Test Results of Key and Spherical Pivot Five-Shoe Tilt Pad Journal Bearings—Part II: Dynamic Measurements

Tests were conducted to measure the dynamic characteristics for two types of five-shoe tilt pad journal bearings having a load between pads configuration. Dynamic measurements of the two bearings are presented for 5000, 7000 and 12,000 rpm, and unit loads ranging from 126 kPa to 1724 kPa. Comparisons are made between the measured data and the theoretical prediction. Test data includes fluid film stiffness and damping, with calculated uncertainties and the measured radial stiffness of the pivots. The measured dynamic characteristics show the bearings differ with respect to stiffness and damping, while the measured radial pivot stiffnesses were nearly identical     

Severity Estimation of Cracked Shaft Vibrations within Fluid Film Bearings

The equations of motion, with four degrees of freedom, taking into consideration the flexibility, damping and cross coupling of the fluid film bearings are derived for a cracked Jeffcott rotor supported on fluid film bearings.

Dimensionless equations are developed for dynamic radial load, dynamic pressure developed in the fluid film bearings and coefficient of dissipation considering the journal vibrations in two harmonics; bearing fluid film stiffness and damping coefficients. These are applied to a cracked Jeffcott rotor supported on different types of bearings, i.e., cylindrical journal bearings, offset cylindrical bearings, tilting pad journal bearings and three-lobe bearings. Based on the allowable dynamic pressure developed in the fluid bearings, the severity of cracked shaft and allowable crack depths are estimated in this study. Measurement of dynamic pressure and dissipation for monitoring the crack growth is suggested. However, 2x vibration is the best indicator of cracks in the shafts.     

偏载下大长径比水润滑轴承分布式动特性参数识别方法

为了解决偏载下大长径比水润滑轴承分布式动特性参数识别问题,建立水润滑轴承的两支点分布式动力学模型,提出水润滑轴承分布式动特性参数的识别方法,并验证该动特性识别方法的可靠性。通过仿真试验,引入加载和位移信号扰动对轴承分布式动特性参数识别的精度进行分析,开展偏载水润滑轴承分布式动特动特性试验。结果表明：随着激振力振幅扰动的增加,刚度和阻尼系数的识别误差线性增加;激振力相位扰动对刚度系数的影响较小,对阻尼系数的影响较大;随着位移幅值扰动的增加,刚度和阻尼系数的识别误差增加;位移信号的相位扰动对刚度系数的影响较小,对阻尼系数的影响较大;若要求刚度和阻尼的识别误差小于10%,则激振力和位移信号的扰动幅值应小于10%;若要求刚度和阻尼的识别误差小于20%,则这2个信号的扰动相位扰动偏差应小于1°。     

A Method for Identification of Bearing Force Coefficients and Its Application to a Squeeze Film Damper with a Bubbly Lubricant

A general formulation of the instrumental variable filter (IVF) method for parameter identification of a n-DOF (Degrees Of Freedom) mechanical linear system is presented. The IVF is a frequency domain method and an iterative variation of the least-squares approximation to the system flexibilities. Weight functions constructed with the estimated flexibilities are introduced to reduce the effect of noise in the measurements, thus improving the estimation of dynamic force coefficients. The IVF method is applied in conjunction to impact force excitations to estimate the mass, stiffness, and damping coefficients of a test rotor supported on a squeeze film damper (SFD) operating with a bubbly lubricant. The amount of air in the lubricant is varied from nil to 100 percent to simulate increasing degrees of severity of air entrainment into the damper film lands. The experimental results and parameter estimation technique show that the SFD damping force coefficients increase as the air volume fraction in the mixture increases to about 50 percent in volume content. The damping coefficients decrease rapidly for mixtures with larger air concentrations. The unexpected increase in direct damping coefficients indicates the complexity of the SFD bubbly flow field and warrants further experimental verification.     

On the Dynamics of Lubricated Cylindrical Roller Bearings,Part I: Evaluation of Stiffness and Damping Characteristics

The stiffness and damping coefficients of a single roller-to-race contact of lubricated cylindrical roller bearings are numerically evaluated using a linearized perturbation method for both elastohydrodynamic lubrication (EHL) finite and infinite contact theories. A steady-state pressure equation is solved by a multilevel method and the elastic deformation is evaluated with the multilevel multi-integration method. Dynamic pressures are obtained by solving a set of perturbed pressure equations and are used to calculate the stiffness and damping coefficients. The influence of various nondimensional parameters (load parameter, speed parameter, material parameter, edge radius, and geometrical parameter) on the stiffness and damping are studied. The results show that a finite line contact gives higher values of stiffness and damping coefficients compared to an infinite contact, particularly at higher load and lower speed values. Based on the numerically evaluated data, curve-fitted relations for the stiffness and damping coefficients of a single roller-to-race contact are developed that can be used in the dynamic analysis of rotor–bearing systems.     

