On Flexural Vibrations of Shear Deformable Laminated Beams

2012 ◽  
Author(s):  
Stefano Lenci ◽  
Francesco Clementi
Keyword(s):  
Mechanical Characteristics ◽  
Natural Frequencies ◽  
Normal Direction ◽  
Rotational Inertia ◽  
Shear Deformations ◽  
Various Boundary Conditions ◽  
Elastic Interface ◽  
Shear Deformable ◽  
Perfect Adherence ◽  
Tangential Directions

The natural frequencies of a two-layer beam with an elastic interface are investigated. Each beam is modeled by the Timoshenko kinematics, and the interface coupling is linear in both normal and tangential directions. Attention is focused on the shear deformation, axial and rotational inertia, and interface normal stiffness. The dependence of the natural frequencies on these mechanical characteristics is investigated by solving the associated eigenvalue problem. The convergence of the solution toward that of a simplified problem obtained by neglecting axial and rotational inertia, shear deformations and by considering interface perfect adherence in the normal direction, is studied. Various boundary conditions are investigated to extend the generality of the proposed results.

A Limit Model for the Linear Dynamics of a Two-Layer Beam

2013 ◽  
Author(s):  
Stefano Lenci ◽  
Giuseppe Rega
Keyword(s):  
Original Problem ◽  
Natural Frequencies ◽  
Rotational Inertia ◽  
Asymptotic Approach ◽  
Shear Deformations ◽  
Development Method ◽  
First Order ◽  
Limit Model ◽  
Asymptotic Development ◽  
Perfect Adherence

The asymptotic development method is used to study the effects of axial and rotational inertia, of shear deformations and of interface normal compliance in the vibrations of a two-layer beam. Starting from a model with Timoshenko kinematics, a model with Euler-Bernoulli kinematics for each layer, and with normal perfect adherence at interface, is obtained as a limit case. The limit model has only 1 unknown, versus the 8 of the original problem, and has only 2 dimensionless parameters, versus the 14 of the original problem, and thus it is much more easy. The asymptotic approach used to obtain the limit model guarantees on its reliability. Both the limit natural frequencies and their first order corrections are computed; the latter, in particular, permit to determine the sensitivity with respect to the considered parameters.

A Nonlinear Shear Deformable Nanoplate Model Including Surface Effects for Large Amplitude Vibrations of Rectangular Nanoplates with Various Boundary Conditions

2015 ◽  
Vol 07 (05) ◽  
pp. 1550076 ◽  
Author(s):  
Reza Ansari ◽  
Mostafa Faghih Shojaei ◽  
Vahid Mohammadi ◽  
Raheb Gholami ◽  
Mohammad Ali Darabi
Keyword(s):  
Boundary Conditions ◽  
Surface Stress ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Free Vibrations ◽  
Model Parameters ◽  
Geometrically Nonlinear ◽  
Residual Surface Stress ◽  
Various Boundary Conditions ◽  
Shear Deformable

In this paper, a geometrically nonlinear first-order shear deformable nanoplate model is developed to investigate the size-dependent geometrically nonlinear free vibrations of rectangular nanoplates considering surface stress effects. For this purpose, according to the Gurtin–Murdoch elasticity theory and Hamilton's principle, the governing equations of motion and associated boundary conditions of nanoplates are derived first. Afterwards, the set of obtained nonlinear equations is discretized using the generalized differential quadrature (GDQ) method and then solved by a numerical Galerkin scheme and pseudo arc-length continuation method. Finally, the effects of important model parameters including surface elastic modulus, residual surface stress, surface density, thickness and boundary conditions on the vibration characteristics of rectangular nanoplates are thoroughly investigated. It is found that with the increase of the thickness, nanoplates can experience different vibrational behavior depending on the type of boundary conditions.

