A New Approach to Compute Natural Frequencies and Mode Shapes of Non-Uniform Continuous Bar, Circular Shaft and Beam Vibration

2019 ◽  
Author(s):  
Alok Sinha
Keyword(s):  
Natural Frequencies ◽  
Linear Time ◽  
Longitudinal Vibration ◽  
Mode Shapes ◽  
Engineering Structures ◽  
Cross Sectional ◽  
Closed Form Solutions ◽  
Continuous Structure ◽  
Circular Shaft ◽  
Spatial Parameters

Abstract The wave equation governing longitudinal vibration of a bar and torsional vibration of a circular shaft, and the Euler-Bernoulli equation governing transverse vibration of a beam were developed in the eighteenth century. Natural frequencies and mode shapes are easily obtained for uniform or constant spatial parameters (cross sectional area, material property and mass distribution). But, real engineering structures seldom have constant parameters. For non-uniform continuous structure, a large number of papers have been written for more than 100 years since the publication of Kirchhoff’s memoir in 1882. There are analytical solutions only in few cases, and there are approximate numerical methods to deal with other (almost all) cases, most notably Stodola, Holzer and Myklestad methods in addition to Rayleigh-Ritz and finite element methods. This paper presents a novel approach to compute natural frequencies and mode shapes for arbitrary variations of spatial parameters on the basis of linear time-varying system theory. The advantage of this approach is that now it can be claimed that “almost” closed-form solutions are available to find natural frequencies and mode shapes of any non-uniform, linear and one-dimensional continuous structure.

A New Approach to Compute Natural Frequencies and Mode Shapes of One-Dimensional Continuous Structures With Arbitrary Nonuniformities

2020 ◽  
Vol 15 (11) ◽  
Author(s):  
Alok Sinha
Keyword(s):  
Natural Frequencies ◽  
Linear Time ◽  
Longitudinal Vibration ◽  
Numerical Approach ◽  
Mode Shapes ◽  
Cross Sectional ◽  
One Dimensional ◽  
Dirac Delta ◽  
Mass Distributions ◽  
Jump Discontinuities

Abstract One-dimensional continuous structures include longitudinal vibration of bars, torsional vibration of circular shafts, and transverse vibration of beams. Using the linear time-varying system theory, algorithms are developed in this paper to compute natural frequencies and mode shapes of these structures with nonuniform spatial parameters (mass distributions, material properties and cross-sectional areas) which can have jump discontinuities. A general numerical approach has been presented to include Dirac-delta functions and their spatial derivatives due to jump discontinuities. Numerical results are presented to illustrate the application of these techniques to the solution of different types of spatial variations of parameters and boundary conditions.

Exact Solutions for Longitudinal Vibration of Multi-Step Bars with Varying Cross-Section

1999 ◽  
Vol 122 (2) ◽  
pp. 183-187 ◽  
Author(s):  
Q. S. Li
Keyword(s):  
Natural Frequencies ◽  
Longitudinal Vibration ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Transmission Tower ◽  
Power Functions ◽  
Closed Form Solutions ◽  
High Rise ◽  
One Step ◽  
Area Variation

Using appropriate transformations, the equation of motion for free longitudinal vibration of a nonuniform one-step bar is reduced to an analytically solvable equation by selecting suitable expressions, such as power functions and exponential functions, for the area variation. Exact analytical solutions to determine the longitudinal natural frequencies and mode shapes for a one step nonuniform bar are derived and used to obtain the frequency equation of multi-step bars. The new exact approach is presented which combines the transfer matrix method and closed form solutions of one step bars. A numerical example demonstrates that the calculated natural frequencies and mode shapes of a television transmission tower are in good agreement with the corresponding experimental data, and the selected expressions are suitable for describing the area variation of typical high-rise structures. [S0739-3717(00)00302-0]

Vibrational Analysis of Human Femur Bone

2018 ◽  
Author(s):  
Reza Sadeghi ◽  
Firooz Bakhtiari-Nejad ◽  
Taha Goudarzi
Keyword(s):  
Natural Frequencies ◽  
Vibrational Analysis ◽  
Bone Geometry ◽  
The Body ◽  
Mode Shapes ◽  
Pelvic Bone ◽  
Cross Sectional ◽  
Femur Bone ◽  
Bone Structures ◽  
Elderly Falls