On the Dynamics of Lubricated Cylindrical Roller Bearings,Part II: Linear and Nonlinear Vibration Analysis

This article is the second part of two companion papers. In the first article, curve-fitted relations of stiffness and damping coefficients of a single roller-to-race contact of lubricated roller bearings were developed. In the present work, these relations are applied to a rotor–bearing system. Two cases are studied to investigate the influence of lubricated cylindrical roller bearings on the vibration characteristics of the rotor system. In the first case, lubricated contacts are simulated as a linear spring–damper model. The overall stiffness and damping matrices are calculated by using the dynamic coefficients of individual load sharing rollers. These matrices are used in the finite element analysis of flexible rotor. In the second case, the nonlinear structural vibration of a lubricated cylindrical roller bearing is studied. Equations of motion of bearing elements are derived using the Lagrange equation. A nonlinear load–deflection contact model developed through the derived curve-fitted relations of dynamic coefficients is used in the equations of motion. Equations of motion are solved by a fourth-order Runge-Kutta integration method. The response of bearing elements under free vibration and due to rotating unbalance is studied for damped and undamped cases. Furthermore, results obtained using elastohydrodynamic finite and infinite contact theories are compared.     

Experimental Study of a New Compliant Foil Air Bearing with Elastic Support

A new air-lubricated compliant foil journal bearing with elastic support, which has uniform surface stiffness and is much simpler in structure than previous compliant foil bearings (CFBs), is introduced in this article. Experiments have been conducted on the application of this type of CFB to a high-speed test rig, and this CFB can operate stably at 151,000 rpm. From the tests it is clear that the radial clearance C has a direct impact on the performance of this CFB, so the numerical relationship of structural parameters is listed in this article. Experimental results indicate that the CFB presented here offers preferable system dynamic and stability performance and has adequate damping to effectively reduce the possibility of self-excited and fractional frequency whirl.     

Structural Stiffness and Coulomb Damping in Compliant Foil Journal Bearings: Theoretical Considerations

Compliant foil bearings operate on either gas or liquid, which makes them very attractive for use in extreme environments such as in high-temperature aircraft turbine engines and cryogenic turbopumps. However, a lack of analytical models to predict the dynamic characteristics of foil bearings forces the bearing designer to rely on prototype testing, which is time-consuming and expensive. In this paper, the authors present a theoretical model to predict the structural stiffness and damping coefficients of the bump foil strip in a journal bearing or damper. Stiffness is calculated based on the perturbation of the journal center with respect to its static equilibrium position. The equivalent viscous damping coefficients are determined based on the area of a closed hysteresis loop of the journal center motion. The authors found, theoretically, that the energy dissipated from this loop was mostly contributed by the frictional motion between contact surfaces. In addition, the source and mechanism of the nonlinear behavior of the bump foil strips were examined. With the introduction of this enhanced model, the analytical tools are now available for the design of compliant foil bearings.     

Effects of Three-Lobe Bearing Geometries on the Unbalance Response of a Flexible Rotor

The synchronous response due to unbalance was studied both theoretically and experimentally for a three-mass rotor mounted in three-lobe and axial-groove bearings. Offset factor, preload factor and load orientation were parametrically varied for the three-lobe bearings. Theoretical predictions indicate that low preload factors (with offset factors of 0.5) result in the smallest critical speed responses. Also, for a preload factor of 0.75, offset factors of approximately 0.65 minimize the critical speed response. Experimental data for peak responses agree within 42 percent (usually much less) of predictions. Reasons for the differences are attributed to the predicted bearing stiffness and damping coefficients.     

Dynamic Stiffness and Damping Characteristics of a High-Temperature Air Foil Journal Bearing

Using a high-temperature optically based displacement measurement system, a foil air bearing s stiffness and damping characteristics were experimentally determined. Results were obtained over a range of modified Sommerfeld Number from 1.5E6 to 1.5E7, and at temperatures from 25° to 538°C.

