Dynamics of a rectangular plate resting on an elastic foundation and partially in contact with a quiescent fluid

In this study, a method of analysis is presented for investigating the effects of elastic foundation and fluid on the dynamic response characteristics (natural frequencies and associated mode shapes) of rectangular Kirchhoff plates. For the interaction of the Kirchhoff plate–Pasternak foundation, a mixed-type finite element formulation is employed by using the Gâteaux differential. The plate finite element adopted in this study is quadrilateral and isoparametric having four corner nodes, and at each node four degrees of freedom are present (one transverse displacement, two bending moments and one torsional moment). Therefore, a total number of 16 degrees-of-freedom are assigned to each element. A consistent mass formulation is used for the eigenvalue solution in the mixed finite element analysis. The plate structure considered is assumed clamped or simply supported along its edges and resting on a Pasternak foundation. Furthermore, the plate is fully or partially in contact with fresh water on its one side. For the calculation of the fluid–structure interaction effects (generalized fluid–structure interaction forces), a boundary element method is adopted together with the method of images in order to impose an appropriate boundary condition on the fluid's free surface. It is assumed that the fluid is ideal, i.e., inviscid, incompressible, and its motion is irrotational. It is also assumed that the plate–elastic foundation system vibrates in its in vacuo eigenmodes when it is in contact with fluid, and that each mode gives rise to a corresponding surface pressure distribution on the wetted surface of the structure. At the fluid–structure interface, continuity considerations require that the normal velocity of the fluid is equal to that of the structure. The normal velocities on the wetted surface of the structure are expressed in terms of the modal structural displacements, obtained from the finite element analysis. By using the boundary integral equation method the fluid pressure is eliminated from the problem, and the fluid–structure interaction forces are calculated in terms of the generalized hydrodynamic added mass coefficients (due to the inertial effect of fluid). To asses the influences of the elastic foundation and fluid on the dynamic behavior of the plate structure, the natural frequencies and associated mode shapes are presented. Furthermore, the influence of the submerging depth on the dynamic behavior is also investigated.     

Purely axial vibration of thin cylindrical shells with shear-diaphragm boundary conditions

This work presents the free vibration characteristics of a thin walled cylindrical shell at the zeroth axial mode number. The cylindrical shell has shear-diaphragm boundary conditions at each end. The thin shell equations by Flügge are used as these equations of motion lead to more accurate results at low frequencies. The zeroth axial mode number is found to occur at the cut-on of the second class of waves. The mode shapes at these natural frequencies result in a purely axial displacement of the middle surface of the shell. High modal density for the first class of waves occurs before the cutting-on of the second class of waves. Beyond this frequency, the dynamic response is dominated by the latter modes.     

Natural frequencies of shallow cylindrically curved panels

The natural frequencies of cylindrically curved panels are available in closed form for only two boundary condition sets. This paper demonstrates how Sewall's shallow shell formation can be recast in a relatively simple form to allow direct computation of the natural frequencies and mode shapes of cylindrical panels with a wide range of boundary conditions.     

Vibration frequency of a curved blade with weighted edge

The natural vibration frequencies and mode shapes of a curved cylindrical blade with a weighted edge are investigated. A finite element method is used, in which curved cylindrical shell finite elements are utilized to model the blade. The weighted edge is modelled as a beam with its stiffness and mass added into the stiffness and mass of the blade. Vibration frequencies and mode shapes for blades with different boundary conditions and with different radii of curvature are obtained. Finite element results are compared with experimental results.     

Dynamic behaviour of a cylindrical shell with a cutout

This paper presents the results of analytical and experimental investigations connected with the dynamic behaviour of a cylindrical shell with a rectangular cutout. The finite element method is used to predict the vibration frequencies and mode shapes. The resulting eigenvalue problems are solved by using a simultaneous iteration technique. The analytical study shows the influence of the cutout on the natural frequencies and mode shapes of the shell. The subtended angle of the cutout ranges from 40° to 120°. Experimental verification was performed on a machined mild steel shell having welded end rings bolted on to sturdy supports. A reasonably good agreement is obtained, with the discrepancies of the order of less than 10 %. The cutout is found to have very little influence on the natural frequencies.     

Free vibration analysis of a hanged clamped-free cylindrical shell partially submerged in fluid: The effect of external wall,internal shaft,and flat bottom

The free flexural vibration of a hanged clamped-free cylindrical shell with various boundary conditions partially submerged in a fluid is investigated. Specifically, the effects of the boundary conditions such as the existence of the external wall, internal shaft, and bottom on the natural vibration characteristics of the partially submerged cylindrical shell are investigated both theoretically and experimentally. The fluid is assumed to be inviscid and irrotational. The cylindrical shell is modeled by using the Rayleigh–Ritz method based on the Sanders shell theory. The kinetic energy of the fluid is derived by solving a boundary-value problem related to the fluid motion. The theoretical predictions were in good agreement with the experimental results validating the theoretical approach developed in this study. The effects of the external wall, internal shaft, and bottom on the natural vibration characteristics can be neglected when its boundaries are not very close to the shell structure.     

