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Quadrature of Differential Equations

Quadrature of Differential Equations

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Quadrature of differential equations refers to the process of finding exact or approximate solutions to differential equations by integrating them. This method often involves transforming the equations into a form that can be solved through integration techniques, thereby determining the function that satisfies the given differential relationship.
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Quadrature of differential equations refers to the process of finding exact or approximate solutions to differential equations by integrating them. This method often involves transforming the equations into a form that can be solved through integration techniques, thereby determining the function that satisfies the given differential relationship.
Reducing Effects of Initial Imperfection by Investment in the Orthotropic Characteristics of Laminated Composite Plate
by Civil Engineering Journal (civilejournal.org)

2025, Civil Engineering Journal

The target of this study is to reduce the impact of initial imperfection on the nonlinear dynamical performance of laminated composite plates by taking advantage of the orthotropic characteristics of laminated composite plates by changing... more
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Dynamic analysis of composite cylindrical shells using differential quadrature method (DQM)
by mansour darvizeh

2025, Composite Structures

Free vibration analysis of composite cylindrical shells with different boundary conditions is presented in this paper using differential quadrature method (DQM). Equations of motion are derived based on first order shear deformation... more
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Haar wavelet discretization approach for frequency analysis of the functionally graded generally doubly-curved shells of revolution
by R. Talebitooti

2025, Applied Mathematical Modelling

This paper aims to investigate the free vibrational analysis of the generally doubly-curved shells of revolution made of functionally graded (FG) materials and constrained with different boundary conditions by means of an efficient,... more
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Buckling analysis of generally laminated composite plates (generalized differential quadrature rules versus Rayleigh–Ritz method)
by mansour darvizeh

2025, Composite Structures

Due to the importance of buckling analysis of composite structures in various industrial applications a comparative study of buckling behavior of composite plates is presented here. Mathematical modelling developed in the present work for... more
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Using Chebyshev polynomial’s zeros as point grid for numerical solution of nonlinear PDEs by differential quadrature- based radial basis functions
by Sajad Kosari

2025, International Journal of Mathematical Modelling & Computations

Radial Basis Functions (RBFs) have been found to be widely successful for the interpolation of scattered data over the last several decades. The numerical solution of nonlinear Partial Differential Equations (PDEs) plays a prominent role... more
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Using Chebyshev polynomial’s zeros as point grid for numerical solution of nonlinear PDEs by differential quadrature- based radial basis functions
by Sajad Kosari

2025, International Journal of Mathematical Modelling & Computations

Radial Basis Functions (RBFs) have been found to be widely successful for the interpolation of scattered data over the last several decades. The numerical solution of nonlinear Partial Differential Equations (PDEs) plays a prominent role... more
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Modelling of Concrete Gravity Dam Including Dam-Water-Foundation Rock Interaction
by sh kh

2024

A hybrid meta-heuristic optimization method is introduced to efficiently find the optimal shape of concrete gravity dams including dam-water-foundation rock interaction. For this purpose model of two-dimensional finite element that is... more
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Effect of Discontinuities and Uncertainties on the Response and Failure of Composite Structures
by Ahmed Noor

2024

The overall goal of this research was to assess the effect of discontinuities and uncertainties on
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Mohamad S. Qatu, Vibration of Laminated Shells and Plates , Elsevier, Academic Press, Oxford, UK (2004) ISBN 0-08044271-4 Hard cover xvii + 409 pages
by Sebastian Ghinet

2024, Applied Acoustics - APPL ACOUST

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Performance and optimum design of convective–radiative rectangular fin with convective base heating, wall conduction resistance, and contact resistance between the wall and the fin base
by Abdul Aziz

2024, Energy Conversion and Management

This paper investigates the performance and optimum design of a longitudinal rectangular fin attached to a convectively heated wall of finite thickness. The exposed surfaces of the fin lose heat to the environmental sink by simultaneous... more
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Anizotropik katmanlı kompozit eksenel simetrik kabukların serbest titreşim analizi
by Erdem Yavuzbalkan

