Showing papers in "Journal of Vibration and Acoustics in 2005"

Measurement and Modeling of Normal Contact Stiffness and Contact Damping at the Meso Scale

Abstract: Modeling of contact inteffaces that inherently include roughness such as joints, clamping devices, and robotic contacts, is very important in many engineering applications. Accurate modeling of such devices requires knowledge of contact parameters such as contact stiffness and contact damping, which are not readily available. In this paper, an experimental method based on contact resonance is developed to extract the contact parameters of realistic rough surfaces under lightly loaded conditions. Both Hertzian spherical contacts and flat rough surfaces in contact under normal loads of up to 1000 mN were studied. Due to roughness, measured contact stiffness values are significantly lower than theoretical values predicted from smooth surfaces in contact. Also, the measured values favorably compare with theoretical values based on both Hertzian and rough contact surfaces. Contact damping ratio values were found to decrease with increasing contact load for both Hertzian and flat surfaces. Furthermore, Hertzian contacts have larger damping compared to rough flat surfaces, which also agrees with the literature. The presence of minute amount of lubricant and wear debris at the interface was also investigated. It was found that both lubricant and wear debris decrease the contact stiffness significantly though only the lubricant significantly increases the damping.

Optimization of the Individual Stiffness and Damping Parameters in Multiple-Tuned-Mass-Damper Systems

Abstract: The characteristics of multiple tuned-mass-dampers (MTMDs) attached to a single-degree-of-freedom primary system have been examined by many researchers. Several papers have included some parameter optimization, all based on restrictive assumptions. In this paper, we propose an efficient numerical algorithm to directly optimize the stiffness and damping of each of the tuned-mass dampers (TMDs) in such a system. We formulate the parameter optimization as a decentralized H2 control problem where the block-diagonal feedback gain matrix is composed of the stiffness and damping coefficients of the TMDs. The gradient of the root-mean-square response with respect to the design parameters is evaluated explicitly, and the optimization can be carried out efficiently. The effects of the mass distribution, number of dampers, total mass ratio, and uncertainties in system parameters are studied. Numerical results indicate that the optimal designs have neither uniformly spaced tuning frequencies nor identical damping coefficients, and that optimization of the individual parameters in the MTMD system yields a substantial improvement in performance. We also find that the distribution of mass among the TMDs has little impact on the performance of the system provided that the stiffness and damping can be individually optimized.

Tunable Microelectromechanical Filters that Exploit Parametric Resonance

Abstract: Background: This paper describes an analytical study of a bandpass filter that is based on the dynamic response of electrostatically-driven MEMS oscillators. Method of Approach: Unlike most mechanical and electrical filters that rely on direct linear resonance for filtering, the MEM filter presented in this work employs parametric resonance. Results: While the use of parametric resonance improves some filtering characteristics, the

Wave Reflection and Transmission in Timoshenko Beams and Wave Analysis of Timoshenko Beam Structures

Abstract: This paper concerns wave reflection, transmission, and propagation in Timoshenko beams together with wave analysis of vibrations in Timoshenko beam structures. The transmission and reflection matrices for various discontinuities on a Timoshenko beam are derived. Such discontinuities include general point supports, boundaries, and changes in section. The matrix relations between the injected waves and externally applied forces and moments are also derived. These matrices can be combined to provide a concise and systematic approach to vibration analysis of Timoshenko beams or complex structures consisting of Timoshenko beam components. The approach is illustrated with several numerical examples.

