Showing papers in "Journal of Vibration and Acoustics in 2002"

Differential Diagnosis of Gear and Bearing Faults

Abstract: This paper deals with the vibration-based diagnosis of rolling element bearings in the presence of strong interfering gear signals, such as is typical of helicopter gearboxes. The key idea consists in recognizing gear signals as purely periodic, whereas bearing signals experience some randomness and are close to cyclostationary, i.e. with a periodic bivariate autocorrelation function. This assertion is demonstrated by introducing a comprehensive model for the vibration generating process of bearing faults: distinctions are made between localized and distributed faults, between cyclostationary and pseudocyclostationary processes, and between additive and multiplicative interactions with gear signals. Finally, an original diagnostic procedure is proposed and its performance illustrated using simulated, experimental and actual cases.

Analytical Solutions to H∞ and H2 Optimization of Dynamic Vibration Absorbers Attached to Damped Linear Systems

Abstract: H∞ and H 2 optimization problems of the Voigt type dynamic vibration absorber (DVA) are classical optimization problems, which have been already solved for a special case when the primary system has no damping. However, for the general case including a damped primary system, no one has solved these problems by algebraic approaches. Only the numerical solutions have been proposed until now. This paper presents the analytical solutions for the H∞and H 2 optimization of the DVA attached to the damped primary systems. In the H∞ optimization the DVA is designed such that the maximum amplitude magnification factor of the primary system is minimized; whereas in the H 2 optimization the DVA is designed such that the squared area under the response curve of the primary system is minimized. We found a series solution for the H∞ optimization and a closed-form algebraic solution for the H 2 optimization. The series solution is then compared with the numerical solution in order to check the accuracy in connection with the truncation error of the series. The exact solution presented in this paper is too complicated to handle by a hand-held calculator, so we proposed an approximate solution for the practical object.

Embedded Self-Sensing Piezoelectric Active Sensors for On-Line Structural Identification

Abstract: The benefits and limitations of using embedded piezoelectric active sensors for structural identification at ultrasonic frequency are highlighted. An analytical model based on structural vibration theory and theory of piezoelectricity was developed and used to predict the electro-mechanical (E/M) impedance response, as it would be measured at the piezoelectric active sensor's terminals. The model considers one-dimension structures and accounts for both axial and flexural vibrations. Experiments were conducted on simple specimens in support of the theoretical investigation, and on realistic turbine blade specimen to illustrate the method's potential. It was shown that E/M impedance spectrum recorded by the piezoelectric active sensor accurately represents the mechanical response of a structure. It was further proved that the response of the structure is not modified by the presence of the sensor, thus validating the latter's noninvasive characteristics. It is shown that such sensors, of negligible mass, can be permanently applied to the structure creating a nonintrusive sensor array adequate for on-line automatic structural identification and health monitoring. The sensor calibration procedure is outlined. Numerical estimation of the noninvasive properties of the proposed active sensors in comparison with conventional sensors is presented. Self-diagnostics capabilities of the proposed sensors were also investigated and methods for automatic self-test implementation are discussed. The paper underlines that the use of piezoelectric wafer active sensors is not only advantageous, but, in certain situations, may be the sole investigative option, as in the case of precision machinery, small but critical turbine-engine parts, and computer industry components.

Stick-Slip Whirl Interaction in Drillstring Dynamics

Abstract: A Stick-slip Whirl Model is presented which is a simplification of an oilwell drillstring confined in a borehole with drilling fluid. The disappearance of stick-slip vibration when whirl vibration appears is explained by bifurcation theory. The numerical results are compared with the experimental data from a full-scale drilling rig.

Autoregressive Model-Based Gear Fault Diagnosis

Abstract: This paper presents a model-based technique for the detection and diagnosis of gear faults. Based on the signal averaging technique, the proposed technique first establishes an autoregressive (AR) model on the vibration signal of the gear of interest in its healthy-state. The model is then used as a linear prediction error filter to process the future-state signal from the same gear. The health condition of the gear is diagnosed by characterizing the error signal between the filtered and unfiltered signals. The technique is validated using both numerical simulation and experimental data. The results show that the AR model technique is an effective tool in the detection and diagnosis of gear faults and it may lead to an effective solution for in-flight diagnosis of helicopter transmissions.

