General Closed-form Solution Research Articles

A General Exact Closed-Form Solution for Nonlinear Differential Equation of Pendulum

In the present paper, the nonlinear differential equation of pendulum is investigated to find an exact closed form solution, satisfying governing equation as well as initial conditions. The new concepts used in the suggested method are introduced. Regarding the fact that the governing equation for any arbitrary system represents its inherent properties, it is shown that the nonlinear term causes that the system to have a variable identity. Hence, the original function is included as a variable in the solution to can take into account the variation of governing equation. To find the exact closed form solution, the variation of the nonlinear differential equation tends to zero, where in this case the system with a local linear differential equation has a definite identity with a definite local answer. It is shown that the general answer is an arbitrary curve on a surface, a newly developed concept known as super function, and different initial conditions give different curves as particular solutions. The comparison of the results with those of finite difference shows an exact agreement for any arbitrary amplitude and initial conditions.

Planning Trajectory of Anthropomorphic Walking Robot (Biped Robot)

Humanoid robots have been a topic of great interest for a long time. Among the various motions of a humanoid robot, the most basic and important motion is bipedal walking. Bipedal walking is probably the most appropriate way for robots to move around in a real environment. However Building trajectories for biped robot walking is a complex task considering all degrees of freedom (DOFs) commonly bound within the mechanical structure. To study the stable motion of any bipedal robot we should study the Kinematics, Dynamics, design a control system and of most important the trajectories. This work deals with planning trajectory of humanoid robots (AK Biped robot), especially for planning trajectory of center of mass (CoM), zero moment point (ZMP)a and planning trajectory of swing foot. diffident method used in planning trajectory like 3D linear inverted pendulum (LIMP), polynomial interpolation. The object of study is the process of automatic control walking of an anthropomorphic walking robot (AK biped robot), for that this paper proposes an approach of joint trajectory generation for the biped robot using the three Three-dimensional linear inverted pendulum for planning trajectory of CoM which keep the ZMP to be inside the support polygon which and guarantee stable dynamic walking of AK biped robot, then trajectory of swing foot derived using the polynomial interpolation function for smooth motion. The aim of this paper is to present a full 3D walking strategy using dynamic model of 3D inverted pendulum for generation references joint trajectories with simulation also design a control system that granites a stable walking of Biped Robot. This is done by first reviewing the literature about different walking strategies. During this literature review the 3D-Linear Inverted Pendulum Model appeared to be the most interesting strategy for further research. The idea of the strategy is simple. It models the human as a linear inverted pendulum with massless rods, which represent the legs, and a point mass at the end of the rods representing the total mass of the body. During walking, there is always at least one foot on the ground which can be seen as stance leg. This stance leg is then modeled as an inverted pendulum. The general closed form solution of the dynamics of the linear inverted pendulum are used to design a trajectory for the center of mass (CoM ) for stance leg. In the trajectory generator, the general solution of the 3D-LIPM is used to prescribe a trajectory. for the CoM of the biped robot AK. The trajectory of the swing leg is designed by a cosine velocity profile interpolation function. The joint trajectories are used as the input for a dynamic model of biped robot in SimMechanics. The proposed control system in simulation are carried out to tune the trajectory such that the dynamic model is able to walk balanced and the robustness of the gait was verified by adding of disturbances. For example, the position of the CoM was increased or decreased and the steps were simulated with different ground levels. These simulations showed that the trajectory is relatively robust. Ref. 10, fig. 4.

A closed-form general solution for the distance of point-to-ellipse in two dimensions

This paper resolved a closed-form general solution for the distance of one point and one ellipse in a two dimensional plane. Even such problem is simple as apparently to be understood, the solutions provided from literature are all about iterative algorithms or approximating approaches. None of O(1) methods or general roots solutions are found. The author takes several geometrical transformations to reduce the power of the equation to simplify the problem. Eventually, a closed-form general solution is resolved in this paper. This can be a contribution both for academic and for in practice and should be archived. Based on this general solution, we can develop the algorithm computing the shortest path touring n sequenced ellipses soon. Since the results of this paper is sufficiently fundamental, it can be conducted in a variety of applications: in the fields of computer-aided design and manufacturing, computer graphics, layered manufacturing, prototyping, robot motion (path) planning, wireless sensor networking, multimedia animation, geodesy, astronomy, physics, molecular geometry, electromagnetics, and Fluid mechanics.

