MICRO-SENSING CHARACTERISTICS AND MODAL VOLTAGES OF LINEAR/NON-LINEAR TOROIDAL SHELLS

Toroidal shells belong to the shells of revolution family. Dynamic sensing signals and their distributed characteristics of spatially distributed sensors or neurons laminated on thin toroidal shell structures are investigated in this study. Spatially distributed modal voltages and signal patterns are related to the meridional and circumferential membrane/bending strains, based on the direct piezoelectricity, the Gauss theorem, the Maxwell principle and the open-circuit assumption; linear and non-linear toroidal shells are defined based on the thin shell theory and the von Karman geometric non-linearity. With the simplified mode shape functions defined by the Donnell-Mushtari-Vlasov theory, modal-dependent distributed signals and detailed signal components of spatially distributed sensors or neurons are defined and these signals are quantitatively illustrated. Signal distributions basically reveal distinct modal characteristics of toroidal shells. Parametric studies suggest that the dominating signal component results from the meridional membrane strains. Shell dimensions, materials, boundary conditions, natural modes, sensor locations/distributions/sizes, modal strain components, etc., all influence the spatially distributed modal voltages and signal generations.     

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.     

Distributed shell control with a new multi-DOF photostrictive actuator design

With the photovoltaic effect and the converse piezoelectric effect, the lanthanum-modified lead zirconate titanate (PLZT) actuator can transform the narrow-band photonic energy to mechanical strain/stress—the photodeformation effect. This photodeformation process can be further used for non-contact precision actuation and control in various structural, biomedical and electromechanical systems. Although there are a number of design configurations of distributed actuators, e.g., segmentation and shaping, been investigated over the years, this study is to explore a new actuator configuration spatially bonded on the surface of shell structures to enhance the spatial modal controllability. A mathematical model of a new multiple degree-of-freedom (multi-DOF) photostrictive actuator configuration is presented first, followed by the photostrictive/shell coupling equations of a cylindrical shell structure laminated with the newly proposed multi-DOF distributed actuator. Distributed microscopic photostrictive actuation and its contributing components of a one-piece actuator and the multi-DOF actuator are evaluated in the modal domain. Effects of shell's curvature and actuator's size are also evaluated. Parametric analyses suggest that the new multi-DOF distributed actuator, indeed, provides better performance and control effect to shell actuation and control. This multi-DOF configuration can be further applied to actuation and control of various shell and non-shell structures.     

Semi-analytical study of free vibration characteristics of shear deformable filament wound anisotropic shells of revolution

Free vibration characteristics of filament wound anisotropic shells of revolution are investigated by using multisegment numerical integration technique in combination with a modified frequency trial method. The applicability of multisegment numerical integration technique is extended to the solution of free vibration problem of anisotropic composite shells of revolution through the use of finite exponential Fourier transform of the fundamental shell equations. The governing shell equations comprise the full anisotropic form of the constitutive relations, including first-order transverse shear deformation, and all components of translatory and rotary inertia. The variation of the stiffness coefficients along the axis of the shell is also incorporated into the solution method. Filaments are assumed to be placed along the geodesic fiber path on the shell of revolution resulting in the variation of the stiffness coefficients along the axis of the composite shell of revolution with general meridional curvature. Sample solutions have been performed on the effect of the variation of the stiffness coefficients on the free vibration behavior of filament wound truncated conical and spherical shells of revolution.     