A Numerical Scheme for Dynamic Systems Containing Fractional Derivatives

Abstract: This paper presents a numerical scheme for dynamic analysis of mechanical systems subjected to damping forces that are proportional to fractional derivatives of displacements. These equations appear in the modeling of frequency dependent viscoelastic damping of materials. In the scheme presented, the fractional differential equation governing the dynamics of a system is transformed into a set of differential equations with no fractional derivative terms. Using Laguerre integral formula, this set is converted to a set of first order ordinary differential equations, which are integrated using a numerical scheme to obtain the response of the system. In contrast to other numerical techniques, this method does not require one to store the past history of the response. Numerical studies show that the solution converges as the number of Laguerre node points increase. Further, results obtained using this scheme agree well with those obtained using analytical techniques.

Nonlinear Control of a Shape Memory Alloy Actuated Manipulator

Abstract: This paper presents a nonlinear robust control algorithm for accurate positioning of a single degree of freedom rotary manipulator actuated by Shape Memory Alloy (SMA) A model for an SMA actuated manipulator is presented The model includes nonlinear dynamics of the manipulator, a constitutive model of Shape Memory Alloy, and electrical and heat transfer behavior of SMA wire This model is used for open and closed loop motion simulations of the manipulator Experiments are presented that show results similar to both closed and open loop simulation results Due to modeling uncertainty and nonlinear behavior of the system, classic control methods such as Proportional-Integral-Derivative control are not able to present fast and accurate performance Hence a nonlinear, robust control algorithm is presented based on Variable Structure Control This algorithm is a control gain switching technique based on the weighted average of position and velocity feedbacks This method has been designed through simulation and tested experimentally Results show fast, accurate, and robust performance of the control system Computer simulation and experimental results for different stabilization and tracking situations are also presented

Mesh Stiffness Variation Instabilities in Two-Stage Gear Systems

Abstract: Mesh stiffness variation, the change in stiffness of meshing teeth as the number of teeth in contact changes, causes parametric instabilities and severe vibration in gear systems. The operating conditions leading to parametric instability are investigated for two-stage gear chains, including idler gear and countershaft configurations. Interactions between the stiffness variations at the two meshes are examined. Primary, secondary, and combination instabilities are studied. The effects of mesh stiffness parameters, including stiffness variation amplitudes, mesh frequencies, contact ratios, and mesh phasing, on these instabilities are analytically identified. For mesh stiffness variation with rectangular waveforms, simple design formulas are derived to control the instability regions by adjusting the contact ratios and mesh phasing. The analytical results are compared to numerical solutions.

Closed-Form Solutions to the Exact Optimizations of Dynamic Vibration Absorbers (Minimizations of the Maximum Amplitude Magnification Factors)

Abstract: A typical design problem for which the fixed-points method was originally developed is that of minimizing the maximum amplitude magnification factor of a primary system by using a dynamic vibration absorber This is an example of usual cases for which their exact solutions are not obtained by the well-known heuristic approach. In this paper, more natural formulation of this problem is studied, and algebraic closed-form exact solutions to both the optimum tuning ratio and the optimum damping coefficient for this classic problem are derived under assumption of undamped primary system. It is also proven that the minimum amplitude magnification factor, resonance and anti-resonance frequencies are entirely algebraic.

A Comparative Study and Analysis of Semi-Active Vibration-Control Systems

Abstract: Semi-active (SA) vibration-control systems are those which otherwise passively generated damping or spring forces are modulated according to a parameter tuning policy with only a small amount of control effort. SA units, as their name implies, fill the gap between purely passive and fully active vibration-control systems and offer the reliability of passive systems, yet maintain the versatility and adaptability of fully active devices. During recent years there has been considerable interest towards practical implementation of these systems for their low energy requirement and cost. This paper briefly reviews the basic theoretical concepts for SA vibration-control design and implementation, and surveys recent developments and control techniques for these systems. Some related practical applications in vehicle suspensions are also presented.

Transmissibility as a Differential Indicator of Structural Damage

Abstract: This article discusses the use of frequency domain transmissibility functions for detecting, locating, and quantifying damage in linear and nonlinear structures. Structural damage affects both the system poles and zeros; however, zeros are much more sensitive than poles to localized damage. This is because zeros depend on the input and output locations whereas poles do not. It is demonstrated here that since transmissibility functions are determined solely by the system zeros, they are potentially better indicators of localized linear and nonlinear types of damage. Furthermore, excitation measurements are not required to compute transmissibility functions so damage indices can be calculated directly from response measurements. It is also demonstrated that sensor arrays can sometimes be used to yield mixed transmissibility functions that are differential in nature, that is, they are less sensitive to gross fluctuations in the dynamic loading or environmental variables.

