Parametric Instability Behavior Research Articles

Dynamic Instability Characteristics of Spherical Panels Subjected to linearly varying In-plane Loading

The present paper reports the parametric instability phenomenon of spherical shell panels under linearly varying harmonic load employing finite element technique. An isoparametric curved element with eight nodes is considered in the formulation.. The sanders’ theory is used to model the spherical panel incorporating the properties of transverse shear and rotary inertia. A computer programme is created based on the formulation to carry out all essential calculations.The correctness of the formulation is established by comparing with results available in the literature.The impact of different parameters such as load factors, shallowness ratio on the instability regions of spherical panels with three support conditions such as SSSS (all edges simply supported), CCCC (all sides clamped) and CFCF (two opposite edges clamped and other two edges free) are studied. The study demonstrates that the parametric instability behavior of the spherical shell panels is significantly influenced by its shallowness parameter, boundary conditions and type of loads. These results will help the designers for deciding the dimensions of the spherical panels exposed to linearly varying periodic load at the time of the preliminary design.

Dynamic instability analysis of perforated stiffened laminated composite panels subjected to non-uniform in-plane edge load under hygrothermal condition

A robust and computationally efficient finite element (FE) formulation has been developed in an effort to understand the parametric instability behaviour of a perforated stiffened composite panel subjected to a pulsating non-uniform edge load under a hygrothermal environment. The panel is modelled using a 9-noded heterosis element to avoid shear locking, whereas the stringer uses a 3-noded isoparametric beam element with a displacement compatibility condition employed at the skin–stringer​ interface. Due to the non-uniform distribution of stress in the perforated panel as a result of exposure to environmental and mechanical loading conditions, a unique dynamic technique has been implemented to account for the instability performance by employing two types of boundary conditions. In the current investigation, an unstiffened panel is initially obtained based on the highest value of the dynamic instability opening (DIO) and the unstable excitation frequency of the panel for different lamina schemes, cutout shapes, and loading conditions. The obtained unstiffened panel is further investigated for various stiffener configurations to determine a highly unstable perforated stiffened panel based on the DIO value and instability zone frequency range. Detailed parametric studies are then carried out considering the cutout sizes, cutout orientation, panel’s aspect ratio, and stiffener depth to understand the parametric instability behaviour of the panel in a hygrothermal environment. It is found that moisture significantly influences the dynamic instability behaviour of the perforated stiffened panel compared to the temperature exposure by shifting the instability zone towards the lower frequency. The width of the instability region of the stiffened panel broadens abruptly as the magnitude of the applied static in-plane edge load enhances. Also, for certain combinations of the dynamic and static load factors, the lower bound of the instability region reduces to zero, resulting in fluttering. In general, it is observed that stiffer the panel, wider is the instability zone with a higher unstable excitation frequency range.

Dynamic instability of trapezoidal composite plates under non-uniform compression using moving kriging based meshfree method

Meshfree formulation based on the element free Galerkin method (EFGM) in conjunction with moving kriging (MK) shape function is employed to investigate buckling and parametric instability behaviour of shear deformable isotropic and laminated composite trapezoidal plates subjected to different types of non-uniform periodic edge compressive loads. Hamilton’s principle is used to derive the governing equations, which are transformed into the discretized form using the EFG method. The actual pre-buckling stresses are determined from static analysis to evaluate the accurate buckling loads of isotropic and laminated composite trapezoidal plates under non-uniform edge compression. The ordinary differential equations of Mathieu–Hill type are solved using Bolotin’s method to determine regions of dynamic instability. The accuracy of the present formulation is examined first by comparing results with those available in the literature. Thereafter, the influence of geometric parameters, lamination scheme, boundary conditions, static pre-load, and various types of non-uniform edge compression on the critical buckling loads and dynamic instability behaviour of both isotropic and laminated composite trapezoidal plate is investigated. The new results on dynamic stability behaviour of trapezoidal plates under non-uniform edge loads are presented for the first time, which may serve as benchmark results for future research. Furthermore, the time history response and corresponding phase plots are also presented for a better understanding of the dynamic behaviour of the trapezoidal plates.

Effect of edge plasma density on hot spot in LHCD plasma in EAST

Abstract Hot spot is a serious challenge limiting long pulse operation with LHCD (lower hybrid current drive) in tokamak. In order to mitigate the hot spot in guard limiter and improve LHCD capability, effect of edge plasma density on inducing hot spot and current drive has been studied in EAST. The temperature in the guard limiter of the LH antenna, inducing hot spot directly, increases with edge density and LH power. Studies show that the hot spot is mainly ascribed to the heat flux in front of LH antenna. Further simulation indicates that such spot corresponds to the peak position of edge density due to local LH electric field. In addition, due to the stronger parametric instability (PI) behavior in the case of higher edge density, the current drive capability decreases with edge density. Strike-point splitting behaviors appear with density increase, in agreement with current profiles in the edge region and the reduction of total driven current, suggesting that more power is deposited in edge region, which does more contribution to hot spot. Above studies offer one possible idea to optimize the edge density so as to satisfy the coupling, mitigate the heat flux in the guard limiter, and improve current drive capability for fusion device.

