Measured Frequency Response Functions Research Articles

Modeling of the coupled dynamics of damping particles filled in the cells of a honeycomb sandwich plate and experimental validation

Equipment panels of a spacecraft are made up of a sandwich composite with aluminum face sheets and a honeycomb (HC) core. The HC sandwich plate responds to the launch vibration loads subjecting the equipment mounted on it to a high level of accelerations at resonances owing to a lower natural damping. Damping particles (DPs) when inserted in the empty cells of a HC core improve the damping characteristics and reduce the resonance responses. In this work, we present a mathematical model governing the motion of the cell walls, DPs and HC plate under dynamic loading. The discrete element method (DEM) has been used to model the dynamics of the DPs wherein the contacts are modeled using modified nonlinear dissipative Hertz contact theory in conjunction with Coulomb friction. The effect of DPs on the responses at resonances, damping, and frequency response function (FRF) of the HC plate is obtained. Numerical and experimental studies were conducted on a HC plate where a selected portion of the plate was filled with DPs. The HC plate was subjected to sine sweep base acceleration at the edges to study the effect of DPs on the dynamic characteristic of the plate. The damping ratios and resonance peaks of the lower four modes of the HC plate, excited up to 1000 Hz, obtained experimentally from the FRF measurements and numerically from the DEM model compare well. The damping ratios, response at resonances and the FRF profiles are also similar. Significant improvement in damping ratios and attenuation of vibration level has been observed.

Suitability of integrated sensors for the determination of chatter characteristics in a cylindrical grinding machine

Wheel-sided chatter vibration is a limiting factor in terms of productivity and surface finish regarding the application of conventional grinding wheels. Due to the formation of waviness on the grinding wheel during the process, many expensive trueing cycles are necessary. Therefore, a system for chatter detection and elimination for external cylindrical grinding machines was presented in previous work. It was demonstrated that an electromagnetic actuator reduces the generation of the grinding wheel waviness. The control of this actuator requires precise knowledge about frequency, amplitude, and phase of the chatter vibration. This information is extracted from two eddy-current sensors, which measure the workpiece displacement. However, a direct measurement of the workpiece displacement restricts the working area and is inflexible. Hence, this paper explores the qualification of sensors integrated into the grinding machine for extracting the aforementioned information with high accuracy of phase. For this purpose, two acceleration sensors mounted on the workpiece spindle and the tailstock, an acoustic emission (AE) sensor applied to the grinding wheel, and the motor current of the directly driven headstock spindle are considered. To determine the suitability of these sensors, the frequency response function (FRF) for each sensor is calculated under three conditions: no contact between the workpiece and the grinding wheel, the grinding wheel tensed to the workpiece without a grinding process, and during the process. The method for the in-process measurement of the sensor FRFs and the influence of the contact conditions on these FRFs is shown in this paper.

Structural parameter estimation and damage detection using experimental transfer function data

ABSTRACTIn this paper, the capabilities of an enhanced sensitivity relation for the structural model updating are validated using a subset of the measured frequency response function (FRF) and natural frequency data of a laboratory concrete beam. The sensitivity relation of FRF is expressed implicitly as a function of the change of the mode shapes and incomplete measured natural frequencies of the damaged structure. In order to achieve more accurate parameters estimation results, an enhanced sensitivity relation is proposed by incorporating the measured natural frequencies of the damaged structures in the mathematical formulation. Solution of sensitivity equations by Least Square algorithm yields to the damage parameters. The method is applied to experimental data of a pin-roller concrete beam subjected to the extensive damage cases. The experimental results prove robust model updating results by the proposed method.

