Transverse Vibrations of Cantilever Beams Having Unequal Breadth and Depth Tapers

1977 ◽  
Vol 44 (4) ◽  
pp. 737-742 ◽  
Author(s):  
B. Downs
Keyword(s):  
Stress Distribution ◽  
Cross Section ◽  
Degrees Of Freedom ◽  
Dynamic Behaviour ◽  
Natural Frequencies ◽  
Distribution Patterns ◽  
Continuous Systems ◽  
Cantilever Beams ◽  
Transverse Vibrations ◽  
Discretization Technique

Natural frequencies of doubly symmetric cross section, isotropic cantilever beams, based on both Euler and Timoshenko theories, are presented for 36 combinations of linear depth and breadth taper. Results obtained by a new dynamic discretization technique include the first eight frequencies for all geometries and the stress distribution patterns for the first four (six) modes in the case of the wedge. Comparisons are drawn wherever possible with exact solutions and with other numerical results appearing in the literature. The results display outstanding accuracy and demonstrate that it is possible to model with high precision the dynamic behaviour of continuous systems by discretization on to a strictly limited number of degrees of freedom.

scholarly journals Higher-order accurate compact difference solutions for vibration problems of one dimensional continuous systems

2011 ◽  
pp. 771-794
Author(s):  
Mondher Yahiaoui
Keyword(s):  
Natural Frequencies ◽  
Continuous Systems ◽  
One Dimensional ◽  
First Order ◽  
Compact Difference ◽  
Transverse Vibrations ◽  
Seventh Order ◽  
One Step ◽  
Vibration Problems ◽  
Difference Methods

In this paper, we present a fourth-order accurate and a seventh-order accurate, one-step compact difference methods. These methods can be used to solve initial or boundaryvalue problems which can be modeled by a first-order linear system of differential equations. It is then shown in detail how these methods can be used to solve vibration problems of onedimensional continuous systems. Natural frequencies of a cantilever beam in transverse vibrations are computed and the results are compared to analytical ones to prove the high accuracy and efficiency of both methods. A comparison was also made to a finite element solution and the results have shown that both compact-difference methods yield more accurate values even with a reduced number of intervals.

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 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.

Improving Spot Weld Models in Structural Dynamics

2003 ◽  
Author(s):  
Matteo Palmonella ◽  
Michael I. Friswell ◽  
Cristinel Mares ◽  
John E. Mottershead
Keyword(s):  
Finite Element ◽  
Structural Dynamics ◽  
Degrees Of Freedom ◽  
Dynamic Behaviour ◽  
Natural Frequencies ◽  
Spot Weld ◽  
Physical Parameters ◽  
Finite Element Models ◽  
Spot Welds ◽  
Coarse Meshes

This paper gives an overview of the finite element modelling of spot welds for the analysis of the dynamic response of structures. In particular models for dynamic analysis that use coarse meshes and equivalent parameters are considered. A major requirement for these models is their accuracy in predicting the dynamic behaviour of spot welded structures despite the low number of degrees of freedom. Three different models of spot welds are investigated [1–3] and for each model physical parameters have to be assigned based on engineering insight. The aim of the present paper is to improve the accuracy of these three models by searching for the optimum values of the parameters characterising the spot weld models using experimental data. For this purpose a benchmark structure has been analysed, consisting of a thin walled hat section beam made of two plates welded together by twenty spot welds. The predicted natural frequencies and modes of the benchmark structure have been compared to the experimental modes. Updating of the finite element models has been performed and the accuracy of the three models has been significantly improved.

Dynamical Stress Distribution Analysis of a Non-Uniform Cross-Section Beam Under Moving Mass

2006 ◽  
Author(s):  
M. T. Ahmadian ◽  
E. Esmailzadeh ◽  
M. Asgari
Keyword(s):  
Stress Distribution ◽  
Cross Section ◽  
Natural Frequencies ◽  
Moving Load ◽  
Variable Cross Section ◽  
Moving Mass ◽  
Variable Cross ◽  
Distribution Analysis ◽  
Concentrated Force ◽  
Direct Integration Method

One of the engineers concern in designing bridges and structures under moving load is the uniformity of stress distribution. In this paper the analysis of a variable cross-section beam subjected to a moving concentrated force and mass is investigated. Finite element method with cubic Hermitian interpolation functions is used to model the structure based on Euler-Bernoulli beam and Wilson-Θ direct integration method is implemented to solve time dependent equations. Effects of cross-section area variation, boundary conditions, and moving mass inertia on the deflection, natural frequencies and longitudinal stresses of beam are investigated. Results indicates using a beam of parabolically varying thickness with constant mass can decrease maximum deflection and stresses along the beam while increasing natural frequencies of the beam. The effect of moving mass inertia of moving load is found to be significant at high velocity.

Energy Transfer From High-to Low-Frequency Modes in a Flexible Structure via Modulation

1994 ◽  
Vol 116 (2) ◽  
pp. 203-207 ◽  
Author(s):  
S. A. Nayfeh ◽  
A. H. Nayfeh
Keyword(s):  
Cross Section ◽  
High Frequency ◽  
Natural Frequencies ◽  
Circular Cross Section ◽  
Low Frequency ◽  
Cantilever Beams ◽  
Axially Symmetric ◽  
Internal Resonances ◽  
Planar Motions ◽  
High Frequency Modes

An experimental study of the response of axially-symmetric (i.e., circular cross-section) cantilever beams to planar external excitations is presented. Because of the axial symmetry, one-to-one internal resonances occur at each natural frequency. These resonances cause the planar motions to lose stability and nonplanar (whirling) motions are observed. Under certain conditions, periodically-and chaotically-modulated motions may occur. In addition, when the beam is excited near one of its high natural frequencies, large first-mode responses accompanied by slow modulations of the amplitudes and phases of high-frequency modes are observed. This interaction between high-and low-frequency modes may be extremely dangerous because the amplitudes of the responses of the low-frequency modes can be very large compared with those of the directly excited high-frequency modes.

