Primary resonance excitation of electrically actuated clamped circular plates

We investigate the response of an electrically actuated clamped circular plate to a primary resonance excitation of its first axisymmetric mode using an analytical reduced-order model (macromodel). We discuss the inf luence of the number of modes retained in the discretization on the predicted solutions. The reduced-order model, which is a system of coupled nonlinear ordinary-dif ferential equations, accounts for general residual stress and strain hardening and allows for general material and geometric design variables. Our reduced-order model is robust up to the pull-in instability and is general enough to be an ef fective design tool for capacitive micromachined ultrasonic transducers.     

Nonlinear oscillations of an elastic two-degrees-of-freedom pendulum

Nonlinear oscillations of the vertical plane swinging spring pendulum in the resonance case are studied (frequencies ratio regarding horizontal and vertical directions is equal to 1:2). Square and cubic terms of the Hamiltonian are taken into account. Novel normal form method, i.e., the so called invariant normalization is applied to solve the stated problem. Full system of integrals exhibits equations of the normal form, and solution for the pendulum coordinates is expressed via elementary functions. Frequencies of modes of oscillations are proportional to the first power of amplitude, and not to the second power as it is exhibited by one dimensional Duffing oscillator. Amplitudes of the modes are changed periodically, and energy from one mode is transited to energy of the second one, whereas the period of oscillations depends on the initial conditions. It is illustrated that asymptotic solution with small amplitudes approximates well numerical solution of the governing equations. In addition, an example of a periodic stable solution with constant amplitudes of the oscillation modes is given. Stability of this solution is proved.     

Resonant many-mode periodic and chaotic self-sustained aeroelastic vibrations of cantilever plates with geometrical non-linearities in incompressible flow

The plates interacting with inviscid, incompressible, potential gas flow are analyzed. Many modes interaction is considered to describe self-sustained vibrations of plates. The singular integral equation is solved to obtain gas pressures acting on the plate. The Von Karman equations with respect to three displacements are used to describe the plate geometrical non-linear vibrations. The Galerkin method is applied to each partial differential equation to obtain the finite-degree-of-freedom model of the plate vibrations. Self-sustained vibrations, which take place due to the Hopf bifurcation, are investigated. These vibrations undergo the Naimark?CSacker bifurcation and the periodic motions are transformed into the almost periodic ones. If the stream velocity is increased, almost periodic motions are transformed into chaotic ones. As a result of the internal resonance, the saturation of the vibration mode is observed. The non-linear dynamics of low- and high-aspect-ratio plates is analyzed.     

Coupled, forced response of an axially moving strip with internal resonance

In this paper, the forced response of a non-linear axially moving strip with coupled transverse and longitudinal motions is studied. In particular, the response of the system is examined in the neighborhood of a 3 : 1 internal resonance between the first two transverse modes. The equations of motion are derived using the Hamilton's Principle and discretized by the Galerkin's method. First, with the longitudinal motion neglected, the forced transverse response is investigated by applying the method of multiple scales to assess the effects of speed and the internal resonance. In general, the speed is shown to affect each mode differently. The internal resonance results in the constant solutions having transition to instability of both a saddle-node type and a Hopf bifurcation. In the region where the Hopf bifurcation occurs, steady-state periodic motion does not exist. Instead the stable motion is amplitude- and phase-modulated. When the coupled system with longitudinal motion is examined with internal resonance, results reveal that the modulated motions disappear. Thus, the presence of the longitudinal motion has a stabilizing effect on the transverse modes in the Hopf bifurcation region. The second longitudinal mode is shown to drift due primarily to a direct excitation of the first transverse mode. Effects of the longitudinal motion on the transverse response are shown to be significant for speeds both away from and close to the critical speed.     

