TRANSIENT ANALYSIS OF RING-STIFFENED COMPOSITE CYLINDRICAL SHELLS WITH BOTH EDGES CLAMPED

A theoretical method is developed to investigate the effects of ring stiffeners on vibration characteristics and transient responses for the ring-stiffened composite cylindrical shells subjected to the step pulse loading. Love's thin shell theory combined with the discrete stiffener theory to consider the ring stiffening effect is adopted to formulate the theoretical model. The ring stiffeners are laminated with a composite material and have a uniform rectangular cross-section. The Rayleigh-Ritz procedure is applied to obtain the frequency equation. The modal analysis technique is used to develop the analytical solutions of the transient response. The analysis is based on an expansion of the loads, displacements in the double Fourier series that satisfy the boundary conditions. The effect of stiffener's eccentricity, number, size, and position on transient response of the shells is examined. The theoretical results are verified by comparison with FEM results.     

FREE VIBRATION ANALYSIS OF COMPOSITE RECTANGULAR MEMBRANES WITH AN OBLIQUE INTERFACE

The natural frequencies and mode shapes of a composite rectangular membrane with no exact solutions are found by using an analytical method appropriate for the geometric feature of the title problem membrane presented here. The method has a key feature in which the theoretical development is very simple and only a small amount of numerical calculation is required. Example studies show that the natural frequencies and their associated modes obtained from the method are found to be very accurate compared with the results by the FEM (SYSNOISE) or exact solutions. Furthermore, the natural frequencies converge rapidly and accurately to the exact values or the numerical results obtained from the finite element model using meshes sufficient to yield already converging natural frequencies, even when a small number of series functions are used in the proposed method.     

VIBRATION OF TWISTED AND CURVED CYLINDRICAL PANELS WITH VARIABLE THICKNESS

A numerical procedure, with an exact strain-displacement relationship of twisted and curved cylindrical panels having variable thickness derived by considering the Green strain tensor on general shell theory, is presented using the principle of virtual work and the Rayleigh-Ritz method with algebraic polynomials as in-plane and transverse displacement functions. The accuracy and applicability of the procedure are verified by comparing the present results with previous experimental and theoretical results for several panels. The effects of variation ratio of thickness in chordwise and lengthwise directions, twist, and curvature both in two directions aforementioned on vibrations of cylindrical panels are studied in detail, and typical vibration mode shapes are plotted to demonstrate the effects.     

A GROUP THEORETIC APPROACH TO THE LINEAR FREE VIBRATION ANALYSIS OF SHELLS WITH DIHEDRAL SYMMETRY

This paper deals with a group theoretic approach to the finite element analysis of linear free vibrations of shells with dihedral symmetry. Examples of such shell structures are cylindrical shells, conical shells, shells with circumferential stiffeners, corrugated shells, spherical shells, etc. The group theoretic approach is used to exploit the inherent symmetry in the problem. For vibration analysis, the group theoretic results give the correct symmetry-adapted basis for the displacement field. The stiffness matrix K and the mass matrix M are identically block diagonalized in this basis. The generalized linear eigenvalue problem of free vibration gets split into independent subproblems due to this block diagonalization. The Simo element is used in the finite element formulation of the shell equilibrium equations. Numerical results for natural frequencies and natural modes of vibration of several dihedral shell structures are presented. The results are shown to be in very good agreement with those reported in the literature. The computational advantages and physical insights due to the group theoretic approach are also discussed.     

ON THE FREE VIBRATION OF STIFFENED SHALLOW SHELLS

A finite element analysis for free vibration behaviour of doubly curved stiffened shallow shells is presented. The stiffened shell element is obtained by the appropriate combinations of the eight-/nine-node doubly curved isoparametric thin shallow shell element with the three-node curved isoparametric beam element. The shell types examined are the elliptic and hyperbolic paraboloids, the hypar and the conoidal shells. The accuracy of the formulation is established by comparing some of the authors' results of specific problems with those available in the literature. Numerical results of additional stiffened shells are also presented to study the effects of various parameters of shells and stiffeners such as orientation (i.e., along x -/y -/both x and y directions), type (concentric, eccentric at top and eccentric at bottom) and number of stiffeners, stiffener depth to shell thickness ratio, and aspect ratio, shallowness and boundary conditions of shells on free vibration characteristics.     

