EIGENFREQUENCIES OF OSCILLATIONS OF A PLATE WHICH SEPARATES A TWO-LAYER IDEAL FLUID WITH A FREE SURFACE IN A RECTANGULAR CHANNEL

The frequency spectrum of plane vibrations of an elastic plate separating a two-layer ideal fluid with a free surface in a rectangular channel is investigated analytically and numerically. For an arbitrary fixing of the contours of a rectangular plate, it is shown that the frequency spectrum of the problem under consideration consists of two sets of frequencies describing the vibrations of the free surface of the liquid and the elastic plate. The equations of coupled vibrations of the plate and the fluid are presented using a system of integro-differential equations with the boundary conditions for fixing the contours of the plate and the condition for the conservation of the volume of the fluid. When solving a boundary value problem for eigenvalues, the shape of the plate deflection is represented by the sum of the fundamental solutions of a homogeneous equation for a loose plate and a partial solution of an inhomogeneous equation by expanding in terms of eigenfunctions of oscillations of an ideal fluid in a rectangular channel. The frequency equation of free compatible vibrations of a plate and a liquid is obtained in the form of a fourth-order determinant. In the case of a clamped plate, its simplification is made and detailed numerical studies of the first and second sets of frequencies from the main mechanical parameters of the system are carried out. A weak interaction of plate vibrations on vibrations of the free surface and vice versa is noted. It is shown that with a decrease in the mass of the plate, the frequencies of the second set increase and take the greatest value for inertialess plates or membranes. A decrease in the frequencies of the second set occurs with an increase in the filling depth of the upper liquid or a decrease in the filling depth of the lower liquid. Taking into account two terms of the series in the frequency equation, approximate formulas for the second set of frequencies are obtained and their efficiency is shown. With an increase in the number of terms in the series of the frequency equation, the previous roots of the first and second sets are refined and new ones appear.

On a Novel Approximate Solution to the Inhomogeneous Euler–Bernoulli Equation with an Application to Aeroelastics

This paper focuses on the development of an explicit finite difference numerical method for approximating the solution of the inhomogeneous fourth-order Euler–Bernoulli beam bending equation with velocity-dependent damping and second moment of area, mass and elastic modulus distribution varying with distance along the beam. We verify the method by comparing its predictions with an exact analytical solution of the homogeneous equation, we use the generalised Richardson extrapolation to show that the method is grid convergent and we extend the application of the Lax–Richtmyer stability criteria to higher-order schemes to ensure that it is numerically stable. Finally, we present three sets of computational experiments. The first set simulates the behaviour of the un-loaded beam and is validated against the analytic solution. The second set simulates the time-dependent dynamic behaviour of a damped beam of varying stiffness and mass distributions under arbitrary externally applied loading in an aeroelastic analysis setting by approximating the inhomogeneous equation using the finite difference method derived here. We compare the third set of simulations of the steady-state deflection with the results of static beam bending experiments conducted at Cranfield University. Overall, we developed an accurate, stable and convergent numerical framework for solving the inhomogeneous Euler–Bernoulli equation over a wide range of boundary conditions. Aircraft manufacturers are starting to consider configurations with increased wing aspect ratios and reduced structural weight which lead to more slender and flexible designs. Aeroelastic analysis now plays a central role in the design process. Efficient computational tools for the prediction of the deformation of wings under external loads are in demand and this has motivated the work carried out in this paper.

Elastic Analysis of Steel-Concrete Composite Beams with Partial Interaction

The paper presents an exact analytical method for the elastic analysis of steel-concrete composite beams with partial interaction. Accepting the basic assumptions of the Newmark analytical model and adopting the axial force in the concrete slab as the main unknown, the second order nonhomogeneous differential equation of the steel-concrete composite element with partial interaction is derived. Further, the complete solutions for simply supported and fixed-ended composite beams subjected to concentrated and uniform loads respectively, are developed. The solution of the homogeneous equation is determined by imposing proper Dirichlet or Neumann boundary conditions depending on the static scheme of the element. The particular solutions are then derived for the considered loading conditions. It is shown that the internal axial force in concrete slab associated to composite beams with partial interaction can be expressed as a fraction of the axial force in concrete slab under full interaction through a non-dimensional function f(aL) which takes into account the connection’s stiffness, the mechanical properties and also the length of the element. Moreover, the solutions are included in a flexibility-based approach to derive the force-displacement relations of the beam element with partial interaction. For the resulted 2-noded beam-column element with 6DOF, the stiffness matrix is derived, showing that the partial composite action may be included at the element level by means of a series of correction factors applied to the standard full-interaction stiffness matrix coefficients. A numerical example is provided to demonstrate the accuracy and performance of the proposed method. Within the elastic range, the predicted load-midspan deflection curve is in very good agreement with both experimental and other numerical results retrieved from international literature. A parametric study was conducted to investigate the influence of the shear connection degree on the beam’s midspan deflection and the results were compared with those computed by using code provisions.

Semi-nonoscillation intervals and sign-constancy of Green’s functions of two-point impulsive boundary-value problems

UDC 517.9 We consider the second order impulsive differential equation with delays where for In this paper, we obtain the conditions of semi-nonoscillation for the corresponding homogeneous equation on the interval Using these results, we formulate theorems on sign-constancy of Green's functions for two-point impulsive boundary-value problems in terms of differential inequalities.

