Nonlinear dynamics of mistuned bladed disks with ring dampers

In turbomachinery applications, rotating bladed disks (blisks) are often subject to high levels of dynamic loading, such as traveling wave excitations, which result in large response amplitudes at resonance. To prevent premature high cycle fatigue, various dry friction dampers are designed for blisk systems to reduce the forced responses. Ring dampers are located in the disk, underneath the blades, and are held in contact with the blisk by centrifugal loading. Energy is dissipated by nonlinear friction forces when relative motions between the ring damper and the blisk take place. To investigate the dynamic responses of blisk–damper systems in the presence of the nonlinear frictional contacts, conventional methods based on numerical time integration are not suitable since they are computationally expensive. This paper presents a reduced-order modeling technique to efficiently capture the nonlinear dynamic responses of the blisk–damper systems. Craig–Bampton component mode synthesis (CB-CMS) serves as the first model reduction step. A novel mode basis that mimics the contact behavior under sliding and sticking conditions is developed to further reduce the CB-CMS model while maintaining its accuracy. The resulting reduced nonlinear equations of motion are solved by a hybrid frequency/time domain (HFT) method. In the HFT method, the contact status and friction forces are determined in the time domain by a three-dimensional contact model at each contact point, whereas the reduced equations of motion are solved in the frequency domain according to a harmonic balance formulation. Moreover, to investigate the effects of blade mistuning, which can lead to drastic increase of forced responses, an extension of the reduced-order models (ROMs) is developed based on component mode mistuning. Forced responses computed by the proposed ROMs are validated for both tuned and mistuned systems. A statistical analysis is performed to study the effectiveness of ring dampers under random blade mistuning patterns.     

A friction contact stiffness model of fractal geometry in forced response analysis of a shrouded blade

To investigate the nonlinear vibration behavior of a shrouded blade with friction dynamic contact interface, a friction contact stiffness model is proposed to describe the friction force at different rough interfaces and different normal loads. In the proposed model, the friction contact interface is discretized to a series of friction contact pairs and each of them can experience stick, slip, or separate states. Fractal geometry is used to simulate the topography of contact surfaces. The contact stiffness is calculated using the Hertz contact theory and fractal geometry, which is related to contact interfaces parameters including normal load, roughness, Young??s modulus, and Poisson??s ratio. The trajectory tracking method is used to predict the friction force and it is not necessary to judge the transition condition among stick, slip, and separate states. It is suitable for complicated periodic motion of the contact interfaces. The forced response of a real shrouded blade is predicted using the proposed model and the multi-harmonic balance method. The effect of surface roughness, initial normal load, and contact area on the forced response of a shrouded blade is studied. It is shown that contact stiffness increases with normal load and fractal dimension. The resonant amplitude is sensitive to the initial normal load and contact surface roughness. The response can be influenced by the contact area, which is an important parameter for blade designers.     

A direct experimental–numerical method for investigations of a laboratory under-platform damper behavior

Under-platform dampers are commonly adopted in order to mitigate resonant vibration of turbine blades. The need for reliable models for the design of under-platform dampers has led to a considerable amount of technical literature on under-platform damper modeling in the last three decades.Although much effort has been devoted to the under-platform damper modeling in order to avail of a predictive tool for new damper designs, experimental validation of the modeling is still necessary. This is due to the complexity caused by the interaction of the contacts at the two damper-platform interfaces with the additional complication of the variablity of physical contact parameters (in particularly friction) and their nonlinearity. The traditional experimental configuration for evaluating under-platform damper behavior is measuring the blade tip response by incorporating the damper between two adjacent blades (representing a cyclic segment of the bladed disk) under controlled excitation. The effectiveness of the damper is revealed by the difference in blade tip response depending on whether the damper is applied or not. With this approach one cannot investigate the damper behavior directly and no measurements of the contact parameters can be undertake. Consequently, tentative values for the contact parameters are assigned from previous experience and then case-by-case finely tuned until the numerical predictions are consistent with the experimental evidence. In this method the physical determination of the contact parameters is obtained using test rigs designed to produce single contact tests which simulate the local damper-platfom contact geometry. However, the significant limitation of single contact test results is that they do not reveal the dependence of contact parameters on the real damper contact conditions. The method proposed in this paper overcomes this problem.In this new approach a purposely developed test rig allows the in-plane forces transferred through the damper between the two simulated platforms to be measured, while at the same time monitoring in-plane relative displacements of the platforms. The in-plane damper kinematics are reconstructed from the experimental data using the contact constraints and two damper motion measurements, one translational and one rotational. The measurement procedures provide reliable results, which allow very fine details of contact kinematics to be revealed. It is demonstrated that the highly satisfactory performance of the test rig and the related procedures allows fine tuning of the contact parameters (local friction coefficients and contact stiffness), which can be safely fed into a direct time integration numerical model.The numerical model is, in turn, cross-checked against the experimental results, and then used to acquire deeper understanding of the damper behavior (e.g. contact state, slipping and sticking displacement at all contact points), giving an insight into those features which the measurements alone are not capable of producing. The numerical model of the system is based on one key assumption: the contact model does not take into account the microslip effect that exists in the experiments.Although there is room for improvement of both experimental configuration and numerical modeling, which future work will consider, the results obtained with this approach demonstrate that the optimization of dampers can be less a matter of trial and error development and more a matter of knowledge of damper dynamics.     

