scholarly journals Small oscillations of an ideal liquid contained in a vessel closed by an elastic circular plate, in uniform rotation

2017 ◽  
Vol 44 (1) ◽  
pp. 35-49
Author(s):  
Hilal Essaouini ◽  
Bakkali El ◽  
Pierre Capodanno
Keyword(s):  
Circular Plate ◽  
Small Amplitude ◽  
Variational Equation ◽  
Equations Of Motion ◽  
Coupled System ◽  
Ideal Liquid ◽  
Existence And Uniqueness Theorem ◽  
Uniform Rotation ◽  
Small Oscillations ◽  
Elastic Circular Plate

The problem of the small oscillations of an ideal liquid contained in a vessel in uniform rotation has been studied by Kopachevskii and Krein in the case of an entirely rigid vessel [3]. We propose here, a generalization of this model by considering the case of a vessel closed by an elastic circular plate. In this context, the linearized equations of motion of the system plateliquid are derived. Functional analysis is used to obtain a variational equation of the small amplitude vibrations of the coupled system around its equilibrium position, and then two operatorial equations in a suitable Hilbert space are presented and analyzed. We show that the spectrum of the system is real and consists of a countable set of eigenvalues and an essential continuous spectrum filling an interval. Finally the existence and uniqueness theorem for the solution of the associated evolution problem is proved by means the semigroups theory.

scholarly journals Mathematical study of the small oscillations of a finite cylindrical column liquid-gas under zero gravity

2019 ◽  
Vol 7 (1) ◽  
pp. 4
Author(s):  
Hilal Essaouini ◽  
Pierre Capodanno
Keyword(s):  
Variational Equation ◽  
Equations Of Motion ◽  
Geometric Condition ◽  
Zero Gravity ◽  
Mathematical Study ◽  
Small Oscillations ◽  
Cylindrical Column ◽  
Barotropic Gas ◽  
Circular Rings ◽  
Classical Vibration

This paper deals with the mathematical study of the small motions of a system formed by a cylindrical liquid column bounded by two parallel circular rings and an internal cylindrical column constituted by a barotropic gas under zero gravity. From the equations of motion, the authors deduce a variational equation. Then, the study of the small oscillations depends on the coerciveness of a hermitian form that appears in this equation. It is proved that this last problem is reduced to an auxiliary eigenvalues problem. The discussion shows that, under a simple geometric condition, the problem is a classical vibration problem.  

scholarly journals Dynamic interaction between elastic plate and transversely isotropic poroelastic medium

2019 ◽  
Vol 258 ◽  
pp. 05016
Author(s):  
Suraparb Keawsawasvong ◽  
Teerapong Senjuntichai
Keyword(s):  
Circular Plate ◽  
Half Space ◽  
Contact Surface ◽  
Equations Of Motion ◽  
Admissible Function ◽  
Transversely Isotropic ◽  
Vertical Displacement ◽  
Generalized Coordinates ◽  
Vertical Loading ◽  
Elastic Circular Plate

In this paper, dynamic response of an elastic circular plate, under axisymmetric time-harmonic vertical loading, resting on a transversely isotropic poroelastic half-space is investigated. The plate-half-space contact surface is assumed to be smooth and fully permeable. The discretization techniques are employed to solve the unknown normal traction at the contact surface based on the solution of flexibility equations. The vertical displacement of the plate is represented by an admissible function containing a set of generalized coordinates. Solutions for generalized coordinates are obtained by establishing the equation of motion of the plate through the application of Lagrange’s equations of motion. Selected numerical results corresponding to the deflections of a circular plate, with different degrees of flexibility, resting on a transversely isotropic poroelastic half-space are presented.

Transient Interaction of a Circular Plate and a Fluid Medium

1979 ◽  
Vol 46 (1) ◽  
pp. 26-30 ◽  
Author(s):  
J. W. Berglund
Keyword(s):  
Boundary Conditions ◽  
Circular Plate ◽  
Applied Pressure ◽  
Plate Boundary ◽  
Fluid Medium ◽  
Fluid Field ◽  
Elastic Circular Plate ◽  
Unstrained State ◽  
State Of Rest ◽  
Acoustic Fluid

The transient dynamic response of an elastic circular plate subjected to a suddenly applied pressure is determined for several edge boundary conditions. The plate boundary is attached to a semi-infinite, radially rigid tube which is filled with an acoustic fluid, and pressure is applied to the in-vacuo side of the plate. The transient solution is determined by using a technique in which the plate is subjected to a periodic pressure function constructed of appropriately signed and time-shifted Heaviside step functions, and by relying on a physical mechanism which returns the plate and fluid near the plate to an unstrained state of rest between pulses. The plate response is presented for a number of radius-to-thickness ratios and edge boundary conditions when interacting with water. Comparisons are also made with solutions obtained using a plane wave approximation to the fluid field.

