Rational Fraction Polynomial Method Research Articles

Effects of particle damper design parameters on the damping performance of laser powder bed fused structures

Particle dampers (PD), a passive damping technology, absorb energy from particle-particle and particle-cell wall interactions originating from friction and collision. PDs offer advantages such as design simplicity, low cost, applicability in harsh conditions, and flexibility to be used in a wide frequency band range. Additive manufacturing, specifically the powder bed fusion process, can fabricate structures with integrated PDs in a single printing process, eliminating the need to implement external dampers. However, the dynamic behavior of PDs must be determined to utilize their full potential. In this study, we examined 16 cases of integrated PDs by varying specific parameters including size, number, and locations on the structure to understand the effects of these parameters on the dynamic behavior of the first and second modes of the structure. Modal tests were conducted on additively manufactured samples to extract frequency response functions and calculate modal parameters (natural frequency and damping ratio) using the rational fraction polynomial method, studying the effects of PDs. The results showed that the damping performance of the parts was increased by a factor of up to 10 using body-integrated PDs compared with the fully fused specimen. The effectiveness of body-integrated PDs was shown to be strongly dependent on their volume and location. For instance, the damping generally increased as the volume fraction increased, which also reduced the total weight of the specimens by up to 60 g. Furthermore, the damping performance significantly increased for a specific mode when the PDs were located near the maximum displacement regions.

Developing a method for determining the natural frequency of a structure using multiple combined measurements in the frequency domain

Modal identification of structures using vibration measurement data has attracted considerable attention. Current identification methods can be divided into several groups: single-degree-of-freedom and multi-degrees-of-freedom methods; time domain identification, frequency domain identification. In which, the simultaneous determination of many natural frequencies of the structure from a single measurement data remains a complicated problem, in particular in the presence of noise. The article develops the rational fraction polynomial method using combined measurements in the frequency domain to identify the natural frequencies of structures. The effectiveness of the proposed method is validated by numerical simulations and experimental tests of steel beam.

A 3D finite element model updating of spinal lumber segment applying experimental modal data and particle swarm optimization algorithm

The dynamic loadings can be more harmful than static ones for the human lumbar spine health. Consequently, the prediction of the spine behavior under dynamic and vibration loads is vital and can be achieved by creating a precise finite element model in which the dynamic mechanical properties of the components need to be estimated. For this purpose, noninvasive experimental modal analysis can be applied to evaluate the dynamic mechanical properties of the spine in the numerical simulation by supplying the structural dynamic characteristics (natural frequencies, mode shapes, and damping ratio) of the system. Since the most adequate model for the human lumbar segment is a sheep model, in this paper, a 3D finite element model of a fresh spinal lumber segment of a sheep is generated based on the poroelasticity theory via Abaqus and is updated utilizing particle swarm optimization (PSO) algorithm and experimental modal analysis. In this regard, the frequency response function (FRF) of the specimen is obtained by performing the experimental modal test and the modal parameters are derived by using the rational fraction polynomial method. Afterward, the sensitivity analysis is carried out to determine the appropriate design variables. Finally, the PSO algorithm using experimental data is employed to update the design variables, including the elastic material properties and the structural damping factors of the specimen components, and the stiffness and damping coefficient of the suspension system. According to the results, the error percentage between the numerical FRF and the experimental one decreases remarkably after model updating indicating the high efficiency of the methods used.

Eigenvalue sensitivity minimisation for robust pole placement by the receptance method

The problem of robust pole placement in active structural vibration control by the method of receptance is considered in this paper. Expressions are derived for the eigenvalue sensitivities to parametric perturbations, which are subsequently minimised to improve performance robustness of the control of a dynamical system. The described approach has application to a vibrating system where variations are present due to manufacturing and material tolerances, damages and environment variabilities. The closed-loop eigenvalue sensitivities are expressed as a linear function of the velocity and displacement feedback gains, allowing their minimisation with carefully calculated feedback gains. The proposed algorithm involves curve fitting perturbed frequency response functions, FRFs, using the rational fraction polynomial method and implementation of a polynomial fit to the individual estimated rational fraction coefficients. This allows the eigenvalue sensitivity to be obtained entirely from structural FRFs, which is consistent with the receptance method. This avoids the need to evaluate the M,C,K matrices which are typically obtained through finite element modelling, that produces modelling uncertainty. It is also demonstrated that the sensitivity minimisation technique can work in conjunction with the pole placement and partial pole placement technique using the receptance method. To illustrate the working of the proposed algorithm, the controller is first implemented numerically and then experimentally.