Shear deformable super-convergent finite element for steel beams strengthened with glass-fiber reinforced polymer (GFRP) plate

The present study investigates the flexural behaviour of steel beams strengthened by adhesively bonding a glass-fiber reinforced polymer (GFRP) plate to one of the flanges. The model captures shear deformation effects and partial interaction between the steel and GFRP owing to the relative flexibility of the adhesive. A general closed form solution is first developed for the governing coupled system of differential equations. The solution is then used to formulate mechanics-based shape functions and develop a finite element with superior convergence characteristics. The model is used to investigate the response of multi-span continuous beams, determine the strength gained by GFRP strengthening, and quantify shear deformation effects on the response of strengthened beams. A technique capturing partial interaction effects is devised to characterize the flexural strength of Class 3 strengthened beams. A classification limit for strengthened Class 3 sections is also proposed within the framework of the Canadian Standard CAN-CSA S16 (2014).

Globally optimal volume-trap arrangements for the narrow-capture problem inside a unit sphere

The determination of statistical characteristics for particles undergoing Brownian motion in constrained domains has multiple applications in various areas of research. This work presents an attempt to systematically compute globally optimal configurations of traps inside a three-dimensional domain that minimize the average of the mean first passage time (MFPT) for the narrow capture problem, the average time it takes a particle to be captured by any trap. For a given domain, the mean first passage time satisfies a linear Poisson problem with Dirichlet-Neumann boundary conditions. While no closed-form general solution of such problems is known, approximate asymptotic MFPT expressions for small traps in a unit sphere have been found. These solutions explicitly depend on trap parameters, including locations, through a pairwise potential function. After probing the applicability limits of asymptotic formulas through comparisons with numerical and available exact solutions of the narrow capture problem, full three-dimensional global optimization was performed to find optimal trap positions in the unit sphere for 2≤N≤100 identical traps. The interaction energy values and geometrical features of the putative optimal trap arrangements are presented.

An Anylatical approach for space–time fractal order nonlinear dynamics of microtubules

ABSTRACTHere, A new fractional sub-equation method is proposed for constructing the exact solutions of space–time fractional partial differential equations arising in nonlinear dynamics of microtubules in the sense of modified Riemann-Liouville derivative, which is the fractional version of the known method.The proposed approach is efficient and powerful for solving wide classes of nonlinear evoluation fractional order equations. It is observed that the performance of the proposed method is reliable and will be used to establish new general closed form solutions for any other NPDEs of fractional order. The solutions obtained here are new and have not been reported in former literature.

General inverse solution of six-degrees-of-freedom serial robots based on the product of exponentials model

PurposeDeveloping general closed-form solutions for six-degrees-of-freedom (DOF) serial robots is a significant challenge. This paper thus aims to present a general solution for six-DOF robots based on the product of exponentials model, which adapts to a class of robots satisfying the Pieper criterion with two parallel or intersecting axes among its first three axes.Design/methodology/approachThe proposed solution can be represented as uniform expressions by using geometrical properties and a modified Paden–Kahan sub-problem, which mainly adopts the screw theory.FindingsA simulation and experiments validated the correctness and effectiveness of the proposed method (general resolution for six-DOF robots based on the product of exponentials model).Originality/valueThe Rodrigues rotation formula is additionally used to turn the complex problem into a solvable trigonometric function and uniformly express six solutions using two formulas.

Exact solutions of the vertical gravitational anomaly for a polyhedral prism with vertical polynomial density contrast of arbitrary orders

We present general closed-form solutions for the vertical gravitational anomaly caused by a polyhedral prism with mass density contrast varying with depth. Our equations are the first ones to imple ...

Influence of model type, bias and input parameter variability on reliability analysis for simple limit states with two load terms

A general closed-form solution to compute the reliability index of a simple limit state function with two load terms and one resistance term is derived. The formulation considers contributions to margins of safety expressed in probabilistic terms due to the choice of load and resistance models, bias values, dependencies between nominal values and bias, uncertainty in estimates of nominal values for uncorrelated load and resistance terms at time of design, and average margin of safety expressed as the operational factor of safety. A sensitivity analysis and example application demonstrate the quantitative influence of the contributing random variables on reliability index.

Traveling wave solutions to some nonlinear fractional partial differential equations through the rational (G′/G)-expansion method

In this article, the analytical solutions to the space-time fractional foam drainage equation and the space-time fractional symmetric regularized long wave (SRLW) equation are successfully examined by the recently established rational (G′/G)-expansion method. The suggested equations are reduced into the nonlinear ordinary differential equations with the aid of the fractional complex transform. Consequently, the theories of the ordinary differential equations are implemented effectively. Three types closed form traveling wave solutions, such as hyperbolic function, trigonometric function and rational, are constructed by using the suggested method in the sense of conformable fractional derivative. The obtained solutions might be significant to analyze the depth and spacing of parallel subsurface drain and small-amplitude long wave on the surface of the water in a channel. It is observed that the performance of the rational (G′/G)-expansion method is reliable and will be used to establish new general closed form solutions for any other NPDEs of fractional order.

