Nonlinear dynamics of a support-excited flexible rotor with hydrodynamic journal bearings

The major purpose of this study is to predict the dynamic behavior of an on-board rotor mounted on hydrodynamic journal bearings in the presence of rigid support movements, the target application being turbochargers of vehicles or rotating machines subject to seismic excitation. The proposed on-board rotor model is based on Timoshenko beam finite elements. The dynamic modeling takes into account the geometric asymmetry of shaft and/or rigid disk as well as the six deterministic translations and rotations of the rotor rigid support. Depending on the type of analysis used for the bearing, the fluid film forces computed with the Reynolds equation are linear/nonlinear. Thus the application of Lagrange's equations yields the linear/nonlinear equations of motion of the rotating rotor in bending with respect to the moving rigid support which represents a non-inertial frame of reference. These equations are solved using the implicit Newmark time-step integration scheme. Due to the geometric asymmetry of the rotor and to the rotational motions of the support, the equations of motion include time-varying parametric terms which can lead to lateral dynamic instability. The influence of sinusoidal rotational or translational motions of the support, the accuracy of the linear 8-coefficient bearing model and the interest of the nonlinear model for a hydrodynamic journal bearing are examined and discussed by means of stability charts, orbits of the rotor, time history responses, fast Fourier transforms, bifurcation diagrams as well as Poincaré maps.     

Dynamic analysis of double-row self-aligning ball bearings due to applied loads,internal clearance,surface waviness and number of balls

In this paper, a three degrees of freedom (dof) model was established for a double-row self-aligning ball bearing (SABB) system, and was applied to study the dynamic behavior of the system during starting process and constant speed rotating process. A mathematical model was developed concerning stiffness and damping characteristics of the bearing, as well as three-dimensional applied load, rotor centrifugal force, etc. Balls and races were all considered as nonlinear springs, and the contact force between ball and race was calculated based on classic Hertzian elastic contact deformation theory and deformation compatibility theory. The changes of each ball?s contact force and loaded angle of each row were taken into account. In order to solve the nonlinear dynamical equilibrium equations of the system, these equations were rewritten as differential equations and the fourth order Runge–Kutta method was used to solve the equations iteratively. In order to verify accuracy of the dynamical model and correctness of the numerical solution method, a kind of SABB-BRF30 was chosen for case studies. The effects of several important governing parameters, such as radial and axial applied loads, normal internal, inner and outer races waviness, and number of balls were investigated. These parametric studies led to a complete characterization of the shaft-bearing system vibration transmission. The research provided a theoretical reference for new type bearing design, shaft-bearing system kinetic analysis, optimal design, etc.     

Accurate identification of the frequency response functions for the rotor-bearing-foundation system using the modified pseudo mode shape method

In this paper, an identification technique in the dynamic analyses of rotor-bearing-foundation systems called the pseudo mode shape method (PMSM) was improved in order to enhance the accuracy of the identified dynamic characteristic matrices of its foundation models. Two procedures, namely, phase modification and numerical optimisation, were proposed in the algorithm of PMSM to effectively improve its accuracy. Generally, it is always necessary to build the whole foundation model in studying the dynamics of a rotor system through the finite element analysis method. This is either unfeasible or impractical when the foundation is too complicated. Instead, the PMSM uses the frequency response function (FRF) data of joint positions between the rotor and the foundation to establish the equivalent mass, damping, and stiffness matrices of the foundation without having to build the physical model. However, the accuracy of the obtained system's FRF is still unsatisfactory, especially at those higher modes. In order to demonstrate the effectiveness of the presented methods, a solid foundation was solved for its FRF by using both the original and modified PMSM, as well as the finite element (ANSYS) model for comparisons. The results showed that the accuracy of the obtained FRF was improved remarkably with the modified PMSM based on the results of the ANSYS. In addition, an induction motor resembling a rotor-bearing-foundation system, with its housing treated as the foundation, was taken as an example to verify the algorithm experimentally. The FRF curves at the bearing supports of the rotor (armature) were obtained through modal testing to estimate the above-mentioned equivalent matrices of the housing. The FRF of the housing, which was calculated from the equivalent matrices with the modified PMSM, showed satisfactory consistency with that from the modal testing.     

