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.     

Resonances of an in-extensional asymmetrical spinning shaft with speed fluctuations

In this study, main and parametric resonances of an asymmetrical spinning shaft with in-extensional nonlinearity and large amplitude are simultaneously investigated. The main resonance is due to inhomogeneous part of the equations of motion, which is due to dynamic imbalances of shaft whereas the parametric resonances are due to parametric excitations due to speed fluctuations and a shaft asymmetry. The shaft is simply supported with unequal mass moments of inertia and flexural rigidities in the direction of principal axes. The equations of motion are derived by the extended Hamilton principle. The stability and bifurcations are obtained by multiple scales method, which is applied to both partial and ordinary differential equations of motion. The influences of asymmetry of shaft, speed fluctuations, inequality between two eccentricities corresponding to the principal axes and external damping on the stability and bifurcation are studied. To investigate the effect of speed fluctuations on the bifurcations and stability the loci of bifurcation points are plotted as function of damping coefficient. The numerical solutions are used to verify the results of multiple scales method. The results of multiple scales method show a good agreement with those of numerical solutions.     

Transverse vibrations of rotating shafts: Probability density and first-passage time of whirl radius

Transverse vibrations are considered for a single mass/two-degrees-of-freedom rotating shaft with linear internal or “rotating” damping and nonlinear external damping. The shaft is excited by external random forces. Analysis of resulting random vibrations is based on stochastic averaging method which yields separated (in the linear approximation) equations for complex amplitudes of forward and backward whirling motions. The former of these motions is shown to be dominant at rotation speeds in the vicinity of the instability threshold. Using this approximation an analytical solution is obtained for probability density of squared radius of the shaft's whirl. This solution can be used to detect on-line shaft's instability from its observed response. Solution is also obtained for expected time for reaching given level by the squared whirl radius of the shaft.     

Dynamic stability and bifurcation of a nonlinear in-extensional rotating shaft with internal damping

In this paper, stability and bifurcations in a simply supported rotating shaft are studied. The shaft is modeled as an in-extensional spinning beam with large amplitude, which includes the effects of nonlinear curvature and inertia. To include the internal damping, it is assumed that the shaft is made of a viscoelastic material. In addition, the torsional stiffness and external damping of the shaft are considered. To find the boundaries of stability, the linearized shaft model is used. The bifurcations considered here are Hopf and double zero eigenvalues. Using center manifold theory and the method of normal form, analytical expressions are obtained, which describe the behavior of the rotating shaft in the neighborhood of the bifurcations.     

Dynamic stability of a viscoelastic rotating shaft under parametric random excitation

This paper deals with dynamic stability of a viscoelastic rotating shaft subjected to a parametric random axial compressive thrust, by using moment Lyapunov exponents and the largest Lyapunov exponents as indicators. The equation of motion for the shaft is derived, which is a system of gyroscopic stochastic differential equations. The method of stochastic averaging is used to decouple the governing equations into Itô equations, from which the moment Lyapunov exponent is obtained by using mathematical transformations only. The largest Lyapunov exponent is obtained through its relation with moment Lyapunov exponents. The effects of various parameters on the stochastic dynamic stability are discussed. The approximate analytical results are confirmed by Monte Carlo simulation.     

Almost sure stochastic stability of a viscoelastic double-beam system

This paper investigates the dynamic stability of a viscoelastic double-beam system under parametric excitations. It is assumed that the two beams, made from Voigt–Kelvin material, are simply supported and continuously joined by a Winkler elastic layer. Each pair of axial forces consists of a constant part and a time-dependent stochastic function. In the case of “non-white” excitations, by using the direct Liapunov method, bounds of the almost sure stability of the double-beam system as a function of retardation time, bending stiffness, stiffness modulus of the Winkler layer, variances of the stochastic forces and the 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 a harmonic process with a random phase. When the excitations are wideband noises, almost sure stability is obtained within the concept of the Liapunov exponent. White noise and Ornstein–Uhlenbeck processes are considered as models of wideband noises.     

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.     

