Representative Mode Shapes Research Articles

Dynamic Stiffness Formulation for Free Vibration of Truncated Conical Shell and Its Combinations with Uniform Boundary Restraints

This paper presents a dynamic stiffness formulation for the free vibration analysis of truncated conical shell and its combinations with uniform boundary restraints. The displacement fields are expressed as power series, and the coefficients of the series are obtained as recursion formula by substituting the power series into the governing equations. Then, the general solutions can be replaced by an algebraic sum which contains eight base functions, which can diminish the number of degrees of freedom directly. The dynamic stiffness matrix is formulated based on the relationship between the force and displacement along the boundary lines. In the formulation, arbitrary elastic boundary restraints can be realized by introducing four sets of boundary springs along the displacement directions at the boundary lines. The modeling methodology can be easily extended to the combinations of conical shells with different thickness and semivertex angles. The convergence and accuracy of the present formulation are demonstrated by comparing with the finite element method using several numerical examples. Effects of the elastic boundary condition and geometric dimension on the free vibration characteristics are investigated, and several representative mode shapes are depicted for illustrative purposes.

Coupled Bi-Flexural–Torsional Vibration of Fluid-Conveying Pipes Spinning About an Eccentric Axis

This paper presents a dynamical model of a fluid-conveying pipe spinning about an eccentric axis. The coupled bi-flexural–torsional free vibration and stability are analyzed for such a doubly gyroscopic system. The partial differential equations of motions are derived by the extended Hamilton principle, and are then truncated by a 4-term Galerkin technique. The frequency and energy evolutions and representative mode shapes in the two transverse directions and torsional direction are investigated to unveil the essential dynamical attributes of the system. It is indicated that the stability of the present system mainly depends on spinning speed, flow velocity and eccentricity, while the torsional frequencies are almost immune to the flow velocity. The pipe reveals ‘traveling-wave’ modal vibrations in both transverse directions, and a general ‘standing-wave’ modal vibration in the torsional direction.

Free vibration of rotating pretwisted CNTs-reinforced shallow shells in thermal environment

ABSTRACTIn this paper, finite element method is employed to study the free vibration response of the rotating pretwisted carbon nanotube-reinforced composite cylindrical shell in thermal environment. The shells are reinforced by single walled carbon nanotubes (SWCNTs) and the material properties of SWCNTs are considered to be temperature dependent. The dynamic equilibrium equation is derived from Lagrange's equation for moderate rotational speeds for which the Coriolis effect is neglected. Parametric study is performed to investigate the effects of pretwist angles, temperature rise, rotational speed, and volume fraction of CNTs. In addition to this, the representative mode shapes are also presented.

Free vibration characteristics of delaminated composite pretwisted stiffened cylindrical shell

Effects of delamination on free vibration characteristics of laminated stiffened cylindrical shells with pretwist are analyzed by finite element method. The investigation is carried out using an eight-noded quadratic isoparametric shell element, which incorporates the transverse shear deformation and rotary inertia along with a three-noded beam element for the stiffener. The multipoint constraint algorithm has been included to guarantee the compatibility of deformation, equilibrium of resultant forces, and moments at delamination crack tip. The general dynamic equilibrium equation is derived from Lagrange’s equation of motion for moderate rotational speeds for which the Coriolis effect is neglected. The standard eigenvalue problem is solved utilizing QR iteration algorithm. The accuracy of the present formulation is validated with benchmark solutions is available in the literature. The present work concerns about the effects of delamination, fiber orientation, twist angle, stiffener depth-to-shell thickness ratio, and rotational speed on the fundamental frequency of shallow cylindrical shells with stiffener. Representative mode shapes for some typical case of the stiffened shell for different twist angles and rotational speeds are also presented.

Analytic free vibration solutions of rectangular thin plates point-supported at a corner

In a recently published study, we reported the benchmark bending solutions of rectangular thin plates point-supported at a corner by an up-to-date symplectic superposition method. In this paper, for the first time, we extend the approach to the free vibration problems of the same plates and obtain the analytic solutions which cannot be obtained by the conventional symplectic approach. More than three hundred frequencies and the representative mode shapes are presented as the benchmarks for verification of various new approaches developed in future.

