Static analysis of multi-support spindle shafts with nonlinear elastic bearings

In this paper a mathematical model and computational tool are developed for the static analysis of multi-bearing spindle shafts with nonlinear elastic supports. Based on the Timoshenko beam theory a resolving system of equations is obtained that takes into account the nonlinear dependence of the bearing stiffness on the reaction forces acting upon them. A solution method is proposed and appropriate software is developed that implements the static analysis of multi-support spindle shafts with non-linearly elastic bearings in MATLAB environment. Key words: spindle, shaft, nonlinear elastic support, multi-bearing, nonlinear elastic stiffness, Timoshenko beam.

Optimal Controller Design for Ultra-Precision Fast-Actuation Cutting Systems

Fast-actuation cutting systems are in high demand for machining of freeform optical parts. Design of such motion systems requires good balance between structural hardware and controller design. However, the controller tuning process is mostly based on human experience, and it is not feasible to predict positioning performance during the design stage. In this paper, a deterministic controller design approach is adopted to preclude the uncertainty associated with controller tuning, which results in a control law minimizing positioning errors based on plant and disturbance models. Then, the influences of mechanical parameters such as mass, damping, and stiffness are revealed within the closed-loop framework. The positioning error was reduced from 1.19 nm RMS to 0.68 nm RMS with the new controller. Under the measured disturbance conditions, the optimal bearing stiffness and damping coefficient are 1.1×105 N/m and 237.7 N/(m·s−1), respectively. We also found that greater moving inertia helps to reduce all disturbances at high frequencies, in agreement with the positioning experiments. A quantitative understanding of how plant structural parameters affect positioning stability is thus shown in this paper. This is helpful for the understanding of how to reduce error sources from the design point of view.

A New Model for Studying Nonlinear Vibration Behaviors of Gear-Bearing System

A new model for nonlinear vibration behaviors of gear-bearing system is proposed in this work. For presenting the nonlinear excitation from bearing compliance, the enhanced bearing force excitation model containing two kinds of bearing stiffnesses, which are mean stiffness for the load transfer capacity and alternating stiffness for the disturbance resisting ability, is developed. Considering other dynamic excitations including mesh stiffness, contact pressure angle, center distance, unbalance force caused by static and dynamic eccentricity, an advanced iterative numerical method is introduced, which can timely and accurately update the excitations caused by load-dependent and time-varying nonlinearities inside of the system. The constant bearing stiffness and time-varying bearing alternating stiffness models are introduced and compared with the enhanced bearing excitation force model. The parametric resonant regions and system nonlinear periodic motion states are studied and compared for different bearing supporting models. The effects from internal and external excitations on the system nonlinear vibration behaviors are investigated.

Influence of Lubricant Film Cavitation On the Vibration Behaviour of a Semi-Floating Ring Supported Turbocharger Rotor with Thrust Bearing

This contribution investigates the influence of outgassing processes on the vibration behaviour of a hydrodynamic bearing supported turbocharger rotor. The examined rotor is supported radially by floating rings with outer squeeze-film damping and axially by thrust bearings. Due to the highly non-linear bearing properties, the rotor can be excited via the lubricating film, which results in sub-synchronous vibrations known as oil-whirl and oil-whip phenomena. A significant influence on the occurrence of oil-whip phenomena is attributed to the bearing stiffness and damping, which depend both on the kinematic state of the supporting elements and the thermal condition as well as the occurrence of outgassing processes. For modelling the bearing behaviour, the Reynolds equation with mass-conserving cavitation regarding the two-phase model and the 3D energy as well as heat conduction equation is solved. To evaluate the impact of cavitation, run-up simulations are carried out assuming a fully (Half-Sommerfeld) or partially filled lubrication gap. The resulting rotor responses are compared with the shaft motion measurement. Also, the normalized eccentricity, the minimum lubricant fraction and the thermal bearing condition are discussed.

