DYNAMIC STABILITY OF SKEW PLATES SUBJECTED TO AERODYNAMIC AND RANDOM IN-PLANE FORCES

This paper presents an investigation into the dynamic stability of skew plates acted upon simultaneously by an aerodynamic force in the chordwise direction and a random in-plane force in the spanwise direction. Due to this random in-plane force, the plate may become unstable before the aerodynamic force reaches its critical value. In this work, the finite element formulation is applied to obtain the discretized system equations. The system equations are then partially uncoupled and reduced in size by the modal truncation method. Finally, the unsmoothed and the smoothed versions of the stochastic averaging are used to calculate the system response, and the second-moment stability criterion is utilized to determine the stability boundary of the system. Numerical results show that the stability boundary obtained by the smoothed stochastic averaging is less conservative than that obtained by the unsmoothed version, and the former is the tangent of the latter at zero spectral density of the random in-plane force.     

Dynamic stability of spinning pretwisted beams subjected to axial random forces

This paper studies the dynamic stability of a pretwisted cantilever beam spinning along its longitudinal axis and subjected to an axial random force at the free end. The axial force is assumed as the sum of a constant force and a random process with a zero mean. Due to this axial force, the beam may experience parametric random instability. In this work, the finite element method is first applied to yield discretized system equations. The stochastic averaging method is then adopted to obtain Ito's equations for the response amplitudes of the system. Finally the mean-square stability criterion is utilized to determine the stability condition of the system. Numerical results show that the stability boundary of the system converges as the first three modes are taken into calculation. Before the convergence is reached, the stability condition predicted is not conservative enough.     

Dynamic contact analysis of a tensioned beam with a moving mass–spring system

The dynamic contact problem of a tensioned beam with clamped-pinned ends is analyzed when the beam contacts a moving mass–spring system. The contact and contact loss conditions are expressed in terms of constraint equations after considering the dynamic contact between the beam and the moving mass. Using these constraints and equations of motion for the beam and moving mass, dynamic contact equations are derived and then discretized using the finite element method, which is based on the Lagrange multiplier method. The time responses for the contact forces are computed from these discretized equations. The contact force variations and contact loss are investigated for the variations of the moving mass velocity, the beam tension, the moving mass, and the stiffness of the moving mass–spring system. In addition, the possibility of contact loss and safe contact conditions between the moving mass and the tensioned beam are also studied.     

Stability of an axially accelerating string subjected to frictional guiding forces

The dynamic response of an axially translating continuum subjected to the combined effects of a pair of spring supported frictional guides and axial acceleration is investigated; such systems are both non-conservative and gyroscopic. The continuum is modeled as a tensioned string translating between two rigid supports with a time-dependent velocity profile. The equations of motion are derived with the extended Hamilton's principle and discretized in the space domain with the finite element method. The stability of the system is analyzed with the Floquet theory for cases where the transport velocity is a periodic function of time. Direct time integration using an adaptive step Runge-Kutta algorithm is used to verify the results of the Floquet theory. This approach can also be employed in the general case of arbitrary time-varying velocity. Results are given in the form of time history diagrams and instability point grids for different sets of parameters such as the location of the stationary load, the stiffness of the elastic support, and the values of initial tension. This work showed that presence of friction adversely affects stability, but using non-zero spring stiffness on the guiding force has a stabilizing effect. This work also showed that the use of the finite element method and Floquet theory is an effective combination to analyze stability in gyroscopic systems with stationary friction loads.     

带有分数阶阻尼的压电能量采集系统相干共振

研究了含分数阶阻尼的双稳态能量采集系统的相干共振. 建立了带有分数阶阻尼的轴向受压梁压电能量采集系统动力学模型. 对于分数阶方程, 采用Euler-Maruyama-Leipnik方法进行求解, 计算了不同阻尼阶数下的能量采集系统的信噪比、响应均值、跃迁数目等统计物理量. 结果表明: 此压电能量采集系统在随机激励下可以实现相干共振, 阻尼阶数对相干共振的临界噪声强度和相干共振幅值有很大影响. 关键词： 分数阶阻尼 随机激励 能量采集系统 相干共振     

A STOCHASTIC OPTIMAL SEMI-ACTIVE CONTROL STRATEGY FOR ER/MR DAMPERS

A stochastic optimal semi-active control strategy for randomly excited systems using electrorheological/magnetorheological (ER/MR) dampers is proposed. A system excited by random loading and controlled by using ER/MR dampers is modelled as a controlled, stochastically excited and dissipated Hamiltonian system with n degrees of freedom. The control forces produced by ER/MR dampers are split into a passive part and an active part. The passive control force is further split into a conservative part and a dissipative part, which are combined with the conservative force and dissipative force of the uncontrolled system, respectively, to form a new Hamiltonian and an overall passive dissipative force. The stochastic averaging method for quasi-Hamiltonian systems is applied to the modified system to obtain partially completed averaged Itô stochastic differential equations. Then, the stochastic dynamical programming principle is applied to the partially averaged Itô equations to establish a dynamical programming equation. The optimal control law is obtained from minimizing the dynamical programming equation subject to the constraints of ER/MR damping forces, and the fully completed averaged Itô equations are obtained from the partially completed averaged Itô equations by replacing the control forces with the optimal control forces and by averaging the terms involving the control forces. Finally, the response of semi-actively controlled system is obtained from solving the final dynamical programming equation and the Fokker-Planck-Kolmogorov equation associated with the fully completed averaged Itô equations of the system. Two examples are given to illustrate the application and effectiveness of the proposed stochastic optimal semi-active control strategy.     

