A dynamic rotating blade model at an arbitrary stagger angle based on classical plate theory and the Hamilton's principle
04 Mar 2013-Journal of Sound and Vibration (Academic Press Inc.)-Vol. 332, Iss: 5, pp 1355-1371
Abstract: A dynamic model based on classical plate theory is presented to investigate the vibration behavior of a rotating blade at an arbitrary stagger angle and rotation speed. The Hamilton's principle is ...
Abstract: A weak-form formulation for three-dimensional vibration analysis of rotating pre-twisted cylindrical isotropic and functionally graded (FG) shell panels is first developed. The present formulation is established with the three-dimensional (3-D) elasticity shell theory and Carrera unified formulation, which enables the expression of the traditional and high-order shell theories into a unified form. The potential energy stored in the boundary is transformed into a quantification form by using the penalty method. The 3-D coupled displacement fields are constructed by a modified version of the Fourier series with several additional boundary smoothed functions to ensure the exact description of displacement and stress fields. The complete second-order nonlinear strain terms in the frame of the curvilinear coordinate system are presented to take into consideration the initial centrifugal stresses that are exactly determined with a separate static deflection analysis. Numerical verifications and comparisons of the vibration results with the 3-D finite element method (FEM) show the capabilities of the developed high-order hierarchical model to accurately predict the frequency results and mode shapes of rotating pre-twisted cylindrical isotropic and FG shell panels. Parametric studies are then carried out to investigate the effect of the rotating speed, presetting angle, pre-twisted angle, subtended angle and power-law exponent on the dynamic characteristics of the pre-twisted cylindrical shell panels. It is expected that the present formulation for the rotating pre-twisted cylindrical shell panels will serve as benchmarks to assess the accuracy of other numerical and analytical models.
Abstract: In present work, the three-dimensional transient heat transfer equation for a fiber metal laminated (FML) plate is established, and temperature field under dual-ellipse distribution laser processing is obtained through the Newton-Cotes method. Then the three-dimensional constitutive equations are derived, and the transient thermodynamic governing equations are obtained based on the Hamilton variational principle subsequently. Finally, the finite difference method and Newmark method are utilized to solve the thermodynamic equations in both space and time domains, respectively. The research aims at studying three-dimensional thermodynamic behaviors for a FML structure subjected to dual-ellipse distribution laser heat source and giving the effects of Fourier series term, finite difference point, aspect ratio, fiber arrangement angle and laser moving speed on the thermodynamic behaviors for the FML structure.
Abstract: The frequency veering of a metal porous rotating cantilever twisted plate with twist angle and stagger angle is investigated. Metal porous materials may have the characteristics of gradient or uniform distribution along the thickness direction. Based on the classical shell theory, considering the influence of centrifugal force produced by high-speed rotation, the free vibration equations of a rotating cantilever twisted plate are derived. Through the polynomial function and Rayleigh–Ritz method, the natural frequencies and mode shapes of the metal porous cantilever twisted plate in both static and rotating states are derived. The accuracy of the present theory and calculation results is confirmed by a comparison between them and the results available from the literature and those obtained from Abaqus. The influences of the thickness ratio, porosity, twist angle, stagger angle and rotational velocity on the frequency veering and mode shape shift of the rotating cantilever twisted plate with porous material under three different distributions are analyzed. It should be mentioned that the frequency veering accompanied by mode shape shift occurs in both static and dynamic states.
Abstract: In this work, the vibration characteristics of the rotating pretwist functionally graded (FG) sandwich blades operating in the thermal environment are studied for the first time. The FG sandwich blade is composed of a single metallic core combined with two FG surface layers and is assumed to be in a steady-state temperature field. The material properties of the FG sandwich blade are temperature-dependent. Based on the first-shear deformation shell theory (FSDST) and modified Fourier spectral approach (MFSA), a two-stage solving process is presented for the thermo-elastic vibration analysis of the rotating FG sandwich blades. For the first stage, a quasi-static analysis of the pretwist FG sandwich blade enduring both the centrifugal and thermal loadings is carried out to accurately predict the initial thermo-mechanical stresses. For the second stage, the vibration analysis of the rotating pretwist FG sandwich blade is performed taking into consideration the thermo-mechanical prestress effect. The correctness of the theoretical model is verified by comparing the present results with the reference data and solutions of the finite element method (FEM). The vibration results and distinguished mode shapes are given for the rotating pretwist FG sandwich blades with different temperature fields, rotating conditions and material distributions. Parameter studies indicated that the temperature rise, rotating speed, layer thickness ratio, volume fraction index, pretwist angle and presetting angle have significant influences on the dynamic characteristics of the rotating pretwist FG sandwich blades.