柱壳和锥壳结构自由振动特征的数值计算与校验

为了评估柱壳和锥壳结构自由振动特征数值计算的精度,分析不同边界条件、环肋、纵肋以及流体载荷对自由振动特征的影响,计算并校验了典型壳体结构在空气中、浸没以及浸没并充满水情况下的自由振动特征。结果表明,空气中干模态分析在2 kHz内、单面及双面接触水情况下的流固耦合湿模态分析在500 Hz内的计算精度能够控制在10%以内。壳体流固耦合自由振动分析时可以采用实体单元离散也可以采用壳单元离散的方法,前者精度略高,能够有效保证求解收敛的频带范围更宽,但工作量更大。径长比大于0.2时,浸没于水中的自由振动分析可以转换为内部填充水时的自由振动分析,转换时应保证两者流固耦合湿表面积相等,如半浸水和半充满水,能够有效减小计算量;环肋和流体载荷对壳体自由振动特征的影响明显,环肋使柱壳同阶自振频率增加,流体载荷使柱壳同阶自振频率减小且影响幅度更大,两者均会使得柱壳模态振型呈现错序排列;流固耦合效应对无肋柱壳和环肋柱壳自振频率的影响效果相当;柱壳内外有水相对于单面接触水而言,同阶自振频率进一步减小,模态振型基本不变;流体载荷对环肋锥壳的自振频率和模态振型的影响幅度较对环肋柱壳小。     

扬声器辐射体旋转薄壳的谐波失真

研究了扬声器辐射体旋转薄壳几何非线性引起的谐波失真。采用摄动法和有限元法确定了2次和3次谐波振型,计算了薄壳材料参数和几何参数对谐波失真的影响。结果表明扬声器薄壳谐波失真的机制是分割振动模态的基波共振和超谐波共振,以基波共振为主。采用高阻尼、大杨氏模量和低密度的振膜材料可以降低振膜谐波失真;厚度对谐波失真的峰值影响不大;锥壳半顶角过大,可使3次谐波明显增大。     

The vibration and stability of orthotropic conical shells with non-homogeneous material properties under a hydrostatic pressure

In this paper, the vibration and stability of orthotropic conical shells with non-homogeneous material properties under a hydrostatic pressure are studied. At first, the basic relations have been obtained for orthotropic truncated conical shells, Young's moduli and density of which vary continuously in the thickness direction. By applying the Galerkin method to the foregoing equations, the buckling pressure and frequency parameter of truncated conical shells are obtained from these equations. Finally, carrying out some computations, the effects of the variations of conical shell characteristics, the effects of the non-homogeneity and the orthotropy on the critical dimensionless hydrostatic pressure and lowest dimensionless frequency parameter have been studied, when Young's moduli and density vary together and separately. The results are presented in tables, figures and compared with other works.     

Free vibrational characteristics of isotropic coupled cylindrical-conical shells

This paper presents the free vibrational characteristics of isotropic coupled conical-cylindrical shells. The equations of motion for the cylindrical and conical shells are solved using two different methods. A wave solution is used to describe the displacements of the cylindrical shell, while the displacements of the conical sections are solved using a power series solution. Both Donnell-Mushtari and Flügge equations of motion are used and the limitations associated with each thin shell theory are discussed. Natural frequencies are presented for different boundary conditions. The effect of the boundary conditions and the influence of the semi-vertex cone angle are described. The results from the theoretical model presented here are compared with those obtained by previous researchers and from a finite element model.     

Hybrid finite element method applied to supersonic flutter of an empty or partially liquid-filled truncated conical shell

In this study, aeroelastic analysis of a truncated conical shell subjected to the external supersonic airflow is carried out. The structural model is based on a combination of linear Sanders thin shell theory and the classic finite element method. Linearized first-order potential (piston) theory with the curvature correction term is coupled with the structural model to account for pressure loading. The influence of stress stiffening due to internal or external pressure and axial compression is also taken into account. The fluid-filled effect is considered as a velocity potential variable at each node of the shell elements at the fluid-structure interface in terms of nodal elastic displacements. Aeroelastic equations using the hybrid finite element formulation are derived and solved numerically. The results are validated using numerical and theoretical data available in the literature. The analysis is accomplished for conical shells of different boundary conditions and cone angles. In all cases the conical shell loses its stability through coupled-mode flutter. This proposed hybrid finite element method can be used efficiently for design and analysis of conical shells employed in high speed aircraft structures.     