A dynamical systems approach to damage evolution tracking, part 1: Description and experimental application

Abstract: In this two-part paper we present a novel method for tracking a slowly evolving hidden damage process responsible for nonstationarity in a fast dynamical system. The development of the method and its application to an electromechanical experiment is the core of Part 1. In Part 2, a mathematical model of the experimental system is developed and used to validate the experimental results. In addition, an analytical connection is established between the tracking method and the physics of the system based on the idea of averaging and the slow flow equations for the hidden process. The tracking method developed in this study uses a nonlinear, two-time-scale modeling strategy based on the delay reconstruction of a system’s phase space. The method treats damage-induced nonstationarity as evolving in a hierarchical dynamical system containing a fast, directly observable subsystem coupled to a slow, hidden subsystem. The utility of the method is demonstrated by tracking battery discharge in a vibrating beam system with a battery-powered electromagnetic restoring force. Applications to systems with evolving material damage are also discussed. @DOI: 10.1115/1.1456908#

Modal Reduction of a Nonlinear Rotating Beam Through Nonlinear Normal Modes

Abstract: A method for determining reduced-order models for rotating beams is presented. The approach is based on the construction of nonlinear normal modes that are defined in terms of invariant manifolds that exist for the system equations of motion. The beam considered is an idealized model for a rotor blade whose motions are dominated by transverse vibrations in the direction perpendicular to the plane of rotation (known as flapping). The mathematical model for the rotating beam is relatively simple, but contains the nonlinear coupling that exists between transverse and axial deflections. When one employs standard modal expansion or finite element techniques to this system, this nonlinearity causes slow convergence, leading to models that require many degrees of freedom in order to achieve accurate dynamical representations. In contrast, the invariant manifold approach systematically accounts for the nonlinear coupling between linear modes, thereby providing models with very few degrees of freedom that accurately capture the essential dynamics of the system. Models with one and two nonlinear modes are considered, the latter being able to handle systems with internal resonances. Simulation results are used to demonstrate the validity of the approach and to exhibit features of the nonlinear modal responses. @DOI: 10.1115/1.1426071#

Fourier Series Analysis of Epicyclic Gearbox Vibration

Abstract: Synchronous averaging is difficult to use with epicyclic gear systems because the dominant frequencies are slightly skewed and asymmetrical. This Tech Brief uses Fourier series analysis to add new insight to this phenomenon. The results are illustrated on vibration data recorded from a Cobra AH-1S helicopter gearbox.

Nonlinear stochastic drill-string vibrations

Abstract: ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability PMC2000-251AbstractA study of downhole vibrations of drill-strings bottom-hole assemblies is undertaken. The lateral behaviorof the system is of interest. The nonlinear nature of the problem is addressed by considering a lateral clear-ance between the drill-string and the borehole that induces a stiffening of the system when exceeded. Thestochastic input force is defined by its power spectral density and it is applied laterally to the bit. The methodof statistical linearization is used, nd expressions for computing the equivalent linear system of the bottom-hole assembly are presented. The adopted procedure involves a prefiltering of the bit excitation to derive adynamic system under white-noise and colored white-noise excitations. Then, the Lyapunov equation for thecovariance of the linearized system is solved. Further, a Monte-Carlo simulation is conducted by means ofan auto-regressive moving-average digital filter, and the equations of motion are integrated by the Newmarkmethod. Numerical results pertaining to data obtained by measurement-while-drilling tools are presented.The study facilitates the assessment of the appropriateness of the method of statistical linearization for “realworld” problems encountered even in conservative industrial applications such as drilling.

A Dynamical Systems Approach to Damage Evolution Tracking, Part 2: Model-Based Validation and Physical Interpretation

