Variational Principles Of Mechanics Research Articles

Natural Vibrations of the Reinforced Tapered Shell with Taking Into Account the Viscoelastic Properties of the Material

In this paper, the integro-differential equations of natural oscillations of a viscoelastic ribbed truncated conical shell are obtained based on the Lagrange variational equation. The general research methodology is based on the variational principles of mechanics and variational methods. Geometrically nonlinear mathematical models of the deformation of ribbed conical shells are obtained, considering such factors as the discrete introduction of edges. Based on the finite element method, a method for solving and an algorithm for the equations of natural oscillations of a viscoelastic ribbed truncated conical shell with articulated and freely supported edges is developed. The problem is reduced to solving homogeneous algebraic equations with complex coefficients of large order. For a solution to exist, the main determinant of a system of algebraic equations must be zero. From this condition, we obtain a frequency equation with complex output parameters. The study of natural vibrations of viscoelastic panels of truncated conical shells is carried out, and some characteristic features are revealed. The complex roots of the frequency equation are determined by the Muller method. At each iteration of the Muller method, the Gauss method is used with the main element selection. As the number of edges increases, the real and imaginary parts of the eigenfrequencies increase, respectively.

Теория метастабильной кристаллизации переохла-жденного эвтектического расплава

A new theory of crystallization of a supercooled metastable melt, based on the variational principles of mechanics, was developed. The theory takes into account the regularities of the dif-fusion growth of equilibrium crystals, as well as the regularities the diffusionless growth of meta-stable crystals. Mathematical modeling showed that in the supercooled melt Fe83B17 there is sim-ultaneous nucleation and growth of the stable Fe and Fe2B phases with the metastable Fe3B phase. The growth rate of crystals of near-critical sizes of the Fe3B phase exceeds the growth rate of the Fe and Fe2B crystals. For the Fe3B crystals the effect of diffusionless growth has been ob-served; in this case, the rapidly growing surface of the Fe3B crystal captures the boron atoms. On the basis of the developed theory, a quasi-equilibrium phase diagram for a supercooled Fe-B melt was constructed. The phase diagram takes into account both the equilibrium crystal growth and the growth of the metastable phase. The resulting diagram makes it possible to pre-dict the values of the concentration of components at the surface of growing crystals for the Fe and Fe2B crystals, for which the condition of local equilibrium is observed on their surface, and for the metastable Fe3B crystals, whose diffusionless growth is due to the high speed of the crys-tal surface movement. The performed mathematical modeling of the nucleation and growth of crystals in the over-cooled eutectic melt Fe83B17 showed that in a supercooled melt there is simultaneous nucleation and growth of the stable Fe and Fe2B phases with the metastable Fe3B phase. The regularities of nucleation and growth of the metastable Fe3B phase have been studied. The growth mechanism of the Fe3B nuclei differs from the growth of the Fe and Fe2B nuclei due to diffusionless capture of the boron atoms by the growing surface of the Fe3B crystals.

Soil resistances for laterally loaded rigid piles in multi-layered elastic soil

ABSTRACTShort stubby piles like monopiles and large diameter drilled shafts undergo rigid body translation and rotation when subjected to a lateral force and/or a moment at the head. A method of analysis for these piles embedded in multi-layered elastic soil is developed using the variational principles of mechanics. Using this analysis, the soil resistance against pile movement can be rigorously related to the soil elastic constants, and the pile head displacement and rotation can be quickly calculated. The equilibrium equations for pile and soil displacements are obtained using the principle of virtual work and solved using an iterative algorithm. Pile responses obtained from the analysis match well with those obtained from three-dimensional finite element analyses in which the same inputs of loads, geometry, and material properties are given. Based on the new analysis, fitted equations for soil resistance parameters are developed, which can be used to directly calculate the pile head displacement and rotation without the use of the iterative algorithm. Numerical examples are provided that demonstrate how the method can be used to analyse practical problems.

