Divergence Instability Research Articles

Nonlinear forced vibration of a moving paper web with varying density

The study investigates the forced nonlinear vibration characteristics of an axially moving printing paper web with variable density in the lateral direction. The nonlinear governing equations of the web can be obtained by the von Karman large deflection thin plate theory. The vibration equations are discretized using the Bubnov–Galerkin method. The fourth-order Runge–Kutta technique is used to solve the differential equations of the nonlinear system. The phase-plane diagrams, time histories, bifurcation graphs, Poincare maps, and power spectrum are employed to analyze the influence of density coefficient, velocity, and aspect ratio on the nonlinear dynamic behavior of the moving paper web. The stable working region and the divergence instability region of the web are obtained. The research provides a theoretical foundation for improving the dynamic stability of a moving paper web.

An efficient solver for fully coupled solution of interaction between incompressible fluid flow and nanocomposite truncated conical shells

In this research, the dynamic instabilities of nanocomposite truncated conical shells containing a quiescent or a flowing inviscid fluid are scrutinized. Nonlinear dynamic equations are established according to the Novozhilov nonlinear shell theory along with Green’s strains and Hamilton principle. The velocity potential and Bernoulli’s equations are adopted to calculate fluid pressure acting on the conical shell. The nonlinear governing equations are discretized using trigonometric expansion through the circumferential direction and generalized differential quadrature method (GDQM) through the meridional direction. A detailed parametric study is directed to provide an insight into the influence of volume fraction of carbon nanotubes (CNTs), CNT dispersion, geometrical parameters, and boundary conditions on the divergence and flutter instabilities of nanocomposite truncated conical shells. This study shows the superb efficiency of the outlined solution procedure in reducing computational costs and virtual storage. The simulation indicates that the beginning of divergence and flutter instabilities can be significantly postponed by selecting an appropriate dispersion of CNTs through the thickness of the conical shell. Furthermore, the onset of flutter and divergence instabilities are found to be very sensitive to the semi-vertex angle and thickness-to-radius ratio. The results of this research shed light into using ultra-high-strength and low-weight nanocomposite for pressure vessels applications.

Flutter and Divergence Instability of Axially-Moving Nanoplates Resting on a Viscoelastic Foundation

Moving nanosystems often rest on a medium exhibiting viscoelastic behavior in engineering applications. The moving velocity and viscoelastic parameters of the medium usually have an interacting impact on the mechanical properties of nanostructures. This paper investigates the dynamic stability of an axially-moving nanoplate resting on a viscoelastic foundation based on the nonlocal elasticity theory. Firstly, the governing partial equations subject to appropriate boundary conditions are derived through utilizing the Hamilton’s principle with the axial velocity, viscoelastic foundation, nonlocal effect and biaxial loadings taken into consideration. Subsequently, the characteristic equation describing the dynamic characteristics is obtained by employing the Galerkin strip distributed transfer function method. Then, complex frequency curves for the nanoplate are displayed graphically and the effects of viscoelastic foundation parameters, small-scale parameters and axial forces on divergence instability and coupled-mode flutter are analyzed, which show that these parameters play a crucial role in affecting nanostructural instability. The presented results benefit the designation of axially-moving graphene nanosheets or other plate-like nanostructures resting on a viscoelastic foundation.

Fluttering and divergence instability of functionally graded viscoelastic nanotubes conveying fluid based on nonlocal strain gradient theory.

In this paper, a size-dependent viscoelastic pipe model is developed to investigate the size effects on flutter and divergence instability of functionally graded viscoelastic nanotubes conveying fluid. The nonlocal strain gradient theory and the Kelvin-Voigt model are used to consider the significance of nonlocal field, strain gradient field, and viscoelastic damping effects. The dimensionless equation of transverse motion and related classical and non-classical boundary conditions are derived using the variational approach. The partial differential equations are discretized to a system of ordinary differential equations by the use of Galerkin's method. The frequency equation is obtained as a function of dimensionless flow velocity, small-scale parameters, damping coefficient, and power-law parameter. Numerical results are presented to study the dynamical behavior of the system and are compared with experimental and theoretical results reported by other researchers. Coupled and single mode fluttering related to higher vibration modes of fluid-conveying nanotubes supported at both ends are studied for the first time. It is found that coupled mode fluttering can be seen for different vibration modes by increasing the flow velocity in the absence of structural damping. Structural damping changes the dynamical behavior of the system, in which by increasing the flow velocity, single mode fluttering occurs instead of coupled mode fluttering. In addition, the presence of structural damping increases the critical flow velocity and, as a result, increases the stability of the system. The results also show that increasing the nonlocal parameter will have a stiffness-softening effect, while increasing the strain gradient length scale has an opposing effect.

