Thermoactive vibration control of shafts

Flexible long‐span rotating shafts exhibit flutter instability conditioned by internal friction in bending at high rotation speeds. Under usual working conditions a shaft may be additionally subjected to external excitations related to unbalance forcing or edge bearing movements. Flutter occurs at angular speeds exceeding the lowest natural frequency of the shaft as a nonrotating beam. Thus, externally excited resonances mostly appear in the subcritical speed zone although they can interact with flutter vibration as well. The present paper is concerned with resonant vibration control of shafts based on application of thermoactive SMA components in composite shaft structures, as conceptually shown in [1]. The well known unique properties of SMAs consisting in huge changes of the elastic modulus and its loss factor as results of a reversible martensitic phase transformation under slight temperature variations [2] promise to control shaft vibrations through temperature‐induced modal modifications of the structure. The main resonance of a simply supported rotating shaft is considered to be controlled by open‐loop SMA activation. Efficiency of thermoactive vibration control is analysed and a concept of an intelligent self‐controlled shaft structure is introduced. Geometric nonlinearity is assumed in modelling and computer simulations to show the thermoactive resonance suppression including the case when both externally excited and flutter vibrations interact.     

Low-frequency internal friction in complex composite strands

Conclusion A method was proposed which makes it possible to study the low-frequency internal friction of complex fibers. This method was used to study the internal friction of complex composite fibers based on PCA and PE. We observed a relaxation process which is typical of the composite and is absent in the pure materials.Translated from Mekhanika Kompozitnykh Materialov, No. 1, pp. 11–15, January–February, 1989.     

迟滞型材料阻尼转轴的分岔

应用平均法研究迟滞型材料阻尼转轴的分岔.首先用Hamilton原理推导出复数形式的转轴运动微分方程,然后用平均法求出各阶模态主共振时的平均方程,并分析定常解的稳定性,最后用奇异性理论分析正常运动和失稳运动响应(异步涡动)的分岔.研究表明,一定参数条件下,转轴在通过各阶临界转速(主共振)时,可能会因受到冲击而失稳(Hopf分岔).正常运动响应在不平衡量较大时有滞后和跳跃现象,而失稳运动响应是一类余维数较高的非对称分岔.由于内阻尼的非线性,响应随转速增加时还可能产生二次Hopf分岔,对应原系统的双调幅运动.做好动平衡及提高外阻尼水平是避免这种大幅值自激振动的有效措施.     

Kritische Drehzahlen unter Torsion und Druck bei Berücksichtigung des Eigengewichtes

Summary The paper deals with the critical angular velocities of continuously loaded shafts. A method is given for the solution of the problem and an example is presented which shows that the critical angular velocities increase, when, instead of a single concentrated load on the ends, the shaft carries a continuous load of the same magnitude.     

Non-stationary nonlinear analysis of a composite rotating shaft passing through critical speed

In this paper, nonlinear non-stationary dynamics of a nonlinear composite shaft passing through critical speed is studied. The nonlinearity is due to the large amplitude of shaft vibration. The equations of motion are obtained by three-dimensional constitutive relationships of composite materials. The gyroscopic effect, rotary inertia and coupling caused by material anisotropy are considered but shear deformation is neglected. Without any simplification, axial-flexural-flexural-torsional equations of motion (EOM) for the elastic composite shaft with variable rotational speed are obtained. The approximate analytical method namely asymptotic method is applied to analyze the nonstationary behavior of the composite shaft with constant acceleration. First, the EOMs are discretized using one and two-term Galerkin method. Then, the resulted equations are transformed to normal coordinates. Finally, the asymptotic method is applied to equations described in normal coordinates. Analytical expressions governing the amplitude and phase of motion during passage through critical speeds are obtained. By comparing the results obtained from analytical solutions, it is shown that discretization by one mode is not enough due to the existence of coupling in the equations and at least two modes are necessary for this purpose. Effects of damping, eccentricity, initial angular velocity and fiber angle on response amplitude are investigated. For verification, the results of perturbation theory are compared with numerical simulations and it is shown that there is good agreement between both methods.     

Evaluating the potential use of composites in ship shafting 3. Efficient design of the intermediate shaft of ship shafting

This report examines the problems associated with the design of a composite intermediate shaft for a ship shaftline. A comparison is made between shafts of carbon- and glass-fiber-reinforced plastics made by winding with different laws governing the change in winding angle through the thickness of the shaft. An analysis and comparison of results obtained for a specific example show that the use of a composite significantly reduces the weight and stiffness of the shaft compared to steel shafts and thus appreciably reduces all of the additional loads on the bearings and other shafts.See [2] for Report No. 2.Institute of Polymer Mechanics, Latvian Academy of Sciences (Riga, LV-1006 Latvia). Translated from Mekhanika Kompozitnykh Materialov, No. 1, pp. 66–81, January–February, 1997.     