Transition from stick to slip in Hertzian contact with “Griffith” friction: The Cattaneo–Mindlin problem revisited

Classically, the transition from stick to slip is modelled with Amonton–Coulomb law, leading to the Cattaneo–Mindlin problem, which is amenable to quite general solutions using the idea of superposing normal contact pressure distributions – in particular superposing the full sliding component of shear with a corrective distribution in the stick region. However, faults model in geophysics and recent high-speed measurements of the real contact area and the strain fields in dry (nominally flat) rough interfaces at macroscopic but laboratory scale, all suggest that the transition from ‘static’ to ‘dynamic’ friction can be described, rather than by Coulomb law, by classical fracture mechanics singular solutions of shear cracks. Here, we introduce an ‘adhesive’ model for friction in a Hertzian spherical contact, maintaining the Hertzian solution for the normal pressures, but where the inception of slip is given by a Griffith condition. In the slip region, the standard Coulomb law continues to hold. This leads to a very simple solution for the Cattaneo–Mindlin problem, in which the “corrective” solution in the stick area is in fact similar to the mode II equivalent of a JKR singular solution for adhesive contact. The model departs from the standard Cattaneo–Mindlin solution, showing an increased size of the stick zone relative to the contact area, and a sudden transition to slip when the stick region reaches a critical size (the equivalent of the pull-off contact size of the JKR solution). The apparent static friction coefficient before sliding can be much higher than the sliding friction coefficient and, for a given friction fracture “energy”, the process results in size and normal load dependence of the apparent static friction coefficient. Some qualitative agreement with Fineberg's group experiments for friction exists, namely the stick–slip boundary quasi-static prediction may correspond to the arrest of their slip “precursors”, and the rapid collapse to global sliding when the precursors arrest front has reached about half the interface may correspond to the reach of the “critical” size for the stick zone.     

基于LuGre摩擦模型的接触约束法旋转柔性梁斜碰撞研究

基于接触约束法和LuGre摩擦模型对在重力场作用下作大范围旋转运动的柔性梁系统和斜坡发生含摩擦斜碰撞的动力学问题进行研究.首先运用刚柔耦合的多体系统动力学理论对大范围运动的柔性梁进行离散化和动力学建模,在碰撞时采用冲量动量法求出跳跃速度,其次在法向上引入接触约束求解出碰撞力,在切向上采用LuGre摩擦模型分两种方式求解...     

柔性梁斜碰撞问题的非线性动力学建模和实验研究

本文研究柔性梁点面斜碰撞问题。用Hertz接触模型处理法向撞击力,分别用Hertz切向接触模型和Coulomb摩擦力模型处理粘滞状态和滑动状态的摩擦力。从精确的应变与位移的关系出发,用绝对节点坐标法建立了柔性梁的动力学方程。为了准确地处理斜碰撞切向运动的复杂状态,提出滑动-粘滞切换的准则,在此基础上,设计了斜碰撞实验,数值对比了法向撞击力和法向速度的时间历程的仿真计算结果与实验结果,验证了Hertz理论在斜碰撞情况下的正确性。另一方面切向速度的实验与理论的结果对照表明滑动-粘滞切换准则的有效性。     

Micromechanical analysis of friction anisotropy in rough elastic contacts

Computational contact homogenization approach is applied to study friction anisotropy resulting from asperity interaction in elastic contacts. Contact of rough surfaces with anisotropic roughness is considered with asperity contact at the micro scale being governed by the isotropic Coulomb friction model. Application of a micro-to-macro scale transition scheme yields a macroscopic friction model with orientation- and pressure-dependent macroscopic friction coefficient. The macroscopic slip rule is found to exhibit a weak non-associativity in the tangential plane, although the slip rule at the microscale is associated in the tangential plane. Counterintuitive effects are observed for compressible materials, in particular, for auxetic materials.     

