scholarly journals Influence of Impulse Disturbances on Oscillations of Nonlinearly Elastic Bodies

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 819
Author(s):  
Andriy Andrukhiv ◽  
Mariia Sokil ◽  
Bohdan Sokil ◽  
Solomiia Fedushko ◽  
Yuriy Syerov ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Elastic Body ◽  
Elastic Properties ◽  
Degrees Of Freedom ◽  
Nonlinear Differential Equations ◽  
Asymptotic Methods ◽  
Physical And Mechanical Properties ◽  
The Body ◽  
Oscillatory Process ◽  
First Approximation

A method for studying the effect of impulse perturbation on the longitudinal oscillations of a homogeneous constant cross-section of the body and the elastic properties of a material which satisfies the essentially nonlinear law of elasticity has been developed. A mathematical model of the process is presented, which is an equation of hyperbolic type with a small parameter at the discrete right-hand side. The latter expresses the effect of impulse perturbation on the oscillatory process. As for the boundary conditions considered in the work, they are classic of the first, second and third genera. The methodology is based on: the principle of oscillation frequency in nonlinear systems with many degrees of freedom and distributed parameters; basic provisions of asymptotic methods of nonlinear mechanics; the idea of using special periodic Ateb-functions to construct solutions of some classes of nonlinear differential equations; properties of completeness and orthonormality of functions that describe the forms of oscillations of undisturbed motion. In general, the above allowed to obtain relations that describe for the first approximation the defining parameters of the oscillations of an elastic body. Their peculiarity is that even for undisturbed motion, the natural frequency of oscillations depends on the amplitude, and therefore, under the action of a periodic (over time) pulse force on the elastic body, both resonant and nonresonant processes are possible in the latter. It, in contrast to an elastic body with linear or quasilinear elastic properties of the body is determined not only by its basic physical and mechanical properties, but also by the amplitude of oscillations. As a special case, the oscillations of the body under the action of a constant periodic momentum perturbation are considered. It is shown that for the nonresonant case for the first approximation it does not affect the laws of change of amplitude and frequency of the process. As for the resonant is the amplitude of origin through the main resonance significantly depends not only on the speed but also on the points of action of the pulsed perturbation. Moreover, the closer the point of application of the pulsed force to the middle of the elastic body under boundary conditions of the first kind is greater (for boundary conditions of the second kind closer to the end).

Mechanics of a thin anisotropic elastic layer and a layer that is bonded to an anisotropic elastic body or bodies

2007 ◽  
Vol 463 (2085) ◽  
pp. 2223-2239 ◽  
Author(s):  
T.C.T Ting
Keyword(s):  
Boundary Conditions ◽  
Thin Layer ◽  
Elastic Body ◽  
Elastic Layer ◽  
Elastic Half Space ◽  
The Body ◽  
Thin Layers ◽  
Effective Boundary ◽  
Effective Boundary Conditions ◽  
Anisotropic Elastic

When a very thin elastic layer is bonded to an elastic body, it is desirable to have effective boundary conditions for the interface between the layer and the body that take into account the existence of the layer. In the literature, this has been done for special anisotropic elastic layers. We consider here the layer that is a general anisotropic elastic material. The mechanics of a thin layer is studied for elastostatics as well as steady state waves. It is shown that one-component surface waves cannot propagate in a semi-infinite thin layer. We then present Love waves in an anisotropic elastic half-space bonded to a thin anisotropic elastic layer. The dispersion equation so obtained is valid for long wavelength. Finally, effective boundary conditions are presented for two thin layers bonded to two surfaces of a plate and a thin layer bonded between two anisotropic elastic half-spaces.