X形超阻尼局域共振声子晶体梁弯曲振动带隙特性

以声子晶体理论为基础,设计了一种具有超阻尼特性的X形局域共振结构,分析了周期性附加X形局域共振的梁弯曲振动传播特性.利用拉格朗日方程分析了X形局域共振结构动力学等效特性,揭示了该结构的阻尼放大的机理,分析了几何结构参数对于带隙特性的影响,并利用有限元法验证了X形局域共振结构的超阻尼特性.研究结果表明,周期性附加X形局域结构能够有效地抑制低频弯曲振动在梁中的传播,产生超阻尼特性,实现低频、宽带的减振效果,为结构的低频减振提供了一个新的设计方案.     

Piezoelectric resonators with mechanical damping and resistance in current conduction

A novel design method for high Q piezoelectric resonators was presented and proposed using the 3-D equations of linear piezoelectricity with quasi-electrostatic approximation which include losses attributed to mechanical damping in solid and resistance in current conduction. There is currently no finite element sofware for estimating the Q of a resonator without apriori assumptions of the resonator impedance or damping. There is a necessity for better and more realistic modeling of resonators and filters due to miniaturization and the rapid advances in frequency ranges in telecommunication. We presented new three-dimensional finite element models of quartz and barium titanate resonators with mechanical damping and resistance in current conduction. Lee, Liu and Ballato’s 3-D equations of linear piezoelectricity with quasi-electrostatic approximation which include losses attributed to mechanical damping in solid and resistance in current conduction were formulated in a weak form and implemented in COMSOL. The resulting finite element model could predict the Q and other electrical parameters for any piezoelectric resonator without apriori assumptions of damping or resistance. Forced and free vibration analyses were performed and the results for the Q and other electrical parameters were obtained. Comparisons of the Q and other electrical parameters obtained from the free vibration analysis with their corresponding values from the forced vibration analysis were found to be in excellent agreement. Hence, the frequency spectra obtained from the free vibration analysis could be used for designing high Q resonators. Results for quartz thickness shear AT-cut and SC-cut resonators and thickness stretch poled barium titanate resonators were presented. An unexpected benefit of the model was the prediction of resonator Q with energy losses via the mounting supports.     

Whirl and whip instabilities in rotor-bearing system considering a nonlinear force model

Linear models and synchronous response are generally adequate to describe and analyze rotors supported by hydrodynamic bearings. Hence, stiffness and damping coefficients can provide a good model for a wide range of situations. However, in some cases, this approach does not suffice to describe the dynamic behavior of the rotor-bearing system. Moreover, unstable motion occurs due to precessional orbits in the rotor-bearing system. This instability is called “oil whirl” or “oil whip”. The oil whirl phenomenon occurs when the journal bearings are lightly loaded and the shaft is whirling at a frequency close to one-half of rotor angular speed. When the angular speed of the rotor reaches approximately twice the natural frequency (first critical speed), the oil whip phenomenon occurs and remains even if the rotor angular speed increases. Its frequency and vibration mode correspond to the first critical speed. The main purpose of this paper is to validate a complete nonlinear solution to simulate the fluid-induced instability during run-up and run-down. A flexible rotor with a central disk under unbalanced excitation is modeled. A nonlinear hydrodynamic model is considered for short bearing and laminar flow. The effects of unbalance, journal-bearing parameters and rotor arrangement (vertical or horizontal) on the instability threshold are verified. The model simulations are compared with measurements at a real vertical power plant and a horizontal test rig.     

Experimental characterization and modeling of microsliding on a small cantilever quartz beam

The design of mechanical systems requires various studies in order to ensure an optimal behavior during operation. In particular, the study of its dynamic behavior makes it possible to evaluate the role of a connection in the energy dissipation mechanisms. In this context, an experimental setup dedicated to small structures has been developed to quantify damping due to microsliding at the beam–clamp interface. The mechanical characterization of the clamped connection is carried out by experimental dynamic tests on a free-clamped structure. The instantaneous frequencies and damping are identified by the wavelet transform technique of a slightly nonlinear system. In parallel, numerical prediction of the equivalent damping is achieved thanks to the implementation of the regularized Coulomb law in a finite element model. A genetic algorithm and artificial neural networks are used to update the stiffness parameter and the friction coefficient. The optimized model is in good agreement with experimental results. It allows for determining the spatial distribution of microsliding and tangential force along the contact interface. The dissipated energy and equivalent damping are finally deduced according to the dynamic deflection of the free part of the beam.     

