Effects of Shear Deformation and Rotary Inertia on the Nonlinear Dynamics of Rotating Curved Beams

1990 ◽  
Vol 112 (2) ◽  
pp. 183-193 ◽  
Author(s):  
Wei-Hsin Gau ◽  
A. A. Shabana
Keyword(s):  
Shear Deformation ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Body Motion ◽  
Rotary Inertia ◽  
Element Formulation ◽  
Timoshenko Beams ◽  
Curved Beams ◽  
Finite Rotations ◽  
Deformable Bodies

In this investigation a method for the dynamic analysis of initially curved Timoshenko beams that undergo finite rotations is presented. The combined effect of rotary inertia, shear deformation, and initial curvature is examined. The kinetic energy is first developed for the curved beam and the beam mass matrix is identified. It is shown that the form of the mass matrix as well as the nonlinear inertia terms that represent the coupling between the rigid body motion and the elastic deformation can be expressed in terms of a set of invariants that depend on the assumed displacement field, rotary inertia, shear deformation, and the initial beam curvature. A nonlinear finite element formulation is then developed for Timoshenko beams that undergo finite rotations. The nonlinear formulation presented in this paper is applied to multibody dynamics where mechanical systems consist of an interconnected set of rigid and deformable bodies, each of which may undergo finite rotations. The equations of motion are developed using Lagrange’s equation and nonlinear algebraic constraint equations that mathematically describe mechanical joints and specified trajectories are adjoined to the system differential equations using the vector of Lagrange multipliers.

scholarly journals Free vibration of prestress Timoshenko beams resting on elastic foundation

2007 ◽  
Vol 29 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Nguyen Dinh Kien
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Elastic Foundation ◽  
Axial Force ◽  
Natural Frequencies ◽  
Mass Matrix ◽  
Element Formulation ◽  
Timoshenko Beams ◽  
Various Boundary Conditions ◽  
Consistent Mass Matrix

This paper presents a finite element formulation for investigating the free vibration of uniform Timoshenko beams resting on a Winkler-type elastic foundation and prestressing by axial force. Taking the effect of prestress, foundation support and shear deformation into account, a stiffness matrix for Timoshenko-type beam element is formulated using the energy method. The element consistent mass matrix is obtained from the kinetic energy using simple linear shape functions. Employing the formulated element, the natural frequencies of the beams having various boundary conditions are determined for different values of the axial force and foundation stiffness. The vibration Characteristics of the beams partially supported on the foundation are also studied and highlighted. Specially, the effects of shear deformation on the vibration frequencies of prestress beams fully and partially supported on the elastic foundation are investigated in detail.

Effect of Slenderness Ratio on the Natural Frequencies of Pre-Twisted Cantilever Beams of Uniform Rectangular Cross-Section

1971 ◽  
Vol 13 (1) ◽  
pp. 51-59 ◽  
Author(s):  
B. Dawson ◽  
N. G. Ghosh ◽  
W. Carnegie
Keyword(s):  
Differential Equations ◽  
Shear Deformation ◽  
Cross Section ◽  
Natural Frequencies ◽  
Rectangular Cross Section ◽  
Slenderness Ratio ◽  
Twist Angle ◽  
Equations Of Motion ◽  
Rotary Inertia ◽  
Cantilever Beams

This paper is concerned with the vibrational characteristics of pre-twisted cantilever beams of uniform rectangular cross-section allowing for shear deformation and rotary inertia. A method of solution of the differential equations of motion allowing for shear deformation and rotary inertia is presented which is an extension of the method introduced by Dawson (1)§ for the solution of the differential equations of motion of pre-twisted beams neglecting shear and rotary inertia effects. The natural frequencies for the first five modes of vibration are obtained for beams of various breadth to depth ratios and lengths ranging from 3 to 20 in and pre-twist angle in the range 0–90°. The results are compared with those obtained by an alternative method (2), where available, and also to experimental results.

