Fourier Expansion Collocation Method Research Articles

Torsional vibration of uniform columns with arbitrarily shaped cross-sections partially submerged in water

This paper presents an analytical-numerical solution to the torsional vibration of uniform columns with arbitrarily shaped cross-sections partially submerged in water, considering the effects of surface waves and the compressibility of water. First, the exact expression for the velocity potential of water motion is given by the method of separation of variables, whose coefficients are decided by utilizing the orthogonality of a group of generalized trigonometric series and the Fourier expansion collocation method. Then, the analytical solution of the differential equation of torsional motion of the column created by water pressure is obtained. The calculated formulae of frequency equations and mode shapes of column-water coupling torsional vibration, which are suitable for programming, are derived by the use of the boundary conditions of the column. The results may be numerically calculated by means of a computer.

Free vibration of arbitrarily shaped plates with concentric ring elastic and/or rigid supports

In the work reported in this paper the problem of free vibration of arbitrarily shaped plates with an arbitrary number of concentric ring supports, elastic and/or rigid, has been considered. A general analytical method is presented. In this analysis, the reaction forces of concentric ring supports, elastic and/or rigid, are regarded as the unknown external forces acting on the plate; the analytical solution of dynamic response of the plate is given, and the unknown integral constants in the solution are obtained by utilizing the Fourier expansion collocation method. The frequency equation which is a determinant whose order is equal to the number of the concentric ring supports, is obtained by using the linear relationships between the displacements of the plate at the supports and the reaction forces of the supports. The eigenfrequencies can be given by the searching root method of determinants. To verify the present method, numerical calculations are carried out for a clamped circular plate with a concentric ring, elastic support, and the results obtained are compared with ones obtained by other methods.

Vibration of uniform columns with arbitrarily shaped cross-sections partially submerged in water considering the effects of surface wave and compressibility of water

This paper is concerned with a method for solving the vibration of uniform columns with arbitrarily shaped cross-sections partially submerged in water considering the effects of surface wave and compressibility of water. First, the exact expression for the velocity potential of water motion is obtained by the method of separation of variables, whose coefficients are decided by utilizing the orthogonality of a group of generalized trigonometric series and the Fourier expansion collocation method. Then, the analytical solution of the differential equation of motion of the column in water is obtained, frequency equation and mode shapes are derived by the use of the boundary conditions of the column. The results may be numerically given by means of a computer.

Method for solving eigenvalue problems of the Helmholtz equation with an arbitrarily shaped outer boundary and a number of eccentric inner boundaries of arbitrary shape

This paper presents a method for solving eigenvalue problems of various phenomena that are governed by the Helmholtz equation with an arbitrarily shaped outer boundary and a number of eccentric boundaries of arbitrary shape. In the analysis, the outer boundary condition is satisfied by means of the Fourier expansion collocation method using the exact solution of the equation of motion. To satisfy the inner boundary conditions, the point restraints are added to the equation of motion, and the inner boundary conditions are satisfied by making use of a point matching procedure. The equation for finding the eigenvalues are derived, and numerical calculations are carried out for cases of elliptical outer and inner boundaries and polygonal outer and inner boundaries.

Wave propagation in a piezoelectric bone with a cylindrical cavity of arbitrary shape

This paper is concerned with the vibration of a piezoelectic cylinder of arbitrary cross section with a cylindrical cavity of arbitrary shape belonging to class 6. The boundary conditions for the inner and the outer arbitrary surfaces are satisfied by using the Fourier expansion collocation method. The frequency equations for the symmetric and the antisymmetric cases are obtained for the cylinder of general cross section of the inner and the outer boundaries. The analysis of the frequency equations are carried out for the material bone. The longitudinal and flexural vibrations are considered and the numerical results are tabulated. The frequency spectra are also presented.

Wave propagation in a piezoelectric human bone of arbitrary cross section with a circular cylindrical cavity

A theoretical study of wave propagation in a piezoelectric cylinder of infinite length of arbitrary cross section with a circular cylindrical cavity of class 6 is investigated. The frequency equations are obtained by using the Fourier expansion collocation method and are analyzed numerically. The frequencies are evaluated for circular, elliptic, and cardioidal sections of bone and are tabulated. A plot of frequency spectrum is also presented for the cardioidal cross-section bar.

Wave propagation in a piezoelectric solid cylinder of arbitrary cross section of class 6 (bone)

This problem deals with the wave propagation in a solid cylinder of arbitrary cross section of class 6. The Fourier expansion collocation method by Nagaya is used to satisfy the boundary conditions. The frequency equation which is applicable for the irregular shape is derived. Numerical solutions have been carried out for the circular, elliptic and cardioidal cross sections for bones which belong to class 6. The results are tabulated and the frequency spectrum is also presented.

