Dynamic Instability of a Cantilever Column Subjected to a Follower Force Including Thermomechanical Coupling Effect

Within the context of the linear thermoelasticity theory, including thermomechanical coupling effect, the dynamic instability of equilibrium of an elastic cantilever column subjected to a follower-type force at its free end is studied. The surfaces of the column are either kept at constant temperature (isothermal) or thermally insulated (adiabatic). The boundary-value problem is first formulated, and then it is solved in general and in particular for the depth-length ratio much less than unity. Numerical results for the critical loads are given for various values of the thermal parameters. It is shown that, for a tangential follower force, the critical dynamic (oscillatory) instability load may be reduced through the coupling effect to approximately one half of that of the corresponding uncoupled isothermal problem (which was solved by Beck [14] in 1952). Analytical results are also obtained for the damping coefficients for both adiabatic and isothermal boundary conditions. From a comparison with available experiment for aluminum, it appears that material damping, at least for this material, is almost entirely due to thermomechanical coupling.

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Thermoelastic Vibration and Damping for Circular Timoshenko Beams

Journal of Applied Mechanics ◽

10.1115/1.3423590 ◽

1975 ◽

Vol 42 (2) ◽

pp. 405-410 ◽

Cited By ~ 17

Author(s):

R.-C. Shieh

Keyword(s):

Boundary Conditions ◽

Circular Cross Section ◽

Timoshenko Beams ◽

Thermoelastic Damping ◽

Governing Equations ◽

Damping Coefficients ◽

Thermal Boundary Conditions ◽

Transverse Vibrations ◽

Thermoelastic Vibration ◽

Linear Thermoelasticity

Within the framework of the theories of coupled linear thermoelasticity and Timoshenko beams, the vibration and thermoelastic damping (with emphasis on the transverse ones) of circular cross-section beams are studied. The governing equations are derived for the case of general mechanical boundary conditions and special thermal boundary conditions that follow the Newton surface heat transfer law. A variational principle governing the eigenfunctions associated with an eigenvalue is formulated. An exact solution, together with the thermoelastic damping coefficient, is obtained for the case of transverse vibrations of a simply supported beam with lateral surfaces thermally insulated and end surfaces kept at constant temperature. Numerical results, together with the discussion for the first two eigenvalues and the thermoelastic damping coefficients, are also presented.

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Identification of the string boundary conditions using natural frequencies

Proceedings of the Mavlyutov Institute of Mechanics ◽

10.21662/uim2016.1.007 ◽

2016 ◽

Vol 11 (1) ◽

pp. 38-52

Author(s):

I.M. Utyashev ◽

A.M. Akhtyamov

Keyword(s):

Inverse Problems ◽

Power Series ◽

Boundary Value Problem ◽

Boundary Conditions ◽

Natural Frequencies ◽

Boundary Value ◽

External Environment ◽

Direct And Inverse Problems ◽

Symmetric Potential ◽

Modified Method

The paper discusses direct and inverse problems of oscillations of the string taking into account symmetrical characteristics of the external environment. In particular, we propose a modified method of finding natural frequencies using power series, and also the problem of identification of the boundary conditions type and parameters for the boundary value problem describing the vibrations of a string is solved. It is shown that to identify the form and parameters of the boundary conditions the two natural frequencies is enough in the case of a symmetric potential q(x). The estimation of the convergence of the proposed methods is done.

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The inverse problem of recovering the coefficients of a differential equations on a graph

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2020-0070 ◽

2020 ◽

Vol 28 (5) ◽

pp. 727-738

Author(s):

Victor Sadovnichii ◽

Yaudat Talgatovich Sultanaev ◽

Azamat Akhtyamov

Keyword(s):

Inverse Problem ◽

Inverse Problems ◽

Differential Equations ◽

Boundary Value Problem ◽

Boundary Conditions ◽

Finite Number ◽

Characteristic Equation ◽

Arbitrary Number ◽

New Class ◽

Finite Set

AbstractWe consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the second, third, and fourth orders on a graph with three, four, and five edges. The inverse problem with an arbitrary number of edges is solved similarly.

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Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

Mediterranean Journal of Mathematics ◽

10.1007/s00009-021-01822-5 ◽

2021 ◽

Vol 18 (5) ◽

Author(s):

Francesco Aldo Costabile ◽

Maria Italia Gualtieri ◽

Anna Napoli

Keyword(s):

Boundary Value Problem ◽

Boundary Conditions ◽

Existence And Uniqueness ◽

Interpolation Problem ◽

Boundary Value ◽

Computational Tools ◽

Numerical Examples ◽

Odd Order ◽

Uniqueness Of The Solution ◽

Type Boundary

AbstractGeneral nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.

