Influence of rotatory inertia on dynamic stability of the viscoelastic symmetric cross-ply laminated plates

The dynamic stability problem of the symmetrically laminated cross-ply plates made of the viscoelastic Voigt–Kelvin material, compressed by time-dependent stochastic membrane forces, is investigated. The effect of rotatory inertia is included in the present formulation. It is assumed that all elastic moduli have the same retardation times. By using the direct Liapunov method, bounds of the almost sure stability of cross-ply plates as a function of viscous damping coefficient, retardation time, variances of the stochastic forces, ratio of the principal lamina stiffnesses, number of layers, plate aspect ratio, cross-ply ratio and intensity of the deterministic components of axial loading are obtained. Numerical calculations are performed for the Gaussian process with a zero mean, as well as an harmonic process with random phase.     

Stochastic stability of a rotating shaft

The stochastic stability problem of an elastic, balanced rotating shaft subjected to action of axial forces at the ends is studied. The shaft is of circular cross-section, it rotates at a constant rate about its longitudinal axis of symmetry. The effect of rotatory inertia of the shaft cross-section is included in the present formulation. Each force consists of a constant part and a time-dependent stochastic function. Closed form analytical solutions are obtained for simply supported boundary conditions. By using the direct Liapunov method almost sure asymptotic stability conditions are obtained as the function of stochastic process variance, damping coefficient, damping ratio, angular velocity, mode number and geometric and physical parameters of the shaft. Numerical calculations are performed for the Gaussian process with a zero mean and as well as an harmonic process with random phase.     

Influence of rotatory inertia and transverse shear on stochastic instability of the cross-ply laminated beam

The stochastic instability problem associated with an axially loaded cross-ply laminated beam is formulated. The effects of shear deformation and rotatory inertia are included in the present formulations. The beam is subjected to time-dependent deterministic and stochastic forces. By using the direct Liapunov method, bounds for the almost sure instability of beams as a function of viscous damping coefficient, variance of the stochastic force, ratio of principal lamina stiffnesses, shear correction factor, number of layers, mode numbers and geometrical ratio, are obtained. Numerical calculations are performed for the Gaussian process with a zero mean and variance σ² as well as for harmonic process with an amplitude A.     

Stabilization of beam parametric vibrations with shear deformations and rotary inertia effects

The purpose of this theoretical work is to present a stabilization problem of beam with shear deformations and rotary inertia effects. A velocity feedback and particular polarization profiles of piezoelectric sensors and actuators are introduced. The structure is described by partial differential equations with time-dependent coefficient including transverse and rotary inertia terms, general deformation state with interlaminar shear strains. The first order deformation theory is utilized to investigate beam vibrations. The beam motion is described by the transverse displacement and the slope. The almost sure stochastic stability criteria of the beam equilibrium are derived using the Liapunov direct method. If the axial force is described by the stationary and continuous with probability one process the classic differentiation rule can be applied to calculate the time-derivative of functional. The particular problem of beam stabilization due to the Gaussian and harmonic forces is analyzed in details. The influence of the shear deformations, rotary inertia effects and the gain factors on dynamic stability regions is shown.     

Influence of Transverse Shear on Stochastic Instability of the Elastic Beam

In the case when Kirchhoff–Love hypotheses do not give satisfactory results, we have to take the rotatory inertia and transverse shear into account. This paper studies the elastic beam subjected to stochastic axial load, when transverse shear is taken into account. By using the direct Liapunov method, the bounds of the almost sure instability of beams as the function of the damping coefficient, variance of the stochastic force, geometric parameters, mode number, section shape factor and intensity of the deterministic component of axial loading, are obtained. Calculations are performed for the Gaussian process with zero mean and variance ², as well as for the harmonic process with amplitude A.     

Moment Lyapunov exponents of the stochastic parametrical Hill’s equation

The Lyapunov exponent and moment Lyapunov exponents of Hill’s equation with frequency and damping coefficient fluctuated by white noise stochastic process are investigated. A perturbation approach is used to obtain explicit expressions for these exponents in the presence of small intensity noises. The results are applied to the study of the almost-sure and the moment stability of the stationary solutions of the thin simply supported beam subjected to axial compressions and time-varying damping which are small intensity stochastic excitations.     