Adaptive Singularity-Free Control Moment Gyroscopes

Design considerations of agile, precise, and reliable attitude control for a class of small spacecraft and satellites using adaptive singularity-free control moment gyroscopes (ASCMG) are presented. An ASCMG differs from that of a conventional control moment gyroscopes (CMG) because it is intrinsically free from kinematic singularities and so does not require a separate singularity avoidance control scheme. Furthermore, ASCMGs are adaptive to the asymmetries in the structural members (gimbal and rotor) as well as misalignments between the center of mass of the gimbal and rotors. Moreover, ASCMG clusters are highly redundant to failure and can function as variable and constant-speed CMGs without encountering singularities. A generalized multibody dynamics model of the spacecraft–ASCMG system is derived using the variational principles of mechanics, relaxing the standard set of simplifying assumptions made in prior literature on CMG. The dynamics model so obtained shows the complex nonlinear coupling between the internal degrees of freedom associated with an ASCMG and the spacecraft attitude motion. The general dynamics model is then extended to include the effects of multiple ASCMG, called the ASCMG cluster, and the sufficient conditions for nonsingular cluster configurations are obtained. The adverse effects of the simplifying assumptions that lead to the intricate design of the conventional CMG, and how they lead to singularities, become apparent in this development. A bare-minimum hardware prototype of an ASCMG using low-cost commercial off-the-shelf components is developed to show the design simplicity and scalability. A geometric variational integration scheme is obtained for this multibody spacecraft–ASCMG system for numerical and embedded implementation. Attitude pointing control of a CubeSat with three ASCMGs in the absence of external torques is numerically simulated to demonstrate the singularity-free characteristics and redundancy of the ASCMG cluster.

Development of Finite-Element Modeling of Processing Metals Under Pressure Using Variational Principles of Mechanics Proposed by V. L. Kolmogorov

On the basis of the variational principle of metal virtual stresses and flow rates in high plastic metal deformation with complex processes of hardening and softening, we propose an iterative procedure using finite-element simulation of processing metals under pressure, taking into account the heterogeneity of mechanical properties in the volume of the deformation zone. The proposed method of solving the boundary value problem increases the computational precision for the workpiece shape deformation, satisfying the condition of ideal plasticity for the material particles at any arbitrary point in time, as well as improving the accuracy of determining the field of the average normal stress in the deformation zone.

Роль вариационных принципов механики в исследовании эволюции неконсервативных динамических систем

Роль вариационных принципов механики в исследовании эволюции неконсервативных динамических систем

Математическое обеспечение конечно-элементного моделирования микромеханических датчиков инерциальной информации в рамках неклассической теории изгиба

A complete mathematical support of the 3D finite beam element for modeling of the micromechanical inertial measurement sensors and their components has been developed. The mathematical support includes mass matrix, stiffness matrix, Coriolis matrix, and centrifugal matrix. The mathematical support takes full account of the gyroscopic effect and the theory of Timoshenko. Employing of the Variational principles of mechanics and Lagrange's equation makes the process of derivation of the mathematical support clear, accurate and well-founded. The developed software was verified by a numerical simulation of the influence of the gyroscopic effect on the dynamics of the simplest model of the vibrating gyro. The results obtained due to the numerical simulation by using the developed mathematical support were compared with the results obtained in ANSYS, well-known engineering simulation software. The difference between these results was less than 5 %. This difference can be explained by the dissimilarity of the elements used in ANSYS and in the developed software. This paper shows that the developed mathematical support can be used for development of special software, which ensures, in contrast to the universal proprietary closed-source software such as ANSYS, a transparent implementation of the algorithms, a complete control of the progress of computing and significantly lower cost. Thus, the developed mathematical support for the three-dimensional finite element based on the theory of Timoshenko can be used to solve a wide range of problems of statics and dynamics, including the gyroscopic effect, e.g. in the area of research and development of the microelectromechanical sensors of the inertial information.