Aeroelastic Instability of Flexible Rocket Bodies on the Basis of a Simplified Mechanical Model

The present paper deals with the applicability of simplified mechanical models to discuss the aeroelastic instability behavior of flexible rocket bodies. A suitable mechanical model to demonstrate body divergence and flutter has been discussed and the stability analysis of the model has been presented. The flutter criterion requires a computation of the characteristic polynomial. Sylvester’s dialytic method of elimination is used to compute the required discriminant. A mechanical model composed of three bars, connected together by two elastic rotational springs, having five degrees-of-freedom is chosen for demonstrating both divergence and flutter instabilities. Effects of mass distribution, stiffness distribution, and location of stabilizer fin on instability behavior are discussed.

Divergence and flutter instabilities of nanobeams in moving state accounting for surface and shear effects

Transverse vibrations of elastically rested moving beam-like nanostructures accounting for surface effect are of high concern. The role of nonlocality on the free dynamic response of moving nanobeams has been revealed in recent years; nevertheless, the influence of the surface energy on the mechanical behavior of such elements has not been explained yet. In this paper, equations of motion of rested nanoscaled beams in the moving state are derived carefully via surface energetic-shear deformable beam models. Subsequently, the transverse vibrations of the nanostructure are evaluated using Galerkin-based assumed mode method. The explicit expressions of divergence velocities are obtained analytically, and these are successfully verified with the results of a numerical approach. The roles of crucial parameters on the first divergence velocity are addressed in some detail. Additionally, the stable and unstable regions are determined systematically and the influence of both surface energy and shear energy on the stability of moving nanostructure is discussed.

Stability analysis of pipes conveying fluid with fractional viscoelastic model

Divergence and flutter instabilities of pipes conveying fluid with fractional viscoelastic model has been investigated in the present work. Attention is concentrated on the boundaries of the stability. Based on the Euler–Bernoulli beam theory for structural dynamics, viscoelastic fractional model for damping and, plug flow model for fluid flow, equation of motion has been derived. The effects of gravity, and distributed follower forces are also considered. By transferring the equation of motion to the Laplace domain and using the Galerkin method, the characteristic equations are obtained. By solving the eigenvalue problem, frequencies and dampings of the system have been obtained versus flow velocity. Some numerical test cases have been studied with viscoelastic fractional model and the effect of the fractional derivative order and the retardation time is investigated for various boundary conditions.

Structural instability of non-conservative functionally graded micro-beams tunable with piezoelectric layers

This investigation aims to explore the non-conservative instability of a functionally graded material micro-beam subjected to a subtangential force. The functionally graded material micro-beam is integrated with piezoelectric layers on the lower and upper surfaces. To take size effect into account, the mathematical derivations are expanded in terms of three length scale parameters using the modified strain gradient theory in conjunction with the Euler–Bernoulli beam model. However, the modified strain gradient theory includes modified couple stress theory and classical theory as special cases. Applying extended Hamilton’s principle and Galerkin method, the governing equation and corresponding boundary conditions are obtained and then solved numerically by the eigenvalue analysis, respectively. The results illustrated effects of non-conservative parameter, length scale parameter, different material gradient index, and various values of piezoelectric voltage on the natural frequencies, flutter and divergence instabilities of a cantilever functionally graded material micro-beam. It is found that both the material gradient index and applied piezoelectric voltage have significant influence on the vibrational behaviors, divergence and flutter instability regions. Furthermore, a comparison between the various micro-beam theories on the basis of modified couple stress theory, modified strain gradient theory, and classical theory are presented.

