scholarly journals Unsteady dynamics of a sandwich plate under the influence of a cylindrical wave in an elastic medium

2021 ◽  
Vol 13 (S) ◽  
pp. 117-132
Author(s):  
Natalia A. LOKTEVA ◽  
Nguyen Duong PHUNG
Keyword(s):  
Sandwich Plate ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Problem Formulation ◽  
Cylindrical Wave ◽  
Physical Principle ◽  
Ground Movement ◽  
Contact Conditions ◽  
The Laplace Transform ◽  
Cauchy Relations

The interaction of a sandwich plate with a damped cylindrical wave in the ground has been investigated. A sandwich plate is considered as a model of a barrier in the ground, described by a system of equations by V. N. Paimushin, placed in the ground dividing it into two parts. The plane problem formulation is considered. The boundary conditions correspond to the hinge attachment of the barrier, and the initial conditions are zero. A cylindrical damped wave is considered as an external influence. To describe the ground movement, the equations of the elasticity theory, the Cauchy relations and the physical principle, or equivalent displacements in potentials and the Lame equations are used. The problem is solved in a related formulation, where the movement of the plate and its surrounding media is considered together. All components of the equations of motion of the plate and media are decomposed into trigonometric series and the Laplace transform is applied to them. As the conditions for the contact of the plate and the ground, the equality of normal displacements at the boundary of the medium and the plate is assumed. It is also assumed that the pressure amplitudes and normal stresses coincide. After determining the constants from the contact conditions, the displacement values and the values of normal and tangential stresses are found, after which their originals are found.

scholarly journals The Nonlinear Dynamic Behavior Analysis and Research for Sheet Steel Stamping Forming Problems Based on Finite Element Method

2015 ◽  
Vol 9 (1) ◽  
pp. 198-202
Author(s):  
Li Zhang ◽  
Tao Zhang ◽  
Kaiteng Wu
Keyword(s):  
Finite Element ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Sheet Steel ◽  
Element Model ◽  
Stress Strain ◽  
Contact Conditions ◽  
Deformation Stages ◽  
Element Algorithm ◽  
The Finite Element Model

A numerical framework for the simulation of sheet steel stamping forming is presented. The main problems, the equations of motion, the constitutive relation, the initial conditions, boundary conditions and contact conditions, are presented in detail. Based on this, the finite element model is established and solved for exploring the changes in laws of stress, strain and so on. The information on stress, strain and load displacement is obtained at different deformation stages. The numerical results show that the finite element algorithm can effectively simulate the deformation process of sheet steel which helps to explain that the numerical framework is feasible for sheet steel stamping forming problems.

scholarly journals Dynamics of Space Particles and Spacecrafts Passing by the Atmosphere of the Earth

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Vivian Martins Gomes ◽  
Antonio Fernando Bertachini de Almeida Prado ◽  
Justyna Golebiewska
Keyword(s):  
Initial Conditions ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
The Sun ◽  
Three Body Problem ◽  
The Earth ◽  
Before And After ◽  
Three Body ◽  
Body Energy ◽  
Spacecraft Position

The present research studies the motion of a particle or a spacecraft that comes from an orbit around the Sun, which can be elliptic or hyperbolic, and that makes a passage close enough to the Earth such that it crosses its atmosphere. The idea is to measure the Sun-particle two-body energy before and after this passage in order to verify its variation as a function of the periapsis distance, angle of approach, and velocity at the periapsis of the particle. The full system is formed by the Sun, the Earth, and the particle or the spacecraft. The Sun and the Earth are in circular orbits around their center of mass and the motion is planar for all the bodies involved. The equations of motion consider the restricted circular planar three-body problem with the addition of the atmospheric drag. The initial conditions of the particle or spacecraft (position and velocity) are given at the periapsis of its trajectory around the Earth.

scholarly journals Numerical solution for consistent initial conditions of constrained mechanical systems

2003 ◽  
Vol 25 (3) ◽  
pp. 170-185
Author(s):  
Dinh Van Phong
Keyword(s):  
Numerical Solution ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Flexible Approach ◽  
Constrained Mechanical Systems ◽  
Consistent Initial Values

The article deals with the problem of consistent initial values of the system of equations of motion which has the form of the system of differential-algebraic equations. Direct treating the equations of mechanical systems with particular properties enables to study the system of DAE in a more flexible approach. Algorithms and examples are shown in order to illustrate the considered technique.