Asymptotic analysis of nonlinear dynamics of simply supported cylindrical shells

Donnell equations are used to simulate free nonlinear oscillations of cylindrical shells with imperfections. The expansion, which consists of two conjugate modes and axisymmetric one, is used to analyze shell oscillations. Amplitudes of the axisymmetric motions are assumed significantly smaller, than the conjugate modes amplitudes. Nonlinear normal vibrations mode, which is determined by shell imperfections, is analyzed. The stability and bifurcations of this mode are studied by the multiple scales method. It is discovered that stable quasiperiodic motions appear at the bifurcations points. The forced oscillations of circular cylindrical shells in the case of two internal resonances and the principle resonance are analyzed too. The multiple scales method is used to obtain the system of six modulation equations. The method for stability analysis of standing waves is suggested. The continuation algorithm is used to analyze fixed points of the system of the modulation equations.     

A finite element formulation for global linear stability analysis of a nominally two‐dimensional base flow

A stabilized finite element method, to carry out the linear stability analysis of a two‐dimensional base flow to three‐dimensional perturbations that are periodic along span, is presented. The resulting equations for the time evolution of the disturbance requires a solution to the generalized eigenvalue problem. The analysis is global in nature and is also applicable to non‐parallel flows. Equal‐order‐interpolation functions for velocity and pressure are utilized. Stabilization terms are added to the Galerkin formulation to admit the use of equal‐order‐interpolation functions and to eliminate node‐to‐node oscillations that might arise in advection‐dominated flows. The proposed formulation is tested on two flow problems. First, the mode transitions in the circular Couette flow are investigated. Two scenarios are considered. In the first one, the outer cylinder is at rest, while the inner one spins. Two linearly unstable modes are identified. The primary mode is real and represents the axisymmetric Taylor vortices. The second mode is complex and consists of spiral vortices. For the counter‐rotating cylinders, the primary transition is via the appearance of spiral vortices. Excellent agreement with results from earlier studies is observed. The formulation is also utilized to investigate the parallel and oblique modes of vortex shedding past a cylinder for the Re = 100 flow. It is found that the flow is associated with a large number of unstable oblique shedding modes. The parallel mode of vortex shedding is a special case of this family of modes and is associated with the largest growth rate. Copyright © 2014 John Wiley & Sons, Ltd.     

Combination Internal Resonances in Heated Annular Plates

Nonlinear modal interactions in the forced vibrations of a thermally loaded pre-buckled annular plate with clamped–clamped immovable boundary conditions are investigated. The mechanism responsible for the interaction is a combination internal resonance involving the natural frequencies of the three lowest axisymmetric modes. The in-plane thermal load acting on the plate is assumed to be axisymmetric and the plate is externally excited by a harmonic force. The nonlinear von Kármán plate equations along with the heat conduction equation are combined to model the behavior of the system. An analytical/numerical approach is used to examine the plate vibrations to a harmonic excitation near primary resonance of one of the modes.     

Non-linear modal analysis of a rotating beam

The free non-linear vibration of a rotating beam has been considered in this paper. The von Karman strain-displacement relations are implemented. Non-linear equations of motion are obtained by Hamilton’s principle. Results are obtained by applying the method of multiple scales to a set of discretized ordinary differential equations which obtained by using the Galerkin discretization method. This set contains coupling between transverse and axial displacements as quadratic and cubic geometric non-linearities. Non-linear normal modes and non-linear natural frequencies with or without internal resonance are observed. In the internal resonance case, the internal resonance between two transverse modes and between one transverse and one axial mode are explored. Obtained results in this study are compared with those obtained from literature. The stability and some dynamic characteristics of the non-linear normal modes such as the phase portrait, Poincare section and power spectrum diagrams have been inspected. It is shown that, for the first internal resonance case, the beam has one stable or degenerate uncoupled mode and either: (a) one stable coupled mode, (b) one unstable coupled mode, (c) two stable and one unstable coupled modes, (d) three stable coupled modes, and (e) one stable coupled mode. On the other hand, for the second internal resonance case, the beam has one stable or unstable or degenerate uncoupled mode and either: (a) two stable coupled modes, (b) two unstable coupled modes, and (c) one stable coupled mode depending on the parameters.     