Femur bone is the longest and largest bone in the human skeleton. This bone connects the pelvic bone to the knee and carries most of the body weight. The static behavior of femur bone has been a center of investigation for many years while little attention has been given to its dynamic and vibrational behavior, which is of great importance in sports activities, car crashes and elderly falls. Investigation of natural frequencies and mode shapes of bone structures are important to understand the dynamic and vibrating behaviors. Vibrational analysis of femoral bones is presented using finite element method. In the analysis, the bone was modeled with isotropic and orthotropic mechanical properties. The effect of surrounding bone muscles has also been accounted for as a viscoelastic medium embedding the femur bone. Natural frequencies extracted considering the effects of age aggravated by weakening the elastic modulus and density loss. The effects of real complex bone geometry on natural frequencies are studied and are compared with a simple circular cross-sectional model.

scholarly journals Improved Riccati Transfer Matrix Method for Free Vibration of Non-Cylindrical Helical Springs Including Warping

2012 ◽  
Vol 19 (6) ◽  
pp. 1167-1180 ◽  
Author(s):  
A.M. Yu ◽  
Y. Hao
Keyword(s):  
Free Vibration ◽  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Axial Deformation ◽  
Cross Sectional ◽  
Helical Springs

Free vibration equations for non-cylindrical (conical, barrel, and hyperboloidal types) helical springs with noncircular cross-sections, which consist of 14 first-order ordinary differential equations with variable coefficients, are theoretically derived using spatially curved beam theory. In the formulation, the warping effect upon natural frequencies and vibrating mode shapes is first studied in addition to including the rotary inertia, the shear and axial deformation influences. The natural frequencies of the springs are determined by the use of improved Riccati transfer matrix method. The element transfer matrix used in the solution is calculated using the Scaling and Squaring method and Pad'e approximations. Three examples are presented for three types of springs with different cross-sectional shapes under clamped-clamped boundary condition. The accuracy of the proposed method has been compared with the FEM results using three-dimensional solid elements (Solid 45) in ANSYS code. Numerical results reveal that the warping effect is more pronounced in the case of non-cylindrical helical springs than that of cylindrical helical springs, which should be taken into consideration in the free vibration analysis of such springs.

Computing Natural Frequencies and Mode Shapes of an Axially Moving Non-Uniform Beam

2020 ◽  
Author(s):  
Alok Sinha
Keyword(s):  
Cross Sections ◽  
Critical Speed ◽  
Natural Frequencies ◽  
Linear Time ◽  
Mode Shapes ◽  
Axially Moving Beam ◽  
Time Varying Systems ◽  
Axially Moving ◽  
The Galerkin Method ◽  
Spatially Varying

Abstract The partial differential equation of motion of an axially moving beam with spatially varying geometric, mass and material properties has been derived. Using the theory of linear time-varying systems, a general algorithm has been developed to compute natural frequencies, mode shapes, and the critical speed for stability. Numerical results from the new method are presented for beams with spatially varying rectangular cross sections with sinusoidal variation in thickness and sine-squared variation in width. They are also compared to those from the Galerkin method. It has been found that critical speed of the beam can be significantly reduced by non-uniformity in a beam’s cross section.

Natural Frequencies and Critical Buckling Loads of Marine Risers

1988 ◽  
Vol 110 (1) ◽  
pp. 2-8 ◽  
Author(s):  
Y. C. Kim
Keyword(s):  
Closed Form ◽  
Natural Frequencies ◽  
Simple Procedure ◽  
Variable Cross Section ◽  
Mode Shapes ◽  
Variable Cross ◽  
Bending Rigidity ◽  
Buckling Loads ◽  
Closed Form Solutions ◽  
Marine Risers

Natural frequencies, mode shapes and critical buckling loads of marine risers simply supported at both ends are given in closed form by using the WKB method. These solutions allow variable cross section, bending rigidity, tension and mass distribution along the riser length. Furthermore, a simple procedure to predict natural frequencies for other boundary conditions is described. Some special forms of these closed-form solutions are compared with existing solutions in the literature.

scholarly journals Linear Finite Element Modeling of Joined Structures With Riveted Connections

2020 ◽  
Vol 142 (2) ◽  
Author(s):  
J. S. Kim ◽  
Y. F. Xu ◽  
W. D. Zhu
Keyword(s):  
Finite Element ◽  
Natural Frequencies ◽  
Modeling Method ◽  
Test Structure ◽  
Mode Shapes ◽  
Fe Model ◽  
Fe Modeling ◽  
Engineering Structures ◽  
Modal Assurance Criterion ◽  
Linear Finite Element

