Exact Vibration Solution for Exponentially Tapered Cantilever With Tip Mass

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
C.Y. Wang ◽  
C. M. Wang
Keyword(s):  
Numerical Methods ◽  
Free Vibration ◽  
Cantilever Beam ◽  
Technical Note ◽  
Mode Shapes ◽  
Constant Thickness ◽  
Vibration Frequencies ◽  
Tapered Cantilever Beam ◽  
Tip Mass ◽  
First Time

This technical note is concerned with the free vibration problem of a cantilever beam with constant thickness and exponentially decaying width. Existing analytical results for such a vibration beam problem are found to be incomplete because lower frequencies could not be obtained. Presented herein is the exact characteristic equation for generating the complete vibration frequencies for the considered vibrating beam problem. Also the note treated for the first time such a tapered cantilever beam with a tip mass. The exact solutions (frequencies and mode shapes) are important to engineers designing such tapered beams and the results serve as benchmarks for assessing the validity, convergence and accuracy of numerical methods and solutions.

In-Plane Vibration of a Deployable Segmented Ring

2020 ◽  
Vol 20 (07) ◽  
pp. 2071007
Author(s):  
C. Y. Wang
Keyword(s):  
Vibration Mode ◽  
Mode Shapes ◽  
Space Structures ◽  
Graphene Sheets ◽  
Vibration Frequencies ◽  
Elastic Ring ◽  
Nonlinear Dynamical ◽  
Classical Vibration ◽  
First Time ◽  
Number Of Segments

The in-plane vibrations of regular polygonal rings composed of rigid segments joined by torsional springs are studied for the first time. The nonlinear dynamical difference equations are formulated and solved by perturbation about the equilibrium state. As the number of segments increase, the frequencies, if aptly normalized, converge to the classical vibration frequencies of a continuous elastic ring. The vibration mode shapes are illustrated. The tiling of many identical polygons is discussed. Possible applications include the vibrations of space structures and graphene sheets.

Free Vibration Analysis of Shafts on Resilient Bearings Using Timoshenko Beam Theory

1999 ◽  
Vol 121 (2) ◽  
pp. 256-258 ◽  
Author(s):  
S. Karunendiran ◽  
J. W. Zu
Keyword(s):  
Free Vibration ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Frequency Equation ◽  
Timoshenko Beam Theory ◽  
Mode Shapes ◽  
Complex Eigenvalues ◽  
First Time

This paper presents an analytical method adopted for the free vibration analysis of a shaft, both ends of which are supported by resilient bearings. The shaft is modeled by Timoshenko beam theory. Based on this model exact frequency equation to calculate complex eigenvalues is derived and presented in complex compact form for the first time. Explicit expressions to compute the corresponding mode shapes are also presented.

Estimation of Structural Displacements for Cantilever Beam Using Mode Shapes and Accelerometers Under Free Vibration

2017 ◽  
Vol 45 (5) ◽  
pp. 376-385
Author(s):  
Kyung Jong Kim ◽  
Yong Hwan Lee ◽  
Kyu Beom Lee ◽  
Cheol Soon Lee ◽  
Jin Yeon Cho ◽  
...  
Keyword(s):  
Free Vibration ◽  
Cantilever Beam ◽  
Mode Shapes

Three-Dimensional Vibrations of Thick, Circular Rings with Isosceles Trapezoidal and Triangular Cross-Sections

1999 ◽  
Vol 122 (2) ◽  
pp. 132-139 ◽  
Author(s):  
Jae-Hoon Kang ◽  
Arthur W. Leissa
Keyword(s):  
Free Vibration ◽  
Cross Sections ◽  
Three Dimensional ◽  
Circular Ring ◽  
Mode Shapes ◽  
Vibration Frequencies ◽  
Algebraic Polynomials ◽  
Free Boundaries ◽  
Circular Rings ◽  
Time Periodic

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, circular rings with isosceles trapezoidal and triangular cross-sections. Displacement components us,uz, and uθ in the meridional, normal, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the ϕ and z directions. Potential (strain) and kinetic energies of the circular ring are formulated, and upper bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Novel numerical results are presented for the circular rings with isosceles trapezoidal and equilateral triangular cross-sections having completely free boundaries. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the rings. The method is applicable to thin rings, as well as thick and very thick ones. [S0739-3717(00)00702-9]

A rigid multibody method for free vibration analysis of beams with variable axial parameters

2016 ◽  
Vol 23 (1) ◽  
pp. 131-146 ◽  
Author(s):  
Aleksandar Nikolić ◽  
Slaviša Šalinić
Keyword(s):  
Multibody System ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Constant Thickness ◽  
Flexible Beam ◽  
Cantilever Beams ◽  
New Approach ◽  
Transverse Vibrations ◽  
Tip Mass ◽  
Rigid Multibody

This paper presents a new approach to the problem of determining the frequencies and mode shapes of Euler–Bernoulli tapered cantilever beams with a tip mass and a spring at the free end. The approach is based on the replacement of the flexible beam by a rigid multibody system. Beams with constant thickness and exponentially and linearly tapered width, as well as double-tapered cantilever beams are considered. The influence of the tip mass, stiffness of the spring, and taper on the frequencies of the free transverse vibrations of tapered cantilever beams are examined. Numerical examples with results confirming the convergence and accuracy of the approach are given.