HMM-Based Fault Detection and Diagnosis Scheme for Rolling Element Bearings

Abstract: In this paper, we introduce a new bearing fault detection and diagnosis scheme based on hidden Markov modeling (HMM) of vibration signals. Features extracted from amplitude demodulated vibration signals from both normal and faulty bearings were used to train HMMs to represent various bearing conditions. The features were based on the reflection coefficients of the polynomial transfer function of an autoregressive model of the vibration signals. Faults can be detected online by monitoring the probabilities of the pretrained HMM for the normal case given the features extracted from the vibration signals. The new technique also allows for diagnosis of the type of bearing fault by selecting the HMM with the highest probability. The new scheme was also adapted to diagnose multiple bearing faults. In this adapted scheme, features were based on the selected node energies of a wavelet packet decomposition of the vibration signal. For each fault, a different set of nodes, which correlates with the fault, is chosen. Both schemes were tested with experimental data collected from an accelerometer measuring the vibration from the drive-end ball bearing of an induction motor (Reliance Electric 2 HP IQPreAlert) driven mechanical system and have proven to be very accurate.

An Adaptive Semiactive Control Algorithm for Magnetorheological Suspension Systems

Abstract: In this paper, we will present a nonlinear-model-based adaptive semiactive control algorithm developed for magnetorheological (MR) suspension systems exposed to broadband nonstationary random vibration sources that are assumed to be unknown or not measurable. If there exist unknown and/or varying parameters of the dynamic system such as mass and stiffness, then the adaptive algorithm can include on-line system identification such as a recursive least-squares method. Based on a nonparametric MR damper model, the adaptive system stability is proved by converting the hysteresis inherent with MR dampers to a memoryless nonlinearity with sector conditions. The convergence of the adaptive system, however, is investigated through a linearization approach including further numerical illustration of specific cases. Finally the simulation results for a magnetorheological seat suspension system with the suggested adaptive control are presented. The results are compared with low-damping and high-damping cases, and such comparison further shows the effectiveness of the proposed nonlinear model-based adaptive control algorithm for damping tuning.

Extending Den Hartog’s Vibration Absorber Technique to Multi-Degree-of-Freedom Systems

Abstract: The most common method to design tuned dynamic vibration absorbers is still that of Den Hartog, based on the principle of invariant points. However, this method is optimal only when attaching the absorber to a single-degree-of-freedom undamped main system. In the present paper, an extension of the classical Den Hartog approach to a multi-degree-of-freedom undamped main system is presented. The Sherman-Morrison matrix inversion theorem is used to obtain an expression that leads to invariant points for a multi-degree-of-freedom undamped main system. Using this expression, an analytical solution for the optimal damper value of the absorber is derived. Also, the effect of location of the absorber in the multi-degree-of-freedom system and the effect of the absorber on neighboring modes are discussed.

Nonlinear Dynamics of High-Speed Milling—Analyses, Numerics, and Experiments

Abstract: High-speed milling is often modeled as a kind of highly interrupted machining, when the ratio of time spent cutting to not cutting can be considered as a small parameter. In these cases, the classical regenerative vibration model, playing an essential role in machine tool vibrations, breaks down to a simplified discrete mathematical model. The linear analysis of this discrete model leads to the recognition of the doubling of the so-called instability lobes in the stability charts of the machining parameters. This kind of lobe-doubling is related to the appearance of period doubling vibrations originated in a flip bifurcation. This is a new phenomenon occurring primarily in low-immersion high-speed milling along with the Neimark-Sacker bifurcations related to the classical self-excited vibrations or Hopf bifurcations. The present work investigates the nonlinear vibrations in the case of period doubling and compares this to the well-known subcritical nature of the Hopf bifurcations in turning processes. The identification of the global attractor in the case of unstable cutting leads to contradiction between experiments and theory. This contradiction draws the attention to the limitations of the small parameter approach related to the highly interrupted cutting condition.

