In-Plane Vibration Analysis of Rotating Tapered Timoshenko Beams

2016 ◽  
Vol 08 (05) ◽  
pp. 1650064 ◽  
Author(s):  
Jianshi Fang ◽  
Ding Zhou
Keyword(s):  
Chebyshev Polynomials ◽  
Coupling Effect ◽  
Mode Shapes ◽  
Transverse Motion ◽  
Timoshenko Beams ◽  
Rotational Angle ◽  
Plane Vibration ◽  
Present Solution ◽  
Boundary Functions ◽  
Geometric Boundary

Modal analysis of rotating tapered cantilevered Timoshenko beams undergoing in-plane vibration is investigated. The coupling effect of axial motion and transverse motion is considered. The Kane dynamic method is applied to deriving the governing eigenvalue equations. The displacement and rotational angle components are approximately described by the products of Chebyshev polynomials and corresponding boundary functions. Chebyshev polynomials guarantee the numerical robustness while the boundary functions guarantee the satisfaction of the geometric boundary conditions. The excellent convergence of the present solution is exhibited. The results are compared with those available in literature, good agreement is observed. The parametric studies on modal characteristics are presented in detail. The tuned rotational speed is examined and the eigenvalue loci veering phenomenon along with the corresponding mode shapes is investigated.

Free Vibration Analysis of Rotating Mindlin Plates with Variable Thickness

2017 ◽  
Vol 17 (04) ◽  
pp. 1750046 ◽  
Author(s):  
J. S. Fang ◽  
D. Zhou
Keyword(s):  
Variable Thickness ◽  
Chebyshev Polynomials ◽  
Vibration Analysis ◽  
Present Method ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Energy Functional ◽  
Boundary Functions ◽  
Geometric Boundary ◽  
Admissible Functions

The modal analysis of rotating cantilevered rectangular Mindlin plates with variable thickness is studied. The Ritz method is used to derive the governing eigenfrequency equation by minimizing the energy functional of the plate. The admissible functions are taken as a product of the Chebyshev polynomials multiplied by the boundary functions, which enable the displacements and rotational angles to satisfy the geometric boundary conditions of the plate. The Chebyshev polynomials guarantee the numerical robustness, while the Ritz approach provides the upper bound of the exact frequencies. The effectiveness of the present method is confirmed through the convergence and comparison studies. The effects of the dimensionless rotational speed, taper ratio, aspect ratio and thickness ratio on modal characteristics are investigated in detail. The frequency loci veering phenomenon along with the corresponding mode shape switching is exhibited and discussed.

scholarly journals Semi-Analytical Solution for Free Vibration Differential Equations of Curved Girders

2017 ◽  
Vol 63 (1) ◽  
pp. 115-132
Author(s):  
Y. Song ◽  
X. Chai
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Free Vibration ◽  
Coupling Effect ◽  
Longitudinal Vibration ◽  
Synthesis Method ◽  
Variable Separation ◽  
Element Analysis ◽  
Plane Vibration ◽  
Curved Girders

Abstract In this paper, a semi-analytical solution for free vibration differential equations of curved girders is proposed based on their mathematical properties and vibration characteristics. The solutions of in-plane vibration differential equations are classified into two cases: one only considers variable separation of non-longitudinal vibration, while the other is a synthesis method addressing both longitudinal and non-longitudinal vibration using Rayleigh’s modal assumption and variable separation method. A similar approach is employed for the out of- plane vibration, but further mathematical operations are conducted to incorporate the coupling effect of bending and twisting. In this case study, the natural frequencies of a curved girder under different boundary conditions are obtained using the two proposed methods, respectively. The results are compared with those from the finite element analysis (FEA) and results show good convergence.

Free Flexural Vibrations Response of Composite Mindlin Plates or Panels With a Centrally Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint)

2005 ◽  
Author(s):  
U. Yuceoglu ◽  
O. Gu¨vendik ◽  
V. O¨zerciyes
Keyword(s):  
Natural Frequencies ◽  
Mode Shapes ◽  
Solution Technique ◽  
Shear Stresses ◽  
Lap Joint ◽  
Adhesive Layers ◽  
Present Solution ◽  
Flexural Vibrations ◽  
Support Conditions ◽  
Interpolation Polynomials

The present study is concerned with the “Free Flexural Vibrations Response of Composite Mindlin Plates or Panels with a Centrally Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint). The plate “adherends” and the plate “doublers” are considered as dissimilar, orthotropic “Mindlin Plates” with the transverse and the rotary moments of inertia. The relatively, very thin adhesive layers are taken into account in terms of their transverse normal and shear stresses. The mid-center of the bonded region of the joint is at the mid-center of the entire system. In order to facilitate the present solution technique, the dynamic equations of the plate “adherends” and the plate “doublers” with those of the adhesive layers are reduced to a set of the “Governing System of First Order ordinary Differential Equations” in terms of the “state vectors” of the problem. This reduced set establishes a “Two-Point Boundary Value Problem” which can be numerically integrated by making use of the “Modified Transfer Matrix Method (MTMM) (with Interpolation Polynomials)”. In the adhesive layers, the “hard” and the “soft” adhesive cases are accounted for. It was found that the adhesive elastic constants drastically influence the mode shapes and their natural frequencies. Also, the numerical results of some parametric studies regarding the effects of the “Position Ratio” and the “Joint Length Ratio” on the natural frequencies for various sets of support conditions are presented.