A general closed-form solution to a Timoshenko beam on elastic foundation under moving harmonic line load

In this paper, a general closed-form solution for evaluating the dynamic behavior of a Timoshenko beam on elastic foundation under a moving harmonic line load is formulated in the frequency-wavenumber domain and in a moving coordinate system. It is found that the characteristic equation is quartic with real coefficients only, and its poles can be presented explicitly. This enables the substitution of these poles into Cauchy\'s residue theorem, leading to the general closed-form solution. The solution can be reduced to seven existing closed-form solutions to different sub-problems and a new closed-form solution to the subproblem of a Timoshenko beam on an elastic foundation subjected to a moving quasi-static line load. Two examples are included to verify the solution.

Elastic Analysis of Steel Beams Strengthened with GFRP Plates Including Preexisting Loading Effects

The present study develops a theory for the elastic analysis of a preloaded wide flange steel beam, strengthened with two glass fiber-reinforced polymer (GFRP) plates bonded to both flanges, then subjected to additional loads. Starting with the principle of stationary potential energy, the governing equilibrium equations and corresponding boundary conditions are formulated prior to and after GFRP strengthening. The resulting theory involves four coupled equilibrium equations and 10 boundary conditions. A general closed form solution is then provided for general loading and boundary conditions. Detailed comparisons with three-dimensional finite-element solutions show that the theory provides reliable predictions for the displacements and stresses. A parametric study is then developed to quantify the effects of strengthening, GFRP plate thicknesses, and preexisting loads on the capacity of the strengthened beam.

Multiple closed form wave solutions to the KdV and modified KdV equations through the rational (G′/G)-expansion method

The purpose of this article is to generate the closed form traveling wave solutions of some nonlinear partial differential equations namely; the KdV equation and the modified KdV equation using the recently established rational (G′/G)-expansion method. By this method, new and further general closed form solutions expressed by three types of functions as for instance hyperbolic, trigonometric and rational function solutions are found. The basic ideas presented in this article and the obtained solutions might be useful to analyze the shallow water waves, ion acoustic waves in plasma, acoustic waves on a crystal lattice and the role of nonlinear dispersion in the formation of structures like liquid drops.

Surface-admittance equivalence principle for nonradiating and cloaking problems

In this paper, we address non-radiating and cloaking problems exploiting the surface equivalence principle, by imposing at any arbitrary boundary the control of the admittance discontinuity between the overall object (with or without cloak) and the background. After a rigorous demonstration, we apply this model to a non-radiating problem, appealing for anapole modes and metamolecules modeling, and to a cloaking problem, appealing for non-Foster metasurface design. A straightforward analytical condition is obtained for controlling the scattering of a dielectric object over a surface boundary of interest. Previous quasi-static results are confirmed and a general closed-form solution beyond the subwavelength regime is provided. In addition, this formulation can be extended to other wave phenomena once the proper admittance function is defined (thermal, acoustics, elastomechanics, etc.).

LRFD Calibration of Simple Soil-Structure Limit States Considering Method Bias and Design Parameter Variability

A general closed-form solution for the reliability index (or probability of failure) of a simple linear limit-state design function with one load term and one resistance term is used to compute the resistance factor expressed in a load and resistance factor design (LRFD) format. The solution considers method bias, bias dependencies, and uncertainties in choice of nominal values of load and resistance determined as part of the project-specific design process. Uncertainty in the choice of nominal values for design is linked quantitatively to the concept of project level of understanding that has been recently adopted in Canadian design practice. All random variables are assumed to be lognormally distributed. Parametric analyses are carried out to show that ignoring possible correlations between random variables can lead to conservative (safe) values of resistance factor and in other cases to nonconservative (unsafe) values. Example LRFD calibrations are carried out using different load and resistance models for the pullout internal stability limit state of steel-reinforced soil walls together with matching bias data reported in the literature. The results demonstrate the practical influence of model type, method bias statistics including dependencies, and operational factor of safety on computed resistance factors.