The Physical Reason and the Analytical Condition for the Onset of Dry Whip in Rotor-to-Stator Contact Systems

Abstract: Dry whip is an instability of rotor-to-stator contact systems and may lead to a catastrophic failure of rotating machinery. The physical reason for the onset of dry whip in rotor/stator systems with imbalance is not yet well understood. This paper explores the development of the rotor response into dry whip of a specific rotor-to-stator contact model and finds that the rotor in resonance, at a negative (natural) frequency of the coupled nonlinear rotor/stator system is the physical reason for the onset of dry whip with imbalance. Based on this find, the equations of motion of the rotor/stator system are formulated in a different way that includes the dynamic characteristics in the vicinity of the onset point of dry whip. The onset condition of dry whip with imbalance is then derived by using the multiple scale method. As shown by examples, the analytical onset condition of dry whip agrees well with the numerically simulated one. In addition, the results are consistent with phenomena observed in tests.

Patch Transfer Functions as a Tool to Couple Linear Acoustic Problems

Abstract: A method to couple acoustic linear problems is presented in this paper. It allows one to consider several acoustic subsystems, coupled through surfaces divided in elementary areas called patches. These subsystems have to be studied independently with any available method, in order to build a database of transfer functions called patch transfer functions, which are defined using mean values on patches, and rigid boundary conditions on the coupling area. A final assembly, using continuity relations, leads to a very quick resolution of the problem. The basic equations are developed, and the acoustic behavior of a cavity separated in two parts is presented, in order to show the ability of the method to study a strong-coupling case. Optimal meshing size of the coupling area is then discussed, some comparisons with experiments are shown, and finally a complex automotive industrial case is presented.

Dynamic Response Optimization of Piezoelectrically Excited Thin Resonant Beams

Abstract: Piezoelectrically excited, resonant, elastic beams find wide use as piezoelectric fans, optical choppers, MEMS sensors, and piezoelectric motors. The devices consist of either one piezoelectric ceramic patch (piezopatch) bonded on one side (asymmetric configuration), or of two oppositely poled patches placed symmetrically on either side of a thin, flexible elastic beam (symmetric configuration). Field equations of the coupled structure governing the coupled longitudinal and bending motions of the resonator are derived using linear constitutive equations, slender beam approximations, and Hamilton's principle. Analytical solutions are found to the coupled eigenvalue problem. Eigenvalues and eigenfunctions for the short-circuited and open-circuited configurations are predicted analytically and are found to be in excellent agreement with results from three-dimensional finite element simulations. Electromechanical coupling factors (EMCF) are computed using the analytical and finite element model and optimal resonator geometries are identified for maximal EMCF. The EMCF predictions are also compared with experiments for an asymmetrically configured resonator. The analytical solution provides a convenient tool for the optimal design of such devices.

Analysis of Static Deformation, Vibration and Active Damping of Cylindrical Composite Shells with Piezoelectric Shear Actuators

Abstract: An analytical solution is presented for the static deformation and steady-state vibration of simply supported hybrid cylindrical shells consisting of fiber-reinforced layers with embedded piezoelectric shear sensors and actuators. The piezoelectric shear actuator, which is poled in the circumferential direction, will induce transverse shear deformation of the hybrid shell when it is subjected to an electric field in the radial direction. Suitable displacement and electric potential functions that identically satisfy the boundary conditions at the simply supported edges are used to reduce the governing equations of static deformation and steady-state vibrations of the hybrid laminate to a set of coupled ordinary differential equations in the radial coordinate, which are solved by employing the Frobenius method. Natural frequencies, mode shapes, displacements, electric potential, and stresses are presented for four-layer hybrid laminates consisting of a piezoelectric shear sensor and actuator sandwiched between fiber-reinforced composite layers. Active vibration damping is implemented using a positive position feedback controller. Frequency response curves for different controller frequencies, controller damping ratio, and feedback gain demonstrate that the embedded shear actuator can be used for active damping of the fundamental flexural mode. In addition, it is demonstrated that vibration suppression of thickness modes is also feasible using the piezoelectric shear actuator.

