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.

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.

Rotary Inertia and Shear in Beam Vibration Treated by the Ritz Method

1968 ◽  
Vol 72 (688) ◽  
pp. 341-344 ◽  
Author(s):  
B. Dawson
Keyword(s):  
Shear Deformation ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Normal Modes ◽  
Rotary Inertia ◽  
Characteristic Functions ◽  
Cantilever Beams ◽  
Dynamic Displacement ◽  
Modes Of Vibration ◽  
Good Agreement

Summary The natural frequencies of vibration of a cantilever beam allowing for rotary inertia and shear deformation are obtained by the approximate Ritz method. The workability of the method is dependent upon the approximating functions chosen for the dynamic displacement curves. A series of characteristic functions representing the normal modes of vibration of cantilever beams in simple flexure is used as the approximating functions for both deflections due to flexure and shear deformation. Good agreement is shown between frequencies obtained by the Ritz method and those resulting from an analytical solution. The effect upon the natural frequencies of allowing for rotary inertia alone is shown and it is seen to increase rapidly with mode number.

Dynamic Analysis of the Rotating Composite Thin-Walled Cantilever Beams with Shear Deformation

2013 ◽  
Vol 690-693 ◽  
pp. 309-313
Author(s):  
Yong Sheng Ren ◽  
Qi Yi Dai
Keyword(s):  
Shear Deformation ◽  
Cross Section ◽  
Fiber Orientation ◽  
Model Comparison ◽  
Natural Frequencies ◽  
Theoretical Study ◽  
Equations Of Motion ◽  
Cantilever Beams ◽  
The Finite Element Method ◽  
Rotating Speed

This paper presents a theoretical study of the dynamic characteristics of rotating composite cantilever beams. Considering shear deformation and cross section warping, the equations of motion of the rotating cantilever beams are derived using Hamilton’s principle. The Galerkin’s method is used in order to analysis the free vibration behaviors of the model. Comparison of the theoretical solutions has been made with the results obtained from the finite element method, which prove the validity of the model presented in this paper. Natural frequencies are obtained for circular tubular composite beams. The effects of fiber orientation, rotating speed and structure parameters on modal frequencies are investigated.

Shear Deformation in Heterogeneous Anisotropic Plates

1970 ◽  
Vol 37 (4) ◽  
pp. 1031-1036 ◽  
Author(s):  
J. M. Whitney ◽  
N. J. Pagano
Keyword(s):  
Shear Deformation ◽  
Plate Theory ◽  
Flexural Vibration ◽  
Laminated Plates ◽  
Transverse Load ◽  
Laminated Plate ◽  
Anisotropic Plates ◽  
Reinforced Composite ◽  
Classical Elasticity Theory ◽  
Good Agreement

A bending theory for anisotropic laminated plates developed by Yang, Norris, and Stavsky is investigated. The theory includes shear deformation and rotary inertia in the same manner as Mindlin’s theory for isotropic homogeneous plates. The governing equations reveal that unsymmetrically laminated plates display the same bending-extensional coupling phenomenon found in classical laminated plate theory based on the Kirchhoff assumptions. Solutions are presented for bending under transverse load and for flexural vibration frequencies of symmetric and nonsymmetric lamninates. Good agreement is observed in numerical results for plate bending as compared to exact solutions obtained from classical elasticity theory. For certain fiber-reinforced composite materials, radical departure from classical laminated plate theory is indicated.

PROPAGATION AND RESONANCE OF COMPOSITE WAVES IN PRISMATIC RODS

1936 ◽  
Vol 14a (3) ◽  
pp. 66-70
Author(s):  
R. Ruedy
Keyword(s):  
Boundary Conditions ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Experimental Results ◽  
Cubic Equation ◽  
Resonance Frequencies ◽  
Good Agreement

By taking into account the three main terms of the equation of motion of the prismatic rod, there is obtained for the frequency a cubic equation which is in good agreement with the experimental results when the thickness of the rod is not negligible compared with its length but does not exceed about one-fifth of the length. It corresponds to the equation obtained for a system with three degrees of freedom.For a composite vibration consisting of a wave of dilatation and a wave of distortion in the direction of the smallest dimension of the rod, and waves of dilatation in the two other directions, the equations of motion combined with some of the boundary conditions yield another cubic equation for the resonance frequencies.

