scholarly journals Free flexural vibrations of homogeneous beams with symmetrically variable depths

2021 ◽  
Author(s):  
Krzysztof Magnucki ◽  
Ewa Magnucka-Blandzi ◽  
Szymon Milecki ◽  
Damian Goliwąs ◽  
Leszek Wittenbeck
Keyword(s):  
Differential Equations ◽  
Natural Frequency ◽  
Cross Sections ◽  
Equations Of Motion ◽  
Hamilton’S Principle ◽  
Variable Depth ◽  
Hamilton's Principle ◽  
Fundamental Natural Frequency ◽  
The Subject ◽  
Flexural Vibrations

AbstractThe subject of the paper are homogeneous beams of symmetrically variable depth and bisymmetrical cross sections. Free flexural vibrations of these beams are analytically and numerically studied. Based on Hamilton’s principle, the differential equations of motion of these beams are obtained. The equations of motion are analytically solved with consideration of the bending lines of these beams subjected to their own weight. The fundamental natural frequency for exemplary beams is derived and presented in Tables and Figures.

Modelling and vibration of composite thin-walled rotating blades featuring extension-twist elastic coupling

2005 ◽  
Vol 109 (1095) ◽  
pp. 233-246 ◽  
Author(s):  
S-Y. Oh ◽  
L. Librescu ◽  
O. Song
Keyword(s):  
Composite Material ◽  
Boundary Conditions ◽  
Cross Sections ◽  
Equations Of Motion ◽  
Physical Characteristics ◽  
Elastic Coupling ◽  
Hamilton’S Principle ◽  
Rotating Blades ◽  
Advanced Composite ◽  
Thin Walled

Abstract The modelling and vibration of composite thin-walled pre-twisted rotating blades of non-uniform cross-sections along their span, and featuring the extension-twist elastic coupling are addressed. To this end, Hamilton’s principle is used to derive the equations of motion and the associated boundary conditions. In addition to the pretwist and warping restraint, the exotic properties of advanced composite material are used, and the efficiency of implementing the tailoring technique toward the enhancement, without weight penalties, of the vibratory behaviour of rotating blades is illustrated. Comparisons between the predictions by both Wagner’s and Washizu’s approaches are presented, and pertinent conclusions regarding the implications of the various geometrical and physical characteristics of the blade are outlined.

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.

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.

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.

scholarly journals Differential Equations of Motion for Naturally Curved and Twisted Composite Space Beams

2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Ying Hao ◽  
Wei He ◽  
Yanke Shi
Keyword(s):  
Differential Equations ◽  
Cross Section ◽  
Cross Sections ◽  
Design Principle ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Coupled System ◽  
Variable Coefficients ◽  
Curve Beam ◽  
Design Factor

The differential equations of motion for naturally curved and twisted elastic space beams made of anisotropic materials with noncircular cross sections, being a coupled system consisting of 14 second-order partial differential equations with variable coefficients, are derived theoretically. The warping deformation of beam’s cross section, as a new design factor, is incorporated into the differential equations in addition to the anisotropy of material, the curvatures of the rod axis, the initial twist of the cross section, the rotary inertia, and the shear and axial deformations. Numerical examples show that the effect of warping deformation on the natural frequencies of the beam is significant under certain geometric and boundary conditions. This study focuses on improving and consummating the traditional theories to build a general curve beam theory, thereby providing new scientific research reference and design principle for curve beam designers.

Hamilton’s Principle for External Viscous Fluid-Structure Interaction

2000 ◽  
Author(s):  
Haym Benaroya ◽  
Timothy Wei
Keyword(s):  
Variational Principle ◽  
Equation Of Motion ◽  
External Flow ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Structure Interaction ◽  
Future Paper ◽  
The Subject ◽  
End Times ◽  
Transport Theorem

Abstract In this paper, Hamilton’s principle is extended so as to be able to model external flow-structure interaction. This is accomplished by using Reynold’s Transport theorem. In this form, Hamilton’s principle is hybrid in the sense that it has an analytical part as well as a part that depends on experimentally derived functions. Examples are presented. A discussion on implications and extensions is extensive. In this work, the general theory is developed for the case where the configuration is not prescribed at the end times of the variational principle. This leads to a single governing equation of motion. This limitation can be removed by prescribing the end times, as is usual. This is outlined in the present paper, and will be the subject of a future paper.

PDE Modeling of a Flexible Two-Link Manipulator

2001 ◽  
Author(s):  
X. Zhang ◽  
S. S. Nair ◽  
V. Chellaboina
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Experimental Studies ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Partial Differential

Abstract The partial differential equations of a flexible two-link manipulator are derived using the Hamilton’s Principle. The model is validated by simulation as well as experimental studies using a two-link setup in the laboratory.

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