QUANTITATIVE IDENTIFICATION OF ROTOR CRACKS BASED ON FINITE ELEMENT OF B-SPLINE WAVELET ON THE INTERVAL

2009 ◽  
Vol 07 (04) ◽  
pp. 443-457 ◽  
Author(s):  
XUEFENG CHEN ◽  
BING LI ◽  
JIAWEI XIANG ◽  
ZHENGJIA HE
Keyword(s):  
Finite Element ◽  
Natural Frequencies ◽  
Beam Element ◽  
Rotational Inertia ◽  
Spline Wavelet ◽  
B Spline ◽  
Rotor Systems ◽  
Crack Location ◽  
Contour Lines ◽  
Quantitative Identification

Based on finite element of B-spline wavelet on the interval (BSWI), the quantitative identification method of transverse crack for rotor systems was studied. The new model of BSWI Rayleigh–Euler rotary beam element considering gyroscopic effect and rotational inertia was constructed to solve the first three natural frequencies of the cracked rotor with high precision, and the first three frequencies solution surfaces of normalized crack location and size were obtained by using surface-fitting technique. Then the first three metrical natural frequencies were employed as inputs of the solution curve surfaces. The intersection of the three frequencies contour lines predicted the normalized crack location and size. The numerical and experimental examples were given to verify the validity of the beam element for crack quantitative identification in rotor systems. The new method can be applied to prognosis and quantitative diagnosis of cracks in the rotor system.

Development of an Analytical Model for Beams With Two Dimples in Opposing Direction

2017 ◽  
Author(s):  
Mofareh Ghazwani ◽  
Kyle Myers ◽  
Koorosh Naghshineh
Keyword(s):  
Differential Equations ◽  
Opposite Direction ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Modal Strain Energy ◽  
Noise And Vibration ◽  
Various Boundary Conditions ◽  
Strategic Approach ◽  
Value Model ◽  
Good Agreement

Structures such as beams and plates can produce unwanted noise and vibration. An emerging technique can reduce noise and vibration without any additional weight or cost. This method focuses on creating two dimples in the same and opposite direction on a beam’s surface where the effect of dimples on its natural frequencies is the problem of interest. The change in the natural frequency between both cases have a different trend. The strategic approach to calculate natural frequencies is as follows: first, a boundary value model (BVM) is developed for a beam with two dimples and subject to various boundary conditions using Hamilton’s Variational Principle. Differential equations describing the motion of each segment are presented. Beam natural frequencies and mode shapes are obtained using a numerical solution of the differential equations. A finite element method (FEM) is used to model the dimpled beam and verify the natural frequencies of the BVM. Both methods are also validated experimentally. The experimental results show a good agreement with the BVM and FEM results. A fixed-fixed beam with two dimples in the same and opposite direction is considered as an example in order to compute its natural frequencies and mode shapes. The effect of dimple locations and angles on the natural frequencies are investigated. The natural frequencies of each case represent a greater sensitivity to change in dimple angle for dimples placed at high modal strain energy regions of a uniform beam.

Node Control of Uniform Beams Subject to Various Boundary Conditions

1992 ◽  
Vol 59 (4) ◽  
pp. 983-990 ◽  
Author(s):  
L. Weaver ◽  
L. Silverberg
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Controlled System ◽  
Free Boundary Conditions ◽  
Various Boundary Conditions ◽  
Natural Modes ◽  
Modes Of Vibration ◽  
Control Forces ◽  
Direct Feedback

This paper introduces node control, whereby discrete direct feedback control forces are placed at the nodes of the N+1th mode (the lowest N modes participate in the response). Node control is motivated by the node control theorem which states, under certain conditions, that node control preserves the natural frequencies and natural modes of vibration of the controlled system while achieving uniform damping. The node control theorem is verified for uniform beams with pinned-pinned, cantilevered, and free-free boundary conditions, and two cases of beams with springs on the boundaries. A general proof of the node control theorem remains elusive.

Use of "digital core" module in SAE Fidesis to determine effective parameters of fractured porous media

2020 ◽  
Author(s):  
Vladimir Levin
Keyword(s):  
Heterogeneous Media ◽  
Effective Properties ◽  
Displacement Vector ◽  
Fractured Reservoirs ◽  
Periodicity Cell ◽  
Shear Deformations ◽  
Fractured Medium ◽  
Digital Core ◽  
The Relationship ◽  
Tangential Directions

