Comparison of 1D and 2D Theories of Thermoelastic Damping in Flexural Microresonators

2007 ◽  
Vol 1052 ◽  
Author(s):  
Sairam Prabhakar ◽  
Srikar Vengallatore
Keyword(s):  
Boundary Conditions ◽  
Two Dimensions ◽  
Temperature Gradients ◽  
Mode Shapes ◽  
Thermoelastic Damping ◽  
Flexural Mode ◽  
Aspect Ratios ◽  
Curvature Information ◽  
Structural Boundary ◽  
2D Model

AbstractThermoelastic damping (TED) represents the lower limit of material damping in flexural mode micro- and nanoresonators. Current predictive models of TED calculate damping due to thermoelastic temperature gradients along the beam thickness only. In this work, we develop a two dimensional (2D) model by considering temperature gradients along the thickness and the length of the beam. The Green's function approach is shown to be a robust means of solving the coupled heat conduction equation in one and two dimensions. In the 1D model, curvature information is lost and, hence, the effects of structural boundary conditions and mode shapes on TED are not captured. In contrast, the 2D model retains curvature information in the expression for TED and can account for beam end conditions and higher order modes. The differences between the 1D and 2D models are systematically explored over a range of beam aspect ratios, frequencies, boundary conditions, and flexural mode shapes.

scholarly journals SHEAR BUCKLING OF THIN PLATES USING THE SPLINE COLLOCATION METHOD

2008 ◽  
Vol 08 (04) ◽  
pp. 645-664 ◽  
Author(s):  
LAI-YUN WU ◽  
CHENG-HUNG WU ◽  
HSU-HUI HUANG
Keyword(s):  
Boundary Conditions ◽  
Collocation Method ◽  
Buckling Load ◽  
Mode Shapes ◽  
Lagrangian Multiplier ◽  
Spline Collocation ◽  
Buckling Loads ◽  
Spline Collocation Method ◽  
Aspect Ratios ◽  
Shear Buckling

This paper presents a highly accurate method for analyzing the critical shear buckling load of thin elastic rectangular plates. The solutions are approximated by the extended spline collocation method (SCM). Using the quintic table in place of the complex quintic B-spline functions, one can easily formulate the field equation of shear buckling loads for a thin elastic rectangular plate. Through the generalized eigenvalue analysis, the shear buckling loads and mode shapes for the plate can be determined precisely. Numerical examples are given for the critical shear buckling load of plates with various combinations of boundary conditions, aspect ratios, and uni- and bi-directional compressive/tensile loadings. The solutions obtained by the SCM are compared with those by the finite element method, the Lagrangian multiplier method, and the extended Kantorovich method under several types of boundary conditions. Compared with the other methods, the proposed SCM is not only more accurate, but also easier for computation.

Buckling Behavior of Non-Uniformly Heated Tapered Laminated Composite Plates with Ply Drop-Off

2018 ◽  
Vol 18 (04) ◽  
pp. 1850059 ◽  
Author(s):  
Shushanth Ashok ◽  
Jeyaraj Pitchaimani
Keyword(s):  
Boundary Conditions ◽  
Temperature Profile ◽  
Laminated Composite ◽  
Composite Plates ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Buckling Mode ◽  
Buckling Behavior ◽  
Parametric Studies ◽  
Structural Boundary

The thermal buckling characteristics of non-uniformly heated tapered laminated composites plates with ply drop-off have been investigated numerically. Detailed parametric studies have been carried out for the effects of taper configuration, temperature variation, aspect ratio and structural boundary conditions on critical buckling temperatures and buckling mode shapes. It is found that the nature of taper as well as the applied temperature field have considerable effects on the critical buckling temperatures of laminated composite tapered plates. Square plates buckle at the highest temperature when subjected to the decreasing temperature profile. Additionally, it is noted that Taper B and Taper C plates show the best behavior under buckling for most structural boundary conditions. Moreover, the change in buckling mode shapes with respect to temperature profile and taper configuration is significant for rectangular plates in comparison with square plates.

Coupled Dynamic Instability Analysis of Thin-Walled Columns Subjected to Harmonic Axial Loading

2018 ◽  
Vol 10 (05) ◽  
pp. 1850051 ◽  
Author(s):  
Amit Yadav ◽  
Sarat Kumar Panda ◽  
Tanish Dey
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Cross Section ◽  
Dynamic Instability ◽  
Axial Loading ◽  
Mode Shapes ◽  
Instability Analysis ◽  
Aspect Ratios ◽  
Instability Regions ◽  
Thin Walled

This paper presents triply coupled vibrations and instability analysis of thin-walled columns having a non-symmetrical open cross-section. Vlasov’s theory is used to derive the governing differential equations for coupled flexural and torsional vibrations. A numerical method is presented to determine the exact natural frequencies and corresponding mode shapes in terms of real functions. Employing Galerkin’s method, the coupled partial differential equations are reduced to a set of coupled Mathieu type equations. Following Bolotin’s method, the principal instability regions of thin-walled column with a non-symmetrical cross-section having various boundary conditions are determined. Numerical examples are presented to examine the effect of different boundary conditions, aspect ratios, static and dynamic load factors on principal instability regions. The study of response and corresponding phase plot in stable and unstable regions are carried out to identify the dynamic instability behavior.