磁流变液阻尼器在转子振动控制中的应用

设计了一种转子振动控制用的剪切式磁流变液阻尼器,建立了磁流变液阻尼器－悬臂转子系统的分析模型,理论和实验研究了转子系统的不平衡响应特性。研究表明,随着施加磁场强度的增加,磁流变液阻尼和刚度增大,转子系统的临界振幅明显下降,系统的临界转速也明显提高。通过简单的开关控制,可抑止转子通过临界转速过程中的振动。     

Rotor–stator contact dynamics using a non-ideal drive—Theoretical and experimental aspects

The possible contact between rotor and stator is considered a serious malfunction that may lead to catastrophic failure. Rotor rub is seen as a secondary phenomenon caused by a primary source, i.e. sudden mass unbalance, instabilities generated by aerodynamic and hydrodynamic forces in seals and bearings among others. The contact event gives rise to normal and friction forces exerted on the rotor at impact events. The friction force plays a significant role by transferring some rotational energy of the rotor to lateral motion. A mathematical model has been developed to capture this for a conventional backup annular guide setup. It is reasonable to superpose an impact condition to the rub, where the rotor spin energy can be fully transformed into rotor lateral movements. Using a nonideal drive, i.e. an electric motor without any kind of velocity feedback control, it is even possible to stop the rotor spin under rubbing conditions. All the rotational energy will be transformed in a kind of “self-excited” rotor lateral vibration with repeated impacts against the housing. This paper studies the impact motion of a rotor impacting a conventional backup annular guide for the case of dry and lubricated inner surface of the guide. For the dry surface case, the experimental and numerical analysis shows that the rotational energy is fully transformed into lateral motion and the rotor spin is stopped. Based on this study this paper proposes a new unconventional backup bearing design in order to reduce the rub related severity in friction and center the rotor at impact events. The analysis shows that the rotor at impacts is forced to the center of the backup bearing and the lateral motion is mitigated. As a result of this, the rotor spin is kept constant.     

IDENTIFICATION OF SPEED-DEPENDENT BEARING PARAMETERS

Bearing dynamic characteristics have been a major unknown in the modelling and analysis of large turbo-generators. An identification algorithm for bearing dynamic characterization by using unbalance response measurements is developed for multi-degree-of-freedom (m.d.o.f.) flexible rotor-bearing systems. The algorithm identifies the bearing dynamic parameters, consisting of four effective stiffness and four damping coefficients for each bearing, utilizing frequency domain synchronous unbalance response measurements from the accelerometers attached to the bearing housings in the horizontal and vertical directions, for a minimum two different unbalance configurations. The procedure of identifying bearing dynamic coefficients by using the proposed algorithm is presented and demonstrated through a numerical example. Adding noise to the simulated signal checks the robustness of the algorithm against measurement noise. Combinations of regularization and the generalized singular value decomposition (SVD) are used to tackle an ill-posed problem due to the nearly circular orbit of the rotor at the bearings, as a special case for nearly isotropic bearings. It is demonstrated that by measuring noisy bearing responses with the direction of rotation of the rotor both in the clockwise and anticlockwise directions, the bearing estimation problem for circular orbit becomes well-conditioned. The regularization algorithm is tested for an experimental rotor-bearing rig. The response reproduction capabilities are excellent even in the presence of measurement noise.     

Nonlinear damping calculation in cylindrical gear dynamic modeling

The nonlinear dynamic problem posed by cylindrical gear systems has been extensively covered in the literature. Nonetheless, a significant proportion of the mechanisms involved in damping generation remains to be investigated and described. The main objective of this study is to contribute to this task. Overall, damping is assumed to consist of three sources: surrounding element contribution, hysteresis of the teeth, and oil squeeze damping. The first two contributions are considered to be commensurate with the supported load; for its part however, squeeze damping is formulated using expressions developed from the Reynolds equation. A lubricated impact analysis between the teeth is introduced in this study for the minimum film thickness calculation during contact losses. The dynamic transmission error (DTE) obtained from the final model showed close agreement with experimental measurements available in the literature. The nonlinear damping ratio calculated at different mesh frequencies and torque amplitudes presented average values between 5.3 percent and 8 percent, which is comparable to the constant 8 percent ratio used in published numerical simulations of an equivalent gear pair. A close analysis of the oil squeeze damping evidenced the inverse relationship between this damping effect and the applied load.     

