Vibration Analysis of Continuous Systems by Dynamic Discretization

1980 ◽  
Vol 102 (2) ◽  
pp. 391-398 ◽  
Author(s):  
B. Downs
Keyword(s):  
Power Series ◽  
Shear Deformation ◽  
Vibration Analysis ◽  
Dynamic Properties ◽  
Mass Matrix ◽  
Continuous System ◽  
Mode Shapes ◽  
Continuous Systems ◽  
Stress Distributions ◽  
Equivalent Mass

An equivalent mass matrix may be defined, for a segment of a continuous system, as one which retains precisely the dynamic properties of the original segment in discretized form. Dynamic Discretization, which makes use of a particular form of Stodola iteration, progressively generates the equivalent mass matrix in ascending powers of frequency squared, whilst simultaneously generating deformation functions in a similar power series. The method is quasi-static and readily copes with shear deformation, rotary inertia and quite complex segment geometry. Accurate vibration analysis in terms of frequencies, mode shapes and corresponding stress distributions is achieved using an extremely coarse system subdivision for a variety of geometries.

Non-Dimensional Vibration of Timoshenko Beams with Axial Loads

2011 ◽  
Vol 368-373 ◽  
pp. 1577-1582
Author(s):  
Yi Zhang ◽  
Li Chen ◽  
Guo Liang Cheng ◽  
Qing Hua Li
Keyword(s):  
Mass Matrix ◽  
Separation Of Variables ◽  
Beam Theory ◽  
Mode Shapes ◽  
Timoshenko Beams ◽  
Axial Loads ◽  
Load Vector ◽  
Equivalent Mass ◽  
Axially Loaded ◽  
Equivalent Load

The paper is supposed to establish a non-dimensional approach to analyze the free vibration of axially loaded beams with different kinds of boundary conditions, based on the Timoshenko’s beam theory and the method of separation of variables. When the mode-shapes and frequencies are known, it is convenient to get the equivalent load vector, equivalent mass matrix and the frequencies matrix, then, the characteristics of vibration of axially loaded beams will be obtained by the approach. The results and conclusions were also helpful to the anti-blast analysis of columns

A two-dimensional free vibration analysis of functionally graded sandwich beams under thermal environment

2012 ◽  
Vol 226 (12) ◽  
pp. 2860-2873 ◽  
Author(s):  
AR Setoodeh ◽  
M Ghorbanzadeh ◽  
P Malekzadeh
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Vibration Analysis ◽  
Thermal Environment ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Geometrical Parameters ◽  
Sandwich Beams ◽  
Two Dimensional

In this article, free vibration analysis of elastically supported sandwich beams with functionally graded face sheets subjected to thermal environment is presented. In order to accurately include the transverse shear deformation and the inertia effects, two-dimensional elasticity theory is used to formulate the problem. The layerwise theory in conjunction with the differential quadrature method is employed to discretize the governing equations in the thickness and axial directions, respectively. The material properties of functionally graded face sheets are assumed to be temperature-dependent and graded in the thickness direction according to a power-law distribution. For the purpose of comparison, the problem under consideration is also solved using two-dimensional finite element method and the first-order shear deformation theory. The accuracy, convergence, and versatility of the method are demonstrated by comparing the results with those of the two aforementioned approaches and also with the existing solutions in literature. Eventually, some new numerical results are presented which depict the effects of different material and geometrical parameters on natural frequencies and mode shapes of the beam.

Finite Element Analysis on Modal Parameters of Anisotropic Laminated Plates

1988 ◽  
Vol 110 (4) ◽  
pp. 473-477 ◽  
Author(s):  
C. Z. Xiao ◽  
D. X. Lin ◽  
F. Ju
Keyword(s):  
Finite Element ◽  
Shear Deformation ◽  
Transverse Shear ◽  
Natural Frequencies ◽  
Dynamic Properties ◽  
Shear Deformation Theory ◽  
Laminated Plates ◽  
Mode Shapes ◽  
Transverse Shear Deformation ◽  
Element Technique

This paper is concerned with the finite element technique for predicting the dynamic properties of anisotropic fiber-reinforced composite laminated plates. Considering the effect of transverse shear deformation, a higher order shear deformation theory which satisifes the zero shear stress conditions at the upper and bottom surfaces is assumed. The natural frequencies and mode shapes of a rectangular plate with all free edges are obtained by finite element method and the modal damping values by finite damped element technique. An equivalent stiffness method is introduced to reduce computation time. Four different theoretical predictions of natural frequencies and damped values of a laminated plate are compared with experimental results. Discussions on the effect of transverse shear deformation to the dynamic properties of composite plates are given.