Abstract: In this paper, starting with the thin shell theory, the governing partial differential equation of motion for the transverse deflection of a rotating pre-twisted plate is derived. Strain–displacement relationships include the effect of warping of the cross-section due to twist–bend coupling effect introduced as a result of pre-twist in the plate of non-circular (rectangular) cross-section. Then the equation of motion, thus derived, is used to formulate the free vibration of a typical turbo-machinery cantilevered airfoil blade by considering it as a plate of an equivalent rectangular cross-section subjected to a quasi-static load due to centrifugal force field. The analytical derivation considers both the stress-stiffening as well as stress-softening effects of the centrifugal forces on the spinning airfoil. The partial differential equation governing the flexural motion of the plate is transformed into a matrix-eigenvalue form using a Rayleigh–Ritz technique. The plate deformations are represented by a set of ‘admissible’ sinusoidal trial functions, which fully satisfy all the clamped-end constrains as well as the free-edge boundary conditions. The results of the analytical model exhibit an excellent agreement with the previously published test data both for thin and thick plate geometries and even in highly twisted configurations. The results of the eigenvalue solution are presented in a non-dimensional form for plates of varying aspect ratios and different amounts of pre-twist in the plate. The numerical results are directly applicable in determining the static and running frequencies of typical blades used in turbo-machinery.
01 Jan 2011
Abstract: A dynamic model of a straight, rotating blade is used and the equations of motion as well as the boundary conditions are extracted from the corresponding variational formulas. The Chebyshev collocation method is applied to discretize the two-dimension equations of motion on a 2D mesh grid. With an appropriate implementation on the dynamic boundary conditions, a coupled 2 nd order ordinary differential equation is obtained for the numerical simulation. A validation study with the convergence analysis is performed showing an acceptable agreement and a parametric analysis is implemented giving a Campbell diagram. Furthermore, the results archived by this method can be applied to both forced response and transient analyses.
01 Jan 2010
Abstract: In order to compare numerical and analytical results for the free vibration analysis of Kirchoff plates with both mixed and damaged boundaries, the Chebyshev collocation and perturbation methods are utilized in this paper, where the damaged boundaries are represented by distributed translational and torsional springs. In the Chebyshev collocation method, the convergence studies are performed to determine the sufficient number of the grid points used. In the analytical method, the small perturbation parameter is defined in terms of the damage parameter of the plate, and a sequence of recurrent linear boundary value problems is obtained which is further solved by the separation of variables technique. The results of the two methods are in good agreement for small values of the damage parameter as well as with the results in the literature for the undamaged case. This study can lead to an efficient technique for structural health monitoring (SHM).Copyright © 2010 by ASME
Abstract: This paper proposes a novel approach, the matched interface and boundary (MIB) method, for the vibration analysis of rectangular plates with simply supported, clamped and free edges, and their arbitrary combinations. In previous work, the MIB method was developed for three-dimensional elliptic equations with arbitrarily complex material interfaces and geometric shapes. The present work generalizes the MIB method for eigenvalue problems in structural analysis with complex boundary conditions. The MIB method utilizes both uniform and non-uniform Cartesian grids. Fictitious values are utilized to facilitate the central finite difference schemes throughout the entire computational domain. Boundary conditions are enforced with fictitious values—a common practice used in the previous discrete singular convolution algorithm. An essential idea of the MIB method is to repeatedly use the boundary conditions to achieve arbitrarily high-order accuracy. A new feature in the proposed approach is the implementation of the cross derivatives in the free boundary conditions. The proposed method has a banded matrix. Nine different plates, particularly those with free edges and free corners, are employed to validate the proposed method. The performance of the proposed method is compared with that of other established methods. Convergence and comparison studies indicate that the proposed MIB method works very well for the vibration analysis of plates. In particular, modal bending moments and shear forces predicted by the proposed method vanish at boundaries for free edges. Copyright © 2008 John Wiley & Sons, Ltd.
Abstract: Problems involving the modeling and free vibration of pre-twisted rotating blades made of functionally graded materials (FGMs) and operating in a high-temperature field are considered. The blade, mounted on a rigid hub, is modeled as a thin-walled beam that incorporates the warping restraint and the pre-twist effects. As a result of the latter feature, an extension-twist elastic coupling is induced. Consistent with the concept of the FGM structures, the two constituent materials, ceramic and metal, experience a continuous variation across the beam wall thickness, and, as a result, the adverse effects featured by the standard laminated structures, such as delamination/debonding, are precluded to occur. Numerical results highlighting the effects of the extension-twist elastic coupling considered in conjunction with the volume fraction of the two constituent phases and of the thermal degradation of material properties on eigenfrequencies are presented, and pertinent conclusions are outlined. Comparisons of predictions, as well as validations of results against those obtained in some special cases, which are available in the specialized literature, are also supplied. In addition to a better understanding of the implication of incorporation of FGMs, the results of this research can be instrumental toward the reliable design of advanced turbomachinery blades that operate in a high-temperature environment.
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