Hydraulic Modal Analysis in Theory and Practice

Theoretical and experimental modal analyses are treated for hydraulic systems modeled by discrete capacities, inductances, resistances, and fluid lines with dynamic laminar flow. Based on an approximate multi-degrees-of-freedom description, it is shown how hydraulic natural frequencies, damping ratios, and mode shapes can be identified from measured frequency response functions between flow rate excitation and pressure response. Experiments are presented for a pipeline system that includes three side branches and an accumulator. In view of practical applications, two different types of servovalve excitation as well as impact hammer excitation are considered. Pressure is measured by 19 sensors throughout the system. Results are compared in terms of frequency response functions between 50 and 850 Hz, the first five hydraulic modes, and weighted auto modal assurance criteria of experimental mode shapes. Out of the tested excitation devices, the servovalve is clearly preferred; if valves cannot be used, the impact hammer offers a reasonable workaround. For a reduced number of five sensors, different sensor arrangements are assessed by the respective weighted auto modal assurance criteria of experimental mode shapes. A theoretical hydraulic modal model provides a similar assessment. The quality of the theoretical model is confirmed by the weighted modal assurance criterion of theoretical and experimental mode shapes from servovalve excitation.

The sliding friction contact frequency response function

Abstract This paper details frequency response function (FRF) measurement and simulation for a dynamic oscillator with a sliding (Coulomb) friction contact. The oscillator is approximated physically using the friction measuring machine (FMM), a parallelogram leaf-type flexure with a sliding friction contact. The flexure-based FMM provides linear motion between a polytetrafluouroethylene pin and (lubricated) polished steel counterface. Both the input force and response velocity are measured by impact testing to determine the FRF. Numerical simulations are completed to identify the best-fit friction coefficient for three input force levels (450 N, 1000 N, and 1450 N). It is shown that: 1) the FRF magnitude increases linearly with impulse (i.e., the area under the time domain force profile); and 2) a single friction coefficient (0.113) is sufficient to describe the dynamic response for the three force levels.

Optimal Experiment Design for Multi-variable Motion Systems: with Application to a Next-Generation Wafer Stage

Frequency Response Functions (FRFs) are essential for motion control design of complex mechatronic systems. The aim of this paper is to develop Optimal Experiment Design (OED) approaches for accurate identification of FRF models, with the particular focus on Multiple Input Multiple Output (MIMO) systems in closed-loop. The resulting optimization problem is shown to be highly challenging, especially for complex systems. A practical and feasible numerical approach based on a convex relaxation is developed. Experimental results on a next-generation wafer stage confirm that the quality of FRF measurements can be improved significantly using the proposed techniques.

Prediction of the Frequency Response Function of a Tool Holder-Tool Assembly Based on Receptance Coupling Method

Regenerative chatter has a fatal influence on machine performance in high-speed milling process. Basically, machine condition without chattering can be selected from the stability lobes diagram, which is estimated from the tool point frequency response function (FRF). However, measurements of the tool point FRF would be a complicated and time-consuming task with less efficiency. Therefore prediction of the tool point FRF is of importance for further calculation of the machining stability. This study employed the receptance coupling analysis method to predict the FRF of a tool holder-tool module, which is normally composed of substructures, tool holder and cutter with different length. In this study, the angular components of FRFs of the substructures required for coupling operation were predicted by finite element analysis, apart from the translational components measured by vibration experiments. Using this method, the effects of the overhang length of the cutter on the dynamic characteristics have been proven and successfully verified by the experimental measurements. The proposed method can be an effective way to accurately predict the dynamic behavior of the spindle tool system with different tool holder-tool modules.

Beyond equidistant sampling for performance and cost: A loop‐shaping approach applied to a motion system

SummaryNonequidistant sampling potentially enhances the performance/cost trade‐off that is present in traditional equidistant sampling schemes. The aim of this paper is to develop a systematic feedback control design approach for systems that go beyond equidistant sampling. A loop‐shaping design framework for such nonequidistantly sampled systems is developed that addresses both stability and performance. The framework only requires frequency response function measurements of the LTI system, whereas it appropriately addresses the linear periodically time‐varying behavior introduced by the nonequidistant sampling. Experimental validation on a motion system demonstrates the superiority of the design framework for nonequidistantly sampled systems compared to traditional designs that rely on equidistant sampling.

