Three alternative versions of the theory for a Timoshenko–Ehrenfest beam on a Winkler–Pasternak foundation

2020 ◽  
pp. 108128652094777
Author(s):  
Giulio Maria Tonzani ◽  
Isaac Elishakoff
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Detailed Comparison ◽  
Vibration Frequencies ◽  
Pasternak Foundation ◽  
New Model ◽  
Simply Supported ◽  
Model Based

This paper analyzes the free vibration frequencies of a beam on a Winkler–Pasternak foundation via the original Timoshenko–Ehrenfest theory, a truncated version of the Timoshenko–Ehrenfest equation, and a new model based on slope inertia. We give a detailed comparison between the three models in the context of six different sets of boundary conditions. In particular, we analyze the most common combinations of boundary conditions deriving from three typical end constraints, namely the simply supported end, clamped end, and free end. An interesting intermingling phenomenon is presented for a simply-supported (S-S) beam together with proof of the ‘non-existence’ of zero frequencies for free-free (F-F) and simply supported-free (S-F) beams on a Winkler–Pasternak foundation.

Free vibration analysis of porous laminated rotating circular cylindrical shells

2019 ◽  
Vol 25 (18) ◽  
pp. 2494-2508 ◽  
Author(s):  
Ahmad Reza Ghasemi ◽  
Mohammad Meskini
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Cylindrical Shells ◽  
Free Vibration Analysis ◽  
Equilibrium Equations ◽  
Rotating Speed ◽  
Simply Supported ◽  
Numerical Result ◽  
Love’S Shell Theory ◽  
Circular Cylindrical Shells

In this research, investigations are presented of the free vibration of porous laminated rotating circular cylindrical shells based on Love’s shell theory with simply supported boundary conditions. The equilibrium equations for circular cylindrical shells are obtained using Hamilton’s principle. Also, Navier’s solution is used to solve the equations of the cylindrical shell due to the simply supported boundary conditions. The results are compared with previous results of other researchers. The numerical result of this study indicates that with increase of the porosity coefficient the nondimensional backward and forward frequency decreased. Then the results of the free vibration of rotating cylindrical shells are presented in terms of the effects of porous coefficients, porous type, length to radius ratio, rotating speed, and axial and circumferential wave numbers.

Investigation of Vibration and Thermal Buckling of Nanobeams Embedded in An Elastic Medium Under Various Boundary Conditions

2015 ◽  
Vol 32 (3) ◽  
pp. 277-287 ◽  
Author(s):  
D. S. Mashat ◽  
A. M. Zenkour ◽  
M. Sobhy
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Equations Of Motion ◽  
Thermal Buckling ◽  
Governing Equations ◽  
Linear Temperature ◽  
Simply Supported ◽  
Various Boundary Conditions ◽  
Dynamic Version ◽  
Beam Theories

AbstractAnalyses of free vibration and thermal buckling of nanobeams using nonlocal shear deformation beam theories under various boundary conditions are precisely illustrated. The present beam is restricted by vertically distributed identical springs at the top and bottom surfaces of the beam. The equations of motion are derived using the dynamic version of Hamilton's principle. The governing equations are solved analytically when the edges of the beam are simply supported, clamped or free. Thermal buckling solution is formulated for two types of temperature change through the thickness of the beam: Uniform and linear temperature rise. To validate the accuracy of the results of the present analysis, the results are compared, as possible, with solutions found in the literature. Furthermore, the influences of nonlocal coefficient, stiffness of Winkler springs and span-to-thickness ratio on the frequencies and thermal buckling of the embedded nanobeams are examined.

Exact Solutions for Vibration of Multi-Span Rectangular Mindlin Plates

2002 ◽  
Vol 124 (4) ◽  
pp. 545-551 ◽  
Author(s):  
Y. Xiang ◽  
G. W. Wei
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Analytical Approach ◽  
Solution Method ◽  
Plate Boundary ◽  
Vibration Frequencies ◽  
Type Solution ◽  
Simply Supported ◽  
Space Technique ◽  
Benchmark Solutions

This paper presents the first-known exact solutions for the vibration of multi-span rectangular Mindlin plates with two opposite edges simply supported. The Levy type solution method and the state-space technique are employed to develop an analytical approach to deal with the vibration of rectangular Mindlin plates of multiple spans. Exact vibration frequencies are obtained for two-span square Mindlin plates with varying span ratios and two-, three- and four-equal-span rectangular Mindlin plates. The influence of the span ratios, the number of spans and plate boundary conditions on the vibration behavior of square and rectangular Mindlin plates is examined. The presented exact vibration results may serve as benchmark solutions for such plates.

Analytical Solution for Free Vibration of Laminated Curved Beam with Magnetostrictive Layers

2015 ◽  
Vol 07 (03) ◽  
pp. 1550050 ◽  
Author(s):  
R. Bayat ◽  
A. A. Jafari ◽  
O. Rahmani
Keyword(s):  
Analytical Solution ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Material Properties ◽  
Vibration Suppression ◽  
Radius Of Curvature ◽  
Curved Beam ◽  
Velocity Feedback ◽  
Simply Supported ◽  
Distributed Controller

An analytical model to study the free vibration suppression of laminated curved beam with embedded actuating layers is presented in this study. The magnetostrictive layers are used to control and enhance the vibration suppression. The governing differential equations of the model are derived using the Hamilton principle. Analytical solution of the equations governing laminated curved beam with embedded magnetostrictive layers are obtained for simply-supported boundary conditions. Velocity feedback with constant gain distributed controller, by using Terfenol-D as smart material, is chosen to achieve vibration suppression. The effects of material properties, radius of curvature, lamination scheme, and placement of magnetostrictive layers with respect to laminate midplane on vibration suppression are studied in details.

Vibration Frequencies of Tapered Bars and Circular Plates

1964 ◽  
Vol 31 (2) ◽  
pp. 329-331 ◽  
Author(s):  
H. D. Conway ◽  
E. C. H. Becker ◽  
J. F. Dubil
Keyword(s):  
Free Vibration ◽  
Test Cases ◽  
Cantilever Beams ◽  
Vibration Frequencies ◽  
Circular Plates ◽  
Approximate Methods ◽  
Resonant Frequencies ◽  
Simply Supported ◽  
Tapered Plates ◽  
Special Case

Calculations are made of the values of the transverse vibrationa resonant frequencies for truncated-cone cantilever beams for a number of geometries, the boundary conditions being clamped/free, clamped/simply supported, and clamped/clamped. After noting an analogy which exists between the free vibration of cones and linearly tapered plates for the special case when Poisson’s ratio = 1/3, data are also given for the resonant axisymmetrical frequencies of clamped tapered circular plates. Some data for partially tapered plates are also computed. Aside from their value for design purposes, these data can be used as test cases for assessing the accuracy of various approximate methods of solution.

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