<p>Development of the homogenization algorithms for the heterogeneous periodic and non-periodic materials has applications in different domains and considers different types of upscaling techniques (Fish, 2008, Bagheri, Settari, 2005, Kachanov et al. 1994, Levin et al. 2003).</p><p>The current presentation discusses an algorithm implemented in CAE Fidesys (Levin, Zingerman, Vershinin 2015, 2017) for calculating the effective mechanical characteristics of a porous-fractured medium (Myasnikov et al., 2016) at the scale of a periodicity cell dissected by a group of plane-parallel cracks modeled by elastic bonds with specified stiffnesses in the normal and tangential directions in accordance with the method of modeling cracks based on elastic bonds (Bagheri, Settari, 2005, 2006) In this case, the relationship between the components of the displacement vector and the force vector (normal stresses at the fracture’s boundaries) in the normal and tangential directions will be diagonal, neglecting the effects of dilatancy and shear deformations as a result of normal stresses.<br>The presentation also considers the general case of the relationship between displacements and forces along the fracture’s boundaries, taking into account shear deformations (which leads to an increase in the effective Young's modulus by 30%), and additionally a cell’s geometrical model is generalized by the presence of pores in the matrix’s material. The results of numerical studies on mesh convergence, the influence of periodicity cell sizes and fracture’s thicknesses on the computed effective properties are presented. A comparison between analytical (Kachanov, Tsukrov 1994, 2000) and numerical results obtained in CAE Fidesys for the effective elastic moduli estimation for particular cases of geometrical models of the periodicity cell is shown.<br>The developed algorithm is used to evaluate the effective mechanical properties of a digital core model obtained by the results of CT-scan data interpretation. A comparison is made with the results of laboratory physical core tests. Additionaly an algorithm implemented in CAE Fidesys and the results for the effective thermal conductivity and the effective coefficient of thermal expansion estimation are given for the considered test rock specimen.</p><p>The reported study was funded by Russian Science Foundation project № 19-77-10062. </p><p> </p><p> </p><ol><li>Bagheri, M., Settari, A. Effects of fractures on reservoir deformation and flow modeling // Can. Geotech. J. 43: 574–586 (2006) doi:10.1139/T06-024</li> <li>Bagheri, M., Settari, A. Modeling of Geomechanics in Naturally Fractured Reservoirs – SPE-93083-MS, SPE Reservoir Simulation Symposium, Houston, USA, 2005.</li> <li>Fish J., Fan R. Mathematical homogenization of nonperiodic heterogeneous media subjected to large deformation transient loading // International Journal for Numerical Methods in Engineering. 2008. V. 76. – P. 1044–1064.</li> <li>Kachanov M., Tsukrov I., Shafiro B. Effective moduli of a solid with holes and cavities of various shapes// Appl. Mech. Reviews. 1994. V. 47, № 1, Part 2. P. S151-S174.</li> </ol>

FREE VIBRATION ANALYSIS OF DISCRETELY STIFFENED SKEW PLATES

2001 ◽  
Vol 01 (01) ◽  
pp. 125-144 ◽  
Author(s):  
HUAN ZENG ◽  
CHARLES W. BERT
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Vibration Characteristics ◽  
Skew Angle ◽  
Various Boundary Conditions ◽  
Skew Plate ◽  
Convergence Study

Stiffened skew plates find application in various engineering fields. The free vibration characteristics of such plates have been studied by various methods. An orthogonally stiffened skew plate is a skew plate with stiffeners running orthogonal to two opposite edges. To the best knowledge of the present investigators, no previous work has been done for free vibration characteristics of skew plates of such stiffening geometry. The present work studies the free vibration of such plates. The pb-2 Rayleigh–Ritz method was employed due to its accuracy and computational efficiency. The conventional finite element method was also used as a comparative check. A convergence study was first performed for various boundary conditions. Then the vibration of orthogonally stiffened skew plates with different boundary conditions was studied. Close agreement was found between these two methods. The variations of natural frequencies with different parameters, including skew angle ϕ, edge ratio b/a, and height-thickness ratio f/h, were investigated for three types of boundary conditions.

A Finite Element Approach to Plate Vibration Problems

1965 ◽  
Vol 7 (1) ◽  
pp. 28-32 ◽  
Author(s):  
D. J. Dawe
Keyword(s):  
Finite Element ◽  
Arbitrary Shape ◽  
Natural Frequencies ◽  
Isotropic Plate ◽  
Uniform Thickness ◽  
Flat Plates ◽  
Finite Element Approach ◽  
Various Boundary Conditions ◽  
Element Approach ◽  
Vibration Problems

A method of computing the natural frequencies of vibration of flat plates of arbitrary shape is outlined in which the plate is considered as an assemblage of elements. Both stiffness and inertia matrices are derived for a rectangular isotropic plate element of uniform thickness, and these matrices are used to find the natural frequencies of square plates subject to various boundary conditions. Comparison of finite element frequencies with known exact, experimental and energy solutions shows the method to give good results even for relatively few elements.

Sign in / Sign up

Export Citation Format

Share Document