Coupled longitudinal and bending vibrations of a rotating shaft with an open crack

The coupling of longitudinal and bending vibrations of a rotating shaft, due to an open transverse surface crack is investigated. The assumption of the open crack leads to a system with behaviour similar to that of a rotor with dissimilar moments of inertia along two perpendicular directions. The local flexibility due to the presence of the crack can be represented by way of a 6×6 matrix for six degrees of freedom in a short shaft element which includes the crack. This matrix has off-diagonal terms which cause coupling along the directions which are indicated by these terms. Here shear is not considered and three degrees of freedom are used: bending in the two main directions and extension. This leads to a 3×3 stiffness matrix with coupling terms. The undamped free and forced coupled vibration are first considered. The coupling is investigated and the effects of unbalance and gravity are examined. Then damped coupled vibration is considered for free and forced vibration. The existence of coupling between longitudinal and bending vibration due to the crack is a very useful property which, together with the sub-critical resonance due to crack, can form a basis for crack identification in rotating shafts. New and interesting phenomena of coupled transverse and longitudinal motion are presented and discussed.     

VIBRATION-BASED DIAGNOSTICS OF FATIGUE DAMAGE OF BEAM-LIKE STRUCTURES

The analytical investigation of vibration of damaged structures is a complicated problem. This problem may be simplified if a structure can be represented in the form of a beam with corresponding boundary and loading conditions. In this connection, free vibrations of an elastic cantilever Bernoulli-Euler beam with a closing edge transverse crack is considered in the present work as a model of a structure with a fatigue crack. The modelling of bending vibrations of a beam with a closing crack is realized based on the solutions for an intact beam and for a beam with an open crack. The algorithm of consecutive (cycle-by-cycle) calculation of beam mode shapes amplitudes is presented. It is shown that at the instant of crack opening and closing, the growth of the so-called concomitant mode shapes which differ from the initially given mode shape takes place. Moreover, each of the half-cycles is characterized by a non-recurrent set of amplitudes of concomitant modes of vibration and these amplitudes are heavily dependent on the crack depth.The vibration characteristics of damage based on the estimation of non-linear distortions of the displacement, acceleration and strain waves of a cracked beam are investigated, and the comparative evaluation of their sensitivity is carried out.     

New breathing functions for the transverse breathing crack of the cracked rotor system: Approach for critical and subcritical harmonic analysis

The actual breathing mechanism of the transverse breathing crack in the cracked rotor system that appears due to the shaft weight is addressed here. As a result, the correct time-varying area moments of inertia for the cracked element cross-section during shaft rotation are also determined. Hence, two new breathing functions are identified to represent the actual breathing effect on the cracked element stiffness matrix. The new breathing functions are used in formulating the time-varying finite element stiffness matrix of the cracked element. The finite element equations of motion are then formulated for the cracked rotor system and solved via harmonic balance method for response, whirl orbits and the shift in the critical and subcritical speeds. The analytical results of this approach are compared with some previously published results obtained using approximate formulas for the breathing mechanism. The comparison shows that the previously used breathing function is a weak model for the breathing mechanism in the cracked rotor even for small crack depths. The new breathing functions give more accurate results for the dynamic behavior of the cracked rotor system for a wide range of the crack depths. The current approach is found to be efficient for crack detection since the critical and subcritical shaft speeds, the unique vibration signature in the neighborhood of the subcritical speeds and the sensitivity to the unbalance force direction all together can be utilized to detect the breathing crack before further damage occurs.     

Transient response of a hydraulic engine mount

Linear and non-linear transient responses of a typical hydraulic engine mount are analytically and experimentally studied in this paper. First, a lumped parameter linear model is used to approximate the typical step response and to suggest parameters that must be experimentally determined. Various configurations as related to inertia track and decoupler are analyzed. Two bench experiments are constructed for the identification of non-linear compliances and resistances. One of the main non-linear characteristics, however, comes from the decoupler mechanism. To accurately predict the time events of the decoupler opening and closing, an equivalent viscous damper model is employed along with a multi-staged switching mechanism. Additionally, non-linear behavior arising due to the vacuum formation in the top chamber is studied by defining a bi-linear asymmetric stiffness curve. New transient experiments are conducted on an elastomer test system, and measured transmitted force and top chamber pressure signals are analyzed. Results of the proposed simulation model match well with measured responses when step up, step down and triangular waveforms are applied. Areas for future research are identified.     