An Experimental procedure was developed comparing the error in two curve fitting functions to reveal different modes of physical behavior throughout the operating domain. The maximum change in dimensionless stiffness was 3.0E-2 to 6.5E-2 over the Sommerfeld Number range tested. Stiffness decreased with temperature by as much as a factor of two from 25° to 538°C. Dimensionless damping was a stronger function of Sommerfeld Number ranging from 20 to 300. As the temperature is increased, the damping shifts from a viscous type to a frictional type.     

Identification of the dynamic parameters of active magnetic bearings based on the transfer matrix model updating method

The stiffness and damping coefficients of Active magnetic bearings (AMBs) have a great impact on the dynamics of a high-speed rotor AMB system, from its bending critical speed to the modes of its vibration and stability. To accurately obtain the stiffness and damping coefficients of AMBs, this study proposes a new identification approach based on the transfer matrix model updating method. By minimizing the error between the unbalance response calculated through the transfer matrix approach and the experimental measurements, the stiffness and damping coefficients are obtained using the simplex optimization algorithm based on the updating method of the model. According to the experimental data, we identify the parameters from 20 Hz to 260 Hz (1200 rpm to 15600 rpm). To verify the identified results, a finite element rotor AMBs model is created, and the theoretical unbalance response is predicted using the identified parameters. The theoretical unbalance responses closely coincide with the experimental measurements, indicating the effectiveness of the proposed method.     

Elastohydrodynamically Lubricated Ball Bearings with Couple-Stress Fluids,Part II: Dynamic Analysis and Application

Stiffness and damping coefficients of isothermal elastohydrodynamically lubricated point-contact problems are evaluated numerically with couple-stress fluids. A set of equations under steady-state and dynamic conditions is derived from the modified Reynolds equation using a linearized perturbation method. This paper is the second part of the present study; the modified Reynolds equation derived from the Stokes micro-continuum theory is used in the previous article. Dynamic pressures are found after solving the set of perturbed equations using the previously obtained steady-state pressure from the modified Reynolds equation. The stiffness and damping coefficients of the film are determined using the dynamic pressures. Then the overall stiffness and damping matrices of the ball bearing are obtained from load distribution, coordinate transformation, and compatibility relations. The bearing coefficients are introduced into a rotor system to simulate the response. It has been observed that the influence of couple-stress fluids on the dynamics of a rotor supported on lubricated ball bearings is marginal; hence, Newtonian theory can be used instead for simplicity. However, with increasing content of polymer additives in lubricant, for an accurate analysis the effect of couple stresses in a fluid should not be neglected.     

Stiffness and Damping Coefficient Estimation of Compliant Surface Gas Bearings for Oil-Free Turbomachinery

Gas foil bearings are a key technology in many commercial and emerging oil-free turbomachinery systems. These bearings are nonlinear and have been difficult to analytically model in terms of performance characteristics such as load capacity, power loss, stiffness, and damping. Previous investigations led to an empirically derived method, a rule-of-thumb, to estimate load capacity. This method has been a valuable tool in system development. The current article extends this concept to include rules for stiffness and damping coefficient estimation. It is expected that these rules will further accelerate the development and deployment of advanced oil-free machines operating on gas foil bearings.     

Effect of Fluid Inertia on Journal Bearing Parameters

In the modeling and analysis of rotordynamic systems with journal bearings, the stiffness and damping parameters are usually obtained from the Reynolds equation of hydrodynamic lubrication. The Reynolds equation is derived from the continuity and momentum equations with several assumptions; the principal one among them being that the inertia terms are negligible since the lubricant flow is viscosity-dominated. Some previous work has shown, however, that the effect of fluid inertia on the static and dynamic properties of a bearing is not negligible in many circumstances.

This paper uses a perturbation approach to present a rigorous derivation of the correction terms to be added to account for the effect of inertia in the case of a journal on a short bearing. The governing equation for pressure correction is derived and the corrected stiffness, damping, and inertia coefficients to the first order are displayed as a function of the equilibrium position.     

The Static and Dynamic Characteristics of a Two-Lobe Journal Bearing Lubricated with Couple-Stress Fluid

Static and dynamic characteristics of two-lobe journal bearings lubricated with couple-stress fluids are studied. The load-carrying capacity, the stiffness and damping coefficients, the non-dimensional critical mass, and the whirl ratio are determined for various values of the couple stress parameter l. The results obtained are compared with the characteristics of two-lobe bearings lubricated with Newtonian fluids. It is found that the effect of the couple stress parameter is very significant on the performance of the journal bearing. The stability is improved compared to bearings lubricated with Newtonian fluids.     