Transverse Vibration of a Moving Viscoelastic Hard Membrane Containing Scratches

The transverse vibration and stability of a moving viscoelastic hard printing membrane containing scratches are investigated. Based on the viscoelastic differential constitutive relation, thin plate theory, and d’Alembert principle, the differential equation of a moving viscoelastic hard membrane with a straight scratch through the surface is derived by using the continuity condition of the scratches. The complex characteristic equation is obtained by using the differential quadrature method. The effects of the scratch depth and scratch position on the critical instability speed of the moving hard membrane were highlighted by solving the differential equation and numerical calculation, and the coupling effects of speed and scratch depth on vibration characteristics of the hard membrane are also analyzed. Numerical calculation results show that the hard membrane experiences divergence instability when the actual printing speed is v=23.2 m/s, and the membrane is stable when v<23.2 m/s. The theoretical guidance and method for scratches detection of the precision coated hard membrane are provided.

케이블 캐리어의 작업조건이 동특성에 미치는 영향

This study investigates the dynamics characteristics of a cable carrier with respect to the variations in carrier length and axially moving speed. The characteristic equation of the carrier’s lateral vibration is derived from the configuration of blocks connected through pin joints. Then, the modal characteristics and stability behavior of the carrier are obtained by solving the characteristic equation numerically. Results show that natural frequencies decrease as the carrier speed increases, particularly around the stability boundaries. The instability of divergence and flutter occurs as the carrier speed increases. In addition, the critical speed at the stability boundary is observed to be inversely proportional to the carrier length, which requires a tradeoff between the carrier’s workspace and the productivity of the carrier.

Flutter and divergence instability of rectangular plates under nonconservative forces considering surface elasticity

Nanomaterials are widely used in engineering applications. For nanostructures, surface effects usually have a significant impact on the mechanical properties of micro/nano materials and structures. This paper investigates the dynamic stability of a thin rectangular plate with surface effects under a nonconservative force. Based on the Kirchhoff plate theory incorporating surface elasticity, Hamilton’s principle is employed to derive a governing partial differential equation subject to appropriate boundary conditions. A characteristic equation describing the load–frequency interaction curves is obtained. The load–frequency interaction curves are displayed graphically for various tangency coefficients. The effects of surface stress and surface mass on the frequencies and buckling loads are highlighted. Surface effects on the transition from divergence instability to flutter instability are analyzed. The surface stress and surface mass play a crucial role in affecting flutter loads of nanostructural instability. The obtained results are of benefit to safety design of micro/nano scale plates subjected to compressive loading and generalized follower force in nonconservative systems.

Nonlocal vibration and instability analysis of carbon nanotubes conveying fluid considering the influences of nanoflow and non-uniform velocity profile

In this article, the influences of non-uniform velocity profile attributable to slip boundary condition and viscosity of fluid on the dynamic instability of carbon nanotubes (CNTs) conveying fluid are investigated. The nonlocal elasticity theory and the Euler–Bernoulli beam theory are employed to derive partial differential equation of nanotubes conveying fluid. Furthermore, a dimensionless momentum correction factor (MCF) is obtained as a function of Knudsen number (Kn) so as to insert the effects of non-uniform velocity profile into the equation of motion. In continuation, complex eigen-frequencies of the system are attained with respect to different boundary conditions, the momentum correction factor, slip boundary condition and nonlocal parameter. The results delineate that considering the effects of non-uniform velocity profile could diminish predicted critical velocity of flow. Therefore, the divergence instability occurs in the lower values of flow velocity. In addition, the MCF decreases through enhancement of Kn; hence, the effects of non-uniform velocity profile are more noticeable for liquid fluid than gas fluid.