Dynamic Stability of Timoshenko Beams by Finite Element Method

1976 ◽  
Vol 98 (4) ◽  
pp. 1145-1149 ◽  
Author(s):  
J. Thomas ◽  
B. A. H. Abbas
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stability Analysis ◽  
Shear Deformation ◽  
Dynamic Stability ◽  
Timoshenko Beam ◽  
Dynamic Instability ◽  
Rotary Inertia ◽  
Timoshenko Beams ◽  
Buckling Loads

A Finite Element model is developed for the stability analysis of Timoshenko beam subjected to periodic axial loads. The effect of the shear deformation on the static buckling loads is studied by finite element method. The results obtained show excellent agreement with those obtained by other analytical methods for the first three buckling loads. The effect of shear deformation and for the first time the effect of rotary inertia on the regions of dynamic instability are investigated. The elastic stiffness, geometric stiffness, and inertia matrices are developed and presented in this paper for a Timoshenko beam. The matrix equation for the dynamic stability analysis is derived and solved for hinged-hinged and cantilevered Timoshenko beams and the results are presented. Values of critical loads for beams with various shear parameters are presented in a graphical form. First four regions of dynamic instability for different values of rotary inertia parameters are presented. As the rotary inertia parameter increases the regions of instability get closer to each other and the width of the regions increases thus making the beam more sensitive to periodic forces.

The Dynamics of a Deformable Body Experiencing Large Displacements

1988 ◽  
Vol 55 (3) ◽  
pp. 676-680 ◽  
Author(s):  
W. P. Koppens ◽  
A. A. H. J. Sauren ◽  
F. E. Veldpaus ◽  
D. H. van Campen
Keyword(s):  
Rigid Body ◽  
Mass Matrix ◽  
Deformable Body ◽  
Equations Of Motion ◽  
Body Motion ◽  
General Description ◽  
Large Displacements ◽  
Displacement Fields ◽  
Cpu Time ◽  
Time Savings

A general description of the dynamics of a deformable body experiencing large displacements is presented. These displacements are resolved into displacements due to deformation and displacements due to rigid body motion. The former are approximated with a linear combination of assumed displacement fields. D’Alembert’s principle is used to derive the equations of motion. For this purpose, the rigid body displacements and the displacements due to deformation have to be independent. Commonly employed conditions for achieving this are reviewed. It is shown that some conditions lead to considerably simpler equations of motion and a sparser mass matrix, resulting in CPU time savings when used in a multibody program. This is illustrated with a uniform beam and a crank-slider mechanism.

scholarly journals A concise nodal-based derivation of the floating frame of reference formulation for displacement-based solid finite elements

2019 ◽  
Vol 49 (3) ◽  
pp. 291-313 ◽  
Author(s):  
Andreas Zwölfer ◽  
Johannes Gerstmayr
Keyword(s):  
Finite Element ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Frame Of Reference ◽  
Body Motion ◽  
Reference Formulation ◽  
Lumped Mass ◽  
Floating Frame Of Reference ◽  
Mass Approximation ◽  
Floating Frame

AbstractThe Floating Frame of Reference Formulation (FFRF) is one of the most widely used methods to analyze flexible multibody systems subjected to large rigid-body motion but small strains and deformations. The FFRF is conventionally derived via a continuum mechanics approach. This tedious and circuitous approach, which still attracts attention among researchers, yields so-called inertia shape integrals. These unhandy volume integrals, arising in the FFRF mass matrix and quadratic velocity vector, depend not only on the degrees of freedom, but also on the finite element shape functions. That is why conventional computer implementations of the FFRF are laborious and error prone; they require access to the algorithmic level of the underlying finite element code or are restricted to a lumped mass approximation. This contribution presents a nodal-based treatment of the FFRF to bypass these integrals. Each flexible body is considered in its spatially discretized state ab initio, wherefore the integrals are replaced by multiplications by a constant finite element mass matrix. Besides that, this approach leads to a simpler and concise but rigorous derivation of the equations of motion. The steps to obtain the inertia-integral-free equations of motion (in 2D and 3D spaces) are presented in a clear and comprehensive way; the final result provides ready-to-implement equations of motion without a lumped mass approximation, in contrast to the conventional formulation.

The Effects of Shear Deformation and Rotary Inertia on the Lateral Frequencies of Cantilever Beams in Bending

1972 ◽  
Vol 94 (1) ◽  
pp. 267-278 ◽  
Author(s):  
W. Carnegie ◽  
J. Thomas
Keyword(s):  
Finite Difference ◽  
Shear Deformation ◽  
Matrix Equation ◽  
Equations Of Motion ◽  
Flexural Vibration ◽  
Experimental Results ◽  
Rotary Inertia ◽  
Cantilever Beams ◽  
Algebraic Equations ◽  
Good Agreement

This paper deals with the effect of shear deformation and rotary inertia on the frequencies of flexural vibration of pre-twisted and non-pre-twisted uniform and tapered cantilever beams. The equations of motion are derived and transformed into a set of linear simultaneous algebraic equations by using finite-difference solutions for the derivatives. The resulting eigenvalue matrix equation is solved for the frequency parameters by a QR transformation. The effects of various tapers, depth-to-length ratios and pre-twist angles on the frequencies of vibration are investigated for the first five modes. Results obtained are compared with those presented by other investigators where available and show good agreement. The experimental results presented also show good agreement with the corresponding theoretical values.