Passive control of vibration of thin-walled gears: advanced modelling of ring dampers

Vibrations on gears are mainly induced by the gear mesh contact. Resonance conditions of the gear may occur during service if the mesh frequency is close to the natural frequencies of the system at the designed speed of the shaft. Since detuning is not always possible in gears, the response level must be reduced by increasing the damping of the system. In this paper, a passive approach based on the application of a ring damper to reduce the vibration level is presented. The ring damper is placed in a groove underneath the outer rim of the gear. The contact is guaranteed by the preload due to the elasticity of the ring damper itself and above all by the centrifugal force that presses the damper against the groove during rotation. The relative motion of the two components at the contact interface dissipates energy by friction, and hence damping is generated. The vibration amplitude is reduced by optimizing the material and geometrical properties of the ring damper. One of the most important parameters in the determination of the amount of damping due to friction phenomena is the static normal load at the contact, which depends on the mass, the shape, and the material of the ring damper. A numerical method is presented, which couples the static and dynamic equilibrium equations of the assembly. The core of the proposed method is the contact element that takes into account local stick–slip–lift off of the contact and determines the contact forces in terms of static and dynamic loads, which are then used to solve the coupled static and dynamic equilibrium. Since the ring damper has a cut that breaks its continuous circular shape in order to be fitted on the groove, the hypothesis of cyclic symmetry for the gear/ring–damper assembly fails. As a consequence, an appropriate reduced-order modeling is presented to allow the forced response calculations. The algorithm is applied to a dummy bevel gear and to a ring damper having a flat punch contact area. The forced response calculations are performed to highlight the nonlinear interaction between the gear and damper by varying the parameters that mainly affect the amount and distribution of the contact forces and therefore the response level.     

The effect of underplatform dampers on the forced response of bladed disks by a coupled static/dynamic harmonic balance method

Friction contacts are often used in turbomachinery design as passive damping systems. In particular, underplatform dampers are mechanical devices used to decrease the vibration amplitudes of bladed disks.Numerical codes are used to optimize during designing the underplatform damper effectiveness in order to limit the resonant stress level of the blades. In such codes, the contact model plays the most relevant role in calculation of the dissipated energy at friction interfaces. One of the most important contact parameters to consider in order to calculate the forced response of blades assembly is the static normal load acting at the contact, since its value strongly affects the area of the hysteresis loop of the tangential force, and therefore the amount of dissipation.A common procedure to estimate the static normal loads acting on underplatform dampers consists in decoupling the static and the dynamic balance of the damper. A preliminary static analysis of the contact is performed in order to get the static contact/gap status to use in the calculation, assuming that it does not change when vibration occurs.In this paper, a novel approach is proposed. The static and the dynamic displacements of the system (bladed disk+underplatform dampers) are coupled together during the forced response calculation. Static loads acting at the contacts follow from static displacements and no preliminary static analysis of the system is necessary.The proposed method is applied to a numerical test case representing a simplified bladed disk with underplatform dampers. Results are compared with those obtained with the classical approach.     

Modeling and waveform optimization of stick–slip micro-drives using the method of dimensionality reduction

In this paper, the dynamics of piezo-actuated stick–slip micro-drives are studied experimentally and theoretically. First, the stick–slip-based force-generating test stand is introduced, and experimental results are presented. Then, a numerical model is formulated which explicitly includes the dynamics of normal and tangential properties of the contact areas in the frictional driving elements of the drive. The contact forces are simulated using the method of dimensionality reduction. We show that the experimentally observed behavior can be described without using any fitting parameters or assuming any generalized laws of friction if the explicit contact mechanics of the frictional contacts is taken into account. Furthermore, an even simpler model of the drive is developed to get a qualitative understanding of the system. It is employed to gain a new actuation method, which reduces the vibrations of the drive’s runner and therefore enhances its performance.     