Antifriction basalt-plastics based on polypropylene

A study is made of the dependence of the mechanical and friction-engineering properties of polypropylene reinforced with basalt fibers on the viscosity of the polymer matrix. It is established that the main factors that determine the mechanical properties of the plastics are the quality of impregnation of the fibers by the binder and the residual length of the reinforcing filler in the composite after extrusion and injection molding. The material that was developed has a low friction coefficient and low rate of wear within a relatively brood range of friction conditions. The basalt-plastics can be used in the rubbing parts of machines and mechanisms subjected to dry friction.Ukrainian State Chemicotechnical University, Dnepropetrovsk, Ukraine. Translated from Mekhanika Kompozitnykh Materialov,33, No. 3, pp. 417–421, May–June, 1997.     

Modelling of rotating shafts with flexible disks

This paper deals with the methodology of the modelling of rotating shafts with flexible disks. Rotating shafts are modelled as a one dimensional continuum on the basis of the Bernouli-Euler theory, which assumes that the shaft cross section remains a flat plane and is perpendicular to the centerline during vibration. Disks are modelled as a three dimensional continuum by means of the finite element method. The presented approach allows to introduce centrifugal and gyroscopic effects. The special coupling matrix is used for the connecting a rotating shaft and a mounted flexible disk. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Flexural Vibrations of Shafts with Delaminations

The purpose of this theoretical work is to present a general model of laminated rotating shaft with circumferential delaminations. The shaft is treated as a thin‐walled composite cylindrical shell. The delamination of constant width is parallel to the shell reference surface and it covers the entire circumference. The edge delamination is modeled by changing the effective reduced stiffnesses of debonded parts. The stabilizing effect of external damping and destabilizing effect of internal damping are taken into account in the dynamic stability analysis. The influence of the relative delamination length and configuration on the critical angular velocity of shaft is shown.     

The dynamics of a “flexible-rotor/limited-power-excitation-source” system

A “flexible-rotor/limited-power-excitation-source” dynamical system, which is a model of the vibrations of shafts driven by a motor of limited power is considered. The types of rotation characteristics of the motor and resonance characteristics of the shaft in the resonance parameter zone for different values of the viscosity coefficient of the medium are investigated. These characteristics completely describe the dynamics of the system when its parameters vary.     

Strength Analyses of Laminated Composite Structures Based on Different Theories

Modern materials, such as composite ones, slowly replace conventional materials in structures of different kind and their growing popularity is caused by their multiple advantages. Through selection of parameters, such as number of layers, thickness of layers, direction of arranging fibers, or material from which internal and outside layers are made, it is possible to control properties of a structure. In the result, structures made of composite materials have high ratio of flexural stiffness to weight. The goal of this paper is to compare three theories of laminated composite plates and shells with the help of the multilayered rectangular plate model subjected to crosswise pressure perpendicular to the surface of a plate. Comparison was made for the classical laminated theory (CLT), first-order laminated theory and third-order laminated theory (TSDT). In all these theories, the number of parameters describing the displacement doesn't depend on the number of layers. For each of these theories strains and displacements were determined. Additionally, the computations time for every method were compared. Obtained results are presented in the form of tables. The analysis of the obtained solutions will be used as the base in choosing the best theory in multicriteria optimization process of composite thin-walled structures. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic analysis of short and long journal bearings in laminar and turbulent regimes,application in critical shaft stiffness determination

Linear and non-linear stability of a flexible rotor-bearing system supported on short and long journal bearings is studied for both laminar and turbulent operating conditions. The turbulent pressure distribution and forces are calculated analytically from the modified Reynolds equation based on two turbulent models; Constantinescu's and Ng–Pan–Elrod. Hopf bifurcation theory was utilized to estimate the local stability of periodic solutions near bifurcating operating points. The shaft stiffness was found to play an important role in bifurcating regions on the stable boundaries. It was found that for shafts supported on short journal bearings with shaft stiffness above a critical value, the dangerous subcritical region can be eliminated from a range of operating conditions with high static load. The results presented have been verified by published results in the open literature.     

Application of CFRP as a rotor shaft material

Conclusions Theoretical analysis and tests performed on rotors with composite shaft show that there is a sufficiently wide rotation stability region in the rotor parameter space despite comparatively high damping of a polymeric composite with respect to steel. Optimum parameters of the shaft (lay-up, thickness) and bearing (radial stiffness, damping) can be found within this region for each given rotor ensuring a low vibration level at critical frequencies.If rotor system parameters are far enough from the instability threshold, maximum vibration level is observed when rotor passes the first eigenfrequency zone. Further increase of rotation frequency leads to a rotor self-centering, and vibration level does not change passing the second eigenfrequency zone. The rotor response is not sensitive to small changes in rotor system parameters. If rotor system parameters are close to the instability threshold, vibration level at the second eigenfrequency dominates, and a small variation of bearing parameters causes significant changes in the vibration level.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 2, pp. 227–240, March–April, 1995.     

