Nonlinear Vibrations of Rectangular Plates With Different Boundary Conditions: Theory and Experiments

2005 ◽  
Author(s):  
M. Amabili ◽  
C. Augenti
Keyword(s):  
Boundary Condition ◽  
Good Accuracy ◽  
Continuation Method ◽  
Laser System ◽  
Normal Displacement ◽  
Vibration Modes ◽  
Rectangular Plates ◽  
Geometric Imperfections ◽  
Simply Supported ◽  
Actual Geometry

Large-amplitude vibrations of rectangular plates subjected to harmonic excitation are investigated. The von Ka´rma´n nonlinear strain-displacement relationships are used to describe the geometric nonlinearity. A specific boundary condition, with restrained normal displacement at the plate edges and fully free in-plane displacements, not previously considered, has been introduced as a consequence that it is very close to the experimental boundary condition. Results for this boundary condition are compared to nonlinear results previously obtained for: (i) simply supported plates with immovable edges; (ii) simply supported plates with movable edges, and (iii) fully clamped plates. The nonlinear equations of motion are studied by using a code based on arclength continuation method. A thin rectangular stainless-steel plate has been inserted in a metal frame; this constraint is approximated with good accuracy by the newly introduced boundary condition. The plate inserted into the frame has been measured with a 3D laser system in order to reconstruct the actual geometry and identify geometric imperfections (out-of-planarity). The plate has been experimentally tested in laboratory for both the first and second vibration modes for several excitation magnitudes in order to characterize the nonlinearity of the plate with imperfections. Numerical results are able to follow experimental results with good accuracy for both vibration modes and for different excitation levels once the geometric imperfection is introduced in the model. Effects of geometric imperfections on the trend of nonlinearity and on natural frequencies are shown; convergence of the solution with the number of generalized coordinates is numerically verified.

Effects of Initial Geometric Imperfections on Parametric Instability of Rectangular Plates

2001 ◽  
Author(s):  
Lyne St-Georges ◽  
G. L. Ostiguy
Keyword(s):  
Lateral Displacement ◽  
Plate Thickness ◽  
Parametric Instability ◽  
Experimental Tests ◽  
Rectangular Plates ◽  
Rational Analysis ◽  
Geometric Imperfections ◽  
Simply Supported ◽  
Geometrical Imperfections ◽  
Initial Geometric Imperfections

Abstract The authors present a rational analysis of the effect of initial geometric imperfections on the dynamic behaviour of rectangular plates activated by a parametric excitation. This subject has been extensively investigated theoretically in the past, but no experimental data seems to be complete enough to validate the theory. The main objective of this investigation is to fill this void by performing experimental tests on geometrically imperfect plates, and to highlight the geometric imperfection’s influence on resonance’s curves. The study is carried out for an isotropic, elastic, homogeneous, and thin rectangular plate. The plate under investigation is subjected to the action of an in-plane force uniformly distributed along two opposite edges, is initially stress free and simply supported. Theoretical calculation and experimental tests are performed. In the theoretical approach, a dynamic version of the Von Kármán non-linear theory is used to evaluate the lateral displacement of the plate. The test rig used in the experimentation simulates simply supported edges and can accept plates with different aspect ratio. The test plates are pre-formed with lateral deflection or geometrical imperfections, in a shape corresponding to various vibration modes. Comparison between experimental and theoretical results reveals good agreement and allows the determination of the theory’s limitations. The theory used correctly describes the behaviour of the plate when imperfection amplitude is inferior to the plate thickness.

The Bending of the Clamped Sectorial Plate

1944 ◽  
Vol 11 (3) ◽  
pp. A134-A139
Author(s):  
G. F. Carrier
Keyword(s):  
Boundary Condition ◽  
Integral Equation Method ◽  
Rectangular Plates ◽  
Uniform Pressure ◽  
Bending Moments ◽  
Concentrated Load ◽  
Design Problems ◽  
Simply Supported ◽  
Computational Work ◽  
The Relationship

Abstract The problem of evaluating the bending moments, existing in a uniformly loaded clamped plate having the form of a sector of a ring, is one which arises in connection with the stress analysis of reinforced piston heads and in other design problems. In this paper, expressions are derived for the bending moments along the edges of such a plate. Similar problems, i.e., those of the clamped rectangular plate under uniform pressure, under a central concentrated load, and that of the simply supported sector of a disk under uniform pressure, have been discussed by previous authors. The general approach used in the foregoing problems is adopted in the present case; a considerable reduction in the computational work is achieved, however, by the use of an integral-equation method of solving the boundary-condition equations. Numerical results are obtained for plates of various dimensions, and the edge moment distributions are plotted for these cases. Curves are also plotted which indicate the relationship existing between the maximum bending moments derived for sectorial plates and those previously obtained for clamped rectangular plates of similar size.