Effect of crack on free vibration of a pre-stressed curved plate

2021 ◽  
pp. 095440622199486
Author(s):  
Can Gonenli ◽  
Hasan Ozturk ◽  
Oguzhan Das
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Natural Frequency ◽  
Present Model ◽  
Mode Shapes ◽  
Load Parameter ◽  
Flexible Plate ◽  
Cantilever Plate ◽  
Finite Element Software ◽  
Aspect Ratios

In this study, the effect of crack on free vibration of a large deflected cantilever plate, which forms the case of a pre-stressed curved plate, is investigated. A distributed load is applied at the free edge of a thin cantilever plate. Then, the loading edge of the deflected plate is fixed to obtain a pre-stressed curved plate. The large deflection equation provides the non - linear deflection curve of the large deflected flexible plate. The thin curved plate is modeled by using the finite element method with a four-node quadrilateral element. Three different aspect ratios are used to examine the effect of crack. The effect of crack and its location on the natural frequency parameter is given in tables and graphs. Also, the natural frequency parameters of the present model are compared with the finite element software results to verify the reliability and validity of the present model. This study shows that the different mode shapes are occurred due to the change of load parameter, and these different mode shapes cause a change in the effect of crack.

Free Bending Vibrations of a Centrally Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint) With a Gap in Mindlin Plates or Panels

2007 ◽  
Author(s):  
U. Yuceoglu ◽  
O. Gu¨vendik ◽  
V. O¨zerciyes
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Shear Stresses ◽  
Lap Joint ◽  
Bending Vibrations ◽  
Joint System ◽  
Adhesive Layers ◽  
Bonded Joint ◽  
Free Bending

In this present study, the “Free Bending Vibrations of a Centrally Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint) with a Gap in Mindlin Plates or Panels” are theoretically analyzed and are numerically solved in some detail. The “plate adherends” and the upper and lower “doubler plates” of the “Bonded Joint” system are considered as dissimilar, orthotropic “Mindlin Plates” joined through the dissimilar upper and lower very thin adhesive layers. There is a symmetrically and centrally located “Gap” between the “plate adherends” of the joint system. In the “adherends” and the “doublers” of the “Bonded Joint” assembly, the transverse shear deformations and the transverse and rotary moments of inertia are included in the analysis. The relatively very thin adhesive layers are assumed to be linearly elastic continua with transverse normal and shear stresses. The “damping effects” in the entire “Bonded Joint” system are neglected. The sets of the dynamic “Mindlin Plate” equations of the “plate adherends”, the “double doubler plates” and the thin adhesive layers are combined together with the orthotropic stress resultant-displacement expressions in a “special form”. This system of equations, after some further manipulations, is eventually reduced to a set of the “Governing System of the First Order Ordinary Differential Equations” in terms of the “state vectors” of the problem. Hence, the final set of the aforementioned “Governing Systems of Equations” together with the “Continuity Conditions” and the “Boundary conditions” facilitate the present solution procedure. This is the “Modified Transfer Matrix Method (MTMM) (with Interpolation Polynomials). The present theoretical formulation and the method of solution are applied to a typical “Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint) with a Gap”. The effects of the relatively stiff (or “hard”) and the relatively flexible (or “soft”) adhesive properties, on the natural frequencies and mode shapes are considered in detail. The very interesting mode shapes with their dimensionless natural frequencies are presented for various sets of boundary conditions. Also, several parametric studies of the dimensionless natural frequencies of the entire system are graphically presented. From the numerical results obtained, some important conclusions are drawn for the “Bonded Joint System” studied here.

scholarly journals Interaction between mean flow and turbulence in two dimensions

2016 ◽  
Vol 472 (2191) ◽  
pp. 20160287 ◽  
Author(s):  
Gregory Falkovich
Keyword(s):  
Short Note ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Mean Flow ◽  
Turbulence Statistics ◽  
Planar Domains ◽  
Developed Turbulence ◽  
Aspect Ratios ◽  
Simple Geometry ◽  
Turbulent Cascade

This short note is written to call attention to an analytic approach to the interaction of developed turbulence with mean flows of simple geometry (jets and vortices). It is instructive to compare cases in two and three dimensions and see why the former are solvable and the latter are not (yet). We present the analytical solutions for two-dimensional mean flows generated by an inverse turbulent cascade on a sphere and in planar domains of different aspect ratios. These solutions are obtained in the limit of small friction when the flow is strong while turbulence can be considered weak and treated perturbatively. I then discuss when these simple solutions can be realized and when more complicated flows may appear instead. The next step of describing turbulence statistics inside a flow and directions of possible future progress are briefly discussed at the end.

scholarly journals Detailed Analysis of the Acoustic Mode Shapes of an Annular Combustion Chamber

