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.

scholarly journals Improvement of Transverse-Flux Machine Characteristics by Finding an Optimal Air-Gap Diameter and Coil Cross-Section at the Given Magneto-Motive Force of the PMs

Energies ◽  
2021 ◽  
Vol 14 (3) ◽  
pp. 755
Author(s):  
Grebenikov Viktor ◽  
Oleksandr Dobzhanskyi ◽  
Gamaliia Rostislav ◽  
Rupert Gouws
Keyword(s):  
Cross Section ◽  
Computer Model ◽  
Single Phase ◽  
Air Gap ◽  
Laboratory Measurements ◽  
The Finite Element Method ◽  
Power Characteristics ◽  
Transverse Flux ◽  
Output Characteristics ◽  
The Given

This paper presents analysis and study of the single-phase transverse-flux machine. The finite element method results of the machine are compared with the laboratory measurements to confirm the accuracy of the computer model. This computer model is then used to investigate the effect of the machine’s geometry on its output characteristics. Parametric analysis of the machine is carried out to find the optimal air-gap diameter at which the cogging torque of the machine is minimal. In addition, the influence of the coil cross-section on the torque and output power characteristics of the machine is investigated and discussed.

Influence of Geometric Imperfections on Vibrational Frequencies of Thin Rings

1975 ◽  
Vol 97 (4) ◽  
pp. 1199-1203
Author(s):  
Joseph R. Gartner ◽  
Shrikant T. Bhat
Keyword(s):  
Cross Section ◽  
Natural Frequencies ◽  
Rectangular Cross Section ◽  
Circular Ring ◽  
Body Motion ◽  
Mode Shapes ◽  
Geometric Imperfections ◽  
Modal Coupling ◽  
Truncated Fourier Series ◽  
Energy Formulation

A relatively thin—thickness to radius ratio—circular ring with rectangular cross section has been investigated to numerically evaluate the effect of eccentricity on the in plane bending natural frequencies and mode shapes. The assumed boundary conditions correspond to a ring freely supported in space such that it is free to translate and rotate with rigid body motion. A truncated Fourier series solution is assumed in an energy formulation to obtain numerical approximations of the eigenvalues and the corresponding eigenvectors for different eccentricities. Extensional and inextensional models for both Flu¨gge and Love-Timoshenko ring models were considered with two thickness to radius ratios. Results show different rates of decrease in the magnitudes of the natural frequencies for different mode configurations. Existence of closely spaced frequencies along with modal coupling are noticeable at 50 percent eccentricity.

Effect of Slenderness Ratio on the Natural Frequencies of Pre-Twisted Cantilever Beams of Uniform Rectangular Cross-Section

1971 ◽  
Vol 13 (1) ◽  
pp. 51-59 ◽  
Author(s):  
B. Dawson ◽  
N. G. Ghosh ◽  
W. Carnegie
Keyword(s):  
Differential Equations ◽  
Shear Deformation ◽  
Cross Section ◽  
Natural Frequencies ◽  
Rectangular Cross Section ◽  
Slenderness Ratio ◽  
Twist Angle ◽  
Equations Of Motion ◽  
Rotary Inertia ◽  
Cantilever Beams

This paper is concerned with the vibrational characteristics of pre-twisted cantilever beams of uniform rectangular cross-section allowing for shear deformation and rotary inertia. A method of solution of the differential equations of motion allowing for shear deformation and rotary inertia is presented which is an extension of the method introduced by Dawson (1)§ for the solution of the differential equations of motion of pre-twisted beams neglecting shear and rotary inertia effects. The natural frequencies for the first five modes of vibration are obtained for beams of various breadth to depth ratios and lengths ranging from 3 to 20 in and pre-twist angle in the range 0–90°. The results are compared with those obtained by an alternative method (2), where available, and also to experimental results.

Dynamic Analysis of the Rotating Composite Thin-Walled Cantilever Beams with Shear Deformation

2013 ◽  
Vol 690-693 ◽  
pp. 309-313
Author(s):  
Yong Sheng Ren ◽  
Qi Yi Dai
Keyword(s):  
Shear Deformation ◽  
Cross Section ◽  
Fiber Orientation ◽  
Model Comparison ◽  
Natural Frequencies ◽  
Theoretical Study ◽  
Equations Of Motion ◽  
Cantilever Beams ◽  
The Finite Element Method ◽  
Rotating Speed

This paper presents a theoretical study of the dynamic characteristics of rotating composite cantilever beams. Considering shear deformation and cross section warping, the equations of motion of the rotating cantilever beams are derived using Hamilton’s principle. The Galerkin’s method is used in order to analysis the free vibration behaviors of the model. Comparison of the theoretical solutions has been made with the results obtained from the finite element method, which prove the validity of the model presented in this paper. Natural frequencies are obtained for circular tubular composite beams. The effects of fiber orientation, rotating speed and structure parameters on modal frequencies are investigated.

Sensitivity of Non-Uniform Beams in Depositing Mass Process

2006 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Film Thickness ◽  
Relative Density ◽  
Cross Section ◽  
Natural Frequencies ◽  
Rectangular Cross Section ◽  
Relative Sensitivity ◽  
Constant Thickness ◽  
Thickness Variation ◽  
Slender Beams ◽  
Beam Dimension

This paper deals with the mass deposition influence on the natural frequencies of nonuniform cantilever resonator sensors of linear and parabolic thickness. Resonator sensitivity, defined as fraction of change in frequency per fraction of change in thickness deposition and relative density, was found. A constant thickness mass deposition on all four lateral surfaces of the cantilever of rectangular cross-section was assumed. Euler-Bernoulli theory was used, so only slender beams were considered. Mass deposition on the free end surface of the beams was neglected. The film thickness was considered very small compared to any beam dimension. The film had no contribution to the beam stiffness, only to the mass. Results show that for the same thickness deposition, the sensitivity in the first mode of beams of linear thickness is 2.5 to 3.5 higher when compared to uniform beams. For beams of parabolic thickness variation the relative sensitivity ranges between 1.5 and 2.1.