Dynamic Modeling and Simulation of Satellite Tethered Systems

Abstract: The objective of this paper is to study the dynamic simulation of a tether as it is deployed or retrieved by a winch on a satellite orbiting around earth. In an effort to understand the problem incrementally, the following three models were developed: (a) Model 1: A tether with constant length moves on earth in the plane of constant gravity; (b) Model 2: A tether is deployed from a drum on earth in the plane of constant gravity, i.e., length of the cable changes during deployment; (c) Model 3: A tether is deployed from a drum on an orbiting satellite. These models have been chosen to bring different aspects as well as levels of difficulty in the analysis. For example, in Model 1, the length of cable is fixed and the gravity direction is constant during motion. The equations of motion for this model are derived using Newton's laws and Hamilton's principle to show the equivalence of the two methods. In Model 2, free length of the cable changes during deployment. The changing length of the cable introduces coupled nonlinearities into the motion. Model 3 includes the orbital effect on the motion of deployed cable. Each of these three dynamic models characterized by partial differential equations are first converted to a finite number of ordinary differential equations using Ritz's procedure and are then numerically integrated using Matlab ODE solvers.

Nonlinear Analysis on Hunting Stability for High-Speed Railway Vehicle Trucks on Curved Tracks

Abstract: Based on the heuristic nonlinear creep model, the nonlinear coupled differential equations of the motion of a ten-degree-of-freedom truck system, considering the lateral displacement, the vertical displacement, the roll and yaw angles of the each wheelset, and the lateral displacement and yaw angle of the truck frame, moving on curved tracks, are derived in completeness. To illustrate the accuracy of the analysis, the limiting cases are examined. The influences of the suspension parameters, including those losing in the six-degree-of-freedom system, on the critical hunting speeds evaluated via the linear and nonlinear creep models, respectively, are studied and compared.

Test Response and Nonlinear Analysis of a Turbocharger Supported on Floating Ring Bearings

Abstract: Measurements of casing acceleration on an automotive turbocharger running to a top speed of 115 krpm and driven by ambient temperature pressurized air are reported. Waterfall acceleration spectra versus rotor speed show the effects of increasing lubricant inlet pressure and temperature on turbocharger rotordynamic response. A comprehensive analysis of the test data shows regimes of speed operation with two subsynchronous whirl motions (rotordynamic instabilities). Increasing the lubricant feed pressure delays the onset speed of instability for the most severe subsynchronous motion. However, increasing the lubricant feed pressure also produces larger synchronous displacements. The effect of lubricant feed temperature is minimal on the onset and end speeds of rotordynamic instability. Nevertheless, operation with a cold lubricant exhibits lower amplitudes of motion, synchronous and subsynchronous. The experimental results show the subsynchronous frequencies of motion do not lock (whip) at system natural frequencies but continuously track the rotor speed. No instabilities (subsynchronous whirl) remain for operating speeds above 90 krpm. Linear and nonlinear analysis results for the operation of a small automotive turbocharger supported on floating ring bearings are presented. A comprehensive fluid film bearing model predicting the forced response of floating ring bearings is also described. The linear rotordynamic model predicts well the rotor freefree modes and onset speed of instability using linearized bearing force coefficients. The nonlinear model incorporating instantaneous bearing reaction forces in the numerical integration of the rotor equations of motion predicts the limit cycle amplitudes with two fundamental sub synchronous whirl frequencies. Comparisons of both models to experimental results follow. The predictions evidence two unstable whirl ratios at approximately 1/2 ring speed and 1/2 ring speed plus journal speed. The transient nonlinear responses reveal the importance of rotor imbalance in suppressing the subsynchronous instabilities at large rotor speeds as also observed in the experiments.