Active control of low-frequency sound radiation by cylindrical shell with piezoelectric stack force actuators

A novel active control method of sound radiation from a cylindrical shell under axial excitations is proposed and theoretically analyzed. This control method is based on a pair of piezoelectric stack force actuators which are installed on the shell and parallel to the axial direction. The actuators are driven in phase and generate the same forces to control the vibration and the sound radiation of the cylindrical shell. The model considered is a fluid-loaded finite stiffened cylindrical shell with rigid end-caps and only low-frequency axial vibration modes are involved. Numerical simulations are performed to explore the required control forces and the optimal mounting positions of actuators under different cost functions. The results show that the proposed force actuators can reduce the radiated sound pressure of low-frequency axial modes in all directions.     

Dynamic stability of parametrically-excited linear resonant beams under periodic axial force

The parametric dynamic stability of resonant beams with various parameters under periodic axial force is studied.It is assumed that the theoretical formulations are based on Euler-Bernoulli beam theory.The governing equations of motion are derived by using the Rayleigh-Ritz method and transformed into Mathieu equations,which are formed to determine the stability criterion and stability regions for parametricallyexcited linear resonant beams.An improved stability criterion is obtained using periodic Lyapunov functions.The boundary points on the stable regions are determined by using a small parameter perturbation method.Numerical results and discussion are presented to highlight the effects of beam length,axial force and damped coefficient on the stability criterion and stability regions.While some stability rules are easy to anticipate,we draw some conclusions:with the increase of damped coefficient,stable regions arise;with the decrease of beam length,the conditions of the damped coefficient arise instead.These conclusions can provide a reference for the robust design of parametricallyexcited linear resonant sensors.     

Bending–torsional flutter of a cantilevered pipe conveying fluid with an inclined terminal nozzle

Stability analysis of a horizontal cantilevered pipe conveying fluid with an inclined terminal nozzle is considered in this paper. The pipe is modelled as a cantilevered Euler–Bernoulli beam, and the flow-induced inertia, Coriolis and centrifugal forces along the pipe as well as the follower force induced by the jet-flow are taken into account. The governing equations of the coupled bending–torsional vibrations of the pipe are obtained using extended Hamilton's principle and are then discretized via the Galerkin method. The resulting eigenvalue problem is then solved, and several cases are examined to determine the effect of nozzle inclination angle, nozzle aspect ratio, mass ratio and bending-to-torsional rigidity ratio on flutter speed of the system.     

Higher-order stochastic averaging to study stability of a fractional viscoelastic column

Stochastic stability of a fractional viscoelastic column axially loaded by a wideband random force is investigated by using the method of higher-order stochastic averaging. By modelling the wideband random excitation as Gaussian white noise and real noise and assuming the viscoelastic material to follow the fractional Kelvin–Voigt constitutive relation, the motion of the column is governed by a fractional stochastic differential equation, which is justifiably and uniformly approximated by an averaged system of Itô stochastic differential equations. Analytical expressions are obtained for the moment Lyapunov exponent and the Lyapunov exponent of the fractional system with small damping and weak random fluctuation. The effects of various parameters on the stochastic stability of the system are discussed.     

Coupled nonlinear size-dependent behaviour of microbeams

The nonlinear resonant behaviour of a microbeam, subject to a distributed harmonic excitation force, is investigated numerically taking into account the longitudinal as well as the transverse displacement. Hamilton’s principle is employed to derive the coupled longitudinal-transverse nonlinear partial differential equations of motion based on the modified couple stress theory. The discretized form of the equations of motion is obtained by applying the Galerkin technique. The pseudo-arclength continuation technique is then employed to solve the discretized equations of motion numerically. Different types of bifurcations as well as the stability of solution branches are determined. The numerical results are presented in the form of frequency-response and force-response curves for different sets of parameters. The effect of taking into account the longitudinal displacement is highlighted.     

Semi-implicit surface tension formulation with a Lagrangian surface mesh on an Eulerian simulation grid

We present a method for applying semi-implicit forces on a Lagrangian mesh to an Eulerian discretization of the Navier Stokes equations in a way that produces a sparse symmetric positive definite system. The resulting method has semi-implicit and fully-coupled viscosity, pressure, and Lagrangian forces. We apply our new framework for forces on a Lagrangian mesh to the case of a surface tension force, which when treated explicitly leads to a tight time step restriction. By applying surface tension as a semi-implicit Lagrangian force, the resulting method benefits from improved stability and the ability to take larger time steps. The resulting discretization is also able to maintain parasitic currents at low levels.     