Generalized Kirchhoff equations for a deformable body moving in a weakly non-uniform flow field

1994 ◽  
Vol 446 (1926) ◽  
pp. 169-193 ◽  
Keyword(s):  
Flow Field ◽  
Rigid Body ◽  
Degrees Of Freedom ◽  
Surface Deformation ◽  
Nonlinear Differential Equations ◽  
Deformable Body ◽  
Equations Of Motion ◽  
First Integral ◽  
Uniform Flow ◽  
The Body

The classical Kirchhoff’s method provides an efficient way of calculating the hydrodynamical loads (forces and moments) acting on a rigid body moving with six-degrees of freedom in an otherwise quiescent ideal fluid in terms of the body’s added-mass tensor. In this paper we provide a versatile extension of such a formulation to account for both the presence of an imposed ambient non-uniform flow field and the effect of surface deformation of a non-rigid body. The flow inhomogeneity is assumed to be weak when compared against the size of the body. The corresponding expressions for the force and moment are given in a moving body-fixed coordinate system and are obtained using the Lagally theorem. The newly derived system of nonlinear differential equations of motion is shown to possess a first integral. This can be interpreted as an energy-type conservation law and is a consequence of an anti-symmetry property of the coefficient matrix reported here for the first time. A few applications of the proposed formulation are presented including comparison with some existing limiting cases.

Representing utility and deploying the body

2020 ◽  
Vol 43 ◽  
Author(s):  
David Spurrett
Keyword(s):  
Functional Activity ◽  
Degrees Of Freedom ◽  
The Body ◽  
Target Article ◽  
Processing Demands ◽  
The Many

Abstract Comprehensive accounts of resource-rational attempts to maximise utility shouldn't ignore the demands of constructing utility representations. This can be onerous when, as in humans, there are many rewarding modalities. Another thing best not ignored is the processing demands of making functional activity out of the many degrees of freedom of a body. The target article is almost silent on both.

scholarly journals Nonlinear Differential Equations Modeling the Antarctic Circumpolar Current

2021 ◽  
Vol 23 (4) ◽  
Author(s):  
Jifeng Chu ◽  
Kateryna Marynets
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Topological Degree ◽  
Antarctic Circumpolar Current ◽  
Fixed Point Theorems ◽  
Nonlinear Differential Equations ◽  
Existence Results ◽  
Circumpolar Current ◽  
The Antarctic

AbstractThe aim of this paper is to study one class of nonlinear differential equations, which model the Antarctic circumpolar current. We prove the existence results for such equations related to the geophysical relevant boundary conditions. First, based on the weighted eigenvalues and the theory of topological degree, we study the semilinear case. Secondly, the existence results for the sublinear and superlinear cases are proved by fixed point theorems.

scholarly journals Air mediates the impact of a compliant hemisphere on a rigid smooth surface

Soft Matter ◽  
2021 ◽  
Author(s):  
Siqi Zheng ◽  
Sam Dillavou ◽  
John M. Kolinski
Keyword(s):  
Elastic Body ◽  
Elastic Properties ◽  
Impact Velocity ◽  
Solid Surface ◽  
High Speed ◽  
Smooth Surface ◽  
High Speed Imaging ◽  
The Impact ◽  
Rigid Smooth

When a soft elastic body impacts upon a smooth solid surface, the intervening air fails to drain, deforming the impactor. High-speed imaging with the VFT reveal rich dynamics and sensitivity to the impactor's elastic properties and the impact velocity.

scholarly journals Mechanics of tubular helical assemblies: ensemble response to axial compression and extension

2021 ◽  
Author(s):  
Jacopo Quaglierini ◽  
Alessandro Lucantonio ◽  
Antonio DeSimone
Keyword(s):  
Boundary Conditions ◽  
Elastic Properties ◽  
Central Region ◽  
Mechanical Response ◽  
Different Boundary Conditions ◽  
Kirchhoff Rods ◽  
Model Versus ◽  
The Individual ◽  
Opposite Chirality

Abstract Nature and technology often adopt structures that can be described as tubular helical assemblies. However, the role and mechanisms of these structures remain elusive. In this paper, we study the mechanical response under compression and extension of a tubular assembly composed of 8 helical Kirchhoff rods, arranged in pairs with opposite chirality and connected by pin joints, both analytically and numerically. We first focus on compression and find that, whereas a single helical rod would buckle, the rods of the assembly deform coherently as stable helical shapes wound around a common axis. Moreover, we investigate the response of the assembly under different boundary conditions, highlighting the emergence of a central region where rods remain circular helices. Secondly, we study the effects of different hypotheses on the elastic properties of rods, i.e., stress-free rods when straight versus when circular helices, Kirchhoff’s rod model versus Sadowsky’s ribbon model. Summing up, our findings highlight the key role of mutual interactions in generating a stable ensemble response that preserves the helical shape of the individual rods, as well as some interesting features, and they shed some light on the reasons why helical shapes in tubular assemblies are so common and persistent in nature and technology. Graphic Abstract We study the mechanical response under compression/extension of an assembly composed of 8 helical rods, pin-jointed and arranged in pairs with opposite chirality. In compression we find that, whereas a single rod buckles (a), the rods of the assembly deform as stable helical shapes (b). We investigate the effect of different boundary conditions and elastic properties on the mechanical response, and find that the deformed geometries exhibit a common central region where rods remain circular helices. Our findings highlight the key role of mutual interactions in the ensemble response and shed some light on the reasons why tubular helical assemblies are so common and persistent.