Abstract: In this paper, the hidden variable damage tracking method developed in Part 1 is analyzed using a physics-based mathematical model of the experimental system: a mechanical oscillator with a nonstationary two-well potential. Numerical experiments conducted using the model are in good agreement with the experimental study presented in Part 1, and explicitly show how the tracking metric is related to the slow hidden variable evolution responsible for drift in the fast system parameters. Using the idea of averaging, the slow flow equation governing the hidden variable evolution is obtained. It is shown that the solution to the slow flow equation corresponds to the hidden variable trajectory obtained with the experimental tracking method. Thus we establish in principle the relationship of our algorithm to any underlying physical process. Based on this result, we discuss the application of the tracking method to systems with evolving material damage using the results of some preliminary experiments.@DOI: 10.1115/1.1456907# In Part 1 of this paper, motivated by the need to track damage evolution in machinery, we have developed a nonlinear method for tracking slowly evolving hidden variables. From this perspective, damage is a hidden process causing nonstationarity in a fast, directly observable dynamical system. The method uses a phase space formulation of the damage tracking problem, and uses a tracking metric developed using the short-time reference model prediction error. The method was successfully applied to an electromechanical experimental system consisting of a vibrating beam with a nonlinear potential perturbed by a battery powered electromagnet. The connection between the tracking metric developed in Part 1 and the hidden drift state variable was demonstrated empirically. It was shown that, as expected from the theoretical derivation of the method, the tracking metric is in a one-to-one relationship with the local time average of the measured voltage signal. In this, Part 2, of our paper, a physics based mathematical model of the experimental system is used to study analytically the direct connection between the tracking metric and the hidden drift process. Numerical experiments performed with the model are used to validate the experimental method. The idea of averaging is then used to show that the output of the tracking method is in fact following the solution to the slow flow equation for the drifting process. This provides a physical interpretation for the output of the tracking algorithm, and shows how, in principle, the experimental method can be related to the physics of the damage process. Based on this physical interpretation, we return to the experimental application of the tracking method, and discuss some preliminary results for a system with a crack growing to failure. In the next section, we develop the mathematical model of the battery discharge experiment using a lumped parameter, Lagrangian formulation of the electromechanical system. In Section 3, the tracking method developed in Part 1 is applied to the mathematical model in numerical experiments. Using the output from numerical simulations of the mathematical model, we are able to validate the tracking algorithm. In Section 4, we discuss the method of averaging as it relates to our problem. Finally, in Section 5, we finish with concluding remarks.

A mechanism of low subharmonic response in rotor/stator contact-measurements and simulations

Abstract: There are a variety of abnormal running conditions in rotating machinery that lead to rotor/stator interaction dynamics which, in turn, have a range of effects associated with them. One of these effects is steady vibration response at frequencies which are different from the excitation. This paper describes a mechanism of generating subharmonic vibration frequencies in both numerical simulation and measurements, which are obtained from a study of the relatively new problem of windmilling imbalance in aero-engines. What is different from other nonlinear systems with, say, clearance or squeeze film dampers, is the richness of the frequency spectrum.

Continuous Model for the Transverse Vibration of Cracked Timoshenko Beams

Abstract: A continuous model for the transverse vibrations of cracked beams including the effect of shear deformation is derived. Partial differential equations of motion and associated boundary conditions are obtained via the Hu-Washizu-Barr variational principle, which allows simultaneous and independent assumptions on the displacement, stress and strain fields. The stress and strain concentration caused by the presence of a crack are represented by so-called crack disturbance functions, which modify the kinematic assumptions used in the variational procedure. For the shear stress/strain fields, a quadratic distribution over the beam depth is assumed, which is a refinement of the typical constant shear stress distribution implicit in the Timoshenko model for uncracked beams. The resulting equations of motion are solved by a Galerkin method using local B-splines as test functions. As a numerical verification, natural frequencies of the linear open-crack model are computed and the results are compared to analytical results from similar models based on Euler-Bernoulli assumptions and experimental results found in the literature. For short beams, results from a 2-D finite element model are used to confirm the advantages of the proposed model when compared with previous formulations.

Variable Resistance Type Energy Regenerative Damper Using Pulse Width Modulated Step-up Chopper

Abstract: This paper describes a new technique for improving the damping property and efficiency of an energy regenerative damper It is intended for a linear DC motor type vibration damper to regenerate vibration energy efficiently. Normally a regenerative damper can regenerate vibration energy only at high speed motion. For low speed motion, the damper has nonlinear characteristics with dead zone and cannot regenerate energy. In order to overcome this problem, a step-up chopper is introduced between the actuator and the charging circuit. The energy is regenerated from low speed and low voltage actuator to high voltage charging circuit. This paper also proposes a new control technique to the step-up chopper by using pulse width modulated signals. The damper can change its damping coefficient and the energy can be regenerated more efficiently. The proposed damper is applied to an active mass damper system. A simple experimental setup is used to validate the proposed technique. The results show an increase in performance and energy regeneration as compared to the previously proposed regenerative damper.

