0 degrees, 70 degrees]; while a constant of 1 should be utilized as the normalized power spectral density in the circumferential region [70 degrees, 180 degrees] in structural analyses.

An approach to unified formulae for likelihood ratio calculation in pairwise kinship analysis.

Introduction: The likelihood ratio (LR) can be an efficient means of distinguishing various relationships in forensic fields. However, traditional list-based methods for derivation and presentation of LRs in distant or complex relationships hinder code editing and software programming. This paper proposes an approach for a unified formula for LRs, in which differences in participants' genotype combinations can be ignored for specific identification. This formula could reduce the difficulty of by-hand coding, as well as running time of large-sample-size simulation. Methods: The approach is first applied to a problem of kinship identification in which at least one of the participants is alleged to be inbred. This can be divided into two parts: i) the probability of different identical by descent (IBD) states according to the alleged kinship; and ii) the ratio of the probability that specific genotype combination can be detected assuming the alleged kinship exists between the two participants to the similar probability assuming that they are unrelated, for each state. For the probability, there are usually recognized results for common identification purposes. For the ratio, subscript letters representing IBD alleles of individual A's alleles are used to eliminate differences in genotype combinations between the two individuals and to obtain a unified formula for the ratio in each state. The unification is further simplified for identification cases in which it is alleged that both of the participants are outbred. Verification is performed to show that the results obtained with the unified and list-form formulae are equivalent. Results: A series of unified formulae are derived for different identification purposes, based on which an R package named KINSIMU has been developed and evaluated for use in large-size simulations for kinship analysis. Comparison between the package with two existing tools indicated that the unified approach presented here is more convenient and time-saving with respect to the coding process for computer applications compared with the list-based approach, despite appearing more complicated. Moreover, the method of derivation could be extended to other identification problems, such as those with different hypothesis sets or those involving multiple individuals. Conclusion: The unified approach of LR calculation can be beneficial in kinship identification field.

Load capacity of concrete-filled steel tubular members with multilayer confinement subjected to coupled axial compression-bending and fire

The demands to increase bearing capacity of concrete-filled steel tubular (CFST) members and, thus, cross-sectional size of the members, have caused a series of practical issues, such as the difficulties in manufacturing steel tubes and low quality of concrete pouring. As a solution for these, concrete-filled steel tubular (CFST) members with stiffeners can be used. There is no compression-bending interactive equation for CFST members with multilayer confinement at high temperatures. This paper is mainly based on the compression-bending interactive equations of traditional CFST members recommended by Chinese standards at room temperatures, introduces the reduction factors of material strength and elastic modulus at high temperatures, and further extends the compression-bending interactive equations at room temperatures to high temperatures. To verify the validity of the unified formulae, the calculation results of the theoretical formulae are compared with the numerical and experimental results in this paper. The results of numerical simulation show that the two sets of proposed compression-bending interactive equations of CFST members with multilayer confinement are conservative. In general, the equation computing coefficient is proportional to the fire resistance time. In addition, the experimental results are in good agreement with the compression-bending interactive equation.

Regression Models for Predicting Quantities and Estimates of Steel Reinforcements in Concrete Beams of Frame Buildings

The traditional method of quantifying reinforced concrete steel reinforcements via taking off can be tedious, time consuming and prone to errors which can affect project success due to cost and schedule overruns, disputes and in certain cases, outright abandonment. In Nigeria, some quantity surveyors have developed ‘rule of thumb’ techniques to quantify reinforcements in order to beat pre-contract datelines based on their past experience, but there are still not widely accepted and a unified formulae or empirical basis of generating these quantities is still lacking. This study thus, developed easy-to-apply, time saving regression models for predicting the quantities/weight and material cost estimates of 16mm, 12mm and 8mm diameter high yield reinforcement bars in beams of varying sizes, using the volume of beam concrete as the independent or predictor variable. Data on concrete volume, weight of Y16, Y12 and Y8 reinforcement was collected via taking off/measurement process from 30 structural drawings of frame buildings of varying nature obtained from registered structural engineers and analyzed using correlation and regression statistics. Results indicate high coefficients of determination (R2) ranging from 0.82 to 0.92 which indicate that the predicted values from a forecast models fit with the real-life data. Thus, 3 predictive models were advanced as follows: WY16= -811.265+ 177.339 (Vc) ;WY12= -510.189 + 63.218(Vc); WY8 = -43.273+ 22.533 (Vc), where: W = reinforcement weight and Vc = volume of concrete. The study concludes that concrete volume is a good predictor variable when establishing the weight of reinforcement in beams. The import of these predictive models for construction cost professionals cannot be overemphasized for ease and accuracy of feasibility estimating, preparation of bills of quantities, material ordering, auditing construction costs, vetting consultants’ estimates and contractors’ quotations.

