T. Irie

Free vibration of a conical shell with variable thickness

Abstract An analysis is presented for the free vibration of a truncated conical shell with variable thickness by use of the transfer matrix approach. The applicability of the classical thin shell theory is assumed and the governing equations of vibration of a conical shell are written as a coupled set of first order differential equations by using the transfer matrix of the shell. Once the matrix has been determined by quadrature of the equations, the natural frequencies and the mode shapes of vibration are calculated numerically in terms of the elements of the matrix under any combination of boundary conditions at the edges. The method is applied to truncated conical shells with linearly, parabolically or exponentially varying thickness, and the effects of the semi-vertex angle, truncated length and varying thickness on the vibration are studied.

Free vibration of joined conical-cylindrical shells

Abstract An analysis is presented for the free vibration of joined conical-cylindrical shells. The governing equations of vibration of a conical shell, including a cylindrical shell as a special case, are written as a coupled set of first order differential equations by using the transfer matrix of the shell. Once the matrix has been determined, the entire structure matrix is obtained by the product of the transfer matrices of the shells and the point matrix at the joint, and the frequency equation is derived with terms of the elements of the structure matrix under the boundary conditions. The method has been applied to a joined truncated conical-cylindrical shell and an annular plate-cylindrical shell system, and the natural frequencies and the mode shapes of vibration calculated numerically. The results are presented.

Vibration and stability of a non-uniform Timoshenko beam subjected to a follower force

Abstract An analysis is presented for the vibration and stability of a non-uniform Timoshenko beam subjected to a tangential follower force distributed over the center line by use of the transfer matrix approach. For this purpose, the governing equations of a beam are written in a coupled set of first-order differential equations by using the transfer matrix of the beam. Once the matrix has been determined by numerical integration of the equations, the eigenvalues of vibration and the critical flutter loads are obtained. The method is applied to beams with linearly, parabolically and exponentially varying depths, subjected to a concentrated, uniformly distributed or linearly distributed follower force, and the natural frequencies and flutter loads are calculated numerically, from which the effects of the varying cross-section, slenderness ration, follower force and the stiffness of the supports on them are studied.

Free vibration of polar-orthotropic sector plates

Abstract The free vibration of ring-shaped polar-orthotropic sector plates is analyzed by the Ritz method using a spline function as an admissible function for the deflection of the plates. For this purpose, the transverse deflection of a sector plate is written in a series of the products of the deflection function of a sectorial beam and that of a circular beam satisfying the boundary conditions. The deflection function of the sectorial beam is approximately expressed by a quintic spline function, which satisfies the equation of flexural vibration of the beam at each point dividing the beam into small elements. The frequency equation of the plate is derived by the conditions for a stationary value of the Lagrangian. The present method is applied to ring-shaped polar-orthotropic sector plates with some combination of boundary conditions, and the natural frequencies and the mode shapes are calculated numerically up to higher modes. This method is very effective for the study of vibration problems of variously shaped anisotropic plates including these sector plates.

Free vibration of non-circular cylindrical shells with variable circumferential profile

Abstract An analysis is presented of the free vibration of non-circular cylindrical shells with a variable circumferential profile expressed as an arbitrary function. The applicability of thin-shell theory is assumed and the governing equations of vibration of a non-circular cylindrical shell are written in a matrix differential equation by using the transfer matrix of the shell. Once the transfer matrix has been determined by numerical integration of the matrix equation, the natural frequencies and mode shapes of vibration are calculated numerically in terms of the matrix elements. The method is applied to cylindrical shells of three or four-lobed cross-section, and the effects of the length of the shell and the radius at the lobed corners on the vibration are studied.

Free vibration of a mindlin annular plate of varying thickness

Abstract The free vibration of a Mindlin annular plate of radially varying thickness is analyzed by use of the transfer matrix approach. For this purpose, the Mindlin equations of flexural vibration of an annular plate are written as a coupled set of first-order differential equations by using the transfer matrix of the plate. Once the matrix has been determined by the numerical integration of the equations, the natural frequencies and the mode shapes of the vibration are calculated numerically in terms of the elements of the matrix for a given set of boundary conditions at the edges of the plate. This method is applied to annular plates of linearly, parabolically and exponentially varying thickness, and the effects of the varying thickness are studied.

Determination of the steady state response of a viscoelastically point-supported rectangular plate

The steady state response to a sinusoidally varying force is determined for a viscoelastically point-supported square or rectangular plate. For this purpose, the transverse deflection of the plate is written in a series of the product of the deflection functions of beams parallel to the edges, and the response equation is derived by the generalized Galerkin method. The natural boundary conditions of the plate which cannot be satisfied by the beam functions at the edges and the corners are appropriately compensated by suitable additions to the residual forces and moments. The method is applied to a square plate supported at four points symmetrically located at the corners or on the diagonals; the steady state response of the plate to a point force acting at the centre is calculated numerically, and the effects of the point supports on the vibration are studied.

The steady state out-of-plane response of a Timoshenko curved beam with internal damping

Abstract The steady state out-of-plane response of a Timoshenko curved beam with internal damping to a sinusoidally varying point force or moment is determined by use of the transfer matrix approach. For this purpose, the equations of out-of-plane vibration of a curved beam are written as a coupled set of the first order differential equations by using the transfer matrix of the beam. Once the matrix has been determined by numerical integration of the equations, the steady state response of the beam is obtained. The method is applied to free-clamped non-uniform beams with circular, elliptical, catenary and parabolical neutral axes driven at the free end; the driving point impedance and force or moment transmissibility are calculated numerically and the effects of the slenderness ratio, varying cross-section and the function expressing the neutral axis on them are studied.

Free vibration of a circular cylindrical double-shell system closed by end plates

An analysis is presented for the free vibration of a circular cylindrical double-shell system closed by end plates. The governing equations of vibration of an inner or an outer shell and of an end plate are written as matrix differential equations of the first order by using the transfer matrices of the shell and the plate. Once the matrices have been determined, the entire structure matrix is obtained by forming the product of the transfer matrices of the shell and the plate and the point matrices at the joints, and the frequency equation of the system is derived with terms of the elements of the structure matrix. The method has been applied to a uniform thickness double-shell system composed of two co-axial shells and two annular plates, and results of numerical calculations of the natural frequencies and the mode shapes of vibration are presented.

Determination of the steady state response of a Timoshenko beam of varying cross-section by use of the spline interpolation technique

Abstract The steady state response of an internally damped Timoshenko beam of varying cross-section to a sinusoidally varying point force is determined by use of the spline interpolation technique. For this purpose, with the beam divided into small elements, the response of each element is expressed by a quintic spline function with unknown coefficients. The response is obtained by determining these coefficients so that the spline function satisfies the equation of motion of the beam at each dividing point and also satisfies the boundary conditions at both ends. In this case, the slope due to pure bending of the beam is conveniently adopted as the function essentially expressing the response, from which the transverse deflection, driving point impedance, transfer impedance and force transmissibility of the beam are derived. The method is applied to cantilever beams with linearly, parabolically and exponentially varying rectangular cross-sections; these responses of the beams are calculated numerically and the effects of the varying cross-section on them are studied.