On the stability of spinning thin-walled porous beams

In the current study, the stability analysis of functionally graded (FG) thin-walled porous beams reinforced by nanocomposite graphene platelets (GPLs) under compressive axial load is investigated. It is assumed that the thin-walled porous beam is spinning along its longitudinal axis and both symmetric and asymmetric distributions of porosity are considered. Furthermore, the GPLs are distributed thorough the thickness direction both uniformly and non-uniformly. The effective material properties such as Young's modulus, mass density and Poisson's ratio of the porous beams are computed based on the Halpin-Tsai micromechanics model and the rule of mixture. The extended Hamilton's principle is utilized to establish the governing equations and they are discretized by the extended Galerkin method (EGM). The effects of various parameters such as GPL porous distribution patterns, GPL weight fraction, geometry of GPL nanofillers and porosity coefficient on the frequencies as well as flutter and divergence instabilities of the thin-walled porous beams have been studied. Numerical results demonstrate that the best efficient way to increase the stability region is considering GPL pattern A, with dispersing more GPL fillers near the top and bottom surfaces of the thin-walled porous beam along with symmetric porosity distribution.

Dynamics and Stability of Variable-Length, Vertically-Traveling, Heavy Cables: Application to Tethered Aerostats

The dynamics and stability of variable-length vertically traveling, heavy cables with an end load are investigated. Such cables act as tethers in aerostat systems. The cable is modeled as a heavy string undergoing small, planar vibrations while attached to a rigid, spherical aerostat. Aerodynamic and buoyancy forces act only on the aerostat. An asymptotic analysis for slow deployment rates provides excellent leading-order approximation to the dynamics, which is also obtained from finite element simulations. Stability of the aerostat system is then investigated computationally by considering the linear stability of the system when it is perturbed from its nominal dynamical state at any given time. It is found that, when an aerostat ascends, the cable always goes unstable after a certain time through a divergence instability. In contrast, flutter instability is found in the cable during the aerostat’s descent. These stability analyses help in the development of deployment charts that relate the maximum achievable elevation to deployment rate. Such deployment charts can help in parameter selection for efficient aerostat deployment. The dynamics of the aerostat in the presence of spatiotemporally varying aerodynamic forces are also studied computationally. The paper concludes with two case studies of aerostat deployment that demonstrate the utility of the current analysis.

System modeling and instability control of wind turbine blade based on hydraulic pitch system and radial basic function neural network proportional–integral–derivative controller

System modeling and aeroservoelastic control for divergent instability of stall-induced composite wind turbine blade modeled as thin-walled symmetric layup beam analysis have been investigated based on hydraulic pitch system and radial basic function neural network control. The blade is modeled as single-cell thin-walled beam structure with the circumferentially asymmetric stiffness design, exhibiting flap/lead-lag bending coupling deformation. The stall flutter and aeroservoelastic control of composite blade are investigated based on dynamic stall Beddoes–Leishman aerodynamic model and radial basic function neural network proportional–integral–derivative controller, with pitch actuator performed by hydraulic system. The system motion equations consist of the aeroelastic equations and the six-order pitch equation. The nonlinear aeroelastic responses, including both flap/lead-lag responses and pitch angle responses under different parameters, are solved by Galerkin method and the nonlinear time integration scheme with nonlinear residual analysis. To verify the effectiveness of the control scheme and realize visualized display of large thin-walled blade in the laboratory, experimental platform based on hardware-in-the-loop simulation is built to realize real-time control and virtual simulation. The platform structure consists of PLC hardware, monitoring interface in configuration software, and simulation environment that is connected by the OPC server with PLC system. The platform lays the foundation for vibrational behavior research on visualization of large wind turbine blade under divergent stall situation and verifies the real-time feasibility of the control algorithm proposed.

Effects of the thickness on the stability of axially moving viscoelastic rectangular plates

The differential quadrature method is employed to solve the differential equation governing the in-plane moving rectangular viscoelastic plate with concave or convex cross sections. The boundary conditions of the plate are specified as simply supported-free (SFSF) and clamped-simply supported (CSCS). Plots showing the influence of the cross section's form on the plate’s stability are presented and it is observed that the plate generally undergoes divergence instability but coupled mode instability also occurs for SFSF boundary conditions. Coupled mode instability appears when the cross-section is convex or varies linearly. Also, the increase of the thickness ratio plays an increasing role on the critical instability values of the moving speed.