Hydrostatic Gas Journal Bearings for Micro-Turbomachinery

Abstract: Several years ago an effort was undertaken at MIT to develop high-speed rotating MEMS (Micro Electro-Mechanical Systems) using computer chip fabrication technology. To enable high-power density the micro-turbomachinery must be run at tip speeds of order 500 m/s, comparable to conventional scale turbomachinery. The high rotating speeds (of order 2 million rpm), the relatively low bearing aspect ratios (L/D <0.1) due to fabrication constraints, and the laminar flow regime in the bearing gap place the microbearing designs to an exotic spot in the design space for hydrostatic gas bearings. This paper presents a new analytical model for axially fed gas journal bearings and reports the experimental testing of micro gas bearings to characterize and to investigate their rotordynamic behavior. The analytical model is capable of dealing with all the elements of, (1) micro-devices, (2) dynamic response characteristics of hydrostatic gas bearings, (3) evaluation of stiffness, natural frequency and damping, (4) evaluation of instability boundaries, and (5) evaluation of effects of imbalance and bearing anisotropy. First, a newly developed analytical model for hydrostatic gas journal bearings is introduced. The model consists of two parts, a fluid dynamic model for axially fed gas journal bearings and a rotordynamic model for micro-devices. Next, the model is used to predict the natural frequency, damping ratio and the instability boundary for the test devices. Experiments are conducted using a high-resolution fiber optic sensor to measure rotor speed, and a data reduction scheme is implemented to obtain imbalance-driven whirl response curves. The model predictions are validated against experimental data and show good agreement with the measured natural frequencies and damping ratios. Last, the new model is successfully used to establish bearing operating protocols and guidelines for high-speed operation.

General Interpolated Fast Fourier Transform: A New Tool for Diagnosing Large Rotating Machinery

Abstract: Vibration monitoring and diagnosis of rotating machinery is an important part of a predictive maintenance program to reduce operating and maintenance costs. In order to improve the efficiency and accuracy of diagnosis, the general interpolated fast Fourier transform (GIFFT) is introduced in this paper. In comparison to present interpolated fast Fourier transform, this new approach can deal with any type of window functions and possesses high accuracy and robust performance, especially coping with a small number of sampling points. Then, for the purpose of rotating machinery diagnosis, the harmonic vibration ellipse and orbit is reconstructed based on the GIFFT to extract the features of faults and remove the interference from environmental noise and some irrelevant components. This novel scheme is proving to be very effective and reliable in diagnosing several types of malfunctions in gas turbines and compressors and characterizing of the transient behavior of rotating machinery in the run-up stage.

Sensitivity of Uniform Load Surface Curvature for Damage Identification in Plate Structures

Abstract: In this paper a new sensitivity-based method of using measured modal parameters to locate and quantify damage is developed for plate-like structures. With the measured incomplete modal data for only the few lower modes in both the intact and damaged states, the two-dimensional distributed curvatures of uniform load surface (ULS) over the plate are approximated using the Chebyshev polynomials. Instead of directly comparing the curvatures before and after damage, like many existing damage localization methods using curvature techniques, e.g., mode shape curvature and flexibility curvature, the proposed method analytically studies the sensitivity of the ULS curvature with respect to the element-by-element stiffness parameters. The changes in the elemental stiffness parameters due to damage give the location and magnitude of the damaged plate elements. Based on the first-order Taylor series approximation, the inverse problem is modeled as a linear equation system and solved iteratively using truncated SVD technique. Numerical simulations are performed to verify the effectiveness of the proposed method with different support conditions, measurement noise, and sensor sparsity.