Wave propagation in circular cylindrical shells with periodic axial curvature

The propagation of longitudinal and flexural waves in axisymmetric circular cylindrical shells with periodic circular axial curvature is studied using a finite element method previously developed by the authors. Of primary interest is the coupling of these wave modes due to the periodic axial curvature which results in the generation of two types of stop bands not present in straight circular cylinders. The first type is related to the periodic spacing and occurs independently for longitudinal and flexural wave modes without coupling. However, the second type is caused by longitudinal and flexural wave mode coupling due to the axial curvature. A parametric study is conducted where the effects of cylinder radius, degree of axial curvature, and periodic spacing on wave propagation characteristics are investigated. It is shown that even a small degree of periodic axial curvature results in significant stop bands associated with wave mode coupling. These stop bands are broad and conceivably could be tuned to a specific frequency range by judicious choice of the shell parameters. Forced harmonic analyses performed on finite periodic structures show that strong attenuation of longitudinal and flexural motion occurs in the frequency ranges associated with the stop bands of the infinite periodic structure.     

Non-linear vibrations of free-edge thin spherical shells: modal interaction rules and 1:1:2 internal resonance

This paper is devoted to the derivation and the analysis of vibrations of shallow spherical shell subjected to large amplitude transverse displacement. The analog for thin shallow shells of von Kármán’s theory for large deflection of plates is used. The validity range of the approximations is assessed by comparing the analytical modal analysis with a numerical solution. The specific case of a free edge is considered. The governing partial differential equations are expanded onto the natural modes of vibration of the shell. The problem is replaced by an infinite set of coupled second-order differential equations with quadratic and cubic non-linear terms. Analytical expressions of the non-linear coefficients are derived and a number of them are found to vanish, as a consequence of the symmetry of revolution of the structure. Then, for all the possible internal resonances, a number of rules are deduced, thus predicting the activation of the energy exchanges between the involved modes. Finally, a specific mode coupling due to a 1:1:2 internal resonance between two companion modes and an axisymmetric mode is studied.     

周期激励浅拱分岔研究

研究了一阶和二阶模态在１：２内共振条件下浅拱的复杂动力学行为，指出当周期激励浅拱具有初始静变形时，系统的一阶模态和二阶模态会产生内共振，系统两共振模态之间会产生相互作用，系统的能量会在其低阶和高阶模态之间相互传递，对称破缺后的Ｈｏｐｆ分岔解会通过一系列的倍化周期分岔导致混沌，在混沌域中还会发现稳定的周期解窗口．     

Non-linear dynamics of a system of coupled oscillators with essential stiffness non-linearities

We study the resonant dynamics of a two-degree-of-freedom system composed of a linear oscillator weakly coupled to a strongly non-linear one, with an essential (non-linearizable) cubic stiffness non-linearity. For the undamped system this leads to a series of internal resonances, depending on the level of (conserved) total energy of oscillation. We study in detail the 1:1 internal resonance, and show that the undamped system possesses stable and unstable synchronous periodic motions (non-linear normal modes—NNMs), as well as, asynchronous periodic motions (elliptic orbits—EOs). Furthermore, we show that when damping is introduced certain NNMs produce resonance capture phenomena, where a trajectory of the damped dynamics gets ‘captured’ in the neighborhood of a damped NNM before ‘escaping’ and becoming an oscillation with exponentially decaying amplitude. In turn, these resonance captures may lead to passive non-linear energy pumping phenomena from the linear to the non-linear oscillator. Thus, sustained resonance capture appears to provide a dynamical mechanism for passively transferring energy from one part of the system to another, in a one-way, irreversible fashion. Numerical integrations confirm the analytical predictions.     