Abstract Riveted connections are widely used to join basic components, such as beams and panels, for engineering structures. However, accurately modeling joined structures with riveted connections can be a challenging task. In this work, an accurate linear finite element (FE) modeling method is proposed for joined structures with riveted connections to estimate modal parameters in a predictive manner. The proposed FE modeling method consists of two steps. The first step is to develop nonlinear FE models that simulate riveting processes of solid rivets. The second step is to develop a linear FE model of a joined structure with the riveted connections simulated in the first step. The riveted connections are modeled using solid cylinders with dimensions and material properties obtained from the nonlinear FE models in the first step. An experimental investigation was conducted to study accuracy of the proposed linear FE modeling method. A joined structure with six riveted connections was prepared and tested. A linearity investigation was conducted to validate that the test structure could be considered to be linear. A linear FE model of the test structure was constructed using the proposed method. Natural frequencies and corresponding mode shapes of the test structure were measured and compared with those from the linear FE model. The maximum difference of the natural frequencies was 1.63% for the first 23 out-of-plane elastic modes, and modal assurance criterion values for the corresponding mode shapes were all over 95%, which indicates high accuracy of the proposed linear FE modeling method.

Free vibration analysis of a single edge cracked symmetric functionally graded stepped beams

2020 ◽  
Vol 23 (16) ◽  
pp. 3415-3428
Author(s):  
Yusuf Cunedioglu ◽  
Shkelzen Shabani
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Crack Depth ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Single Edge ◽  
Functionally Graded Beam ◽  
Cross Sectional

Free vibration analysis of a single edge cracked multi-layered symmetric sandwich stepped Timoshenko beams, made of functionally graded materials, is studied using finite element method and linear elastic fracture mechanic theory. The cantilever functionally graded beam consists of 50 layers, assumed that the second stage of the beam (step part) is created by machining. Thus, providing the material continuity between the two beam stages. It is assumed that material properties vary continuously, along the thickness direction according to the exponential and power laws. A developed MATLAB code is used to find the natural frequencies of three types of the stepped beam, concluding a good agreement with the known data from the literature, supported also by ANSYS software in data verification. In the study, the effects of the crack location, crack depth, power law gradient index, different material distributions, different stepped length, different cross-sectional geometries on natural frequencies and mode shapes are analysed in detail.

Explicit Vibration Solutions of a Cable Under Complicated Loads

1997 ◽  
Vol 64 (4) ◽  
pp. 957-964 ◽  
Author(s):  
P. Yu
Keyword(s):  
Vibration Analysis ◽  
Natural Frequencies ◽  
Vibration Mode ◽  
Free Vibration Analysis ◽  
Dynamical Analysis ◽  
Mode Shapes ◽  
Concentrated Loads ◽  
Closed Form Solutions ◽  
Independent Mode ◽  
Complex Arrangement

This paper is concerned with the dynamical analysis of a sagged cable having small equilibrium curvature and horizontal supports under both distributed and concentrated loads. The loads are applied in vertical as well as horizontal directions. Based on a free vibration analysis, a transfer matrix method is generalized for solving coupled, nonhomogeneous differential equations to obtain closed-form solutions for the natural frequencies and the associated vibration mode shapes in vertical, horizontal, and longitudinal directions. It is shown that two sets of independent mode shapes associated with two sets of independent frequencies always exist and can be obtained via an equation of one variable only. This method demonstrates its advantages in dealing with interactions of modes in different directions, complex arrangement of concentrated loads, and high-order modes oscillations.

Longitudinal Vibration of Variable Cross-Sectional Nanorods

2020 ◽  
Vol 64 ◽  
pp. 49-60
Author(s):  
Mustafa Arda ◽  
Metin Aydogdu
Keyword(s):  
Differential Equation ◽  
Longitudinal Vibration ◽  
Mode Shapes ◽  
Variable Cross ◽  
Cross Sectional ◽  
Axial Coordinate ◽  
Atomic Force ◽  
Governing Differential Equation ◽  
Area Variation ◽  
Linear Differential

Vibration problem of variable cross-sectional nanorods have been investigated. Analytical solutions have been determined for the variable cross-sectional nanorods for a family of cross-sectional variation. Cross-sectional area variation has been assumed as power function of the axial coordinate. Nonlocal governing equation of motion has been obtained as a second order linear differential equation. Bessel functions have been used in analytical solution of the governing differential equation. Effect of nonlocal and area variation power parameters on dynamics of nanorods have been analyzed. Mode shapes of nanorod have been depicted in various cases and boundary conditions. Present results could be useful at design of atomic force microscope’s probe tip selection.

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