Second Grade Fluid for Rotating MHD of an Unsteady Free Convection Flow in a Porous Medium

2015 ◽  
Vol 362 ◽  
pp. 100-107 ◽  
Author(s):  
Z. Ismail ◽  
I. Khan ◽  
A.Q. Mohamad ◽  
S. Shafie
Keyword(s):  
Porous Medium ◽  
Free Convection ◽  
Initial Conditions ◽  
Boussinesq Approximation ◽  
Second Grade ◽  
Convection Flow ◽  
Inclined Plate ◽  
Free Convection Flow ◽  
Transform Method ◽  
The Laplace Transform

Rotating effects and magnetohydrodynamic (MHD) free convection flow of second grade fluids in a porous medium is considered in this paper. It is assumed that the bounding infinite inclined plate has ramped wall temperature with the presence of heat and mass diffusion. Based on Boussinesq approximation, the analytical expressions for dimensionless velocity, temperature and concentration are obtained by using the Laplace transform method. All the derived solutions satisfying the involved differential equations with imposed boundary and initial conditions. The influence of various parameters on the velocity has been analyzed in graphs and discussed.

Influence of Bistable Plunge Stiffness On Nonlinear Airfoil Flutter

2021 ◽  
Author(s):  
Renan F. Corrêa ◽  
Flávio D. Marques
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Mechanical Vibration ◽  
Nonlinear Behavior ◽  
Limit Cycle Oscillation ◽  
Restoring Force ◽  
Aeroelastic System ◽  
Technical Literature ◽  
Practical Applications

Abstract Aeroelastic systems have nonlinearities that provide a wide variety of complex dynamic behaviors. Nonlinear effects can be avoided in practical applications, as in instability suppression or desired, for instance, in the energy harvesting design. In the technical literature, there are surveys on nonlinear aeroelastic systems and the different manners they manifest. More recently, the bistable spring effect has been studied as an acceptable nonlinear behavior applied to mechanical vibration problems. The application of the bistable spring effect to aeroelastic problems is still not explored thoroughly. This paper contributes to analyzing the nonlinear dynamics of a typical airfoil section mounted on bistable spring support at plunging motion. The equations of motion are based on the typical aeroelastic section model with three degrees-of-freedom. Moreover, a hardening nonlinearity in pitch is also considered. A preliminary analysis of the bistable spring geometry’s influence in its restoring force and the elastic potential energy is performed. The response of the system is investigated for a set of geometrical configurations. It is possible to identify post-flutter motion regions, the so-called intrawell, and interwell. Results reveal that the transition between intrawell to interwell regions occurs smoothly, depending on the initial conditions. The bistable effect on the aeroelastic system can be advantageous in energy extraction problems due to the jump in oscillation amplitudes. Furthermore, the hardening effect in pitching motion reduces the limit cycle oscillation amplitudes and also delays the occurrence of the snap-through.

scholarly journals New Results on the Motions of Asteroids in Resonances

1992 ◽  
Vol 152 ◽  
pp. 145-152 ◽  
Author(s):  
R. Dvorak
Keyword(s):  
Initial Conditions ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Close Approach ◽  
Integration Time ◽  
Time Interval ◽  
Time Intervals ◽  
3 Dimensional ◽  
Special Cases ◽  
Good Agreement

In this article we present a numerical study of the motion of asteroids in the 2:1 and 3:1 resonance with Jupiter. We integrated the equations of motion of the elliptic restricted 3-body problem for a great number of initial conditions within this 2 resonances for a time interval of 104 periods and for special cases even longer (which corresponds in the the Sun-Jupiter system to time intervals up to 106 years). We present our results in the form of 3-dimensional diagrams (initial a versus initial e, and in the z-axes the highest value of the eccentricity during the whole integration time). In the 3:1 resonance an eccentricity higher than 0.3 can lead to a close approach to Mars and hence to an escape from the resonance. Asteroids in the 2:1 resonance with Jupiter with eccentricities higher than 0.5 suffer from possible close approaches to Jupiter itself and then again this leads in general to an escape from the resonance. In both resonances we found possible regions of escape (chaotic regions), but only for initial eccentricities e > 0.15. The comparison with recent results show quite a good agreement for the structure of the 3:1 resonance. For motions in the 2:1 resonance our numeric results are in contradiction to others: high eccentric orbits are also found which may lead to escapes and consequently to a depletion of this resonant regions.