The parametric instability improvement of fully anisotropic composite plates with embedded shape memory alloy

The parametric resonance or instability challenge in designing laminated composite is crucial in areas such as aeronautical and marine where structures experience dynamic loading. Shape memory alloy (SMA), a type of smart material, has been used to improve the structural behaviours of composite plate using its well-known property of shape memory effect. It is also known that mechanical couplings that exist in unsymmetric composite can increase the instability of the composite. In this study, the SMA property has been exploited to generate recovery stress in the composite to improve its parametric instability problem. The unsymmetric composites were embedded with SMA fibres, and the formulation for the dynamic instability of this composites was developed using finite element method. The third-order shear deformation theory of composite was applied. The results were initially validated for the case of composite without SMA. Following that, the parametric instability behaviour of unsymmetric composites was studied under the effect of several parameters. It was found that the mechanical couplings that exist in the unsymmetric composite have increased the instability of the composite, but the presence of the SMA can significantly reduce this instability.

Dynamic instability response of smart composite material

AbstractLaminated composite is common to replace traditional metals in today's industries due to its high specific strength. Shape memory alloy has been used to improve structural behaviours such as buckling, vibration and fatigue through its well‐ known property of shape memory effect. This ability of shape memory alloy to improve the parametric instability behaviour of laminated composite plate has not been studied in the past and as such, this study is conducted. Here, shape memory alloy wires are embedded within the outer layers of the laminated composite plates. The Mathieu‐Hill equation for the parametric instability of the shape memory alloy composite plate has been developed using finite element method based on the first order shear deformation theory. The formulation is validated and parametric studies have been conducted to investigate the effect of shape memory alloy on the dynamic instability behaviour of the composite that corresponds to factors such as static load factor, thickness of the plate and boundary conditions. The study shows that shape memory alloy improves significantly the dynamic instability behaviour of the laminated composite plate by shifting the instability chart to the right. The effect of shape memory alloy can increase the frequency centre of the instability chart by more than 100 %.

Dynamic Instability of High-Speed Rotating Shaft with Torsional Effect

Today’s design of machine rotor requires the rotor to operate at a high rotational speed to improve the efficiency of the machine. However, the existence of disturbances such as periodic axial load may cause parametric resonance to the rotor system in addition to the common force resonance. Previous studies on this parametric resonance of shaft typically included the element of translational and rotary inertia, gyroscopic moments and bending and shear deformation but surprisingly neglected the effect of the axial torque. This paper investigated the parametric instability behaviour of the shaft rotating at high speed while considering the torsional effect of the shaft. Based on the finite element method, a shaft model that includes torsional deformation as one of its degree of freedom was established. The Mathieu-Hill equation was derived, and then the Bolotin’s method was used to solve the equation by establishing the parametric instability chart. Two types of the rotary system were studied: a shaft with different boundary conditions and shaft with different bearing types. The results were initially validated with past findings. Following that the results were compared to the results correspond to the Timoshenko’s beam formulation that omits the torsional degree of freedom. The effect of axial torsional deformation was found to be very significant especially at high speed. The developed model in this study shows that at the shaft speed of 40000 rpm, the effect of torsional deformation has given the difference of more than 100% in the frequency ratios correspond to the 4DOF and 5DOF models for the case of fix-free boundary condition.

Parametric instability behavior of tow steered laminated quadrilateral plates using isogeometric analysis

For the first time, the parametric instability regions of variable stiffness laminated composite quadrilateral plates subjected to uniform in-plane loadings are studied. The static as well as time-varying inplane loadings are assumed distributed throughout the whole geometry. The isogeometric analysis finite element formulation based on non-uniform rational B-splines is developed in order to address the dynamic instability of quadrilateral panels. The problem has been formulated by utilizing the principle of virtual work based on first order shear deformation plate theory. In terms of tow-steered reinforcements, the fiber orientations in every lamina is assumed to change linearly in the panel longitudinal direction. In order to demonstrate the capabilities of the developed formulation in predicting the structural parametric dynamic behavior, some representative results are obtained and compared with those available in the literature. The effects of geometry layout, loading frequency and amplitude, changes in curvilinear fiber orientations, and material orthogonality on the parametric instability regions are studied by applying Bolotin's first order approximation.