Free vibration analysis of a cracked beam by finite element method

In this paper, the natural frequencies and mode shapes of a cracked beam are obtained using the finite element method. An ‘overall additional flexibility matrix’, instead of the ‘local additional flexibility matrix’, is added to the flexibility matrix of the corresponding intact beam element to obtain the total flexibility matrix, and therefore the stiffness matrix. Compared with analytical results, the new stiffness matrix obtained using the overall additional flexibility matrix can give more accurate natural frequencies than those resulted from using the local additional flexibility matrix. All the elements in the overall additional flexibility matrix are computed by 128-point (1D) or (128×128)-point (2D) Gauss quadrature, and then further best fitted using the least-squares method. The explicit form best-fitted formulas agree very well with the numerical integration results, and are very convenient for use and valuable for further reference. In addition, the authors constructed a shape function that can perfectly satisfy the local flexibility conditions at the crack locations, which can give more accurate vibration modes.     

An electro-mechanical impedance model of a cracked composite beam with adhesively bonded piezoelectric patches

A model of a laminated composite beam including multiple non-propagating part-through surface cracks as well as installed PZT transducers is presented based on the method of reverberation-ray matrix (MRRM) in this paper. Toward determining the local flexibility characteristics induced by the individual cracks, the concept of the massless rotational spring is applied. A Timoshenko beam theory is then used to simulate the behavior of the composite beam with open cracks. As a result, transverse shear and rotatory inertia effects are included in the model. Only one-dimensional axial vibration of the PZT wafer is considered and the imperfect interfacial bonding between PZT patches and the host beam is further investigated based on a Kelvin-type viscoelastic model. Then, an accurate electro-mechanical impedance (EMI) model can be established for crack detection in laminated beams. In this model, the effects of various parameters such as the ply-angle, fibre volume fraction, crack depth and position on the EMI signatures are highlighted. Furthermore, comparison with existent numerical results is presented to validate the present analysis.     

Dynamic instability of supercritical driveshafts mounted on dissipative supports—Effects of viscous and hysteretic internal damping

The case of a rotating shaft with internal damping mounted either on elastic dissipative bearings or on infinitely rigid bearings with viscoelastic suspensions is investigated in order to obtain the stability region. A Euler-Bernoulli shaft model is adopted, in which the transverse shear effects are neglected and the effects of translational and rotatory inertia, gyroscopic moments, and internal viscous or hysteretic damping are taken into account. The hysteretic damping is incorporated with an equivalent viscous damping coefficient. Free motion analysis yields critical speeds and threshold speeds for each damping model in analytical form. In the case of elastic dissipative bearings, the present results are compared with the results of previous studies on finite element models. In the case of infinitely rigid bearings with viscoelastic suspensions, it is established that viscoelastic supports increase the stability of long shafts, thus compensating for the loss of efficiency which occurs with classical bearings. The instability criteria also show that the effect of the coupling which occured between rigid modes introducing external damping and shaft modes are almost more important than damping factor. Lastly, comparisons between viscous and hysteretic damping conditions lead to the conclusion that an appropriate material damping model is essential to be able to assess these instabilities.     

Analytical method for steady state vibration of system with localized non-linearities using convolution integral and Galerkin method

The analytical method using transfer function or impulse response is very effective for analyzing non-linear systems with localized non-linearities. This is because the number of non-linear equations can be reduced to that of the equations with respect to points connected with the non-linear element. In the present paper, analytical method for the steady state vibration of non-linear system including subharmonic vibration is proposed by utilizing convolution integral and the impulse response. The Galerkin method is introduced to solve the non-linear equations formulated by the convolution integral, and then the steady state vibration is obtained. An advantage of the present method is that stability or instability of the steady state vibration can be discriminated by the transient analysis from convolution integral. The three-degree-of-freedom mass-spring system is shown as a numerical example and the proposed method is verified by comparing with the result by Runge-Kutta-Gill method.     