Analytical and Experimental Free Vibration Analysis of Uniform Cracked Thick Beam on Two-Parameter Elastic Partial Foundation

Volume 1 ◽  
2004 ◽  
Author(s):  
Ali Bahc¸ıvan ◽  
Vedat Karadag˘
Keyword(s):  
Free Vibration ◽  
Elastic Foundation ◽  
Cross Section ◽  
Vibration Analysis ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Beam Element ◽  
Thick Beam ◽  
Two Parameter

In this study, the analytical and experimental free vibration analysis of rectangular cross-section uniform cracked thick beam on two-parameter vibration and noise isolating elastic foundation, considering shear deformation and rotatory inertia is made by the finite element method. The beam element in our study is a recently introduced 4 degrees of freedom thick beam element and has two nodes with two degrees of freedom at each node such as transverse displacements and cross-section rotations. Two kinds of end conditions, i.e. clamped-clamped and clamped-free ends are considered in this study. Axial displacement of the beam is also considered. For axial displacement of the beam, linear finite elements are used. The elastic foundation is idealized as a constant two-parameter model characterized by two moduli, i.e. the Winkler foundation modulus k and the shear foundation modulus kG. In the case kG = 0, this model reduces to the Winkler model, i.e. the elastic foundation is idealized as a constant one-parameter model. The effects of foundation stiffness parameters, partial elastic foundation and crack changing its depth on the natural frequencies of the beam are examined. The effect of partial elastic foundation on the natural frequencies of the beam is examined for only half of the beam length. The crack is in the middle of the beam and only on one side of the beam having a form of open crack. In the analytical analysis, the spring coefficients of the crack are calculated in the computer program and then directly added to the stiffness matrix. The crack model used in this study is mentioned as a linear spring model in the literature. The crack modeled is in the middle of the beam and the related spring constants of rotational and extensional springs, which will be used, are added to the global matrix in the process. In the experimental analysis, steel and hard plastic beam are used as the beam material. Moreover, sponge and glass wool, which are manufactured by Petkim Ltd., are used as the isolating elastic foundation material. The results obtained from the analytical and experimental studies are presented by showing in tables and graphs and their importance in design is discussed. The analytical and experimental results and comparisons show the efficiency and effectiveness of the proposed method.

Longitudinal oscillation of a rod with a variable cross section

2019 ◽  
Vol 14 (2) ◽  
pp. 138-141
Author(s):  
I.M. Utyashev
Keyword(s):  
Cross Section ◽  
Natural Frequencies ◽  
Liouville Problem ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Longitudinal Vibrations ◽  
Cross Sectional ◽  
Independent Functions ◽  
The Cross ◽  
The Right

Variable cross-section rods are used in many parts and mechanisms. For example, conical rods are widely used in percussion mechanisms. The strength of such parts directly depends on the natural frequencies of longitudinal vibrations. The paper presents a method that allows numerically finding the natural frequencies of longitudinal vibrations of an elastic rod with a variable cross section. This method is based on representing the cross-sectional area as an exponential function of a polynomial of degree n. Based on this idea, it was possible to formulate the Sturm-Liouville problem with boundary conditions of the third kind. The linearly independent functions of the general solution have the form of a power series in the variables x and λ, as a result of which the order of the characteristic equation depends on the choice of the number of terms in the series. The presented approach differs from the works of other authors both in the formulation and in the solution method. In the work, a rod with a rigidly fixed left end is considered, fixing on the right end can be either free, or elastic or rigid. The first three natural frequencies for various cross-sectional profiles are given. From the analysis of the numerical results it follows that in a rigidly fixed rod with thinning in the middle part, the first natural frequency is noticeably higher than that of a conical rod. It is shown that with an increase in the rigidity of fixation at the right end, the natural frequencies increase for all cross section profiles. The results of the study can be used to solve inverse problems of restoring the cross-sectional profile from a finite set of natural frequencies.

scholarly journals Numerical study on the influence of acoustic natural frequencies on the dynamic behaviour of submerged and confined disk-like structures

2017 ◽  
Vol 73 ◽  
pp. 53-69 ◽  
Author(s):  
Matias Bossio ◽  
David Valentín ◽  
Alexandre Presas ◽  
David Ramos Martin ◽  
Eduard Egusquiza ◽  
...  
Keyword(s):  
Dynamic Behaviour ◽  
Natural Frequencies ◽  
Numerical Study

scholarly journals STRESS DISTRIBUTION IN CROSS-SECTION OF RC COLUMN

2009 ◽  
Vol 74 (637) ◽  
pp. 425-431
Author(s):  
Ippei MARUYAMA ◽  
Masahiro SUZUKI ◽  
Masaomi TESHIGAWARA ◽  
Ryoichi SATO
Keyword(s):  
Stress Distribution ◽  
Cross Section ◽  
Rc Column
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