Principal Instability Regions Research Articles

Fractional Mathieu Differential Equations in Dynamic Stability of Piles

Piles are commonly used in Civil and Mechanical Engineering to provide support to superstructures and to transmit loads to deeper ground layers. During earthquakes, pile instability can result in the collapse of the entire structure. In this paper, the pile is modeled as a column surrounded by Winkler soil foundations with fractional damping. Investigation of the axially loaded pile leads to a fractional Mathieu differential equation of motion. The Bolotin method, employing harmonic balance, is proposed to obtain the approximate instability boundaries of the pile in the stability diagrams. A practical example is presented to conduct a parametric study on pile instability concerning the fractional order. The study observes parametric resonance in the first order, representing the principal instability region, which requires special attention to maintain stability. Furthermore, increasing the fractional damping order leads to a higher critical dynamic load and a slight reduction in the critical frequency ratio in each instability region. Higher-order instability regions exhibit greater sensitivity to changes in the fractional order. These results provided insights into the stability of piles under earthquake conditions.

Static and dynamic instability of functionally graded graphene origami-enabled auxetic metamaterial beams with variable thickness in fluid

Functionally graded (FG) graphene origami (GOri)-enabled auxetic metamaterial (GOEAM) structures have shown great potential for various engineering applications due to their exceptional mechanical and physical properties such as high strength-to-weight ratio, tuneable stiffness and strength, and negative Poisson's ratio (NPR). This paper aims to investigate the buckling and dynamic instability behaviours of FG-GOEAM beams with variable thickness immersed in a fluid, with a particular focus on the influence of NPR. The material properties such as Young's modulus and Poisson's ratio of the GOEAM are determined by using a genetic programming (GP)-based micromechanics model. Within the framework of the first-order shear deformation theory and by employing Hamilton's principle and modelling the fluid effect as added mass, the governing equations of motion are established and are then discretised by means of differential quadrature (DQ) method to obtain a linear system of Mathieu-Hill equations from which the principal instability regions of the FG-GOEAM beam are determined by Bolotin's method. A comprehensive parametric study is conducted to reveal the effects of GOri's folding degree, distribution, weight fraction, as well as fluid density and beam dimensions on the static and dynamic instability behaviours of the FG-GOEAM beam immersed in fluid. Numerical results show that the FG-GOEAM beam with NPR considerably outperforms its pristine metallic counterpart in terms of resistance against static buckling and dynamic stability.

Investigation into the Dynamic Stability of Nanobeams by Using the Levinson Beam Model.

Dynamic stability is an important mechanical behavior of nanobeams, which has been studied extensively using the Euler-Bernoulli and Timoshenko beam theories, while the Levinson-beam-theory-based dynamic instability analysis of nanobeams has not been investigated yet. Shear deformation is not or is not suitably considered in the Euler-Bernoulli and Timoshenko theories, so it is very important to introduce the Levinson beam theory in the dynamic stability analysis of nanobeams, which correctly models the combined action of bending and shear in nanobeams with smaller length/height ratios. In this work, the equation of the transverse vibration of a Levinson beam embedded in an elastic foundation is firstly formulated based on the displacement field of Levinson beam theory, and the nonlocal theory is further applied to the Levinson nanobeam. Then, the governing equation of the dynamic stability of the Levinson nanobeam is derived using Bolotin's method to achieve a generalized eigenvalue problem corresponding to the boundaries of regions of dynamic instability. The principal instability region (PIR) is the most important among all regions, so the boundary of the PIR is focused on in this work to investigate the dynamic stability of the Levinson nanobeam. When the width, length/height ratio, density, Young's modulus, Poisson's ratio, size scale parameter, and medium stiffness increase by about 1.5 times, the width of the PIR changes by about 19%, -57%, -20%, 65%, 0, -9%, and -11%, respectively. If a smaller critical excitation frequency and narrower width of the PIR correspond to the better performance of dynamic stability, the study shows that the dynamic stability of the Levinson nanobeam embedded in an elastic medium improves under a larger length and density and a smaller width, height, and Young's modulus, since these factors are related to the natural frequency of the nanobeam which controls the width of the PIR. Additionally, the local model would overestimate the dynamic stability behavior of the Levinson nanobeam.

Dynamic Stability of Nanobeams Based on the Reddy's Beam Theory.

