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.

On Sensitivity of Capacitive Microresonators Near Three Half Natural Frequency

2010 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Martin Knecht
Keyword(s):  
Natural Frequency ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Constant Thickness ◽  
Microcantilever Sensors ◽  
Electrostatically Actuated ◽  
Resonance Frequencies ◽  
Parametric Oscillations ◽  
Deposition Thickness ◽  
Beam Dimension

This paper deals with sensitivity of electrostatically actuated micro resonator sensors near three half natural frequency. Mass deposition changes resonance frequencies of structures. Resonator sensitivity, defined as a fraction of change in frequency per unit deposited mass, can be found for microcantilever sensors electrostatically actuated to include fringe and Casimir effects. These actuation forces produce nonlinear parametric oscillations. Constant thickness mass deposition on all four lateral surfaces of the cantilever of rectangular cross-section was assumed. The Euler-Bernoulli theory was used under the assumption that the beams are slender. Mass deposition on the free end surface of the cantilever was neglected. The deposition thickness was considered uniform and very small compared to any beam dimension. The deposited mass had no contribution to the stiffness, only to the mass. Analytical expression of the sensitivity of electrostatically actuated uniform microcantilever resonators sensor near natural frequency can be determined.

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.

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.

scholarly journals Hydroelastic waves propagating along a frozen channel with non-uniform thickness of ice

2021 ◽  
Vol 2057 (1) ◽  
pp. 012021
Author(s):  
K N Zavyalova ◽  
K A Shishmarev ◽  
E A Batyaev ◽  
T I Khabakhpasheva
Keyword(s):  
Cross Section ◽  
Elastic Plate ◽  
Rectangular Cross Section ◽  
Normal Modes ◽  
Elastic Beam ◽  
Wave Profile ◽  
Constant Thickness ◽  
Ice Thickness ◽  
Thin Elastic Plate ◽  
Uniform Thickness

Abstract Hydroelastic waves propagating along a channel covered with ice of non-uniform thickness are considered. The channel has a rectangular cross section. The fluid in the channel is inviscid and incompressible. The ice is modeled as a thin elastic plate. The ice thickness changes linearly. The problem is reduced to the problem of the wave profile across the channel, which is solved using the normal modes of an elastic beam with non-uniform thickness. It is shown that with the decrease in the change in the ice thickness, the modes approach the normal modes of an elastic beam with a constant thickness. The behavior of the dispersion relations of the hydroelastic waves depending on the parameter describing the change in the ice thickness is studied.

Natural Frequencies of Pre-Twisted Airfoil Blades

2017 ◽  
Author(s):  
Neeraj Kavan Chakshu ◽  
Sunil K. Sinha
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Free Boundary ◽  
Cross Section ◽  
Cross Sections ◽  
Natural Frequencies ◽  
Rectangular Cross Section ◽  
Element Analysis ◽  
Free Boundary Conditions ◽  
Airfoil Blades

In this paper, the natural frequencies of pre-twisted cantilever blades of various angles of twist having different airfoil cross sections in the NACA 6 series have been determined. The main objectives of this paper are to replicate the results previously published for the similar types of blades but with the assumption of a uniform rectangular cross-section and to compare it with the results obtained for blades with more refined airfoil cross-sections. Cantilevered type clamped-free boundary conditions have been used in this paper for all blades. The comparison of the natural frequencies among different airfoils of the same NACA series has also been described in the paper in order to find out if any parameter of the airfoil such as camber, maximum thickness etc have any significant role in changing the frequencies of the beam. Commonly used commercial codes for finite element analysis have been used to determine these results.

FREE VIBRATION ANALYSIS OF CROSS-PLY LAYERED COMPOSITE BEAMS WITH FINITE LENGTH ON ELASTIC FOUNDATION

2008 ◽  
Vol 05 (01) ◽  
pp. 21-36 ◽  
Author(s):  
RAMAZAN-ALI JAFARI-TALOOKOLAEI ◽  
MOHAMMAD-HOSSEIN KARGARNOVIN ◽  
MOHAMMAD-TAGHI AHMADIAN
Keyword(s):  
Free Vibration ◽  
Elastic Foundation ◽  
Cross Section ◽  
Finite Length ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Rectangular Cross Section ◽  
Free Vibration Analysis ◽  
Composite Beams ◽  
Layered Composite