Frequency Response Function Measurement on Simplified Disc Brake Model

Abstract The paper describes role of non-proportional damping in flutter type instability, demonstrated on simplified disc brake model. The discrete two degrees of freedom system is considered to imply damping induced instability through a system eigenvalues evaluation. The Frequency Response Function (FRF) is further calculated from measurements on the physical disc brake model. From FRF, damping properties are estimated and discussed. Several different loading states of the pad versus disc are considered to show loading impact on FRF and thus damping of the system.

On the treatment of uncertainty in experimentally measured frequency response functions

Frequency response functions (FRFs) are routinely used within experimental structural dynamics, among other fields. The uncertainty associated with their measurement is typically modelled on the simplified assumption that additive uncorrelated noise is present. This treatment of FRF uncertainty is questioned here by demonstrating that the notion of operator uncertainty (i.e. the uncertainty due to inconsistent force excitation) introduces a correlation between measured FRFs which is not accounted for under the assumption of additive noise. The origin of this correlation is illustrated and its experimental presence verified. The importance of this correlation on the propagation of uncertainty is subsequently investigated as part of a numerical and experimental case study, where its influence on the inversion of an uncertain FRF matrix is considered. It is shown that the neglect of such correlations can lead to large errors in uncertainty estimates, for example in dynamic sub-structuring and the solution of inverse problems.

Iterative pole-zero finite element model updating using generic parameters

Prediction and control of unmeasurable performance variables in complex large-scale systems is an important topic in systems and control. The preferred solution to this problem is to have a relatively low-order and accurate standard plant model. A reduced Finite Element (FE) dynamic model may be a good candidate for that purpose apart from the fact that its accuracy is always limited, since it never truly matches the real structure. In this paper, an iterative procedure is proposed to update the FE model using poles and zeros estimated from the measured Frequency Response Functions (FRFs). Assuming the errors in the FE stiffness and/or damping matrix to be dominant over the mass matrix, but the exact physical parameters responsible for these errors to be unknown, the eigenvalues of the (sub)structure stiffness and/or damping matrix may be updated to reduce the differences between the poles and zeros from the model and measured FRFs. The proposed procedure matches poles and zeros sequentially in an iterative manner as to realize convergence to the measured quantities. This procedure, further referred to as Iterative Pole-Zero (IPZ) model updating, facilitates the process of obtaining improved parametric models which can be used for accurate prediction and control of unmeasurable performance variables. The proposed procedure is first illustrated using a mass-spring model and then is validated via a collocated FRF measurement from an experimental beam setup.

Detection of Local Damage of Flexural Member Using Measured Frequency Response Functions

Measurements by sensors provide inaccurate information, including external noises. This study considers a method to reduce the influence of the external noise, and it presents a method to detect local damage transforming the measured frequency response functions (FRFs) to reduce the influence of the external noise. This study is conducted by collecting the FRFs in the first resonance frequency range from the responses in the frequency domain, taking the mean values at two adjacent nodes, and transforming the results to the proper orthogonal decomposition (POD). A damage detection method is provided. The curvature of the proper orthogonal mode (POM) corresponding to the first proper orthogonal value (POV) is utilized as the damage index to indicate the damage region. A numerical experiment and a floor test of truss bridge illustrate the validity of the proposed method for damage detection.

Discrete-time setpoint-triggered reset integrator design with guaranteed performance and stability

According to Bode's gain-phase relationship, in linear time-invariant controllers, introducing an integral action to eliminate the steady-state error has an adverse effect of increased phase delay and overshoot, leading to performance deterioration. Moreover, increasing the bandwidth of the closed-loop system to enhance the low-frequency disturbance rejection invariably amplifies the sensitivity to high-frequency disturbances. Hence, the performance of the linear controllers is always limited due to these fundamental frequency- and time-domain limitations. Motivated by the desire to address the fundamental limitations of linear controllers and improve the time-varying closed-loop performance, we put forward a novel setpoint-triggered reset integrator strategy that varies the integrator cut-off frequency based on the setpoint information. Particularly, to tackle the time-varying disturbances and setpoint profiles, the proposed controller consists of a nominal linear controller and a variable-gain reset integrator. We show the global asymptotic stability of the proposed methodology using positive-real lemma along with the LaSalle's invariance principle and experimentally validate using measured frequency response function. Moreover, the efficacy of the proposed technique compared to that of the linear controller is experimentally demonstrated on a benchmark rotary servo system. Experimental results assessed using the tracking error and cumulative power spectral density substantiate that the proposed control strategy can not only improve the low-frequency disturbance rejection but also augment the high-frequency trajectory tracking performance.

