Free Bending Vibrations of Adhesively Bonded Orthotropic Plates With a Single Lap Joint

1996 ◽  
Vol 118 (1) ◽  
pp. 122-134 ◽  
Author(s):  
U. Yuceoglu ◽  
F. Toghi ◽  
O. Tekinalp
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Plate Theory ◽  
Mode Shapes ◽  
Mindlin Plate ◽  
Lap Joint ◽  
Orthotropic Plates ◽  
Adhesively Bonded ◽  
Bending Vibrations ◽  
Free Bending

This study is concerned with the free bending vibrations of two rectangular, orthotropic plates connected by an adhesively bonded lap joint. The influence of shear deformation and rotatory inertia in plates are taken into account in the equations according to the Mindlin plate theory. The effects of both thickness and shear deformations in the thin adhesive layer are included in the formulation. Plates are assumed to have simply supported boundary conditions at two opposite edges. However, any boundary conditions can be prescribed at the other two edges. First, equations of motion at the overlap region are derived. Then, a Levy-type solution for displacements and stress resultants are used to formulate the problem in terms of a system of first order ordinary differential equations. A revised version of the Transfer Matrix Method together with the boundary and continuity conditions are used to obtain the frequency equation of the system. The natural frequencies and corresponding mode shapes are obtained for identical and dissimilar adherends with different boundary conditions. The effects of some parameters on the natural frequencies are studied and plotted.

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.

Comparison of Free Flexural Vibrations of Composite Plates or Panels Stiffened by a Central and a Non-Central Plate Strip

1999 ◽  
Author(s):  
U. Yuceoglu ◽  
V. Özerciyes
Keyword(s):  
Natural Frequencies ◽  
Plate Theory ◽  
Composite Plates ◽  
Mode Shapes ◽  
Mindlin Plate ◽  
Solution Technique ◽  
Orthotropic Plates ◽  
Bending Vibration ◽  
Central Plate ◽  
Plate Strip

Abstract The natural frequencies and the corresponding mode shapes of two classes of composite base plate or panel stiffened by a central or a non-central plate strip are analyzed and compared with each other. In each case, the base plates and the single, stiffening plate strips are assumed to be dissimilar orthotropic plates connected by a very thin, yet deformable adhesive layer. The free bending vibration problems for the two cases are formulated in terms of the Mindlin Plate Theory for orthotropic plates. The governing equations are reduced to a system of first order equations. The solution technique is the “Modified Version of the Transfer Matrix Method”. The effects of the bonded central and non-central stiffening strip on the mode shapes and the natural frequencies of the composite plate or panel system are investigated. Some important conclusions are drawn from the numerical and parametric studies presented.

Free Flexural (or Bending) Vibrations of Orthotropic Composite Mindlin Plates With a Bonded Non-Central (or Eccentric) Lap Joint

2003 ◽  
Author(s):  
U. Yuceoglu ◽  
O¨. Gu¨vendik ◽  
V. O¨zerciyes
Keyword(s):  
Differential Equations ◽  
Natural Frequencies ◽  
Adhesive Layer ◽  
Mode Shapes ◽  
Mindlin Plate ◽  
Bending Rigidity ◽  
Lap Joint ◽  
Bending Vibrations ◽  
Theoretical Formulation ◽  
Orthotropic Composite

The present study is concerned with the “Free Flexural (or bending) Vibrations of Orthotropic Composite Mindlin Plates with a Bonded Non-Central (or Eccentric) Lap Joint”. The Mindlin plate adherends or panels of dissimilar, orthotropic material are connected by an adhesively bonded non-central (or eccentric) single lap joint. The adhesive layer is considered to be relatively very thin and linearly elastic. The theoretical formulation is based on the combination of the full set of the dynamic plate equations and the adhesive layer stress-displacement equations. Eventually, the system of equations is reduced to a set of the first order governing ordinary differential equations in the “state vector” form. The governing system of the differential equations is numerically integrated by means of the “Modified Transfer Matrix Method (with Interpolation and/or Chebyshev Polynomials)”. The effect of the non-central (or eccentric) location of the bonded lap joint is investigated and presented in detail in terms of natural frequencies and the associated mode shapes. The significant effects of the “hard” or the “soft” adhesive layer constants on the mode shapes and the natural frequencies are also investigated. Some important parametric studies such as the influences of the “Joint Length Ratio”, the “Joint Position Ratio” and the “Bending Rigidity Ratio” on the natural frequencies are computed and presented for the “hard” and the “soft” adhesive cases.