Modeling of a rotor speed transient response with radial rubbing

A rotor-stator model of a turbogenerator is introduced in order to investigate speed transients with rotor-to-stator rubbing caused by an accidental blade-off imbalance. In order to assess the angular deceleration of the rotor due to rubbing, the angular position of its cross-section is considered as an unknown of the problem. Displacement fields are discretized through a finite element formulation. The highly nonlinear equations due to contact conditions are solved through an explicit prediction-correction time-marching procedure combined with the Lagrange multiplier approach dealing with a node-to-line contact strategy. The developed numerical tool is suitable for analyzing rotor-stator interactions in turbomachines as the system passes through critical speeds during an accidental shutdown. The sensitivity of the system response to modeling, physical and numerical parameters is investigated. The results highlight the significant role of the friction coefficient together with the diaphragm modeling, from rigid to fully flexible, in the interaction phenomenon. Rigid models have the advantage of simplicity and provide reasonable estimations of the overall response of the turbine. A flexible model, however, may be more computationally intensive but is more appropriate in order to accurately capture quantities of interest such as shaft eccentricity and bearing loads.     

Three-dimensional nonlinear vibration of gear pairs

This work investigates the three-dimensional nonlinear vibration of gear pairs where the nonlinearity is due to portions of gear teeth contact lines losing contact (partial contact loss). The gear contact model tracks partial contact loss using a discretized stiffness network. The nonlinear dynamic response is obtained using the discretized stiffness network, but it is interpreted and discussed with reference to a lumped-parameter gear mesh model named the equivalent stiffness representation. It consists of a translational stiffness acting at a changing center of stiffness location (two parameters) and a twist stiffness. These four parameters, calculated from the dynamic response, change as the gears vibrate, and tracking their behavior as a post-processing tool illuminates the nonlinear gear response. There is a gear mesh twist mode where the twist stiffness is active in addition to the well-known mesh deflection mode where the translational stiffness is active. The twist mode is excited by periodic back and forth axial movement of the center of stiffness in helical gears. The same effect can occur in wide facewidth spur gears if tooth lead modifications or other factors such as shaft and bearing deflections disrupt symmetry about the axial centers of the mating teeth. Resonances of both modes are shown to be nonlinear due to partial and total contact loss. Comparing the numerical results with gear vibration experiments from the literature verifies the model and confirms partial contact loss nonlinearity in experiments.     

Receptances of non-proportionally and continuously damped plates—Reduced dampers method

This paper presents a method for the dynamic analysis of continuously and non-proportionally damped plates in bending modes. The damping can be in the form of constrained or unconstrained layers. The method is an extension of the equivalent dampers method discussed in a previous paper, in which the damping matrix of a discretized plate is replaced by a diagonal equivalent damping matrix. Each diagonal element represents an equivalent damper inserted between the structure and ground. In this method the number of equivalent dampers is reduced so that the receptance matrix of the damped structure can be obtained economically by a direct matrix method. The receptances of two different partially coated plates in transverse directions are computed by the method suggested. The verification of the results is demonstrated by comparison with the experimental values and also with the theoretical values obtained by the equivalent dampers method. The method presented can also be applied to the transverse vibration analysis of plates with discrete damping inserts.     

Design and construction of shaft-driving type piezoceramic ultrasonic motor

A new approach in design of shaft-driving type piezoceramic ultrasonic motor is proposed. The stator of motor consisted of a commercial available buzzer disk in which a piezoceramic membrane is adhered to a metal sheet. The wave propagation on the metal sheet was generated by extended-shrunk force from piezoceramic oscillation. Driving energy came from the vibration modes by mechanical-electrical oscillation of the metal sheet in corresponding to converse piezoelectric effect using a single-phase AC voltage power. Where the relative elliptic motion was occurred between the bearing seat and rotor in order to kinematical delivery, the rotor being driven was connected directly on the bearing seat to transmit the dynamic power with frictional contact force. In analysis of dynamic features, the system transfer function of admittance and equivalent circuit was obtained. The rotating speed of the prototype motor could be reached as high as 2000 rpm on the driving condition of 72 kHz, +/-10 V(pp), and 0.2 A. The maximum torque was less than 0.003 Nm. It could be utilized in the driver of CD, or the cooling fan in the computer CPU.     