2024

In this thesis, the free vibration analysis of anisotropic laminated composite shells of revolution (ALCSOR) is studied. The governing equations are kinematic, constitutive, and motion equations. Geometrically linear strain-displacement... more
Figure 4.8 Natural Frequency versus circumferential wave number changing from 0 to 10 for fiber orientation angle of 0° to 90° with the increment of 20 degrees when m=2.
Figure 4.9 Natural Frequency versus circumferential wave number changing from 0 to 10 for fibe: orientation angle of 0° to 90° with the increment of 20 degrees when m=3.
Figure 4.10 Variation in strain energy in a particular cylindrical shell with increasing number of
Figure 4.11 Natural Frequency versus fiber orientation angle for the first lowest fundamental mode
Figure 1.1 Specific strength and specific stiffness of advanced composite materials [1]
Figure 1.2 A unidirectional fiber-reinforced lamina with the principal material directions (1, 2, and 3)
Figure 1.3 Deformation of isotropic, orthotropic and anisotropic rectangular block under uniaxial
Figure 2.1 The position vector of a space curve [12].
Figure 2.2 Tangent vector description [12].
Figure 2.3 Parametric curves of a surface and differential change of a position vector on the surface
Figure 2.4 Some examples for surfaces of revolution
Figure 2.6 Ruled Surfaces (a) Hyperboloid of revolution of one sheet, (b) Conoid [15].
Figure 2.7 Geometry and coordinate system of shells of revolution
We have the second fundamental magnitudes, Land N for the shells of revolution bi We get the derivatives of the unit normal vector 71 with respect to ¢ and G as follows: inserting Equations (2.74), (2.75), (2.80), and (2.81) into Equation (2.36).
Recalling Equation (2.21), and inserting Equations (2.76) and (2.77) into it, we get
Figure 2.8 Differential element and coordinate system of a doubly curved thin shell [12]. Incorporating the orthogonality of curvilinear coordinate system, Equations (2.94), (2.48),
From the Figure 2.8 and the Equation (2.106), the length of the shaded surface whose unit and for the one whose unit normal vector is in é, is given by:
T, and T, can be expressed finally as follows:
manipulations, we obtain From Equation (2.122), we obtain two relations which will be used afterwards.
To get the derivatives of the normal vector 7 with respect to é, and és using the position
Substituting, Equation (2.126) and Equation (2.133) into Equation (2.139), C, become Also, substituting Equation (2.123) into Equation (2.140), C, is found to be
Substituting Equations (2.146) and (2.153) into Equation (2.159), we get Inserting Equations (2.125) and (2.126) into Equation (2.66) and expanding that
Collecting the coefficients of TJ, andT,, we can write a similar expression like Equation
The equality in Equation (2.161) holds when the coefficients of T, and a are both zero.
Figure H.7 Mode Shapes in displacements of (a) W (b) U,(c) Ug for (n=0) and (m=1);
Using Equation (3.28) in (E.50) yields
Kutta method for system of ODEs is given in [69]. As a result, IT (G lis given by
The formal solution of (G.26) in the interval (% G ) can be given by
IT. (Gg, ) can be obtained from solutions of 16 initial value problems such that
Kutta Method.
Nonzero elements of iT, (G, )}computed by using Matlab Symbolic Toolbox are listed a:
Figure H.1 Mode Shapes in displacements of (a) W (b) Ug(C) Ug for (n=0) and (m=1);
Figure H.2 Mode Shapes in displacements of (a) W (b) U,(c) Ug for (n=1) and (m=1);
Figure H.3 Mode Shapes in displacements of (a) W (b) Ug (c) Ug for (n=2) and (m=1);
Figure H.4 Mode Shapes in displacements of (a) W (b) U,(c) Ug for (n=0) and (m=1);
Figure H.5 Mode Shapes in displacements of (a) W (b) U,(c) Ug for (n=1) and (m=1);
Figure H.6 Mode Shapes in displacements of (a) W (b) U g(©) Ue for (n=2) and (m=1);
Figure H.9 Mode Shapes in displacements of (a) W (b) Ug(C) Ug for (n=2) and (m=1);
Table 1.1 Advantages and disadvantages of composite materials over metals [3]
Figure H.8 Mode Shapes in displacements of (a) W (b) Ug(C) Ug for (n=1) and (m=1);
Figure 4.12 Natural Frequency versus fiber orientation angle for the second lowest fundamental mode
where @. ior jy? 4ivor j) NA icoy j) ALE, Tespectively, the curvilinear coordinates of the geometry, components of the displacement vector and geometrical scale factor quantities. When the Equations The strain-displacement relations by the theory of elasticity in the orthogonal curvilinear
The normal strain components are: obtained in the shell space.
Naghdi-Reissner Shell Theory, they are neglected in the Equations (2.175), (2.176), and (2.178). It and, the shearing strain components are:
Equations (2.164), (2.165), (2.166), (2.167) and (2.168) are substituted properly into respectively, are neglected. The strain displacement equations turn into the following Equations from (2.181) to (2.186), then the strain displacement relations can be represented as the sun
where C ;, are the elastic constants, and are obtained from C aa DY the following change of subscripts:
where EE sdliys Eo are Young’s moduli in 1, 2, and 3 directions, respectively, Vi is Poisson’s ratio defined as ratio of transverse strain in the a direction to the axial strain in the ith direction, when The elastic coefficients in the stiffness matrixC,.in Equation (2.205) in terms of the stressed in the i-direction —€ j ‘i €, (for O, =O and all other stresses are zero) (i, j=1,2,3), and
Figure 2.9 (a) Specially Orthotropic Lamina, (b) Generally Orthotropic Lamina. and (2.208) are transformed into ones for Generally Orthotropic Lamina.
Figure 2.10 Internal Stresses on a Doubly Curved Shell Element The normal and shear force resultants acting on the face perpendicular to the é, coordinat
Figure 2.11 Force resultants in the shell coordinates.
Figure 2.12 Moment resultants in the shell coordinates. A laminate of doubly curved shell element is shown in Figure 2.13. For a laminated shell, the stress components can be integrated across each lamina, then must be added together. In the lamination, laminae are assumed to be bonded perfectly to each other so that the displacements are stresses will be discontinuous across the interface of two laminae.
Figure 2.13 N-layered laminate of a doubly curved shell element.
Combining Equations (2.187) to (2.191), (2.210), (2.211), (2.213) to (2.216) with Equations
As a result, the constitutive equations for the anisotropic laminated composite doubly curvec the force and moments resultants as
where A, , the extensional stiffness matrix which relates the in-plane stress resultants to the reference surface strains, are given by
Substituting the strain-displacement relations, (2.187) to (2.191) into Equation (2.268) yields Thus, the expression for the variation of the strain energy takes the form
can only be satisfied if each of the triple and double integral parts is equal to zero. Furthermore, since
It should be noted that each of the double integrals given in Equation (2.270) is equal to zero only if the coefficients of the variational displacements, variational displacements or one of the two for each term are zero. Since variational displacements are only zero at all times when the
Writing the each element of the left hand side of Equation (2.299) in terms of strain
The equations (3.9) to (3.12) can be written in compact form as:
Up to this point, the displacements and rotations terms of the system of partial differentia!
The elements of the coefficient matrix K are given in the Appendix F. 3.4 NONDIMENSIONALIZATION OF THE SYSTEM OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS The Equation (3.37) can be put into nondimensional form by using the following scheme
Figure 3.1 Division of meridional length of the shell of revolution into segments [71]. Equations (3.71) and (3.72) constitute a system of 2M linear homogeneous matrix equations T'(9., w,(¢)+T? (9, w.(¢)-w,¢.,)=0 T (9, W,(@)4+7;' (@., W.(9)-w,(9.,)=0
