Vibration of Thick Prismatic Structures With Three-Dimensional Flexibilities

1998 ◽  
Vol 65 (3) ◽  
pp. 619-625 ◽  
Author(s):  
K. M. Liew ◽  
K. C. Hung ◽  
M. K. Lim
Keyword(s):  
Three Dimensional ◽  
Small Strain ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Future Research ◽  
Three Dimension ◽  
Cross Sectional ◽  
Practical Applications ◽  
Support Conditions ◽  
Spatial Displacements

This paper presents an investigation on free vibration of thick prismatic structures (thick-walled open sections of L, T, C, and I shapes). The derivation of a linear frequency equation based on an exact three-dimensional small-strain linearly elastic principle is presented. This formulation uses one and two-dimensional polynomial series to approximate the spatial displacements of the thick-walled open sections in three dimension. The proposed technique is applicable to vibration of thick-walled open sections of different cross-sectional geometries and end support conditions. In this study, however, we focus primarily on the cantilevered case which has high value in practical applications. The perturbation of frequency responses due to the variations of cross-sectional geometries and wall thicknesses is investigated. First-known frequency parameters and three-dimensional deformed mode shapes of these thick-walled open sections are presented in vivid graphical forms. The new results may serve as a benchmark reference to future research into the refined beam and plate theories and also for checking the accuracy of new numerical techniques.

Vibration Characteristics of Conical Shell Panels With Three-Dimensional Flexibility

1999 ◽  
Vol 67 (2) ◽  
pp. 314-320 ◽  
Author(s):  
K. M. Liew ◽  
Z. C. Feng
Keyword(s):  
Conical Shell ◽  
Three Dimensional ◽  
Small Strain ◽  
Frequency Equation ◽  
Vibration Characteristics ◽  
Mode Shapes ◽  
Computational Techniques ◽  
Vertex Angle ◽  
Three Dimension ◽  
Spatial Displacements

A first known investigation on the three-dimensional vibration characteristics of conical shell panels is reported. A linear frequency equation is derived based on an exact three-dimensional, small-strain, linearly elastic theory. Sets of one and two-dimensional polynomial series are employed to approximate the spatial displacements of the conical shell panels in three dimension. The perturbation of frequency responses due to the variations of relative thickness L/h, slanted length L/S, vertex angle γv, and subtended angle γo is investigated. First known frequency parameters and three-dimensional deformed mode shapes of the conical shell panels are presented in vivid graphical forms. The new results may serve as benchmark references for validating the new refined shell theories and new computational techniques. [S0021-8936(00)02302-3]

Three-Dimensional Vibration Analysis of Toroidal Sectors With Solid Circular Cross-Sections

2010 ◽  
Vol 77 (4) ◽  
Author(s):  
D. Zhou ◽  
Y. K. Cheung ◽  
S. H. Lo
Keyword(s):  
Cross Sections ◽  
Three Dimensional ◽  
Small Strain ◽  
Boundary Function ◽  
Mode Shapes ◽  
Curvature Radius ◽  
Linear Elasticity Theory ◽  
Energy Principle ◽  
Cross Sectional ◽  
The Cross

This paper studies the free vibration of circular toroidal sectors with circular cross-sections based on the three-dimensional small-strain, linear elasticity theory. A set of orthogonal coordinates, composing the polar coordinate (r,θ) with the origin on the cross-sectional centerline of the sector and the circumferential coordinate φ with the origin at the curvature center of the centerline, is developed to describe the displacements, strains, and stresses in the sector. Each of the displacement components is taken as a product of four functions: a set of Chebyshev polynomials in φ and r coordinates, a set of trigonometric series in θ coordinate, and a boundary function in terms of φ. Frequency parameters and mode shapes have been obtained via a displacement-based extremum energy principle. The upper bound convergence of the first eight frequency parameters accurate up to five figures has been achieved. The present results agree with those from the finite element solutions. The effect of the ratio of curvature radius R to the cross-sectional radius a and the subtended angle φ0 on the frequency parameters of the sectors are discussed in detail. The three-dimensional vibration mode shapes are also plotted.

Anthropometry-Based Surface Modeling of the Human Torso

1994 ◽  
Author(s):  
Peng Li ◽  
Peter R. M. Jones
Keyword(s):  
Human Body ◽  
Computer Aided Design ◽  
Three Dimensional ◽  
Human Growth ◽  
Surface Model ◽  
Cross Sectional ◽  
Practical Applications ◽  
Garment Design ◽  
Aided Design ◽  
Reference Body

Abstract There is an increasing need for computerized surface model of the human body in human growth, garment design and ergonomics. However, there is a shortage of three-dimensional (3-D) models of the human body in practical applications. This paper presents a new approach for constructing a 3-D surface model of the human torso using anthropometry. The torso is created by from a reference body of average shape which is represented by a family of cross-sectional curves. The shape and size of the reference body can be modified according to anthropometric data. This approach has been implemented on a personal computer. The resulting 3-D model is a parametric surface based on non-uniform B-splines and can easily be exported to other computer aided design applications.

