Full-scale Test on Emergency Repair Pavement under Airplane Loading

In order to reveal structural response law of emergency repair pavement under the airplane loading and verify the backfill material and structural applicability, two craters (Crater 1 composed of 2.4 m thick flying objects (FO) + 0.4 m thick graded crushed rocks (GCR) + 0.2 m thick roller compacted concrete + fibre reinforced plastic (FRP) course, and Crater 2 composed of 2.4 m thick FO + 0.6 m thick GCR + FRP course) were backfilled. Static and dynamic loads were applied using two airplanes. Results show that, laying FRP pavement layers reduced the maximum deflection of Crater 2 by 21%. Crater 1 and concrete pavement were both slightly rigid structures with a strong load transfer ability. The dynamic deflection basin curves of Crater 2 could be fit using a Gaussian function; while the curves of Crater 1 and concrete pavement could be fit using a quartic polynomial. Under static loading, the earth pressures of Crater 2 at −0.6 m, −0.4 m, and −0.2 m sites were 4.3, 9, and 9.6 times of those of Crater 1, respectively. At the −0.2 m site, the earth pressure of Crater 1 was 0.11 MPa, while that of Crater 2 reached 1.06 MPa. The research results can guide the rapid quality inspection and optimization design of emergency repair pavement structure and material.

Feasible Trajectories Generation for Autonomous Driving Vehicles

This study presents smooth and fast feasible trajectory generation for autonomous driving vehicles subject to the vehicle physical constraints on the vehicle power, speed, acceleration as well as the hard limitations of the vehicle steering angle and the steering angular speed. This is due to the fact the vehicle speed and the vehicle steering angle are always in a strict relationship for safety purposes, depending on the real vehicle driving constraints, the environmental conditions, and the surrounding obstacles. Three different methods of the position quintic polynomial, speed quartic polynomial, and symmetric polynomial function for generating the vehicle trajectories are presented and illustrated with simulations. The optimal trajectory is selected according to three criteria: Smoother curve, smaller tracking error, and shorter distance. The outcomes of this paper can be used for generating online trajectories for autonomous driving vehicles and auto-parking systems.

Mellin Transform of Logarithm and Quotient Function with Reducible Quartic Polynomial in Terms of the Lerch Function

A class of definite integrals involving a quotient function with a reducible polynomial, logarithm and nested logarithm functions are derived with a possible connection to contact problems for a wedge. The derivations are expressed in terms of the Lerch function. Special cases are also derived in terms fundamental constants. The majority of the results in this work are new.

Existence of Limit Cycles for a Class of Quartic Polynomial Differential System Depending on Parameters

We studied the existence of limit cycles for the quartic polynomial differential systems depending on parameters. To prove that, first, we used the formal series method based on Poincare’ ideas to determine the center-focus. Then, by the Hopf bifurcation theory, we obtained the sufficient condition for the existence of the limit cycles. Finally, we provided some numerical examples for illustration.

Finding Zeros of Quartic Polynomials by Using Radicals

We present a technique for finding roots of a quartic general polynomial equation of a single variable by using radicals. The solution of quartic polynomial equations requires knowledge of lower degree polynomial equations; therefore, we study solving polynomial equations of degree less than four as well. We present self-reciprocal polynomials as a specialization and additionally solve numerical example.

Finding Zeros of Quartic Polynomials by Using Radicals

We present a technique for finding roots of a quartic general polynomial equation of a single variable by using radicals. The solution of quartic polynomial equations requires knowledge of lower degree polynomial equations; therefore, we study solving polynomial equations of degree less than four as well. We present self-reciprocal polynomials as a specialization and additionally solve numerical example.

An Alternative Approach to Solving Cubic and Quartic Polynomial Equations

Aims: The aim of the research study was to develop a more direct and intuitive approach for the solution of polynomial equations of degree 3 and four. Study Design: The study employed equivalent polynomial substitution that is more intuitive and direct to formulate than the traditional formulations and one that is easily solvable. Place and Duration of Study: The study has been undertaken by the author at the university of Eswatini in the period from February to March 2021. Methodology: Two alternative procedures have been presented for the analytical solution of cubic and quartic equations and demonstrated with worked examples. The solution is derived through a direct procedure without involving intermediate variable substitution. Results: For cubic equations, the solution provides explicit expression of an equivalent cubic that is formed directly in terms of the original variable x. As such, the formula is intuitive and simple to derive or understand as well as apply. For the quartic equations, the same decomposition form is used as that of the cubic equation using two quadratic polynomials that have symmetric form thus making it easy to develop the solution as well as solve the equations Conclusion: The alternative formula is easy to formulate and solve and provides a more intuitive basis for understanding and solving polynomial equations.

Research on Cumulative Damage Characteristics of Rock Anchor Beam Concrete Supporting Structure by Blasting Vibration of Underground Powerhouse

Aiming at the dynamic disturbance problem of the upper rock anchor beam concrete support structure caused by the blasting excavation of the lower rock mass of the underground powerhouse of the hydropower station, the dynamic finite element software was used to numerically simulate the cumulative damage characteristics of the rock anchor beam concrete structure under multiple blasting, which aims to study the effect of blasting times and blast center distance on the cumulative damage characteristics of rock anchor beam concrete structures in two directions at different ages. The results show that 7 days after the completion of the concrete pouring of rock anchor beam, the damage and destruction effects are produced under the single blasting action. 28 days after the completion of the concrete pouring, there is basically no damage and destruction under the action of five times of blasting. 14 days after the completion of the concrete pouring, the growth process of the cumulative damage effect of the rock anchor beam concrete structure under the disturbance of the blasting excavation of the underground powerhouse shows nonlinear increasing characteristics with the increase of blasting times and nonlinear decreasing characteristics with the increase of blasting center distance. The cumulative damage of rock anchor beam along and perpendicular to the axis of powerhouse conforms to the relationship of cubic polynomials with the blasting times, which also conform to the relationship of quartic polynomial and quadratic polynomial, respectively, with the blast center distance.