In this paper, free vibration analysis of cross-ply layered composite beams (LCB) with finite length and rectangular cross-section rested on an elastic foundation is investigated by finite element method. Based on the Timoshenko beam theory which includes the shear deformation and rotary inertia, the stiffness and mass matrices of a LCB are obtained using the energy method. Then, the natural frequencies are calculated by employing eigenvalue technique. The obtained results are verified against existing data in the literatures for a LCB with no foundation and uniform cross-section. Good agreements are observed between these cases. In the same way, the natural frequencies of a specific case, i.e. the stepped beam are calculated and finally, free vibrations of a symmetric and non-symmetric LCB are compared with each others.

Fracture Toughness of Welds Using As-Welded-Geometry/SENT Specimens

2016 ◽  
Author(s):  
Mohammed Uddin ◽  
Gery Wilkowski
Keyword(s):  
Cross Section ◽  
Rectangular Cross Section ◽  
Fracture Analysis ◽  
Constant Thickness ◽  
Circumferential Welds ◽  
Ongoing Work ◽  
Specimen Test ◽  
Initial Work ◽  
Edge Notch ◽  
Pipe Geometry

The single-edge notch tension (SENT) test is frequently used for the assessment of the integrity of welds with flaws in them; this is done since the SENT specimen has the same flaw orientation as a surface flaw in the weld, and has similar constraint that affects the brittle-to-ductile transition and upper-shelf value. Traditionally SENT specimens are machined with a rectangular cross-section from the weld, and the thickness might be reduced because of that machining operation. The toughness value of the constant thickness machined specimen is then used in a pipe fracture analysis. Of course real welds have crowns in the roots of the weldment, which are ignored in both the fracture specimen test and the pipe geometry fracture analysis. To assess the importance of the weld crown and root, SENT tests were conducted as an exploratory aspect to determine the effect on toughness. Additionally, assessment of results where the SENT specimen breaks in the weld or base metal outside the weld due to the reinforcing that is usually machined off and ignored were conducted. The use of a full-weldment cross-section in SENT testing can be done for axial seam welds or circumferential welds. The initial work was on axial seam welds, although there is ongoing work for circumferential welds as well.

ANALYSIS OF THE GENERAL EQUATIONS OF THE TRANSVERSE VIBRATION OF A PIECEWISE UNIFORM VISCOELASTIC PLATE

2020 ◽  
Vol 7 (3) ◽  
pp. 52-56
Author(s):  
MMATMATISA JALILOV ◽  
◽  
RUSTAM RAKHIMOV ◽  
Keyword(s):  
Cross Section ◽  
Transverse Vibration ◽  
Three Dimensional ◽  
Viscoelastic Plate ◽  
Constant Thickness ◽  
Regional Problem ◽  
Rigid Contact ◽  
Under Construction ◽  
Derivatives Of ◽  
Theoretical Results

This article discusses the analysis of the general equations of the transverse vibration of a piecewise homogeneous viscoelastic plate obtained in the “Oscillation of inlayer plates of constant thickness” [1]. In the present work on the basis of a mathematical method, the approached theory of fluctuation of the two-layer plates, based on plate consideration as three dimensional body, on exact statement of a three dimensional mathematical regional problem of fluctuation is stood at the external efforts causing cross-section fluctuations. The general equations of fluctuations of piecewise homogeneous viscoelastic plates of the constant thickness, described in work [1], are difficult on structure and contain derivatives of any order on coordinates x, y and time t and consequently are not suitable for the decision of applied problems and carrying out of engineering calculations. For the decision of applied problems instead of the general equations it is expedient to use confidants who include this or that final order on derivatives. The classical equations of cross-section fluctuation of a plate contain derivatives not above 4th order, and for piecewise homogeneous or two-layer plates the elementary approached equation of fluctuation is the equation of the sixth order. On the basis of the analytical decision of a problem the general and approached decisions of a problem are under construction, are deduced the equation of fluctuation of piecewise homogeneous two-layer plates taking into account rigid contact on border between layers, and also taking into account mechanical and rheological properties of a material of a plate. The received theoretical results for the decision of dynamic problems of cross-section fluctuation of piecewise homogeneous two-layer plates of a constant thickness taking into account viscous properties of their material allow to count more precisely the is intense-deformed status of plates at non-stationary external loadings.

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