Directional Frequency Response Functions Research Articles

Wave analysis of rotors with longitudinal periodicity

The dynamics of rotating systems plays an important role in the efficiency and reliability of such machines. For this reason, understanding their dynamics is essential to predict and mitigate the resultant vibrations that can occur in such machines. Considering that vibration is essentially the propagation of waves in a continuous medium, methods of wave analysis can also be employed to investigate the dynamics of rotating systems, especially when the system presents some sort of periodicity. The difficulty lies in the fact that rotating systems are non-reciprocal systems due to the gyroscopic effects, which create forward and backward whirl motions with associated mode waves. The conventional wave methods usually do not tackle this problem. This work presents a methodology for analyzing rotating systems with wave methods based on the combination of conventional modal analysis, Bloch wave propagation theory, and directional frequency response functions. In this case, it is possible to separate the forward and backward modes, thus allowing the analysis of forward and backward wavenumbers separately. Consequently, one can draw separate dispersion diagrams for the wave modes associated with the forward and backward motions of the rotor. The numerical results show the feasibility of the method, and the methodology is applied experimentally to a rotating system in the laboratory.

Identification of in-process machine tool dynamics using forced vibrations in milling process

An accurate description of machine tool dynamics is essential for health monitoring, chatter prevention, and improvement of manufacturing accuracy. The standard identification approach of experimental modal analysis at the standstill of the machine does not realize the possible variations in the dynamics due to operational conditions. The in-process identification methods in the literature, on the other hand, are associated with implementation difficulties in industrial environments due to the required complex and specially-designed setup, limited excitation forms, or excessive measurement efforts. This paper proposes an industrial-friendly method to estimate the in-process structural dynamics of machine tools, considering practical demands and implementation limits. Knowing that the operational dependency of the dynamics is associated with the machine and spindle structure, the machine–spindle and holder–tool structures are considered two distinguished subsystems. The cross Frequency Response Function (FRF) of the coupled structure, between the tooltip and spindle flange, is determined by measuring forced vibrations during milling operations. The milling forces are considered as excitation input and the process is designed to be stable and chatter-free so that the simulated forces can be accurately used in the identification. Then, the receptance coupling theory is utilized to develop an optimization algorithm. Given the model of the coupled structure, the evolutionary optimization tunes the modal parameters of the machine–spindle dynamics and joint parameters until the predicted cross FRFs match the experimentally determined FRFs. The direct FRF at the tooltip is predicted using identified in-process machine–spindle dynamics through the receptance coupling method. The identified in-process dynamics are used to predict stability diagrams for different tooling systems and, moreover, to design an augmented Kalman filter to estimate cutting forces. Comparison of estimated SLDs and cutting forces with chatter test results and dynamometer measurements validate the outcomes of the proposed identification method. This paper demonstrates that the system dynamics under operational conditions can be successfully identified without requiring costly setup, specially-designed workpieces or operations, and destructive chatter tests.

Frequency-based decoupling and finite element model updating in vibration of cable–beam systems

Interactions between cable and structure affect the modal properties of cabled structures such as overhead electricity transmission and distribution line systems. Modal properties of a single in-service pole are difficult to determine. A frequency response function of a pole impacted with a modal hammer will contain information about not only the pole but also the conductors and adjacent poles connected thereby. This article presents a generally applicable method to extract modal properties of a single structural element, within an interacting system of cables and structures, with particular application to electricity poles. A scalable experimental lab-scale pole-line consisting of a cantilever beam and stranded cable and a more complex system consisting of three cantilever beams and a stranded cable are used to validate the method. The frequency response function of a cantilever (“pole”) is predicted by substructural decoupling of measured cable dynamics (known frequency response function matrix) from the measured response of the assembled cable–beam system (known frequency response function matrix). Various amounts of sag can be present in the cable. Comparison of the estimated and directly obtained pole frequency response functions show good agreement, demonstrating that the method can be used in cabled structures to obtain modal properties of an individual structural element with the effects of cables and adjacent structural elements filtered out. A frequency response function–based finite element model updating is then proposed to overcome the practical limitation of accessing some components of the real-world system for mounting sensors. Frequency response functions corresponding to inaccessible points are generated based on the measured frequency response functions corresponding to accessible points. The results verify that the frequency response function–based finite element model updating can be used for substructural decoupling of systems in which some essential points, such as coupling points, are inaccessible for direct frequency response function measurement.

