Exact Solutions for Free Vibrations and Buckling of Double Tapered Columns With Elastic Foundation and Tip Mass

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
R. D. Firouz-Abadi ◽  
M. Rahmanian ◽  
M. Amabili
Keyword(s):  
Power Series ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Free Vibrations ◽  
Power Series Method ◽  
Tapered Columns ◽  
Tip Mass ◽  
Insight Into ◽  
Recursive Relations ◽  
Good Agreement

The present study aims at the free vibration analysis of double tapered columns. Foundation is assumed to be elastic and the effects of self-weight and tip mass with significant moment of inertia are considered. The governing equation of motion is obtained using the Hamilton principle, based on both the Euler–Bernoulli and Timoshenko beam models. Applying the power series method of Frobenius, the base solutions of the governing equations are obtained in the form of a power series via general recursive relations. Applying the boundary conditions, the natural frequencies of the beam/column are obtained using both models. The obtained results are compared with literature and a very good agreement is achieved. Subsequently, comprehensive studies are performed to provide an insight into the variation of the natural frequencies and instability conditions of the beam with respect to the tip mass, self-weight, taper ratio, slenderness, and foundation stiffness and eventually some general conclusions are drawn.

Free Vibrations of Beams on Viscoelastic Pasternak Foundations

2015 ◽  
Vol 744-746 ◽  
pp. 1624-1627
Author(s):  
Li Peng ◽  
Ying Wang
Keyword(s):  
Natural Frequencies ◽  
Free Vibrations ◽  
Mode Analysis ◽  
Analysis Method ◽  
Governing Equations ◽  
Complex Mode ◽  
Transverse Vibrations ◽  
Quadrature Methods ◽  
Foundation Parameters ◽  
Good Agreement

This paper investigates free transverse vibrations of finite Euler–Bernoulli beams resting on viscoelastic Pasternak foundations. The differential quadrature methods (DQ) are applied directly to the governing equations of the free vibrations. Under the simple supported boundary condition, the natural frequencies of the transverse vibrations are calculated, and compared with the results of the complex mode analysis method. The numerical results obtained by using the DQ and the complex mode methods are in good agreement for the first seven order natural frequencies, but with the growth of the orders, the small quantitative differences between them increase. The effects of the foundation parameters on the natural frequencies are also studied in numerical examples.

scholarly journals Green’s function approach to frequency analysis of thin circular plates

2016 ◽  
Vol 64 (1) ◽  
pp. 181-188
Author(s):  
K.K. Żur
Keyword(s):  
Power Series ◽  
Free Vibration ◽  
Influence Function ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Thin Plates ◽  
Circular Plates ◽  
Natural Modes ◽  
Good Agreement ◽  
Analytical Frequency

Abstract The free vibration analysis of homogeneous and isotropic circular thin plates by using the Green’s functions is considered. The formulae for construction of the influence function for all nodal diameters are presented in a closed form. The limited independent solutions of differential Euler equations were expanded in the Neumann power series using the method of successive approximation. This approach allows to obtain the analytical frequency equations as power series rapidly convergent to exact eigenvalues for different number of nodal diameters. The first ten dimensionless frequencies for eight different natural modes of circular plates are calculated. A part of obtained results have not been presented yet in open literature for thin circular plates. The results of investigation are in good agreement with selected results obtained by other methods presented in literature.

scholarly journals An Accurate Estimation of the Eigenfrequency of an Offshore Wind Turbine Considered as the Stepped Euler-Bernoulli Beam in Three-Spring Flexible Foundation Using the Power Series Method

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Mouafo Teifouet Armand Robinson ◽  
Zhenyu Wang
Keyword(s):  
Power Series ◽  
Wind Turbine ◽  
Wall Thickness ◽  
Natural Frequencies ◽  
Offshore Wind ◽  
Series Method ◽  
Bernoulli Beam ◽  
Power Series Method ◽  
Flexible Foundation ◽  
Euler Bernoulli Beam