Parametric Instability of Static Shafts-Disk System Using Finite Element Method

Parametric instability condition is an important consideration in design process as it can cause failure in machine elements. In this study, parametric instability behaviour was studied for a simple shaft and disk system that was subjected to axial load under pinned-pinned boundary condition. The shaft was modelled based on the Nelson’s beam model, which considered translational and rotary inertias, transverse shear deformation and torsional effect. The Floquet’s method was used to estimate the solution for Mathieu equation. Finite element codes were developed using MATLAB to establish the instability chart. The effect of additional disk mass on the stability chart was investigated for pinned-pinned boundary conditions. Numerical results and illustrative examples are given. It is found that the additional disk mass decreases the instability region during static condition. The location of the disk as well has significant effect on the instability region of the shaft.

Parametric Instability of Shafts Based on the Nelson’s Beam Model Using Finite Element Method

Dynamic instability condition may cause failure in machines and as such it is an important consideration in design process. In this work, parametric instability behavior was studied for a simple shaft system that was subjected to periodic axial loading. The beam is modelled based on the Nelson’s beam model that considered translational and rotary inertia, transverse shear deformation and the torsional effect. Finite element method has been used to develop the Mathieu-Hill equation of the dynamic instability of the shaft where with Nelson’s beam formulation, 10 dof per element was required. The Bolotin’s method was used to solve the Mathieu-Hill equation where then FEM codes were developed using MATLAB to establish the instability chart. The effect of the torsional motion on the stability chart was investigated for different conditions where numerical results and illustrative example are given. It is found that torsional motion has significant effect on instability region for the shaft system.

Atypical parametric instability in linear and nonlinear systems

The linear parts of the equations of motion of parametrically excited mechanical systems are characterized by the M, D, G, K, N matrices which may all be time-periodic (mass, damping, gyroscopic, stiffness and circulatory matrices, respectively). The stability of these systems can be studied via Floquet theory. A typical property of parametric instability behavior is the existence of combination resonances. It has been known for a long time that the type of parametric resonance depends very much on whether the excitation is in the in the K or in the N matrices, or simultaneously in both of them, the other matrices being constant. In general, problems of parametric excitation are studied for the case in which all the excitation terms are in phase. If this is not the case, an atypical behavior may occur: The linear system may then be unstable for all frequencies of the parametric excitation, and not only in the neighborhood of certain discrete frequencies. Examples of differential equations of this type were first given about 70 years ago by Lamberto Cesari, but seem largely to have fallen into oblivion since then. It was recently observed that the linearized equations of motion for a minimal model of a squealing disk brake have such out of phase parametric excitation. Additional nonlinearities are introduced in the equations and the corresponding limit cycles are calculated using normal form theory.

Linear and nonlinear parametric instability behavior of cylindrical sandwich panels subjected to various mechanical edge loadings

ABSTRACTLinear and nonlinear dynamic instability behavior of cylindrical sandwich panels subjected to combined static and dynamic nonuniform in-plane loadings is studied in this article. The core compressibility effects are considered in the model by assuming fourth and fifth order expansions for the transverse and tangential displacement of the core. The exact stress distributions within the panel are determined by panel prebuckling analysis for the applied parabolic and partial edge loadings. Galerkin's method is used to reduce the governing partial differential equations of the shell panel into a set of nonlinear ordinary differential equations. Dropping the nonlinear term, dynamic instability regions are obtained by solving the Mathieu-type differential equation by the method of Fourier series. The characteristics feature of the stable and unstable regions are investigated by linear and nonlinear time history responses and phase plots of the shell panel in those regions using Newmark's time integration. Incremental harmonic balance (IHB) method is used to study the nonlinear frequency amplitude responses of the cylindrical sandwich panels.

Parametric Study on the Dynamic Instability Behavior of Laminated Composite Stiffened Plate

This paper deals with the study of dynamic or parametric instability behavior of laminated composite stiffened plates with step-uniform and concentrated in-plane harmonic edge loading. The eight-noded isoparametric degenerated shell element and a compatible three-noded curved beam element are used to model the plate and the stiffeners, respectively. The method of Hill’s infinite determinant is applied to analyze the dynamic instability regions. Numerical results are presented through convergence and comparison with the published results from the literature. The effects of parameters like loading type, stiffening scheme, lamination scheme, dynamic load factor, and boundary conditions are considered in the dynamic instability analysis of laminated composite stiffened plate. It has been shown that the type of loading and the width of loading have remarkable effect on the dynamic instability characteristics of the stiffened plate.