Free vibrations of laminated composite shells with uniformly distributed attached mass using higher order shell theory including stiffness effect

In this paper, free vibrations of a cross-ply composite shell with or without a uniformly distributed attached mass are analyzed using higher order shell theory. The results of free vibrations without distributed attached mass are validated by previous literatures. The stiffness effect of this distributed attached mass are also considered and compared with those well-known published results in which this effect is ignored. Various results for composite shells under a variety of conditions such as variations in the thickness of the shell, variation in the thickness of the distributed attached mass, variation in the radii of curvatures and various elasticity moduli are presented in this paper. In some cases, to verify the novel results, first-order shear deformation theory (FSDT) is also used. In this paper, parameters which influence the natural frequencies of the shells with attached mass including the stiffness of the mass are investigated. Parameters which are investigated in this paper are thickness of the shell, thickness of the distributed attached mass, elasticity moduli of the distributed attached mass and radius of curvatures of shells. Increasing the thickness or elasticity moduli of the distributed attached mass will increase the fundamental natural frequency of the shell. The effect of the stiffness of the distributed attached mass is decreased by decreasing the radii of curvatures or increasing the thickness of the shells.     

Simplified prediction of the modal characteristics of ring-stiffened cylindrical shells

Two simplified means of predicting the modal characteristics of ring-stiffened circular cylindrical shells are developed. These approaches are based on the receptance method and use the modal characteristics of the unstiffened cylinder and of the rings as inputs. In the first approach simplicity is achieved while retaining accuracy by confining the geometry to a cylindrical shell with simply supported ends stiffened by identical, equally-spaced rings. For this not uncommon configuration, a single non-matrix frequency equation is developed which can be solved surely and rapidly by a numerical method. The second approach is more general and provides insight into the effects of ring stiffeners. The results are not as precise as the first approach, but the calculations are simple enough to be carried out at one's desk given the modal characteristics of the unstiffened cylindrical shell and of the rings.     

Wave interpretation of numerical results for the vibration in thin conical shells

The dynamic behaviour of thin conical shells can be analysed using a number of numerical methods. Although the overall vibration response of shells has been thoroughly studied using such methods, their physical insight is limited. The purpose of this paper is to interpret some of these numerical results in terms of waves, using the wave finite element, WFE, method. The forced response of a thin conical shell at different frequencies is first calculated using the dynamic stiffness matrix method. Then, a wave finite element analysis is used to calculate the wave properties of the shell, in terms of wave type and wavenumber, as a function of position along it. By decomposing the overall results from the dynamic stiffness matrix analysis, the responses of the shell can then be interpreted in terms of wave propagation. A simplified theoretical analysis of the waves in the thin conical shell is also presented in terms of the spatially-varying ring frequency, which provides a straightforward interpretation of the wave approach. The WFE method provides a way to study the types of wave that travel in thin conical shell structures and to decompose the response of the numerical models into the components due to each of these waves. In this way the insight provided by the wave approach allows us to analyse the significance of different waves in the overall response and study how they interact, in particular illustrating the conversion of one wave type into another along the length of the conical shell.     

VIBRATION ANALYSIS OF TWISTED CANTILEVERED CONICAL COMPOSITE SHELLS

The finite element method based on the Hellinger-Reissner principle with independent strain is applied to the vibration problem of cantilevered twisted plates and cylindrical, conical laminated shells. With a small number of elements, the present assumed strain finite element method is validated by convergence tests and numerical tests, comparing with the previous published vibration results for cantilevered conical shell. Computational effort and virtual storage reduce significantly due to good convergence. This study presents the twisting angle effect on vibration characteristics of conical laminated shells. Parameter studies with varying shallowness of cylindrical and conical shells are carried out. As the curvature increases, the fundamental mode shape changes from twisting mode to bending mode. For shells with a large curvature, the fundamental frequency, which is always characterized to bending mode, is almost constant independent of twisting angle. The twisting angle affects greatly twisting frequency and mode shape.     