Stability of whirling shafts with internal and external damping

Necessary and sufficient conditions for the stability of motion of whirling shafts are established using the direct method of Liapunov. The non-linear mathematical model employed is based on the work of V. V. Bolotin and includes the effects of both internal and external damping. A coordinate transformation is used to facilitate the analysis. In effect, this transformation establishes a mathematical equivalence between the governing equations for a whirling shaft with both internal and external damping, and the governing equations for a whirling shaft with internal damping only.     

Extension of Lagrangian–Hamiltonian mechanics for continuous systems—investigation of dynamics of a one-dimensional internally damped rotor driven through a dissipative coupling

In this paper, the extended Lagrangian formulation for a one-dimensional continuous system with gyroscopic coupling and non-conservative fields has been developed. Using this formulation, the dynamics of an internally and externally damped rotor driven through a dissipative coupling has been studied. The invariance of the extended or so-called umbra-Lagrangian density is obtained through an extension of Noether’s theorem. The rotor shaft is modeled as a Rayleigh beam. The dynamic behavior of the rotor shaft is obtained and validated through simulation studies. Results show an interesting phenomenon of limiting behavior of the rotor shaft with internal damping beyond certain threshold speeds which are obtained theoretically and affirmed by simulations. It is further observed that there is entrainment of whirling speeds at natural frequencies of the rotor shaft primarily depending on the damping ratio.     

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.     

Dynamics and stability of elastic shaft-bearing systems with nonlinear bearing parameters

The dynamics and stability of a continuously elastic spinning shaft mounted on two dissimilar end bearings possessing nonlinear anisotropic and cross coupling stiffness and damping coefficients are analyzed. Sufficient conditions of system stability in the sense of Liapunov are derived. The developed stability criteria of the considered system have been shown to reduce to that of simpler models found in the pertinent literature. The effects of nonlinearity of bearing stiffness and damping parameters together with shaft stiffness parameter and other system parameters on the dynamic stability of the system are investigated. Several graphs demonstrating parametrically the influence of various system nondimensionalized parameters on system stability boundaries for typical cases are presented. Received on 26 October 1998     

The moment Lyapunov exponent of a Timoshenko beam under bounded noise excitation

The moment Lyapunov exponents and the Lyapunov exponent of a two-dimensional system under bounded noise excitation are studied in this paper. The method regular perturbation is applied to obtain the small noise expansion of the pth moment Lyapunov exponent and the Lyapunov exponent. The results are applied to the study of the almost-sure and moment stability of the stationary solutions of the elastic beam subjected to the stochastic axial load. The boundaries of the almost-sure and moment stability of the elastic beam as the function of the damping coefficient and characteristics of the stochastic force are obtained.     

具有非轴对称刚度转轴的分岔

研究具有非轴对称刚度转轴的１／２亚谐共振和分岔,首先用Ｈａｍｉｌｔｏｎ原理导出运动微分方程,这是刚度系数周期性变化的参数激励方程,然后用多尺度法求得平均方程分岔响应方程和定常解,最后用奇异性理论分析分岔响应方程和定常解的稳定性,得到了局部分岔集和不同区域的不同分岔响应曲线。     

Stationary response of strongly non-linear oscillator with fractional derivative damping under bounded noise excitation

The stochastic averaging method for strongly non-linear oscillators with lightly fractional derivative damping of order α (0<α≤1) subject to bounded noise excitations is proposed by using the generalized harmonic function. The system state is approximated by a two-dimensional time-homogeneous diffusion Markov process of amplitude and phase difference using the proposed stochastic averaging method. The approximate stationary probability density of response is obtained by solving the reduced Fokker–Planck–Kolmogorov (FPK) equation using the finite difference method and successive over relaxation method. A Duffing oscillator is taken as an example to show the application and validity of the method. In the case of primary resonance, the stochastic jump of the Duffing oscillator with fractional derivative damping and its P-bifurcation as the system parameters change are examined for the first time using the stationary probability density of amplitude.     

Nonlinear vibrations of a rotating shaft with broadband random variations of internal damping

A simple Jeffcott rotor is considered with broadband temporal random variations of internal damping which are described using the theory of Markov processes. Transverse response of the rotor with stiffening nonlinearity either in external damping or in restoring force is studied by stochastic averaging method. This method reduces the problems to stochastic differential equations (SDEs) for which analytical solutions are obtained for the Fokker–Planck–Kolmogorov (FPK) equations for stationary probability density functions (PDFs) of the squared whirl radius of the shaft. These PDFs do exist beyond the dynamic instability threshold and they correspond to forward whirl of the rotor. At rotation speeds just slightly above the instability threshold, the response PDF has integrable singularity at zero which corresponds to intermittency in the response.     