Structure and properties of ceramic composites based on ZrO2

Monodisperse fine powders of high purity obtained by sol-gel method are used for production of high technical data ceramics. The fiber reinforcement is used for hardening of composite materials. It was of interest to study production possibility of reinforced composite material based on ZrO₂ obtained by sol-gel method with filler from fibers of partially stabilized zirconia. ZrO₂ powders were obtained by precipitation of its hydrated gel from aqueous zirconium oxychloride solution by ammonium hydroxide followed by thermal treatment. For composite reinforcement ceramic partially stabilized (8 mole Y₂O₃) ZrO₂ fibers 0.16–0.67 mm in length and 5–7 m in diameter were used. Content of the fibers in composite was 20 wt.%. From powders and their mixtures with fibers, the samples were pressed as disks, beams and cylinders, and anneal in air at 1100–1600°C temperature range. The investigation has shown that the fibers of partially stabilized zirconia change the composite structure, increase the content of tetragonal modification that promotes its hardening. Treatment temperature of precursor determines physical chemical properties of compositions with fibers. Their high specific surface and reaction ability provides a workability of forming and sintering processes into strong composite material. The ceramics was increased by 2.5–3 times as strength after fibrous filler introduction into ZrO₂ hydrogel matrix.Presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October 1995).Presented at the Ninth International conference on the Mechanics of Composite Materials (Riga, October 1995). Institute of General and Inorganic Chemistry of the Academy Sciences of the Belarus. Minsk. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 3, pp. 418–427, May–June. 1996.     

Calculation and Experimental Study of a Two-Way Reinforced Rubber-Cord Composite in Tension

A rubber-cord composite, reinforced in two directions with fibers of polyamide cord, under large tensile deformations is investigated based on calculations of a rubber-cord composite material and on tensile tests of specimens made of the casing of a diagonal truck tire. A method of the experimental tensile testing of rubber-cord composite specimens is described. The calculations are based on the carcass theory of composite materials. The calculated and experimental parameters of the macroscopic strains of the rubber tire cord and of its structure in a deformed configuration are given. The manifestation of edge effects in relation to the reinforcement angle is described.     

Dynamisches Verhalten einer einfach besetzten rotierenden Welle mit Kardangelenken

Summary If a rotating, massless, elastic shaft carrying a disk is supported at the ends by Cardan links, the motion of the disk depends on the angles at the joints and the torques transmitted by the joints. The system is considered for constant angular velocity and constant torques of the driving shafts. The investigation of this nonstationary system leads to two second order differential equations with periodic coefficients. In order to establish conditions for instability the characteristics exponents are calculated by means of generalized Hills determinants. It is found that there exist critical intervals for the angular velocity.     

Deformation and Damaging of Composites with Transversally-isotropic Components under Compressive Loading

In the present paper, the problem of deformation and damage of composites with a porous isotropic matrix and transversally-isotropic unidirectional fibers under compressive loading is considered when microdamages are accumulated in the fiber. Fractured micro-volumes are modelled by a system of randomly distributed quasi spherical pores. The Shleicher-Nadai fracture criterion is used as a condition for the origin of micro-pores (micro-damage) based on the assumption of a rigid material. The limit value of the strength of the material is assumed as a stochastic function of coordinates. By using a numerical procedure, the solution of the above problem is found. The nonlinear stress-strain diagrams for a transversally-isotropic composite are obtained for the case of uniaxial compression-tension along the fibers. The nonlinearity of the deformations of the composite is caused by accumulation of micro-damages in the matrix. The influence of the physical-mechanical properties of materials, of the volume concentration, of the porosity of components, of the geometrical parameters of the structure, and of the character of the scatter of the strength in the material on the micro-damage of the material and, as a consequence, the influence on the macro-stress-macro-strain diagram is analyzed. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

嵌入弹性半空间的弹性迴转轴的扭转*

本文用线载荷积分方程法(LLIEM)研究嵌在弹性半空间的弹性迴转轴的扭转问题.将“点环力偶(PRC)”和“半空间点环力偶(PRCHS)”分别分布于迴转轴内和外的轴线上,就能将本问题归结为一维的Fredholm第一种积分方程组.直接用离散法求解时,会发现有时解是不稳定的,也就是病态情形.本文采用以带小参数的Fredholm第二种积分方程代替病态的Fredholm第一种积分方程的方法可以得到稳定的解,此法比Tikhonov正规化法简单,易于在计算机上运行.文中给出圆维、圆柱、圆锥-圆柱、抛物线轴等数值例子.     

The stability of the rotation of a heavy body with a viscous filling

The stability of the permanent rotation of a symmetrical heavy body with a viscous filling is investigated. A finite-dimensional phenomenological model of the “internal friction” with which the filling acts on the wall of the cavity is constructed based on the Helmholtz equations for a vortex. The boundaries of the stability limit are constructed and the interaction between the internal friction and the external damping is tracked. It is shown that the cases of a cavity that is oblate and prolate along the axis of rotation lead to the existence of different forms of stability regions.