The dynamic stability of nanobeams has been investigated by the Euler-Bernoulli and Timoshenko beam theories in the literature, but the higher-order Reddy beam theory has not been applied in the dynamic stability evaluation of nanobeams. In this work, the governing equations of the motion and dynamic stability of a nanobeam embedded in elastic medium are derived based on the nonlocal theory and the Reddy's beam theory. The parametric studies indicate that the principal instability region (PIR) moves to a lower frequency zone when length, sectional height, nonlocal parameter, Young's modulus and mass density of the Reddy nanobeam increase. The PIR shifts to a higher frequency zone only under increasing shear modulus. Increase in length makes the width of the PIR shrink obviously, while increase in height and Young's modulus makes the width of the PIR enlarge. The sectional width and foundation modulus have few effects on PIR.

A theoretical and experimental study on in-plane parametric resonance of laminated composite circular arches under a vertical base excitation

For the first time, analytical solutions for the in-plane dynamic instability of laminated composite circular arches under a vertical base excitation are developed and experimentally verified in this paper. Hamilton principle is used to derive the governing equation of motion of the laminated arch from which the principal instability region corresponding to parameter resonance can be determined. An experimental system is designed and set up to verify the developed solutions and investigate the dynamic performance of laminated arches. Comprehensive analytical and experimental results are presented, with a particular focus on the effects of rise-to-span ratio and stacking sequence on the dynamic instability region of the laminated arch. It is found that dynamic instability region mainly occurs when the excitation frequency is nearly twice the fundamental frequency. For an arch with a lower rise-to-span ratio, both the critical excitation frequency and the width of its dynamic instability region increase significantly. Under the same excitation amplitude, the dynamic instability region of an angle-ply laminated arch becomes wider as the ply-angle increases. Results also show that the critical excitation frequency is highly sensitive to the stacking sequence and quickly grows as the zero-degree ply is placed further away from the midplane.

Isogeometric technique for dynamic instability analysis of nanocomposite folded plates based on higher-order shear deformation theory

This paper proposes a dynamic instability isogeometric analysis (IGA) of functionally graded (FG) folded plates strengthened by carbon nanotubes (CNTs) under a uniform in-plane loading. The folded plates are modeled by two patches to overcome the existing discontinuity in the geometry. A recently developed logarithmic higher-order shear deformation theory (HSDT) is used to describe the kinematic relations of each patch. Then, IGA in conjunction with Hamilton’s principle is used to discretize the governing equations of the patch. Afterwards, an appropriate coordinate transformation is applied to transfer the patch stiffness, mass and geometric stiffness matrices to the global coordinates. Moreover, the continuity conditions along the joined line are enforced by means of the bending strip method (BSM). The final form of the governing equations, describing the dynamic instability behavior of folded plates, is solved by Bolotin’s method in order to estimate the boundaries of the principal instability regions with the first and second-order approximations. Various types of numerical examples are investigated to verify the precision and reliability of the present solution method. These comparative studies confirm that the proposed logarithmic HSDT-based isogeometric formulation can precisely predict natural frequencies, critical buckling loads and principal instability regions of FG carbon nanotube-reinforced composite (CNTRC) folded plates. The effects of first and second-order approximations, various configurations of CNTRCs, in-plane loading behavior and geometrical parameters on the aforementioned mechanical analyses of the plate are investigated.

Dynamic Stability of a Cylindrical Shell Reinforced by Longitudinal Ribs of Piecewise-Constant Thickness under Axial Loading

A model is proposed for studying the simultaneous parametric and longitudinal vibrations of an orthotropic cylindrical shell reinforced by longitudinal ribs of piecewise-constant thickness under the action of an axial time-harmonic load. The discrete location and changes in the thickness of the ribs are taken into account using generalized functions. The basic stages of solving the problem are considered with the combination of methods proposed. Using a numerical example, the dependences of the principal instability region on the geometric parameters of the ribs are plotted for the first time, and a comparison is made with the instability regions for the ribs with a mean-integral thickness. The model developed makes it possible to correct significantly the solutions of problems on calculation of the dynamic stability of orthotropic cylindrical shells with the ribs of piecewise-constant thickness.