Aerodynamic stability analysis of geometrically nonlinear orthotropic membrane structure with hyperbolic paraboloid in sag direction

This paper studies the aerodynamic stability of a tensioned, geometrically nonlinear orthotropic membrane structure with hyperbolic paraboloid in sag direction. Considering flow separation, the wind field around membrane structure is simulated as the superposition of a uniform flow and a continuous vortex layer. By the potential flow theory in fluid mechanics and the thin airfoil theory in aerodynamics, aerodynamic pressure acting on membrane surface can be determined. And based on the large amplitude theory of membrane and D\'Alembert\'s principle, interaction governing equations of wind-structure are established. Then, under the circumstance of single-mode response, the Bubnov-Galerkin approximate method is applied to transform the complicated interaction governing equations into a system of second-order nonlinear differential equation with constant coefficients. Through judging the frequency characteristic of the system characteristic equation, the critical velocity of divergence instability is determined. Different parameter analysis shows that the orthotropy, geometrical nonlinearity and scantling of structure is significant for preventing destructive aerodynamic instability in membrane structures. Compared to the model without considering flow separation, it\'s basically consistent about the divergence instability regularities in the flow separation model.

Stability analysis of thin-walled spinning reinforced pipes conveying fluid in thermal environment

Abstract The divergence and flutter instabilities of the thin-walled spinning pipes reinforced by single-walled carbon nanotubes in thermal environment are investigated. The material properties of carbon nanotube-reinforced composites are assumed to be uniform distribution as well as two types of functionally graded distribution patterns. The thermal effects are also considered and the material properties of carbon nanotube-reinforced composites are assumed to be temperature-dependent. The cantilever pipe conveying fluid is spinning along its longitudinal axis and subjected to an axial force at the free end. Based on the thin-walled Timoshenko beam theory, the governing equations of motion are derived using the extended Hamilton's principle and discretized via the Galerkin method. The resulting thermal-structural-fluid eigenvalue problem is solved and the frequency and the critical fluid velocities are calculated. The effects of carbon nanotubes distributions, volume fraction of carbon nanotubes, compressive axial force, spinning speed, gravity and fluid mass ratio on the critical divergence and flutter velocities of the thin-walled spinning pipe conveying fluid are studied.

An isogeometric analysis approach to the stability of curved pipes conveying fluid

Stability analysis of curved pipes conveying fluid is of significant interest in many engineering aplications such as floating and moored system dynamics. The aim of this paper is to develop a new formulation based on the Isogeometric Analysis (IGA) for vibration and stability analyses of curved pipes conveying fluid. Both divergence and flutter instabilities of curved pipes are investigated. IGA uses B-Splines and Non-Uniform Rational B-Splines (NURBS) as basis functions. The main feature of IGA, as required in dynamic analysis of curved structures, is the ability of the NURBS functions to represent the exact geometry of the problem with fewer control points. This method provides several advantages including high-order continuity of the solution and better accuracy. The governing differential equations of the problem are obtained using the Hamiltonian's principle. The effects of rotary inertias of both pipe and fluid are also included in the mathematical formulation. It is shown that the present formulation can provide accurate results with small number of degrees of freedom. It is concluded that IGA can be used efficiently to predict the instability of curved pipes conveying fluid with the advantage of considering the exact curvature of the pipe.

Vibration of deploying rectangular cross-sectional beam made of functionally graded materials

Transverse vibration and stability of deploying rectangular cross-sectional cantilever beam made of functionally graded material are investigated. The functionally graded material beam is assumed to be constructed with ceramics and metal phases, and the corresponding equivalent parameters of functionally graded material are found to continuously vary across the cross-sectional height with a simple power law. The differential equations of motion of deploying functionally graded material cantilever beam are derived by Hamilton’s principle. Based on the assumed modal method, the beam deflection function is expanded into a series, in which each term is expressed to admissible function multiplied by generalized coordinate. The eigenfunctions of cantilever beam in which the length of the beam is time-dependent are chosen as admissible functions. Galerkin method is employed to discretize the differential equation to a set of time-coordinate-dependent ordinary differential equations, and then the eigenvalue problem depending on time is obtained. The deployment motion of beam is considered as a constant speed in this study, and some numerical results, which are variations of tip deflection response and complex frequencies, are obtained. Finally, the effects of gradient index of functionally graded material, deploying speed, initial length and protruded length, the cross-sectional height on dynamical characteristics, and divergence instability of the deploying functionally graded material beam are discussed.