Solving for Dynamic Problems in Flexible Manufacturing Systems

2010 ◽  
Vol 156-157 ◽  
pp. 1501-1504
Author(s):  
Yunn Lin Hwang ◽  
Shen Jenn Hwang
Keyword(s):  
Flexible Manufacturing ◽  
Flexible Manufacturing Systems ◽  
Manufacturing Systems ◽  
Equations Of Motion ◽  
Open Loop ◽  
Body Motion ◽  
Recursive Formulation ◽  
Deformable Bodies ◽  
System Equations ◽  
Time Invariant

Generally speaking, the flexible manufacturing systems can be classified into two main groups: open-loop and closed-loop systems. In this investigation, a recursive formulation is developed for the dynamic analysis of open-loop flexible manufacturing systems. The nonlinear generalized Newton-Euler equations are developed for rigid and deformable bodies that undergo large translational and rotational displacements. These equations are formulated in terms of a set of time invariant scalars, vectors and matrices that depend on the spatial coordinates as well as the assumed displacement fields, and these time invariant quantities represent the dynamic coupling between the rigid body motion and elastic deformation. The method to solve equations of motion for open-loop systems consisting of interconnected rigid and deformable bodies is presented in this paper. This method applies recursive method with the Newton-Euler method for deformable bodies to obtain a large, loosely coupled system equations of motion. The solution techniques used to solve for the system equations of motion can be more efficiently implemented in the modern computer systems. The algorithms presented in this paper are demonstrated by using cylindrical joints that can be easily extended to revolute, slider and rigid joints. The recursive formulation developed in this investigation is illustrated by a practical numerical example.

Modelling lateral manoeuvres in fish

2012 ◽  
Vol 697 ◽  
pp. 1-34 ◽  
Author(s):  
K. Singh ◽  
T. J. Pedley
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Rigid Body Dynamics ◽  
The Body ◽  
Body Motion ◽  
Dynamic State ◽  
Parameter Study ◽  
Element Formulation ◽  
Turn Angle ◽  
Impulsive Input

AbstractWe propose a method to model manoeuvres in self-propelled flexible-bodied fish by modelling the hydrodynamics coupled to the body inertia. Flexible body motion is prescribed and the equations of motion are solved for the position of the centre of mass and rotation of the body. The governing equations are formulated by applying the conservation of linear and angular momentum. Two independent methods to model the fluid dynamics are pursued: Model 1 is an extension of elongated-body theory, modified for self-propulsion and flexible motion. Model 2 applies a numerical boundary-element formulation with the fish modelled as an infinitely thin rectangular body. The manoeuvring response to an impulsive input is first examined to understand the rigid-body characteristics of the fish. A flexible bend action is included to model C-bends of the type observed during escapes in fish. Models 1 and 2 are used to cross-verify the respective implementations as well as to develop physical insights into manoeuvring. A parameter study shows that fish of intermediate body depths are best adapted to rapid turns whereas the initial dynamic state of the fish is instrumental in affecting the sign as well as the magnitude of the turn angle, for a prescribed bend deflection. Computations for combined swimming and turning show that the initial rigid-body dynamics of the fish is much more effective than the induced effect of the prior shed wake in enhancing the turning response.

FREE VIBRATIONS OF SHEAR DEFORMABLE CIRCULAR CURVED BEAMS RESTING ON ELASTIC FOUNDATIONS

2002 ◽  
Vol 02 (01) ◽  
pp. 77-97 ◽  
Author(s):  
BYOUNG KOO LEE ◽  
SANG JIN OH ◽  
KWANG KYOU PARK
Keyword(s):  
Shear Deformation ◽  
Natural Frequencies ◽  
Free Vibrations ◽  
Rotary Inertia ◽  
Transverse Shear Deformation ◽  
Curved Beams ◽  
System Parameters ◽  
Elastic Foundations ◽  
Out Of Plane ◽  
Shear Deformable

The governing differential equations for the out-of-plane, free vibration of circular curved beams resting on elastic foundations are derived and solved numerically. The formulation takes into consideration the effects of rotary inertia and transverse shear deformation. The lowest three natural frequencies are calculated for beams with hinged–hinged, hinged-clamped, and clamped–clamped end constraints. The effects of various system parameters as well as rotary inertia and shear deformation on the natural frequencies are investigated.

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