Wave propagation in a piezoelectric ceramic cylinder of arbitrary cross section with a circular cylindrical cavity

Wave propagation in an infinite piezoelectric cylinder of arbitrary cross section with a circular cylindrical cavity is discussed. The boundary conditions on the inner surface of the cylinder are satisfied directly and on the outer surface they are satisfied using the Fourier expansion collocation method. The frequency equation that is applicable for the general cross section is obtained. Numerical calculations have been carried out for circular, elliptic, hexagonal, and parabolic cross sections for the ceramic material PZT-4. The results are tabulated and compared with the results obtained in the present case for the circular cylinder. The frequency versus the aspect ratio is plotted and the dispersion curves are also presented.

The torsional vibration of a piezoelectric solid cylinder of arbitrary cross section of (622) class

The torsional vibration of a piezoelectric solid cylinder of arbitrary cross section of class (622) crystals is investigated. The boundary conditions are satisfied using the Fourier expansion collocation method, and the frequency equation which is applicable for the general cross section is derived. Numerical calculations have been carried out for the circular cylinder and elliptic cylinders of β-quartz. The frequencies obtained in the present method for the circular cylinder are compared with those of an exact method.

Wave propagation in a piezoelectric solid cylinder of arbitrary cross section

In this article, the wave propagation in an infinite piezoelectric solid cylinder of an arbitrary cross section is studied. In the analysis, the boundary conditions are satisfied by the use of Nagaya’s Fourier expansion collocation method. The frequency equations for symmetric and antisymmetric cases are obtained and analyzed numerically for longitudinal and flexural waves for the piezoceramic material PZT-4. The vibrations of circular and elliptical cylinders are investigated and the results are tabulated. The results obtained for a circular cross section are compared with the available results. The frequency versus the aspect ratio (the ratio of the semimajor and semiminor axes of the elliptical cylinder) is plotted and the dispersion curve is also presented.

Stresses in a Helical Spring of Arbitrary Cross Section With Consideration of End Effects

This paper presents expressions for stresses in the spring due to compression or tension in the direction of the axis of the cylindrical coil. In deriving stresses in the spring, it is necessary to first obtain resultant forces and resultant moments which occur in the coil spring. Then the analysis first derives exact results based on three-dimensional curved beam theory in which all displacements and all forces in three orthogonal directions of the coil spring are included with consideration of the constraints of the ends of the spring. Since the boundary shape of the spring is irregular, it is also difficult to satisfy stress-free boundary conditions along the surface of the coil, so that this paper applies the Fourier expansion collocation method and obtains stresses in the coil spring based on the theory of elasticity.

Stress analysis of a viscoelastic solid of arbitrary cylindrical shape having a circular cavity and subject to internal vibration pressures

This paper is concerned with a method for solving stress problems of a viscoelastic solid of arbitrary shape with a circular cavity subjected to a vibration pressure along the cavity. In the analysis the exact solutions which satisfy the boundary conditions along the cavity have been derived from the equations of motion of elastic solids based on the two-dimensional theory of elasticity. The boundary conditions along the outer surface of arbitrary shape have been satisfied by means of the Fourier expansion collocation method. The analysis takes account of the viscoelasticity and the structural damping theory is used. As examples, stresses in the solids of circular, hexagonal, square, elliptical and truncated elliptical shapes have been calculated, and curves for the forced vibration response are presented.

A rotary eddy-current brake or damper consisting of several sector magnets and a plate conductor of arbitrary shape

A method for analyzing a rotary magnetic damper (eddy-current brake) consisting of several sector magnets and a plate conductor of arbitrary shape is presented. The analytical solution to obtain the eddy current, braking force, and damping coefficient is obtained by dividing the magnetic flux into the narrow sector bands, and the unit step function is applied to solve the differential equation of the electromagnetic fields. The boundary condition of the plate conductor of arbitrary shape is satisfied directly by making use of the Fourier expansion collocation method. Numerical calculations have been carried out for the conductors of square plates, hexagonal plates, and elliptical plates, and the damping coefficients versus the variation of the flux range, the position, and the number of the magnets have been investigated in detail.

Transient response of plates with arbitrary shape in contact with a fluid subjected to general dynamic pressures on a fluid surface

This paper is concerned with a method for solving dynamic response problems of arbitrarily shaped plates in contact with a fluid. By utilizing the Fourier expansion and the Laplace transform methods for disposing of general dynamic pressures, and by applying the Fourier expansion collocation method for satisfying the boundary conditions, expressions for the dynamic response of displacement and bending moment are obtained in a general form which is applicable to general dynamic pressures and arbitrarily shaped plates. Numerical calculations have been carried out for circular plates, rectangular plates, rectangular plates with round corners, and oval plates. The results obtained in certain plates are compared with the exact ones.