1999 ◽  
Author(s):  
Günther Walz ◽  
Werner Krebs ◽  
Stefan Hoffmann ◽  
Hans Judith
Keyword(s):  
Boundary Conditions ◽  
Acoustic Mode ◽  
Mode Shapes ◽  
Maximum Deviation ◽  
Minor Importance ◽  
Velocity Of Sound ◽  
Fe Method ◽  
Shift Frequency ◽  
Space Dependence ◽  
Eigenfrequency Spectrum

To get a better understanding of the formation of thermoacoustic oscillations in an annular gasturbine combustor, an analysis of the acoustic eigenmodes has been conducted using the Finite Element (FE) method. The influence of different boundary conditions and a space dependent velocity of sound has been investigated. The boundary conditions actually define the eigenfrequency spectrum. Hence, it is crucial to know e.g. the burner impedance. In case of the combustion system without significant mixing air addition considered in this paper, the space dependence of the velocity of sound is of minor importance for the eigenfrequency spectrum leading to a maximum deviation of only 5% in the eigenvalues. It is demonstrated that the efficiency of the numerical eigenvalue analysis can be improved by making use of symmetry, by splitting the problem into several steps with alternate boundaries conditions, and by choosing the shift frequency ωs in the range of frequencies one is interested in.

scholarly journals Boundary states for chiral symmetries in two dimensions

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Philip Boyle Smith ◽  
David Tong
Keyword(s):  
Boundary Conditions ◽  
Ground State ◽  
Zero Mode ◽  
Central Charge ◽  
Two Dimensions ◽  
Dirac Fermions ◽  
Chiral Symmetries ◽  
Boundary States ◽  
Ground State Degeneracy ◽  
Left And Right

Abstract We study boundary states for Dirac fermions in d = 1 + 1 dimensions that preserve Abelian chiral symmetries, meaning that the left- and right-moving fermions carry different charges. We derive simple expressions, in terms of the fermion charge assignments, for the boundary central charge and for the ground state degeneracy of the system when two different boundary conditions are imposed at either end of an interval. We show that all such boundary states fall into one of two classes, related to SPT phases supported by (−1)F , which are characterised by the existence of an unpaired Majorana zero mode.

Approximate Formulas for Cylindrical Shell Free Vibration Based on Vlasov’s and Enhanced Vlasov’s Semi-Momentless Theory

2018 ◽  
Author(s):  
Igor Orynyak ◽  
Yaroslav Dubyk
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Middle Surface ◽  
Axial Direction ◽  
Mode Shapes ◽  
Natural Mode ◽  
Transverse Deflection ◽  
Approximate Formulas

Simple approximate formulas for the natural frequencies of circular cylindrical shells are presented for modes in which transverse deflection dominates. Based on the Donnell-Mushtari thin shell theory the equations of motion of the circular cylindrical shell are introduced, using Vlasov assumptions and Fourier series for the circumferential direction, an exact solution in the axial direction is obtained. To improve the results assumptions of Vlasov’s semimomentless theory are enhanced, i.e. we have used only the hypothesis of middle surface inextensibility to obtain a solution in axial direction. Nonlinear characteristic equations and natural mode shapes, are derived for all type of boundary conditions. Good agreement with experimental data and FEM is shown and advantage over the existing formulas for a variety of boundary conditions is presented.

EFFECTS OF THERMAL CREEP ON COOLING OF MICROFLOWS IN SHORT MICROCHANNELS WITH CONSTANT WALL TEMPERATURE

2012 ◽  
Vol 23 (11) ◽  
pp. 1250072 ◽  
Author(s):  
ALI AMIRI-JAGHARGH ◽  
HAMID NIAZMAND ◽  
METIN RENKSIZBULUT
Keyword(s):  
Wall Temperature ◽  
Temperature Jump ◽  
Control Volume ◽  
Velocity Slip ◽  
Temperature Gradients ◽  
Thermal Creep ◽  
Inlet Temperature ◽  
Constant Wall Temperature ◽  
Aspect Ratios ◽  
Wide Range

Fluid flow and heat transfer in the entrance region of rectangular microchannels of various aspect ratios are numerically investigated in the slip-flow regime with particular attention to thermal creep effects. Uniform inlet velocity and temperature profiles are prescribed in microchannels with constant wall temperature. An adiabatic section is also employed at the inlet of the channel in order to prevent unrealistically large axial temperature gradients due to the prescribed uniform inlet temperature as well as upstream diffusion associated with low Reynolds number flows. A control-volume technique is used to solve the Navier–Stokes and energy equations which are accompanied with appropriate velocity slip and temperature jump boundary conditions at the walls. Despite the constant wall temperature, axial and peripheral temperature gradients form in the gas layer adjacent to the wall due to temperature jump. The simultaneous effects of velocity slip, temperature jump and thermal creep on the flow and thermal patterns along with the key flow parameters are examined in detail for a wide range of cross-sectional aspect ratios, and Knudsen and Reynolds numbers. Present results indicate that thermal creep effects influence the flow field and the temperature distribution significantly in the early section of the channel.

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