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

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.     

A note on vibration of a cantilever plate immersed in water

The added mass of the fluid surrounding it plats an important role in the dynamic behaviour of a submerged structure. The first few mode shapes and the respective natural frequencies of a submerged cantilever plate are found by using a finite element procedure, eigenvalues being obtained by a simultaneous iteration technique. The influence of the water depth below the plate and also of the water's lateral extent is considered, in order to test the convergency of the results. Results on the effects of the depth of immersion on the natural frequencies and mode shapes of the cantilever plate for different aspect ratios are presented.     

Coupled vibrations of cantilever cylindrical shells partially submerged in fluids with continuous, simply connected and non-convex domain

The approach developed in the present paper is applied for the coupled-vibration analysis of a cantilever cylindrical shell partially submerged in a fluid with a continuous, simply connected and non-convex domain. The shell is partially and concentrically submerged in a rigid cylindrical container partially filled by a fluid which is assumed to be incompressible and inviscid. The velocity potential for fluid motion is formulated in terms of eigenfunction expansions using the collocation method. The interaction between the fluid and the structure takes into account by using the compatibility requirement along the wet surface of the shell and the Rayleigh-Ritz method is used to calculate natural frequencies and modes of the coupled system. The validity of the developed theoretical method is verified by comparing the results with those obtained from the finite element analysis. Furthermore, the effects of submergence depth, radial distance between shell and container, and circumferential wavenumbers on the natural frequencies and modes of the coupled system are investigated.     

Generalized asymptotic expansions for the wavenumbers in infinite flexible in vacuo orthotropic cylindrical shells

Analytical expressions are found for the wavenumbers and resonance frequencies in flexible, orthotropic shells using the asymptotic methods. These expressions are valid for arbitrary circumferential orders n. The Donnell-Mushtari shell theory is used to model the dynamics of the cylindrical shell. Initially, an in vacuo cylindrical isotropic shell is considered and expressions for all the wavenumbers (bending, near-field bending, longitudinal and torsional) are found. Subsequently, defining a suitable orthotropy parameter ?, the problem of wave propagation in an orthotropic shell is posed as a perturbation on the corresponding problem for an isotropic shell. Asymptotic expressions for the wavenumbers in the in vacuo orthotropic shell are then obtained by treating ? as an expansion parameter. In both cases (isotropy and orthotropy), a frequency-scaling parameter (η) and Poisson's ratio (ν) are used to find elegant expansions in the different frequency regimes. The asymptotic expansions are compared with numerical solutions in each of the cases and the match is found to be good. The main contribution of this work lies in the extension of the existing literature by developing closed-form expressions for wavenumbers with arbitrary circumferential orders n in the case of both, isotropic and orthotropic shells. Finally, we present natural frequency expressions in finite shells (isotropic and orthotropic) for the axisymmetric mode and compare them with numerical and ANSYS results. Here also, the comparison is found to be good.     

Free vibration of joined conical-cylindrical shells

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.     

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.     

Buckling and free vibration analysis of functionally graded cylindrical shells subjected to a temperature-specified boundary condition

Linear thermal buckling and free vibration analysis are presented for functionally graded cylindrical shells with clamped-clamped boundary condition based on temperature-dependent material properties. The material properties of functionally graded materials (FGM) shell are assumed to vary smoothly and continuously across the thickness. With high-temperature specified on the inner surface of the FGM shell and outer surface at ambient temperature, 1D heat conduction equation along the thickness of the shell is applied to determine the temperature distribution; thereby, the material properties based on temperature distribution are made available for thermal buckling and free vibration analysis. First-order shear deformation theory along with Fourier series expansion of the displacement variables in the circumferential direction are used to model the FGM shell. Numerical studies involved the understanding of the influence of the power-law index, r/h and l/r ratios on the critical buckling temperature. Free vibration studies of FGM shells under elevated temperature show that the fall in natural frequency is very drastic for the mode corresponding to the lowest natural frequency when compared to the lowest buckling temperature mode.     

Free vibration of non-circular cylindrical shells with longitudinal interior partitions

An analysis is presented for the free vibration of a simply supported non-circular cylindrical shell with longitudinal interior partitions. For this purpose, the governing equations of vibration of a non-circular cylindrical shell including a plate as special case are written in a matrix differential equation by using the transfer matrix in the circumferential direction. Once the matrix has been determined, the entire structure matrix is obtained by the product of the transfer matrix of the shell and the point matrix at the joint of the structure matrix. The method is applied to approximately elliptical cylindrical shells with an interior plate, and the natural frequencies and the mode shapes of vibration are calculated numerically, giving the results.     

