Analysis of nonlinear vibrations of a two-degree-of-freedom mechanical system with damping modelled by a fractional derivative

Free damped vibrations of a mechanical two-degree-of-freedom system are considered under the conditions of one-to-one or two-to-one internal resonance, i.e., when natural frequencies of two modes – a mode of vertical vibrations and a mode of pendulum vibrations – are approximately equal to each other or when one natural frequency is nearly twice as large as another natural frequency. Damping features of the system are defined by the fractional derivatives with fractional parameters (the orders of the fractional derivatives) changing from zero to one. It is assumed that the amplitudes of vibrations are small but finite values, and the method of multiple scales is used as a method of solution. The model put forward allows one to obtain the damping coefficient dependent on the natural frequency of vibrations, so it has been shown that the amplitudes of vertical and pendulum vibrations attenuate by an exponential law with damping ratios which are exponential functions of the natural frequencies. Damped soliton-like solutions have been found analytically.     

多自由度裂纹转子系统非线性动力学特性分析

大型旋转机械转轴裂纹具有复杂的高维非线性特征, 由此产生的机械故障时有发生。本文根据某航空发动机低压转子系统,利用拉格朗日原理建立了具有26个自由度的含有裂纹故障的高维非线性动力学模型。引进改进的POD方法成功将该系统降为具有两个自由度含有特征的低维非线性系统。数值模拟结果显示降维系统具有与原系统一致的非线性动力学特征,表明本方法对解决高维非线性问题具有较好的有效性。此外,本文还利用C-L方法对其进行分岔分析,讨论了系统参数与系统动态行为之间的关系,得到了裂纹转子各种不同分岔模式,得到了裂纹二分之一亚谐共振条件下的非线性动力学特性, 准确反映了裂纹转子的动力学特征。     

Numerical study on deformations in a cracked viscoelastic body with the extended finite element method

Deformations such as crack opening and sliding displacements in a cracked viscoelastic body are numerically investigated by the extended finite element method (XFEM). The solution is carried out directly in time domain with a mesh not conforming to the crack geometry. The generalized Heaviside function is used to reflect the displacement discontinuity across a crack surface while the basis functions extracted from the viscoelastic asymptotic fields are used to manifest the gradient singularity at a crack tip. With these treatments, the XFEM formulations are derived in an incremental form. In evaluating the stiffness matrix, a selective integration scheme is suggested for problems with high Poisson ratios often encountered in viscoelastic materials over different element types in the XFEM. Numerical examples show that the crack opening displacement and crack sliding displacement are satisfactory.     

Elastic moduli of a cracked body

Statistical solutions for the internal stress and displacement fields are constructed for a body in an external applied stress field, and containing a prescribed distribution of penny shaped cracks. From the fields, a stress-strain relationship is calculated for the body, and effective elastic moduli are derived. The fields are constructed as expansions in the parameter \(\overline {Na^3 }\) wich is assumed small (N is the crack density, anda is the crack radius). Angular integrals over the distribution function are performed for two special cases that represent a bedded crack distribution, and an isotropic crack distribution. A criterion for the validity of the use of these quasistatic results in dynamical calculations is presented.     

Numerical analysis of strongly nonlinear extensional vibrations in elastic rods

In the framework of transduction, nondestructive testing, and nonlinear acoustic characterization, this article presents the analysis of strongly nonlinear vibrations by means of an original numerical algorithm. In acoustic and transducer applications in extreme working conditions, such as the ones induced by the generation of high-power ultrasound, the analysis of nonlinear ultrasonic vibrations is fundamental. Also, the excitation and analysis of nonlinear vibrations is an emergent technique in nonlinear characterization for damage detection. A third-order evolution equation is derived and numerically solved for extensional waves in isotropic dissipative media. A nine-constant theory of elasticity for isotropic solids is constructed, and the nonlinearity parameters corresponding to extensional waves are proposed. The nonlinear differential equation is solved by using a new numerical algorithm working in the time domain. The finite-difference numerical method proposed is implicit and only requires the solution of a linear set of equations at each time step. The model allows the analysis of strongly nonlinear, one-dimensional vibrations and can be used for prediction as well as characterization. Vibration waveforms are calculated at different points, and results are compared for different excitation levels and boundary conditions. Amplitude distributions along the rod axis for every harmonic component also are evaluated. Special attention is given to the study of high-amplitude damping of vibrations by means of several simulations. Simulations are performed for amplitudes ranging from linear to nonlinear and weak shock.     