Dynamics of windscreen wiper blades: Squeal noise,reversal noise and chattering

The motion of a windshield wiper blade is modelled by a mass-spring-damper system on a moving frictional surface. The system dynamics is time-varying, since three different regimes of motion, characterized by different degrees of freedom, are possible. Indeed the system, which schematizes a blade cross-section, can experience stick and slip motions when it is in contact with the glass surface, and free-flight motion when it is detached. The contact between the system and the surface is governed by Stribeck׳s friction law and Poisson׳s impact law, which make the dynamics non-smooth. The model is numerically implemented in an event-driven code, and simulations are performed which reproduce the three basic classes of undesired oscillations observed in the motion of real windscreen wipers, i.e., squeal, reversal and chattering noises. Attention is focused on the causes of these vibrations, and remedies for reducing or avoiding them are proposed.     

A numerical analysis of partial slip problems under Hertzian contacts

In this study the tangential partial slip problems in Hertzian contact regions are treated by a numerical technique. The tangential loading may include tangential forces in the contact plane and a twisting moment normal to the contact plane. The Coulomb’s law of friction and the property that the direction of friction must oppose the relative motion lead to nonlinear equations. The Newton-Raphson method is utilized to solve these nonlinear equations. Numerical results for tangential tractions and sizes of stick and slip zones may be determined, and they agree with existing analytical results for circular contacts.     

A comprehensive study on the behavior of a rigid block on an oscillating ground with friction,elastic and viscous forces

This paper investigates the behavior of a non-linear mechanical model where a block is driven by an oscillating ground through Coulomb friction, a linear viscous damper and a linear spring. The governing equation is solved analytically for different partial configurations: friction only, friction with viscous damping, friction with a linear restoring force, and for the complete model. Using dimensionless groups, the analysis of the block motion provides a comprehensive set of information on the motion regime (stick, stick-slip or permanent sliding), on the dominant energies or forces, on the resonance and on the amplification of the ground oscillation by the system. The limit between the stick-slip regime and the permanent slipping regime is found either analytically or numerically. It is also shown that there exists a set of parameters for which the friction force, the viscous dissipative force and the elastic restoring force are equal.     

Hertzian fracture at unloading

Hertzian fracture through indentation of flat float glass specimens by steel balls has been examined experimentally. Initiation of cone cracks has been observed and failure loads together with contact and fracture radii determined at monotonically increasing load but also during unloading phases. Contact of dissimilar elastic solids under decreasing load may cause crack inception triggered by finite interface friction and accordingly the coefficient of friction was determined by two different methods. In order to make relevant predictions of experimental findings, a robust computational procedure has been developed to determine global and local field values in particular at unloading at finite friction. It was found that at continued loading it is possible to specify in advance how the contact domain divides into invariant regions of stick and slip. The maximum tensile stress was found to occur at the free surface just outside the contact contour, the relative distance depending on the different elastic compliance properties and the coefficient of friction. In contrast, at unloading invariance properties are lost and stick/slip regions proved to be severely history dependant and in particular with an opposed frictional shear stress at the contact boundary region. This causes an increase of the maximum tensile stress at the contour under progressive unloading. Predictions of loads to cause crack initiation during full cycles were made based on a critical stress fracture criterion and proved to be favourable as compared to the experimental results.     

On the Analytical Approach to Galin’s Stick-Slip Problem. A Survey

Galin’s classical work (PMM J Appl Math Mech 9:413–424, 1945) on the contact of a rigid flat-ended indenter with an elastic half-plane with partial slip was the first successful attempt to take into account friction in the problem of normal contact. As Galin was unable to find an exact solution of the formulated problem, the problem of contact with partial slip of a rigid punch with an elastic half-plane was challenged by many researchers. At the same time Galin’s seminal work stimulated development of solutions for other contact problems with friction that feature different punch geometries and different material responses. This paper presents an overview of the developments in the area of elastic contact with partial slip. In the spirit of Galin’s work the focus is placed on contributions with substantial analytical merit.      

New method for oblique impact dynamics research of a flexible beam with large overall motion considering impact friction force

A flexible beam with large overall rotating motion impacting with a rigid slope is studied in this paper. The tangential friction force caused by the oblique impact is analyzed. The tangential motion of the system is divided into a stick state and a slip state. The contact constraint model and Coulomb friction model are used respectively to deal with the two states. Based on this hybrid mod-eling method, dynamic equations of the system, which include all states (before, during, and after the collision) are obtained. Simulation results of a concrete example are compared with the results obtained from two other models: a nontangential friction model and a modified Coulomb model. Differences in the results from the three models are discussed. The tangential friction force cannot be ignored when an oblique impact occurs. In addition, the results obtained from the model proposed in this paper are more consistent with real movement.     