Effects of Varying Dynamics of Flexible Workpieces in Milling Operations

Abstract In this study, surface error calculations and stability conditions are presented for milling operations in case of slender parts. The dynamic behavior of the flexible beam-type workpiece is modeled by means of finite element method (FEM), while the varying dynamical properties related to the feed motion as well as the material removal process are incorporated in the model. The FEM-generated direct frequency response function is verified through a closed-form solution based on the distributed transfer function method. Relative errors and convergence of the FEM are investigated based on the analytical solutions of the continuum model, from which appropriate element size and mode number can be selected for modal coordinate transformations. The pattern in the variation of the natural frequencies is explored using the analytical model in case of high radial depth of cut relative to the original cross section of the beam-like workpiece. Both the stability conditions and the resulted surface errors are predicted as a function of the tool position. The presented approach and the results are validated by laboratory tests.

Output-Only Modal Analysis of Micromilling Systems

An output-only modal analysis (OMA) approach is presented to obtain the direct frequency response function (FRF) at the tip of the tool in micromilling setups. White noise input is provided using acoustic excitation and the resulting vibrations are measured using a laser Doppler vibrometer (LDV). Autoregressive (AR) identification is used to extract the natural frequencies and damping ratios of the structural modes of the milling setup, and mass-sensitivity analysis is used to obtain modal stiffness values. The accuracy of the tool tip FRFs that are constructed using OMA is verified by comparing them against the FRFs that are measured using impulse hammer tests. The direct FRF at the tool tip is an essential component in predicting and avoiding excessive and unstable vibrations in milling operations, and the presented approach provides a practical method for the direct measurement of the tool tip FRF in micromilling where the application of traditional hammer tests is not possible.

Effects of Tuned Rail Damper on Track Dynamic Characteristics Optimization

Egg-III system track is widely used in the vibration-abatement-sections of Beijing Metro, while DTVI2 fastener track is a common used form that frequently applied in the sections without vibration reduction requirements. These two railway track forms are favored in the field of urban rail transit for the respective superiorities. Recently after several months of operation, a series of abnormal problems occur, including rail corrugation problem on Egg-III fasteners section and clips fracture problem on DTVI2 rail fastening section. To provide a feasible solution for both problems, the presented paper introducing the laboratory tests which were conducted to optimize the dynamic properties of Egg-III fastener track and DTVI2 fastener track using the device of Tuned Rail Damper (‘TRD’ for short).The tests were carried out on the experimental platform in Beijing Jiaotong university, which provides dynamic testing conditions for varying track forms. To evaluate the effects of TRD on track dynamic properties, tests of direct frequency response function, vibration decay rate and vibration acceleration level were done for Egg-III system track and DTVI2 fastener track (installed/uninstalled with TRD).Experimental results indicate that TRD obviously optimizes the dynamic characteristics of both Egg-III system track and DTVI2 fastener track. The optimization effects are listed as follows: (1) The resonance peaks of frequency response function curve are significantly reduced and moved in the range of 100-400Hz, which is identified as the cause of the rail corrugation occurred on Egg-III system track sections. (2) The vertical and transverse vibration decay rates on Egg-III system are obviously increased with TRD， suppressing the vibration transited along the rail. (3) Vibration reduction ability of Egg-III system track is not weakened with TRD, but strengthened up to 1.2-6 times at most frequencies. (4) The amplitudes of frequency response function curve are reduced when TRD was installed on the rail of DTVI2 fastener track. (5) The vertical and transverse vibration decay rates of DTVI2 fastener track are raised markedly with TRD, that the vibration energy transmitted to rail fastening clips is reduced.

Experimental wear parameters identification in hydrodynamic bearings via model based methodology

When a rotor operates in very low rotating speeds, the fluid-film in hydrodynamic bearings is not completely formed, which allows direct metal to metal contact between the bearing and the journal. Numerous start/stop cycles also may lead to bearing wear. In this context, the present work aims to evaluate a model-based method for identification of hydrodynamic bearing wear parameters: more specifically, wear depth and angular position. The proposed method is based on minimizing an objective function, which compares the directional Frequency Response Function (dFRF) of the physical system with the response of the model developed here. It contemplates the numerical model of the rotor and the worn bearings, in this case, represented by linearized coefficients of damping and stiffness. Through experimental results, a deeper evaluation of the identification method is presented, such as the sensitivity to the number of points of the dFRF used in the objective function, different meshes of starting points, and the region of the dFRF used – around the resonance peak or around the oil-whirl peak. The analyses are conducted for different levels of bearing wear. The precision obtained in the results indicates that the presented method is a promising tool in the identification of bearing wear.

Dynamic study of viscoelastic rotor: Modal analysis of higher order model considering various asymmetries

The primary aim of this paper is to study modal analysis of higher order finite element model for a generally viscoelastic rotor supported by the journal bearing. The gyroscopic couple, internal damping and fluid film forces of journal bearings have a significant asymmetric influence on rotor dynamics. A complex modal coordinate creates a platform to indicate the directivity of modes and provides better information about the direction of the whirl. In which, study of natural modes and directional frequency response function are obtained for free and force vibration analysis. It also distinguishes the importance of higher order model over conventional second order model.

