Numerical Simulation of Mechanical Characteristics of Roadway Surrounding Rock under Dynamic and Static Loading

Mechanical characteristics of roadway surrounding rock under different stress wave disturbances are the key to design roadway supporting scheme. In this study, the 2802 transportation roadway in Zhangcun Coal Mine is selected as the engineering background. The distribution of stress, displacement, and plastic zone in surrounding rock under the impact of different stress waves is studied. Results show that the stress and displacement of the roof, floor, and coal walls present fluctuating change with time during the stress wave loading process. With the increase of disturbing intensity of stress wave, the resistance ability for stress wave disturbance of the roof is lower than that of the floor, while the resistance ability of two sides is the same. The volume of plastic zone in roadway surrounding rock was calculated by the self-compiled FISH code. The relationship between the plastic zone volume and the stress wave disturbing intensity in different states is explored. The cubic polynomial relationship between the volume and the disturbing intensity in the state of shear_past and tension_past is obtained. Under the simulated condition, the disturbing intensity of stress wave has the greatest influence on the increase of shear_past volume when it equals 11 MPa. While the disturbing intensity of stress wave has the greatest influence on the increase of tension_past volume, it equals 7 MPa. Meanwhile, the relation between stress wave disturbing intensity and surrounding rock stress and displacement is obtained respectively. The achievements provide a theoretical base for roadway surrounding rock support under dynamic and static loading.

The quantum dynamics of a mesoscopic driven RLC circuit consisting of a linear inductor, a linear resistor and a nonlinear capacitor

The nonlinear capacitor that obeys of a cubic polynomial voltage–charge relation (usually a power series in charge) is introduced. The quantum theory for a mesoscopic electric circuit with charge discreteness is investigated, and the Hamiltonian of a quantum mesoscopic electrical circuit comprised by a linear inductor, a linear resistor and a nonlinear capacitor under the influence of a time-dependent external source is expressed. Using the numerical solution approaches, a good analytic approximate solution for the quantum cubic Duffing equation is found. Based on this, the persistent current is obtained antically. The energy spectrum of such nonlinear electrical circuit has been found. The dependency of the persistent current and spectral property equations to linear and nonlinear parameters is discussed by the numerical simulations method, and the quantum dynamical behavior of these parameters is studied.

Fifteen Limit Cycles Bifurcating from a Perturbed Cubic Center

In this work, we study the bifurcation of limit cycles from the period annulus surrounding the origin of a class of cubic polynomial differential systems; when they are perturbed inside the class of all polynomial differential systems of degree six, we obtain at most fifteenth limit cycles by using the averaging theory of first order.

SPLINES SOLUTIONS OF HIGHER-ORDER BVPs THAT ARISE IN CONSISTENT MAGNETIZED FORCE FIELD

In this paper, cubic polynomial and nonpolynomial splines are developed to solve solutions of 10th- and 12th-order nonlinear boundary value problems (BVPs). Such types of BVPs occur when a consistent magnetized force field is applied crosswise the fluid in the substance of gravitational force. We will amend our problem into such a form that converts the system of [Formula: see text]th- [Formula: see text] [Formula: see text]th-order BVPs into a new system of [Formula: see text]nd-order BVPs. The appropriate outcomes by using CP Spline and CNP Spline are compared with the exact root. To show the efficiency of our results, absolute errors calculated by using CP Spline and CNP Spline have been compared with other methods like differential transform method, Adomian decomposition method, variational iteration method, cubic B-spline, homotopy perturbation method, [Formula: see text]th- and [Formula: see text]th-order B-spline and our results are very encouraging. Graphs and tables are also presented in the numerical section of this paper.

A New Beam Finite Element for Static Bending Analysis of Slender Transversely Cracked Beams on Two-Parametric Soils

This paper derives an original finite element for the static bending analysis of a transversely cracked uniform beam resting on a two-parametric elastic foundation. In the simplified computational model based on the Euler–Bernoulli theory of small displacements, the crack is represented by a linear rotational spring connecting two elastic members. The derivations of approximate transverse displacement functions, stiffness matrix coefficients, and the load vector for a linearly distributed load along the entire beam element are based on novel cubic polynomial interpolation functions, including the second soil parameter. Moreover, all derived expressions are obtained in closed forms, which allow easy implementation in existing finite element software. Two numerical examples are presented in order to substantiate the discussed approach. They cover both possible analytical solution forms that may occur (depending on the problem parameters) from the same governing differential equation of the considered problem. Therefore, several response parameters are studied for each example (with additional emphasis on their convergence) and compared with the corresponding analytical solution, thus proving the quality of the obtained finite element.

Cubic Polynomial for the Series of Consecutive Cubes under Alternating Signs

This paper aims to develop an elegant formula for the series of consecutive cubes of natural numbers under alternating signs. In addition, this paper investigates the formula under odd and even number of terms and discuss some important findings.Keywords: Consecutive cubes; alternating signs; odd and even terms.2010 Mathematics Subject Classification: 11B13, 11B50, 97I30AbstrakPaper ini bertujuan membangun formula yang elegan untuk deret berganti tanda bilangan-bilangan kubik berurutan. Paper ini juga menyelidiki formula untuk banyak suku ganjil dan untuk banyak suku genap, dan mendiskusikan beberapa temuan penting.Kata kunci:. Bilangan kubik berurutan, berganti tanda, suku ganjil dan suku genap

Modelling the water droplet motion on a leaf surface

The main object of the research given in this paper is modelling the water droplet movements on the leaf surface which is an important factor in realising how pesticide, nutrient or water engrossed through the surface. Therefore, a physical model based on mathematics proposed here for producing a realistic trajectory of droplet traversing a leaf surface. A vital feature of our methodology is to build the leaf surface using a recently developed method, by the author, so-called a hybrid CloughTocher cubic polynomial interpolation (CT-CPI) method. The leaf surface consisted of a mesh of triangles over which the hybrid CT-CPI method is build from 3D real life data gathered using a laser scanner. The droplet motion in our model affected by friction, resistance and gravity forces. The model verified using Matlab programming; the outcomes are promising and seem to capture reality well.

Shape functions for high-order shear deformation beam element

In this article, shape functions for higher-order shear deformation beam theory are derived. For the two nodded beam element, transverse deflection is assumed as cubic polynomial. By using equations of equilibrium of high-order theory that are already derived by J. N. Reddy in 1997, equation for slope of high- order theory is found. Finally with the boundary conditions of beam element and assumed kinematics of high-order theory, shape functions are derived.