Experiments on the instability modes of buoyant diffusion flames and effects of ambient atmosphere on the instabilities

Large scale dynamic behavior of buoyant diffusion flames were studied experimentally. It was found that buoyant diffusion flames originating from circular nozzles exhibit two different modes of flame instabilities. The first mode results in a sinuous meandering of the diffusion flame, characteristic of flames originating from small diameter nozzles. This instability originates at some distance downstream of the nozzle exit in the contraction region of the buoyant flame envelope and develops into a sinuous motion of the flame. The second mode is the varicose mode which develops very close to the nozzle exit as axisymmetric perturbations of a contracting flame surface. In this mode, flame oscillations result in the formation of toroidal vortical structures that convect through the flame and cause periodic burn out at the flame top resulting in the observed flame height fluctuations. The average flame heights are found to be typically shorter for these flames. The oscillation frequencies and their scaling for the two modes are also different with the sinuous mode having higher frequencies than the varicose mode. It was also observed that the instability can switch from one mode to the other and the probability of observing the varicose mode appears to increase with increasing Richardson number. Additionally, the feasibility of altering the behavior of buoyant diffusion flames was explored through variation of the oxidizer medium density. It was found that the flame oscillations can be completely suppressed for flames burning in helium rich helium–oxygen mixtures. At lower helium concentrations, the oscillation frequency can be significantly reduced. In order to enhance the buoyancy effect, CO₂–O₂ mixtures were also studied. However, the density increase and its effects on flame oscillation frequency were found to be small compared to those flames burning in air. These experiments point towards the feasibility of altering buoyant flame behavior under earth gravity and studying the large scale dynamical aspects of buoyant flames without the need of variable gravity environment. Received: 2 March 1999/Accepted: 6 August 1999     

Complicated nonlinear responses of a simply supported FGM rectangular plate under combined parametric and external excitations

In this paper, we use the asymptotic perturbation method based on the Fourier expansion and the temporal rescaling to investigate the nonlinear oscillations and chaotic dynamics of a simply supported rectangular plate made of functionally graded materials (FGMs) subjected to a through-thickness temperature field together with parametric and external excitations. Material properties are assumed to be temperature-dependent. Based on the Reddy’s third-order plate theory, the governing equations of motion for the plate are derived using the Hamilton’s principle. The Galerkin procedure is employed to obtain a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms. The resonant case considered here is 1:2 internal resonance, principal parametric resonance-1/2 subharmonic resonance. Based on the averaged equation in polar coordinate form, the stability of steady state solutions is analyzed. The phase portrait, waveform and Poincaré map are used to analyze the periodic and chaotic motions of the FGM rectangular plate. It is found that the FGM rectangular plate exhibits the chaotic motions under certain circumstances. It is seen that the nonlinear dynamic responses of the FGM rectangular plate are more sensitive to transverse excitation. The excitation force can be used as a controlling factor which can change the response of the FGM rectangular plate from periodic motion to the chaotic motion.     

Non-linear vibrations of free-edge thin spherical shells: Experiments on a 1:1:2 internal resonance

This study is devoted to the experimental validation of a theoretical model of large amplitude vibrations of thin spherical shells described in a previous study by the same authors. A modal analysis of the structure is first detailed. Then, a specific mode coupling due to a 1:1:2 internal resonance between an axisymmetric mode and two companion asymmetric modes is especially addressed. The structure is forced with a simple-harmonic signal of frequency close to the natural frequency of the axisymmetric mode. The experimental setup, which allows precise measurements of the vibration amplitudes of the three involved modes, is presented. Experimental frequency response curves showing the amplitude of the modes as functions of the driving frequency are compared to the theoretical ones. A good qualitative agreement is obtained with the predictions given by in the model. Some quantitative discrepancies are observed and discussed, and improvements of the model are proposed.     