混合轴承动特性简化计算方法

用一维流计算原理分析了多腔混合轴承的动态特性；计算了典型的毛细管节流四腔轴承在各种设计参数下的线性化刚度系数和阻尼系数，得到了轴承相应的刚度和稳定性速度阈值；比较了简化计算方法得到的数据与传统的雷诺方程数值解法得到的数据，且讨论了前者的误差影响。在常用工况条件下，其计算结果满足工程需要。     

Effects of dynamic transmission errors and vibration stability in helical gears

Transmission error is an important reason for instability in helical gears. A six-degree-of-freedom dynamic model coupled flexional, torsional and axial motion of a helical gear transmission system, which includes time varying mesh stiffness, bearing supporting stiffness, mesh damping and backlash, is developed, after taking into account the dynamic characteristics and vibration responses of helical gear in three dimensions. Influences of involute contact ratio, bearing supporting stiffness, mesh damping and backlash on the dynamic transmission errors and vibration stability of the helical gear system are investigated using numerical simulation technique. The effects on dynamic transmission errors and stabilities by contact ratio, supporting stiffness and mesh damping as well as gear backlash are analyzed. The intrinsic relationship between above parameters and dynamic transmission errors and stabilities for helical gear system are presented. The stable and unstable regions under different parameters are given. The results in this paper can be helpful to the dynamic and stable design of a helical gear transmission system.     

Spacecraft attitude determination using a decoupling filter

In this paper, an algorithm for real-time attitude estimation of spacecraft motion is investigated. For efficient computation, the decoupling filter presented in this paper is accomplished by a derived pseudo-measurement from the given measurement and the decoupled state in the original system. However, the proposed decoupling filter contains model errors due to coupling terms in the system. Therefore, we develope an attitude determination algorithm in which coupling terms are compensated through an error analysis. The attitude estimation algorithm using the state decoupling technique for real-time processing provides accurate attitude determination capability under a highly maneuvering dynamic environment, because the algorithm does not have any bias errors from a truncation, and the covariance of the estimator is compensated by nonlinear terms in the system. To verify the performance of the proposed algorithm vis-a-vis the EKF (extended Kalman filter), and the nonlinear filter, simulations have been performed by varying the initial values of the state and covariance, and measurement covariance. Results show that the proposed algorithm has consistently better performance than the EKF in all of the ranges of initial state values and covariance values of measurement, and it is as accurate as the nonlinear filter. However, the convergence speed of the nonlinear filter is faster than the proposed algorithm because of the pseudo-measurement model errors in the proposed algorithm. We show that the computational time of the proposed algorithm is improved by about 23% over the nonlinear filter.     

Identification of the dynamic performance of a gas foil journal bearing operating at high temperatures

This paper presents an experimental investigation of the dynamic force performance of gas foil bearings (GFBs) at high temperatures. A dynamic performance test rig with a GFB mounted on a rotating hollow shaft, heated by a cartridge heater inside the hollow shaft, and excited by two orthogonally positioned electromagnetic shakers determines the frequency dependent stiffness and damping coefficients of the test GFB for increasing shaft temperatures. The test heater temperatures are 21°C (room temperature without heating), 100°C, 200°C, 300°C, and 400°C, and the excitation frequencies are 120 Hz, 140 Hz, 160 Hz, and 180 Hz. The test rotating speed and static load are 12 krpm and 30 N, respectively. The vibration amplitude of the test GFB is adjusted to approximately 30 μm by controlling the power amplifier connected to the electromagnetic shakers throughout the series of experiments. The test results show that both the direct stiffness and damping coefficients of the test GFB increase with increasing excitation frequencies. As the shaft temperature increases, the direct stiffness coefficients decrease by ～ 8%, and the direct damping coefficients decrease by approximately 30%. A model prediction benchmarked against the test data reveals that the cross-coupled stiffness coefficients are smaller than the direct stiffness coefficients for the test GFB.     

动压气体轴承的动态刚度和动态阻尼系数

采用偏导数法求解动压气体润滑Reynolds方程,给出动压气体轴承动态刚度和动态阻尼系数普遍适应的计算方法.进行有限元数值仿真,计算动压气体轴承的动态刚度和动态阻尼系数,并重点研究轴颈扰动频率和不同轴颈扰动频率下的轴承静态载荷及轴承数对这些系数的影响.从理论上解释动压气体轴承的动态刚度和动态阻尼系数与轴颈扰动频率的密切相关性.