Die Corioliskopplung im quantenmechanischen N-Körperproblem

In describing a system ofN particles (e.g. nucleons) in the body-fixed coordinate system of instantaneous principal axes of inertia, in the rotation-vibration part of the Hamiltonian there occur the hydrodynamical moments of inertia. It is, however, not possible to use them as effective moments of inertia as long as the Coriolis term in the Hamiltonian does not vanish. In order to obtain a quantitative picture of the Coriolis coupling influence upon the moments of inertia, we show by means of a unitary transformation how the Coriolis coupling can be eliminated in the limiting case of a rigid body and how by such a procedure the hydrodynamical moments of inertia are transformed into the rather different moments of the rigid body.     

BIFURCATIONS AND CHAOS OF AN IMMERSED CANTILEVER BEAM IN A FLUID AND CARRYING AN INTERMEDIATE MASS

The concern of this work is the local stability and period-doubling bifurcations of the response to a transverse harmonic excitation of a slender cantilever beam partially immersed in a fluid and carrying an intermediate lumped mass. The unimodal form of the non-linear dynamic model describing the beam-mass in-plane large-amplitude flexural vibration, which accounts for axial inertia, non-linear curvature and inextensibility condition, developed in Al-Qaisia et al. (2000Shock and Vibration7 , 179-194), is analyzed and studied for the resonance responses of the first three modes of vibration, using two-term harmonic balance method. Then a consistent second order stability analysis of the associated linearized variational equation is carried out using approximate methods to predict the zones of symmetry breaking leading to period-doubling bifurcation and chaos on the resonance response curves. The results of the present work are verified for selected physical system parameters by numerical simulations using methods of the qualitative theory, and good agreement was obtained between the analytical and numerical results. Also, analytical prediction of the period-doubling bifurcation and chaos boundaries obtained using a period-doubling bifurcation criterion proposed in Al-Qaisia and Hamdan (2001 Journal of Sound and Vibration244, 453-479) are compared with those of computer simulations. In addition, results of the effect of fluid density, fluid depth, mass ratio, mass position and damping on the period-doubling bifurcation diagrams are studies and presented.     

On certain time characteristics of dynamical systems driven by noise

The moments of the transition time of non-linear dynamical systems driven by noise are introduced. It is demonstrated that the well-known recurrence formula for the moments of the first passage time (FTP) of the absorbing boundary by a Brownian particle can also be used to obtain the exact values of the moments of the transition time in non-linear dynamical systems with noise, described by a symmetric potential profile.     

Dynamic rotational characteristics of actinides on empirical approach

In the paper calculation of the moments of inertia for nuclei from the region 87 ≤ Z ≤ 100 and 130 ≤ N ≤ 156 was made in dependence on the angular momentum of their rotational states. The experimental values of the moments of inertia were calculated for rotational energy of the classic rotor in its quantum form, with the use of a simple formula. The moment of inertia term appearing in the formula was treated as a variable. The calculations were carried out on the basis of experimental data for the energies of the rotational levels for 51 bands built on ground states for even-even nuclei and for nuclei with odd mass number A. In addition, 30 rotational bands built on excited states were also analysed in the investigated region in case of even-even nuclei. For many bands and nuclei the considered dependence of the moment of inertia on angular momentum has been found in the analytical form by fitting polynomials to the experimental data. It turned out that obtained results for the moments of inertia made it possible to describe the energies of rotational levels with a relative deviation not greater or only slightly greater than 1%. In general, in the case of 12 bands of ground level the maximum relative deviation of obtained level energies is smaller than 1%.      

An insight into the breathing mechanism of a crack in a rotating shaft

The time history of local flexibilities associated with a breathing crack in a rotating shaft is the concern of this paper. Considering quasi-static approximation, the deflections of a circular cross-section beam presenting a crack of different depths, due to bending or torsion loads are analyzed with the aid of a refined nonlinear contact-finite element procedure in order to predict accurately the time-variant flexibility of the fractured shaft. This method predicts the partial contact of crack surfaces, and it is appropriate to evaluate the instantaneous crack flexibilities. The bending load is applied in several aperture angles, in order to simulate a rotating load on a fixed beam. Results obtained for the rotating beam can then be used for the analysis of cracked, horizontal axis rotors. The effect of friction is also considered in the cracked area. Portions of crack surfaces in contact are predicted, the direct and the cross-coupled flexibility coefficients are calculated by applying energy principles. The numerical results compared with relevant previously published results, show high consistency.     

