Higher Order Dynamic Equations of Motion for Soft Core Sandwich Beams Using Hamilton's Principle

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.

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Dynamic Modeling of Satellite Tether Systems Using Newton’s Laws and Hamilton’s Principle

Journal of Vibration and Acoustics ◽

10.1115/1.2776342 ◽

2007 ◽

Vol 130 (1) ◽

Cited By ~ 11

Author(s):

Kalyan K. Mankala ◽

Sunil K. Agrawal

Keyword(s):

Dynamic Modeling ◽

Continuous System ◽

Variable Length ◽

Variable Domain ◽

Dynamic Equations ◽

Hamilton’S Principle ◽

Hamilton's Principle ◽

Newton’S Laws

The objective of this paper is to derive the dynamic equations of a tether as it is deployed or retrieved by a winch on a satellite orbiting around Earth using Newton’s laws and Hamilton’s principle and show the equivalence of the two methods. The main feature of this continuous system is the presence of a variable length domain with discontinuities. Discontinuity is present at the boundary of deployment because of the assumption that the stowed part of the cable is unstretched and the deployed part is not. Developing equations for this variable domain system with discontinuities, specially using Hamilton’s principle, is a nontrivial task and we believe that it has not been adequately addressed in the literature.

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Dynamics of 3D Micropolar Gyroelastic Beams

Volume 1: Advances in Aerospace Technology ◽

10.1115/imece2014-39259 ◽

2014 ◽

Cited By ~ 4

Author(s):

Soroosh Hassanpour ◽

G. R. Heppler

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Dynamic Modeling ◽

Dynamic Equations ◽

Hamilton’S Principle ◽

The Finite Element Method ◽

Hamilton's Principle ◽

Modeling And Analysis ◽

Element Method

This paper is devoted to the dynamic modeling of micropolar gyroelastic beams and explores some of the modeling and analysis issues related to them. The simplified micropolar beam torsion and bending theories are used to derive the governing dynamic equations of micropolar gyroelastic beams from Hamilton’s principle. Then these equations are solved numerically by utilizing the finite element method and are used to study the spectral and modal behaviour of micropolar gyroelastic beams.

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

Volume 7A: 17th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc99/vib-8144 ◽

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.

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

Journal of Glaciology ◽

10.3189/002214310792447761 ◽

2010 ◽

Vol 56 (197) ◽

pp. 497-513 ◽

Cited By ~ 11

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.

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

Volume 4: 7th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B and C ◽

10.1115/detc2009-87780 ◽

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.

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

Journal of Applied Mechanics ◽

10.1115/1.2899493 ◽

1992 ◽

Vol 59 (2S) ◽

pp. S224-S229 ◽

Cited By ~ 1

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.

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