Effects of Piezo Actuated Damping on Parametrically Excited Laminated Composite Plates

The present day aerospace vehicles are becoming lighter and more flexible and are subjected to different kinds of longitudinal forces, namely thermal and pulsating inertial loads. While designing such structures, parametric instability plays a very important role. In the present work, a study is undertaken to understand the parametric instability behavior of laminated composite plates integrated with active piezoelectric layers used for feedback damping. An eight-noded isoparametric finite element model has been developed with structural and electrical degrees as degree of freedom and coded in FORTRAN to obtain the stiffness, mass, and the feedback damping matrices. The resulting Mathieu—Hill equation is then solved using the strained parameter method. Results for different geometry and fiber orientation are obtained and presented.

Tension buckling and parametric instability characteristics of doubly curved panels with circular cutout subjected to nonuniform tensile edge loading

This paper deals with the study of tensile buckling, vibration, and parametric instability behaviour of doubly curved panels with central circular cutout subjected to uniaxial in-plane partially distributed tensile edge loadings using finite element method. First order shear deformation theory is used to model the curved panels, to consider the effects of the transverse shear deformation and rotary inertia. The vibration analysis for this problem shows that for certain parameters of the tensile loading, the frequencies of the panel initially rise with the load, but begin to decrease with increasing tension, showing the onset of tension buckling. The parametric instability behaviour under tensile periodic edge loading with different load parameters shows that instability regions are influenced by the cutout size, load width and its location.

DYNAMIC STABILITY OF CURVED PANELS WITH CUTOUTS

The parametric instability behaviour of curved panels with cutouts subjected to in-plane static and periodic compressive edge loadings are studied using finite element analysis. The first order shear deformation theory is used to model the curved panels, considering the effects of transverse shear deformation and rotary inertia. The theory used is the extension of dynamic, shear deformable theory according to Sanders' first approximation for doubly curved shells, which can be reduced to Love's and Donnell's theories by means of tracers. The effects of static and dynamic load factors, geometry, boundary conditions and the cutout parameters on the principal instability regions of curved panels with cutouts are studied in detail using Bolotin's method. Quantitative results are presented to show the effects of shell geometry and load parameters on the stability boundaries. Results for plates are also presented as special cases and are compared with those available in the literature.

Parametric instability of edge cracked plates

The presented paper deals with the parametric instability behavior of a simply supported rectangular plate with a crack emanating from one edge, subjected to in-plane compressive periodic edge loading. The problem is reduced to computing the free vibration frequencies and the corresponding mode shapes and substituting them into an integral equation based formula, which leads to a compact matrix form. Once the components of this matrix are found, the rest of the computation, i.e., establishing regions of instability, buckling loads and modified frequencies, is straightforward and fast. Several plates, each with a different dimension and crack length size are analyzed using this approach. The comparison of results with those of finite element models is found to be in close agreement.

PARAMETRIC INSTABILITY OF A COLUMN WITH AN AXIALLY OSCILLATING MASS

This study investigates the dynamic instability behavior of a column carrying a concentrated mass with oscillating motion along the column axis. The dynamic equation of the column was derived based on the assumed-modes method. The derived dynamic equation, which contains parametrically excited terms associated with modal accelerations, modal velocities, and modal displacements, is a general form of Mathieu's equation. A new analytical method used to determine the instability regions of the column was directly applied to the transition state. This method is different from the traditional perturbation method in which a criterion, involving the determination of the characteristic exponents, is used to yield the transition curves. The principal vibration frequencies, the ratio of principal amplitudes, and the phase difference between the parametrically excited force and the principal frequency response on the transition state were obtained systematically. The parametric instability behavior of a column carrying a periodically moving concentrated mass is different from that of a column subjected to a periodic tangential inertia force. The present case contains the simple resonances and combination resonances of sum type only, while the case with tangential inertia force may contain the combination resonances of the difference type additionally. Four examples are given to demonstrate the instability behavior of various columns carrying concentrated oscillating mass along the column axis at varying positions.

Vibration, buckling and parametric instability behaviour of plates with centrally located cutouts subjected to in-plane edge loading (Tension or compression)

The vibration, buckling and parameterc instability behaviour of a plate with internal opening subjected to in-plane compressive or tensile periodic edge loading are studied. Different shapes and sizes of the cutout are considered. Compression and tension buckling results are discussed. The vibration analysis for this problem shows that for certain parameters of tensile loading and the opening, the frequency of the plate initially rises' with the load, but then begins to decrease with increasing tension showing the onset of tension buckling. Dynamic instability behaviour of the plate under compressive and tensile loading shows that the instability regions are affected by the size, shape of the opening and the loading parameters. Instability behaviour due to all the parameters are discussed.

Experiments on the parametric vibration response of plates under tensile loading

Results of an experimental investigation of the parametric vibration response characteristics of plates subjected to tensile edge loads are presented. Two distinct types of parametric instability behavior are observed: principal resonance and secondary resonance. The principal resonance zones are found to be more dominant. It was observed that the natural frequencies and the boundaries of the instability regions of the plate are considerably influenced by the nature of the edge loading.