Bifurcation analysis of rotor/bearing system using third-order journal bearing stiffness and damping coefficients

Hydrodynamic journal bearings are used in many applications which involve high speeds and loads. However, they are susceptible to oil whirl instability, which may cause bearing failure. In this work, a flexible Jeffcott rotor supported by two identical journal bearings is used to investigate the stability and bifurcations of rotor bearing system. Since a closed form for the finite bearing forces is not exist, nonlinear bearing stiffness and damping coefficients are used to represent the bearing forces. The bearing forces are approximated to the third order using Taylor expansion, and infinitesimal perturbation method is used to evaluate the nonlinear bearing coefficients. The mesh sensitivity on the bearing coefficients is investigated. Then, the equations of motion based on bearing coefficients are used to investigate the dynamics and stability of the rotor-bearing system. The effect of rotor stiffness ratio and applied load on the Hopf bifurcation stability and limit cycle continuation of the system are investigated. The results of this work show that evaluating the bearing forces using Taylor’s expansion up to the third-order bearing coefficients can be used to profoundly investigate the rich dynamics of rotor-bearing systems.

Vibration Mode and Motion Trajectory Simulations of an Articulated Robot by a Dynamic Model Considering Joint Bearing Stiffness

Articulated robots are widely used in industries because they can perform manufacturing tasks with complicated movements. Higher speed and accuracy of motions are always required to improve the quality and productivity of products. The vibration characteristics of the robots are an important factor to achieve higher speed and accuracy motions. Robots are increasingly being used for machining. The vibration characteristics must also be considered when designing proper cutting conditions for the machining. To design control and cutting strategies for higher speed and accuracy motions or higher productivity of the machining process, it is effective to investigate the vibration characteristics of the robot and develop a mathematical model which can represents the vibration characteristics. The aim of this study is to investigate the vibration characteristics of an architectural robot and develop a mathematical model which can represent the dynamic behavior of the robot. To achieve this, vibration mode of an industrial architectural robot is analyzed based on measured frequency characteristics. According to the results of the modal analysis, it was clarified that the axial and angular stiffness of bearings of each joint of the robot has a significant impact on the vibration characteristics. Therefore, in this study, a mathematical model of the robot is developed considering the joint bearing stiffness. The mathematical model that also considers the kinematics of the robot, stiffness of reduction gears, control system for motors, and disturbance, such as friction and gravity, is introduced into the model. The control system is precisely modeled based on actual control algorithm in accordance with the implemented source codes. Although mass and inertia of the links are obtained from the 3D-CAD model, stiffness and damping parameters of the bearings and reduction gears are identified by matching the measured and simulated frequency responses. It has been confirmed that the model can adequately represents the vibration mode of the actual robot. Circular motion tests were performed to verify the model. Motion trajectories of the end effector were measured and simulated. As a result, it has been confirmed that the developed model is effective to analyze the dynamic behaviors.

A Critical Analysis of Finite-Element Modeling Procedures for Radial Bearing Stiffness Estimation

Different strategies are commonly employed by researchers in order to decrease the computational effort associated with the finite-element analysis of rolling-element bearings. The purpose of this work is to review and analyze the procedures and hypotheses that may be exploited to evaluate the nonlinear radial stiffness of these components. Techniques are utilized to develop a meshing procedure aimed at balancing the computational effort and the accuracy of the results, to define a robust approach to the problem. The geometry is reduced by taking advantage of the available symmetry planes, by removing unloaded rollers, and by substituting the shaft with an equivalent sinusoidal load. In addition, the element dimensions are adapted to the applied load depending on the size of the contact area as computed by means of the Hertz theory. The proposed methodology may be applied to all bearing types provided that symmetry conditions and contact area dimensions are properly assessed. The estimated stiffness is compared against analytical formulae retrieved from the literature. Influence of different element types, roller position, cage, and clearance on accuracy and computational time is discussed.

Parameters research on time-varying stiffness of the ball bearing system without race control hypothesis

When the ball bearing serving under the combined loading conditions, the ball will roll in and out of the loaded zone periodically. Therefore the bearing stiffness will vary with the position of the ball, which will cause vibration. In order to reveal the vibration mechanism, the quasi static model without raceway control hypothesis is modeled. A two-layer nested iterative algorithm based on Newton–Raphson (N-R) method with dynamic declined factors is presented. The effect of the dispersion of bearing parameters and the installation errors on the time-varying carrying characteristics of the ball-raceway contact and the bearing stiffness are investigated. Numerical simulation illustrates that besides the load and the rotating speed, the dispersion of bearing parameters and the installation errors have noticeable effect on the ball-raceway contact load, ball-inner raceway contact state and bearing stiffness, which should be given full consideration during the process of design and fault diagnosis for the rotor-bearing system.