arXiv subjects

Blaine A. Chronik

Publications and source records attributed to Blaine A. Chronik.

4 recordsLinked to original sources

arXiv2020

Harmonic Standing-Wave Excitations of Simply-Supported Thick-Walled Hollow Elastic Circular Cylinders: Exact 3D Linear Elastodynamic Response

The forced-vibration response of a simply-supported isotropic thick-walled hollow elastic circular cylinder subjected to two-dimensional harmonic standing-wave excitations on its curved surfaces is studied within the framework of linear elastodynamics. Exact semi-analytical solutions for the steady-state displacement field of the cylinder are constructed using recently-published parametric solutions to the Navier-Lamé equation. Formal application of the standing-wave boundary conditions generates three parameter-dependent $6\times6$ linear systems, each of which can be numerically solved in order to determine the parametric response of the cylinder's displacement field under various conditions. The method of solution is direct and demonstrates a general approach that can be applied to solve many other elastodynamic forced-response problems involving isotropic elastic cylinders. As an application, and considering several examples, the obtained solution is used to compute the steady-state frequency response in a few specific low-order excitation cases. In each case, the solution generates a series of resonances that are in exact correspondence with a unique subset of the natural frequencies of the simply-supported cylinder. The considered problem is of general theoretical interest in structural mechanics and acoustics and more practically serves as a benchmark forced-vibration problem involving a thick-walled hollow elastic cylinder.

physics.class-ph↗

arXiv2019

Constructing Separable Non-$2π$-Periodic Solutions to the Navier-Lamé Equation in Cylindrical Coordinates Using the Buchwald Representation: Theory and Applications

In a previous paper [Adv. Appl. Math. Mech. 10 (2018), pp. 1025-1056], we used the Buchwald representation to construct several families of separable cylindrical solutions to the Navier-Lamé equation; these solutions had the property of being $2π$-periodic in the circumferential coordinate. In this paper, we extend the analysis and obtain the complementary set of separable solutions whose circumferential parts are elementary $2π$-aperiodic functions. Collectively, we construct eighteen distinct families of separable solutions; in each case, the circumferential part of the solution is one of three elementary $2π$-aperiodic functions. These solutions are useful for solving a wide variety of dynamical problems that involve cylindrical geometries and for which $2π$-periodicity in the angular coordinate is incompatible with the given boundary conditions. As illustrative examples, we show how the obtained solutions can be used to solve certain forced-vibration problems involving open cylindrical shells and open solid cylinders where (by virtue of the boundary conditions) $2π$-periodicity in the angular coordinate is inappropriate. As an addendum to our prior work, we also include an illustrative example of a certain type of asymmetric problem that can be solved using the particular $2π$-periodic subsolutions that ensue when there is no explicit dependence on the circumferential coordinate.

math-ph↗

arXiv2019

Harmonic Standing-Wave Excitations of Simply-Supported Isotropic Solid Elastic Circular Cylinders: Exact 3D Linear Elastodynamic Response

The vibration of a solid elastic cylinder is one of the classical applied problems of elastodynamics. Many fundamental forced-vibration problems involving solid elastic cylinders have not yet been studied or solved using the full three-dimensional (3D) theory of linear elasticity. One such problem is the steady-state forced-vibration response of a simply-supported isotropic solid elastic circular cylinder subjected to two-dimensional harmonic standing-wave excitations on its curved surface. In this paper, we exploit certain previously-obtained particular solutions to the Navier-Lamé equation of motion and exact matrix algebra to construct an exact closed-form 3D elastodynamic solution to the problem. The method of solution is direct and demonstrates a general approach that can be applied to solve other similar forced-vibration problems involving elastic cylinders. Two complete analytical solutions are in fact constructed corresponding to two different but closely-related families of harmonic standing-wave excitations. The second of these analytical solutions is evaluated numerically in order to study the steady-state frequency response in some example excitation cases. In each case, the solution generates a series of resonances that are in correspondence with a subset of the natural frequencies of the simply-supported cylinder. The considered problem is of general interest both as an exactly-solvable 3D elastodynamics problem and as a benchmark forced-vibration problem involving a solid elastic cylinder.

math-ph↗

arXiv2018

Solving the Navier-Lame Equation in Cylindrical Coordinates Using the Buchwald Representation: Some Parametric Solutions with Applications

Using a separable Buchwald representation in cylindrical coordinates, we show how under certain conditions the coupled equations of motion governing the Buchwald potentials can be decoupled and then solved using well-known techniques from the theory of PDEs. Under these conditions, we then construct three parametrized families of particular solutions to the Navier-Lame equation in cylindrical coordinates. In this paper, we specifically construct solutions having 2pi-periodic angular parts. These particular solutions can be directly applied to a fundamental set of linear elastic boundary value problems in cylindrical coordinates and are especially suited to problems involving one or more physical parameters. As an illustrative example, we consider the problem of determining the response of a solid elastic cylinder subjected to a time-harmonic surface pressure that varies sinusoidally along its axis, and we demonstrate how the obtained parametric solutions can be used to efficiently construct an exact solution to this problem. We also briefly consider applications to some related forced-relaxation type problems.

math-ph↗