An Unsymmetrical Motion in a Horizontal Impact Oscillator

Abstract: Stability and bifurcation for the unsymmetrical, periodic motion of a horizontal impact oscillator under a periodic excitation are investigated through four mappings based on two switch-planes relative to discontinuities. Period-doubling bifurcation for unsymmetrical period-1 motions instead of symmetrical period-1 motion is observed. A numerical investigation for symmetrical, period-1 motion to chaos is completed. The numerical and analytical results of periodic motions are in very good agreement. The methodology presented in this paper is applicable to other discontinuous dynamic systems. This investigation also provides a better understanding of such an unsymmetrical motion in symmetrical discontinuous systems.

Free Vibration Analysis of an Inflated Toroidal Shell

Abstract: Free vibration analysis of a free inflated torus of circular cross-section is presented. The shell theory of Sanders, including the effect of pressure, is used in formulating the governing equations. These partial differential equations are reduced to ordinary differential equations with variable coefficients using complete waves in the form of trigonometric functions in the longitudinal direction. The assumed mode shapes are divided into symmetric and antisymmetric groups, each given by a Fourier series in the meridional coordinate. The solutions (natural frequencies and mode shapes) are obtained using Galerkin's method and verified with published results. The natural frequencies are also obtained for a circular cylinder with shear diaphragm boundary condition as a special case of the toroidal shell. Finally, the effects of aspect ratio, pressure, and thickness on the natural frequencies of the inflated torus are studied.

Subspace-Based System Identification for an Acoustic Enclosure

Abstract: This paper is aimed at identifying a dynamical model for an acoustic enclosure, a duct with rectangular cross section, closed ends, and side-mounted speaker enclosures. Acoustic enclosures are known to be resonant systems of high order In order to design a high performance feedback controller for an acoustic enclosure, one needs to have an accurate model of the system. Subspace-based system identification techniques have proven to be an efficient means of identifying dynamics of high order highly resonant systems. In this paper a frequency domain subspace-based method together with a second iterative optimization step minimizing a frequency domain least-squares criterion is successfully employed to identify a dynamical model for an acoustic enclosure.

Spectral Finite Element Analysis of Sandwich Beams With Passive Constrained Layer Damping

Abstract: We present a spectral finite element model (SFEM) for sandwich beams with passive constrained layer damping (PCLD) treatments. The viscoelastic core has a complex modulus that varies with frequency. The SFEM is formulated in the frequency domain using dynamic shape functions based on the exact displacement solutions from progressive wave methods, where we implicitly account for the frequency dependent complex modulus of the viscoelastic core. The SFEM results of natural frequencies and frequency response functions are compared to those calculated using conventional finite element (CFEM), where the Golla-Hughes-McTavish method is used to account for the frequency dependent complex modulus of a viscoelastic core. Also experimental data are used to validate both analyses using frequency response functions measured for two cantilevered sandwich beams with PCLD treatments having 50% and 75% coverage of the beam length. SFEM shows improved computational efficiency and accuracy, because many more elements must be incorporated into the CFEM for comparable accuracy.

H2 Optimization of the Three-Element Type Dynamic Vibration Absorbers

Abstract: The dynamic vibration absorber (DVA) is a passive vibration control device which is attached to a vibrating body (called a primary system) subjected to exciting force or motion. In this paper, we will discuss an optimization problem of the three-element type DVA on the basis of the H 2 optimization criterion. The objective of the H 2 optimization is to reduce the total vibration energy of the system for overall frequencies; the total area under the power spectrum response curve is minimized in this criterion. If the system is subjected to random excitation instead of sinusoidal excitation, then the H 2 optimization is probably more desirable than the popular H∞ optimization. In the past decade there has been increasing interest in the three-element type DVA. However, most previous studies on this type of DVA were based on the H∞ optimization design, and no one has been able to find the algebraic solution as of yet. We found a closed-form exact solution for a special case where the primary system has no damping. Furthermore, the general case solution including the damped primary system is presented in the form of a numerical solution. The optimum parameters obtained here are compared to those of the conventional Voigt type DVA. They are also compared to other optimum parameters based on the H∞ criterion.