Figure 3.3 Finding first natural frequency of the example problem with (a)traditional Fourier Decomposition Method, and (b) Method of Finite Exponential Fourier Transform.
Figure 4.1 General Configuration and Coordinate System of Laminated Composite Circular Cylindrical Shell and its Laminate. where R is the mean radius of laminated composite circular cylindrical shell.
Figure 4.2 Circumferential Nodal Patterns and Axial Nodal Patterns for a Simply Supported Circular
Figure 4.3 The comparison of natural frequencies calculated by the exact method of solution [13] with those calculated by the present method of solution from n=0 to n=10
variation of stretching stiffness matrix elements with respect to alpha Figure 4.4 Effect of the fiber orientation angle on the elements of stretching stiffness matrix.
Figure 4.5 Effect of the fiber orientation angle on the elements of transverse shear stiffness matrix.
variation of flexural stiffness matrix elements with respect to alpha Figure 4.6 Effect of the fiber orientation angle on the elements of flexural stiffness matrix.
Figure 4.7 Natural Frequency versus circumferential wave number changing from 0 to 10 for fiber orientation angle of 0° to 90° with the increment of 20 degrees when m=1.
Figure 4.13 Natural Frequency versus fiber orientation angle for the third lowest fundamental mode
Figure 4.14 Mode shapes for the first fundamental transverse mode when Q changes from 0° to 90° with the increment of 10° (n=1).
Figure 4.15 Mode shapes for the first fundamental transverse mode when Q@ changes from 0° to 90°
Figure 4.16 Mode shapes for the first fundamental transverse mode when Q@ changes from 0° to 90°
Table 4.15 Six symmetric layups used in the case study on the stacking sequence. The geometrical, mechanical and laminate properties of the clamped laminated composite
Figure 4.17 Effect of the stacking sequence on the elements of stretching stiffness matrix (h/R=0.01). variation of stretching stiffness matrix elements
Figure 4.18 Effect of the stacking sequence on the elements of transverse shear stiffness matrix variation of transverse shear stiffness matrix elements
Figure 4.19 Effect of the stacking sequence on the elements of flexural stiffness matrix (h/R=0.01) variation of flexural stiffness matrix elements
Figure 4.20 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-clamped (CC) boundary condition (m=1 and h/R=0.01).
Figure 4.21 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-clamped (CC) boundary condition (m=2 and h/R=0.01).
Figure 4.22 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-clamped (CC) boundary condition (m=3 and h/R=0.01).
Figure 4.23 Natural frequencies versus circumferential wave numbers for all stacking sequence for
Figure 4.24 Natural frequencies versus circumferential wave numbers for all stacking sequence for
Figure 4.25 Natural frequencies versus circumferential wave numbers for all stacking sequence for
Figure 4.26 Comparison of the natural frequencies of the laminated composite circular cylindrical shell having CC boundary conditions with the ones having CF boundary conditions (m=1 and
Figure 4.27 Comparison of the natural frequencies of the laminated composite circular cylindrical shell having CC boundary conditions with the ones having CF boundary conditions (m=2 and
Figure 4.28 Comparison of the natural frequencies of the laminated composite circular cylindrical shell having CC boundary conditions with the ones having CF boundary conditions (m=3 and
Figure 4.29 Effect of the stacking sequence on the elements of stretching stiffness matrix (h/R=0. 1).
Figure 4.30 Effect of the stacking sequence on the elements of transverse shear stiffness matrix variation of transverse shear stiffness matrix elements
Figure 4.31 Effect of the stacking sequence on the elements of flexural stiffness matrix (h/R=0.1) variation of flexural stiffness matrix elements
Figure 4.32 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-clamped (CC) boundary condition (m=1 and h/R=0.1).
Figure 4.33 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-clamped (CC) boundary condition (m=2 and h/R=0.1).
Figure 4.34 Natural frequencies versus circumferential wave numbers for all stacking sequence for
Figure 4.35 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-free (CF) boundary condition (m=1 and h/R=0.1).
Figure 4.36 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-free (CF) boundary condition (m=2 and h/R=0.1).
Figure 4.37 Natural frequencies versus circumferential wave numbers for all stacking sequence for clamped-free (CF) boundary condition (m=3 and h/R=0.1).
Figure 4.38 Comparison of the natural frequencies of the laminated composite circular cylindrical shell having CC boundary conditions with the ones having CF boundary conditions (m=1 and
‘igure 4.39 Comparison of the natural frequencies of the laminated composite circular cylindrical shell having CC boundary conditions with the ones having CF boundary conditions (m=2 and
Figure 4.40 Comparison of the natural frequencies of the laminated composite circular cylindrical shell having CC boundary conditions with the ones having CF boundary conditions (m=3 and
Figure 4.41 Comparison of the natural frequencies of the thick and thin shell configurations
Figure 4.42 Comparison of the natural frequencies of the thick and thin shell configurations for clamped-clamped (CC) boundary condition (m=2).
Figure 4.43 Comparison of the natural frequencies of the thick and thin shell configurations for clamped-clamped (CC) boundary condition (m=3).
Figure 4.44 Comparison of the natural frequencies of the thick and thin shell configurations
Figure 4.45 Comparison of the natural frequencies of the thick and thin shell configurations for clamped-free (CF) boundary condition (m=2).
Figure 4.46 Comparison of the natural frequencies of the thick and thin shell configurations
Figure 4.47 Effect of the stacking sequence on the elements of stretching stiffness matrix
Figure 4.48 Effect of the stacking sequence on the elements of transverse shear stiffness matrix variation of transverse shear stiffness matrix elements for three different layups in the case study on effect of coupling terms.
Figure 4.49 Effect of the stacking sequence on the elements of stretching-flexural coupling stiffness matrix for three different layups in the case study on effect of coupling terms.
variation of flexural stiffness matrix elements Figure 4.50 Effect of the stacking sequence on the elements of flexural stiffness matrix for three
‘igure 4.51 Effect of coupling terms on the natural frequencies of the simply supported laminated number in Figures 4.51, 4.52, and 4.53 for m=1,2, and 3, respectively.
Figure 4.52 Effect of coupling terms on the natural frequencies of the simply supported laminated composite circular cylindrical shell (m=2).
Figure 4.53 Effect of coupling terms on the natural frequencies of the simply supported laminated
variation of stretching stiffness matrix elements Figure 4.54 Elements of stretching stiffness matrix for the layups used in the case study on
Figure 4.55 Elements of transverse shear stiffness matrix for the layups used in the case study on variation of transverse shear stiffness matrix elements the existence of coupling terms.
‘igure 4.56 Elements of stretching-flexural coupling stiffness matrix for the layups used variation of stretching-flexural coupling stiffness matrix elements in the case study on the existence of coupling terms.
Figure 4.57 Elements of flexural stiffness matrix for the layups used variation of flexural stiffness matrix elements in the case study on the existence of coupling terms.
Figure 4.58 Effect of the coupling terms on the natural frequencies over n range of 0-9 (m=1). omitted is seen to be higher.
Figure 4.59 Effect of the coupling terms on the natural frequencies over n range of 0-9 (m=2).
Figure 4.60 Effect of the coupling terms on the natural frequencies over n range of 0-9 (m=3).
Figure 4.61 The side and perspective views of the laminated composite spherical shell.