Identification and Defect Detection of Continuous Dynamic Systems

2006 ◽  
Author(s):  
A. Siami ◽  
M. Farid
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Defect Detection ◽  
Equations Of Motion ◽  
Continuous System ◽  
Least Square Method ◽  
Least Square ◽  
Continuous Systems ◽  
Algebraic Equations ◽  
Continuous Dynamic

This paper presents a systematic and efficient algorithm using a coupled finite element - finite difference - least square method for identification and defect detection of continuous system using dynamic response of such systems. First the governing partial differential equations of motion of continuous systems such as beams are reduced to a set of ordinary differential equations in time domain using finite elements. Then finite difference method is used to convert these equations into a set of algebraic equations. This set of equations is considered as a set of equality constraints of an optimization problem in which the objective function is the summation of the squares of differences between measured data at specific points and the predicted data obtained by the solution of the governing system of differential of equations. This method has been successfully applied to find mechanical properties of aforementioned systems in an iterative procedure.

Nonlinear Dynamic Modelling of a Revolute-Prismatic Flexible Composite-Material Robot Arm

1991 ◽  
Vol 113 (4) ◽  
pp. 461-468 ◽  
Author(s):  
F. Gordaninejad ◽  
A. Azhdari ◽  
N. G. Chalhoub
Keyword(s):  
Shear Deformation ◽  
Nonlinear Dynamic ◽  
Fibrous Composite ◽  
Equations Of Motion ◽  
Dynamic Modelling ◽  
Beam Theory ◽  
Rotary Inertia ◽  
Robot Arm ◽  
Nonlinear Dynamic Model ◽  
End Effector

In this work, a nonlinear dynamic model is derived to study the motion of a planar robot arm consisting of one revolute and one prismatic joint. Both links of the arm are considered to be flexible and are assumed to be constructed from either isotropic conventional metallic materials or anisotropic laminated fibrous composite materials. The model is derived based on the Timoshenko beam theory in order to account for the rotary inertia and shear deformation. In addition, a nonlinear strain-displacement field is implemented to consider the large deformation of the arm. The deflections of the links are discretized by using a shear-deformable beam finite element. The governing equations of motion are derived from Hamilton’s principle. The digital simulation studies examine the combined effects of geometric nonlinearity, rotary inertia, and shear deformation on the arm’s end effector displacements. Furthermore, effects of the fiber’s angle and material orthotropy on the end effector displacements and maximum normal bending stress are studied.

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.

Effects of Shear Deformation, Rotary Inertia, and Elasticity of the Support on the Resonance Frequencies of Short Cantilever Beams

1976 ◽  
Vol 98 (1) ◽  
pp. 79-87 ◽  
Author(s):  
V. Beglinger ◽  
U. Bolleter ◽  
W. E. Locher
Keyword(s):  
Shear Deformation ◽  
Elastic Half Space ◽  
Rotary Inertia ◽  
Finite Element Program ◽  
Influence Coefficients ◽  
Resonance Frequencies ◽  
Element Program ◽  
Turbomachinery Blades ◽  
The Individual ◽  
Good Agreement

The purpose of this paper is to contribute to the assessment of the influence of shear deformation, rotary inertia, and elasticity of the support on the resonance frequencies of short turbomachinery blades. To obtain defined test conditions and to model the stiffest possible root design, the study is made on beams and supports machined from one piece. Experimental results show good agreement with results obtained from the finite element program NASTRAN. An estimate of the combined effects obtained by taking the individual effects multiplicatively, with the effect of the elasticity of the support being computed on the basis of known static influence coefficients of the elastic half space, results in a frequency reduction larger than measured.

Nonlinear Deformation of a Shear-Deformable Laminated Composite-Material Flexible Robot Arm

1992 ◽  
Vol 114 (1) ◽  
pp. 96-102 ◽  
Author(s):  
F. Gordaninejad ◽  
N. G. Chalhoub ◽  
A. Ghazavi ◽  
Q. Lin
Keyword(s):  
Shear Deformation ◽  
Geometric Nonlinearity ◽  
General Procedure ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Element Model ◽  
Rotary Inertia ◽  
Robot Arm ◽  
Flexible Robot ◽  
Nonlinear Dynamic Model

In this work a general procedure to derive a nonlinear dynamic model for a three-link revolute flexible robot arm constructed from laminated fiber-reinforced composite materials is presented. The effects of geometric nonlinearity as well as rotary inertia and shear deformation are included to study the dynamic response of robotic manipulators made of moderately thick beams under large deformations. Hamilton’s principle is used to derive the equations of motion. A displacement finite element model based on the Timoshenko beam theory is implemented to approximate the solution. The digital simulation studies examine the combined effects of geometric nonlinearity, rotary inertia, and shear deformation on the arm’s end effector displacements. Furthermore, the effects of angle of fiber orientation and material orthotropy on the end-of-the-arm displacements and maximum normal bending stresses, are assessed.

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