二阶线性随机微分方程的渐近稳定性

对刚度系数是遍历过程的二阶线性随机微分方程，本文研究了其平凡解几乎处处渐近稳定性问题。利用刚度系数导数过程的性质，给出了平凡解几乎处处渐近稳定的充分条件。当刚度系数是遍历高斯过程或周期过程时，还具体计算了其渐进稳定区域。结果表明，本文结果改进了目前有关的渐近稳定性的条件。     

Stabilization of Parametric Vibrations of a Nonlinear Continuous System

The purpose of the present paper is to solve an active control problem of nonlinear continuous system parametric vibrations excited by the fluctuating force. The problem is solved using the concept of distributed piezoelectric sensors and actuators with a sufficiently large value of velocity feedback. The direct Liapunov method is proposed to establish criteria for the almost sure stochastic stability of the unperturbed (trivial) solution of the shell with closed-loop control. The distributed control is realized by the piezoelectric sensor and actuator, with the changing widths, glued to the upper and lower shell surface. The relation between the stabilization of nonlinear problem and a linearized one is examined. The fluctuating axial force is modeled by the physically realizable ergodic process. The rate velocity feedback is applied to stabilize the shell parametric vibrations.     

Stability of elastic systems under a stochastic parametric excitation

An efficient method to investigate the stability of elastic systems subjected to the parametric force in the form of a random stationary colored noise is suggested. The method is based on the simulation of stochastic processes, numerical solution of differential equations, describing the perturbed motion of the system, and the calculation of top Liapunov exponents. The method results in the estimation of the almost sure stability and the stability with respect to statistical moments of different orders. Since the closed system of equations for moments of desired quantities y _j (t) cannot be obtained, the statistical data processing is applied. The estimation of moments at the instant t _n is obtained by statistical average of derived from the solution of equations for the large number of realizations. This approach allows us to evaluate the influence of different characteristics of random stationary loads on top Liapunov exponents and on the stability of system. The important point is that results found for filtered processes, are principally different from those corresponding to stochastic processes in the form of Gaussian white noises.     

Almost sure stability condition of weakly coupled linear nonautonomous random systems

In this study, the sufficient condition of almost sure stability of twodimensional oscillating systems under parametric excitations is investigated. The systems considered are assumed to be composed of two weakly coupled subsystems. The driving actions are considered to be stationary stochastic processes satisfying ergodic properties. The properties of quadratic forms are used in conjunction with the bounds for the eigenvalues to obtain, in a closed form, the sufficient condition for the almost sure stability of the systems.     

Stochastic stability of quasi-integrable and non-resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises

A procedure for calculating the largest Lyapunov exponent and determining the asymptotic Lyapunov stability with probability one of multi-degree-of-freedom (MDOF) quasi-integrable and non-resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises is proposed. The averaged stochastic differential equations (SDEs) of quasi-integrable and non-resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises are first derived by using the stochastic averaging method for quasi-Hamiltonian systems and the stochastic jump-diffusion chain rule. Then, the expression for the largest Lyapunov exponent is obtained by generalizing Khasminskii's procedure to the averaged SDEs and the stochastic stability of the original systems is determined approximately. An example is given to illustrate the application of the proposed procedure and its effectiveness is verified by comparing with the results from Monte Carlo simulation.     

Stochastic stability of quasi-partially integrable and non-resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises

The asymptotic Lyapunov stability with probability one of multi-degree-of freedom quasi-partially integrable and non-resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises is studied. First, the averaged stochastic differential equations for quasi partially integrable and non-resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises are derived by means of the stochastic averaging method and the stochastic jump-diffusion chain rule. Then, the expression of the largest Lyapunov exponent of the averaged system is obtained by using a procedure similar to that due to Khasminskii and the properties of stochastic integro-differential equations. Finally, the stochastic stability of the original quasi-partially integrable and non-resonant Hamiltonian systems is determined approximately by using the largest Lyapunov exponent. An example is worked out in detail to illustrate the application of the proposed method. The good agreement between the analytical results and those from digital simulation show that the proposed method is effective.     

Effect of rotary inertia and shear on vibration and buckling of a double beam system under compressive axial loading

Free transverse vibration and buckling of a double-beam continuously joined by a Winkler elastic layer under compressive axial loading with the influence of rotary inertia and shear are considered in this paper. The motion of the system is described by a homogeneous set of two partial differential equations, which is solved by using the classical Bernoulli?CFourier method. The boundary value and initial value problems are solved. The natural frequencies and associated amplitude ratios of an elastically connected double-beam complex system and the analytical solution of the critical buckling load are determined. The presented theoretical analysis is illustrated by a numerical example, in which the effect of physical parameters characterizing the vibrating system on the natural frequency, the associated amplitude ratios and the critical buckling load are discussed.     