Figure 4.62 Dominant mode shapes (transverse displacement) of the corresponding natural
Figure 5.1 Cross-sections of the circular cylindrical shells with
Let us split the Equation (A.3) and do the variational computations term by term, anc
Carrying out the Finite exponential Fourier transform to Equation (E. 1) (c,, Yu’
Separating the real and imaginary parts of Equation (E.4)
We now separate the real and imaginary parts of Equation (E.10). First, writing real parts Then, writing imaginary parts of Equation (E.10) term by term.
We separate the real and imaginary parts of Equation (E. 16)
We separate the real and imaginary parts of Equation (E.22). First, writing real parts of Then, writing imaginary parts of Equation (E.22) term by term.
Carrying out the Finite exponential Fourier transform to Equation (E.25). Using Equation (3.28) in Equation (E.26) yields
We separate the real and imaginary parts of Equation (E.28).
Separating the real and imaginary parts of Equation (E.34)
Using Equation (3.28) in Equation (E.38) yields Carrying out the Finite exponential Fourier transform to Equation (E.37)
We separate the real and imaginary parts of Equation (E.40). First, writing real parts of Equation (E.40) term by term.
Using Equation (3.28) in Equation (E.44) yields
We separate the real and imaginary parts of Equation (E.46)
The infinitesimal distance P,P, between two arbitrary points on the reference surface of the other words, the Gauss-Codazzi conditions are clearly satisfied for shells of revolution. expressions for A, A, JR, R, obtained for shells of revolution into Gauss-Codazzi conditions. Ir where A, and A, are sometimes called Lamé parameters which are related with the first fundamental
frequency of the example problem with traditional Fourier Decomposition procedure.
frequency of the example problem with method of Finite Exponential Fourier Transform. Based on Figure 3.3, it can be said that in order to determine the root by the application of Table 3.2 Trial natural frequencies and their computed determinants in order to find first natural
Figure 4.3 shows the comparison of natural frequencies calculated by the exact method of cylindrical shell used in comparison study of the method of solution.
for m=1-3 and n=0-10.
laminated composite circular cylindrical shell used for the case study on the fiber orientation angle. When ply stacking sequence, material, and geometry (ply thickness) are symmetric about the Table 4.3 Geometrical, material properties, and laminate properties of the simply supported
the increment of 10 degrees for n=0, and m=1,2, and 3. Table 4.4 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
Table 4.5 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with Table 4.6 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
the increment of 10 degrees for n=2, and m=1,2, and 3.
the increment of 10 degrees for n=3, and m=1,2, and 3. Table 4.7 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
Table 4.8 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
Table 4.9 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
the increment of 10 degrees for n=6, and m=1,2, and 3. Table 4.10 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
the increment of 10 degrees for n=7, and m=1,2, and 3. Table 4.11 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
the increment of 10 degrees for n=8, and m=1,2, and 3. Table 4.12 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
the increment of 10 degrees for n=9, and m=1,2, and 3. Table 4.13 Natural Frequencies (rad/sec) when the fiber orientation angle changing from 0 to 90 with
the increment of 10 degrees for n=10, and m=1,2, and 3.
laminated composite circular cylindrical shell used for the case study on stacking sequence. The results of the six different stacking sequence for the lowest three natural frequencies
Table 4.17 Natural frequencies (rad/sec) for n 0 to 9 for the layup 1 in Table 4.15 for CC (h/R=0.01). Table 4.18 Natural frequencies (rad/sec) for n 0 to 9 for the layup 2 in Table 4.15 for CC (h/R=0.01).
Table 4.19 Natural frequencies (rad/sec) for n 0 to 9 for the layup 3 in Table 4.15 for CC (h/R=0.01). Table 4.20 Natural frequencies (rad/sec) for n 0 to 9 for the layup 4 in Table 4.15 for CC (h/R=0.01).
Table 4.21 Natural frequencies (rad/sec) for n 0 to 9 for the layup 5 in Table 4.15 for CC (h/R=0.01)
Table 4.22 Natural frequencies (rad/sec) for n 0 to 9 for the layup 6 in Table 4.15 for CC (h/R=0.01).
Table 4.23 Natural frequencies (rad/sec) for n 0 to 9 for the layup 1 in Table 4.15 for CF (h/R=0.01) Table 4.24 Natural frequencies (rad/sec) for n 0 to 9 for the layup 2 in Table 4.15 for CF (h/R=0.01).
Table 4.25 Natural frequencies (rad/sec) for n 0 to 9 for the layup 3 in Table 4.15 for CF (h/R=0.01). Table 4.26 Natural frequencies (rad/sec) for n 0 to 9 for the layup 4 in Table 4.15 for CF (h/R=0.01).
Table 4.27 Natural frequencies (rad/sec) for n 0 to 9 for the layup 5 in Table 4.15 for CF (h/R=0.01)
Table 4.28 Natural frequencies (rad/sec) for n 0 to 9 for the layup 6 in Table 4.15 for CF (h/R=0.01).
Table 4.29 Natural frequencies (rad/sec) for n 0 to 9 for the layup 1 in Table 4.15 for CC (h/R=0.1). Table 4.30 Natural frequencies (rad/sec) for n 0 to 9 for the layup 2 in Table 4.15 for CC (h/R=0.1).
Table 4.31 Natural frequencies (rad/sec) for n 0 to 9 for the layup 3 in Table 4.15 for CC (h/R=0.1)
Table 4.32 Natural frequencies (rad/sec) for n 0 to 9 for the layup 4 in Table 4.15 for CC (h/R=0.1).
Table 4.33 Natural frequencies (rad/sec) for n 0 to 9 for the layup 5 in Table 4.15 for CC (h/R=0.1). Table 4.34 Natural frequencies (rad/sec) for n 0 to 9 for the layup 6 in Table 4.15 for CC (h/R=0.1).
Table 4.35 Natural frequencies (rad/sec) for n 0 to 9 for the layup 1 in Table 4.15 for CF (h/R=0. 1). Table 4.36 Natural frequencies (rad/sec) for n 0 to 9 for the layup 2 in Table 4.15 for CF (h/R=0.1).
Table 4.37 Natural frequencies (rad/sec) for n 0 to 9 for the layup 3 in Table 4.15 for CF (h/R=0.1).
Table 4.38 Natural frequencies (rad/sec) for n 0 to 9 for the layup 4 in Table 4.15 for CF (h/R=0.1)
Table 4.39 Natural frequencies (rad/sec) for n 0 to 9 for the layup 5 in Table 4.15 for CF (h/R=0. 1). Table 4.40 Natural frequencies (rad/sec) for n 0 to 9 for the layup 6 in Table 4.15 for CF (h/R=0.1).
Table 4.41 Three different layups used in the case study on effect of coupling terms.
Stretching, transverse shear, stretching-flexural coupling, and flexural stiffness coefficients laminated composite circular cylindrical shell used for the case study on effect of coupling terms
Table 4.43 Natural frequencies (rad/sec) of circular cylindrical shell with symmetric layup 10/30/ 90/60 | ; Table 4.44 Natural frequencies (rad/sec) of circular cylindrical shell with antisymmetric layup
Natural frequencies for the three layup cases are plotted with respect to circumferential wave Table 4.45 Natural frequencies (rad/sec) of circular cylindrical shell with unsymmetric layup
Table 4.46 Dominant mode shapes for three layups for n=0, and n=1. As one can see from Table 4.46 for n=0, axisymmetric case, the dominant modes are all
The variation of the natural frequencies versus circumferential wave number is given in
layup case and unsymmetric layup with coupling terms omitted for n=0,1, and 2. The results given in Table 4.48 also confirm that at low circumferential modes (low n) the Table 4.48 Comparison of the natural frequencies (rad/sec) between antisymmetric influence of bending stiffness D, is very little. Although these two layups have different bending
Table 4.50 Natural Frequencies for the laminated
composite spherical shell (n=1). As one can see from the mode shape plots, first three modes for n=1 correspond to transverse
(G.30) is expanded to let us carry out single step numerical integration as follows:
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80 Gb/s optical quadrature phase shift keying modulation using an integrated LiNbO<inf>3</inf> modulator
by Junichiro Ichikawa

