One-Dimensional Impact Waves in Inhomogeneous Elastic Media

1969 ◽  
Vol 36 (4) ◽  
pp. 803-808 ◽  
Author(s):  
Edward L. Reiss
Keyword(s):  
Asymptotic Methods ◽  
Variable Coefficients ◽  
Elastic Media ◽  
One Dimensional ◽  
Klein Gordon ◽  
Inhomogeneous Elastic Media ◽  
Impact Problems ◽  
Impact Data ◽  
The Impact ◽  
Impact Problem

Several one-dimensional impact problems for inhomogeneous elastic media are reduced by suitable transformations of variables to three different canonical forms. One of them is the impact problem for the wave equation with variable sound speed. Another, which we call Problem I, is the impact problem for the Klein-Gordon equation with variable coefficients. Several methods of approximating the solution of Problem I are discussed. The wave front approximation is obtained by assuming that sufficiently near the discontinuity propagating from the impacted boundary the solution has a Taylor series in time. The coefficients in the series are the time derivatives of the solution evaluated on the discontinuity. They are obtained from an analysis of the propagation of the discontinuity using the theory of weak solutions. A formal asymptotic expansion of the solution is obtained for oscillatory impact data. Reflections from boundaries are also considered. Perturbation methods for media with slowly varying inhomogeneities and asymptotic methods for media with rapidly varying inhomogeneities are briefly discussed.

scholarly journals Impact of a rigid body against a half-infinite elastic bar

1988 ◽  
Vol 10 (1) ◽  
pp. 28-33
Author(s):  
Nguyen Dang To
Keyword(s):  
Rigid Body ◽  
Longitudinal Vibration ◽  
Vibration Theory ◽  
Elastic Bar ◽  
Impact Problems ◽  
The Impact ◽  
Precise Theory ◽  
Impact Problem

The solution of the impact - problems of a rigid body against an elastic bar arc before based on the classical longitudinal vibration theory of elastic bar. In this paper, we are using a more precise theory to deal with the impact problem of a rigid body against a half -infinite elastic bar with an elastic gasket. This problem is solved completely.

scholarly journals Wave concept of motion in mathematical models of the dynamics of two-dimensional media studying

2019 ◽  
Vol 5 (3-4) ◽  
pp. 8-15
Author(s):  
Andrij Andrukhiv ◽  
◽  
Bohdan Sokil ◽  
Mariia Sokil ◽  
◽  
...  
Keyword(s):  
Mathematical Models ◽  
Asymptotic Methods ◽  
Wave Theory ◽  
Single Frequency ◽  
Two Dimensional ◽  
Klein Gordon ◽  
Basic Parameters ◽  
The Impact ◽  
The Mathematical Model ◽  
The Waves

The methodology of the studying of dynamic processes in two-dimensional systems by mathematical models containing nonlinear equation of Klein-Gordon was developed. The methodology contains such underlying: the concept of the motion wave theory; the single - frequency fluctuations principle in nonlinear systems; the asymptotic methods of nonlinear mechanics. The aggregate content allowed describing the dynamic process for the undisturbed (linear) analogue of the mathematical model of movement. The value determining the impact of nonlinear forces on the basic parameters of the waves for the disturbed analogue is defined.

scholarly journals Impact of a rigid body against a finite elastic bar set on a rigid foundation

1988 ◽  
Vol 10 (3) ◽  
pp. 13-19
Author(s):  
Nguyen Dang To
Keyword(s):  
Rigid Body ◽  
Longitudinal Vibration ◽  
Rigid Foundation ◽  
Vibration Theory ◽  
Elastic Bar ◽  
Impact Problems ◽  
The Impact ◽  
Precise Theory ◽  
Impact Problem

The solutions of the impact problems of a rigid body against an elastic bar formerly are based on the classical longitudinal vibration theory of an elastic bar. In this paper, we based on the more precise theory to deal with the impact problem of a rigid body against a finite elastic bar set on a foundation.