Effects of Position (or Location) of Non-Centrally Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint) on Bending Vibrations of Composite Mindlin Plates or Panels

2010 ◽  
Author(s):  
U. Yuceoglu ◽  
O¨. Gu¨vendik
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Shear Stresses ◽  
Lap Joint ◽  
Joint System ◽  
Adhesive Layers ◽  
Present Solution ◽  
Stress Resultant ◽  
Bonded Joint

In the present study, the “Effects of Position (or Location) of Non-Centrally Bonded Symmetric Double Doubler Joint in Composite 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. 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 “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 analysis and the present method of solution are applied to a typical “Non-Centrally Positioned (or Located) Symmetric Double Lap Joint (or Symmetric Double Doubler Joint) System”. The effects of the location (or position) of the “Bonded Joint System” and also of the relatively “Stiff (or “Hard”) and the relatively “Flexible” (or “Soft”) adhesive properties, on the natural frequencies and mode shapes are considered in some detail. The very interesting mode shapes with their dimensionless natural frequencies are presented for various sets of “Boundary Conditions”. From the numerical results obtained, some important conclusions are drawn for the “Bonded Joint System” studied here.

Rotation of Material Axes Effects on Free Bending Vibrations Response of Composite Mindlin Base Plates or Panels Stiffened by Three Bonded Plate Strips

2011 ◽  
Author(s):  
U. Yuceoglu ◽  
J. Javanshir ◽  
T. Farsadi ◽  
O¨. Gu¨vendik
Keyword(s):  
Base Plate ◽  
Mode Shapes ◽  
Mindlin Plate ◽  
Governing Equations ◽  
Bending Vibrations ◽  
Plate Elements ◽  
System 2 ◽  
Interpolation Polynomials ◽  
System 1 ◽  
Free Bending

In the present study, the rotation of material axes on the free bending vibrations response of a certain type of composite “Bonded and Stiffened System” is theoretically analyzed and numerically solved with some numerical results. The composite “Bonded and Stiffened System” is composed of a “Mindlin Base Plate or Panel” reinforced by three “Bonded Stiffening Plate Strips”. In the analysis, the 90° rotation effects of the material axes on the natural frequencies and the mode shapes of the entire “System” are investigated. The aforementioned “Bonded and Stiffened System” is considered in terms of the “System.1” and the “System.2”. In the “System.1”, the material axes of the “Base Plate” are rotated 90° (about z-axis), while there is no change in the material axes of the “Bonded Plate Strips”. In the “System.2”, there is no change in the material directions of the “Base Plate”, while the material axes of the “Bonded Plate Strips” are rotated 90° degrees. The “Base Plate or Panel” and the three “Bonded Plate Strips” are assumed to be dissimilar “Orthotropic Mindlin Plates”. The in-between, relatively very thin, linearly elastic adhesive layers are considered with different material characteristics. All “Mindlin Plate Elements” of both “Systems.1 and 2” are included in the analysis with the transverse (or bending) moments of inertia and rotary moments of inertia. The dynamic equations of the “Mindlin Plate Elements” and the in-between adhesive layer expressions (with the transverse normal and shear stresses) are combined togather. After some algebraic manipulations and combinations, they are eventually reduced to a set of the “Governing System of the First Order O.D.E’s” in compact matrix forms with the “state vectors” for each case of the “System.1” and “System.2”. The aforementioned “Governing Equations” facilitate direct application of the present method of solution that is the “Modified Transfer Matrix Method (MTMM) (with Interpolation Polynomials)”. The “Governing Equations” are numerically Integrated by means of the “ (MTMM) (with Interpolation Polynomials)”. The natural frequencies and the mode shapes of the “Systems.1 and 2” are computed and graphically presented for some “Support Conditions” of the “Systems” under consideration. The comparison of the numerical results led to some important conclusions.

Free Vibration Analysis and Design of an Adhesively Bonded Composite Single Lap Joint

2007 ◽  
Author(s):  
M. Kemal Apalak ◽  
Recep Ekici ◽  
Mustafa Yildirim
Keyword(s):  
Fiber Volume Fraction ◽  
Natural Frequencies ◽  
Volume Fraction ◽  
Plate Thickness ◽  
Mode Shapes ◽  
Lap Joint ◽  
Adhesively Bonded ◽  
Fiber Volume ◽  
Fiber Angle ◽  
Single Lap Joint

In this study the three dimensional vibration analysis of an adhesively bonded cantilevered composite single lap joint was carried out. The first four bending natural frequencies and mode shapes were considered. The back-propagation Artificial Neural Network (ANN) method was used to determine the effects of the fiber angle, fiber volume fraction, overlap length and plate thickness on the bending natural frequencies and the mode shapes of the adhesive joint. The bending natural frequencies and modal strain energies of the composite adhesive lap joint were calculated using the finite element method for random values of the fiber angle, the fiber volume fraction, the overlap length and the plate thickness. Later, the proposed neural network models were trained and tested with the training and testing data. The fiber angle was more dominant parameter than the fiber volume fraction on the natural bending frequencies and corresponding bending mode shapes, and the plate thickness and the overlap length were also important geometrical design parameters whereas the adhesive thickness had a minor effect. In addition, the present ANN models were combined with Genetic Algorithm to search a joint design satisfying maximum natural frequency and minimum modal strain energy conditions for each natural bending frequency and mode shape.