2024, 2006 Conference on Lasers and Electro-Optics and 2006 Quantum Electronics and Laser Science Conference

We demonstrated 80 Gb/s optical differential quadrature-shift-keying modulation, by using a single-chip integrated LiNbO 3 modulator with two sub Mach-Zehnder interferometers, where 40 Gb/s in-phase and quadrature data were,... more
80 Gb/s optical DQPSK transmission based on 40 Gsample/s vector modulation was demonstrated by usin an optical QPSK modulator consisting of parallel two MZIs, where the 3 dB optical bandwidth of tk modulator was 27 GHz. To the best of our knowledge, this is the first demonstration of 80 Gb/s DQPS! modulation by an integrated single-chip modulator. 4 Conclusion of the central wavelength peak was 60 GHz. At the DQPSK demodulator shown in Fig. 1 (b), the DQPSK signal generated destructive port experiment, and bit-error-ratio ( at the modulator was decoded by a one-bit delay interferometer whose constructive anc s were connected to a balanced photodetector. However no precoder was employed for ou hence there was a deterministic mapping of data from input to output. In order to allov BER) measurements, the error detector was programmed with the expected data sequence We used a single receiver to decode each 40 Gb/s tributary by adjusting the differential optical phase in the one-bit delay in terferometer (A@) at 7/4 or —7/4. Fig. 2 (b) shows eye diagrams measured at the electric output of the ba anced photodetector. In back-to-back transmission, clear eye openings were observed for the two tributaries whose symbol rate was 40 Gsample/s. A back-to-back BER curve of a sub channel extractec from a tributary of A¢é = +7/4 by a 1:4 demultiplexer is shown in Fig. 2 (c), where the receiver sensitivity at the BER of 1 0-° was -17 dBm.
References
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Optical FSK transmission with group delay compensated balance detection
by Junichiro Ichikawa

2024, IEICE Electronics Express

A novel balance detection scheme for high-speed optical frequency-shift-keying (FSK) transmission was proposed. Fiber dispersion can be compensated by a balanced photodetector with an optical delay. 10 Gbit/s optical FSK transmission over... more
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Optical FSK/IM Signal Generation Using an Integrated Optical FSK Modulator
by Junichiro Ichikawa

2024, IEICE Electronics Express

We demonstrate simultaneous optical modulation by frequency shift keying (FSK) and intensity modulation (IM) using a newly developed high-speed FSK modulator, for optical packet systems where payload and label signals are, respectively,... more
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LiNbO3 high-speed optical FSK modulator
by Junichiro Ichikawa

2024, Electronics Letters

An optical frequency-shift-keying (FSK) modulator with a pair of Mach-Zehnder structures that can switch optical frequency is developed. The operation is demonstrated at a 10 Gbit/s FSK transmission conducted on a standard singlemode... more
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40-Gb/s BPSK modulation using a silicon modulator
by Keren Bergman

2024

We demonstrate BPSK modulation using a silicon traveling-wave modulator at a data rate of 40 Gb/s with record 10.5 pJ/bit energy efficiency, and compare its performance with a commercial Lithium Niobate phase modulator. I.
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A differential quadrature algorithm for simulations of nonlinear Schrödinger equation
by idris dağ

2024, Computers & mathematics with applications

Numerical simulations of Nonlinear Schrödinger Equation are studied using differential quadrature method based on cosine expansion. Propogation of a soliton, interaction of two solitons, birth of standing and mobile solitons and bound... more
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Non-linear bending analysis of laminated sector plates using Generalized Differential Quadrature
by محمد اقدم