Sensitivity analysis of optically preamplified Stokes-vector receivers using analytically derived formulae for bit-error rate.

This study analyzes the bit-error-rate characteristics of Stokes-vector modulation-demodulation systems employing optically preamplified receivers. We derive unified analytical formulae for the bit-error rate of binary, quad, and octal modulation formats, in which Stokes vectors are modulated in a binary manner per dimension. The receiver sensitivities estimated from the derived bit-error-rate formulae are examined by analyzing the probability-density function of noise in the Stokes-vector direct-detection receiver. In addition, the bit-error-rate formulae enable a comprehensive comparison of the sensitivity among various optical receivers, such as coherent, direct-detection, optically differential direct-detection, and Stokes-vector direct-detection receivers.

Unified formulae for critical load and unified DSM expressions for pinned-end and fixed-end columns exhibiting pure distortional buckling

Abstract Unified formulae of elastic critical distortional buckling load and unified DSM expressions for columns undergoing pure distortional buckling, are proposed in this study for both pinned-end and fixed-end columns with a C/Z/hat sections. The normal stress distribution of pre-buckling and post-buckling for C-section columns undergoing distortional buckling is presented based on theoretical analysis and literature review. The basic purpose of the DSM expressions is to express the strength problem as a stability problem to determine the ultimate strength of columns experiencing pure distortional buckling. To understand the effects of both pinned-end and fixed-end conditions in the columns undergoing pure distortional buckling, the effective length factor of flexural buckling about the strong axis of a cross-section, is introduced into elastic distortional buckling load formulae and the DSM expressions. Numerical results computed by computer software (CUFSM), along with numerical and experimental results in literature reviews are used to assess the accuracy of the proposed formulae. It is shown that the proposed formulae are accurate and universal.

Analytical calculations of self‐ and mutual inductances for rectangular coils with lateral misalignment in IPT

Rectangular coils are widely used in inductive power transfer (IPT), and the self- and mutual inductances of them are key system parameters. However, the study of the rectangular coil is not enough. In this study, a series of unified analytical formulae for the self- and mutual inductance calculations of a variety of rectangular coils including filamentary coils, pancake coils and thin- or thick-walled solenoids are presented. Various positions of the rectangular coils above are discussed, and the mutual inductance formulae in the case of co-axis, co-plane, misalignment and even overlapping are given. Finally, the experimental measurements are carried out on various types of rectangular coils. The experimental results agree well with the calculated ones, which indicate the effectiveness of the proposed method. Therefore, this study provides an interesting tool for designing the transceiver of rectangular coils in IPT system.

Evaluation of axial strength in circular STCC columns using UHPC and UHPFRC

This study investigates the axial strength of circular steel tube confined concrete (STCC) columns with the employment of plain ultra high performance concrete (UHPC) and ultra high performance fiber reinforced concrete (UHPFRC) having compressive strengths of cylinders varying from 178.9 MPa and 198 MPa. UHPFRC used in this study had 1% and 2% volume fraction of steel fibers in the mixture. Eighteen specimens including both short and intermediate columns were cast and tested under concentric loading on the concrete core to generate the maximal confinement effect. Experimental results showed that the effect of steel fibers on the axial load versus axial strain responses, and on the enhancement of strength and ductility was insignificant, while the parameters of the steel tube and the concrete core length were found to play a major role. The ultimate loads of the tests presented herein were compared with those obtained from the predictions by seven analytical models and five existing design codes. The design expressions from these analytical models and codes were assessed. For the prediction of the ultimate loads of both short and intermediate column, the model by De Oliveira et al. [13] was the most suitable approach among seven selected models, whereas the ACI provision [1] was the best among five existing design codes. Furthermore, the experimental ultimate loads obtained from this study were combined with those from previous studies to examine the influence of different concrete strength ranges on the strength enhancement. Finally, unified formulae to predict the ultimate strength with a wide range of concrete strength up to 200 MPa were also proposed for both short and intermediate columns.

Enumeration of Wiener indices in random polygonal chains

The Wiener index of a connected graph (molecule graph) G is the sum of the distances between all pairs of vertices of G, which was reported by Wiener in 1947 and is the oldest topological index related to molecular branching. In this paper, simple formulae of the expected value of the Wiener index in a random polygonal chain and the asymptotic behavior of this expectation are established by solving a difference equation. Based on the results above, we obtain the average value of the Wiener index of all polygonal chains with n polygons. As applications, we use the unified formulae to obtain the expected values of the Wiener indices of some special random polygonal chains which were deeply discussed in the context of organic chemistry or statistical physics.