scholarly journals Identification of Mode-shapes and Eigen-frequencies of Bi-hinge Beam with Distributed Mass and Stiffness

2020 ◽  
Vol 9 (3) ◽  
pp. 119-126
Author(s):  
Triantafyllos K. Makarios
Keyword(s):  
Differential Equation ◽  
Mass Matrix ◽  
Degree Of Freedom ◽  
Local Network ◽  
Mode Shapes ◽  
Specific Mass ◽  
Concentrated Mass ◽  
Equivalent Mass ◽  
Eigen Frequencies ◽  
Three Degree Of Freedom

 In the present paper, an equivalent Three Degree of Freedom (DoF) system of a bi-hinge beam, which has infinity number of degree of freedoms because possesses distributed mass and stiffness along its length, is presented. Based on the vibration partial differential equation of the abovementioned bi-hinge beam, an equivalent, mathematically, three-degree of freedom system, where the equivalent mass matrix is analytically formulated with reference on specific mass locations. Using the Three DoF model, the first three fundamental mode-shapes of the real beam can be identified. Furthermore, taking account the 3x3 mass matrix, it is possible to estimate the possible beam damages using a known technique of identification mode-shapes via records of response accelerations. Moreover, the way of instrumentation with a local network by three accelerometers is shown. It is worth noting this technique can be applied on bridges consist of bays with two hinges at its end sections, supported on elastometallic bearings, where the sense of concentrated mass is fully absent from the beam.

Effective Modelling of Sticking Motions in Vibro-Impacting Continuous Systems

2010 ◽  
Author(s):  
Chandrika P. Vyasarayani ◽  
Sukhpreet Singh Sandhu ◽  
John McPhee
Keyword(s):  
Numerical Simulations ◽  
Closed Form ◽  
Lagrange Multiplier ◽  
Continuous System ◽  
Coefficient Of Restitution ◽  
Mode Switching ◽  
Mode Shapes ◽  
Continuous Systems ◽  
Switching Method

In this work we propose a new methodology to simulate sticking motions occurring in vibro-impacting continuous systems. We have developed this method in the framework of the coefficient of restitution approach. During the sticking phase, the sticking constraints are imposed exactly using a Lagrange multiplier, which represents the reaction between the continuous system and the obstacle. The expression for the Lagrange multiplier is developed in closed form. The proposed method does not require the mode-shapes during the sticking phase, unlike the mode-switching method. The developed methodology is supported by illustrative numerical simulations.

The Use of Modan 3D in Experimental Modal Analysis

2013 ◽  
Vol 486 ◽  
pp. 36-41 ◽  
Author(s):  
Róbert Huňady ◽  
František Trebuňa ◽  
Martin Hagara ◽  
Martin Schrötter
Keyword(s):  
Modal Analysis ◽  
Vibration Analysis ◽  
High Speed ◽  
Mechanical Vibration ◽  
Steel Sample ◽  
Experimental Modal Analysis ◽  
Image Correlation ◽  
Optical Methods ◽  
Mode Shapes ◽  
Vibration Modes

Experimental modal analysis is a relatively young part of dynamics, which deals with the vibration modes identification of machines or their parts. Its development has started since the beginning of the eighties, when the computers hardware equipment has improved and the fast Fourier transform (FFT) could be used for the results determination. Nowadays it provides an uncountable set of vibration analysis possibilities starting with conventional contact transducers of acceleration and ending with modern noncontact optical methods. In this contribution we mention the use of high-speed digital image correlation by experimental determination of mode shapes and modal frequencies. The aim of our work is to create a program application called Modan 3D enabling the performing of experimental modal analysis and operational modal analysis. In this paper the experimental modal analysis of a thin steel sample performed with Q-450 Dantec Dynamics is described. In Modan 3D the experiment data were processed and the vibration modes were determined. The reached results were verified by PULSE modulus specialized for mechanical vibration analysis.

