Dynamic analysis of circular and sector thick,layered plates

The finite element method is extended to the free vibration analysis of laminated thick plates with curved boundaries. Two elements are developed on the basis of Mindlin's thick plate theory in which the effects of thickness-shear deformation and rotary inertia are included. Both elements are derived in polar co-ordinates and can be joined together to handle annular as well as circular laminated anisotropic plate problems. Since axisymmetry has not been assumed, variations in material properties in the tangential direction can be dealt with. Numerical results are presented to demonstrate the influence of geometrical shape as well as that of thickness-shear deformation on the free vibrations of both homogeneous and layered plates. Comparisons between the numerical results obtained and those presented by other investigators confirm the accuracy of the new elements. The elements also can be used in the analysis of rectangular plates by assuming very large radii and very small subtended angle values.     

The dynamic transient analysis of axisymmetric circular plates by the finite element method

The linear elastic, dynamic transient, analysis of some circular plate bending problems is considered by using axisymmetric, parabolic isoparametric, elements with an explicit time marching scheme. The effects of rotatory inertia and transverse shear deformation are included. A special mass lumping scheme and the use of a reduced integration technique allow the treatment of thin as well as thick plates. Several numerical examples are presented and compared with results from other sources.     

A finite element analysis of multiple ion receiving plates for ionizer balance monitoring

This paper reports a finite element analysis of multiple ion receiving plates to investigate the optimum number of plates for achieving fine-grained resolution in ionizer balance measurement. Both square and circular plates, also subdivided into 4, 9, 16, 25, and 36 segment plates, were modeled in an electrostatic field. The potential distribution of each model was further analyzed by simple linear regression to assess the measurement resolution. The results indicate that the segmented plates provide improved measurement resolution.     

A finite element analysis of the harmonic response of damped three-layer plates

Solutions have been obtained for the vibration response under harmonic excitation of three-layer plates with a constrained viscoelastic layer (e.g., plates with two metallic outer layers and a viscoelastic core) by means of a finite element method. Damping has been introduced by replacing the real modulus of the viscoelastic material by a complex equivalent which accounts for the phase difference between strain and stress. Triangular finite elements were used with different numbers of degrees of freedom and the dynamic stiffness of the overall structure was calculated. The present method allows for the nonlinear stress-strain behaviour of the viscoelastic material, the effects of the rotatory inertia and the extension within the viscoelastic core. In addition, the use of triangular elements allows for a great variety of shapes and boundary conditions. The finite element computation has been verified by comparison with experimental results for circular three-layer plates and for sandwich beams.     

Three-dimensional finite element for thick shells of general shape

When the thickness of a shell increases in comparison with other dimensions as seen in arch dams, pressure vessels and shelter roofs, the effect of the unit deformations and shear deformations along the thickness have to be taken into account for achieving an acceptable approximation. This however, may entail the use of three dimensional elasticity solutions in developing suitable finite elements. In the present study, a twenty node isoparametric finite element is developed for analysis of thick shells having a general shape by using the three-dimensional theory of elasticity in orthogonal curvilinear coordinates. Shell geometric properties and their derivatives, which are necessary for obtaining a displacement-strain matrix, are determined by means of the shape functions. The accuracy of the presented formulation is assessed by comparing the numerical results with those given in the three dimensional elasticity solutions as well as with those obtained by the use of three dimensional finite elements.     

Non-linear dynamic analysis of rotors by finite element method

Non-linear natural vibration characteristics and the dynamic response of hingeless and fully articulated rotors of rectangular cross-section are studied by using the finite element method. In the formulation of response problems, the global variables are augmented with appropriate additional variables, facilitating direct determination of sub-harmonic response. Numerical results are given showing the effect of the geometric non-linearity on the first three natural frequencies. Response analysis of typical rotors indicates a possibility of substantial sub-harmonic response especially in the fully articulated rotors widely adopted in helicopters.     

A modified axisymmetric finite element for the 3-D vibration analysis of piezoelectric laminated circular and annular plates

A finite element is presented to analyze the three-dimensional (3-D) vibration of piezoelectric coupled circular and annular plates. The proposed finite element is a modification of a conventional axisymmetric finite element and is capable of conducting both axisymmetric and nonaxisymmetric vibration analysis of circular and annular laminated plates, with piezoelectric layers therein. The present formulation, a two-dimensional model itself, can investigate 3-D vibration of those plates for a preselected number of nodal diameters, and is therefore more economical than the conventional 3-D finite element analysis, yet still has almost the same accuracy and versatility as the 3-D analysis. In cases such as analysis of stators of traveling wave ultrasonic motors where only vibration modes with particular numbers of nodal diameters are of interest, the proposed approach is very convenient and useful.     