The present study employs the power series method (PSM) to accurately predict the natural frequencies of eleven offshore wind turbines (OWT). This prediction is very important as it helps in the quick verification of experimental or finite element results. This study idealizes the OWT as a stepped Euler-Bernoulli beam carrying a top mass and connected at its bottom to a flexible foundation. The first part of the beam represents a monopile and the transition piece while its second part is a tower. The foundation is modeled using three springs (lateral, rotational, and cross-coupling springs). This work’s aim is at improving therefore the previous researches, in which the whole wind turbine was taken as a single beam, with a tower being tapered and its wall thickness being negligible compared to its diameter. In order to be closer to real-life OWT, three profiles of the tapered tower are explored: case 1 considers a tower with constant thickness along its height. Case 2 assumes a tower’s thickness being negligible compared to its mean diameter, while case 3 describes the tower as a tapered beam with varying thickness along its height. Next, the calculated natural frequencies are compared to those obtained from measurements. Results reveal that case 2, used by previous researches, was only accurate for OWT with tower wall thickness lower than 15 mm. Frequencies produced in case 3 are the most accurate as the relative error is up to 0.01%, especially for the OWT with thicknesses higher or equal to 15 mm. This case appears to be more realistic as, practically, wall thickness of a wind tower varies with its height. The tower-to-pile thickness ratio is an important design parameter as it highly has impact on the natural frequency of OWT, and must therefore be taken into account during the design as well as lateral and rotational coupling springs.

A Homogenization Procedure for Nonlinear Free Vibration Analysis of Functionally Graded Beams Resting on Nonlinear Elastic Foundations

2012 ◽  
Vol 232 ◽  
pp. 427-431
Author(s):  
Ahmed Zerkane ◽  
Khalid El Bikri ◽  
Rhali Benamar
Keyword(s):  
Geometrical Nonlinearity ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Free Vibrations ◽  
Functionally Graded ◽  
Axial Stiffness ◽  
Nonlinear Frequency ◽  
Homogenization Procedure ◽  
Nonlinear Elastic ◽  
Good Agreement

The present work deals with a homogenization procedure (HP), which is developed to reduce the problem of geometrically nonlinear free vibrations of functionally graded beams (FGB) resting on elastic nonlinear foundation with immovable ends to that of isotropic homogeneous beams with effective bending stiffness and axial stiffness parameters. The material properties of the functionally graded composites examined are assumed to be graded in the thickness direction and estimated through the rule of mixture. The theoretical model is based on the Euler-Bernouilli beam theory and the Von Kármán geometrical nonlinearity assumptions. Hamilton’s principle is applied and a multimode approach is derived to calculate the fundamental nonlinear frequency parameters, which are found to be in a good agreement with the published results.

scholarly journals Free Vibrations of a Reddy-Bickford Multi-Span Beam Carrying Multiple Spring-Mass Systems

2011 ◽  
Vol 18 (5) ◽  
pp. 709-726 ◽  
Author(s):  
Yusuf Yesilce
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Structural Elements ◽  
Mass System ◽  
Assembly Technique

The structural elements supporting motors or engines are frequently seen in technological applications. The operation of machine may introduce additional dynamic stresses on the beam. It is important, then, to know the natural frequencies of the coupled beam-mass system, in order to obtain a proper design of the structural elements. The literature regarding the free vibration analysis of Bernoulli-Euler and Timoshenko single-span beams carrying a number of spring-mass system and multi-span beams carrying multiple spring-mass systems are plenty, but the free vibration analysis of Reddy-Bickford multi-span beams carrying multiple spring-mass systems has not been investigated by any of the studies in open literature so far. This paper aims at determining the exact solutions for the natural frequencies and mode shapes of Reddy-Bickford beams. The model allows analyzing the influence of the shear effect and spring-mass systems on the dynamic behavior of the beams by using Reddy-Bickford Beam Theory (RBT). The effects of attached spring-mass systems on the free vibration characteristics of the 1–4 span beams are studied. The natural frequencies of Reddy-Bickford single-span and multi-span beams calculated by using the numerical assembly technique and the secant method are compared with the natural frequencies of single-span and multi-span beams calculated by using Timoshenko Beam Theory (TBT); the mode shapes are presented in graphs.

Multistage Coupling of Eight Mistuned Bladed Disk on a Solid Shaft: Part 1—Free Vibration Analysis

2012 ◽  
Author(s):  
Romuald Rzadkowski ◽  
Artur Maurin
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Free Vibrations ◽  
Rotor Blades ◽  
Mode Shapes ◽  
Bladed Disk ◽  
Centrifugal Stiffening

Considered here was the effect of multistage coupling on the dynamics of a rotor consisting of eight mistuned bladed discs on a solid shaft. Each bladed disc had a different number of rotor blades. Free vibrations were examined using finite element representations of rotating single blades, bladed discs, and the entire rotor. In this study the global rotating mode shapes of eight flexible mistuned bladed discs on shaft assemblies were calculated, taking into account rotational effects such as centrifugal stiffening. The thus obtained natural frequencies of the blade, shaft, bladed disc and entire shaft with discs were carefully examined to discover resonance conditions and coupling effects. This study found that mistuned systems cause far more intensive multistage coupling than tuned ones. The greater the mistuning, the more intense the multistage coupling.