同轴脉冲形成线的运输振动环境适应性研究

为了准确评估某MV级Tesla型脉冲功率源的运输振动环境适应性水平,针对脉冲功率源中采用悬臂绝缘支撑结构的同轴脉冲形成线,通过仿真和试验结合的方法开展研究。针对叠层结构式内外磁芯,提出一种通过结构元胞等效材料参数和坐标变换的等效建模方法进行有限元建模,通过模态试验修正有限元模型,首次对比研究了绝缘油对形成线的模态频率和阻尼的影响,仿真分析了形成线车载运输典型工况的应力及响应,设计实施了大尺寸形成线等效件振动试验进行验证。通过振动试验发现了形成线结构存在非线性,通过仿真分析和等效件试验验证,同轴脉冲形成线现有结构设计基本满足车载运输振动环境适应性要求。     

A theoretical-experimental comparison of the synchronous response of a bowed rotor in five different sets of fluid film bearings

A comparison of theoretical and experimental synchronous unbalance responses of a bowed Jeffcott rotor in fluid film bearings has been completed. A transfer matrix method was used to predict theoretically the response of a 25·4 mm shaft in fluid film bearings and results were compared to data from a previous experimental study. Four bearing types were used: two axial groove, pressure dam, tilting pad and four-lobe. Very good agreement was found for all bearing types at the rotor critical speed (3000 rpm). Differences less than 15% in peak response were found and the theoretical and experimental peaks were found to occur within 200 rpm. Worst agreement was found for the preloaded four-lobe bearings and this disagreement was found for speeds other than the critical speed. Also, for equal bow and unbalance the tilting pad and four-lobe bearings were found to produce the least and most damping at the critical speed, respectively. Previous to this time a comparison of theoretical and experimental synchronous responses of a rotor system representing industrial turbomachines has not been published, nor has a comparative study of the different bearing types.     

单框架控制力矩陀螺转子动不平衡对遥感卫星成像的影响

分析了单框架控制力矩陀螺（SGCMG）转子在高速旋转时产生的动不平衡干扰力矩引起的星体颤振角和颤振角速度对TDI CCD相机成像的影响。通过坐标变换将转子坐标系下的干扰力矩转换至星体坐标系下的干扰力矩,将卫星姿态动力学方程计算出干扰力矩引起的星体颤振角位移和颤振角速度代入TDI CCD相机成像像移补偿模型;利用TDICCD相机像点与物点对应模型的仿真系统仿真了陀螺转子在不同转速下引起星体颤振角位移和角速度对相机成像的影响;最后,利用图像对比度和互相关相似性测度分析仿真成像质量。仿真显示：SGCMG转子在转速为3 000 r/min时,横向调制传递函数为0.997,图像互相关相似性测度为0.996 1;转速为6 000 r/min时,横向调制传递函数为0.928 3,图像互相关相似性测度为0.974 8。结果表明：SGCMG转子在高速旋转过程中引起的星体颤振角位移和角速度严重影响了TDI CCD相机的成像质量,应依据颤振的影响对SGCMG实施减震措施。     

某型内转子光电雷达稳定平台的模态分析与优化

为了更好地设计某型内转子光电雷达稳定平台,运用Pro-E软件对其进行了三维实体建模,合理简化后,进入Ansys软件建立有限元模型.应用有限元理论和Ansys软件对该平台进行了模态分析,得到稳定平台前八阶模态的固有频率和振型.其中对比了不同网格、不同材料产生的不同模态,对同类分析提供了参考,也为此稳定平台的设计优化提供了...     

Finite element modelling for the investigation of in-plane modes and damping shims in disc brake squeal

Squeal propensity of the in-plane modes and the constrained-layer type damping shims for disc brake system is investigated by using the finite element method. Theoretical formulation is derived for a rotating disc in contact with two stationary vibrating pads attached to the damping shim components. By the conversion from the theoretical to FE brake model, the full equations of motion for the actual disc brake system describes the disc rotation, the in-plane friction characteristics and damping shims in association with squeal vibration. It is concluded from the results that the in-plane torsion modes can be generated by the negative friction slope, but they cannot be controlled by the damping shims. The in-plane radial mode is also investigated and found to be very insensitive in squeal generation.