A generic double-curvature piezoelectric shell energy harvester: Linear/nonlinear theory and applications

Converting vibration energy to useful electric energy has attracted much attention in recent years. Based on the electromechanical coupling of piezoelectricity, distributed piezoelectric zero-curvature type (e.g., beams and plates) energy harvesters have been proposed and evaluated. The objective of this study is to develop a generic linear and nonlinear piezoelectric shell energy harvesting theory based on a double-curvature shell. The generic piezoelectric shell energy harvester consists of an elastic double-curvature shell and piezoelectric patches laminated on its surface(s). With a current model in the closed-circuit condition, output voltages and energies across a resistive load are evaluated when the shell is subjected to harmonic excitations. Steady-state voltage and power outputs across the resistive load are calculated at resonance for each shell mode. The piezoelectric shell energy harvesting mechanism can be simplified to shell (e.g., cylindrical, conical, spherical, paraboloidal, etc.) and non-shell (beam, plate, ring, arch, etc.) distributed harvesters using two Lamé parameters and two curvature radii of the selected harvester geometry. To demonstrate the utility and simplification procedures, the generic linear/nonlinear shell energy harvester mechanism is simplified to three specific structures, i.e., a cantilever beam case, a circular ring case and a conical shell case. Results show the versatility of the generic linear/nonlinear shell energy harvesting mechanism and the validity of the simplification procedures.     

The effective dielectric response of nonlinear coated granular composites

The dielectric response of a nonlinear composite system, which is composed of coated granular cylinders with nonlinear cores and linear shells or with linear cores and nonlinear shells randomly embedded in a linear host matrix, is investigated. For the concentric cylinders with weak nonlinear core and linear shell embedded in a linear host, the system can be replaced by solid cylinders with weak nonlinearity embedded in the same host under certain conditions. One of the linear partially resonant conditions is ϵ_s + ϵ_h = 0, which associated with two linear components can be extended to this nonlinear composite, the equivalent nonlinear solid cylinders have the same dielectric function as the original cores and radii larger than those of the original coating shells. Another is the linear partially resonant condition ϵ_c + ϵ_s = 0, which associated with one linear component and another nonlinear component cannot be extended to this nonlinear composite, but under this condition the nonlinear dielectric response of whole system will still be greatly enhanced. For another case of concentric cylinders with linear cores and nonlinear shells, we cannot find the equivalent nonlinear solid cylinders embedded in the same host, and the nonlinear partially resonant condition cannot be realized.     

Effect of heterogeneity on the axisymmetric vibrations of cylindrical shells

The influence of large heterogeneity on the axisymmetric vibration characteristics of thin, composite cylindrical shells is studied, both analytically and numerically. In the neighborhood of the axisymmetric breathing mode, frequency spectra for shells of infinite and finite length are shown to be influenced qualitatively as well as quantitatively by large deviations from material and geometric symmetry in layer arrangement. A study of mode coupling in a semi-infinite shell is made for both end modes and modes with stationary frequency with real finite wave number, the latter being uniquely generated by a special class of heterogeneity. In each case, analytical estimates are given for frequencies, wave numbers, and modal amplitudes as functions of material and geometric properties of the shell.     

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.     

扬声器锥壳的全频段振动通解

为更好地理解扬声器数值计算结果,解析研究了扬声器锥壳轴对称振动在整个工作频段一致有效的通解,这些解以特殊函数的形式给出。在典型低频段,弯曲解仅表现为边界层效应;在典型转点频段,弯曲解存在于转点的外侧域;在典型高频段,弯曲解覆盖了整个扬声器锥壳。无矩解和弯曲解在转点频段是耦合的,该耦合特性表明在此频段不可能消除分割振动。     

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.     

The measurement of structural mobilities of a circular cylindrical shell

Structural mobility is useful for the estimation of structural power flows in coupled systems. Although the methods of measuring structural mobilities are easily found for one-dimensional beam structures, few are available for cylindrical shells. In this paper, a new method is proposed for the measurement of the structural mobilities of a circular cylindrical shell. A point force excitation is used instead of circumferential modal forces which are difficult to implement in practice. This method utilizes the least squares technique to obtain the transfer function components of different circumferential modes from the measured data. Experiments were carried out on a circular cylindrical shell with different end conditions excited by a point force to verify the feasibility of this proposed method.