一般各向异性单侧接触界面上波的反射和折射

研究简谐弹性波在一般各向异性介质单侧接触界面上的反射和折射问题．利用Fouier分析方法将非线性Coulomb摩擦接触边界波动问题化为一组代数方程．给出了确定局部分离、滑移和粘着区的思路和方法及各区域的解；讨论了出现界面局部分离和滑移的条件．对特定材料组合情况进行了详细数值计算，给出了界面力、相对滑移速度、张开位移、高频谐波的反射折射系数等特征参量；考察了平面和反平面波动的耦合及整体滑移等．其中关于高频谐波的结果可对已有实验结果给出很好的定性解释．在大多数情况下，即使对摩擦系数无穷大的粘滞接触界面，分离区端部也总是存在一个很小的滑移区。     

High-speed frictional slip at metal-on-metal interfaces

In this study plate-impact pressure-shear friction experiments are employed to investigate dynamic slip resistance and time-resolved growth of molten metal films during dry metal-on-metal slip under extreme interfacial conditions. By employing a tribo-pair comprising of a hard tool-steel against a relatively low melt-point metal (7075-T6 Al alloy), interfacial normal stress of up to 5 GPa and slip speeds of approximately 250 m/s have been achieved. These extreme interfacial conditions are conducive to the development of molten metal film at the tribo-pair interface. A Lagrangian finite element code is developed to understand the evolution of the thermo-mechanical fields and their relationship to the observed slip response. The code accounts for dynamic effects, heat conduction, contact with friction, and full thermo-mechanical coupling. At temperatures below the melting point of the tribo-pair materials are described as isotropic, thermally softening, elastic–viscoplastic solids. For material elements with temperatures in excess of the melt temperature a purely Newtonian fluid constitutive model is employed.     

Direct simulation of multiphase flows with modeling of dynamic interface contact angle

We describe a modeling technique for dynamic contact angle between a phase interface and a solid wall using a generalized Navier boundary condition in the context of a front-tracking-based multiphase method. The contact line motion is determined by the generalized Navier slip boundary condition in order to eliminate the infinite shear stress at the contact line. Applying this slip boundary condition only to the interface movement with various slip ratios shows good agreement with experimental results compared to allowing full fluid slip along the solid surface. The interface slip model performs well on grid convergence tests using both the slip ratio and slip length models. A detailed energy analysis was performed to identify changes in kinetic, surface, and potential energies as well as viscous and contact line dissipation with time. A friction coefficient for contact line dissipation was obtained based on the other computed energy terms. Each energy term and the friction coefficient were compared for different grid resolutions. The effect of varying the slip ratio as well as the contact angle distribution versus contact line speed was analyzed. The behavior of drop impact on a solid wall with different advancing and receding angles was investigated. Finally, the proposed dynamic contact model was extended to three dimensions for large-scale parallel calculations. The impact of a droplet on a solid cylinder was simulated to demonstrate the capabilities of the proposing formulation on general solid structures. Widely different contact angles were tested and showed distinctive characteristic behavior clearly.     

On the dynamic vibrations of an elastic beam in frictional contact with a rigid obstacle

Existence and uniqueness results are established for weak formulations of initial-boundary value problems which model the dynamic behavior of an Euler-Bernoulli beam that may come into frictional contact with a stationary obstacle. The beam is assumed to be situated horizontally and may move both horizontally and vertically, as a result of applied loads. One end of the beam is clamped, while the other end is free. However, the horizontal motion of the free end is restricted by the presence of a stationary obstacle and when this end contacts the obstacle, the vertical motion of the end is assumed to be affected by friction. The contact and friction at this end is modelled in two different ways. The first involves the classic Signorini unilateral or nonpenetration conditions and Coulomb's law of dry friction; the second uses a normal compliance contact condition and a corresponding generalization of Coulomb's law. In both cases existence and uniqueness are established when the beam is subject to Kelvin-Voigt damping. In the absence of damping, existence of a solution is established for a problem in which the normal contact stress is regularized.The work of the last two authors was supported in part by Oakland University Research Fellowships.