Dynamic response of a two-dimensional elastic solid of arbitrary shape having a circular cavity subjected to a transient pressure along the cavity

This paper is concerned with a method for solving response problems of an elastic solid of arbitrary shape with a circular cylindrical cavity subjected to a transient pressure along the cavity. In the analysis, the exact solutions which satisfy the boundary conditions along the cavity are obtained by applying the Laplace transformation to the equations of motion of elastic solids based on the two-dimensional theory of elasticity. The boundary conditions along the outer surface of arbitrary shape are satisfied by means of the Fourier expansion collocation method. The Laplace transform inversion integral is obtained from the residue theorem. Reasonably accurate results are obtained when the integration for finding the coefficients of the Fourier series and the numerical differentiation are carried out appropriately. To discuss the accuracy of the present analysis, results are compared with those from an exact method for circular rings. Numerical calculations are carried out for sample cases.

A Rotary Magnetic Damper Consisting of Several Sector Magnets and an Arbitrarily Shaped Conductor With a Circular Cavity

In this paper, the damping force and the damping coefficient of a rotary magnetic damper consisting of several sector magnets and an arbitrarily shaped plate conductor with a circular cavity have been obtained theoretically. The unit step function is applied to solve the differential equations of the electromagnetic field, and the boundary condition of the outer arbitrarily shaped boundary of the plate conductor is satisfied directly by making use of the Fourier expansion collocation method. Numerical calculations have been carried out for the dimensionless damping coefficients with the variations of various factors such as the magnetic flux range, the outer shape and the radius of the inner circular cavity of the conductor, the position and the number of the magnets.

A method for solving forced vibration problems of arbitrarily shaped sandwich plates

This paper is concerned with a method for solving forced vibration problems of solid sandwich plates with irregular boundaries. The exact, general solution of the equation of motion in terms of Bessel functions is found. The boundary problem is solved by using the Fourier expansion collocation method. Damping properties of an intermediate, viscoelastic layer are taken into consideration by means of a concept of a complex shear modulus. As an application, the numerical calculations for circular, truncated elliptical, and parabolic sandwich plates with clamped edges are carried out. Miscellaneous phenomena occurring in sandwich plates are discussed.

Free vibration of a plate with arbitrary shape on a finite elastic foundation in consideration of effects of masses of foundation

AbstractThis paper presents a method for solving vibration problems of a plate with irregular shape lying on an elastic foundation in consideration of effects of masses of foundation. For the foundation, the equation of motion of longitudinal vibrations of a solid is utilized, and the solutions of both the foundation and the plate are joined together by use of the conditions of continuity between the plate and the foundation. The analysis uses the Fourier expansion collocation method given by Nagaya to satisfy the irregular boundary conditions. Numerical calculations have been carried out for general polygonal plates on elastic foundations. The results for the plate on Winkler's foundation are also given to compare both results. It is clarified that there are some discrepancies between the present result which includes foundation inertias and that of Winkler's foundation.

Stresses Analysis of a Cylindrical Coil Spring of Arbitrary Cross Section : 1st Report, Solution by Fourier Expansion Collocation Method

This paper presents a theoretical analyis for obtaining stresses of a spring of arbitrary cross section. In the analysis the stresses of a straight rod subjected to torsion, shear and bending are derived based on the two-dimensional theory of elasticity by means of the Fourier expansion collocation method which has been reported by the author. The results for a coil spring are derived by including the curved rod theory in the expressions for a straight rod. The present result is applicable to the problem of coil springs by merely including the geometrical data on the spring, whatever the shape. The numerical results obtained by this analysis are in very good agreement with the previously published ones for some typical cases.

Stress analysis of a viscoelastic solid of arbitrary shape having a circular cavity subjected to a vibration pressure along the cavity.

This paper is concerned with a method for solving the stress problems of a viscoelastic solid of arbitrary shape with a circular cavity subjected to a vibration pressure along the cavity. In the analysis the exact solutions which satisfy the boundary conditions along the cavity have been derived from the equations of motion of elastic solids based on the two dimensional theory of elasticity. The boundary conditions along the outer surface of arbitrary shape have been satisfied by means of the Fourier expansion collocation method. The analysis considers the viscoelasticity and uses the structural damping theory. As examples, stresses in the solids of circular, hexagonal, square, elliptical and truncated elliptical shapes have been calculated.