Prediction of Railway-Induced Ground Vibrations in Tunnels

Abstract: The authors of this paper present the results of a study concerned with the assessment of the vibrational impact induced by the passage of commuter trains running in a tunnel placed underground the city of Rome. Since the railway line is not yet operational, it was not possible to make a direct measurement of the ground vibrations induced by the railway traffic and the only way to make predictions was by means of numerical simulations. The numerical model developed for the analyses was calibrated using the results of a vibration measurement campaign purposely performed at the site using as a vibration source a sinusoidal vibration exciter operating in a frequency-controlled mode. The problem of modeling the vibrational impact induced by the passage of a train moving in a tunnel is rather complex because it requires the solution of a boundary value problem of three-dimensional elastodynamics in a generally heterogeneous, nonsimply connected continuum with a moving source. The subject is further complicated by the difficulties of modeling the source mechanism, which constitutes itself a challenge even in the case of railway lines running at the surface. At last, the assessment of the vibrational impact at a receiver placed inside a building (e.g., a human individual or a sensitive instrument) requires an evaluation of the role played by the structure in modifying the computed free-field ground motion. So far, few attempts have been made to model the whole vibration chain (from the source to the receiver) of railway-induced ground vibrations, with results that have been only moderately successful. The numerical simulations performed in this study were made by using a simplified numerical model aimed to capture the essence of the physical phenomena involved in the above vibration chain including the influence of the structural response as well as the dependence of the predicted vibration spectra on the train speed.

Numerical Study of Some Nonlinear Dynamics of a Rotor Supported on a Three-Pad Tilting Pad Journal Bearing (TPJB)

Abstract: The use of tilting pad journal bearings (TPJBs) has increased in the recent past due to their stabilizing effects on the rotor bearing system. However, in this paper two mechanisms capable of producing instabilities in terms of subharmonic and chaotic motions are suggested. The first one is that of a centrally loaded pad with rotor unbalance excitations. The second one represents a concentric rotor (or a vertical rotor) acted upon by centering sprigs and large unbalance excitations. Extensive numerical experimentation shows, for certain parameters, subharmonic, quasi-periodic, and chaotic motions. The pad state space trajectory, in many cases, resembles that of the two-well potential case as in Duffing’s oscillator. Time trajectories, Poincare maps, fast Fourier transform (FFT) plots, and the max Lyapunov exponent are utilized to examine the periodicity (order) of the nonsynchronous rotor orbits and pad trajectories. The TPJB problem belongs to a family of nonlinear rotor-dynamical phenomena that are potentially of a considerable value as diagnostic tools in assessing rotating machinery condition monitoring.

Active vibration isolation of metrology frames; a modal decoupled control design

Abstract: For a six degree-of-freedom active vibration isolation system, a control strategy based on modal decoupling is proposed. This has the advantage of controlling the modal directions on a centralized single-input single-output basis. As a consequence, stability and performance can be imposed in each of the modal directions separately. An experimental demonstration is given using a dummy metrology frame. That is, a 1600 kg payload mass supported by three combined pneumatic and Lorentz controlled isolators. With this setup, two unstable modal directions resulting from a high center of gravity are stabilized without compromising performance in any of the remaining directions. In fact, performance in the remaining directions is enhanced using manual loop shaping.

Friction-Compensating Command Shaping for Vibration Reduction

Abstract: Fast and accurate point-to-point motion is a common operation for industrial machines, but vibration will frequently corrupt such motion. This paper develops commands that can move machines without vibration, even in the presence of Coulomb friction. Previous studies have shown that input shaping can be used on linear systems to produce point-to-point motion with no residual vibration. This paper extends command-shaping theory to nonlinear systems, specifically systems with Coulomb friction. This idea is applied to a PD-controlled mass with Coulomb friction to ground. The theoretical developments are experimentally verified on a solder cell machine. The results show that the new commands allow the proportional gain to be increased, resulting in reduced rise time, settling time, and steady-state error.

A Temperature-based Controller for a Shape Memory Alloy Actuator

Abstract: This paper presents a robust nonlinear control that uses a state variable estimator for control of a single degree of freedom rotary manipulator actuated by shape memory alloy (SMA) wire. A model for SMA actuated manipulator is presented. The model includes nonlinear dynamics of the manipulator, a constitutive model of the shape memory alloy, and the electrical and heat transfer behavior of SMA wire. The current experimental setup allows for the measurement of only one state variable which is the angular position of the arm. Due to measurement difficulties, the other three state variables, arm angular velocity and SMA wire stress and temperature, cannot be directly measured. A model-based state estimator that works with noisy measurements is presented based on the extended Kalman filter (EKF). This estimator estimates the state vector at each time step and corrects its estimation based on the angular position measurements. The estimator is then used in a nonlinear and robust control algorithm based on variable structure control (VSC). The VSC algorithm is a control gain switching technique based on the arm angular position (and velocity) feedback and EKF estimated SMA wire stress and temperature. Using simulation it is shown that the state vector estimates help reduce or avoid the undesirable and inefficient overshoot problem in SMA one-way actuation control.