Revisiting the Slichter mode of Mercury in the context of differentiated models

We revisit the Slichter mode of a set of differentiated Mercury models that were considered previously. We apply a different and independent formalism, based on variational principles of mechanics, to analytically model the internal translations of a body with a global internal perfect fluid layer limited by two differentiated rigid solid constituents that have a spherical symmetric mass distribution. The Slichter mode is specified by providing the time evolution of the barycenters of the solid constituents, the amplitude ratio of the mantle to the inner core, and the period of the oscillatory motion. All these parameters only depend on the mass of the body, the mass of the inner core, the density of the fluid, and the mean density of the inner core. For previously developed Mercury models we find, in contrast to other results, that there are no discontinuities in the Slichter period when passing from undifferentiated to differentiated inner cores. Hence, in a general situation, a potential detection of the Slichter mode cannot determine the differentiation of the inner core. We also find that the BepiColombo mission may be able to detect the Slichter mode caused by an impactor meteroid with a mass of about 1012 kg in the most favorable of circumstances. This shows that a measurable excitation of the mode by this mechanism is even more unlikely than was previously established, where a mass about 1010 kg was found.

Dynamics and Control of Spacecraft With a Generalized Model of Variable Speed Control Moment Gyroscopes

The attitude dynamics model for a spacecraft with a variable speed control moment gyroscope (VSCMG) is derived using the principles of variational mechanics. The resulting dynamics model is obtained in the framework of geometric mechanics, relaxing some of the assumptions made in prior literature on control moment gyroscopes (CMGs). These assumptions include symmetry of the rotor and gimbal structure, and no offset between the centers of mass of the gimbal and the rotor. The dynamics equations show the complex nonlinear coupling between the internal degrees-of-freedom associated with the VSCMG and the spacecraft base body's rotational degrees-of-freedom. This dynamics model is then further generalized to include the effects of multiple VSCMGs placed in the spacecraft base body, and sufficient conditions for nonsingular VSCMG configurations are obtained. General ideas on control of the angular momentum of the spacecraft using changes in the momentum variables of a finite number of VSCMGs are provided. A control scheme using a finite number of VSCMGs for attitude stabilization maneuvers in the absence of external torques and when the total angular momentum of the spacecraft is zero is presented. The dynamics model of the spacecraft with a finite number of VSCMGs is then simplified under the assumptions that there is no offset between the centers of mass of the rotor and gimbal, and the rotor is axisymmetric. As an example, the case of three VSCMGs with axisymmetric rotors, placed in a tetrahedron configuration inside the spacecraft, is considered. The control scheme is then numerically implemented using a geometric variational integrator (GVI). Numerical simulation results with zero and nonzero rotor offset between centers of mass of gimbal and rotor are presented.

Response of Laterally Loaded Rectangular and Circular Piles in Soils with Properties Varying with Depth

Continuum-based analyses for laterally loaded piles with rectangular and circular cross sections are presented using solutions that can be obtained quickly without requiring any elaborate inputs for the geometry and numerical mesh. The analysis is developed by solving the differential equations governing the displacements of the pile-soil system derived using the variational principles of mechanics. Parametric studies are performed to investigate the influence of the pile cross-sectional shape, soil layering, pile slenderness ratio, and pile-soil modulus ratio on the response of laterally loaded piles in heterogeneous soil in which the soil shear modulus varies continuously or discretely with depth. The results show that piles with the same second moment of inertia have similar lateral-load response. The lateral responses of piles in two-layer systems were mainly affected by the thickness and stiffness of the top soil layer. Soil layering also influences the lateral response of piles in three-layer soil deposits consisting of two thin layers overlying the third layer. Algebraic equations for estimating the pile-head deflection and maximum bending moment are proposed that can be readily used in design. A user-friendly spreadsheet program is developed as a tool to perform calculations of pile response using the analysis. Numerical examples demonstrating the use of the analysis are provided.