scholarly journals Elastic shocks in relativistic rigid rods and balls

2019 ◽  
Vol 475 (2225) ◽  
pp. 20180858 ◽  
Author(s):  
João L. Costa ◽  
José Natário
Keyword(s):  
Free Boundary ◽  
Minkowski Space ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Time Derivative ◽  
Space Time ◽  
Spherically Symmetric ◽  
Soft Phase ◽  
Minkowski Space Time ◽  
Dimensional Minkowski Space

We study the free boundary problem for the ‘hard phase’ material introduced by Christodoulou in (Christodoulou 1995 Arch. Ration. Mech. Anal. 130 , 343–400), both for rods in (1 + 1)-dimensional Minkowski space–time and for spherically symmetric balls in (3 + 1)-dimensional Minkowski space–time. Unlike Christodoulou, we do not consider a ‘soft phase’, and so we regard this material as an elastic medium, capable of both compression and stretching. We prove that shocks must be null hypersurfaces, and derive the conditions to be satisfied at a free boundary. We solve the equations of motion of the rods explicitly, and we prove existence of solutions to the equations of motion of the spherically symmetric balls for an arbitrarily long (but finite) time, given initial conditions sufficiently close to those for the relaxed ball at rest. In both cases we find that the solutions contain shocks if and only if the pressure or its time derivative do not vanish at the free boundary initially. These shocks interact with the free boundary, causing it to lose regularity.

Natural frequency, damping and forced responses of sandwich plates with viscoelastic core and graphene nanoplatelets reinforced face sheets

2020 ◽  
Vol 26 (15-16) ◽  
pp. 1165-1177 ◽  
Author(s):  
Ali Mohseni ◽  
Meisam Shakouri
Keyword(s):  
Sandwich Plate ◽  
Effective Properties ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Graphene Nanoplatelets ◽  
Complex Eigenvalue ◽  
Viscoelastic Core ◽  
Forced Vibration Analysis ◽  
Forced Responses

The free and forced vibration analysis of a sandwich plate with the viscoelastic core and face layers reinforced functionally with multilayered graphene nanoplatelets is presented. Different graphene nanoplatelet distributions are considered through the thickness, and the effective properties of the graphene reinforced nanocomposite are obtained by the rule of mixture. The equations of motion are extracted using Hamilton’s principle and assuming the classical thin plate theory for face layers and the first-order shear deformation theory for the thick viscoelastic core. Assuming the simply-supported boundary condition for all edges, the displacement components are proposed by Fourier series and the complex eigenvalue problem is solved to obtain the natural frequencies as well as the loss factors. The results are validated with available investigations, and effects of some important parameters on the free and forced responses of the sandwich plate are studied.

Initial conditions and the Laplace transform

1965 ◽  
Vol 11 (11) ◽  
pp. 385
Author(s):  
J.H. Brodie ◽  
C. Jones ◽  
S.E. Tweedy ◽  
E. Besag
Keyword(s):  
Laplace Transform ◽  
Initial Conditions ◽  
The Laplace Transform

Vibrational Characteristics of Ball Bearings

1977 ◽  
Vol 99 (2) ◽  
pp. 284-287 ◽  
Author(s):  
P. K. Gupta ◽  
L. W. Winn ◽  
D. F. Wilcock
Keyword(s):  
Initial Conditions ◽  
Equations Of Motion ◽  
Vibrational Analysis ◽  
Dominant Frequency ◽  
Hertzian Contact ◽  
Analytical Simulation ◽  
Contact Frequency ◽  
Ball Contact ◽  
Vibrational Characteristics ◽  
General Motion

The classical differential equations of motion of the ball mass center in an angular contact thrust loaded ball bearing are integrated with prescribed initial conditions in order to simulate the natural high frequency vibrational characteristics of the general motion. Two distinct frequencies are identified in the analytical simulation and their existence is also confirmed experimentally. One of the frequencies is found to be associated with the Hertzian contact spring at the ball race contact and it is therefore defined as the “elastic contact frequency”, Ωe. The other dominant frequency corresponding to oscillatory motion of the ball in the raceway groove appears to be kinematic in nature and it is, therefore, termed as the “bearing kinematic frequency”, Ωk. It is shown that for a given bearing Ωe and Ωk, vary as, respectively, 1/6 and 1/2 powers of the ball contact load and, therefore, for a given load these frequencies correspond to the natural frequencies of the bearing as applied in any vibrational analysis or simulation.

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