Quasi-Green’s function method for free vibration of simply-supported trapezoidal shallow spherical shell

The idea of quasi-Green’s function method is clarified by considering a free vibration problem of the simply-supported trapezoidal shallow spherical shell. A quasi-Green’s function is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the prob-lem. The mode shape differential equations of the free vibration problem of a simply-supported trapezoidal shallow spherical shell are reduced to two simultaneous Fredholm integral equations of the second kind by the Green formula. There are multiple choices for the normalized boundary equation. Based on a chosen normalized boundary equa-tion, a new normalized boundary equation can be established such that the irregularity of the kernel of integral equations is overcome. Finally, natural frequency is obtained by the condition that there exists a nontrivial solution to the numerically discrete algebraic equations derived from the integral equations. Numerical results show high accuracy of the quasi-Green’s function method.     

Quasi-Green’s function method for free vibration of clamped thin plates on Winkler foundation

The quasi-Green’s function method is used to solve the free vibration problem of clamped thin plates on the Winkler foundation. Quasi-Green’s function is established by the fundamental solution and the boundary equation of the problem. The function satisfies the homogeneous boundary condition of the problem. The mode-shape differential equation of the free vibration problem of clamped thin plates on the Winkler foundation is reduced to the Fredholm integral equation of the second kind by Green’s formula. The irregularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. The numerical results show the high accuracy of the proposed method.     

Evaluation of the fatigue life of a tractor’s transmission spiral bevel gear

Conventional tractor transmission spiral bevel gears are designed and evaluated based on the engine rated load, which is significantly higher than the load conditions in the field. In this study, the fatigue life of a spiral bevel gear is evaluated to obtain data for design optimization. The equivalent load was calculated using the field load data, and the integrated equivalent load was calculated based on the annual usage of major field operations in Korea. The fatigue life of three spiral bevel gear samples was evaluated using the accelerated life test (ALT) under an engine rated load condition of 120%. It was also evaluated under engine rated, plow equivalent, and integrated equivalent load. Fatigue life was estimated using the ALT results and the fatigue damage exponent based on the ALT equation. We observed that the fatigue life of the spiral bevel gear under the plow equivalent and integrated equivalent loads is higher than that under the rated load by 214 and 9,400 times, respectively. The results of this study can provide useful information for the design optimization of tractor transmission spiral bevel gears considering the field equivalent load.     

Quasi-Green＇s function method for free vibration of simply-supported trapezoidal shallow spherical shell

The idea of quasi-Green＇s function method is clarified by considering a free vibration problem of the simply-supported trapezoidal shallow spherical shell. A quasi- Green＇s function is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the prob- lem. The mode shape differential equations of the free vibration problem of a simply- supported trapezoidal shallow spherical shell are reduced to two simultaneous Fredholm integral equations of the second kind by the Green formula. There are multiple choices for the normalized boundary equation. Based on a chosen normalized boundary equa- tion, a new normalized boundary equation can be established such that the irregularity of the kernel of integral equations is overcome. Finally, natural frequency is obtained by the condition that there exists a nontrivial solution to the numerically discrete algebraic equations derived from the integral equations. Numerical results show high accuracy of the quasi-Green＇s function method.     

Use of Tikhonov’s regularization method for solving identification problem for elastic systems

We present a method for solving nonlinear inverse problems, which also include identification problems for elastic systems. The problems whose initial data contain an error are usually solved by regularization methods [1–5]. In the present paper, we give preference to Tikhonov’s regularization method, which has been widely used in the recent years in practice to increase the stability of computational algorithms for solving problems in various areas of mechanics [6–9].     

Mathematical modeling of Maklakoff’s method for measuring the intraocular pressure

The physical content of Maklakoffs tonometric (based on the loading of the cornea) method of measuring the intraocular pressure, widely used in medical practice, is discussed. For this purpose, we employ both the results of physical modeling of the eye described in the literature and the results of our own mathematical modeling based on the representation of the eyeball as a thin shell. The effect of the physical properties of the shell on the results of the modeling is investigated. Qualitative conclusions that follow from our study and may be of practical interest in measuring the intraocular pressure are discussed.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2005, pp. 24–39. Original Russian Text Copyright © 2005 by Bauer, Lyubimov, and Tovstik.     

Kirchhoff’s equations for the rigid body motion revisited

The purpose of the note is to relate Kirchhoff’s equations with Euler’s equations of rigid body dynamics and to establish the conservation laws arising from given external forces. This is done in vector form, for an inhomogeneous rigid body, by considering the kinetic energy and showing how the terms occurring in the Euler’s equations may be given the form of partial derivatives of the kinetic energy. Next it is shown how the vector form, via derivatives of the kinetic energy, provides the conservation laws and the corresponding first integrals along with their immediate mechanical meaning. An outline is given of different approaches in the literature to the derivation of Kirchhoff’s equations.     