Numerical modeling of the disturbances of the separated flow in a rounded compression corner

Numerical modeling of the time-dependent supersonic flow over a compression corner with different roundness radii is performed on the basis of the solution of the two-dimensional Navier-Stokes equations in the regimes corresponding to local boundary layer separation. The development of unstable disturbances generated by local periodic injection/suction in the preseparated boundary layer is calculated. The results are compared with those of similar calculations for a flat plate. It is shown that the natural oscillations of the boundary-layer second mode stabilize in the separation zone and grow intensely downstream of the reattachment point. The acoustic modes excited within a separation bubble are studied using numerical calculations and an asymptotic analysis.     

Multi-parameter bifurcation study of shimmy oscillations in a dual-wheel aircraft nose landing gear

We develop and investigate a mathematical model of an aircraft nose landing gear with a dual-wheel configuration. The main aim here is to study the influence of a dual-wheel configuration on the existence of shimmy oscillations. To this end, we consider a model that describes the torsional and lateral vibrational modes and the non-linear interaction between them via the tyre-ground contact. More specifically, we perform a bifurcation analysis (with the software package auto) of the model in the two-parameter plane of forward velocity of the aircraft and vertical load on the nose landing gear. This two-parameter bifurcation diagram allows one to identify regions of different dynamics, and the question addressed here is how it depends on two key parameters of the dual-wheel configuration. Namely, we consider the influence of, first, the separation distance between the two wheels and, second, of gyroscopic effects arising from the inertia of the wheels. For both cases, we find that with increasing separation distance and wheel inertia, respectively, the lateral mode becomes more stable and the torsional mode becomes less stable. More specifically, we present associated bifurcation scenarios that explain the transitions between qualitatively different two-parameter bifurcation diagrams. Overall, we find that the separation distance and gyroscopic effects due to wheel inertia may have a significant influence on the quantitative and qualitative nature of shimmy oscillations in aircraft nose landing gears. In particular, the torsional and the lateral modes of a dual-wheel nose landing gear may interact in a quite complicated fashion.     

Stability of a Liquid Film Flowing Down an Oscillating Inclined Surface

The stability of flow of a liquid film along an inclined plate subject to periodic oscillations under the action of the gravity force is investigated with allowance for the surface tension. An equation of the Orr-Sommerfeld type with time-periodic coefficients is used. A method for determining the eigenvalues of the linear stability problem is developed on the basis of Floquet theory, spectral representation of the variables, and multistep methods of integration of ordinary differential equations. The bifurcation spectrum of the resonance modes is investigated, and the amplification coefficients and phase velocities are calculated for the surface waves, Tollmien-Schlichting waves, and resonance waves. The influence of external parameters, namely, the inclination, the surface tension, and the layer thickness, on the resonance modes and the steady-state flow modes is studied.     

Vibrations of Circular Sandwich Plates under Resonance Loads

The paper studies axisymmetric resonance vibrations of an elastic circular sandwich plate under local periodic surface loads of rectangular, sinusoidal, and parabolic forms. The hypotheses of broken normal are used to describe the kinematics of the plate, which is asymmetric in thickness. The core is assumed to be light. The initial–boundary-value problems are solved analytically. The solutions are analyzed     

磁场中旋转运动圆板的分叉与混沌研究

应用数值模拟方法研究磁场中旋转运动圆板的分叉与混沌问题。首先,基于薄板理论和麦克斯韦电磁场方程组,给出了动能、应变势能、外力虚功以及电磁力的表达式,再利用哈密顿原理,得到磁场中旋转运动圆板横向振动的非轴对称非线性磁弹性振动微分方程组。其次,采用贝塞尔函数作为圆板的振型函数进行伽辽金积分,得到了轴对称情况下横向振动的常微分方程组表达式。最后,针对主共振,取周边夹支边界条件的圆板作为算例,得到了当振型函数取一阶时,将磁感应强度、外激励振幅和激励频率作为控制参数的分叉图及庞加莱映射图等计算结果,并讨论了分叉参数对系统的分叉与混沌的影响。数值计算结果表明,这些控制参数的变化影响系统稳定性,在分叉参数逐渐变化的过程中,系统经历从混沌到多倍周期运动再到混沌的往复过程。