scholarly journals Revisited on the Free Vibration of a Cantilever Beam with an Asymmetrically Attached Tip Mass

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Xiangsheng Lei ◽  
Yanfeng Wang ◽  
Xinghua Wang ◽  
Gang Lin ◽  
Shihong Shi
Keyword(s):  
Eigenvalue Problem ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Laplace Transform ◽  
Traditional Method ◽  
Separation Of Variables ◽  
Vertical Direction ◽  
Mode Shapes ◽  
Approximate Analytical Solutions ◽  
Tip Mass

Cantilever with an asymmetrically attached tip mass arises in many engineering applications. Both the traditional method of separation of variables and the method of Laplace transform are employed in the present paper to solve the eigenvalue problem of the free vibration of such structures, and the effect of the eccentric distance along the vertical direction and the length direction of the tip mass is considered here. For the traditional method of separation of variables, tip mass only affects to the boundary conditions, and the eigenvalue problem of the free vibration is solved based on the nonhomogeneous boundary conditions. For the method of Laplace transform, the effect of the tip mass is introduced in the governing equation with the Dirac function, and the eigenvalue problem then can be solved through Laplace transform with homogeneous boundary conditions. The computed results with these two methods are compared well with the numerical solution obtained by finite element method and approximate analytical solutions, and the effect of tip mass dimensions on the natural frequencies and corresponding mode shapes is also given.

Vibration Modes of Centrifugally Stiffened Beams

1982 ◽  
Vol 49 (1) ◽  
pp. 197-202 ◽  
Author(s):  
A. D. Wright ◽  
C. E. Smith ◽  
R. W. Thresher ◽  
J. L. C. Wang
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Cantilever Beam ◽  
Mass Distribution ◽  
Flexural Rigidity ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Finite Element Code ◽  
Rotating Beams ◽  
Tip Mass

The method of Frobenius is used to solve for the exact frequencies and mode shapes for rotating beams in which both the flexural rigidity and the mass distribution vary linearly. Results are tabulated for a variety of situations including uniform and tapered beams, with root offset and tip mass, and for both hinged root and fixed root boundary conditions. The results obtained for the case of the uniform cantilever beam are compared with other solutions, and the results of a conventional finite-element code.

scholarly journals Effects of Cable on the Dynamics of a Cantilever Beam with Tip Mass

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Yi-Xin Huang ◽  
Hao Tian ◽  
Yang Zhao
Keyword(s):  
Cantilever Beam ◽  
Spectral Element Method ◽  
Natural Frequencies ◽  
Spectral Element ◽  
Mode Shapes ◽  
Beam System ◽  
Double Beam ◽  
Tip Mass ◽  
Beam Mode ◽  
Element Method

The dynamic effects of cable attachment on a cantilever beam with tip mass are investigated by an improved Chebyshev spectral element method. The cabled beam is modeled as a double-beam system connected by springs at several discrete locations. By utilizing high order Chebyshev polynomials as basis functions and meshing the system at the locations of connections, precise numerical results of the natural frequencies and mode shapes can be obtained using only a few elements. The accuracy of this method is validated through comparing the results of finite element method and those of spectral element method in literature. The validated method is implemented to investigate the effects of parameters, including spring stiffness, number of connections, density, and Young’s modulus of cable. The results show that the mode shapes of the cabled beam system can be classified into two types: beam mode shapes and cable mode shapes, according to their main deformation. Their corresponding natural frequencies change in very different ways with the variation of system parameters. This work can be applied to optimize the dynamic characteristics of precise spacecraft structures with cable attachments.

Application of the Nondimensional Dynamic Influence Function Method for Free Vibration Analysis of Arbitrarily Shaped Membranes

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
Sang Wook Kang ◽  
Satya N. Atluri ◽  
Sang-Hyun Kim
Keyword(s):  
Numerical Methods ◽  
Free Vibration ◽  
Influence Function ◽  
Vibration Analysis ◽  
Function Method ◽  
Free Vibration Analysis ◽  
Frequency Parameter ◽  
Mode Shapes ◽  
System Matrix ◽  
Influence Function Method

A new formulation for the NDIF method (the nondimensional dynamic influence function method) is introduced to efficiently extract eigenvalues and mode shapes of arbitrarily shaped, homogeneous membranes with the fixed boundary. The NDIF method, which was developed by the authors for the accurate free vibration analysis of arbitrarily shaped membranes and plates including acoustic cavities, has the feature that it yields highly accurate solutions compared with other analytical methods or numerical methods (the finite element method and the boundary element method). However, the NDIF method has the weak point that the system matrix of the method is not independent of the frequency parameter and as a result the method needs the inefficient procedure of searching eigenvalues by plotting the values of the determinant of the system matrix in the frequency parameter range of interest. An improved formulation presented in the paper does not require the above-mentioned inefficient procedure because a newly developed system matrix is independent of the frequency parameter. Finally, the validity of the proposed method is shown in several case studies, which indicate that eigenvalues and mode shapes obtained by the proposed method are very accurate compared to those calculated by exact, analytica, or numerical methods.

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