Directional frequency response applied to wear identification in hydrodynamic bearings

This work analyzes the influence of the wear in hydrodynamic bearings in the directional frequency response of a rotating system. Previous works presented a numerical approach to identify the bearing wear parameters using the unbalance response of the system in different rotational speeds. In continuity, this paper analyzes the directional frequency response function (dFRF) to identify the wear model, prompting a feasible application of the method, taken into account that it is not possible to operate the rotor near the critical speeds. The results show that the dFRF matrix is sensitive to the wear parameters and more suitable for bearing wear identification, based on the cross-coupled terms of the dFRF matrix, which present a higher sensitivity to the bearing wear.

An identification method for damping ratio in rotor systems

Centrifugal compressor testing with magnetic bearing excitations is the last step to assure the compressor rotordynamic stability in the designed operating conditions. To meet the challenges of stability evaluation, a new method combining the rational polynomials method (RPM) with the weighted instrumental variables (WIV) estimator to fit the directional frequency response function (dFRF) is presented. Numerical simulation results show that the method suggested in this paper can identify the damping ratio of the first forward and backward modes with high accuracy, even in a severe noise environment. Experimental tests were conducted to study the effect of different bearing configurations on the stability of rotor. Furthermore, two example centrifugal compressors (a nine-stage straight-through and a six-stage back-to-back) were employed to verify the feasibility of identification method in industrial configurations as well.

Numerical and Experimental Stability Investigation of a Flexible Rotor on Two Different Tilting Pad Bearing Configurations

Rotordynamic stability is crucial for high performance centrifugal compressors. In this paper, the weighted instrumental variable (WIV) based system identification method for rotating machinery stability is investigated based on a sine sweep forward excitation with an electromagnetic actuator. The traditional multiple input multiple output (MIMO) frequency response function (FRF) is transformed into a directional frequency response function (dFRF). The rational polynomial method (RPM) combined with WIV is developed to identify the rotor’s first forward mode parameters. This new approach is called the COMDYN method. Experimental work using the COMDYN method is carried out under different rotating speeds, oil inlet temperatures, and pressure conditions. Two sets of bearings with preloads 0.1 and 0.3 are investigated. A numerical rotor-bearing model is also built. The numerical results correlate reasonably well with the experimental results. The investigation results indicate that the new method satisfies the desired features of rotating machine stability identification. Furthermore, the system log decrement was improved somewhat with the increase of oil inlet temperature. The increase of oil supply pressure affects the rotor-bearing system stability very slightly. The results of this paper provide new and useful insights for potentially avoiding instability faults in centrifugal compressors.

Mixed Mode Precession Display Based on Normal Directional Frequency Response Functions of Rotor Dynamic Systems

Mixed Mode Precession Display Based on Normal Directional Frequency Response Functions of Rotor Dynamic Systems

Chatter Stability Modeling and Simulation for Long Slender Shaft Turning Operation

Dynamic vibration and static deformation are two main factors that affect the machining quality in a long slender shaft turning operation. By simplifying the turning system of a slender shaft into a one-section beam with a clamped-pinned constraint condition, the direct receptance at any arbitrary cutting point is derived. On the basis, the regenerative stability lobes diagram (SLD) for a long slender shaft turing operation is achieved. With the proposed modeling methodology and simulation algorithm, the effect of cutting position on the direct frequency response function (FRF) and the predicated SLD, as well as the effect of the cutting conditions on the predicated SLD are investigated. The predicated direct FRF at the cutting point is validated by hammer tests.

Rotordynamic Evaluation of Centrifugal Compressor Using Electromagnetic Exciter

Since heavier gases exert larger effects on rotordynamic stability, stability evaluation is important in developing or designing high-pressure compressors. To evaluate the rotor stability during operation, an excitation test using a magnetic bearing is the most practical method. In stability analysis, labyrinth seals can produce significant cross coupling forces, which particularly reduce the damping ratio of the first forward mode. Therefore, forward modes should be distinguished from backward modes in the excitation test. One method that excites only the forward modes, not the backward modes (and vice versa), is the use of a rotating excitation. In this method, the force is simultaneously applied to two axes to excite the rotor in circular orbits. Two trigonometric functions, i.e., cosine and sine functions, are used to generate this rotation force. Another method is the use of a unidirectional excitation and a mathematical operation to distinguish the forward whirl from the backward whirl. In this method, a directional frequency response function that separates the two modes in the frequency domain is obtained from four frequency response functions by using a complex number expression for the rotor motion. In this study, the latter method was employed to evaluate the rotor stability of a high-pressure compressor. To obtain the frequencies and damping ratios of the eigenvalues, the curve fitting based on system identification methods, such as the prediction error method, was introduced for the derived frequency response functions. Firstly, these methods were applied to a base evaluation under a low-pressure gas operation, in which the stability mainly depends on the bearing property. Using the obtained results, the bearing coefficients were estimated. Next, the same methods were applied to stability evaluations under high-pressure gas operations. The destabilizing forces were also estimated from the test results and compared with the calculation results.

