Free Vibration of the Cracked Non-uniform Beam with Cross Section Varying as Polynomial Functions

2018 ◽  
Vol 22 (11) ◽  
pp. 4530-4546 ◽  
Author(s):  
Guojin Tan ◽  
Yang Liu ◽  
Yafeng Gong ◽  
Yangfan Shen ◽  
Ziyu Liu
Keyword(s):  
Free Vibration ◽  
Cross Section ◽  
Polynomial Functions ◽  
Uniform Beam

scholarly journals Isospectrals of non-uniform Rayleigh beams with respect to their uniform counterparts

2018 ◽  
Vol 5 (2) ◽  
pp. 171717 ◽  
Author(s):  
Srivatsa Bhat K ◽  
Ranjan Ganguli
Keyword(s):  
Boundary Condition ◽  
Cross Section ◽  
Natural Frequencies ◽  
Rectangular Cross Section ◽  
Beam Equation ◽  
The Finite Element Method ◽  
Rayleigh Beam ◽  
Uniform Beam ◽  
Governing Differential Equation ◽  
The Given

In this paper, we look for non-uniform Rayleigh beams isospectral to a given uniform Rayleigh beam. Isospectral systems are those that have the same spectral properties, i.e. the same free vibration natural frequencies for a given boundary condition. A transformation is proposed that converts the fourth-order governing differential equation of non-uniform Rayleigh beam into a uniform Rayleigh beam. If the coefficients of the transformed equation match with those of the uniform beam equation, then the non-uniform beam is isospectral to the given uniform beam. The boundary-condition configuration should be preserved under this transformation. We present the constraints under which the boundary configurations will remain unchanged. Frequency equivalence of the non-uniform beams and the uniform beam is confirmed by the finite-element method. For the considered cases, examples of beams having a rectangular cross section are presented to show the application of our analysis.

Free Vibration of Thin-Walled Open Section Beams With Unconstrained Damping Treatment

1981 ◽  
Vol 48 (1) ◽  
pp. 169-173 ◽  
Author(s):  
S. Narayanan ◽  
J. P. Verma ◽  
A. K. Mallik
Keyword(s):  
Free Vibration ◽  
Cross Section ◽  
Transverse Vibration ◽  
Natural Frequencies ◽  
Vibration Characteristics ◽  
Numerical Results ◽  
Simply Supported ◽  
Clamped Beams ◽  
Thin Walled ◽  
Open Cross Section

Free-vibration characteristics of a thin-walled, open cross-section beam, with unconstrained damping layers at the flanges, are investigated. Both uncoupled transverse vibration and the coupled bending-torsion oscillations, of a beam of a top-hat section, are considered. Numerical results are presented for natural frequencies and modal loss factors of simply supported and clamped-clamped beams.

scholarly journals Phononic Band Gap and Free Vibration Analysis of Fluid-Conveying Pipes with Periodically Varying Cross-Section

2021 ◽  
Vol 11 (21) ◽  
pp. 10485
Author(s):  
Hao Yu ◽  
Feng Liang ◽  
Yu Qian ◽  
Jun-Jie Gong ◽  
Yao Chen ◽  
...  
Keyword(s):  
Free Vibration ◽  
Band Gap ◽  
Cross Section ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Critical Parameters ◽  
Mode Shapes ◽  
Beam Model ◽  
The Impact ◽  
Fluid Conveying Pipes

Phononic crystals (PCs) are a novel class of artificial periodic structure, and their band gap (BG) attributes provide a new technical approach for vibration reduction in piping systems. In this paper, the vibration suppression performance and natural properties of fluid-conveying pipes with periodically varying cross-section are investigated. The flexural wave equation of substructure pipes is established based on the classical beam model and traveling wave property. The spectral element method (SEM) is developed for semi-analytical solutions, the accuracy of which is confirmed by comparison with the available literature and the widely used transfer matrix method (TMM). The BG distribution and frequency response of the periodic pipe are attained, and the natural frequencies and mode shapes are also obtained. The effects of some critical parameters are discussed. It is revealed that the BG of the present pipe system is fundamentally induced by the geometrical difference of the substructure cross-section, and it is also related to the substructure length and fluid–structure interaction (FSI). The number of cells does not contribute to the BG region, while it has significant effects on the amplitude attenuation, higher order natural frequencies and mode shapes. The impact of FSI is more evident for the pipes with smaller numbers of cells. Moreover, compared with the conventional TMM, the present SEM is demonstrated more effective for comprehensive analysis of BG characteristics and free vibration of PC dynamical structures.