Design for Desired Mode Shapes: Preliminary Results

1998 ◽  
Author(s):  
Elizabeth K. Lai ◽  
G. K. Ananthasuresh
Keyword(s):  
Finite Element ◽  
Feasibility Study ◽  
Mode Shape ◽  
Longitudinal Axis ◽  
Mode Shapes ◽  
Future Research ◽  
Structural Members ◽  
Element Analysis ◽  
Cross Sectional ◽  
Made In

Abstract This paper is concerned with the shape optimization of structures to attain prescribed normal mode shapes. Optimizing structural members in order to have desired mode shapes, besides the desired natural frequencies, is of interest in some applications at both macro and micro scales. After reviewing the relevant past work on the “inverse mode shape” problem, a feasibility study using the lumped spring-mass models and finite element models of an axially vibrating bar is presented. Based on the observations made in the feasibility study with bars, a meaningful optimization problem is formulated and solved. Using finite element analysis and numerical optimization, a method for designing beam-like structures for prescribed mode shapes is developed. The method is demonstrated with an example of designing the cross-sectional area profile of a beam along its longitudinal axis to get a desired fundamental mode shape. The nonuniqueness of the solution is noted and avenues for future research are identified.

Vibration of Stress-Free Hollow Cylinders of Arbitrary Cross Section

1995 ◽  
Vol 62 (3) ◽  
pp. 718-724 ◽  
Author(s):  
K. M. Liew ◽  
K. C. Hung ◽  
M. K. Lim
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Three Dimensional ◽  
Arbitrary Cross Section ◽  
Mode Shapes ◽  
Width Ratio ◽  
Cross Sectional ◽  
Hollow Cylinders ◽  
Displacement Based ◽  
Bending Modes

A three-dimensional elasticity solution to the vibrations of stress-free hollow cylinders of arbitrary cross section is presented. The natural frequencies and deformed mode shapes of these cylinders are obtained via a three-dimensional displacement-based energy formulation. The technique is applied specifically to the parametric investigation of hollow cylinders of different cross sections and sizes. It is found that the cross-sectional property of the cylinder has significant effects on the normal mode responses, particularly, on the transverse bending modes. By varying the length-to-width ratio of these elastic cylinders, interesting results demonstrating the dependence of frequencies on the length of the cylinder have been concluded.

Analytical Piezoelasticity Solution for Natural Frequencies of Levy-Type Piezolaminated Plates

2019 ◽  
Vol 11 (03) ◽  
pp. 1950023 ◽  
Author(s):  
Susanta Behera ◽  
Poonam Kumari
Keyword(s):  
Natural Frequencies ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Single Layer ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Kantorovich Method ◽  
Extended Kantorovich Method ◽  
Aspect Ratios ◽  
Support Conditions

First time, an analytical solution based on three-dimensional (3D) piezoelasticity is developed for the free vibration analysis of Levy-type piezolaminated plates using 3D extended Kantorovich method (EKM). Extended Hamilton principle (which is extended from elastic to piezoelectric case) is further extended to the dynamic version of mixed form containing contributions from the electrical terms. Multi-term multi-field extended Kantorovich method in conjunction with Fourier series (along [Formula: see text]-direction) is employed to obtain two sets of first-order homogeneous ordinary differential equations (8[Formula: see text] along [Formula: see text]- and [Formula: see text]-axes). A robust algorithm is designed (Fortran Code) to extract the natural frequencies and mode shapes of Levy-type piezolaminated plates. The accuracy and efficacy of this technique are verified thoroughly by comparing it with the existing results in the literature and with the 3D finite element (FE) solutions. Numerical results are presented for single-layer piezoelectric and smart sandwich plates considering five different boundary support conditions, three aspect ratios (length to thickness ratio) and electric open and close circuit conditions. The present results shall be used as a benchmark to assess various two-dimensional (2D) and 3D numerical solutions (e.g., FEM, DQM, etc.).

scholarly journals An Analysis of Network Structure in Mazda’s Yokokai using the DEC Spatial Model

2021 ◽  
Vol 15 ◽  
pp. 74-78
Author(s):  
T. Ito ◽  
S. Tagawa ◽  
S. Matsuno ◽  
Y. Uchida ◽  
Rajiv Mehta ◽  
...  
Keyword(s):  
Time Series ◽  
Spatial Model ◽  
Euclidean Distance ◽  
Time Series Data ◽  
Three Dimensional ◽  
Series Data ◽  
Future Research ◽  
Cross Sectional ◽  
Corporate Group ◽  
The Relationship

By examining networks is possible to understand the nature of inter-firm relationships among organizational entities in any given corporate group, such as Toyota’s, Nissan’s or Mazda’s Keiretsu. Recently, a new three-dimensional spatial model has been developed that allows organizational scholars to ascertain the structure of a corporate group, the position of the individual firms, and the determinants of the firm performance. This new spatial paradigm –called the DEC spatial model– composed of degree, effective size and capacity that assessed the relationship between Euclidean distance and sales. Although it advances our understanding of networks, the bulk of the research is based on cross-sectional data, it is not possible ascertain the real nature of the relationship between the distance and sales. Instead, the analysis of networks requires using time series data as all the corporate members of a network are ongoing- concerns. To augment our understanding of the nature of inter-firms networks, the interrelationship between distance and sales is examined using time series data drawn from Mazda’s Yokokai in 1986, 2004 and 2005. More specifically, in this paper the data on transactions were collected and used to calculate the Euclidean distance using the DEC spatial model. The position and its determinants of all individual firms are identified and the trend of structure changes is discussed. Based on the findings of offered and avenues of future research are suggested.

