Non-linear parametric vibration and stability analysis for two dynamic models of axially moving Timoshenko beams

Non-linear parametric vibration and stability of an axially moving Timoshenko beam are considered for two dynamic models; the first one, with considering only the transverse displacement and the second one, with considering both longitudinal and transverse displacements. The set of non-linear partial-differential equations of both models are derived using an energy approach. The method of multiple scales is applied directly to both models, and using the equation order one, the mode shape equations and natural frequencies are obtained. Then, for the equation order epsilon, the solvability conditions are considered for the resonance case and the stability boundaries are formulated analytically via Routh–Hurwitz criterion. Eventually, some numerical examples are provided to show the differences in the behavior of the above-mentioned non-linear models.     

Dynamic response of clamped axially moving beams: Integral transform solution

The generalized integral transform technique (GITT) is employed to obtain a hybrid analytical-numerical solution for dynamic response of clamped axially moving beams. The use of the GITT approach in the analysis of the transverse vibration equation leads to a coupled system of second order differential equations in the dimensionless temporal variable. The resulting transformed ODE system is then solved numerically with automatic global accuracy control by using the subroutine DIVPAG from IMSL Library. Excellent convergence behavior is shown by comparing the vibration displacement of different points along the beam length. Numerical results are presented for different values of axial translation velocity and flexural stiffness. A set of reference results for the transverse vibration displacement of axially moving beam is provided for future co-validation purposes.     

Vibrations and stability of axially traveling laminated beams

In this paper, vibrations and stability of an axially traveling laminated composite beam are investigated analytically via the method of multiple scales. Based on classical laminated beam theory, the governing equations of motion for a time-variant axial speed are obtained using Newton’s second law of motion and constitutive relations. The method of multiple scales, an approximate analytical method, is applied directly to the gyroscopic governing equations of motion and complex eigenfunctions and natural frequencies of the system are obtained. The stability boundaries of the system near resonance are determined via the Routh-Hurwitz criterion. Finally, a parametric study is conducted which considers the effects of laminate type and configuration as well as the mean speed and amplitude of speed fluctuations on the vibration response, natural frequencies and stability boundaries of the system.     

Vibration behavior of simply supported inclined single-walled carbon nanotubes conveying viscous fluids flow using nonlocal Rayleigh beam model

A mathematical model is proposed to investigate the dynamic response of an inclined single-walled carbon nanotube (SWCNT) subjected to a viscous fluid flow. The tangential interaction of the inside fluid flow with the equivalent continuum structure (ECS) of the SWCNT is taken into account via a slip boundary condition. The dimensionless equations of motion describing longitudinal and lateral vibrations of the fluid-conveying ECS are obtained in the context of nonlocal elasticity theory of Eringen. The unknown displacement fields are expressed in terms of admissible mode shapes associated with the ECS under simply supported conditions with immovable ends. Using Galerkin method, the discrete form of the equations of motion is derived. The time history plots of the normalized longitudinal and transverse displacements as well as the nonlocal axial force and bending moment of the midspan point of the SWCNT are provided for different levels of the fluid flow speed, small-scale parameter, and inclination angle of the SWCNT. The effects of small-scale parameter, inclination angle, speed and density of the fluid flow on the maximum dynamic amplitude factors of longitudinal and transverse displacements as well as those of nonlocal axial force and bending moment of the SWCNT are then studied in some detail.     

Vibration and stability analysis of thick orthotropic plates using hybrid-Trefftz elements

This paper discusses the vibration and stability analysis of thick orthotropic plate structures using finite elements based on the hybrid-Trefftz formulation. While the formulation can be used for elements of arbitrary geometry, the paper concentrates on the use of a simple and robust triangular element. The key feature of the formulation is to use element interpolations that are consistent for all values of the plate thickness, including the limit when it goes to zero. This eliminates the locking problem automatically and ensures a robust approximation for thick and thin plates. Results for various problems are included to demonstrate the accuracy and efficiency of the element.     

Modified Rayleigh conjecture method and its applications

The Rayleigh conjecture about convergence up to the boundary of the series representing the scattered field in the exterior of an obstacle D$D$ is widely used by engineers in applications. However this conjecture is false for some obstacles. AGR introduced the Modified Rayleigh Conjecture (MRC), which is an exact mathematical result. In this paper we present the theoretical basis for the MRC method for 2D and 3D obstacle scattering problems, for static problems, and for scattering by periodic structures. We also present successful numerical algorithms based on the MRC for various scattering problems. The MRC method is easy to implement for both simple and complex geometries. It is shown to be a viable alternative for other obstacle scattering methods. Various direct and inverse scattering problems require finding global minima of functions of several variables. The Stability Index Method (SIM) combines stochastic and deterministic method to accomplish such a minimization.     

