The Equations of Motion for the Torsional and Bending Vibrations of a Stranded Cable

1992 ◽  
Vol 59 (2S) ◽  
pp. S224-S229 ◽  
Author(s):  
Warren N. White ◽  
Srinivasan Venkatasubramanian ◽  
P. Michael Lynch ◽  
Chi-Lung D. Huang
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Hamilton’S Principle ◽  
Attached Mass ◽  
Hamilton's Principle ◽  
Bending Vibrations ◽  
Aerodynamic Loading ◽  
Simplifying Assumptions ◽  
Rotational Degrees Of Freedom

Equations of motion of a thin, stranded elastic cable with an eccentric, attached mass and subject to aerodynamic loading are derived using Hamilton’s principle. Coupling between the translational and rotational degrees of freedom owing to inertia, elasticity, and stranded geometry are considered. By invoking simplifying assumptions, the equations of motion are reduced to those obtained previously by other researchers.

Formulation of Equations of Motion for a Chain of Flexible Links Using Hamilton’s Principle

1994 ◽  
Vol 116 (1) ◽  
pp. 81-88 ◽  
Author(s):  
M. Benati ◽  
A. Morro
Keyword(s):  
Boundary Conditions ◽  
Finite Number ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Dynamic Equations ◽  
Relative Deformation ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Flexible Links ◽  
A Chain

The dynamic equations of a chain of flexible links are determined by means of Hamilton’s principle. First a continuous model is adopted and the boundary conditions are determined, along with the partial differential equations of motion. Then a model with a finite number of degrees of freedom is set up. The configuration of each link is described through the line which joins the end points and the relative deformation is described in terms of appropriate trial functions. The boundary conditions are incorporated into a set of basic trial functions. The time-dependent coefficients of the remaining shape functions play the role of Lagrangian coordinates. The dynamic equations are then derived and the procedure is contrasted with other methods for reduction of a system of links to a system with a finite number of degrees of freedom.

ROTATIONS AND VIBRATIONS OF FULLERENES IN THE MOLECULAR COMPLEX C20@C80

2021 ◽  
pp. 35-48
Author(s):  
M.A. Bubenchikov ◽  
◽  
A.M. Bubenchikov ◽  
D.V. Mamontov ◽  
◽  
...  
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Rotation Frequency ◽  
Dynamic State ◽  
Large Molecules ◽  
Rotational Degrees Of Freedom ◽  
Rotational Motions ◽  
Centers Of Mass ◽  
The Moment

The aim of this work is to apply classical mechanics to a description of the dynamic state of C20@C80 diamond complex. Endohedral rotations of fullerenes are of great interest due to the ability of the materials created on the basis of onion complexes to accumulate energy at rotational degrees of freedom. For such systems, a concept of temperature is not specified. In this paper, a closed description of the rotation of large molecules arranged in diamond shells is obtained in the framework of the classical approach. This description is used for C20@C80 diamond complex. Two different problems of molecular dynamics, distinguished by a fixing method for an outer shell of the considered bimolecular complex, are solved. In all the cases, the fullerene rotation frequency is calculated. Since a class of possible motions for a single carbon body (molecule) consists of rotations and translational displacements, the paper presents the equations determining each of these groups of motions. Dynamic equations for rotational motions of molecules are obtained employing the moment of momentum theorem for relative motions of the system near the fullerenes’ centers of mass. These equations specify the operation of the complex as a molecular pendulum. The equations of motion of the fullerenes’ centers of mass determine vibrations in the system, i.e. the operation of the complex as a molecular oscillator.

SYMBOLIC EQUATIONS FOR A FULL-CAR MODEL

2000 ◽  
Vol 24 (3-4) ◽  
pp. 493-514
Author(s):  
Natalie Baddour ◽  
K. A. Morris
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
A Priori ◽  
Active Suspension ◽  
Lagrangian Mechanics ◽  
Full Generality ◽  
Active Suspensions ◽  
Car Model ◽  
Simplifying Assumptions ◽  
Improved Performance

Active suspensions provide improved performance over conventional, passive suspensions. In this paper, modelling issues for an active suspension are considered. Symbolic equations for a full car model are derived using Lagrangian mechanics. The model has ten degrees of freedom instead of the usual seven. Furthermore, many of the usual simplifying assumptions are not made a priori so that the model retains its full generality. The model is developed so that modifications to any of the assumptions might easily be made and so that the equations of motion can be easily altered to satisfy more restrictive assumptions.

