Vibration and stability of cracked hollow-sectional beams

This paper presents simple tools for the vibration and stability analysis of cracked hollow-sectional beams. It comprises two parts. In the first, the influences of sectional cracks are expressed in terms of flexibility induced. Each crack is assigned with a local flexibility coefficient, which is derived by virtue of theories of fracture mechanics. The flexibility coefficient is a function of the depth of a crack. The general formulae are derived and expressed in integral form. It is then transformed to explicit form through 128-point Gauss quadrature. According to the depth of the crack, the formulae are derived under two scenarios. The first is for shallow cracks, of which the penetration depth is contained within the top solid-sectional region. The second is for deeper penetration, in which the crack goes into the middle hollow-sectional region. The explicit formulae are best-fitted equations generated by the least-squares method. The best-fitted curves are presented. From the curves, the flexibility coefficients can be read out easily, while the explicit expressions facilitate easy implementation in computer analysis. In the second part, the flexibility coefficients are employed in the vibration and stability analysis of hollow-sectional beams. The cracked beam is treated as an assembly of sub-segments linked up by rotational springs. Division of segments are made coincident with the location of cracks or any abrupt change of sectional property. The crack's flexibility coefficient then serves as that of the rotational spring. Application of the Hamilton's principle leads to the governing equations, which are subsequently solved through employment of a simple technique. It is a kind of modified Fourier series, which is able to represent any order of continuity of the vibration/buckling modes. Illustrative numerical examples are included.     

Vibration analysis of horizontal self-weighted beams and cables with bending stiffness subjected to thermal loads

This paper aims at proposing an analytical model for the vibration analysis of horizontal beams that are self-weighted and thermally stressed. Geometrical nonlinearities are taken into account on the basis of large displacement and small rotation. Natural frequencies are obtained from a linearization of equilibrium equations. Thermal force and thermal bending moment are both included in the analysis. Torsional and axial springs are considered at beam ends, allowing various boundary conditions. A dimensionless analysis is performed leading to only four parameters, respectively, related to the self-weight, thermal force, thermal bending moment and torsional spring stiffness. It is shown that the proposed model can be efficiently used for cable problems with small sag-to-span ratios (typically , as in Irvine's theory). For beam problems, the model is validated thanks to finite element solutions and a parametric study is conducted in order to highlight the combined effects of thermal loads and self-weight on natural frequencies. For cable problems, solutions are first compared with existing results in the literature obtained without thermal effects or bending stiffness. Good agreement is found. A parametric study combining the effects of sag-extensibility, thermal change and bending stiffness is finally given.     

Gyroscopic behaviour of stressed rotating shafts

An analysis is presented of the secondary effects due to deformation as a result of the shearing force, to the kinetic energy of rotation, and to the gyroscopic coupling, on the linearized equation of a straight shaft turning at a constant angular velocity and which is subjected to an axial torque and a compressive force. The local equations of motion of a section of shaft are derived and the magnitudes of the coupling terms are studied, non-dimensional variables being used. Results obtained by different authors in certain particular cases are rediscovered, and, in particular, the case of a shaft on short bearings is studied. Some comparisons of the theoretical results with those from experiments are given.     

On the cracked Bell

A simple model based on energy methods of applied mechanics is developed and employed to study the effect of clapper location on the initiation and growth of cracks in bells. The origins of cracks in the Liberty Bell and Big Ben are discussed, and the simple model is employed to explain why remedial measures on Big Ben, though perhaps not optimal, have prolonged its life.     

Whirling of flexible shafts with intermediate supports

The present paper deals with an approximate solution of the title problem using the Rayleight-Schmidt procedure. It is shown that rotating shafts of non-uniform cross section can be analyzed in a straightforward manner. Experimental and analytical results are obtained for the case of uniform shafts with intermediate supports, and good engineering agreement is shown to exist.     