Generalization of Joukowski’s Solution for a Bubble in a Channel

Fluid Dynamics - A new exact solution for the problem of potential flow of a capillary fluid past a two-dimensional bubble in a rectilinear channel is derived; the solution generalizes the known...     

Development of a fully Lagrangian MPS-based coupled method for simulation of fluid–structure interaction problems

A fully Lagrangian particle-based method is developed for simulating the FSI (Fluid–Structure Interaction) problems corresponding to incompressible fluid flows and elastic structures. First, the developed elastic structure model is verified by static and dynamic tests corresponding to a simple cantilever beam. The simulation results are compared with analytical and other researchers׳ numerical solutions. Then, the structure model is carefully coupled with a fluid model comprising of the so-called PNU-MPS (Pusan-National-University-modified Moving Particle Simulation) method and several recently developed enhanced schemes. The coupled fluid–structure method is applied to a dam break with an elastic gate and a violent sloshing flow with a hanging rubber baffle. The results of simulations are compared with those of the experiments by Antoci et al. (2007) and Idelsohn et al. (2008).     

An extension of Neumann’s method for shape control of force-induced elastic vibrations by eigenstrains

This paper is concerned with the force-induced vibrations of linear elastic solids and structures. We seek a transient distribution of actuating stresses produced by additional eigenstrain, such that the vibrations produced by a given set of imposed forces are exactly compensated. This problem, known as dynamic shape control problem in structural engineering, or as dynamic displacement compensation problem in automatic control, is inverse to the usual direct problem of determining displacements due to imposed forces and actuation stresses. In the present paper, we extend a method, which was introduced by F.E. Neumann for demonstrating the uniqueness of direct elastodynamic problems. We use this extended Neumann method in order to show that the distribution of the actuating stresses for shape control must be equal to any statically admissible stress distribution that is in temporal equilibrium with the imposed forces. We furthermore discuss the role of stresses corresponding to this class of solutions in some detail, emphasizing the non-unique nature of a statically admissible stress. As an analytical justification of our formulations, we show that our method reveals some static results by J.M.C. Duhamel and by W. Voigt and D.E. Carlson. Particularly, our method can be interpreted as a dynamic extension of the Duhamel body-force analogy. We moreover present numerical results for a dynamically loaded, irregularly shaped domain in a state of plane strain. These finite element computations give excellent evidence for the validity of the presented method of shape control for both, the case of a step-input and the case of a harmonic excitation.     

Path Integral approach via Laplace’s method of integration for nonstationary response of nonlinear systems

Meccanica - In this paper the nonstationary response of a class of nonlinear systems subject to broad-band stochastic excitations is examined. A version of the Path Integral (PI) approach is...     

Development and experimental investigation of a Quadrotor’s robust generalized dynamic inversion control system

Nonlinear Dynamics - The development of a two-loops Quadrotor’s robust generalized dynamic inversion (RGDI)-based control system is presented. The outer (position) loop utilizes PD position...     

A Ritz’s method based solution for the contact problem of a deformable rectangular plate on an elastic quarter-space

In this article, Ritz’s method is used to calculate with unprecedented accuracy the displacements related to a deformable rectangular plate resting on the surface of an elastic quarter-space. To achieve this required three basic steps. The first step involved the study of Green’s function describing the vertical displacements of the surface of an elastic quarter-space due to vertical force applied on its surface. For this case, an explicit formula was obtained by analytically resolving a complicated integral that did not previously have an analytical solution. The second step involved the study of the coupled system of a plate and an elastic quarter-space. This portion focused on determining reactive forces in the contact zone based on Hetenyi’s solution. After determination of the reactive forces, certain features were attributed to the plate’s edges. The final step involved the application of Ritz’s method to determine the deflections of the plate resting on the surface of the quarter-space. Finally, an example calculation and validation of results are given. This is the first semi-analytical solution proposed for this type of contact problem.     

Radial basis functions collocation and a unified formulation for bending,vibration and buckling analysis of laminated plates,according to a variation of Murakami’s zig-zag theory

In this paper, we propose a variation of the use of Murakami’s zig-zag theory for the analysis of laminated plates. The new theory accounts for through-the-thickness deformation, by considering a quadratic evolution of the transverse displacement with the thickness coordinate. The equations of motion and the boundary conditions are obtained by the Carrera’s Unified Formulation, and further interpolated by collocation with radial basis functions. This paper considers the analysis of static deformations, free vibrations and buckling loads on laminated composite plates.