Study on dynamic instability characteristics of functionally graded material sandwich conical shells with arbitrary boundary conditions

This paper presents analytical studies on the dynamic instability of functionally graded material (FGM) sandwich conical shell subjected to time dependent periodic parametric axial and lateral load. In the analysis, four kinds of sandwich distributions, including FGM core and isotropic skins, and isotropic core and FGM skins are considered and the power law distribution is employed to estimate the volume of the constituents. The arbitrary boundary conditions of the FGM sandwich conical shell are achieved by using the artificial spring boundary. The governing equations of conical shell subjected to parametric excitation are established by the Hamilton’s principle considering first order shear deformation shell theory. Then the Mathieu-Hill equations describing the parametric stability of conical shell are obtained by generalized differential quadrature (GDQ) method, and the Bolotin’s method is utilized to obtain the first-order approximations of principal instability regions of shell structure. By comparing the numerical results with the existing solutions in open literature, the validity of the proposed theoretical model is verified. Finally, the influences of sandwich distribution types, gradient indexes, skin-core-skin ratio and load forms on the dynamic stability of FGM sandwich conical shell have been investigated.

Dynamic instability analysis of porous FGM conical shells subjected to parametric excitation in thermal environment within FSDT

The conical shell structure is prone to dynamic instability when subjected to time dependent periodic axial loads and it can cause structural damage. Based on that this paper presents an accurate and semi-analytical method for investigation the dynamic instability of porous functionally graded material (PFGM) conical shell in thermal environment. In the analysis, three common types of PFGM conical shells, namely, uniform, symmetric and asymmetric distribution are considered assuming that material properties are related to temperature. The governing equations of conical shell subjected to parametric excitation are established by the Hamilton's principle considering first order shear deformation theory. Then the Mathieu-Hill equations describing the parametric stability of conical shell are obtained by generalized differential quadrature (GDQ) method, and the Bolotin's method is utilized to obtain the first-order approximations of principal instability regions of shell structure. By comparing the numerical results with the existing solutions in open literature, the validity of the proposed theoretical model is verified. Finally, the influences of porosity distribution type, gradient index, radius-to-thickness ratio, porosity volume fraction and temperature fields on the dynamic stability of PFGM conical shell have been investigated, wherein for different porosity distribution types, the UPD type is more sensitive to gradient index as compared to other three types, while the SPD has the minimum relative width.

Dynamic instability analysis of FG-CNTRC laminated conical shells surrounded by elastic foundations within FSDT

A comprehensive understanding of the dynamic instability of shell structure is critical to avoid resonance damage. On the basis of that, an accurate and analytical method for investigation the dynamic instability of laminated functionally graded carbon nanotube reinforced composite (FG-CNTRC) conical shell surrounded by the elastic foundations is presented in this work based on the first-order shear deformation theory. In the analysis, uniform or functionally graded distributions of reinforcements across the shell thickness are considered and the extended Voigt model is employed to estimate the CNTRC material properties. The governing equations of conical shell subjected to parametric excitation are established by the Hamilton's principle considering first order shear deformation shell theory. Then the Mathieu-Hill equations describing the parametric stability of conical shell are obtained by generalized differential quadrature (GDQ) method, and the Bolotin's method is utilized to obtain the first-order approximations of principal instability regions of shell structure. By comparing the numerical results with the existing solutions in open literature, the validity of the proposed theoretical model is verified. Finally, the influences of volume fractions and types of CNTs, lamination angle, elastic foundations stiffness and lamination angle on the dynamic stability of laminated FG-CNTRC conical shell have been investigated.

Investigation on nonlinear lateral-torsional coupled vibration of a rotor system with substantial unbalance

Substantial unbalance may be caused by fan blade off during the operation period of gas turbine engines, and related dynamic problems are very critical to the safety design of rotor system in aero-engine. This article aims to understand lateral-torsional coupled vibration of the rotor system with substantial unbalance. The governing equation of a modified unbalanced rotor system is established based on Lagrangian approach. Then, a mathematical analytical method is proposed in which a linear approximation is derived and the Floquet theory and Hill’s method are incorporated, from which the modal characteristics of the unbalanced rotor are obtained. The modal characteristics of the unbalanced rotor system are revealed comprehensively for the first time. Furthermore, the relation between the modes and responses of the unbalanced rotor is discussed in detail. The results show that the lateral vibration and torsional vibration of the unbalanced rotor are coupled through the inertial terms in the governing equations. Due to the coupling, veering and lock-in phenomena occur between the frequencies of the forward whirl mode and the torsional mode. Furthermore, lock-in can lead to a kind of principal instability. With regard to the response of the unbalanced rotor, both natural vibration components and enforced vibration components appear in the lateral response, while only natural vibration components appear during torsional vibration. Moreover, natural vibration components play a crucial role in the response within the principal instability region and cause divergence of the vibration amplitudes in the lateral and torsional directions.