Non-linearly parametric resonances of an axially moving viscoelastic sandwich beam with time-dependent velocity

Non-linearly parametric resonances of an axially moving viscoelastic sandwich beam are investigated in this paper. The beam is moving with a time-dependent velocity, namely a harmonically varied velocity about the mean velocity. The partial differential equation is discretized into nonlinear ordinary differential equations via the method of Galerkin truncation and then the steady-state response is obtained using the method of multiple scales, an approximate analytical method. The tuning equations are obtained by eliminating secular terms and the amplitude of the vibration is derived from the tuning equations expressed in polar form, and two bifurcation points are obtained as well. Additionally, the stability conditions of trivial and nontrivial solutions are analyzed using the Routh–Hurwitz criterion. Eventually, the effects of various parameters such as the thickness of core layer, mean velocity, initial tension, and the amplitude of axially moving velocity on amplitude–frequency response curves and unstable regions are investigated.     

Basis Property of a Rayleigh Beam with Boundary Stabilization

A Rayleigh beam equation with boundary stabilization control is considered. Using an abstract result on the Riesz basis generation of discrete operators in Hilbert spaces, we show that the closed-loop system is a Riesz spectral system; that is, there is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis in the state Hilbert space. The spectrum-determined growth condition, distribution of eigenvalues, as well as stability of the system are developed. This paper generalizes the results in Ref. 1.     

Vibration and stability of ring-stiffened Euler–Bernoulli tie-bars

Tie-bars are frequently used in structural and mechanical engineering applications. To satisfy requirements like weight reduction, stability improvement, etc., the tie-bars are stiffened with rings. In this paper a method is developed to calculate the natural frequencies, buckling axial force, etc., of the ring-stiffened tie-bars. The dynamics of the ringed and the unringed portions of the beam are treated separately. The mode shape of the first portion was expressed as the superposition of two functions multiplied by constants. Consideration of continuity of deflection and of slope and compatibility of bending moment and shearing force at the first step enabled the mode shape of the second portion to be expressed as the superposition of two functions but multiplied by the same constants as in the first portion. This procedure was recursively carried out up to the last portion. The frequency equation was then derived from the boundary conditions at the end. Buckling of the tie-bar was considered as the case when one of the natural frequencies is zero. The first three frequency parameters and the first two buckling dimensionless axial forces are tabulated for tie-bars stiffened with various number of rings and for various combinations of boundary conditions. The calculation procedure can handle any number and any type of ring-stiffeners.     

Vibration of a Reissner–Mindlin–Timoshenko plate–beam system

In this paper, we consider a plate–beam system in which the Reissner–Mindlin plate model is combined with the Timoshenko beam model. Natural frequencies and vibration modes for the system are calculated using the finite element method. The interface conditions at the contact between the plate and beams are discussed in some detail. The impact of regularity on the enforcement of certain interface conditions is an important feature of the paper.     

Dynamic stability of a curved rail under a moving load

The stability of a curved rail under a constant moving load has been investigated using a linear theory; critical speeds of the moving load, and the dynamic rail deflections and rotation were calculated. The effect of the foundation was included through distributed linear springs. It was assumed that the moving load remains in constant contact with the rail and travels along a fixed path on the rail head.     

Transverse vibration of viscoelastic sandwich beam with time-dependent axial tension and axially varying moving velocity

Nonlinearly parametric resonances of axially accelerating moving viscoelastic sandwich beams with time-dependent tension are investigated in this paper. Based on the Kelvin differential constitutive equation, the controlling equation of the transverse vibration of a beam with large deflection is established. The system has been subjected to a time varying velocity and a harmonic axial tension. Here the governing equation of motion contains linear parametric terms and two frequencies, one is the frequency of axially moving velocity and the other one is the frequency of varying tension. The method of multiple scales is applied directly to the governing equation to obtain the complex eigenfunctions and natural frequencies of the system. The elimination of secular terms leads to the steady-state response and amplitude of vibrations. The influence of various parameters such as initial tension on natural frequencies and the amplitude of axial fluctuation, the phase angle between the two frequencies on response curves has been investigated for two different resonance conditions. With the help of numerical results, it has been shown that by using suitable initial tension, the amplitude of axial fluctuation, the phase angle, the vibration of the sandwich beam can be significantly controlled.     

Transverse free vibration and stability of axially moving nanoplates based on nonlocal elasticity theory

Transverse dynamical behaviors of axially moving nanoplates which could be used to model the graphene nanosheets or other plate-like nanostructures with axial motion are examined based on the nonlocal elasticity theory. The Hamilton's principle is employed to derive the multivariable coupling partial differential equations governing the transverse motion of the axially moving nanoplates. Subsequently, the equations are transformed into a set of ordinary differential equations by the method of separation of variables. The effects of dimensionless small-scale parameter, axial speed and boundary conditions on the natural frequencies in sub-critical region are discussed by the method of complex mode. Then the Galerkin method is employed to analyze the effects of small-scale parameter on divergent instability and coupled-mode flutter in super-critical region. It is shown that the existence of small-scale parameter contributes to strengthen the stability in the super-critical region, but the stability of the sub-critical region is weakened. The regions of divergent instability and coupled-mode flutter decrease even disappear with an increase in the small-scale parameter. The natural frequencies in sub-critical region show different tendencies with different boundary effects, while the natural frequencies in super-critical region keep constants with the increase of axial speed.     