2024, Composite Structures

Linear and non-linear bending analysis of laminated plates with different boundary conditions is presented using generalized differential quadrature (GDQ) method. Governing equations are based on the first-order shear deformation theory... more
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Vibrations of Three-Layered Cylindrical Shells with FGM Middle Layer Resting on Winkler and Pasternak Foundations
by abdul sattar shah

2024, Advances in Acoustics and Vibration

Vibrations of a cylindrical shell composed of three layers of different materials resting on elastic foundations are studied out. This configuration is formed by three layers of material in thickness direction where the inner and outer... more
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Parametric instability of delaminated composite spherical shells subjected to in-plane pulsating forces
by Prof Pravat Parhi

2024, Composite Structures

The present study deals with numerical results of parametric instability of delaminated woven fibre Glass/Epoxy composite curved panels at elevated moisture contents. A composite shell model based on first order shear deformation theory... more
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Mechanical Vibration: Where do we Stand?
by isaac elishakoff

2024, International Centre for Mechanical Sciences

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Performance and optimum design of convective–radiative rectangular fin with convective base heating, wall conduction resistance, and contact resistance between the wall and the fin base
by Abdul Aziz

2024, Energy Conversion and Management

This paper investigates the performance and optimum design of a longitudinal rectangular fin attached to a convectively heated wall of finite thickness. The exposed surfaces of the fin lose heat to the environmental sink by simultaneous... more
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Transverse dipolar chaining in binary suspensions induced by rf fields
by Ronald Pethig

2024, Journal of Physics D: Applied Physics

In this paper we present several topologies of simple non-reflective filters using transmission lines and resistive elements. These filters are well-matched at all frequencies, making them especially suitable for gain slope equalizers... more
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10Gbit/s FSK transmission over 95km SMF using a LiNbO3 modulator
by Junichiro Ichikawa

2024, IEICE Electronics Express

We demonstrate high-speed optical frequency-shift-keying (FSK) transmission over a 95 km single-mode fiber (SMF) at 10 Gbit/s using a LiNbO 3 optical FSK modulator and an optical interleaver. The measured eye diagrams, bit error rate... more
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The radial basis function-differential quadrature method for elliptic problems in annular domains
by Daniel Watson

2024, Journal of Computational and Applied Mathematics

We propose a simple approach to improve the accuracy of the Radial Basis Function Differential Quadrature (RBF-DQ) method for the solution of elliptic boundary value problems. While the traditional RBF-DQ method places the centers... more
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Hydrodynamic Pressures on Dams During Earthquakes - Approximate Formulation
by Giovanni de Marinis

2024

Whenever carrying out safety inspections on dams under construction in officially declared seismic areas or in any zones of this ilk, one must always take into account not only the normal static effects, but also those triggered by the... more
Particular attention has been lent to the various simplified hypotheses introduced by the writers as a basis for their studies (two-dimensional motion, non-viscous liquid, compressible or incompressible liquid, small amplitude motion of the liquid particles, infinite or finite tank length, rigid dam, reservoir- dam interaction). To this end, a sensitivity test was performed on_ the parameters that the above inertial effects depend on, both in the case of, rigid dam (with liquid taken to be both incompressible and compressible) as well as in that of dynamic interaction of coupled reservoir-dam system (elastic structure) [3].
) } Earthquake Resistant Engineering Structures VAIUCS UGE UIUOe ALUIOULG UY CHOI o LULA. In figure 4 we have drawn up an abstract of the results of our analytical study. In particular, in the above mentioned graph we have written out, with reference to a gravity dam with a vertical upstreame face, the values of the ratio K/C as a function of Q (in the range 1>Q>0), both with reference to the case of rigid dam and compressible liquid (relationship (3)) and with reference to structure-reservoir interaction (relationship (10) and (11)), as well as the K/C value (which is, obviously, independent from 2) drawn from the Italian Regulations in force (K/C=1.074) in the case of rigid dam and incompressible liquid.
As can be gathered from figure 5. in which the ratio K/C has been written out as a function of the water depth to the maximum reservoir storage capacity (and therefore, with good accuracy. of the height of the barrage), the deviations contained in the values deducible from Westergaard’s method and those provided by (12) are quite small, and moreover contributes to higher safety standards as already has been made clear in figure 2. written out as:
In figure 5. we have also written out the constant value of the ratio K/ deducible from the current Italian Regulations (based, as already mentioned. o the hypothesis of rigid dam and incompressible liquid), equal to 1.074. whic for dams of a height greater than 100 m comes to less (and increasingly so as ) goes up) compared to those provided in Westergaard’s method, and therefore by the approximate formula proposed. Bearing in mind that for dams (eve earth ones) the height goes over 300 m, the hydrodynamic thrust can reac values equal to even 2+3 times those deducible from the hypothesis of rigi structure and incompressibie liquid.
T=1 s:
for values of y, ranging from 275 to 300 m. In figure 6 we have written out the ratio K/C as a function of the wate depth deduced from formulae (13) and (14). as well as the constant valu (K/C=1.074) deducible from Zangar’s method (which the Italian Regulation 1 force are based on). From this figure, apart from noting the low deviation between the values of the thrust S, computed on the basis of approximat formulae and those assessable using Chopra’s methods, starting with dam higher than 100 m, the considerable difference between the values provided b (13) and (14) and those deduced from the hypothesis of rigid dam an incompressible liquid, should become evident to us. For example, for a 300 1 high gravity dam, the S, obtains a value equal to roughly 5 times that calculate with the Zangar’s method.
From the data in the table above, we can easily see that when the angle of the upstream face is not greater than approximately 15°, the deviations between the values of the derating factor provided by the methods quoted here as well as those contained in the Italian Regulations are quite small. On the other hand, for © >15°, Kulmaci provides derating coefficient values which are still greater than those which may be inferred from the formulae (4) and (5). To this end, on the basis of the values given by the derating coefficient as a function of & and written out in the table, we propose the following simple a. ' a . — peopel > at _
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Coupled FE–BE method for eigenvalue analysis of elastic structures submerged in an infinite fluid domain
by Changjun Zheng

2024, International Journal for Numerical Methods in Engineering

SummaryFor thin elastic structures submerged in heavy fluid, e.g., water, a strong interaction between the structural domain and the fluid domain occurs and significantly alters the eigenfrequencies. Therefore, the eigenanalysis of the... more
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Shock wave simulations using Sinc Differential Quadrature Method
by idris dağ