Modal Analysis for the Active Multi-Layer Beams by Using Spectrally Formulated Exact Eigenfunctions

1999 ◽  
Author(s):  
Usik Lee ◽  
Joohong Kim
Keyword(s):  
Modal Analysis ◽  
Closed Form ◽  
Dynamic Characteristics ◽  
Equations Of Motion ◽  
Hamilton’S Principle ◽  
Analysis Method ◽  
Hamilton's Principle ◽  
Numerical Examples ◽  
Coupled Equations ◽  
Fully Coupled

Abstract In this paper, a modal analysis method (MAM) is introduced for the active multi-layer laminate beams. Two types of active multi-layer laminate beams are considered: the elastic-viscoelastic-piezoelectric three-layer beams and the elastic-piezoelectric two-layer beams. The dynamics of the multi-layer laminate beams are represented by a set of fully coupled equations of motion, derived by using Hamilton’s principle. The exact eigenfunctions are spectrally formulated and the orthogonality of eigenfunctions is derived in a closed form. The present MAM is evaluated through some numerical examples. It is shown that the dynamic characteristics obtained by the present MAM certainly converge to the exact ones obtained by SEM as the number of eigenfunctions superposed in MAM is increased. The modal analysis results are also compared with the results obtained by FEM.

Many-Body Algebraic Theory

1995 ◽  
Author(s):  
F. Iachello ◽  
R. D. Levine
Keyword(s):  
Alternative Treatment ◽  
Degrees Of Freedom ◽  
Point Group ◽  
Algebraic Theory ◽  
Benzene Molecule ◽  
Polyatomic Molecules ◽  
Bending Vibrations ◽  
Simultaneous Treatment ◽  
Symmetry Coordinates ◽  
Rotational Degrees Of Freedom

For molecules with many atoms, the simultaneous treatment of rotations and vibrations in terms of vector coordinates r1, r2, r3, . . . , quantized through the algebra . . .G ≡ U1(4) ⊗ U2(4) ⊗ U3(4). . . . . . .(6.1). . . becomes very cumbersome. Each time a U(4) algebra is added one must go through the recoupling procedure using Racah algebra, which, although feasible, is in practice very time consuming. An alternative treatment, which can be carried out for molecules with any number of atoms, is that of separating vibrations and rotations as already discussed in Sections 4.2-4.5 for triatomic molecules. For nonlinear molecules, there are three rotational degrees of freedom, described by the Euler angles α, β, γ of Figure 3.1, and thus there remain 3n − 6 independent vibrational degrees of freedom, where n is the number of atoms in the molecule. For linear molecules, there are two rotational degrees of freedom, described by the angles α, β, and thus there remain 3n − 5 independent vibrational degrees of freedom, some of which (the bending vibrations) are doubly degenerate. In this alternative treatment, the algebraic theory of polyatomic molecules consists in the separate quantization of rotations and vibrations. Each bond coordinate is then a scalar, and the corresponding algebra is that of U(2). In polyatomic molecules, the geometric symmetry of the molecule also plays a very important role. For example, the benzene molecule, which is the example we discuss in this book has the point group symmetry D6h. A consequence of the symmetry of the molecule is that states must transform according to representations of the appropriate symmetry group. In terms of coordinates, this implies that one must form internal symmetry coordinates. These are linear combinations of the internal coordinates.

scholarly journals Hamilton-type principles applied to ice-sheet dynamics: new approximations for large-scale ice-sheet flow

2010 ◽  
Vol 56 (197) ◽  
pp. 497-513 ◽  
Author(s):  
J.N. Bassis
Keyword(s):  
Variational Principle ◽  
Large Scale ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Ice Sheet ◽  
Alternative Formulation ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Ice Sheet Dynamics ◽  
Large Scale Flow

AbstractIce-sheet modelers tend to be more familiar with the Newtonian, vectorial formulation of continuum mechanics, in which the motion of an ice sheet or glacier is determined by the balance of stresses acting on the ice at any instant in time. However, there is also an equivalent and alternative formulation of mechanics where the equations of motion are instead found by invoking a variational principle, often called Hamilton’s principle. In this study, we show that a slightly modified version of Hamilton’s principle can be used to derive the equations of ice-sheet motion. Moreover, Hamilton’s principle provides a pathway in which analytic and numeric approximations can be made directly to the variational principle using the Rayleigh–Ritz method. To this end, we use the Rayleigh–Ritz method to derive a variational principle describing the large-scale flow of ice sheets that stitches the shallow-ice and shallow-shelf approximations together. Numerical examples show that the approximation yields realistic steady-state ice-sheet configurations for a variety of basal tractions and sliding laws. Small parameter expansions show that the approximation reduces to the appropriate asymptotic limits of shallow ice and shallow stream for large and small values of the basal traction number.