Analysis of pile subjected to torsion in multi‐layered soil

SummaryA new method of analysis of piles in multi‐layered elastic soil subjected to a torque at the head is developed. The differential equation governing the angle of twist in the pile is derived using the variational principles of mechanics. The method of initial parameters is used to obtain closed‐form solutions of the angle of twist and torque in the pile as a function of depth. The inputs required for the analysis are shear moduli of pile and soil, pile geometry and thickness of soil layers. Copyright © 2013 John Wiley & Sons, Ltd.

Response of Euler–Bernoulli beam on spatially random elastic soil

A new method is developed for analysis of flexible foundations (beams) on spatially random elastic soil. The elastic soil underneath the beams is treated as a continuum, characterized by spatially random Young’s modulus and constant Poisson’s ratio. The randomness of the soil Young’s modulus is modeled using a two-dimensional non-Gaussian, homogeneous random field. The beam geometry and Young’s modulus are assumed to be deterministic. The total potential energy of the beam-soil system is minimized, and the governing differential equations and boundary conditions describing the equilibrium configuration of the system are obtained using the variational principles of mechanics. The differential equations are solved using the finite element and finite difference methods to obtain the beam and soil displacements. Four different beam lengths, representing moderately short, moderately long and long beams are analyzed for beam deflection, differential settlement, bending moment and beam shear force. The statistics of the beam responses are investigated using Monte Carlo simulations for different beam-soil modulus ratios and for different variances and scales of fluctuations of the soil Young’s modulus. Suggestions regarding the use of the analysis in design are made. A novelty in the analysis is that the two-dimensional random heterogeneity of soil is taken into account without the use of traditional two-dimensional numerical methods, which makes the new approach computationally efficient.

On the variational principles of mechanics

Variational principles, generalizing the classical d’Alembert–Lagrange, Hölder, and Hamilton–Ostrogradskii principles, are established. After the addition of anisotropic dissipative forces and taking the limit, when the coefficient of viscous friction tends to infinity, these variational principles transform into the classical principles, which describe the motion of systems with constraints. New variational relations are established for searching for the periodic trajectories of the reversible equations of dynamics.

Sasaki’s Pivotal Contribution: Calculus of Variations Applied to Weather Map Analysis

Abstract Yoshikazu Sasaki developed a variational method of data assimilation, a cornerstone of modern-day analysis and prediction in meteorology. Fundamentally, he formulated data assimilation as a constrained minimization problem with equality constraints. The generation of this idea is tracked by analyzing his education and research at the University of Tokyo in the immediate post–World War II (WWII) period. Despite austere circumstances—including limited financial support for education, poor living conditions, and a lack of educational resources—Sasaki was highly motivated and overcame these obstacles on his path to developing this innovative method of weather map analysis. The stages of his intellectual development are followed where information comes from access to his early publications, oral histories, and letters of reminiscence. It has been argued that Sasaki’s unique contribution to meteorological data assimilation stems from his deterministic view of the problem—a view founded on the principles of variational mechanics. Sasaki’s approach to the problem is compared and contrasted with the stochastic view that was pioneered by Arnt Eliassen. Both of these optimal approaches are viewed in the context of the pragmatic–operational objective analysis schemes that were developed in the 1950s–1960s. Finally, current-day methods [e.g., three- and four-dimensional variational data assimilation (3DVAR and 4DVAR)] are linked to the optimal methods of Eliassen and Sasaki.