Certain Fractional Derivative Formulae Involving the Product of a General Class of Polynomials and the Multivariable Gimel-Function

In the present paper, we obtain three unified fractional derivative formulae. The first involves the product of a general class of polynomials and the multivariable Gimel-function. The second involves the product of a general class of polynomials and two multivariable Gimel-functions and has been obtained with the help of the generalized Leibniz rule for fractional derivatives.The last fractional derivative formulae also involves the product of a general class of polynomials and the multivariable Gimel-function but it is obtained by the application of the first fractional derivative formulae twice and, it involve two independents variables instead of one.The polynomials and the functions involved in all our fractional derivative formulae as well as their arguments which are of the type The formulae are the very general character and thus making them useful in applications. In the end, we shall give a particular case.

Recursive super-convergence computation for multi-dimensional problems via one-dimensional element energy projection technique

This paper presents a strategy for computation of super-convergent solutions of multi-dimensional problems in the finite element method (FEM) by recursive application of the one-dimensional (1D) element energy projection (EEP) technique. The main idea is to conceptually treat multi-dimensional problems as generalized 1D problems, based on which the concepts of generalized 1D FEM and its consequent EEP formulae have been developed in a unified manner. Equipped with these concepts, multi-dimensional problems can be recursively discretized in one dimension at each step, until a fully discretized standard finite element (FE) model is reached. This conceptual dimension-by-dimension (D-by-D) discretization procedure is entirely equivalent to a full FE discretization. As a reverse D-by-D recovery procedure, by using the unified EEP formulae together with proper extraction of the generalized nodal solutions, super-convergent displacements and first derivatives for two-dimensional (2D) and three-dimensional (3D) problems can be obtained over the domain. Numerical examples of 3D Poisson’s equation and elasticity problem are given to verify the feasibility and effectiveness of the proposed strategy.

Full forward kinematics of redundant kinematic hybrid manipulator

• Establish unified recursive mathematic model for full forward kinematics analysis of redundant kinematic hybrid manipulators. • Discover relationship between general input velocity/acceleration and general output velocity/ acceleration. • Solve general output acceleration of hybrid manipulator by general input velocity/acceleration of every parallel manipulator. • Construct novel (4SPS + SPR) + (4SPS/SP) + 3SPR hybrid manipulator, derive its forward velocity/ acceleration formulae. A unified recursive mathematic model is established for full forward kinematics analysis of various redundant kinematic hybrid manipulators which are formed by several different parallel manipulators connected in series. In the established model, the relationships between the general input velocity/acceleration and the general output velocity/acceleration are discovered. The Jacobian matrices for mapping the general output velocities to the general input velocity and the Hessian matrices for mapping the general output velocities to the general input acceleration are derived. The unified iteration recursive formulae are derived for solving the general output velocity/acceleration of the end moving platform of the redundant kinematic hybrid manipulator by only giving the general input velocity/acceleration of every parallel manipulator. A 3D model of a novel (4SPS + SPR )+ (4SPS/SP) + 3SPR type hybrid manipulator is constructed and its full forward velocity/acceleration formulae are derived from the established unified recursive mathematic model and are proved to be correct by utilizing a simulation solution of the novel redundant kinematic hybrid manipulator.

Unified formulae for evaluating load reduction by change in stiffness of circumferential crack considering general piping systems

The conservatism of leak-before-break (LBB) analysis depends on the methodologies for evaluation of the crack opening displacement (COD) and crack stability analysis, in which the applied load at the cracked section is an important variable. In current procedure, the applied loads are obtained from an uncracked pipe analysis, and the effect of restraint is not assumed in the calculations of COD and the allowable applied moment. When the presence of a crack and the pipe restraint are considered, however, the applied load can be reduced because of the change in the piping system stiffness. In this regard, there have been several proposed analytical expressions to evaluate the restraint effects on COD or crack stability analysis, but the expressions cannot be applied to practical cases because of the limitation of the pipe geometries and applied loading conditions. This study develops unified formulae to evaluate the effective applied loads that enable balanced analyses of both the COD and stability of circumferential cracked pipes. The formulae were developed for a piping system containing a circumferential crack based on the compliance approach. By comparing this with finite element analysis results, the validity of the formulae was demonstrated. It is expected that the proposed formulae can be used as a simple approach to detailed LBB analysis to secure the safety margin before implementing the complex non-linear finite element analysis.