DYNAMICS AND DISTRIBUTED CONTROL OF CONICAL SHELLS LAMINATED WITH FULL AND DIAGONAL ACTUATORS

Nozzles, rocket fairings and many engineering structures/components are often made of conical shells. This paper focuses on the finite element modelling, analysis, and control of conical shells laminated with distributed actuators. Electromechanical constitutive equations and governing equations of a generic piezo(electric)elastic continuum are defined first, followed by the strain-displacement relations and electric field-potential relations of laminated shell composites. Finite element formulation of a piezoelastic shell element with non-constant Lamé parameters is briefly reviewed; element and system matrix equations of the piezoelastic shell sensor/actuator/structure laminate are derived. The system equation reveals the coupling of mechanical and electric fields, in which the electric force vector is often used in distributed control of shells. Finite element eigenvalue solutions of conical shells are compared with published numerical results first. Distributed control of the conical shell laminated with piezoelectric shell actuators is investigated and control effects of three actuator configurations are evaluated.     

GENERALIZED DIFFERENTIAL QUADRATURE METHOD FOR THE FREE VIBRATION OF TRUNCATED CONICAL PANELS

This paper presents an improved generalized differential quadrature (GDQ) method for the investigation of the effects of boundary conditions on the free vibration characteristics of truncated conical panels. The truncated conical panel is an important geometrical shape in the fields of aerospace, marine and structural engineering. However, despite this importance, few works in free vibration analysis have dealt with this particular geometry. In this work, the vibration characteristics of clamped and simply supported truncated conical shells are obtained for various circumferential wave numbers. Further, the effects of the vertex and subtended angles on the frequency parameters are also examined in detail. Due to limited published results in the open literature, results for a range of cases are compared with those generated from the commercial finite element solver McNeal-Schwendler Corporation Nastran, and excellent agreement is observed.     

THE EFFECTS OF LARGE VIBRATION AMPLITUDES ON THE MODE SHAPES AND NATURAL FREQUENCIES OF THIN ELASTIC SHELLS. PART II: A NEW APPROACH FOR FREE TRANSVERSE CONSTRAINED VIBRATION OF CYLINDRICAL SHELLS

The non-linear dynamic behaviour of infinitely long circular cylindrical shells in the case of plane strains is examined and results are compared with previous studies. A theoretical model based on Hamilton's principle and spectral analysis previously developed for non-linear vibration of thin straight structures (beams and plates) is extended here to shell-type structures, reducing the large-amplitude free vibration problem to the solution of a set of non-linear algebraic equations. In the present work, the transverse displacement is assumed to be harmonic and is expanded in the form of a finite series of functions corresponding to the constrained vibrations, which exclude the axisymmetric displacements. The non-linear strain energy is expressed by taking into account the non-linear terms due to the considerable stretching of the shell middle surface induced by large deflections. It has been shown that the model presented here gives new results for infinitely long circular cylindrical shells and can lead to a good approximation for determining the fundamental longitudinal mode shape and the associated higher circumferencial mode shapes (n>3) of simply supported circular cylindrical shells of finite length. The non-linear results at small vibration amplitudes are compared with linear experimental and theoretical results obtained by several authors for simply supported shells. Numerical results (non-linear frequencies, vibration amplitudes and basic function contributions) of infinite shells associated to the first four mode shapes of free vibrations, are obtained, using a multi-mode approach and are summarized in tables. Good agreement is found with results from previous studies for both small and large amplitudes of vibration. The non-linear mode shapes are plotted and discussed for different thickness to radius ratios. The distributions of the bending stresses associated with the mode shapes are given and compared with those obtained via the linear theory.     

VIBRATION AND BUCKLING OF COMPOSITE THIN-WALLED BEAMS WITH SHEAR DEFORMABILITY

In this paper, a theoretical model is developed for the dynamic analysis of composite thin-walled beams with open or closed cross-sections. The present model incorporates, in a full form, the shear flexibility (bending and warping shear) as well as a state of initial stresses. This allows to study the free vibration and buckling problems in a unified fashion. An analytical solution of the developed equations is obtained for the case of simply supported thin-walled beams. Numerical examples are given to demonstrate the importance of the shear flexibility on the vibration and buckling behavior of the considered structures.     

MODAL ANALYSIS OF ROTATING COMPOSITE CANTILEVER PLATES

A modelling method for the modal analysis of a rotating composite cantilever plate is presented in this paper. A set of linear ordinary differential equations of motion for the plate is derived by using the assumed mode method. Two in-plane stretch variables are employed and approximated to derive the equations of motion. The equations of motion include the coupling terms between the in-plane and the lateral motions as well as the motion-induced stiffness variation terms. Dimensionless parameters are identified and the explicit mass and the stiffness matrices for the modal analysis are obtained with the dimensionless parameters. The effects of the dimensionless angular velocity and the fiber orientation angles of rotating composite cantilever plates on their modal characteristics are investigated. Natural frequency loci veering and crossing along with associated mode shape variations are observed.     