On the steady-state transverse vibrations of a cracked spherical shell

Using an integral formulation, the problem of a spherical shell containing a through crack of length 2c and subjected to periodic transverse vibrations of frequency is solved for the in-plane and Kirchhoff bending stresses. The usual inverse square root singular behavior characteristic to crack problems is recovered. Furthermore, it is found that the transverse vibrations reduce the stresses in the vicinity of the crack tip, except when the forcing frequency reaches the natural frequency of the uncracked shell in which case they become infinite.

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Zusammenfassung Das Problem einer sphärischen Hülle, mit einem sich über die gesamte Dicke der Hülle hinziehenden Riß der Länge 2c, welche Querschwingungen mit einer Pulsierung unterworfen ist, wurde für die Fälle von koplanaren and von Kirchhoff-Biegebeanspruchungen mit Hilfe einer Integralformulierung gelöst. Hierbei ergab sich wiederum das fur Riß-probleme charakteristische Gesetz der umgekehrten Quadratwurzel.Außerdem zeigte sich, das Querschwingungen die Spannungen in der Umgebung der Rißspitze vermindern, ausgenommen der Fall, wo die Frequenz der aufgezwungenen Schwingung mit der Eigenfrequenz der unbeschädigten Hülle übereinstimmt, wo sie dann ins Unendliche ansteigen.

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Résumé On analyse le problème de l'enveloppe sphérique ayant une fissure de longeur 2c de part en part de son épaisseur, et sujette à des vibrations transversales de pulsation ; on résoud ré problème à l'aide de fonctions intégrales, pour les contraintes coplanaires et les contraintes de flexion de Kirchhoff.On retrouve le comportement singulier habituel d'ordre 1/2, caractéristique des problèmes de fissuration. En outre, on trouve que des vibrations transversales ont tendance à réduire les contraintes au voisinage de l'extrémitée des fissures, sous réserve que leur fréquence atteigne la fréquence naturelle de l'enveloppe non fissurée; dans ces conditions les contraintes deviennent en effet infinies.

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List of Symbols and Notations c half crack length - D Eh ³/[12(1 – v ²)] = flexural rigidity - E Young's modulus of elasticity - F (x, y, t), (x,y) stress functions - ^(c)(x,y) complementary stress function - G shear modulus - h thickness - K _n modified Bessel function of the third kind of order n - L _i kernels as defined in text - m ₀ constant as defined in text - M _x ^(c) , M _y ^(c) , M _xy ^(c) complementary bending forces - M _x ^(p) , M _y ^(p) , M _xy ^(p) particular bending forces - n ₀ constant as defined in text - N _x ^(c) , N _y ^(c) , N _xy ^(c) complementary membrane forces - N _x ^(p) , N _y ^(p) , N _xy ^(p) Particular membrane forces - N _n Newman function of order n - q(x, y, t), q(x, y) internal pressure - r - R radius of curvature of the shell - R - t time variables - U(s–), U(–s) the unit step function - V _y equivalent shear - W(x, y, t), (x, y, t) displacement functions - ^(c)(x,y) complementary displacement function - dimensionless rectangular coordinates - X, Y, Z rectangular cartesian coordinates     

Numerical investigation of the stability of forced nonlinear vibrations of a shallow cylindrical shell

Kiev Civil-Engineering Institute. Translated from Problemy Prochnosti, No. 4, pp. 74–78, April, 1989.     

Free transverse vibrations of cracked nanobeams with surface effects

The flexural vibrations of cracked micro- and nanobeams in the presence of surface effects are studied. The cracked-beam model is set up by dividing the classical cracked beam element into two segments connected by a rotational spring located at the cracked section. This model promotes a discontinuity in bending slope, which is proportional to the second derivative of the displacements. Numerical examples demonstrate the effects of beam length, and crack position and severity on the calculated values of natural frequencies of an anodic alumina nanowire in the presence of surface effects. Limiting cases are considered and good agreements with the data available in the literature are obtained.     