A numerical solution for vibration analysis of composite laminated conical, cylindrical shell and annular plate structures

This paper focuses on the free vibration analysis of composite laminated conical, cylindrical shells and annular plates with various boundary conditions based on the first order shear deformation theory, using the Haar wavelet discretization method. The equations of motion are derived by applying the Hamilton’s principle. The displacement and rotation fields are expressed as products of Fourier series for the circumferential direction and Haar wavelet series and their integral along the meridional direction. The constants appearing from the integrating process are determined by boundary conditions, and thus the equations of motion as well as the boundary condition equations are transformed into a set of algebraic equations. Then natural frequencies of the laminated shells are obtained by solving algebraic equations. Accuracy, stability and reliability of the current method are validated by comparing the present results with those in the literature and very good agreement is observed. Effects of some geometrical and material parameters on the natural frequencies of composite shells are discussed and some representative mode shapes are given for illustrative purposes. Some new results for laminated shells are presented, which may serve as benchmark solutions.

Free vibration of a paraboloidal shell of revolution including shear deformation and rotary inertia

The governing strain-displacement and curvature-displacement equations for paraboloidal shells including shear deformation and rotary inertia are solved for free vibration of closed shells. The finite element method is used to obtain three-dimensional frequency of vibration solutions for a variety of boundary conditions, free, fixed and simply supported. Assumptions concerning the circumferential vibrational behavior are incorporated that reduce the analysis to a single coordinate and the element shape function is formulated using the meridional coordinate. The results for frequency of vibration compare favorably with the available literature. Selected results for frequency of vibration are presented in tabular form for several shell parameters, including free, pinned and fixed boundary conditions. Representative mode shapes are plotted for a fixed boundary condition.

Aerodynamics of guyed masts

Guyed masts response to wind action, ice load and thermal action has been analysed in this paper. Computational model of wind action based on quasi-steady theory has been applied. It has been assumed that nodal displacements may be expressed as a linear combination of representative mode shapes. Special computer program has been made to allow analysis of such structures. It has been found that the selection of representative mode shapes is very influential on the calculations outcome. The obtained results have been compared with the ones calculated according to patch loads method presented in Eurocode 3. The dynamic part of forces calculated with own method are about half of the ones coming from the standard.

A Comparative Aerodynamic Analysis of Two Cable-Stayed Bridges Built in Poland

The two longest cable-stayed bridges in Poland (230m and 250m main span lengths) have been subjected to the aerodynamic analyses. This paper presents the main topics in these calculation processes: modal analysis, estimation of internal forces generated by the static load action, analysis of vibrations excited by the atmospheric turbulence and feedbacks, determination of bridge vibrations generated by the vortex excitation, calculation of stresses produced with the dynamic wind action. The dynamic structure response has been calculated using the quasi-steady method, which allowed analysis of just a few representative mode shapes, with the results precision acceptable in the engineering practice. Finally, both bridges have been found safe and resistant to the dynamic wind action, however the Warsaw bridge is more susceptible to the aerodynamic influence, especially its deck.

Natural Frequencies and Mode Shapes of Beams Carrying a Two Degree-of-Freedom Spring-Mass System

An exact solution for the natural frequencies and mode shapes for a beam elastically constrained at its ends and to which a rigid mass is elastically mounted is obtained. The attached mass can both translate and rotate. The general solution is obtained using the Laplace transform with respect to the spatial variable and yields the exact solutions to several previously published simpler configurations that were obtained using approximate methods. Numerous numerical results are presented for the natural frequency coefficients that extend previously reported results and that show the transition between various limiting cases. In addition, values are presented for the lowest two natural frequency coefficients for a beam that is clamped at both ends and is carrying a two dof spring-mass system. Representative mode shapes at selected values of the system’s parameters are also given.

Axisymmetric vibrations of a cylindrical- hemispherical tank partially filled with a liquid.

A mathematical analysis and evaluation is made of the axisymmetric vibration modes of a cylindrical-hemispherical tank partially filled with a liquid. The procedure used is similar to that employed previously by Miles. It consists of formulating generalized functions that describe compatible deformations of the shell and liquid. Kinetic and potential energy expressions are then formed and the equations governing modal motion are derived by application of Lagrange's equations. A number of calculations are made for study of the modal behavior of the tank. The results are presented in the form of modal frequency vs liquid level, with cylinder-hemisphere stiffness ratio as a parameter. Representative mode shapes showing displacements of the tank wall and free surface of the liquid are also presented. Finally, a check on the accuracy of the calculations is obtained by study of the pressure matching of the liquid-shell interfaces. The results show that the bulging behavior of the tank can be visualized as a coupling of the motions of two simpler tank configurations: 1) a tank with rigid cylindrical walls and a flexible hemispherical bottom and 2) a tank with flexible cylindrical walls and a rigid bottom.