Unified recursive derivation and analysis of complex associated linkages with various links and type synthesis of complex robot mechanisms

It has been a significant and challenging issue to synthesize novel complex robot mechanisms to create the equipments/devices such as flexible hand-finger, flexible leg-feet, waist, the propeller generator of submarine or helicopter, artificial heart, flexible airfoil, and bionic mechanism. In this paper, an approach for recursively deriving the complex-associated linkages (ALs) with various links is proposed, and novel mechanisms with high complexity coefficient c are synthesized using derived ALs and the topology graph. First, the concepts of c and the various links Li with i connection joints are explained, and various kinematic pairs with multi-DoF are replaced with binary links L2 and connection joints J. Second, the unified recursive formulae are derived for solving c, Li (2 ≤ i) and the maximum number of Li as 0 ≤ c. Third, the 839 ALs with different groups of Li (2 ≤ i ≤ 8) as to 0 ≤ c ≤ 10 are derived utilizing the unified recursive formulae and a relative algorithm. The relations among c, AL, Li (2 ≤ i), redundant constraint, passive DoF and prescript DoF of the mechanisms are discovered. Finally, several examples are given to illustrate the approach of type synthesis of novel complex mechanisms using derived ALs and the topology graph. The recursively derived formulae and proposed approach are verified to be correct and effective.

Fracture assessment based on unified constraint parameter for pressurized pipes with circumferential surface cracks

The fracture assessment capability and methodology based on the unified constraint parameter Ap have been comparably investigated with the conventional procedures without considering constraint effect and the assessment approach based on the constraint parameter Q for pressurized pipes with circumferential surface cracks. The results show that the Ap can be used in failure assessment diagram based fracture assessment by modifying the fracture toughness using the unified correlation formulae of fracture toughness with Ap. A fracture assessment methodology based on the parameter Ap for incorporating both in-plane and out-of-plane constraints has been proposed. The constraint characterization and fracture assessment using the parameter Ap for the pipe cracks may get the best accuracy and can reduce excessive conservatism of the conventional assessments. The benefits from the unified constraint assessments increase with decreasing crack depth and length.

Explicitly self-similar solutions for the Euler/Navier–Stokes–Korteweg equations in RN

Abstract In this paper, we present a unified formulae for explicit self-similar solutions of the Euler/Navier–Stokes–Korteweg equations arising in the modeling of capillary fluids. The technique used here is to reduce Euler/Navier–Stokes–Korteweg equations into a series of solvable ordinary differential equations by making use of multi-dimensional self-similar ansatz and variable separation method.

Exact cone beam reconstruction formulae for functions and their gradients for spherical and flat detectors

We derive unified inversion formulae for the cone beam transform similar to the Radon transform. Reinterpreting Grangeat’s formula we find a relation between the Radon transform of the gradient of the searched-for function and a quantity computable from cone beam data. This gives a uniqueness result for the cone beam transform of compactly supported functions under much weaker assumptions than the Tuy–Kirillov condition. Furthermore this relation leads to an exact formula for the direct calculation of derivatives of the density distribution; but here, similar to the classical Radon transform, complete Radon data are needed, hence the Tuy–Kirillov condition has to be imposed. Numerical experiments reported in Hahn B N et al (2013 Meas. Sci. Technol. 24 125601) indicate that these calculations are less corrupted by beam-hardening noise. Finally, we present flat detector versions for these results, which are mathematically less attractive but important for applications.

Dynamics model of redundant hybrid manipulators connected in series by three or more different parallel manipulators with linear active legs

A general redundant kinematic hybrid manipulator connected in series by three or more different parallel manipulators with linear active legs is constructed. Its kinematics and dynamics are studied systematically. First, the mapped relations between the velocities of the moving links and the input velocities are discovered; the displacement, velocity, and acceleration of the general redundant hybrid manipulator are derived. Second, the inertial wrench and gravity of all the moving links are transformed into the fixed coordinate system; a unified recursive dynamics formula of the general redundant hybrid manipulator is derived. Third, a novel 3SPR+3SPR+3RPS type manipulator is designed and illustrated, its kinematics and dynamics formulae are derived based on the derived unified recursive dynamics formulae of the general redundant hybrid manipulator. Finally, the kinematics and dynamic active/constrained forces of the 3SPR+3SPR+3RPS type manipulator are solved and verified using its simulation mechanism.

Dynamics analysis of 3-leg 6-DoF parallel manipulator with multi different-DoF finger mechanisms

A dynamics of a 3-leg 6-DoF parallel manipulator with multi different-DoF fingers is studied systematically. First, the unified kinematics formulae of the parallel manipulator with multi different-DoF fingers are established for solving the velocity/acceleration of the moving links of parallel manipulator and finger mechanisms and the Jacobian/Hessian matrices. Second, their unified dynamics formulae are established for solving the dynamic input wrench of the parallel manipulator and the finger mechanisms. Third, the special kinematics/ dynamics formulae of a novel 6-DoF 3-UPS type parallel manipulator with three one-DoF finger mechanisms are derived for illustrating application of dynamics formulae. Finally, its analytic solutions are analyzed and verified by the simulation solutions.