Free Vibration Analysis of Angle-ply Laminated Shallow Cylindrical Shell with Clamped Edges

2000 ◽  
Vol 123 (2) ◽  
pp. 188-197 ◽  
Author(s):  
Kenji Hosokawa ◽  
Minehiro Murayama ◽  
Toshiyuki Sakata
Keyword(s):  
Cylindrical Shell ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Numerical Approach ◽  
Free Vibrations ◽  
Numerical Results ◽  
Mode Shapes ◽  
Shallow Cylindrical Shell ◽  
Plastic Composite ◽  
Vibration Tests

In a previous paper, the authors proposed a numerical approach for analyzing the free vibrations of a laminated FRP (fiber reinforced plastic) composite plate. In the present paper, this approach is modified for application to a symmetrically laminated shallow cylindrical shell having a rectangular planform. First, the natural frequencies of the shell are calculated for discussion of the convergence and accuracy of the solution. Next, the effects of the curvature ratio and stacking sequence on the natural frequencies and mode shapes of the shell are studied. Furthermore, to justify the numerical results, vibration tests of the clamped symmetrically laminated shallow cylindrical shell having a square planform are carried out. These experimental results are found to agree well with the numerical results computed using the measured material properties of the lamina.

On the Free Vibration Analysis of a Sandwich Beam With Tip Mass

2018 ◽  
Author(s):  
E. F. Joubaneh ◽  
O. R. Barry
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Sandwich Beam ◽  
Free Vibration Analysis ◽  
Design Parameters ◽  
Governing Equations ◽  
Tip Mass

This paper presents the free vibration analysis of a sandwich beam with a tip mass using higher order sandwich panel theory (HSAPT). The governing equations of motion and boundary conditions are obtained using Hamilton’s principle. General Differential Quadrature (GDQ) is employed to solve the system governing equations of motion. The natural frequencies and mode shapes of the system are presented and Ansys simulation is performed to validate the results. Various boundary conditions are also employed to examine the natural frequencies of the sandwich beam without tip mass and the results are compared with those found in the literature. Parametric studies are conducted to examine the effect of key design parameters on the natural frequencies of the sandwich beam with and without tip mass.

scholarly journals An Investigation on the Nonlinear Free Vibration Analysis of Beams with Simply Supported Boundary Conditions Using Four Engineering Theories

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
R. A. Jafari-Talookolaei ◽  
M. H. Kargarnovin ◽  
M. T. Ahmadian ◽  
M. Abedi
Keyword(s):  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Free Vibrations ◽  
Geometrically Nonlinear ◽  
Timoshenko Beams ◽  
Analysis Method ◽  
Nonlinear Free Vibration ◽  
End Conditions ◽  
Homotopy Analysis ◽  
Transverse Deflection

The objective of this study is to present a brief survey on the geometrically nonlinear free vibrations of the Bernoulli-Euler, the Rayleigh, shear, and the Timoshenko beams with simple end conditions using the Homotopy Analysis Method (HAM). Expressions for the natural frequencies, the transverse deflection, postbuckling load-deflection relation to, and critical buckling load are presented. The results of nonlinear analysis are validated with the published results, and excellent agreement is observed. The effects of some parameters, such as slender ratio, the rotary inertia, and the shear deformation, are examined as other parameters are fixed.

Free Vibrations of Spinning Beams Under Nonclassical Boundary Conditions Using Adomian Modified Decomposition Method

2014 ◽  
Vol 14 (07) ◽  
pp. 1450027 ◽  
Author(s):  
Qibo Mao
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Decomposition Method ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Equation System ◽  
Free Vibrations ◽  
Arbitrary Boundary ◽  
Various Boundary Conditions ◽  
Straightforward Method

This study employs the Adomian modified decomposition method (AMDM) for the dynamic analysis of Euler–Bernoulli beams spinning about their longitudinal axes under various boundary conditions. Based on the AMDM, the governing differential equations for the spinning beam become a recursive algebraic equation system. By using the boundary condition equations, the natural frequencies can be readily obtained. The computed results under different classical and nonclassical boundary conditions as well as spinning speeds are presented. The accuracy is assured from comparison with published results. It is shown that the AMDM offers an accurate and straightforward method of free vibration analysis of spinning beams with arbitrary boundary conditions.

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