2024, Engineering Computations

PurposeThis paper aims to present a numerical solution of non‐linear Burger&#39;s equation using differential quadrature method based on sinc functions.Design/methodology/approachSinc Differential Quadrature Method is used for space... more
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An iterative BEM for the dynamic analysis of interface crack contact problems
by Oleksandr Menshykov

2024, Engineering Analysis with Boundary Elements

The multiparametric algorithm for solving a fracture dynamics problem is presented in the paper. The linear crack between two dissimilar elastic isotropic half-spaces under normal tension-compression load is considered. The system of... more
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Analytical and differential quadrature results for vibration analysis of damaged circular arches
by EDOARDO ARTIOLI

2024, Journal of Sound and Vibration

The present paper focuses on in-plane linear free vibrations of circular arches, in undamaged and damaged configurations. For the model herein utilized, the equations of motion, in terms of displacements and rotation, take into account... more
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Partial response signaling for improved chromatic dispersion tolerance in intensity modulation optical transmissions
by Nebojša Stojanović

2024, Optics Express

We investigate partial response signaling (PRS) as a way of increasing the transmission length achievable by direct detection optical systems. The performance of the duobinary and PRS modulations is evaluated against that of conventional... more
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Performance and optimum design of convective–radiative rectangular fin with convective base heating, wall conduction resistance, and contact resistance between the wall and the fin base
by Abdul Aziz

2024, Energy Conversion and Management

This paper investigates the performance and optimum design of a longitudinal rectangular fin attached to a convectively heated wall of finite thickness. The exposed surfaces of the fin lose heat to the environmental sink by simultaneous... more
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Integrated Lithium Niobate Mach-Zehnder Interferometers for Advanced Modulation Formats
by Tetsuya Kawanishi

2024, IEICE Transactions on Electronics

Tetsuya KAWANISHI †a) , Takahide SAKAMOTO †b) , and Akito CHIBA †c) , Members SUMMARY We present recent progress of high-speed Mach-Zehnder modulator technologies for advanced modulation formats. Multi-level quadrature amplitude... more
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Optical filter characterization by using optical frequency sweep technique with a single sideband modulator
by Tetsuya Kawanishi

2024, IEICE Electronics Express

We demonstrated optical filter characterization using optical frequency sweep with optical single sideband modulation technique, where the frequency shift range was 2 GHz-10 GHz. Undesired sidebands generated in the single sideband... more
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10Gbit/s FSK transmission over 95km SMF using a LiNbO3 modulator
by Tetsuya Kawanishi

2024, IEICE Electronics Express

We demonstrate high-speed optical frequency-shift-keying (FSK) transmission over a 95 km single-mode fiber (SMF) at 10 Gbit/s using a LiNbO 3 optical FSK modulator and an optical interleaver. The measured eye diagrams, bit error rate... more
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Optical FSK transmission with group delay compensated balance detection
by Tetsuya Kawanishi

2024, IEICE Electronics Express

A novel balance detection scheme for high-speed optical frequency-shift-keying (FSK) transmission was proposed. Fiber dispersion can be compensated by a balanced photodetector with an optical delay. 10 Gbit/s optical FSK transmission over... more
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Optical FSK/IM Signal Generation Using an Integrated Optical FSK Modulator
by Tetsuya Kawanishi

2024, IEICE Electronics Express

We demonstrate simultaneous optical modulation by frequency shift keying (FSK) and intensity modulation (IM) using a newly developed high-speed FSK modulator, for optical packet systems where payload and label signals are, respectively,... more
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A bias condition monitor technique for the nested Mach-Zehnder modulator
by Tetsuya Kawanishi

2024, IEICE Electronics Express

A nested Mach-Zehnder (MZ) modulator, which consists of two sub MZ interferometers embedded in a main MZ interferometer, is a multifunctional modulator. It can work as an optical single side-band (SSB) modulator, a differential quadrature... more
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Vibration of locally cracked pre-loaded parabolic arches
by Ekrem Tufekci

2024, Journal of Sound and Vibration

We study linear dynamics of an initially parabolic arch deformed by a uniform 'dead' load. The arch is seen as a fully deformable one-dimensional continuum with rigid cross-sections, one of which suffers from a small local crack at its... more
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On the use of a roving body with rotary inertia to locate cracks in beams
by Ivo Caliò

2024, Journal of Sound and Vibration

Identifying cracks and damages in structures using measured vibrational characteristics has received considerable attention in the past few decades. The possibility of using frequency changes due to the application of a mass appended to... more
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Implementation and Verification of RKPM in the Sierra/SolidMechanics Analysis Code
by Jiun-Shyan Chen

2024

The reproducing kernel particle method (RKPM) is a meshfree method for computational solid mechanics that can be tailored for an arbitrary order of completeness and smoothness. The primary advantage of RKPM relative to standard... more
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Buckling of thin skew isotropic plate resting on Pasternak elastic foundation using extended Kantorovich method
by Ahmed Hassan