On a sphere performing linear and torsional oscillations in a viscous fluid

1988 ◽  
Vol 66 (7) ◽  
pp. 576-579
Author(s):  
G. T. Karahalios ◽  
C. Sfetsos
Keyword(s):  
Boundary Layer ◽  
Viscous Fluid ◽  
Secondary Flow ◽  
Small Amplitude ◽  
Equations Of Motion ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Time Varying ◽  
Torsional Oscillations

A sphere executes small-amplitude linear and torsional oscillations in a fluid at rest. The equations of motion of the fluid are solved by the method of successive approximations. Outside the boundary layer, a steady secondary flow is induced in addition to the time-varying motion.

scholarly journals The formation of vortices from a surface of discontinuity

1931 ◽  
Vol 134 (823) ◽  
pp. 170-192 ◽  
Keyword(s):  
Steady State ◽  
Plane Surface ◽  
Equations Of Motion ◽  
Steady Motion ◽  
Small Oscillations ◽  
Approximate Form ◽  
History Of ◽  
State Increase ◽  
Liquid Surfaces

Helmholtz was the first to remark on the instability of those “liquid surfaces” which separate portions of fluid moving with different velocities, and Kelvin, in investigating the influence of wind on waves in water, supposed frictionless, has discussed the conditions under which a plane surface of water becomes unstable. Adopting Kelvin’s method, Rayleigh investigated the instability of a surface of discontinuity. A clear and easily accessible rendering of the discussion is given by Lamb. The above investigations are conducted upon the well-known principle of “small oscillations”—there is a basic steady motion, upon which is superposed a flow, the squares of whose components of velocity can be neglected. This method has the advantage of making the equations of motion linear. If by this method the flow is found to be stable, the equations of motion give the subsequent history of the system, for the small oscillations about the steady state always remain “small.” If, however, the method indicates that the system is unstable, that is, if the deviations from the steady state increase exponentially with the time, the assumption of small motions cannot, after an appropriate interval of time, be applied to the case under consideration, and the equations of motion, in their approximate form, no longer give a picture of the flow. For this reason, which is well known, the investigations of Rayleigh only prove the existence of instability during the initial stages of the motion. It is the object of this note to investigate the form assumed by the surface of discontinuity when the displacements and velocities are no longer small.

scholarly journals Numerical Integration of Multibody Dynamic Systems Involving Nonholonomic Equality Constraints

2021 ◽  
Author(s):  
Sotirios Natsiavas ◽  
Panagiotis Passas ◽  
Elias Paraskevopoulos
Keyword(s):  
Dynamic Systems ◽  
Equations Of Motion ◽  
Time Integration ◽  
Nonholonomic Constraints ◽  
Coupled System ◽  
Weak Form ◽  
Integration Scheme ◽  
Motion Constraints ◽  
Multibody Dynamic ◽  
Time Integration Scheme

Abstract This work considers a class of multibody dynamic systems involving bilateral nonholonomic constraints. An appropriate set of equations of motion is employed first. This set is derived by application of Newton’s second law and appears as a coupled system of strongly nonlinear second order ordinary differential equations in both the generalized coordinates and the Lagrange multipliers associated to the motion constraints. Next, these equations are manipulated properly and converted to a weak form. Furthermore, the position, velocity and momentum type quantities are subsequently treated as independent. This yields a three-field set of equations of motion, which is then used as a basis for performing a suitable temporal discretization, leading to a complete time integration scheme. In order to test and validate its accuracy and numerical efficiency, this scheme is applied next to challenging mechanical examples, exhibiting rich dynamics. In all cases, the emphasis is put on highlighting the advantages of the new method by direct comparison with existing analytical solutions as well as with results of current state of the art numerical methods. Finally, a comparison is also performed with results available for a benchmark problem.

Resistive Ballooning Modes in Three-Dimensional Configurations

1982 ◽  
Vol 37 (8) ◽  
pp. 848-858 ◽  
Author(s):  
D. Correa-Restrepo
Keyword(s):  
Growth Rates ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Coupled System ◽  
Small Constant ◽  
Ballooning Mode ◽  
The Stability ◽  
The Magnetic Field ◽  
Symmetric Systems ◽  
The Ideal

Resistive ballooning modes in general three-dimensional configurations are studied on the basis of the equations of motion of resistive MHD. Assuming small, constant resistivity and perturbations localized transversally to the magnetic field, a stability criterion is derived in the form of a coupled system of two second-order differential equations. This criterion contains several limiting cases, in particular the ideal ballooning mode criterion and criteria for the stability of symmetric systems. Assuming small growth rates, analytical results are derived by multiple-length-scale expansion techniques. Instabilities are found, their growth rates scaling as fractional powers of the resistivity

scholarly journals Mechanical aspects of the semicircular ducts in the vestibular system

2020 ◽  
Vol 114 (4-5) ◽  
pp. 421-442
Author(s):  
Mees Muller
Keyword(s):  
Brownian Motion ◽  
Vestibular System ◽  
Equations Of Motion ◽  
Complex Structure ◽  
Coupled System ◽  
Continuity Equations ◽  
Mathematical Expressions ◽  
Semicircular Ducts ◽  
Evolutionary Aspects ◽  
Instrumental Role

Abstract The semicircular ducts (SCDs) of the vestibular system play an instrumental role in equilibration and rotation perception of vertebrates. The present paper is a review of quantitative approaches and shows how SCDs function. It consists of three parts. First, the biophysical mechanisms of an SCD system composed of three mutually connected ducts, allowing endolymph to flow from one duct into another one, are analysed. The flow is quantified by solving the continuity equations in conjunction with the equations of motion of the SCD hydrodynamics. This leads to mathematical expressions that are suitable for further analytical and numerical analysis. Second, analytical solutions are derived through four simplifying steps while keeping the essentials of the coupled system intact. Some examples of flow distributions for different rotations are given. Third, the focus is on the transducer function of the SCDs. The complex structure of the mechano-electrical transduction apparatus inside the ampullae is described, and the consequences for sensitivity and frequency response are evaluated. Furthermore, both the contributions of the different terms of the equations of motion and the influence of Brownian motion are analysed. Finally, size limitations, allometry and evolutionary aspects are taken into account.

Sign in / Sign up

Export Citation Format

Share Document