Comments on “finite element model for dynamic analysis of Timoshenko beam”

An analysis is presented of the effect of dry friction on vibrations of two self-excited systems. Attention is directed to cases where practical quenching of self-excited vibrations cannot be achieved merely by the action of linear additional damping. It is shown that at correct tuning of the systems discussed, a combination of linear additional damping with dry friction is so effective in limiting their amplitudes that self-excited vibrations may be regarded as practically suppressed.     

A tapered beam finite element for rotor dynamics analysis

The stiffness, mass and gyroscopic matrices of a rotating beam element are developed, a cubic function being used for the transverse displacement. Shear deflection is included by use of end nodal variables of shear strain, along with transverse displacement and cross-section rotation; rotatory inertia effects are included in the energy functional to provide a Timoshenko beam formulation. The gyroscopic effects for small perturbations are linearized as a skew symmetric damping matrix. The formulation is implemented by numerical integration for a linearly tapered circular beam. A technique of reduction of the shear nodal variable prior to global assembly is shown to provide little loss in accuracy with reduced system bandwidth. Numerical comparisons for three previously published beam models are included, with results presented for the case of forward and reverse precession to verify the gyroscopic effects. The utility of the element in a general program for rotor dynamics analysis is identified.     

A finite element for the vibration analysis of Timoshenko beams

A Timoshenko beam finite element is presented which has three nodes and two degrees of freedom per node, namely the values of the lateral deflection and the cross-sectional rotation. The element properties are based on a coupled displacement field; the lateral deflection is interpolated as a quintic polynomial function and the cross-sectional rotation is linked to the deflection by specifying satisfaction of the governing differential equation of moment equilibrium in the absence of the rotary inertia term. Numerical results confirm that this procedure does not preclude convergence to true Timoshenko theory solutions since rotary inertia is included in lumped form at element ends. The new Timoshenko beam element has good convergence characteristics and where comparison can be made in numerical studies it is shown to be generally more efficient than previous elements.     

谱有限元数值分析各向异性复合板中导波色散特性

提出谱有限元方法研究层状各向异性复合板中导波的色散特性和波结构。基于三维弹性动力学方程,用有限元方法离散波导截面,波传播方向的位移用简谐波表示,得到了导波色散的特征方程。分析了单层和双层复合板中导波沿不同方向传播的色散特性和波结构,讨论了双层复合板中层厚比对相速度的影响。数值研究结果表明:导波的对称模态沿纤维方向传播时在较宽的频率范围内保持弱色散状态。双层复合板中导波基本模态的相速度在低频时受层厚比的影响较明显,随着频率的增加趋向于相速度较低的材料。数值模拟结果为导波用于复合材料定量无损检测和性能评价提供理论依据。     

Accurate vibration analysis of thick, cracked rectangular plates

This work applies the Ritz method to accurately determine the frequencies and nodal patterns of thick, cracked rectangular plates analyzed using Mindlin plate theory. Two types of cracked configuration are considered, namely, side crack and internal crack. To enhance the capabilities of the Ritz method in dealing with cracked plates, new sets of admissible functions are proposed to represent the behaviors of true solutions along the crack. The proposed admissible functions appropriately describe the stress singularity behaviors around a crack tip and the discontinuities of transverse displacement and bending rotations across the crack. The present solutions monotonically converge to the exact frequencies as upper bounds when the number of admissible functions increases. The validity and accuracy of the present solutions are confirmed through comprehensive convergence studies and comparison with the published results based on the classical thin plate theory. The proposed approach is further employed to investigate the effects of the length, location, and orientation of crack on frequencies and nodal patterns of simply supported and cantilevered cracked rectangular plates. The results shown are the first ones available in the published literature.     

A semi-analytic solution for free vibration of annular sector plates

The paper describes a semi-analytical method in which the basic function in the circumferential direction satisfying the boundary conditions of the radial edges is substituted into the free vibration equation of the curved plate. By a suitable transformation, an ordinary differential equation is obtained. The resulting equation is solved by a finite difference technique. Tabulated results have been presented for annular sector plates possessing different boundary conditions. Excellent accuracy has been obtained wherever comparisons have been possible.