Numerical and Experimental Investigations on Flexible Multi-bearing Rotor Dynamics

Abstract: An experimental test rig is built to verify the dynamics of a multi-bearing rotor It consists of two flexibly coupled shafts and is connected to a motor at one end via a flexible coupling. Each of the shafts is supported at the ends by two hydrodynamic bearings and is attached with two disks with equal and unequal masses, respectively. The mathematical model of the test rig is developed and is simulated numerically. The non-stationary dynamic responses of the system during speed-up with a constant angular acceleration are shown, respectively, by the non-stationary bifurcation diagrams, the selected time flows, and the spectrum cascades. Experiments are then carried out on the test rig. Generally, the numerical results are verified qualitatively by the experiments. Both results indicate that the non-synchronous whirls of the two shafts influence each other when flexibly coupled together. In particular, a new phenomenon is found for the four-bearing rotor system: the pre-existing non-synchronous whirl/whip resulted from the instability of one shaft can activate the onset of oil instability of another shaft. In the theoretical simulation, this phenomenon represents the rapid increase of the non-synchronous whirl orbit, whereas in the experiment, it represents the simultaneous existence of two whirl/ whip frequencies in the spectra.

A Frequency Domain Technique for Characterizing Nonlinearities in a Tire-Vehicle Suspension System

Abstract: Characterization of tire and suspension system nonlinearities in measured data is the first step in developing input-output quarter car models; however, system identification procedures, which require a priori knowledge of all nonlinearities within a system, often receive more attention in the research community. Furthermore, relatively few investigations have focused on nonlinear characterization and identification in the absence of input measurements. A new method for characterizing nonlinearities, in the absence of an input measurement, using transmissibility functions and ordinary coherence functions between response measurement degrees of freedom is discussed here. It is shown that the nonlinear nature of a vehicle system provides information about the nominal linear system when the input is unknown. Nonlinear frequency permutations, which create drops in the ordinary coherence function, serve to characterize the associated nonlinearities. In the absence of input measurements, coherence functions of the response transmissibility between the vehicle spindle and body allow the nonlinearities in the suspension system, but not the tires, to be characterized. Simulation results are discussed and the method is applied to experimental laboratory and operating data to validate the approach.

Hopf Bifurcation of a Magnetic Bearing System with Time Delay

Abstract: This paper is concerned with a study of the influence of a time delay occurring in a PD feedback control on the dynamic stability of a rotor suspended by magnetic bearings. In the presence of geometric coordinate coupling and time delay, the equations of motion governing the response of the rotor are a set of two-degree-of-freedom nonlinear differential equations with time delay coupling in nonlinear terms. It is found that as the time delay increases beyond a critical value, the equilibrium position of the rotor motion becomes unstable and may bifurcate into two qualitatively different kinds of periodic motion. The resultant Hopf bifurcation is associated with two coincident pairs of complex conjugate eigenvalues crossing the imaginary axis. Based on the reduction of the infinite dimensional problem to the flow on a four-dimensional center manifold, the bifurcating periodic solutions are investigated using a perturbation method.