Crack detection in hollow section structures through coupled response measurements

Detection of a circumferential crack in a hollow section beam is investigated using coupled response measurements. The crack section is represented by a local flexibility matrix connecting two undamaged beam segments. This matrix defines the relationship between the displacements and forces across the crack section and is derived by applying fundamental fracture mechanics theory. The suitability of the mode coupling methodology is first demonstrated analytically. Laboratory test results are then presented for circular hollow section beams with artificially generated cracks of varying severity. It is shown that this method has the potential as a damage detection tool for mechanical structures.     

对关联在反磁转动中的作用（英文）

采用基于推转壳模型的粒子数守恒方法对105Cd 和106Cd 中的反磁转动带进行了研究,在计算当中,粒子数严格守恒,并且堵塞效应也是严格考虑的。计算结果很好地再现了实验上观测到的I-Ω 关系、转动惯量以及约化跃迁几率B（E2）。通过检验双剪角,即两个质子空穴角动量的合拢,对反磁转动中的双剪刀机制进行了分析。研究表明剪刀角的合拢非常敏感地依赖于对关联。The antimagnetic rotation bands in 105;106Cd are investigated by the cranked shell model with pairing correlations treated by a particle-number conserving method, in which the blocking effects are taken into account exactly. The experimental moments of inertia, I-Ω relation and the reduced B(E2) transition probabilities are well reproduced. The two-shears-like mechanism for the antimagnetic rotation is investigated by examining the shears angle, i.e., the closing of the two proton hole angular momenta. The sensitive dependence of the shears angle on the nuclear pairing correlations is revealed.     

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.     

Random excitation of a vibratory system with autoparametric interaction

The paper is concerned with the broad band random excitation of a two degree of freedom vibratory system with non-linear coupling of autoparametric type. A general equation for the evolution of the moments of any order of the response co-ordinates is derived by using stochastic calculus and found to represent an infinite hierarchy set. Consideration is given to the determination of the mean square stability boundary for unimodal response with no transverse motion of the coupled system. Two approximate solutions are obtained. These are first of all a solution based on a Gaussian closure technique applied to the system moment equations which allows the stability condition to be determined from the eigenvalues of a four by four matrix, and secondly a perturbation solution which leads to a simple analytical expression for the stability boundary. The two methods give results in close agreement for low values of system damping, but which differ appreciably at high damping levels. Finally, results are obtained from an investigation of the response regions of a laboratory model excited from a random noise generator. The experimental results are found to give excellent correlation with the predicted instability boundaries in the close neighbourhood of internal resonance but show a distinct indication of a wider instability region than predicted by both analytical methods.     

Effect of in-plane inertia on buckling of imperfect plates with large deformations

The influence of in-plane inertia on non-linear dynamic buckling of rectangular initially-imperfect plates is studied. A regular, consistent perturbation technique is used, and both the equations of motion and the boundary conditions are perturbed, yielding a relatively simple procedure for solving an otherwise very involved non-linear problem. Both pulse and vibration buckling during parametric resonance are analyzed. It is shown that, in the former case, the in-plane inertia can be disregarded whereas in the latter it plays an important role for non-slender plates.     

Variation in the stability of a rotating blade disk with a local crack defect

The effect of a near root local blade crack on the stability of a grouped blade disk is investigated in this paper. A bladed disk comprised of periodically shrouded blades is used to simulate the coupled periodic structure. The blade crack is modeled using the local flexibility with coupling terms. The mode localization phenomenon introduced by the blade crack on the longitudinal and bending vibrations in the rotating blades has also been considered. Using the Galerkin's method, the imperturbation equations of a bladed disk in which one of the blades is cracked, subject to fluctuations in the rotation speed, can be derived. Employing the multiple scales method, the boundaries of the instability zones in the mistuned turbo blade system are approximated. Numerical results indicate that an additional unstable zone is introduced near the localization frequency and the regions of unstable zones are varied with the crack size and fluctuations in disk speed.