Frequency domain response of a parametrically excited riser under random wave forces

Floating Production, Drilling, Storage and Offloading units represent a new technology with a promising future in the offshore oil industry. An important role is played by risers, which are installed between the subsea wellhead and the Tension Leg Deck located in the middle of the moon-pool in the hull. The inevitable heave motion of the floating hull causes a time-varying axial tension in the riser. This time dependent tension may have an undesirable influence on the lateral deflection response of the riser, with random wave forces in the frequency domain. To investigate this effect, a riser is modeled as a Bernoulli–Euler beam. The axial tension is expressed as a static part, along with a harmonic dynamic part. By linearizing the wave drag force, the riser's lateral deflection is obtained through a partial differential equation containing a time-dependent coefficient. Applying the Galerkin method, the equation is reduced to an ordinary differential equation that can be solved using the pseudo-excitation method in the frequency domain. Moreover, the Floquet–Liapunov theorem is used to estimate the stability of the vibration system in the space of parametric excitation. Finally, stability charts are obtained for some numerical examples, the correctness of the proposed method is verified by comparing with Monte-Carlo simulation and the influence of the parametric excitation on the frequency domain responses of the riser is discussed.     

COUPLED AXIAL-LATERAL-TORSIONAL DYNAMICS OF A ROTOR-BEARING SYSTEM GEARED BY SPUR BEVEL GEARS

The coupled lateral-torsional dynamics of parallel rotor-bearing systems has been intensively investigated. However, little attention has been paid to the analysis of coupled vibrations of angled rotor-bearing systems so that the torsional and the lateral vibrations of those systems are usually analyzed separately. In this paper, the coupled axial-lateral-torsional dynamics of a rotor-bearing system geared by bevel gears is studied. The meshing of two spur bevel gears is analyzed on the basis of a pair of virtual cylindrical gears, and thereafter the constraint condition describing the relationship between the generalized displacements of bevel gears is derived under some assumptions. The coupled dynamic model is established by using Lagrange's equation under this constraint condition. The numerical results of a number of case studies show that the critical speeds of the coupled model are different from those of the uncoupled model both in values and modes, and the threshold speed of stability is fairly less than that of the uncoupled model. The effects of system parameters, such as the pitch cone angles, on the coupling behavior are also discussed.     

Coupled longitudinal–transverse dynamics of an axially accelerating beam

The coupled longitudinal–transverse nonlinear dynamics of an axially accelerating beam is numerically investigated; this problem is classified as a parametrically excited gyroscopic system. The axial speed is assumed to be comprised of a constant mean value along with harmonic fluctuations. Hamilton’s principle is employed to derive the equations of motion of the system which are in the form of two coupled partial differential equations. The equations are discretized using the Galerkin method, which yields a set of coupled second-order nonlinear ordinary differential equations with time-dependent coefficients. The sub-critical dynamics of the system is examined via the pseudo-arclength continuation technique, while the global dynamics is investigated using direct time integration. The mean axial speed and the amplitude of the speed variations are varied so as to construct the bifurcation diagrams of Poincaré maps. The vibration specifications of the system are investigated more detailed via plotting time histories, phase-plane portraits, and fast Fourier transforms (FFTs).     

The Influence of Inlet Disturbances on the Post-Stall Behaviors in Compression System

The object of this paper is to provide a reliable tool to carry out the parametrical studies of post-stall behaviors in multistage axial compression systems. An adapted version of the 1.5D Euler equations with additional source terms is discretized with a finite volume method and are solved in time by a fourth-order Runge–Kutta scheme. The equations are discretized at mid-span both inside the blade rows and the non-bladed regions. The source terms express the blade-flow interactions and are estimated by calculating the velocity triangles for each blade row. Additional source terms are introduced to represent the effects of inlet disturbances on post-stall behaviors and the physical analysis is therefore proposed to explain the phenomenon.     

The quantum transverse spin-2 Ising model with a bimodal random-field in the pair approximation

In this paper, we have investigated the bimodal random-field spin-2 Ising system in a transverse field by combining the pair approximation with the discretized path-integral representation. The exact equations for the second-order phase transition lines and tricritical points are obtained in terms of the random field H, the transverse field G and the coordination number z. It is found that there are some critical values for H and G where the tricritical points disappear for given z. We have also observed that the system presents reentrant behavior which may be caused by the quantum effects and randomness. The phase diagram with respect to the random field and the second-order phase transition temperature are studied extensively for given values of the transverse field and the coordination number.     

A sensitivity analysis of the effects of interconnection joint size,flexibility, and inertia on the natural frequencies of Timoshenko frames

The free vibrations of frame structures are influenced by the geometry, stiffness, and inertia of interconnection joints. The effects of generalized joint properties on the natural frequencies and mode shapes are studied for a wide range of natural frequencies by modeling the structure as a Timoshenko continuous system with discretized joints. Dynamic slope-deflection equations are used in the analysis, adapted to the boundary conditions imposed by joints with axial length, axial and rotary stiffness, and inertia. Beam/column axial deformation is also included. Frequency curves are presented for a wide range of beam/column and joint properties to establish the relative importance of model parameters on system free vibrations.