A nonlinear mathematical model for dynamic analyses of a cantilevered beam with a tip-mass under support excitation

The dynamic response of a cantilevered beam subjected to horizontal, vertical and angular support excitation is numerically investigated. An extensible Bernoulli–Euler beam model is assumed, and the governing equations of motion are derived using the extended Hamilton’s principle and the Galerkin’s method. Modal-amplitude time histories, as well as maps of post-critical steady-state vibration amplitudes, are obtained and discussed. It is pointed out the existence of similarities with the principal Mathieu’s instability observed in a one degree-of-freedom system, i.e., the condition in which the frequency of stiffness variation is twice the structural natural frequency. Particularly, the angular support excitation may lead to a principal region of parametric instability and the presence of simultaneous vertical and angular support excitation increased a second region of important responses. Another finding is that the horizontal support excitation leads to an increase in the maximum lateral response near the frequencies associated with the parametric resonance.

Dynamic Stability of Piezoelectric Plate Reinforced with BNNTs Considering Temperature Effect

Subjected to axial dynamic load, the dynamic stability of piezoelectric plate reinforced with boron nitride nanotube (BNNT) is investigated. Recurring to classical thin plate theory and piezoelectric theory containing thermal effects, the constitutive equation of the piezoelectric plate reinforced with BNNTs is set up. Adopting ’XY’ rectangle model, the material constants are reckoned. Afterwards the dynamic control equation of the structure is deduced through Lagrangian equation and resolved by using the Bolotin method. In numerical calculations, the influence of temperature, voltage, boundary condition, volume fraction, material property and static loading parameter on the principal instability region of BNNTs reinforced piezoelectric plate are discussed in detail.

Assessment of dynamic instability of laminated composite-sandwich plates

The current work deals with the assessment of dynamic stability behavior of laminated composite and sandwich plates subjected to in-plane static and periodic compressive loads based on a recently developed zigzag theory by the authors. This theory satisfies the traction-free boundary conditions at top and bottom surfaces of the laminate as well as the inter-laminar stress continuity at layer interfaces. Also, it obviates the need of artificial shear correction factor. The theory is based upon shear strain shape function assuming non-linear distribution of transverse shear stresses. An efficient C0 continuous, eight-noded isoparametric element with seven field variables is employed for the dynamic stability analysis of laminated composite and sandwich plates. The boundaries of principal instability domains are obtained following Bolotin's approach and are represented either in the non-dimensional load amplitude-excitation frequency plane or load amplitude-load frequency plane. A series of numerical examples on the dynamic stability analysis of laminated composite and sandwich plates are studied to demonstrate the effects of modular ratio, span to thickness ratio, boundary conditions, thickness ratio, static load factor and various load parameters on the principal instability regions. The predicted results are compared with the available existing results in order to ensure the performance of the proposed model.

An interval algorithm for uncertain dynamic stability analysis

Analysis on the stability of dynamic systems is a very important field in structural elasticity theory. For possible engineering situations where the system parameters are uncertain‑but‑bounded, the propagation from the initial uncertainties of parameters to the consequent uncertainty of dynamic stability of structure is elaborately investigated by interval analysis for the first time. The interval dynamic stability issue is studied through the column structure as a generalized interval eigenvalue problem, and an analytic theorem is rigorously derived and numerically verified for solving the generalized interval eigenvalue problem. Moreover, to avoid the exaggeration on actual bounds of solution, an effective approach is established to limit the dependency phenomenon in the interval dynamic stability analysis. The interval relationships of the excitation parameter and the critical load frequency with five types of system parameters are further bridged according to the interval arithmetic to propose a three-stage interval scheme for evaluating the effects of interval system parameters on the dynamic stability of structures. Numerical studies demonstrate that the uncertainty of the constant part of load influences the boundaries of principal instability region much more than that of the variable part of load. Within all structural parameters, the uncertainty of column length has the most effect on the boundaries, which could make the critical frequency of load be magnified about ten times the uncertainty of parameter. In particular, if all the system parameters are interval with an identical uncertainty degree, the uncertainty of parameter would be propagated in the system of dynamic stability and enlarged as high as 20 times the uncertainty of parameter. Consequently, the impacts of parameter uncertainties on the dynamic stability of structures are fairly significant.