Free vibration of a toroidal shell with elliptical cross-section

1989 ◽  
Vol 135 (3) ◽  
pp. 411-425 ◽  
Author(s):  
G. Yamada ◽  
Y. Kobayashi ◽  
Y. Ohta ◽  
S. Yokota
Keyword(s):  
Free Vibration ◽  
Cross Section ◽  
Elliptical Cross Section ◽  
Toroidal Shell ◽  
Elliptical Cross

Free Vibration Response of Thin and Thick Nonhomogeneous Shells by Refined One-Dimensional Analysis

2014 ◽  
Vol 136 (6) ◽  
Author(s):  
Alberto Varello ◽  
Erasmo Carrera
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Cross Section ◽  
Vibration Analysis ◽  
Plane Deformation ◽  
Free Vibration Analysis ◽  
Higher Order ◽  
Variable Order ◽  
Vibrational Modes ◽  
One Dimensional

The free vibration analysis of thin- and thick-walled layered structures via a refined one-dimensional (1D) approach is addressed in this paper. Carrera unified formulation (CUF) is employed to introduce higher-order 1D models with a variable order of expansion for the displacement unknowns over the cross section. Classical Euler–Bernoulli (EBBM) and Timoshenko (TBM) beam theories are obtained as particular cases. Different kinds of vibrational modes with increasing half-wave numbers are investigated for short and relatively short cylindrical shells with different cross section geometries and laminations. Numerical results of natural frequencies and modal shapes are provided by using the finite element method (FEM), which permits various boundary conditions to be handled with ease. The analyses highlight that the refinement of the displacement field by means of higher-order terms is fundamental especially to capture vibrational modes that require warping and in-plane deformation to be detected. Classical beam models are not able to predict the realistic dynamic behavior of shells. Comparisons with three-dimensional elasticity solutions and solid finite element solutions prove that CUF provides accuracy in the free vibration analysis of even short, nonhomogeneous thin- and thick-walled shell structures, despite its 1D approach. The results clearly show that bending, radial, axial, and also shell lobe-type modes can be accurately evaluated by variable kinematic 1D CUF models with a remarkably lower computational effort compared to solid FE models.

A New Methodology for Modeling and Free Vibrations Analysis of Rotating Shaft Based on the Timoshenko Beam Theory

2016 ◽  
Vol 138 (2) ◽  
Author(s):  
S. H. Mirtalaie ◽  
M. A. Hajabasi
Keyword(s):  
Free Vibration ◽  
Cross Section ◽  
Timoshenko Beam ◽  
Variable Coefficient ◽  
Slenderness Ratio ◽  
Beam Theory ◽  
Free Vibrations ◽  
Timoshenko Beam Theory ◽  
Euler Angles ◽  
Shear Deformations

The linear lateral free vibration analysis of the rotor is performed based on a new insight on the Timoshenko beam theory. Rotary inertia, gyroscopic effects, and shear deformations are included, but the torsion is neglected and a new dynamic model is presented. It is shown that if the total rotation angle of the beam cross section is considered as one of the degrees-of-freedom of the Timoshenko rotor, as is common in the literature, some terms are missing in the modeling of the global dynamics of the system. The total deflection of the beam cross section is divided into two steps, first the Euler angles relations are employed to establish the curved geometry of the beam due to the elastic deformation of the beam centerline and then the shear deformations was superposed on it. As a result of this methodology and the mutual interaction of shear and Euler angles some variable coefficient terms appeared in the kinetic energy of the system which makes the problem be classified as the parametrically excited systems. A linear coupled variable coefficient system of differential equations is derived while the variable coefficient terms have been missing in all previous studies in the literature. The free vibration behavior of parametrically excited system is investigated by perturbation method and compared with the common Rayleigh, Timoshenko, and higher-order shear deformable spinning beam models in the rotordynamics. The effects of rotating speed and slenderness ratio are studied on the forward and backward natural frequencies and the critical speeds of the system are examined. The study demonstrates that the shear and Euler angles interaction affects the high-frequency free vibrations behavior of the spinning beam especially for higher slenderness ratio and rotating speeds of the rotor.

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