Influence of the mode number on the stochastic stability regions of the elastic beam

The moment Lyapunov exponents and Lyapunov exponent of a two-dimensional system under stochastic parametric excitation are studied. A perturbation approach is used to obtain explicit expressions for these exponents in the presence of small intensity noises. Approximate analytical results for the pth moment Lyapunov exponents are compared with the numerical values obtained by the Monte Carlo simulation approach. The results are applied to the study of the almost-sure and the moment stability of the stationary solutions of the elastic beam subjected to the stochastic axial load. The boundaries of the almost-sure and the moment stability of the elastic beam as the function of the damping coefficient, spectral density of the stochastic force and mode number are obtained.     

A new approach to almost-sure asymptotic stability of stochastic systems of higher dimension

The almost sure asymptotic stability of higher-dimensional linear stochastic systems and of a special class of nonlinear stochastic systems with homogeneous drift and diffusion coefficients of order one is studied. Based on the well-known Khasminskii's theorem, a new approach for obtaining the regions of almost sure asymptotic stability and instability without evaluating the stationary probability density of the diffusion process defined on unit hypersphere is proposed. Two examples of two and three degree-of-freedom linear stochastic systems are given to illustrate the application and effectiveness of the proposed approach combined with stochastic averaging.     

Almost sure cluster synchronization of Markovian switching complex networks with stochastic noise via decentralized adaptive pinning control

This paper investigates the issue of almost sure cluster synchronization in nonlinearly coupled complex networks with nonidentical nodes and time-varying delay. These networks are modulated by a continuous-time Markov chain and disturbed by a Brownian movement. The decentralized adaptive update law and pinning control protocol are employed in designing controllers for guaranteeing almost sure cluster synchronization. By constructing a novel stochastic Lyapunov–Krasovskii function and using the stochastic Lasalle-type invariance theorem, some sufficient conditions for almost sure cluster synchronization of the networks are derived. Finally, a numerical example is given to testify the effectiveness of the theoretical results.     

Dynamic stability of weak equations of continuous systems

The stability analysis method is developed for distributed dynamic problems with relaxed assumptions imposed on solutions. The problem is motivated by structural vibrations with external time-dependent parametric excitations which are controlled using surface-mounted or -embedded actuators and sensors. The strong form of equations involves irregularities which lead to computational difficulties for estimation and control problems. In order to avoid irregular terms resulting from differentiation of the force and moment terms the dynamics equations are written in a weak form. The weak form of dynamics equations of linear mechanical structures is obtained using variational calculus. The study of stability of stochastic weak system is based on examining properties of Liapunov functional along a weak solution.     

二自由度耦合线性随机系统的最大Lyapunov指数和稳定性

利用摄动方法和Fokker-Planck算子及其伴随算子的特征函数展开法,讨论了两个模态都处于临介状态的耦合二自由度振动系统,在小强度的非高斯噪声参数激励下系统运动的稳定性,获得了系统扩散过程的稳态概率密度的渐近表达式,建立了系统最大Lyapunov指数的渐近表达式,由此获得了系统运动模态几乎必然稳定的充分必要条件。     

Stochastic averaging of quasi-integrable and non-resonant Hamiltonian systems under combined Gaussian and Poisson white noise excitations

A stochastic averaging method for predicting the response of quasi-integrable and non-resonant Hamiltonian systems to combined Gaussian and Poisson white noise excitations is proposed. First, the motion equations of a quasi-integrable and non-resonant Hamiltonian system subject to combined Gaussian and Poisson white noise excitations is transformed into stochastic integro-differential equations (SIDEs). Then $n$ -dimensional averaged SIDEs and generalized Fokker–Plank–Kolmogrov (GFPK) equations for the transition probability densities of $n$ action variables and $n$ - independent integrals of motion are derived by using stochastic jump–diffusion chain rule and stochastic averaging principle. The probability density of the stationary response is obtained by solving the averaged GFPK equation using the perturbation method. Finally, as an example, two coupled non-linear damping oscillators under both external and parametric excitations of combined Gaussian and Poisson white noises are worked out in detail to illustrate the application and validity of the proposed stochastic averaging method.