On cubic nonlinearity of cracked media

An equation of state with cubic nonlinearity has been obtained for the model of a cracked medium. It is shown that cracks in a solid can lead not only to a strong change of the cubic nonlinearity parameter but also to the sign reversal of the latter. We describe the experiment on the measurement of a nonlinear shift in the resonance frequency of a bar resonator of glass with cracks where the positive shift of the resonance frequency was observed. We estimate the effective parameter of the cubic nonlinearity for such a resonator by comparing theoretical and experimental dependences. Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 40, No. 7, pp. 897–902, July. 1997.     

Dynamic behavior analysis of cracked rotor

In the present study, the additional slope is used to consider the crack breathing, and is expressed explicitly in the equation of motion as one of the inputs to produce the bending moment at the crack position. Inversely, the additional slope is calculated by integrating on the crack region based on a fracture mechanics concept. The response of a cracked rotor is formulated based on the transfer matrix method. The transient behavior due to the crack breathing is considered by introducing a ‘moving’ Fourier-series expansion concept to the additional slope. The time-varying harmonic components of the additional slope are used to calculate the harmonic responses. The application considered is a general rotor model composed of multiple shafts, disks and cracks, and resilient bearings at both ends. Verification analysis is carried out for a simple rotor model similar to those found in the literature. Using the additional slope, the cracked rotor behavior is explained by the crack depth and rotation speed increase. It is shown that region on the crack front line having the dominant stress intensity factor value moves from the central area to both ends, as the crack depth increases. The result matches well with the crack propagation pattern shown in a bench mark test in the literature. Whirl orbits near the critical and sub-critical speed ranges of the rotor are discussed. It is shown that there exists some speed range near the critical speed, where the temporary whirl direction reversal and phase shift exist. When an unbalance is applied, the peculiar features, such as the whirl direction reversal and phase shift, disappear.     

A worst balance analysis of flexible rotating shafts

A worst balance analysis is formulated for a rotating shaft. The analysis provides the maximum deflection of the shaft resulting from the worst combination of errors in placing balance weights along the axis of the rotor. The analysis is formulated in general as a non-linear programming problem and in particular cases as a linear programming problem. Both formulations require computational implementation for multi-mass rotors. Curves illustrating the trade-off between the error in balance weight placement and the corresponding worst disturbance are presented.     

Heat propagation due to torsional vibration of shafts

Torsional vibration of rotating shafts can yield substantial temperatures in the shaft due to heat generated from material damping. It has been recently observed that such a situation in electric generators can lead to insulation failures and machine outages. In the study reported here forced torsional vibration is assumed, and both lumped mass and continuous systems are considered. The hysteretic model for material damping is used to yield the heat generation in the elastic deformation range and an elastoplastic material is assumed in the plastic range. The heat conduction equation is solved for a cylindrical shaft with surface cooling. Closed form solutions and expressions for the maximum temperatures and the maximum surface temperatures are obtained and tabulated for design purposes. It is shown that substantial temperatures can develop in shafts undergoing torsional vibration.     

Stability control of flexible shafts supported on oil-film bearings

This paper describes research carried out to establish instability criteria for flexible rotors and shafts supported on oil-film bearings. In particular, the effect of external point damping in delaying the onset of instability is investigated, both theoretically and experimentally.     

含裂纹弹性结构声相互作用研究

应用分形有限元和边界元相结合的方法研究了含裂纹的结构声的耦合作用问题。利用二级分形有限元方法离散含裂纹结构,这将使得自由度大为减少;使用边界元方法计算外域辐射声场,这将自动满足无限远辐射边界条件。数值结果初步表明:(1)随着裂纹深度的增加,结构声耦合系统的共振频率将下降;(2)裂纹附近的声场所受的影响更为明显。     

Lattice bending in x-ray interferometers

This paper investigates both theoretically and experimentally the effect of lattice bending on the output signals of a two-crystal x-ray interferometer of the Laue LLL type. The cross section intensity of the outgoing beams is modulated by the moiré effect produced by the overlapping of the analysing lattice on the x-ray standing field in front of it. Since the intensities of the transmitted and diffracted beams are integrated, the moiré pattern causes loss of visibility in the x-ray fringes and a non-linear phase shift, which depends on the pitch alignment of the analysing crystal with respect to the fixed crystal. The analysis of this phase shift allows the lattice curvature to be estimated.