Technical Stability of Dynamic States of Controlled Elastic Aircraft

Sufficient conditions are obtained for the technical stability of the controlled longitudinal vertical motion of elongated elastic aircraft. These aircraft are considered to have a variable cross-section and to be subject to significant transverse deformations and vibrations. The technical-stability criteria formulated depend on a small positive parameter. This parameter is a function of key parameters of the controlled dynamic process such as the increment in the transverse load due to the curvature of the system axis and the aerodynamic forces     

An analytical approach to optimally design of electrorheological fluid damper for vehicle suspension system

This work develops an analytical approach to optimally design electrorheological (ER) dampers, especially for vehicle suspension system. The optimal design considers both stability and ride comfort of vehicle application. After describing the schematic configuration and operating principle of the ER damper, a quasi-static model is derived on the basis of Bingham rheological laws of ER fluid. Based on the quasi-static model, the optimization problem for the ER damper is built. The optimization problem is to find optimal value of significant geometric dimensions of the ER damper, such as the ER duct length, ER duct radius, ER duct gap and the piston shaft radius, that maximize damping force of the ER damper. The two constrained conditions for the optimization problem are: the damping ratio of the damper in the absence of the electric field is small enough for ride comfort and the buckling condition of the piston shaft is satisfied. From the proposed optimal design, the optimal solution of the ER damper constrained in a specific volume is obtained. In order to evaluate performance of the optimized ER damper, simulation result of a quarter-car suspension system installed with the optimized ER damper is presented and compared with that of the non-optimized ER damper suspension system. Finally, the optimal results of the ER damper constrained in different volumes are obtained and presented in order to figure out the effect of constrained volume on the optimal design of ER damper.     

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.     

Stability analysis of a nonlinear rotating asymmetrical shaft near the resonances

An asymmetrical rotating shaft with unequal mass moments of inertia and flexural rigidities in the direction of principal axes is considered. In this system, there are two excitation sources, including a harmonic excitation due to the dynamic imbalances and a parametric excitation due to shaft asymmetry. Nonlinearities are due to the in-extensionality of the shaft and large amplitude. In this study, harmonic and parametric resonances due to the mentioned effects are considered. The influences of inequality of mass moments of inertia and flexural rigidities in the direction of principal axes, inequality between two eccentricities corresponding to the principal axes and external damping on the stability and bifurcation of steady state response of the rotating asymmetrical shaft are investigated. In addition, the characteristic of stable stationary points and loci of bifurcation points as function of damping coefficient are determined. In order to analyze the resonances of the system the multiple scales method is applied to the complex form of partial differential equations of motion. The achieved results show a good agreement with those of numerical computation.     

Lyapunov exponent and stochastic stability of quasi-non-integrable Hamiltonian systems

An n degree-of-freedom (DOF) non-integrable Hamiltonian system subject to light damping and weak stochastic excitation is called quasi-non-integrable Hamiltonian system. In the present paper, the stochastic averaging of quasi-non-integrable Hamiltonian systems is briefly reviewed. A new norm in terms of the square root of Hamiltonian is introduced in the definitions of stochastic stability and Lyapunov exponent and the formulas for the Lyapunov exponent are derived from the averaged Itô equations of the Hamiltonian and of the square root of Hamiltonian. It is inferred that the Lyapunov exponent so obtained is the first approximation of the largest Lyapunov exponent of the original quasi-non-integrable Hamiltonian systems and the necessary and sufficient condition for the asymptotic stability with probability one of the trivial solution of the original systems can be obtained approximately by letting the Lyapunov exponent to be negative. This inference is confirmed by comparing the stability conditions obtained from negative Lyapunov exponent and by examining the sample behaviors of averaged Hamiltonian or the square root of averaged Hamiltonian at trivial boundary for two examples. It is also verified by the largest Lyapunov exponent obtained using small noise expansion for the second example.