Time Impact on Residue in a Non-homogeneous Equation of Statics in the Theory of Elastic Mixture

Residual stress in continuum has not been quantified because time relationship with residues has not been proven analytically. This is achieved in this paper by analyzing a two component mixture with the non-homogeneous equation of statics in the theory of elastic mixture, and second order differential equations with variable coefficients. A dry mixture of sand and cement is transformed into a continuum, which is been determined as an entire or a meromorphic function, as a result of the existence of residues that are contained in the principal component of the mixture obtained directly from the earth. The time relationship with residue, in these two functions are determined. Our result shows that time places a limit on residues, making the meromorphic function prone to implosion..

Asymptotic solution to convolution integral equations on large and small intervals

We consider convolution integral equations on a finite interval with a real-valued kernel of even parity, a problem equivalent to finding a Wiener–Hopf factorization of a notoriously difficult class of 2 × 2 matrices. The kernel function is assumed to be sufficiently smooth and decaying for large values of the argument. Without loss of generality, we focus on a homogeneous equation and we propose methods to construct explicit asymptotic solutions when the interval size is large and small. The large interval method is based on a reduction of the original equation to an integro-differential equation on a half-line that can be asymptotically solved in a closed form. This provides an alternative to other asymptotic techniques that rely on fast (typically exponential) decay of the kernel function at infinity, which is not assumed here. We also consider the problem on a small interval and show that finding its asymptotic solution can be reduced to solving an ODE. In particular, approximate solutions could be constructed in terms of readily available special functions (prolate spheroidal harmonics). Numerical illustrations of the obtained results are provided and further extensions of both methods are discussed.

THE PROBLEM OF ADAPTING THE METHODOLOGY OF TEACHING DIFFERENTIAL EQUATIONS AT SCHOOL

The general properties of the solution of a homogeneous equation are given. Solutions of a homogeneous linear ordinary differential equation of the second order with constant coefficients in three cases related to the coefficients of the equation are investigated. The obtained results are justified in the form of a theorem. These conclusions are proved in the framework of the methods of high school mathematics. This theory, known in general mathematics, is fully adapted to the implementation in secondary school mathematics and developed with the help of new elementary techniques that are understandable to the student. The main purpose of the work was to develop methods for solving a linear homogeneous differential equation of the second-order at a level that a schoolboy can master. The result will be the creation of a special course program on the basics of ordinary differential equations in secondary schools of the natural-mathematical direction, the preparation of appropriate content material, and providing them with a simple teaching method.

UNSYMMETRIC OSCILLATIONS OF ANISOTROPIC PLATEHAVING AN ADDITIONAL MASS

The problem of unsymmetric oscillations of circular plate made from anisotropic material is examined. The plate under consideration has an additional point mass attached offthe center or a system of additional masses. Also the oscillations of anisotropic circular plate with a point support placed offthe center are studied. The exact analytical approach is used for the decision of the above-mentioned problems; the method of compensating loads is applied. For this aim the basic and the compensating solutions are received. The basic solution satisfiesto the resolving differentialequation of the problem under study. The compensating solution satisfiesto the corresponding homogeneous equation and this solution amounting to the basic one also satisfiesto the boundary conditions. The Nielsen’s equation is used for the receiving of the exact solutions expressed in terms of Bessel functions. The equation for determination of frequencies of natural vibrations is obtained.

On Strongly Continuous Resolving Families of Operators for Fractional Distributed Order Equations

The aim of this work is to find by the methods of the Laplace transform the conditions for the existence of a strongly continuous resolving family of operators for a linear homogeneous equation in a Banach space with the distributed Gerasimov–Caputo fractional derivative and with a closed densely defined operator A in the right-hand side. It is proved that the existence of a resolving family of operators for such equation implies the belonging of the operator A to the class CW(K,a), which is defined here. It is also shown that from the continuity of a resolving family of operators at t=0 the boundedness of A follows. The existence of a resolving family is shown for A∈CW(K,a) and for the upper limit of the integration in the distributed derivative not greater than 2. As corollary, we obtain the existence of a unique solution for the Cauchy problem to the equation of such class. These results are used for the investigation of the initial boundary value problems unique solvability for a class of partial differential equations of the distributed order with respect to time.

Free Vibration of Thick FGM Plates under TSDT and Thermal Environment

Three parameters of thermal environment, varied calculated shear correction, and third-order shear deformation theory (TSDT) of displacement are important in the frequency study. These three effects have been studied on the non-dimensional and dimensional frequencies of thick FGM plates. An additional c1 displacement term in nonlinear coefficient of TSDT is used to present the frequency of vibration into the simply homogeneous equation of thick FGM plates. The determinant of the coefficient matrix containing the c1 displacement term in dynamic differential equilibrium equations can be derived into the five degree polynomial free vibration equation. The non-dimensional and dimensional of natural frequency can be obtained. The effects of plate thickness, temperature of environment and power law index of FGM on the non-dimensional and dimensional frequency of FGM plates are investigated.