A general closed-form solution for the static pull-in voltages of electrostatically actuated MEMS/NEMS

In this article, an analytical method for calculating pull-in voltage is proposed. This method can accurately predict pull-in voltage of clamped-free, clamped-clamped and curved micro- and nano-beams. In this study, mid-plane stretching, axial stress, initial deformation and the effect of size are taken into account. To achieve this goal, governing equation of beam based on modified couple stress theory was first derived and then transformed to a single degree of freedom (D.O.F) model by Galerkin method. In this model, electrostatic force appears in integral form which is approximated to non-integral form employing Genetic Algorithm. This single degree of freedom model provides means for obtaining critical deflection of beam and is used to find the general closed-form expression for pull-in voltage. The validation of the method was carried out by comparing the results with the existing literature.

Coding Algorithm for Grayscale Images – Design of Piecewise Uniform Quantizer with Golomb–Rice Code and Novel Analytical Model for Performance Analysis

Scalar quantizer selection for processing a signal with a unit variance is a difficult problem, while both selection and quantizer design for the range of variances is even tougher and to the authors’ best knowledge, it is not theoretically solved. Furthermore, performance estimation of various image processing algorithms is unjustifiably neglected and there are only a few analytical models that follow experimental analysis. In this paper, we analyse application of piecewise uniform quantizer with Golomb-Rice coding in modified block truncation coding algorithm for grayscale image compression, propose design improvements and provide a novel analytical model for performance analysis. Besides the nature of input signal, required compression rate and processing delay of the observed system have a strong influence on quantizer design. Consequently, the impact of quantizer range choice is analysed using a discrete designing variance and it was exploited to improve overall quantizer performance, whereas variable-length coding is applied in order to reduce quantizer’s fixed bit-rate. The analytical model for performance analysis is proposed by introducing Inverse Gaussian distribution and it is obtained by discussing a number of images, providing general closed-form solutions for peak-signal-to-noise ratio and the total average bit-rate estimation. The proposed quantizer design ensures better performance in comparison to the other similar methods for grayscale image compression, including linear prediction of pixel intensity and edge-based adaptation, whereas analytical model for performance analysis provides matching with the experimental results within the range of 1 dB for PSQNR and 0.2 bpp for the total average bit-rate.

Analytical solution of generalized coupled thermoelasticity problem in a rotating disk subjected to thermal and mechanical shock loads

ABSTRACTIn this article, a fully analytical solution of the generalized coupled thermoelasticity problem in a rotating disk subjected to thermal and mechanical shock loads, based on Lord–Shulman model, is presented. The general forms of axisymmetric thermal and mechanical boundary conditions as arbitrary time-dependent heat transfer and traction, respectively, are considered at the inner and outer radii of the disk. The governing equations are solved analytically using the principle of superposition and the Fourier–Bessel transform. The general closed form solutions are presented for temperature and displacement fields. To validate the solutions, the results of this study are compared with the numerical results available in the literature, which show good agreement. For the temperature, displacement and stresses, radial distributions, and time histories are plotted and discussed. The propagation of thermoelastic waves and their reflection from the boundary of the disk are clearly shown. Moreover, effects of relaxation time and angular velocity on temperature, displacement, and stress fields are investigated.

The adjoint neutron transport equation and the statistical approach for its solution

The adjoint equation was introduced in the early days of neutron transport and its solution, the neutron importance, has been used for several applications in neutronics. The work presents at first a critical review of the adjoint neutron transport equation. Afterwards, the adjont model is constructed for a reference physical situation, for which an analytical approach is viable, i.e. an infinite homogeneous scattering medium. This problem leads to an equation that is the adjoint of the slowing-down equation, which is well known in nuclear reactor physics. A general closed-form analytical solution to such adjoint equation is obtained by a procedure that can be used also to derive the classical Placzek functions. This solution constitutes a benchmark for any statistical or numerical approach to the adjoint equation. A sampling technique to evaluate the adjoint flux for the transport equation is then proposed and physically interpreted as a transport model for pseudo-particles. This can be done by introducing appropriate kernels describing the transfer of the pseudo-particles in the phase space. This technique allows estimating the importance function by a standard Monte Carlo approach. The sampling scheme is validated by comparison with the analytical results previously obtained.

Closed-form solution for circular microstructure-dependent Mindlin plates

In this paper, we derive a general closed-form solution for the static, axisymmetric bending of solid and annular microstructure-dependent Mindlin plates based on the modified couple-stress theory. The solution is developed by employing a suitable change in displacement variables. The couple-stress solution contains modified Bessel functions which do not appear in the classical case. The stress distributions of the annular microstructure-dependent plate are obtained in terms of load resultants by using the general closed-form solution. Analytical and numerical examples considering solid and annular circular plates are presented. The examples include simply supported and clamped solid and annular plates subjected to a uniformly distributed load and annular plates attached to a point-loaded rigid shaft.