Hydrodynamic Phenomena During High-Speed Collision Between Liquid Droplet and Rigid Plane

1973 ◽  
Vol 95 (2) ◽  
pp. 276-292 ◽  
Author(s):  
Yen C. Huang ◽  
F. G. Hammitt ◽  
W-J Yang
Keyword(s):  
High Speed ◽  
Liquid Droplet ◽  
Lateral Flow ◽  
Maximum Pressure ◽  
Two Dimensional ◽  
Droplet Shape ◽  
One Dimensional ◽  
Liquid Impact ◽  
The Impact ◽  
Impact Problem

The dynamics of high-speed impact between a compressible water droplet and a rigid solid surface is investigated analytically. The purpose of the study is to examine the mechanism leading to the erosion of a material due to liquid impingement. A Compressible-Cell-and-Marker (ComCAM) numerical method is developed to solve the differential equations governing the unsteady, two-dimensional liquid-solid impact phenomena. The method is designed to solve this unsteady portion up until the flow reasonably approaches the steady-state solution. The validity of the method is confirmed by comparing its numerical results with the idealized exact solution for the classical one-dimensional liquid impact problem. The accuracy of the numerical reresults is found to be very good in that only slight numerical oscillations occur. Viscosity and surface tension are neglected as seems resaonable with the relatively large drops and high velocities considered. Pressure and velocity distributions are solved as a function of time. The deformation of a drop is also recorded for three different shapes: cylindrical, spherical, and a combination of the two, which may more closely model the actual droplet shapes to be encountered in such impacts. Typical liquid impact Mach numbers of 0.2 and 0.5 (sonic velocity referred to water) were studied. Thus impact velocities of about 980 and 2450 fps are considered. Compression predominates during the early stages of the impact, while rarefaction governs later, during which time the radial lateral flow velocity exceeds the initial impact velocity. The reflection of compression waves and the lateral flow leads to the possibility of cavitation within the drop, due to the consequent generation of negative pressures, exists. The maximum pressure calculated in this two-dimensional liquid impact problem is found to be less than the one-dimensional maximum pressure for all three different droplets in various degrees. It is found that droplet shape impact angle and liquid impact Mach number are the only important parameters of the problem for a flat fully-rigid target surface. As more time elapses, i.e., up to 2–3 μsec for a 2.0 mm-dia drop, the maximum pressure shifts from the center of the contact area radially outward, while the pressure at the center attenuates rapidly toward conventional stagnation pressure.

scholarly journals Bound states of a Klein–Gordon particle in the presence of a smooth potential well

2020 ◽  
Vol 35 (23) ◽  
pp. 2050140
Author(s):  
Eduardo López ◽  
Clara Rojas
Keyword(s):  
Bound States ◽  
Gordon Equation ◽  
Bound State ◽  
Potential Well ◽  
One Dimensional ◽  
Smooth Potential ◽  
Klein Gordon ◽  
The One ◽  
Potential Parameters ◽  
Klein Gordon Equation

We solve the one-dimensional time-independent Klein–Gordon equation in the presence of a smooth potential well. The bound state solutions are given in terms of the Whittaker [Formula: see text] function, and the antiparticle bound state is discussed in terms of potential parameters.

Altitude Effects on the Performance of Small Radial Turbomachinery: Part I — Compressor Impellers

2016 ◽  
Author(s):  
Dries Verstraete ◽  
Kjersti Lunnan
Keyword(s):  
Reynolds Number ◽  
Performance Analysis ◽  
Rotational Speed ◽  
Pressure Ratio ◽  
Unmanned Aircraft ◽  
Good Match ◽  
One Dimensional ◽  
Design Changes ◽  
Current Article ◽  
The Impact

Small unmanned aircraft are currently limited to flight ceilings below 20,000 ft due to the lack of an appropriate propulsion system. One of the most critical technological hurdles for an increased flight ceiling of small platforms is the impact of reduced Reynolds number conditions at altitude on the performance of small radial turbomachinery. The current article investigates the influence of Reynolds number on the efficiency and pressure ratio of two small centrifugal compressor impellers using a one-dimensional meanline performance analysis code. The results show that the efficiency and pressure ratio of the 60 mm baseline compressor at the design rotational speed drops with 6–9% from sea-level to 70,000 ft. The impact on the smaller 20 mm compressor is slightly more pronounced and amounts to 6–10%. Off-design changes at low rotational speeds are significantly higher and can amount to up to 15%. Whereas existing correlations show a good match for the efficiency drop at the design rotational speed, they fail to predict efficiency changes with rotational speed. A modified version is therefore proposed.

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