Unified Approach for Holonomic and Nonholonomic Systems Based on the Modified Hamilton’s Principle

2009 ◽  
Author(s):  
Keisuke Kamiya ◽  
Junya Morita ◽  
Yutaka Mizoguchi ◽  
Tatsuya Matsunaga
Keyword(s):  
Equations Of Motion ◽  
Time Derivative ◽  
Nonholonomic Systems ◽  
Hamilton’S Principle ◽  
Virtual Power ◽  
Unified Approach ◽  
Hamilton's Principle ◽  
Basic Principles ◽  
Principle Of Virtual Power ◽  
Holonomic And Nonholonomic Systems

As basic principles for deriving the equations of motion for dynamical systems, there are d’Alembert’s principle and the principle of virtual power. From the former Hamilton’s principle and Langage’s equations are derived, which are powerful tool for deriving the equation of motion of mechanical systems since they can give the equations of motion from the scalar energy quantities. When Hamilton’s principle is applied to nonholonomic systems, however, care has to be taken. In this paper, a unified approach for holonomic and nonholonomic systems is discussed based on the modified Hamilton’s principle. In the present approach, constraints for both of the holonomic and nonholonomic systems are expressed in terms of time derivative of the position, and their variations are treated similarly to the principle of virtual power, i.e. time and position are fixed in operation with respect to the variations. The approach is applied to a holonomic and a simple nonholonomic systems.

Planar Motion of a Flexible Beam With a Tip Mass Driven by Two Kinematic Rotational Degrees of Freedom

1998 ◽  
Vol 120 (1) ◽  
pp. 206-213
Author(s):  
D. C. Winfield ◽  
B. C. Soriano
Keyword(s):  
Finite Element Method ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Planar Motion ◽  
Flexible Beam ◽  
The Finite Element Method ◽  
Rotational Degrees Of Freedom ◽  
Tip Mass

The objective was to model planar motion of a flexible beam with a tip mass that is driven by two kinematic rotational degrees of freedom which are (1) at the center of the hub and (2) at the point the beam is attached to the hub. The equations of motion were derived using Lagrange’s equations and were solved using the finite element method. The results for the natural frequencies of the beam especially at high tip masses and high rotational velocities of the hub were calculated and compared to results obtained using the Raleigh-Ritz method. The dynamic response of the beam due to a specified hub rotation was calculated for two cases.

Dynamics of a system of articulated pipes conveying fluid - I.Theory

1961 ◽  
Vol 261 (1307) ◽  
pp. 457-486 ◽  
Keyword(s):  
Degrees Of Freedom ◽  
Lateral Displacement ◽  
Elastic Tube ◽  
Collective Effects ◽  
Previous Attempt ◽  
Hamilton’S Principle ◽  
Centrifugal Pumps ◽  
Hamilton's Principle ◽  
Elementary Model ◽  
Conditions Of Stability

A general theory is derived to account for the free motions of a chain of articulated pipes through which there is a constant flow of incompressible fluid. It is assumed that the position of the joint at the inlet end of the chain is fixed, and that the pipes are subject to conservative forces, which might include their weights and the stresses in resilient joints between them. The motion is observed to be approximately independent of the effects of fluid friction, even when they are of considerable magnitude, and accordingly the fluid is taken to be frictionless in the theoretical model. The Lagrangian method of analysis is used, and a slightly unusual aspect of the method arises in that the complete physical system has unbounded energy; this approach to the problem incidentally provides a clear physical interpretation of the collective effects of the fluid upon the pipes. Lagrangian equations are established in a form where the main dependent variables are the potential and kinetic energies of the ‘finite part’ of the system, i. e. the pipes and the space enclosed by them; hence an appropriate statement of Hamilton’s principle is deduced. An elementary model for centrifugal pumps and turbines is noted to be a system of the present class though having only a single degree of feedom, and this is briefly considered as an illustration of energy transfer to or from the fluid. Infinitesimal motions about a state of equilibrium in which the pipes are alined are next investigated, and the stability of the equilibrium is discussed. In this connexion there appear some remarkable properties owing essentially to the fact that the hydrodynamical forces on the pipes are conservative when the outlet end of the chain is simply supported, whereas they are in general non-conservative when the end is free. Two different forms of instability are recognized; one is termed ‘buckling’, being similar to the failure of structures under static loading, and the other consists of self-excited oscillations. It is pointed out that a continuously flexible elastic tube comprises an extreme example of the present type of system in which the number of degrees of freedom is infinite, and Hamilton’s principle is shown in this case to imply that a certain partial differential equation is satisfied by the lateral displacement of a tube during bending motions; this derivation of the equation is a fairly delicate matter, and the only previous attempt at it is believed to be seriously in error. In §3 the conditions of stability for a particular system with two degrees of freedom are examined in detail.

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