scholarly journals Numerical Investigation of Discrepancies Between Two-Dimensional and Three-Dimensional Acoustic Metamaterials

2021 ◽  
Vol 8 ◽  
Author(s):  
Wenchao Jin ◽  
Hui Guo ◽  
Pei Sun ◽  
Yansong Wang ◽  
Tao Yuan
Keyword(s):  
Band Structure ◽  
Boundary Conditions ◽  
Three Dimensional ◽  
Experimental Models ◽  
Mode Shapes ◽  
Periodic Lattice ◽  
Two Dimensional ◽  
Acoustic Metamaterials ◽  
Band Structures ◽  
Practical Applications

In order to get insight information of the band structure of acoustic metamaterials (AMMs) in condensed matter, periodic lattice structures are analyzed using Bloch’s theorem. Typical approaches of the band structure computation methods, topology optimization, and tunable abilities cannot overcome the gap between the two-dimensional (2D) AMMs theoretical and three-dimensional (3D) specimens’ experimental data yet. In this work, the variation in the results of the band structure obtained from the 2D mathematical model computed with respect to the 3D experimental models, and related cause of the variation is explored. The band structures and mode shapes of the 2D AMMs, quasi-2D models, and 3D specimen models are followed to reveal the boundary conditions and source for the observed differences in band structures. The cause for the discrepancies is verified by using the finite element method (FEM) with corresponding boundary conditions. It is found that outcomes from computational data of the 2D AMMs model are diverted significantly by means of bandgap, band structure, and stress distribution in counterparts of the 3D specimen model. This approach can provide assistance for computing the band structure of 2D AMMs for practical applications.

Three-Dimensional Vibration Analysis of a Truncated Quadrangular Pyramid

1987 ◽  
Vol 54 (1) ◽  
pp. 115-120 ◽  
Author(s):  
T. Irie ◽  
G. Yamada ◽  
Y. Tagawa
Keyword(s):  
Vibration Analysis ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Three Dimensional ◽  
Frequency Equation ◽  
The Body ◽  
Mode Shapes ◽  
Algebraic Polynomials ◽  
Transformation Of Variables ◽  
Unit Edge

An analysis is presented for the three-dimensional vibration problem of determining the natural frequencies and the mode shapes of a truncated quadrangular pyramid. For this purpose, the body is transformed into a right quadrangular prism with unit edge lengths by a transformation of variables. With the displacements of the transformed prism assumed in the forms of algebraic polynomials, the dynamical energies of the prism are evaluated, and the frequency equation is derived by the Ritz method. This method is applied to quadrangular pyramids in which the base is clamped and the other sides are free, and the natural frequencies (the eigenvalues of vibration) and the mode shapes are calculated numerically, from which the vibration characteristics arising in the pyramids are studied.

A Spectral-Tchebychev Solution for Three-Dimensional Vibrations of Parallelepipeds Under Mixed Boundary Conditions

2012 ◽  
Vol 79 (5) ◽  
Author(s):  
Sinan Filiz ◽  
Bekir Bediz ◽  
L. A. Romero ◽  
O. Burak Ozdoganlar
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Mixed Boundary ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Mixed Boundary Conditions ◽  
Integral Form ◽  
Mode Shapes ◽  
Different Boundary Conditions ◽  
Practical Applications

Vibration behavior of structures with parallelepiped shape—including beams, plates, and solids—are critical for a broad range of practical applications. In this paper we describe a new approach, referred to here as the three-dimensional spectral-Tchebychev (3D-ST) technique, for solution of three-dimensional vibrations of parallelepipeds with different boundary conditions. An integral form of the boundary-value problem is derived using the extended Hamilton’s principle. The unknown displacements are then expressed using a triple expansion of scaled Tchebychev polynomials, and analytical integration and differentiation operators are replaced by matrix operators. The boundary conditions are incorporated into the solution through basis recombination, allowing the use of the same set of Tchebychev functions as the basis functions for problems with different boundary conditions. As a result, the discretized equations of motion are obtained in terms of mass and stiffness matrices. To analyze the numerical convergence and precision of the 3D-ST solution, a number of case studies on beams, plates, and solids with different boundary conditions have been conducted. Overall, the calculated natural frequencies were shown to converge exponentially with the number of polynomials used in the Tchebychev expansion. Furthermore, the natural frequencies and mode shapes were in excellent agreement with those from a finite-element solution. It is concluded that the 3D-ST technique can be used for accurate and numerically efficient solution of three-dimensional parallelepiped vibrations under mixed boundary conditions.

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