Boundaries Influence on the Flow Field Around an Oscillating Sphere

2013 ◽  
Author(s):  
Domenica Mirauda ◽  
Antonio Volpe Plantamura ◽  
Stefano Malavasi
Keyword(s):  
Free Surface ◽  
Flow Field ◽  
Boundary Conditions ◽  
Damping Ratio ◽  
Water Channel ◽  
Open Water ◽  
Free Surface Flows ◽  
The Body ◽  
Sphere Diameter ◽  
Oscillating Sphere

This work analyzes the effects of the interaction between an oscillating sphere and free surface flows through the reconstruction of the flow field around the body and the analysis of the displacements. The experiments were performed in an open water channel, where the sphere had three different boundary conditions in respect to the flow, defined as h* (the ratio between the distance of the sphere upper surface from the free surface and the sphere diameter). A quasi-symmetric condition at h* = 2, with the sphere equally distant from the free surface and the channel bottom, and two conditions of asymmetric bounded flow, one with the sphere located at a distance of 0.003m from the bottom at h* = 3.97 and the other with the sphere close to the free surface at h* = 0, were considered. The sphere was free to move in two directions, streamwise (x) and transverse to the flow (y), and was characterized by values of mass ratio, m* = 1.34 (ratio between the system mass and the displaced fluid mass), and damping ratio, ζ = 0.004. The comparison between the results of the analyzed boundary conditions has shown the strong influence of the free surface on the evolution of the vortex structures downstream the obstacle.

Numerical Solution of a Hydrofoil Moving Near an Interface

1996 ◽  
Vol 40 (04) ◽  
pp. 269-277
Author(s):  
G. X. Wu ◽  
T. Miloh ◽  
G. Zilman
Keyword(s):  
Boundary Conditions ◽  
Numerical Solution ◽  
Body Surface ◽  
Iteration Scheme ◽  
The Body ◽  
Two Fluids ◽  
Linearized Theory

The problem of a hydrofoil moving near an interface of two fluids of different densities is analyzed. An iteration scheme is proposed which imposes the boundary conditions on the body surface and on the interface alternately. The numerical solution is obtained by using the linearized theory and a Glauert-type expansion for the vortex distribution. Results are provided for various cases with different densities and different speeds.

Problems of Nonlinear Multiple-Mode Vibrations of Thin Elastic Shells of Revolution

2000 ◽  
Author(s):  
Veniamin D. Kubenko ◽  
Piotr S. Kovalchuk
Keyword(s):  
Degrees Of Freedom ◽  
Asymptotic Methods ◽  
Nonlinear Resonance ◽  
Shells Of Revolution ◽  
Elastic Shells ◽  
Resonance Amplitude ◽  
Multiple Degrees Of Freedom ◽  
Free And Forced Vibrations ◽  
Multiple Mode ◽  
Averaged Equations

Abstract A method is suggested for the calculation of nonlinear free and forced vibrations of thin elastic shells of revolution, which are modeled as dynamic systems of multiple degrees of freedom. Cases are investigated in which the shells are characterized by two or more closely-spaced eigenfrequencies. Based on an analysis of averaged equations, obtained by making use of asymptotic methods of nonlinear mechanics, a number of new first integrals is obtained, which state a regular energy exchange among various modes of cylindrical shells under conditions of nonlinear resonance. Amplitude-frequency characteristics of multiple-mode vibrations are obtained for shells subjected to radial oscillating pressure.

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