Euler and modern science

Introduction Translator's note From the editors 1. Opening speech of the symposium 'Modern Developments of Euler's Ideas' October 24, 1983 A. P. Aleksandrov 2. Leonhard Euler: his life and work A. P. Yushkevich 3. Leonhard Euler, active and honored member of the Petersburg Academy of Sciences Yu. Kh. Kopelevich 4. The part played by the Petersburg Academy of Sciences (the Academy of Sciences of the USSR) in the publication of Euler's collected works E. P. Ozhigova 5. Leonhard Euler and the Berlin Academy of Sciences K. Grau 6. Was Leonhard Euler driven from Berlin by J. H. Lambert? K.-R. Biermann 7. Euler's mathematical notebooks E. Knobloch 8. On Euler's surviving manuscripts and notebooks G. P. Matvievskaya 9. The manuscript materials of Euler on number theory G. P. Matvievskaya and E. P. Ozhigova 10. Euler's contribution to algebra I. G. Bashmakova 11. Diophantine equations in Euler's works T. A. Lavrinenko 12. The foundations of mechanics and hydrodynamics in Euler's works G. K. Mikhailov and L. I. Sedov 13. Leonhard Euler and the variational principles of mechanics V. V. Rumyantsev 14. Leonhard Euler and the mechanics of elastic systems N. V. Banichuk and A. Yu. Ishlinskii 15. Euler's research in mechanics during the first Petersburg period N. N. Polyakhov 16. The significance of Euler's research in ballistics A. P. Mandryka 17. Euler and the development of astronomy in Russia V. K. Abalakin and E. A. Grebenikov 18. Euler and the evolution of celestial mechanics N. I. Nevskaya and K. V. Kholshevnikov 19. New evidence concerning Euler's development as an astronomer and historian of science N. I. Nevskaya 20. Leonhard Euler in correspondence with Clairaut, d'Alembert and Lagrange A. P. Yushkevich and R. Taton 21. Letters to a German Princess and Euler's physics A. T. Grigor'ian and V. S. Kirsanov 22. Euler and I. P. Kulibin N. M. Raskin 23. Euler and the history of a certain musical-mathematical idea E. V. Gertsman 24. Euler's music-rheoretical manuscripts and the formation of his conception of the theory of music S. S. Tserlyuk-Askadskaya 25. An unknown portrait of Euler by J. F. A. Darbes G. B. Andreeva and M. P. Vikturina 26. Eulogy in memory of Leonhard Euler Nikolai Fuss 27. Leonhard Euler's family and descendants I. R. Gekker and A. A. Euler Index.

The Isaacs equation for differential games, totally optimal fields of trajectories and related problems

Some aspects of the Isaacs principle of transition and his main equation for differential games are considered to demonstrate that those equations contain some implicit assumptions, and are valid under certain contiguity condition which is defined and analyzed for differential games. The notion of total optimality is defined, and the totally optimal fields of trajectories and control curves are introduced and studied in relation to the Isaacs principle of transition, Bellman principle of optimality, maximum principle of Pontryagin, and variational principles of mechanics. It is demonstrated that the Isaacs, Bellman and Pontryagin theories are valid if and only if the optimal trajectories and optimal control curves generated by those methods are totally optimal. In this context, the Hamilton–Jacobi partial differential equation can be used for sequential solution of multi-games defined as n -person games with m controls, r cost functionals and multiple min, max, min–max, etc., operators in fixed order of application and not creating multi-objective game problems. Over totally optimal fields, the structure of controls is invariant under time uncertainty. Parallel and series games are considered, so the Isaacs procedure can be reduced to the application of the Bellman equations twice. Control systems with incomplete information or structural limitations on controls do not, in general, satisfy the contiguity condition, thus, are not totally optimal. Game problems for such systems may have optimal solutions which, however, cannot be obtained by the Isaacs equations. This fact is shown in an example of a widely used engineering system for which an optimal trajectory has all its parts non-optimal and non-contiguous to the optimal trajectory. The paper presents theoretical justification of the Isaacs equations for contiguous systems, comparison of optimal control principles with variational principles of mechanics, the consideration of total optimality and totally optimal fields of trajectories as necessary and sufficient conditions for validity of the three major optimal control theories, and some other results important for applications.