Modelling and bifurcation analysis of internal cantilever beam system on a steadily rotating ring

Using Hamilton variation principle, a nonlinear dynamic model of the system with a finite deforming Rayleigh beam clamped radially to the interior of a rotating rigid ring, under the assumption that the constitutive relation of the beam is linearly elastic, is discussed. The bifurcation behavior of the simple system with the Euler-Bernoulli beam is also discussed. It is revealed that these two models have no influence on the critical bifurcation value and buckling solution in the steady state. Then we use the assumption model method to analyse the bifurcation behavior of the steadily rotating Euler-Bernoulli beam and get two different types of bifurcation behavior which physically exist. Finite element method and shooting method are used to verify the analytical results. The numerical results confirm our research conclusion. Project supported by the National Natural Science Foundation of China (Grant No. 19332022) and Space High Technology Foundation of China.     

Global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles

In this paper, we establish existence of global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles by combining variational method, various elliptic estimates and a compensated compactness method. More precisely, it is shown that there exist global subsonic flows in nozzles for incoming mass flux less than a critical value; moreover, uniformly subsonic flows always approach to uniform flows at far fields when nozzle boundaries tend to be flat at far fields, and flow angles for axially symmetric flows are uniformly bounded away from π/2; finally, when the incoming mass flux tends to the critical value, subsonic-sonic flows exist globally in nozzles in the weak sense by using angle estimate in conjunction with a compensated compactness framework.     

中厚悬臂板的固有振动和稳定

本文由设定两个位移函数，应用最小二乘法和能量法，得到中厚悬臂矩形板固有振动和稳定的Ｒｅｉｓｓｎｅｒ近似解。     

Linearized stability of traveling cell solutions arising from a moving boundary problem

In 2003, Mogilner and Verzi proposed a one-dimensional model on the crawling movement of a nematode sperm cell. Under certain conditions, the model can be reduced to a moving boundary problem for a single equation involving the length density of the bundled filaments inside the cell. It follows from the results of Choi, Lee and Lui (2004) that this simpler model possesses traveling cell solutions. In this paper, we show that the spectrum of the linear operator, obtained from linearizing the evolution equation about the traveling cell solution, consists only of eigenvalues and there exists such that if is a real eigenvalue, then . We also provide strong numerical evidence that this operator has no complex eigenvalue.

     

Loop-Erased Random Walk on Finite Graphs and the Rayleigh Process

Let (G _n ) _n=1^∞ be a sequence of finite graphs, and let Y _t be the length of a loop-erased random walk on G _n after t steps. We show that for a large family of sequences of finite graphs, which includes the case in which G _n is the d-dimensional torus of size-length n for d≥4, the process (Y _t ) _t=0^∞, suitably normalized, converges to the Rayleigh process introduced by Evans, Pitman, and Winter. Our proof relies heavily on ideas of Peres and Revelle, who used loop-erased random walks to show that the uniform spanning tree on large finite graphs converges to the Brownian continuum random tree of Aldous. Supported in part by NSF Grant DMS-0504882.     

Vibration of carbon nanotube reinforced composite beams based on the first and third order beam theories

This paper investigates the linear free vibration of nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs). Two types of CNT reinforced beams, namely uniformly distributed CNT reinforced (UD-CNT) beams and functionally graded CNT reinforced (FG-CNT) beams, are considered. It is assumed that the SWCNTs are aligned along the beam axial direction and the distribution of the SWCNTs may vary through the thickness of the beam. The virtual strain and kinetic energies of the FG-CNT composite beam are obtained using the classic variational method of Hamilton’s principle and then solved by the p-Ritz method. Vibration frequency parameters for the FG-CNT beams based on the first order and third order beam theories are presented and the effects of CNT filler volume fraction, distribution, beam span to depth ratio and end support conditions on the free vibration characteristics of the beams are discussed. Comparison studies for UD-CNT and FG-CNT beams based on the first order and the third order beam theories are also performed and the differences in vibration frequencies between these two theories are highlighted.     

Uniform stability for the Timoshenko beam with tip load

In this paper we consider a hybrid elastic model consisting of a Timoshenko beam and a tip load at the free end of the beam. We show that uniform stabilization of the model which includes the rotary inertia of the tip load can be obtained when feedback boundary moment and force controls are applied at the point of contact between the beam and the tip load. However, in the presence of the load stabilization is “slower” and subject to a restriction on the boundary data at the free end of the beam.