Monitoring mechanical vibrations using a histogram recorder

The problem of fatigue failures originating from random vibrations in machine tools and process machinery calls for new methods of monitoring these machines before demands of higher productivity and increased performance can be met. A laboratory model of a simple rotor system has been built to study different ways in which ‘normal’ and ‘abnormal’ conditions of random vibrations may be established. A Datamyte histogram recorder was used to collect, digitize and store the results from the rotor when subject to different degrees of imbalance. It was found that the real-time rainflow and time-at-level and level crossing programs could be used to indicate abnormal vibrations arising from a 0.1% change in rotor mass imbalance.     

Shakedown of a cracked body consisting of kinematic hardening material

In this paper, the shakedown behaviour of a cracked body is studied. The main idea is to consider the crack as a notch. Then no singular stresses appear at crack tip. Due to the local character of the problem, Melan's shakedown theorem is used. By solving shakedown as an optimization problem, the limited stress intensity factor (SIF) for shakedown K_sh is obtained. It is found that the shakedown limit SIF of a cracked body is proportional to the initial yield stress σ_y of the material times the square root of the effective crack tip radius π, i.e. K_sh ∝ σ_y√ϱ. Comparison of shakedown limit SIFs with fatigue thresholds for certain materials, so far as can be found in literature, shows that these two quantities agree well with each other. This agreement indicates that shakedown of the cracked body is one of the reasons for arrest of the crack under cyclic loads. Shakedown investigation is then a new method for predicting the fatigue threshold of a cracked body. Thus, a transition from shakedown to cyclic fracture mechanics has been achieved.     

Numerical solution of a parabolic system with coupled nonlinear boundary conditions

A numerical technique has been developed to solve a system that consists of m linear parabolic differential equations with coupled nonlinear boundary conditions. Such a system may represent chemical reactions, chemical lasers and diffusion problems. An implicit finite difference scheme is adopted to discretize the problem, and the resulting system of equations is solved by a novel technique that is a modification of the cyclic odd–even reduction and factorization (CORF) algorithm. At each time level, the system of equations is first reduced to m nonlinear algebraic equations that involve only the m unknown grid points on the nonlinear boundary. Newton's method is used to determine these m unknowns, and the corresponding Jacobian matrix can be computed and updated easily. After convergence is achieved, the remaining unknowns are solved directly. The efficiency of this technique is illustrated by the numerical computations of two examples previously solved by the cubic spline Galerkin method.     

Stability of steady-state vibrations of a nonlinear,imperfectly elastic two-mass system subjected to harmonic excitation

A study is made of the stability of steady-state vibrations of harmonically excited two-mass systems with allowance for the imperfect elasticity of the materials and cubic nonlinearities. It is shown that, in the general case, when imperfect elasticity is accounted for by the model constructed by G. S. Pisarenko, an increase in the inflection of the skeleton curves with an increase in the amplitude of the vibrations may lead to the appearance of regions of instability and amplitude jumps. The averaging method is used to determine the conditions under which the amplitudes will be stable. Resonance curves of amplitude are constructed for a two-mass system modeling an imperfectly elastic dynamic vibration damper. The range of stable amplitudes is also determined for this system. It is shown that a cubic nonlinearity can either reinforce or offset the inflections caused by inelasticity.Translated from Problemy Prochnosti, No. 8, pp. 69–76, August, 1994.     

Improved monitoring of a spectrum of complex mechanical vibrations

This article examines new methods of evaluating a spectrum of mechanical vibrations on the basis of smoothing of the spectrum of the signal being analyzed, spectral analysis, and adaptive averaging. A comparative analysis is made of different methods. Translated from Izmeritel'naya Tekhnika, No. 1, pp. 25–29, January, 1996.     

Three-stage model of an elastoplastic cracked body

We propose a three-stage model of an elastoplastic cracked body, which takes into account the evolution of stresses in the prefracture zone according to the complete tensile stress-strain diagram of the investigated material. The zone of loosening corresponding to the last (descending) part of the tensile stressstrain diagram lies in the immediate vicinity of the crack tip. The zone of plastic deformation of the material is located after the zone of loosening. These zones are simulated by separate rectilinear stress-strain diagrams taking into account the effect of linear hardening of the material. Within the framework of the proposed model, we formulate a deformational fracture criterion and determine the characteristics of crack resistance of D16T steel. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 34, No. 1, pp. 59–64, January–February, 1998.