Probability distribution of extremes of von mises stress in randomly vibrating structures

Abstract: The problem of determining the probability distribution function of extremes of Von Mises stress, over a specified duration, in linear vibrating structures subjected to stationary, Gaussian random excitations, is considered. In the steady state, the Von Mises stress is a stationary, non-Gaussian random process. The number of times the process crosses a specified threshold in a given duration, is modeled as a Poisson random variable. The determination of the parameter of this model, in turn, requires the knowledge of the joint probability density function of the Von Mises stress and its time derivative. Alternative models for this joint probability density function, based on the translation process model, combined Laguerre-Hermite polynomial expansion and the maximum entropy model are considered. In implementing the maximum entropy method, the unknown parameters of the model are derived by solving a set of linear algebraic equations, in terms of the marginal and joint moments of the process and its time derivative. This method is shown to be capable of taking into account non-Gaussian features of the Von Mises stress depicted via higher order expectations. For the purpose of illustration, the extremes of the Von Mises stress in a pipe support structure under random earthquake loads, are examined. The results based on maximum entropy model are shown to compare well with Monte Carlo simulation results.

Effect of Material Coupling on Wave Vibration of Composite Timoshenko Beams

Abstract: This paper presents the effect of coupling between bending and torsional deformations on vibrations of composite Timoshenko beams from the wave standpoint. The dispersion characteristics and the modes of vibrations are in general affected by material coupling, except those of the torsional modes at low frequencies; and higher-frequency modes are normally more sensitive to material coupling. The wave mode transition phenomenon is also investigated. It is found that like their metallic counterparts, composite Timoshenko beams also exhibit wave mode transition. Furthermore, the transition frequency is found to be unaffected by material coupling. Numerical examples for which comparative results are available in the literature are presented.

Scaling Laws for Ultra-Short Hydrostatic Gas Journal Bearings

Abstract: The journal bearings of the MIT micro-devices are located at the outer periphery of the rotor and are designed to operate at rotational speeds of order two million rpm in order to enable high-power densities with turbomachinery tip speeds near 500 m/s. These journal bearings are very short compared to their relatively large bearing diameters such that the bearing L/D is typically less than 0.1, that is at least one order of magnitude smaller than in conventional gas bearings. Thus, the ultra-short micro gas journal bearings essentially act as short annular seals and operate at Reynolds numbers of order 300, two orders of magnitude lower than conventional annular seals. The concepts that hold for turbulent flow, large scale annular seals do not apply to micro bearings and the laminar flow regime sets new challenges in the design, implementation and operation of ultra-short, high-speed gas bearings. In order to reach the goal of operating the MIT micro devices at full design speed, the micro-bearing design must be improved and engineering solutions need to be found to overcome the challenges of high-speed bearing operation. This paper is the first to derive the scaling laws for the dynamics of ultra-short hydrostatic gas journal bearings. The theory is established from first principles and enables a physics based characterization of the dynamic behavior of ultra-short hydrostatic gas bearings. The derived scaling laws for natural frequency and damping ratio show good agreement with experimental data. A simple criterion for whirl instability is found that only depends on bearing geometry. The scaling laws together with this criterion are used to delineate engineering solutions critical for stable high-speed bearing operation. Design charts are developed which provide the link between fabrication tolerances, bearing performance, and the tolerable level of rotor unbalance for a minimum required whirl ratio.

A Technique for Estimating Linear Parameters Using Nonlinear Restoring Force Extraction in the Absence of an Input Measurement

Abstract: Conventional nonlinear system identification procedures estimate the system parameters in nvo stages First, the nominally linear system parameters are estimated by exciting the system at an amplitude (usually low) where the behavior is nominally linear Second, the nominally linear parameters are used to estimate the nonlinear parameters of the system at other arbitrary amplitudes This approach is not suitable for many mechanical systems, which are not nominally linear over a broad frequency range for any operating amplitude A method for nonlinear system identification, in the absence of an input measurement, is presented that uses information about the nonlinear elements of the system to estimate the underlying linear parameters Restoring force, boundary perturbation, and direct parameter estimation techniques are combined to develop this approach The approach is applied to experimental tire-vehicle suspension system data