scholarly journals Bending–torsional-coupled vibrations and buckling characteristics of single and double composite Timoshenko beams

2019 ◽  
Vol 11 (3) ◽  
pp. 168781401983445
Author(s):  
Ma’en S Sari ◽  
Wael G Al-Kouz ◽  
Rafat Al-Waked
Keyword(s):  
Slenderness Ratio ◽  
Equations Of Motion ◽  
Mode Shapes ◽  
Timoshenko Beams ◽  
Engineering Structures ◽  
Axial Forces ◽  
Double Beam ◽  
Tensile Forces ◽  
Torsional Coupling ◽  
The Stability

The stability and free vibration analyses of single and double composite Timoshenko beams have been investigated. The closed-section beams are subjected to constant axially compressive or tensile forces. The double beams are assumed to be connected by a layer of elastic translational and rotational springs. The coupled governing partial differential equations of motion are discretized, and the resulted eigenvalue problem is solved numerically by applying the Chebyshev spectral collocation method. The effects of the elastic layer parameters, the axial forces, the slenderness ratio, the bending–torsional coupling, and the boundary conditions on the critical buckling loads, mode shapes, and natural transverse frequencies have been studied. A parametric study was performed, and the obtained results revealed different features, which hopefully can be useful for single- and double-beam-like engineering structures.

Vibration of Clamped Circular Symmetric Laminates

1994 ◽  
Vol 116 (2) ◽  
pp. 141-145 ◽  
Author(s):  
K. M. Liew
Keyword(s):  
Ritz Method ◽  
Numerical Procedure ◽  
Flexural Vibration ◽  
Boundary Function ◽  
Energy Functional ◽  
Laminated Plates ◽  
Mode Shapes ◽  
Circular Plates ◽  
Present Solution ◽  
Contour Plots

Treated in this paper is the free-flexural vibration analysis of symmetrically laminated thin circular plates. The total energy functional for the laminated plates is formulated where the pb-2 Ritz method is applied for the solution. The assumed displacement is defined as the product of (1) a two-dimensional complete polynomial function and (2) a basic boundary function. The simplicity and accuracy of the numerical procedure will be demonstrated by solving some plate examples. In the present study, the effects of material properties, number of layers and fiber stacking sequences upon the vibration frequency parameters are investigated. Selected mode shapes by means of contour plots for several 16-ply laminated plates with different fiber stacking sequences and composite materials are presented. This study may provide valuable information for researchers and engineers in design applications. In addition, the present solution plays an important role in increasing the existing data base for future references.

scholarly journals Numerical and Experimental Investigations of Coupling Effects in Anisotropic Elastic Rotors

1999 ◽  
Vol 5 (4) ◽  
pp. 263-271 ◽  
Author(s):  
Horst Irretier ◽  
Georges Jacquet-Richardet ◽  
Frank Reuter
Keyword(s):  
Coupling Effect ◽  
Experimental Results ◽  
Mode Shapes ◽  
Experimental Investigations ◽  
Coupling Effects ◽  
Symmetry Approach ◽  
Anisotropic Elastic ◽  
Elastic Modes ◽  
Bending Modes ◽  
The One

It is known that in elastic disc-shaft systems in particular, the one-nodal-diameter mode of the discs can be highly coupled with the bending modes of the shaft. Consequently, when the system rotates, the elastic modes of the flexible discs are coupled with the gyroscopic modes of the flexible shaft equipped with rigid discs. In the paper this coupling effect is investigated numerically and experimentally.A numerical model, based on a finite element cyclic symmetry approach, is presented. This model has been developed for studying the wheel-shaft coupling effects on the global behavior of turbomachinery rotors. In order to better illustrate the phenomenon involved and to validate the model, the method is applied here to a thin tuned and detuned circular disc mounted on an elastic shaft. Related frequency and mode shapes of the rotating assembly are discussed. Additional experimental results, based on an experimental modal analysis technique for rotating structures, are presented. Both numerical and experimental results are compared.

Damping Frame Vibrations Using Anechoic Stubs: Analysis Using an Exact Wave-based Approach

2020 ◽  
pp. 1-21
Author(s):  
Hangyuan Lv ◽  
Michael Leamy
Keyword(s):  
Finite Element ◽  
Finite Difference ◽  
Forced Vibration ◽  
Mode Shapes ◽  
Frame Structures ◽  
Vibration Modes ◽  
Timoshenko Beams ◽  
Modal Damping ◽  
Reflection Matrix ◽  
Resonance Response

Abstract This paper explores the addition of small stubs with anechoic terminations (termed herein ‘anechoic stubs’) as means for damping and/or removing vibration modes from planar frame structures. Due to the difficulties associated with representing anechoic boundary conditions in more traditional analysis approaches (e.g., analytical, finite element, finite difference, finite volume, etc.), the paper employs and further develops an exact wave-based approach, incorporating Timoshenko beams, in which ideal and non-ideal anechoic terminations are simply represented by a reflection matrix. Several numerically-evaluated examples are presented documenting novel effects anechoic stubs have on the vibration modes of a two-story frame, such as eliminated, inserted and exchanged mode shapes. Modal damping ratios are also computed as a function of the location and number of anechoic stubs, illustrating optimal locations and optimal reflection ratios as a function of mode number. Forced vibration studies are then carried-out, demonstrating reduced, eliminated, and inserted resonance response.