Measured natural periods of concrete shear wall buildings: insights for the design of Canadian buildings

2012 ◽  
Vol 39 (8) ◽  
pp. 867-877 ◽  
Author(s):  
Damien Gilles ◽  
Ghyslaine McClure
Keyword(s):  
Seismic Analysis ◽  
Dynamic Properties ◽  
Response Spectrum ◽  
Shear Wall ◽  
Mode Shapes ◽  
Design Stage ◽  
Base Shear ◽  
Period Equation ◽  
Input Load ◽  
Structural Engineers

Structural engineers routinely use rational dynamic analysis methods for the seismic analysis of buildings. In linear analysis based on modal superposition or response spectrum approaches, the overall response of a structure (for instance, base shear or inter-storey drift) is obtained by combining the responses in several vibration modes. These modal responses depend on the input load, but also on the dynamic characteristics of the building, such as its natural periods, mode shapes, and damping. At the design stage, engineers can only predict the natural periods using eigenvalue analysis of structural models or empirical equations provided in building codes. However, once a building is constructed, it is possible to measure more precisely its dynamic properties using a variety of in situ dynamic tests. In this paper, we use ambient motions recorded in 27 reinforced concrete shear wall (RCSW) buildings in Montréal to examine how various empirical models to predict the natural periods of RCSW buildings compare to the periods measured in actual buildings under ambient loading conditions. We show that a model in which the fundamental period of RCSW buildings varies linearly with building height would be a significant improvement over the period equation proposed in the 2010 National Building Code of Canada. Models to predict the natural periods of the first two torsion modes and second sway modes are also presented, along with their uncertainty.

Flexural Vibration of Prestressed Concrete Bridges with Corrugated Steel Webs

2017 ◽  
Vol 17 (02) ◽  
pp. 1750023 ◽  
Author(s):  
Xia-Chun Chen ◽  
Zhen-Hu Li ◽  
Francis T. K. Au ◽  
Rui-Juan Jiang
Keyword(s):  
Finite Element ◽  
Shear Deformation ◽  
Prestressed Concrete ◽  
Natural Frequencies ◽  
Sandwich Beam ◽  
Flexural Vibration ◽  
Concrete Bridges ◽  
Mode Shapes ◽  
Beam Model ◽  
Prestressed Concrete Bridges

Prestressed concrete bridges with corrugated steel webs have emerged as a new form of steel-concrete composite bridges with remarkable advantages compared with the traditional ones. However, the assumption that plane sections remain plane may no longer be valid for such bridges due to the different behavior of the constituents. The sandwich beam theory is extended to predict the flexural vibration behavior of this type of bridges considering the presence of diaphragms, external prestressing tendons and interaction between the web shear deformation and flange local bending. To this end, a [Formula: see text] beam finite element is formulated. The proposed theory and finite element model are verified both numerically and experimentally. A comparison between the analyses based on the sandwich beam model and on the classical Euler–Bernoulli and Timoshenko models reveals the following findings. First of all, the extended sandwich beam model is applicable to the flexural vibration analysis of the bridges considered. By letting [Formula: see text] denote the square root of the ratio of equivalent shear rigidity to the flange local flexural rigidity, and L the span length, the combined parameter [Formula: see text] appears to be more suitable for considering the diaphragm effect and the interaction between the shear deformation and flange local bending. The diaphragms have significant effect on the flexural natural frequencies and mode shapes only when the [Formula: see text] value of the bridge falls below a certain limit. For a bridge with an [Formula: see text] value over a certain limit, the flexural natural frequencies and mode shapes obtained from the sandwich beam model and the classical Euler–Bernoulli and Timoshenko models tend to be the same. In such cases, either of the classical beam theories may be used.

scholarly journals Green’s Function In Free Axisymmetric Vibration Analysis Of Annular Thin Plates With Different Boundary Conditions

2015 ◽  
Vol 20 (4) ◽  
pp. 939-951
Author(s):  
K.K. Żur
Keyword(s):  
Power Series ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Thin Plates ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Axisymmetric Vibration ◽  
Annular Plates ◽  
The Core ◽  
Analytical Frequency

Abstract Free vibration analysis of homogeneous and isotropic annular thin plates by using Green’s functions is considered. The formula of the influence function for uniform thin circular and annular plates is presented in closed-form. The limited independent solutions of differential Euler equation were expanded in the Neumann power series based on properties of integral equations. The analytical frequency equations as power series were obtained using the method of successive approximations. The natural axisymmetric frequencies for singularities when the core radius approaches zero are calculated. The results are compared with selected results presented in the literature.

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