Crack Identification in a Rotating Shaft via the Reverse Directional Frequency Response Functions

A method is proposed for identifying the location of an open transverse crack in flexible rotor systems by modeling the crack as a localized element with rotating asymmetry. It exploits the strong correlations between the modal constants of the reverse directional frequency response functions (r-dFRFs) and the degree and location of asymmetry. A map of the modal constants of the r-dFRFs for all elements is constructed to identify the location of crack by comparing the identified modal constants to those of the reference map. This paper also addresses practical issues associated with measurement noises and limited number of sensors. The proposed crack identification method is finally applied to a flexible rotor system with an open transverse crack in order to demonstrate the identification procedure for detection of the crack location.

Strength of modes in rotating machinery

A new method for the analysis of modal strength of the rotor systems with periodically time-varying parameters due to the presence of either rotating or stationary asymmetry is proposed. The method is based on the complete development of the modal analysis by introducing the modulated coordinates to derive the equivalent infinite-order time-invariant Hill's matrix equation from the finite-order time-varying matrix equation. Depending upon the level of possible contribution of each mode to forced response, the modal strength of a mode is rigorously derived from the norm order analysis of the associated eigenvector; thereby, the modes are classified as strong or weak modes. It is shown that the directional frequency response functions are useful in identifying the strength of modes in detail, or equivalently, the modes of symmetry, anisotropy, asymmetry and coupled asymmetry. Two illustrative examples with a simple, yet general, analysis rotor model and a practical flexible asymmetric rotor finite element (FE) model with an open transverse crack are treated to demonstrate the theoretical findings and effectiveness of the proposed method.

Vibration analysis of periodically time-varying rotor system with transverse crack

This paper proposes the vibration analysis for the periodically time-varying rotor system with transverse crack based on the complex modal analysis by introducing the modulated coordinates. From the feasibility of the crack modeling by harmonically varying stiffness, for which the dynamic behaviors of breathing crack depending on two modeling structures are investigated, the vibration analysis associated with modal characteristics, whirling, instability, directional frequency response functions (dFRFs) and directional spectrum (dS) is carried out. Using the feature of the reverse dFRFs strongly representing the effect of the crack and the interpretation of the dS patterns, the effective method to identify the modes due to the presence or propagation of a transverse crack is suggested.

개폐균열이 존재하는 비대칭 회전부 및 비등방 고정부를 갖는 회전체의 모드해석

This paper describes the new modal analysis method to detect the presence of the breathing crack in a general rotor system with disk asymmetry and stator anisotropy. It is proposed that the modal analysis using directional frequency response functions (dFRFs), which, accounting for the directivity in modes, clears the heavily over-lapping of other harmonics occurring from non-isotropic properties in addition to those due to crack, can provide an effective method to detect the modes by a crack. The simulations from the simple general rotor model show that the r-dFRFs (reverse dFRFs) for asymmetry confirms a good indicator of the presence of the breathing crack and the instability is primarily influenced by the shaft asymmetry than the breathing crack.

Model Correlation and System Health Monitoring using Frequency Domain Measurements

In this paper, algorithmic approaches to enhance model correlation and system health monitoring capability are developed in which direct frequency response function (FRF) data are utilized. The algorithmic approaches utilizes the concept of minimum rank perturbation theory. The use of direct FRFs, as opposed to measured modal parameters, is shown to be one method to address a part of the incomplete measurement problem common to model correlation and system health monitoring; namely the mismatch in the number of measured vibration modes in the measured frequency band in comparison to the number of modes included in the analytical finite element model. Key points made in the development are highlighted using numerical and experimental studies.

Vibration Analysis of Rotors Utilizing Implicit Directional Information of Complex Variable Descriptions

A complex variable description of planar motion incorporates directivity as inherent information which is therefore very convenient in vibration analysis of rotors. This paper proposes to use the directional information explicitly when the equation of motion of a rotor is formulated in complex variables. It is shown that the free vibration solution to the equation of motion formulated as such can be defined as the directional natural mode because it describes not only the shape and frequency but also the direction of the free vibration response. The directional frequency response functions (dFRFs) that have been used recently are obtained as the solution to the forced vibration solution to the equation of motion. Symmetric and anti-symmetric motions of a geometrically symmetric rigid rotor are used as examples to explain these concepts and their practical significances. The proposed approach allows clear understanding and definitions of some unique characteristics of rotor vibrations, such as the forward and backward modes, and forward and backward critical speeds, which have been often used in confusing or incorrect ways.