Natural Frequencies of Thick Annular Plates

1982 ◽  
Vol 49 (3) ◽  
pp. 633-638 ◽  
Author(s):  
T. Irie ◽  
G. Yamada ◽  
K. Takagi
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Plate Theory ◽  
Mindlin Plate ◽  
Annular Plates ◽  
Mindlin Plate Theory

The natural frequencies of vibration based on the Mindlin plate theory are tabulated for uniform annular plates under nine combinations of boundary conditions.

Free Bending Vibrations of Integrally-Stiffened and/or Stepped-Thickness Rectangular Plates or Panels With a Non-Central Plate Stiffener

2007 ◽  
Author(s):  
U. Yuceoglu ◽  
N. Gemalmayan ◽  
O. Sunar
Keyword(s):  
Differential Equations ◽  
Natural Frequencies ◽  
Orthotropic Materials ◽  
Mindlin Plate ◽  
Al Alloy ◽  
Bending Vibrations ◽  
Central Plate ◽  
Plate Elements ◽  
Stress Resultant ◽  
Free Bending

The present study is primarily concerned with the “Free Bending Vibrations of Integrally-Stiffened and/or Stepped-Thickness Plates or Panels with a Non-Central Plate Stiffener”. The general theoretical formulation is based on the “Mindlin Plate Theory”. The plate elements of the system are considered to be made of dissimilar orthotropic materials with unequal thicknesses. The transverse shear deformations and the transverse and the rotary moments of inertia of plate elements are included in the analysis. The damping effects, however, are neglected. The dynamic equations of the orthotropic “Mindlin Plates” in combination with the stress resultant-displacement expressions are algebraically manipulated. They are eventually reduced to a set of the “Governing System of the First Order Ordinary Differential Equations” in the “state vectors” form. The resulting differential equations system is numerically integrated by making use of the “Modified Transfer Matrix Method (MTMM) (with Interpolation Polynomials)”. The mode shapes with their dimensionless natural frequencies are presented for various support conditions in the “isotropic” Al-Alloy and in the “orthotropic” composite cases. Additionally, the effect of some of the important parameters such as (“Stiffener Position Ratio”, “Thickness Ratio”, “Stiffener Length (or Width) Ratio)” on the dimensionless natural frequencies are investigated and plotted. Based on the numerical results, some brief but important conclusions are presented.

Free Flexural Vibrations of Orthotropic Composite Base Plates or Panels With a Bonded Noncentral (or Eccentric) Stiffening Plate Strip

2003 ◽  
Vol 125 (2) ◽  
pp. 228-243 ◽  
Author(s):  
U. Yuceoglu ◽  
V. O¨zerciyes
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Composite System ◽  
Natural Frequencies ◽  
Adhesive Layer ◽  
Base Plate ◽  
Mode Shapes ◽  
Bending Vibrations ◽  
Arbitrary Boundary ◽  
Plate Strip

The problem of the free flexural (or bending) vibrations of a rectangular, composite base plate or panel stiffened by a bonded, noncentral stiffening plate strip is considered. The lower composite base plate and the upper stiffening plate strip are assumed as dissimilar Mindlin Plates connected by a very thin and deformable adhesive layer. In the formulation, the entire composite system is considered to have simply supported edges in one direction while the other two edges may have arbitrary boundary conditions. The set of governing partial differential equations is reduced to a “special form” of a system of the first order ordinary differential equations. Then, they are integrated by the “Modified Transfer Matrix Method (with Interpolation Polynomials).” The mode shapes and the natural frequencies of the composite system are investigated and presented in detail for several boundary conditions. It was also found that the “hardness” and the “softness” of the in-between adhesive layer have significant effects on the mode shapes and the natural frequencies.

Analytical Solution for Free Vibration of Annular Mindlin Plate with a Circumferential Open Crack

2019 ◽  
Vol 24 (3) ◽  
pp. 494-503
Author(s):  
Eshagh Derakhshan ◽  
Mahboobeh Fakhrzarei ◽  
Shahram Derakhshan
Keyword(s):  
Free Vibration ◽  
Natural Frequencies ◽  
Plate Theory ◽  
Crack Depth ◽  
Current Method ◽  
Mode Shapes ◽  
Mindlin Plate ◽  
Open Crack ◽  
Element Analysis ◽  
Radial Location

Mindlin plate theory is employed to obtain the free vibration response of an annular moderately thick plate with a circumferential open crack with fixed-free boundary conditions. To model the crack, a set of continuously distributed rotational springs are employed at the crack location. The corresponding spring stiffness value is a function of the crack depth and is given as a closed-form function. To obtain the vibration behaviour, the eigenvalue problem is solved to obtain the natural frequencies and mode shapes. The current method is verified by comparing the results with those obtained from finite element analysis. Through a parametric study, the effects of the crack depth and its radial location on the natural frequencies and mode shapes are investigated. The results show that for a constant crack depth, the reduction in natural frequency is a strong function of the radial location of the crack.

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