Nonparametric Data-Driven Modeling of Linear Systems: Estimating the Frequency Response and Impulse Response Function

The aim of this article is to give a tutorial overview of frequency response function (FRF) or impulse response (IR) function measurements of linear dynamic systems. These nonparametric system identification methods provide a first view on the dynamics of a system. As discussed in Summary, the article discusses three main points. The first replaces classic FRF measurement techniques based on spectral analysis methods with more advanced, recently developed algorithms. User guidelines will be given to select the best among these methods according to four specific user situations: 1) measurements with a high or low signal-to-noise ratio (SNR), 2) systems with smooth or fast-varying transfer functions as a function of the frequency, 3) batch or realtime processing, and 4) low or high computational cost. The second main point is to store the reference signal together with the data. This will be very useful whenever there are closed loops in the system to be tested, including interactions between the generator and the setup. The final point is to use periodic excitations whenever possible. Periodic excitations provide access to a full nonparametric noise model, even under closed-loop experimental conditions. Combining periodic signals with the advanced methods presented in this article provides access to highquality FRF measurements, while the measurement time is reduced by eliminating disturbing transient effects.

Minimum condition number by orthogonal projection row selection of artificial boundary conditions for finite element model update and damage detection

The method of artificial boundary conditions provides a way of significantly expanding the amount of useable data for performing model update or damage detection. It accomplishes this by artificially applying boundary conditions directly to the measured frequency response functions, which can then be curve fit to obtain the modal parameters. A critical task in the application of this method is the selection of modes from a large available set of candidate modes generated by the application of various artificial boundary conditions. An orthogonal projection procedure is used to select modes from which to build a composite sensitivity matrix of minimum condition number for update and damage detection. Both simulation and experiment are used to investigate this method. The experimental study makes use of two beams, identical except for damage installed in one beam. Issues regarding the number and type of artificial boundary conditions, the synthesis of boundary conditions, and test frequency resolution are studied. Both the simulation and the experiment reveal that model updating and damage detection can be accomplished with high accuracy.

Simulation‐Based Material Characterization of Isotropic and Direction‐Dependent Polymers for Use in Active Structure Assemblies

Piezoceramic sensor and actuator assemblies are of increasing interest for various fields of application and attractive as part of lightweight design. As for the latter, polymers and fiber‐reinforced composites are appropriate as functional materials. Numerical simulations can be performed to predict mechanical and electrical performance of such active structure assemblies. Precise material parameters, however, are indispensable to obtain reliable simulation results. Manufacturer data is often not sufficient since it leads to major deviations between numerical simulation results and measurements. To enable reliable numerical simulations, the authors present a simulation‐based characterization approach for isotropic, transverse isotropic, and orthotropic polymer materials by means of adapting material parameters through an optimization algorithm. Frequency response function measurements of beam‐shaped test samples serve as input values. Thus, it is possible to identify values for linear viscoelastic material models and to provide direction‐dependent material parameters.

Improved FRF Measurements of Lightly Damped Systems Using Local Rational Models

Lightly damped systems exhibit strong resonances which should be accurately characterized to prevent potential harmful damage in real-world applications. Characterizing these resonances using frequency response function (FRF) measurements is challenging due to long transient behavior and spectral leakage. Local modeling techniques exist which remedy these difficulties, but they introduce a bias or do not use an appropriate model structure. In this paper, we solve these problems by developing two local rational modeling techniques that remove the bias on the FRF measurement. The proposed techniques involve the use of the bootstrapped total least squares estimator on the one hand and the incorporation of prior knowledge of the pole locations on the other hand. Furthermore, the performance of both techniques is demonstrated by measuring the flexural vibrations of a steel beam.

