Three-dimensional thermo-electro-elastic field in one-dimensional hexagonal piezoelectric quasi-crystal weakened by an elliptical crack

In this paper, an analytical solution is developed for the problem of an infinite 1D hexagonal piezoelectric quasi-crystal medium weakened by an elliptical crack and subjected to mixed loads on the crack surfaces. The mixed loads comprise the phonon pressure, phason pressure, electric displacement, and temperature increment, and the crack surfaces can be electrically permeable or impermeable. Based on a general solution, combined with the generalized potential theory, the steady-state 3D thermo-electro-elastic field variables in the quasi-crystal are obtained in terms of elliptic integral functions and elementary functions. Several important physical quantities on the cracked plane, such as the generalized crack surface displacements, normal stresses, and stress intensity factors, are derived in closed forms. An illustrative numerical calculation verifies the presented analytical solution and shows the distribution of the 3D thermo-electro-elastic field. It is indicated that the influence of the phason field on the result is pronounced, especially for the electric field variables, and the electric permeability of crack surfaces has a significant effect on the electric displacement intensity factor at the crack tip.

Post-Buckling Spring Vibration Isolator Using Silicone Gel Column: A Theoretical and Experimental Study

Based on the design of a post-buckling silicone gel column (SGC), a novel type of low-frequency vibration isolator is presented, and the vibration isolation performance of this isolator is studied by combining theoretical analysis and experimental verification. The stiffness characteristics of the post-buckling SGC are derived, and its recovery force curves with different parameters are analyzed using two kinds of elliptic integral functions. Displacement transmissibility is formulated using harmonic balance method (HBM), and the influences of the excitation amplitude, damping ratio, SGC section diameter, and Young’s modulus are discussed in terms of the transmissibility. The performance of the SGC system is verified through a series of experimental studies based on the developed experimental prototype. The result shows that the proposed post-buckling spring vibration isolator has a good vibration isolation effect, especially in the low-frequency domain, which may provide a feasible novel design idea for a low-frequency vibration isolator.

Photon boomerang in a nearly extreme Kerr metric

The Kerr rotating black hole metric has unstable photon orbits that orbit around the hole at fixed values of the Boyer-Lindquist coordinate r that depend on the axial angular momentum of the orbit, as well as on the parameters of the hole. For zero orbital axial angular momentum, these orbits cross the rotational axes at a fixed value of r that depends on the mass M and angular momentum J of the black hole. Nonzero angular momentum of the hole causes the photon orbit to rotate so that its direction when crossing the north polar axis changes from one crossing to the next by an angle I shall call ∆φ, which depends on the black hole dimensionless rotation parameter a/M = cJ/(GM2) by an equation involving a complete elliptic integral of the first kind. When the black hole has a/M ≈ 0.994 341 179 923 26, which is nearly maximally rotating, a photon sent out in a constant-r direction from the north polar axis at r ≈ 2.423 776 210 035 73 GM/c2returns to the north polar axis in precisely the opposite direction (in a frame nonrotating with respect to the distant stars), a photon boomerang.

Symmetry Solutions and Conservation Laws for the 3D Generalized Potential Yu-Toda-Sasa-Fukuyama Equation of Mathematical Physics

In this paper we study the fourth-order three-dimensional generalized potential Yu-Toda-Sasa-Fukuyama (gpYTSF) equation by first computing its Lie point symmetries and then performing symmetry reductions. The resulting ordinary differential equations are then solved using direct integration, and exact solutions of gpYTSF equation are obtained. The obtained group invariant solutions include the solution in terms of incomplete elliptic integral. Furthermore, conservation laws for the gpYTSF equation are derived using both the multiplier and Noether’s methods. The multiplier method provides eight conservation laws, while the Noether’s theorem supplies seven conservation laws. These conservation laws include the conservation of energy and mass.

Uniform approximations of the first symmetric elliptic integral in terms of elementary functions

We consider the standard symmetric elliptic integral $$R_F(x,y,z)$$ R F ( x , y , z ) for complex x, y, z. We derive convergent expansions of $$R_F(x,y,z)$$ R F ( x , y , z ) in terms of elementary functions that hold uniformly for one of the three variables x, y or z in closed subsets (possibly unbounded) of $$\mathbb {C}{\setminus }(-\infty ,0]$$ C \ ( - ∞ , 0 ] . The expansions are accompanied by error bounds. The accuracy of the expansions and their uniform features are illustrated by means of some numerical examples.

Frequency–Amplitude Relationship of a Nonlinear Symmetric Panel Absorber Mounted on a Flexible Wall

This study addresses the frequency–amplitude relationship of a nonlinear symmetric panel absorber mounted on a flexible wall. In many structural–acoustic works, only one flexible panel is considered in their models with symmetric configuration. There are very limited research investigations that focus on two flexible panels coupled with a cavity, particularly for nonlinear structural–acoustic problems. In practice, panel absorbers with symmetric configurations are common and usually mounted on a flexible wall. Thus, it should not be assumed that the wall is rigid. This study is the first work employing the weighted residual elliptic integral method for solving this problem, which involves the nonlinear multi-mode governing equations of two flexible panels coupled with a cavity. The reason for adopting the proposed solution method is that fewer nonlinear algebraic equations are generated. The results obtained from the proposed method and finite element method agree reasonably well with each other. The effects of some parameters such as vibration amplitude, cavity depth and thickness ratio, etc. are also investigated.

Exact Solutions and Conserved Vectors of the Two-Dimensional Generalized Shallow Water Wave Equation

In this article, we investigate a two-dimensional generalized shallow water wave equation. Lie symmetries of the equation are computed first and then used to perform symmetry reductions. By utilizing the three translation symmetries of the equation, a fourth-order ordinary differential equation is obtained and solved in terms of an incomplete elliptic integral. Moreover, with the aid of Kudryashov’s approach, more closed-form solutions are constructed. In addition, energy and linear momentum conservation laws for the underlying equation are computed by engaging the multiplier approach as well as Noether’s theorem.

Functional inequalities for Gaussian hypergeometric function and generalized elliptic integral of the first kind

In the article, we present several new functional inequalities for the Gaussian hypergeometric function and generalized elliptic integral of the first kind.

Minimum contact stress modification for profile curve design defects in the beam-spring-cone docking mechanism

A minimum contact stress modification method for profile curve design defects in a beam-spring-cone docking mechanism (BSCDM) based on genetic algorithm is presented in this paper, the profile curve and contact position of BSCDM are optimized. Under low-speed conditions, an improved elastic contact model of semi-space elastic bodies is established to modify and optimize the elliptic profile envelope curve based on Hertz contact theory and two kinds of complete elliptic integral, the improved contact model is used to solve elastic contact problems with the geometric characteristics of the ellipse surface, the optimal profile curve of the docking joint and the optimal docking contact point position are obtained. The results of numerical simulation and the experiment demonstrate the feasibility and validity of above models and methods.