VIBRATION EIGENFREQUENCY ANALYSIS OF A SINGLE-LINK FLEXIBLE MANIPULATOR

We analyse the vibration eigenfrequencies of a flexible slewing beam with a payload attached at one end. A wave propagation method (WPM) is used. There are four types of waves which propagate along a beam—two dispersive waves travelling in opposite directions, and two evanescent waves near the endpoints. We add a fifth time-harmonic function corresponding to oscillation of the beam at the payload end. We show that the large frequencies are asymptotically identical to those for the clamped–free beam, independent of the payload. For small eigenfrequencies, we incorporate WPM with a perturbation iteration procedure, the results of which agree well with “exact” values which result from solving a transcendental equation cited elsewhere in the literature.     

样条混合元法解板壳问题

应用三次B样条插值函数与二类变量广义变分原理建立一种样条混合元法来求解板壳结构的弯曲问题。应用样条函数来构造壳体的二类场函数:一类为位移场函数,另一类为广义力场函数,由二类变量广义变分原理导出样条混合元法的系统方程组。给出若干数值算例,其计算结果与其它方法作了比较。     

GEOMETRICALLY NON-LINEAR FREE VIBRATION OF FULLY CLAMPED SYMMETRICALLY LAMINATED RECTANGULAR COMPOSITE PLATES

The geometrically non-linear free vibration of thin composite laminated plates is investigated by using a theoretical model based on Hamilton's principle and spectral analysis previously applied to obtain the non-linear mode shapes and resonance frequencies of thin straight structures, such as beams, plates and shells (Benamar et al. 1991Journal of Sound and Vibration149 , 179-195; 1993, 164, 295-316; 1990 Proceedings of the Fourth International Conference on Recent Advances in Structural Dynamics, Southampton; Moussaoui et al. 2000 Journal of Sound and Vibration232, 917-943 [1-4]). The von Kármán non-linear strain-displacement relationships have been employed. In the formulation, the transverse displacement W of the plate mid-plane has been taken into account and the in-plane displacements U and V have been neglected in the non-linear strain energy expressions. This assumption, quite often made in the literature has been adopted in reference [2] and (El Kadiri et al. 1999 Journal of Sound and Vibration228, 333-358 [5]), in the isotropic case and has been mentioned here because the results obtained have been found to be in very good agreement with those based on the hierarchical finite element method (HFEM). In a previous study, it was assumed, based on the analogy with the isotropic case, that the fundamental carbon fibre reinforced plastic (CFRP) plate non-linear mode shape could be well estimated, by using nine plate functions, obtained as products of clamped-clamped beam functions in the x and y directions, symmetric in both the length U001and width directions [3]. In the present work, a convergence study has been performed and has shown that, although such an assumption may yield a good estimate for the non-linear resonance frequency, 18 plate functions should be taken into account instead of nine in the first non-linear mode shape and associated bending stress patterns calculations. This allows the anisotropy induced by the fibre orientations to be taken into account. Results are given for the fundamental mode of fully clamped CFRP rectangular plates, for various plate aspect ratios and vibration amplitudes. The non-linear mode shows a higher bending stress near the clamps at large deflection, compared with that predicted by linear theory. Some experimental measurements are presented which are in good qualitative agreement with the theory.     

STRUCTURAL RESPONSE OF LAMINATED COMPOSITE SHELLS SUBJECTED TO BLAST LOADING: COMPARISON OF EXPERIMENTAL AND THEORETICAL METHODS

The results from a theoretical and experimental investigation of the dynamic response of cylindrically curved laminated composite shells subjected to normal blast loading are presented. The dynamic equations of motion for cylindrical laminated shells are derived using the assumptions of Love's theory of thin elastic shells. Kinematically admissible displacement functions are chosen to represent the motion of the clamped cylindrical shell and the governing equations are obtained in the time domain using the Galerkin method. The time-dependent equations of the cylindrically curved laminated shell are then solved by the Runge-Kutta-Verner method. Finite element modelling and analysis for the blast-loaded cylindrical shell are also presented. Experimental results for cylindrically curved laminated composite shells with clamped edges and subjected to blast loading are presented. The blast pressure and strain measurements are performed on the shell panels. The strain response frequencies of the clamped cylindrical shells subjected to blast load are obtained using the fast Fourier transformation technique. In addition, the effects of material properties on the dynamic behaviour are examined. The strain-time history curves show agreement between the experimental and analysis results in the longitudinal direction of the cylindrical panels. However, there is a discrepancy between the experimental and analysis results in the circumferential direction of the cylindrical panels. A good prediction is obtained for the response frequency of the cylindrical shell panels.