Effects of Geometric Imperfections on Frequency-Load Interaction of Biaxially Compressed Antisymmetric Angle Ply Rectangular Plates

1985 ◽  
Vol 52 (1) ◽  
pp. 155-162 ◽  
Author(s):  
David Hui
Keyword(s):  
Biaxial Loading ◽  
Composite Plates ◽  
Laminated Plates ◽  
Rectangular Plates ◽  
Vibration Frequencies ◽  
Geometric Imperfections ◽  
Simply Supported ◽  
Interaction Curves

The present paper deals with the influence of small geometric imperfections on the vibration frequencies of rectangular, simply supported, angle ply, thin composite plates subjected to inplane uniaxial or biaxial compressive preload. Depending on the amount of preload, the frequencies of laminated plates with different imperfection shapes may be significantly higher than those for perfect plates, especially in a certain range of fiber angles. Interaction curves between frequency and applied preload are plotted for various fiber angles and imperfection amplitudes for both the uniaxial and equal biaxial loading cases.

Large-Amplitude Vibrations of Empty and Fluid-Filled Circular Cylindrical Shells With Imperfections: Theory and Experiments

2002 ◽  
Author(s):  
M. Amabili ◽  
M. Pellegrini ◽  
F. Pellicano
Keyword(s):  
Large Amplitude ◽  
Cylindrical Shells ◽  
Continuation Method ◽  
Harmonic Excitation ◽  
Steel Shell ◽  
Nonlinear Phenomena ◽  
Inviscid Fluid ◽  
Simply Supported ◽  
Circular Cylindrical Shells ◽  
Actual Geometry

The large-amplitude response of perfect and imperfect, simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of some of the lowest natural frequencies is investigated. Donnell’s nonlinear shallow-shell theory is used and the solution is obtained by Galerkin method. Several expansions involving 16 or more natural modes of the shell are used. The boundary conditions on the radial displacement (simply supported shell at both ends) and the continuity of circumferential displacement are exactly satisfied. The effect of internal quiescent, incompressible and inviscid fluid is investigated. The nonlinear equations of motion are studied by using a code based on arclength continuation method. A series of accurate experiments on forced vibrations of an empty and water-filled stainless-steel shell have been performed. Several modes have been intensively investigated for different vibration amplitudes. A closed loop control of the force excitation has been used. The actual geometry of the test shell has been measured and the geometric imperfections have been introduced in the theoretical model. Several interesting nonlinear phenomena have been experimentally observed and numerically reproduced, as: softening-type nonlinearity, different types of travelling wave response in the proximity of resonances and amplitude-modulated response. For all the modes investigated, the theoretical and experimental results are in strong agreement.

Deformation of Rectangular Slender Webplates with Boundary Members Flexible in the Webplate Plane

1966 ◽  
Vol 17 (4) ◽  
pp. 371-394 ◽  
Author(s):  
J. Djubek
Keyword(s):  
Linear Problem ◽  
Numerical Results ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Non Linear

SummaryThe paper presents a solution of the non-linear problem of the deformation of slender rectangular plates which are stiffened along their edges by elastically compressible stiffeners flexible in the plane of the plate. The webplate is assumed to be simply-supported along its contour. Numerical results showing the effect of flexural and normal rigidity of stiffeners are given for a square webplate loaded by shear and compression.

Thermal post-buckling of laminated and isotropic rectangular plates with fixed edges: Comparison of experimental and numerical results

2012 ◽  
Vol 226 (10) ◽  
pp. 2393-2401 ◽  
Author(s):  
Marco Amabili ◽  
Mohammad Reza Sareban Tajahmadi
Keyword(s):  
Deformation Theory ◽  
Laboratory Experiments ◽  
Continuation Method ◽  
Shear Deformation Theory ◽  
Laminated Composite ◽  
Numerical Results ◽  
Rectangular Plates ◽  
Arc Length ◽  
Thermal Changes ◽  
Post Buckling

Post-buckling behaviors of laminated composite and isotropic rectangular plates subjected to various thermal changes are studied. Geometric imperfections are taken into account since they play a fundamental role. The plate is modeled using a nonlinear, higher order shear deformation theory. Plates with clamped edges are considered. A pseudo-arc length continuation method is used to obtain numerical results. Laboratory experiments have been performed in order to compare to the numerical calculations.

scholarly journals Dual-series equations formulation for static deformation of plates with a partial internal line support

2007 ◽  
Vol 34 (3) ◽  
pp. 221-248 ◽  
Author(s):  
Yos Sompornjaroensuk ◽  
Kraiwood Kiattikomol
Keyword(s):  
Integral Equation ◽  
Fredholm Integral Equation ◽  
Hankel Transform ◽  
Internal Line ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Distributed Load ◽  
Dual Series Equations ◽  
Finite Hankel Transform ◽  
Standard Techniques

The paper deals with the application of dual-series equations to the problem of rectangular plates having at least two parallel simply supported edges and a partial internal line support located at the centre where the length of internal line support can be varied symmetrically, loaded with a uniformly distributed load. By choosing the proper finite Hankel transform, the dual-series equations can be reduced to the form of a Fredholm integral equation which can be solved conveniently by using standard techniques. The solutions of integral equation and the deformations for each case of the plates are given and discussed in details.

Sign in / Sign up

Export Citation Format

Share Document