Boundary Element Methods for Engineers and Scientists: An Introductory Course With Advanced Topics

11R1. Boundary Element Methods for Engineers and Scientists: An Introductory Course With Advanced Topics. - L Gaul, M Ko¨gl, and M Wagner. Springer-Verlag, New York. 2003. 488 pp. ISBN 3-540-00463-7. $79.95. Reviewed by JT Chen (Dept of Harbor and River Eng, Natl Taiwan Ocean Univ, PO Box 7-59, Keelung 202, Taiwan, ROC).As the title of this book emphasizes, an introductory course to the boundary element method (BEM) and advanced formulations is presented. The book contains four parts: Part I The Direct Boundary Element Method, Part II Dual Reciprocity Method (DRM), Part III Hybrid Boundary Element Methods, and Part IV Appendix. Part I can be seen as an introductory course, while Parts II and III cover advanced topics that contain the authors’ research material. The appendices in Part VI contain some fundamental solutions and particular solutions for the DRM. Exercises and programs are not provided. The reviewer found that an on-line book of BEM by the first author is available on the web site of http://www.bem.uni-stuttgart.de/. This site provides a more friendly and suitable textbook for beginners since exercises are given. As it is common with other BEM books, this text begins with an introduction and mathematically preliminaries. A special chapter on continuum physics is added in Chapter 3: the basic laws and constitutive equations for elastodynamics, heat conduction, electrodynamics, thermoelasticity, acoustics, and piezoelectricity are covered to provide a complete overview on the physical modeling. Chapters 4 and 5 introduce the direct BEM for potential problems of the Laplace and Navier equations with emphases on the issues of anisotropy and piezoelectricity. No indirect formulations in terms of single-layer or double-layer representations are developed. Numerical integration schemes for regular and singular integrals are addressed in Chapter 6. Parts II and III on advanced topics include the dual reciprocity BEM and the hybrid BEM. Chapter 7 basically follows the DRM book by Partridge et al. for a general introduction to the method. Chapter 8 focus on the solution of the DRM equation of motion and Chapters 9 and 10 present the application of the DRM to piezoelectricity and thermoelasticity. The hybrid BEM is derived from variational principles of mechanics that are reviewed in Chapter 11. Hybrid displacement and hybrid stress methods are both addressed in Chapters 12 and 13, respectively. Since the presented hybrid BEM uses the same source of variational principles as for hybrid FEM, one obtains symmetric system matrices. In contrast to symmetric Galerkin BEM, the hybrid BEM does not require a double integration over the boundary. Although this book can be used as a text in a BEM course, it contains some original results regarding the dual reciprocity BEM and the variational formulation of the hybrid BEM. The book is thus recommended to graduate students and engineers. The authors have succeeded in fulfilling their aim of a dual-purpose textbook. In Part I, students as well as practitioners find a clear introduction to the method, whereas Parts II and III can serve as a valuable reference to researchers and engineers. The main distinction of the book in comparison to available works on the BEM may be its focus on the application of the method to anisotropy, piezoelectricity and thermoelasticity, as well as the presentation of the hybrid BEM. The book contains 488 pages with 135 figures. The quality of print and figures is adequate. In general, this is a well-written book and is recommended to individuals and libraries.

Analytical Mechanics. Foundations of Engineering Mechanics Series

Analytical Mechanics. Foundations of Engineering Mechanics Series

Nonlinear vibrations of a slider-crank mechanism

A nonlinear vibration analysis of the slider-crank mechanism used in reciprocating machines such as internal-combustion engines and air compressors is formulated by the principles of variational mechanics. A dynamically equivalent system is employed in which the connecting rod is modelled as a pendulum oscillating about a reciprocating piston and an approximate solution is derived by an application of the Ritz averaging method. Numerical results for a typical example are presented and discussed.

Dynamic Response of Slider-Crank Machines During Startup

An analysis of the dynamic response of slider-crank machines is formulated by the principles of variational mechanics. Velocity ratio constraints rather than loop equation constraints are used resulting in first-order differential equations in the undetermined multipliers. The method is applied to a Kohler K91 engine and experimental verification is presented.