Design of Robust Vibration Controller for a Smart Panel Using Finite Element Model

Abstract: This paper presents a model reduction method and uncertainty modeling for the design of a low-order H∞ robust controller for suppression of smart panel vibration. A smart panel with collocated piezoceramic actuators and sensors is modeled using solid, transition, and shell finite elements, and then the size of the model is reduced in the state space domain. A robust controller is designed not only to minimize the panel vibration excited by applied uniform acoustic pressure, but also to be reliable in real world applications. This paper introduces the idea of Modal Hankel Singular values (MHSV) to reduce the finite element model to a low-order state space model with minimum model reduction error. MHSV measures balanced controllability and observability of each resonance mode to deselect insignificant resonance modes. State space modeling of realistic control conditions are formulated in terms of uncertainty variables. These uncertainty variables include uncertainty in actuators and sensors performances, uncertainty in the knowledge of resonance frequencies of the structure, damping ratio, static stiffness, unmodeled high resonance vibration modes, etc. The simplified model and the uncertainty model are combined as an integrated state space model, and then implemented in the H∞ control theory for controller parameterization. The low-order robust controller is easy to implement in an analog circuit to provide a low cost solution in a variety of applications where cost may be a limiting factor.

Varying the Stiffness of a Beam-Like Neutralizer Under Fuzzy Logic Control

Abstract: Vibration neutralizers are effective vibration control devices at a single frequency. If they can compensate for drift in the excitation frequency by adjusting their stiffness the performance can be improved, and the range of problems to which they can be applied is broadened. This paper considers a beam-like adaptive vibration neutralizer, and it is shown that the stiffness of the device and hence its natural frequency can be significantly altered by varying the beam cross-section. Several different beam configurations are investigated and the rate of change of stiffness as a function of beam separation is calculated for each configuration. The results are validated by some simple experiments. Real-time stiffness control of a beam-like tuneable neutralizer is also demonstrated both by computer simulation and experiment. The neutralizer is subjected to swept sine excitation over a six-second period and the tuned condition is maintained throughout the excitation period. The efficacy of using a nonlinear fuzzy logic controller is compared with the use of a simple proportional controller.

An Active-Passive Piezoelectric Absorber for Structural Vibration Control Under Harmonic Excitations With Time-Varying Frequency, Part 1: Algorithm Development and Analysis

Abstract: It has been shown that piezoelectric materials can be used as passive electromechanical vibration absorbers by shunting them with electrical networks. Semi-active piezoelectric absorbers have also been proposed for suppressing harmonic excitations with varying frequency. However, these semi-active devices have limitations that restrict their practical applications. The approach presented here is a high performance active-passive alternative to semi-active absorbers. By utilizing a combination of a passive electrical circuit and active control actions, the system is synthesized for adaptive variable frequency narrowband disturbance rejection. The active control consists of three parts: an inductor tuning action, a negative resistance action, and a coupling enhancement action. In the current paper (Part 1), the control algorithm is developed and analyzed. Part 2 of the paper contains experimental investigations and parametric studies of the new absorber design.

Adaptive Boundary Control of an Axially Moving String System

Abstract: This paper proposes an adaptive boundary control to an axially moving string system, which couples with a mass-damper-spring (MDS) controller at its right-hand-side (RHS) boundary. Unknown parameters appearing in the system equation are assumed constant and estimated on-line by using adaptation laws. The adaptive computed-torque control algorithm applied to robot manipulators of lumped systems is extended to design the adaptive boundary controller for the coupling system. It is found that the control force and update laws depend only on the displacement, velocity and slope of the string at the RHS boundary. Lyapunov stability guarantees the convergence of the tracking error to zero. Finally, the performance of the proposed controller is demonstrated by numerical simulations.

Adaptive Order Tracking Technique Using Recursive Least-Square Algorithm

Abstract: Order tracking technique is one of the important tools for diagnosis of rotating machinery. Conventional methods of order tracking are primarily based on Fourier analysis with reference to shaft speed. Resampling is generally required in the fast Fourier transform (FFT)-based methods to compromise between time and frequency resolution for varying shaft speeds. Conventional methods suffer from a number of shortcomings. In particular, smearing problems arise when closely spaced orders or crossing orders are present. Conventional methods also are ineffective for the applications involving multiple independent shaft speeds. This paper presents an adaptive order tracking technique based on the Recursive Least-Squares (RLS) algorithm to overcome the problems encountered in conventional methods. In the proposed method, the problem is treated as the tracking of frequency-varying bandpass signals. Order amplitudes can be calculated with high resolution by using the proposed method in real-time fashion. The RLS order tracking technique is applicable whether it is a single-axle or multi-axle system.