A new method for the accurate measurement of higher-order frequency response functions of nonlinear structural systems

Higher-order frequency response functions (FRFs) are important to the analysis and identification of structural nonlinearities. Though much research effort has been devoted recently to their potential applications, practical issues concerning the difficulty and accuracy of higher-order FRF measurement have not been rigorously assessed to date. This paper presents a new method for the accurate measurement of higher-order FRFs. The method is developed based on sinusoidal input, which is ideal for exciting a nonlinear structure into desired regimes with flexible control, and the correlation technique, which is a novel signal processing method capable of extracting accurate frequency components present in general nonlinear responses. The correlation technique adopted is a major improvement over Fourier transform based existing methods since it eliminates leakage and aliasing errors altogether and proves to be extremely robust in the presence of measurement noise. Extensive numerical case studies have been carried out to critically assess the capability and accuracy of the proposed method and the results achieved are indeed very promising. Interesting nonlinear behavior such as frequency shift and jump have been observed in first-, second- and third-order FRFs, as well as solitary islands which have been identified over which higher-order FRFs virtually do not change as input force amplitude varies. Higher-order FRFs over such solitary islands are essentially their theoretical counterparts of Volterra transfer functions which can be measured with very low input force and can be profitably employed for the identification of physical parameters of structural nonlinearities. Subsequently, a nonlinear parameter identification method has also been developed using measured higher-order FRFs and results are presented and discussed.

Comparison of the 3-dim linear viscoelastic behavior of asphalt mixes determined with tension-compression and dynamic tests

In this paper, conventional cyclic tension-compression tests and dynamic measurements have been applied to three cylindrical specimens of asphalt mixes. The results of the two tests have been compared. For the tension-compression tests, the complex modulus was obtained from the measurements of the axial stress and axial strain. For the dynamic testing, an instrumented impact hammer and an accelerometer have been used to obtain the frequency response functions of the specimens at different temperatures. The dynamic complex modulus was then back calculated by optimizing finite element calculated frequency response functions to match the measured frequency response functions. The 2S2P1D linear viscoelastic model was used to estimate master curves of the complex modulus for the two test methods. The two tests give similar results. However, the dynamic measurements give a higher value of the norm of the complex modulus and a lower value of the phase angle compared to the tension–compression results. This result is probably explained by the nonlinearity of asphalt mixes as dynamic tests are performed at a much smaller strain level than the tension-compression tests.

Empirical Evaluation of Variation of Orifice Blocking Ratio in a Tuned Liquid Column Damper using Frequency Response Function Measurement

Tuned liquid column damper (TLCD) is a simple technique used to increase the structure resistance to the external load. This type of damper can effectively decrease the structure response when the TLCD parameters such as the natural frequency and damping factor are well selected. Even though several TLCD models have been proposed and many algorithms to optimize the TLCD parameters have been developed. However, it is very little research has been conducted to evaluate the TLCD damping factor experimentally. A simple method for adjusting the TLCD damping factor empirically is by varying the orifice blocking ratio. In this research, 5 types of blocking ratio were trialled in the TLCD. They were without orifice and with 2, 4, 6 and 8 18 mm diameter orifices. The TLCD is positioned at the second floor of a Two-DOF shear structure. A frequency response function showing the ratio between the response magnitude and the excitation force on the structure in the frequency domain was recorded for each trial. The results show that the TLCDs with orifices damped vibrations more effectively than the one without orifice. The larger the blocking ratio, the larger the TLCD damping factor. Two 18 mm orifices were insufficient to damp the vibration as the blocking ratio is too large and TLCD is less responsive to the vibrations. The optimum condition of a U-Shaped TLCD blocking ratio was found to be 70.77% which corresponded to 4 holes of 18 mm orifice diameter. This type of TLCD dampened up to 80.04% of the vibration magnitude.