Why Is (111) Silicon a Better Mechanical Material for MEMS: Torsion Case

2003 ◽  
Author(s):  
Donghun Kwak ◽  
Jongpal Kim ◽  
Sangjun Park ◽  
Hyoungho Ko ◽  
Dong-Il Cho
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Silicon Wafers ◽  
Torsional Stiffness ◽  
Rotational Inertia ◽  
The Finite Element Method ◽  
Element Type ◽  
Simulation Results

This paper shows that using the Finite Element Method (FEM), the torsional stiffness of silicon varies by the least amount on silicon (111) with respect to crystallographic directions, when compared to silicon (100) and (110). The used simulator is ANSYS 5.7 with the element type of Solid 64. As a simulation model, we use a simple torsion system, in which a rotational inertia is attached to the center of clamped-clamped beam with a rectangular cross-section. From the results of the modal analysis, the torsional stiffness is derived using the formula between the natural frequency and the torsional stiffness. Simulation results show that the maximum variations of the torsional stiffness on silicon (111), (100) and (110) are 2.3%, 26.5%, and 31.2%, respectively. This implies that on <100> and <110> silicon wafers, substantially different physical dimensions are necessary for devices with the same torsional characteristics, but with different orientations. Therefore, <111> silicon wafers represent a more suitable substrate to design and fabricate torsional micro and nano systems.

scholarly journals Optimal preliminary design of variable section beams criterion

2021 ◽  
Vol 3 (8) ◽  
Author(s):  
Raffaele Cucuzza ◽  
Marco Martino Rosso ◽  
Giuseppe Carlo Marano
Keyword(s):  
Cross Section ◽  
Numerical Solutions ◽  
Rectangular Cross Section ◽  
Optimal Solution ◽  
Beam Equation ◽  
Critical Threshold ◽  
Allowable Stress ◽  
Density Maximum ◽  
External Variable ◽  
Arch Effect

AbstractThe present paper discusses about optimal shape solution for a non-prismatic planar beam. The proposed model is based on the standard Timoshenko kinematics hypothesis (i.e., planar cross-section remains planar in consequence of a deformation, but it is able to rotate with respect to the beam center-line). The analytical solution for this type of beam is thus used to obtain deformations and stresses of the beam, under different constraints, when load is assumed as the sum of a generic external variable vertical one and the self-weight. The solution is obtained by numerical integration of the beam equation and constraints are posed both on deflection and maximum stress under the hypothesis of an ideal material. The section variability is, thus, described assuming a rectangular cross section with constant base and variable height which can be described in general with a trigonometric series. Other types of empty functions could also be analyzed in order to find the best strategy to get the optimal solution. Optimization is thus performed by minimizing the beam volume considering the effects of non-prismatic geometry on the beam behavior. Finally, several analytical and numerical solutions are compared with results existing in literature, evaluating the solutions’ sensibility to some key parameters like beam span, material density, maximum allowable stress and load distribution. In conclusion, the study finds a critical threshold in terms of emptying function beyond which it is not possible to neglect the arch effect and the curvature of the actual axis for every different case study described in this work. In order to achieve this goal, the relevance of beam span, emptying function level and maximum allowable stress are investigated.

Modal Analysis Using Implicit Boundary Finite Element Methods

2014 ◽  
Author(s):  
Zhiyuan Zhang ◽  
Ashok V. Kumar
Keyword(s):  
Finite Element ◽  
Modal Analysis ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
The Finite Element Method ◽  
B Spline ◽  
Boundary Method ◽  
Mass Matrices ◽  
Background Mesh ◽  
The Given

Modal analysis is widely used for linear dynamic analysis of structures. The finite element method is used to numerically compute stiffness and mass matrices and the corresponding eigenvalue problem is solved to determine the natural frequencies and mode shapes of vibration. Implicit boundary method was developed to use equations of the boundary to apply boundary conditions and loads so that a background mesh can be used for analysis. A background mesh is easier to generate because the elements do not have to conform to the given geometry and therefore uniform regular shaped elements can be used. In this paper, we show that this approach is suitable for modal analysis and modal superposition techniques as well. Furthermore, the implicit boundary method also allows higher order elements that use B-spline approximations. Several test examples are studied for verification.

Free Vibration of a Thin Shell Structure of Rectangular Cross-Section

2011 ◽  
Vol 486 ◽  
pp. 107-110 ◽  
Author(s):  
Yue Hua Chen ◽  
Guo Yong Jin ◽  
Zhi Gang Liu
Keyword(s):  
Free Vibration ◽  
Cross Section ◽  
Shell Structure ◽  
Rectangular Cross Section ◽  
Ritz Method ◽  
Coupled Model ◽  
Double Series ◽  
General Boundary ◽  
Mode Shapes ◽  
The Finite Element Method

This paper presents an analysis on the free vibration of a shell structure of rectangular cross-section. The shell of rectangular cross-section is modeled by four rectangular panels elastically connected at right angles under general boundary conditions. With the general boundary and coupling conditions accounted for several groups of linear springs, the double series solutions for both flexural and in-plane vibrations are obtained by employing the Rayleigh-Ritz method and the validation of the calculations is proved by comparing the eigenpairs with the Finite Element Method results. It is shown that the mode shapes of the rigidly coupled model perform a symmetrical feature.

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