Coupled Dynamic Instability Analysis of Thin-Walled Columns Subjected to Harmonic Axial Loading

This paper presents triply coupled vibrations and instability analysis of thin-walled columns having a non-symmetrical open cross-section. Vlasov’s theory is used to derive the governing differential equations for coupled flexural and torsional vibrations. A numerical method is presented to determine the exact natural frequencies and corresponding mode shapes in terms of real functions. Employing Galerkin’s method, the coupled partial differential equations are reduced to a set of coupled Mathieu type equations. Following Bolotin’s method, the principal instability regions of thin-walled column with a non-symmetrical cross-section having various boundary conditions are determined. Numerical examples are presented to examine the effect of different boundary conditions, aspect ratios, static and dynamic load factors on principal instability regions. The study of response and corresponding phase plot in stable and unstable regions are carried out to identify the dynamic instability behavior.

Dynamic Stability of a Cylindrical Shell Stiffened with a Cylinder and Longitudinal Diaphragms at External Pressure

A study has been made of the dynamic stability of a cylindrical orthotropic shell stiffened with a hollow cylinder and inhomogeneous longitudinal diaphragms under the action of axial forces and pulsating external pressure. The influence of the cylinder and diaphragms on the stability of the shell was taken account of in the form of elastic foundations whose moduli of subgrade reaction are determined from the equations of a three-dimensional theory of elasticity and the Timoshenko model respectively. A solution to the equation of motion of the shell has been found in the form of a trigonometric circumferential-coordinate series. To construct the principal region of instability of the shell, a binomial approximation was used in the obtained Mathieu–Hill equations. As a result, the problem was reduced to a system of two algebraic equations for normal displacement of the shell at diaphragm installation sites. For uniformly spaced identical diaphragms, a solution has been obtained in explicit form. The dependences of the principal region of instability of the shell on the number and rigidity of the diaphragms have been determined at different radii of the cylinder channel.

Parametric instability analysis of truncated conical shells using the Haar wavelet method

Abstract In this paper, the Haar wavelet method is employed to analyze the parametric instability of truncated conical shells under static and time dependent periodic axial loads. The present work is based on the Love first-approximation theory for classical thin shells. The displacement field is expressed as the Haar wavelet series in the axial direction and trigonometric functions in the circumferential direction. Then the partial differential equations are reduced into a system of coupled Mathieu-type ordinary differential equations describing dynamic instability behavior of the shell. Using Bolotin’s method, the first-order and second-order approximations of principal instability regions are determined. The correctness of present method is examined by comparing the results with those in the literature and very good agreement is observed. The difference between the first-order and second-order approximations of principal instability regions for tensile and compressive loads is also investigated. Finally, numerical results are presented to bring out the influences of various parameters like static load factors, boundary conditions and shell geometrical characteristics on the domains of parametric instability of conical shells.

Non-linear dynamic instability analysis of mono-symmetric thin walled columns with various boundary conditions

This paper presents the nonlinear flexural-torsional vibration and instability analysis of thin-walled column with different mono-symmetric open sections subjected to periodic axial load. Non-linear partial differential equations (PDEs) governing the dynamic instability behavior of the considered sections are derived. Galerkin's method is employed to reduce the coupled partial differential equations to a set of coupled Mathieu equations. The amplitudes of steady state vibrations are determined from the non-trivial solution of Mathieu equations in terms of excitation frequency. Following Bolotin's method, principal instability regions are determined for different mono-symmetric section columns having various boundary conditions. Numerical results for the natural frequencies, instability regions and frequency-amplitude responses for thin-walled columns are presented. Influence of eccentricity, shape of the cross-section and boundary conditions on frequency-amplitude responses and instability regions are investigated.

Dynamic instability analysis of sandwich plates with CNT reinforced facesheets

In this paper, the dynamic instability characteristics of sandwich panels with carbon nanotube (CNT) reinforced facesheets subjected to in-plane periodic load is investigated using a shear flexible QUAD-8 serendipity element. The formulation accounts for the realistic variation of the displacements through the thickness, possible discontinuity in slope at the interface and the thickness stretch affecting the transverse deflection. The influence of the aerodynamic force due to airflow over the plate, the in-plane and the rotary inertia effects are also included in the formulation. The accuracy of the present formulation is demonstrated considering the problems for which solutions are available. A detailed numerical study is carried out to bring out the efficacy of the present higher-order plate theory over the first-order plate theory. The present study examines the influence of the volume fraction of the CNT, core-to-facesheet thickness, the plate thickness and aspect ratio, damping and temperature on the dynamic instability boundaries and its associated origin of instability forcing frequencies. The effect of airflow over such sandwich plate is also investigated. The boundaries of the principal instability region are represented in the non-dimensional excitation frequency-load amplitude plane.