Nonlinear vibration of functionally graded circular cylindrical shells based on improved Donnell equations

In the present work, the study of the nonlinear vibration of a functionally graded cylindrical shell subjected to axial and transverse mechanical loads is presented. Material properties are graded in the thickness direction of the shell according to a simple power law distribution in terms of volume fractions of the material constituents. Governing equations are derived using improved Donnell shell theory ignoring the shallowness of cylindrical shells and kinematic nonlinearity is taken into consideration. One-term approximate solution is assumed to satisfy simply supported boundary conditions. The Galerkin method, the Volmir's assumption and fourth-order Runge–Kutta method are used for dynamical analysis of shells to give explicit expressions of natural frequencies, nonlinear frequency–amplitude relation and nonlinear dynamic responses. Numerical results show the effects of characteristics of functionally graded materials, pre-loaded axial compression and dimensional ratios on the dynamical behavior of shells. The proposed results are validated by comparing with those in the literature.     

Free vibration of a conical shell with variable thickness

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.     

LOCALIZED FAMILIES OF BENDING WAVES IN A THIN MEDIUM-LENGTH CYLINDRICAL SHELL UNDER PRESSURE

The initial-boundary-value problem for the equations describing motion of a thin, medium-length, non-circular cylindrical shell is examined. The shell edges are not necessarily plane curves, with the conditions of a joint support, a rigid clamp or a free edge being considered as the boundary conditions. The shell is supposed to experience normal internal (or external) dynamic pressure which may be non-uniform in the circumferential direction. It is assumed that the initial displacements and velocities of the points at the shell middle surface are functions decreasing rapidly away from some generatrix. Using the complex WKB method the asymptotic solution of the governing equations is constructed by superimposing localized families (wave packets) of bending waves running in the circumferential direction. The dependence of frequencies, group velocities, amplitudes and other dynamic characteristics upon variable pressure and geometrical parameters of the shell are studied. As an example, the wave forms of motion of a circular cylindrical shell with sloping edges under growing dynamic pressure are considered. The effect of localization of bending vibrations near the longest generator as well as the effects of reflection, focusing and increasing amplitude in the running wave packets are revealed.     

External pressure loading,vibration, and acoustic responses at low frequencies of building components exposed to impulsive sound

This paper presents the formulation and experimental validation of a numerical tool that can predict: (1) the external pressure loading due to impulsive sound generated by an outdoor point source and (2) the resulting fully coupled vibro-acoustic response at low frequencies of a two-dimensional building component (e.g. wood-framed wall with windows) backed by a rigid rectangular cavity. The model allows for the building component to have arbitrary geometry and boundary conditions. Assuming linear-acoustic propagation and acoustically rigid surfaces, the external pressure loading is computed with a three-dimensional numerical model, in the time domain, by combining the image-source method for the reflected field (specular reflections) with an extension of the Biot–Tolstoy–Medwin (BTM) method for the diffracted field. The fully coupled vibro-acoustic response of the fluid–structure system is computed in the time-domain using a modal-decomposition approach. The natural frequencies and mode shapes of the structure are computed with a finite-element model based on shell theory. Acoustic natural frequencies and mode shapes of the cavity are computed analytically assuming hard-wall boundary conditions. The experimental validation of the numerical model is also presented. Experiments were conducted on wood-framed walls without and with a window using a speaker to generate N-waves and load the walls structurally. It is found that numerical simulations are in good agreement with experimental data for these two cases.     

Frequencies of Lock Gate Structure Coupled with Reservoir Fluid

This study determines the natural frequencies of the lock gate structure, considering the coupled effect of reservoir fluid on one side using the finite element method (FEM). The gate is assumed to be a uniformly thick plate, and its material is isotropic, homogeneous, and elastic. The reservoir fluid is assumed to be inviscid and incompressible in an irrotational flow field. The length of the reservoir domain is truncated using the far boundary condition by adopting the Fourier series expansion theory. Two different assumptions on the free surface, i.e., undisturbed and linearized, are considered in the fluid domain analysis. The computer code is written based on the developed finite element formulations. The natural frequencies of the lock gate are computed when interacting with and without reservoir fluid. Several numerical problems are studied considering the effects of boundary conditions, aspect ratios, and varying dimensions of the gate and the fluid domain. The frequencies of gate reduce significantly due to the presence of fluid. The frequencies increase when the fluid extends to either side of the gate. The frequencies reduce when the depth of the fluid domain above the top edge of the gate increases. The frequencies drop considerably when the free surface condition is taken into account. The results of frequencies of lock gate structure may be useful to the designer if it is experienced in natural catastrophes.