2024, Heliyon

The extended Kantorovich method (EKM) is implemented to numerically solve the elastic buckling problem of thin skew (parallelogram) isotropic plate under in-plane loading resting on the Pasternak elastic foundation. EKM has never been... more
Fig. 3. Pasternak elastic foundation model. Fig. 1. Coordinate systems convention and edge labeling. introduced and the equation of the total potential energy in rectangular coordinate (x — y) system is transformed to the oblique coordinate sys- tem (é — 7), from which the stability equations and boundary conditions will be later derived.
Fig. 4. A typical skew plate having coarse mesh of 4 elements per edge.
Fig. 5. g,(y) and f "(€) at end of each iteration starting from random g,(v).
Fig. 8. Buckling shapes of uniaxially loaded S.S\SF plate for range of k*. Fig. 9. Percentage relative difference e between 43 obtained using EKM with various number of terms n and the analytical solution of Kitipornchai et al. (1993) for rhombic plates subject to uniaxial loads.
Fig. 10. Percentage relative difference e between 1* obtained using EKM with various number of terms n and the DQM solution of Wang and Yuan (2018) for rhombic plates subject to shear loads.
Fig. 11. Non-dimensional buckling factor 4* of SSSS plate for a range of aspect ratios a/b and skew angles ¢ under different types of loading.
Fig. 12. Comparison of the non-dimensional buckling factor 4* of SSSS plate for a range of skew angles ¢ under different types of loading at different values c aspect ratio a/b.
Fig. 13. Non-dimensional buckling factor a: of (a) rectangular and (b) skew SSSS plate with a range of aspect ratios a/b under uniaxial loading resting on the Pasternak elastic foundation.
Fig. 15. Non-dimensional buckling factor 4; of a rectangular CCCC plate with a range of aspect ratios a/b under uniaxial loading resting on the Pasternak elastic foundation. Fig. 14. Non-dimensional buckling factor 4* of a rectangular SSSS plate with a range of aspect ratios a/b under (a) biaxial and (b) shear loading resting on the Pasternak elastic foundation.
2* of SSSS square plate under uniaxial compression through EKM iterations. Table 1 functions. The first run starts with a random function, while the sec- ond starts with a constant function: g)(7) = 1. Fig. 5 along with Table 1 shows that even with a random trail function go(y), the convergence of EKM is rapid. For all those presented results, the solution is considered converged once the difference /\i* between the non-dimensional buck- ling factor of two successive iterations is less than 10->. For the case in hand, three iterations were enough for EKM to converge starting from a random trail function. Better the initial guess of gy(7) faster the con- vergence. An important point to note is that although the initial trail function does not have to satisfy any of the boundary conditions, using random initial g)(7) may sometimes make EKM not to converge. It is found better to guess a smooth initial function.
Comparison of nondimensional buckling factor 4; of thin square plate under uniaxial compression loading, resting on elastic foundation with different boundary conditions.
Aj, of square plates resting on elastic foundation under equal biaxial compres- sion obtained using EKM compared with the analytical solution of Lam et al. (2000).
was of rectangular plates with various aspect ratios a/b resting on elastic foun- dation obtained using EKM compared with the analytical solution of Akhavan et al. (2009).
Note: The superscripts indicate the number of the EKM terms (n). a of skew plates with various aspect ratios, skew angles and boundary conditions under uniaxial load obtained using EKM compared with the DQM solution of Wang and Yuan (2018), the analytical solution of Kitipornchai et al. (1993), and the FEM solution of ANSYS.
Note: The superscripts indicate the number of the EKM terms (n). 4*i of skew plates with various aspect ratios, skew angles and boundary conditions under uniaxial load obtained using EKM compared with the analytical solution of Kitipornchai et al. (1993) and the FEM solution of ANSYS.
Nondimensional biaxial 4; and shear 4* buckling factors of rhombic plates with various skew angles and boundary conditions obtained using EKM compared with the DQM solution of Wang and Yuan (2018). Note: The superscripts indicate the number of the EKM terms (n).
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Partial response signaling for improved chromatic dispersion tolerance in intensity modulation optical transmissions
by Fotini Karinou

2024, Optics Express

We investigate partial response signaling (PRS) as a way of increasing the transmission length achievable by direct detection optical systems. The performance of the duobinary and PRS modulations is evaluated against that of conventional... more
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Characteristics of the Differential Quadrature Method and Its Improvement
by Rohit Kumar

2024, Mathematical Problems in Engineering

The differential quadrature method has been widely used in scientific and engineering computation. However, for the basic characteristics of time domain differential quadrature method, such as numerical stability and calculation accuracy... more
Since initial grid point cy is usually defined as 0, (8) reduces to
When the grid points have been selected, the weighting coefficients matrices G and Gp are easy to calculate with the above formula. Obviously, the weighting coefficients of the differential quadrature method depend on the test functions and distribution of grid points but are independent of some specific pro distributions: Legen blems. There are four typical grid points’ dre grid points, Chebyshev grid points, Chebyshev-Gauss-Lobatto grid points, and Uniform grid points (also called equally spaced grid points) [18]. This paper will focus on the latter three kinds of commonly used grid points, which are de fined as follows:
Furthermore, from Theorem 343B on page 221 in [18], if the original method satisfies the simplifying assumptions C(s) and B(s), then the adjoint method also satisfies the same simplifying assumptions. Hence, the adjoint method enjoys V-transformation:
and the Butcher tableau of equivalent Runge-Kutta method is In the following, the three-stage differential quadrature method using Uniform grid points will be given as an example. When s = 3, ¢,, &, and c, are given by 1/3, 2/3, 1. It can be worked out that matrices Gy and G are given by
5. Improved Differential Quadrature Method Based on the above deduction, the traditional differential quadrature method is a method of s-stage s-order. Compared with the multistage high-order Runge-Kutta method, for example, Gauss method (s-stage 2s-order method), it has the disadvantage of lower precision. As it is well-known that if the stability function of a numerical method is diagonal Padé approximations to the exponential function, then this method is the method of A-stable and s-stage 2s-order [22]. Inspired by this idea, the stability function of new Runge- Kutta method or new differential quadrature method have been converted into diagonal Padé approximation to the exponential function by using undetermined coefficients method. From (38), it can be seen that the stability function of the equivalent Runge-Kutta method will be determined by A, and A*. Suppose A, = A* = A,, without changing b and
and the Butcher tableau of new Runge-Kutta method is Take the same as above, the improved differential quadra- ture method using Uniform grid points will be given as an example. It can be worked out that the new matrices Gy and G are given by After getting y, = (y,,...,,)', coefficients matrix A or G can also be easily computed through using (46). Therefore, a class of new Runge-Kutta method of s-stage 2s-order has been successfully constructed. In other words, a class of improved differential quadrature method of s-stage 2s-order has been derived. Besides, the adjoint method of new Runge-Kutta method is also s-stage 2s-order.
Ficure 1: Error trajectories comparison of different DQM using Chebyshev grid (s = 3, h = 0.5).
FiGure 2: Error trajectories comparison of different DQM using Chebyshev-Gauss-Lobatto grid (s = 3, h = 0.5s).
Figure 3: Error trajectories comparison of different DQM using Uniform grid (s = 3, h = 0.5s)
TABLE 1: Computational results of the displacement (ft = 60 s). References
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