Anechoic Stubs As a Means for Damping Frame Vibrations: Analysis Using an Exact Wave-Based Approach

2020 ◽  
Author(s):  
Hangyuan Lv ◽  
Michael J. Leamy
Keyword(s):  
Finite Element ◽  
Finite Difference ◽  
Forced Vibration ◽  
Mode Shapes ◽  
Frame Structures ◽  
Vibration Modes ◽  
Timoshenko Beams ◽  
Modal Damping ◽  
Reflection Matrix ◽  
Resonance Response

Abstract This paper explores the addition of small stubs with anechoic terminations (termed herein ‘anechoic stubs’) as means for damping and/or removing vibration modes from planar frame structures. Due to the difficulties associated with representing anechoic boundary conditions in more traditional analysis approaches (e.g., analytical, finite element, finite difference, finite volume, etc.), the paper employs an exact wave-based approach, incorporating Timoshenko beams, in which an anechoic boundary is simply represented by a zero reflection matrix. Several numerically-evaluated examples are presented documenting novel effects anechoic stubs have on the vibration modes of a two-story frame, such as eliminated, inserted and exchanged mode shapes. Modal damping ratios are also computed as a function of the location and number of anechoic stubs, illustrating optimal locations as a function of mode number. Forced vibration studies are then carried-out, demonstrating reduced, eliminated, and inserted resonance response.

scholarly journals Forced Vibration of Delaminated Timoshenko Beams under the Action of Moving Oscillatory Mass

2013 ◽  
Vol 20 (1) ◽  
pp. 79-96 ◽  
Author(s):  
M.H. Kargarnovin ◽  
M.T. Ahmadian ◽  
R.A. Jafari-Talookolaei
Keyword(s):  
Dynamic Response ◽  
Forced Vibration ◽  
Equations Of Motion ◽  
Mode Shapes ◽  
Solution Technique ◽  
Timoshenko Beams ◽  
Modal Expansion ◽  
Forced Vibration Analysis ◽  
First Time ◽  
Moving Oscillator

This paper presents the dynamic response of a delaminated composite beam under the action of a moving oscillating mass. In this analysis the Poisson's effect is considered for the first time. Moreover, the effects of rotary inertia and shear deformation are incorporated. In our modeling linear springs are used between delaminated surfaces to simulate the dynamic interaction between sub-beams. To solve the governing differential equations of motion using modal expansion series, eigen-solution technique is used to obtain the natural frequencies and their corresponding mode shapes necessary for forced vibration analysis. The obtained results for the free and forced vibrations of beams are verified against reported similar results in the literatures. Moreover, the maximum dynamic response of such beam is compared with an intact beam. The effects of different parameters such as the velocity of oscillating mass, different ply configuration and the delamination length, its depth and spanwise location on the dynamic response of the beam are studied. In addition, the effects of delamination parameters on the oscillator critical speed are investigated. Furthermore, different conditions under which the detachment of moving oscillator from the beam will initiate are investigated.

Three-Dimensional Vibrations of Truncated Hollow Cones

1995 ◽  
Vol 1 (2) ◽  
pp. 145-158 ◽  
Author(s):  
Arthur W. Leissa ◽  
Jinyoung So
Keyword(s):  
Boundary Conditions ◽  
Cross Sections ◽  
Ritz Method ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Vibration Frequencies ◽  
Arbitrary Boundary ◽  
Conical Shells ◽  
Hole Radius ◽  
Geometric Boundary

This work presents a three-dimensional (3-D) method of analysis for determining the free vibration frequencies and corresponding mode shapes of truncated hollow cones of arbitrary thickness and having arbitrary boundary conditions. It also supplies the first known numerical results from 3-D analysis for such problems. The analysis is based upon the Ritz method. The vibration modes are separated into their Fourier components in terms of the circumferential coordinate. For each Fourier component, displacements are expressed as algebraic polynomials in the thickness and slant length coordinates. These polynomials satisfy the geometric boundary conditions exactly. Because the displacement functions are mathematically complete, upper bound values of the vibration frequencies are obtained that are as close to the exact values as desired. This convergence is demonstrated for a representative truncated hollow cone configuration where six-digit exactitude in the frequencies is achieved. The method is then used to obtain accurate and extensive frequencies for two sets of completely free, truncated hollow cones, one set consisting of thick conical shells and the other being tori having square-generating cross sections. Frequencies are presented for combinations of two values of apex angles and two values of inner hole radius ratios for each set of problems.

Sign in / Sign up

Export Citation Format

Share Document