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On the Nonlocal Fractional Delta-Nabla Sum Boundary Value Problem for Sequential Fractional Delta-Nabla Sum-Difference Equations

Mathematics ◽

10.3390/math8040476 ◽

2020 ◽

Vol 8 (4) ◽

pp. 476

Author(s):

Jiraporn Reunsumrit ◽

Thanin Sitthiwirattham

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Boundary Value Problem ◽

Boundary Conditions ◽

Difference Equations ◽

Existence And Uniqueness ◽

Existence And Uniqueness Results ◽

Uniqueness Results ◽

Banach Contraction ◽

The Banach Contraction Principle

In this paper, we propose sequential fractional delta-nabla sum-difference equations with nonlocal fractional delta-nabla sum boundary conditions. The Banach contraction principle and the Schauder’s fixed point theorem are used to prove the existence and uniqueness results of the problem. The different orders in one fractional delta differences, one fractional nabla differences, two fractional delta sum, and two fractional nabla sum are considered. Finally, we present an illustrative example.

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On the dependence of the structure of boundary layers on the boundary conditions in a singularly perturbed boundary-value problem with multiple root of the related degenerate equation

Mathematical Notes ◽

10.1134/s0001434616010247 ◽

2016 ◽

Vol 99 (1-2) ◽

pp. 210-221 ◽

Cited By ~ 8

Author(s):

V. F. Butuzov

Keyword(s):

Boundary Value Problem ◽

Boundary Conditions ◽

Boundary Layers ◽

Boundary Value ◽

Multiple Root ◽

Singularly Perturbed ◽

Degenerate Equation

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Investigation of Thermal Resonance in Solution of a Two-Dimensional Singularly Perturbed Boundary-Value Problem of Nonstationary Heat Conduction with Nonlinear Boundary Conditions of the Stefan–Boltzmann Type

Journal of Engineering Physics and Thermophysics ◽

10.1023/b:joep.0000045149.48583.e6 ◽

2004 ◽

Vol 77 (4) ◽

pp. 700-706 ◽

Cited By ~ 1

Author(s):

A. V. Kotovich ◽

G. A. Nesenenko

Keyword(s):

Heat Conduction ◽

Boundary Value Problem ◽

Boundary Conditions ◽

Nonlinear Boundary ◽

Boundary Value ◽

Nonlinear Boundary Conditions ◽

Singularly Perturbed ◽

Nonstationary Heat ◽

Two Dimensional ◽

Nonstationary Heat Conduction

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Dynamic Instability Analysis and Design Charts of Curved Panels with Linearly Varying Periodic In-Plane Load

International Journal of Structural Stability and Dynamics ◽

10.1142/s0219455421501303 ◽

2021 ◽

pp. 2150130

Author(s):

G. Patel ◽

A. N. Nayak ◽

A. K. L. Srivastava

Keyword(s):

Finite Element ◽

Boundary Conditions ◽

Dynamic Instability ◽

Excitation Frequency ◽

Instability Region ◽

Concentrated Load ◽

Simply Supported ◽

Finite Element Software ◽

Design Charts ◽

Curved Panels

The present paper reports an extensive study on dynamic instability characteristics of curved panels under linearly varying in-plane periodic loading employing finite element formulation with a quadratic isoparametric eight nodded element. At first, the influences of three types of linearly varying in-plane periodic edge loads (triangular, trapezoidal and uniform loads), three types of curved panels (cylindrical, spherical and hyperbolic) and six boundary conditions on excitation frequency and instability region are investigated. Further, the effects of varied parameters, such as shallowness parameter, span to thickness ratio, aspect ratio, and Poisson’s ratio, on the dynamic instability characteristics of curved panels with clamped–clamped–clamped–clamped (CCCC) and simply supported-free-simply supported-free (SFSF) boundary conditions under triangular load are studied. It is found that the above parameters influence significantly on the excitation frequency, at which the dynamic instability initiates, and the width of dynamic instability region (DIR). In addition, a comparative study is also made to find the influences of the various in-plane periodic loads, such as uniform, triangular, parabolic, patch and concentrated load, on the dynamic instability behavior of cylindrical, spherical and hyperbolic panels. Finally, typical design charts showing DIRs in non-dimensional forms are also developed to obtain the excitation frequency and instability region of various frequently used isotropic clamped spherical panels of any dimension, any type of linearly varying in-plane load and any isotropic material directly from these charts without the use of any commercially available finite element software or any developed complex model.

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