Surrogate Vibration Modeling Approach for Design Optimization of Electric Machines

This article presents a surrogate modeling approach to predict the vibrational behavior during electric machine design. The methodology used in identifying and implementing the surrogate model allows a time-efficient integration of the surrogate model in multiphysics geometric optimization of electric machines, especially in scenarios of drop-in replacement. Based on the state-of-the-art review, the use of analytical or numerical techniques alone considering motor frame and support are not adequately accurate, time efficient, and flexible. The proposed method uses the 3-D finite-element (FE) structural simulations for identifying characteristic response profiles in the surrogate model and does not use structural FE simulations in the optimization loop. Experimental tests are performed on a 6/10 switched reluctance machine to validate the 3-D FE model used to identify the surrogate model. The rational fraction polynomial (RFP) method is used to fit the vibration impulse responses for a number of machines geometries defined by three stator dimensions. The impulse response of any geometry can be determined using linear interpolation of the RFP coefficients. Using the radial forces determined from the FE electromagnetic simulation, the deformation is determined using superposition.

Modelling and identification of bolted truss structure with the use of design of experiment approach

The paper concerns identification process of the structural model of the multiple degrees of freedom bolted truss structure of complex geometry. In order to properly describe dynamic response of a system, both restraints and connections of the truss structure elements should be modelled as semi-rigid. On the basis of the simplified laboratory model of the overhead power line supporting structure the method of structural model parameters identification is presented. The experimental modal analysis, that involves Rational Fraction Polynomial method, results in modal model for five mode shapes in 72 degrees of freedom. The established finite element model of the construction includes elastic anchor joints and over 150 semi-rigid bolted connections of unknown values of parameters. Applying the enhanced Central Composite Inscribed design for three factors allows to determine the response surfaces and analyze the sensitivity of the individual modal parameters to the input structural parameters. The multi-criteria optimization minimize the quality function and provide the Pareto-optimal solutions. Finally, the identified structural numerical model is verified, comparing its modal parameters with the model determined experimentally.

A comparison among modal parameter extraction methods

An effective comparison among several methods for extraction of the modal parameters from the frequency response functions measurements is presented in this research. In particular, several curve fitting methods, which are the peak amplitude method, the circle fit method, the least square complex exponential method, the eigensystem realization algorithm method and the rational fraction polynomial method were implemented in Matlab environment and compared in terms of natural frequencies, modal damping and mode shapes. Measurements were performed on a carcass of the gearbox in free–free condition. A hammer has been used with a periodic impulsive excitation signal. The natural frequencies values obtained by all methods were very similar and the differences between the results were insignificant. The peak amplitude and the circle fit gave good results for the damping ratios. The rational fraction polynomial method did the best job in detecting the damping and frequency values. The results obtained by the least square complex exponential method and the eigensystem realization algorithm method were reasonable for both frequency and damping.

A versatile method in the space domain to study short-wave rail undulatory wear caused by rail surface defects

This paper examines the development of short-wave undulatory wear caused by vehicles running over defects on rail surfaces. The track model used was obtained using a Rational Fraction Polynomial (RFP) method modified using genetic algorithms. This produces an extremely accurate and simple wheel-track model requiring minimum computational times. As corrugation formation simulations in the time domain require such a large number of thousands of wheelset passovers, it was essential to develop such a simple but precise model. A non-linear Hertzian spring was used to study the interaction between wheel and rail. Wear is considered to be proportional to the friction work, as some other authors assume for this type of problems, emerging upon contact. This model has been validated by comparing its results with experimental results of undulatory wear measured on a track. The developed tool enables the evolution of rail wear to be studied for different ride conditions that are not affordable to study experimentally. Some of the factors studied in the paper are: the influence of the frequency at which the partially worn rail surface profile is updated during the wear evolution simulation; the influence of the train speed dispersion that usually exists around the nominal speed; the influence of the position of the rail defect within a span between two sleepers; and the influence of the train׳s nominal speed.

Comparison of Different Estimation Algorithms Used in the Experimental Determination of Modal Parameters

The paper deals with the estimation of modal parameters and its main purpose is to compare differences in the values of natural frequencies and damping ratios, which were estimated using three different extraction methods: Rational Fraction Polynomial method, Complex Mode Indicator Function and Polyreference Time Domain Technique. These methods are well suited to the more general application to multi-FRF data, both of the SIMO and the MIMO types. The object of measurement was a freely suspended steel rod of circular cross section. The responses of the analyzed structure were measured by accelerometer and laser vibrometer. The results of these measurements are also discussed in the paper.

Modal identification of a GFRP-concrete hybrid footbridge prototype: Experimental tests and analytical and numerical simulations

The use of fibre reinforced polymer (FRP) structural elements, including glass fibre reinforced polymer (GFRP) pultruded profiles, has increased in the last few decades. The high potential of these materials for civil engineering structural applications, particularly for footbridges, stems from their high strength, low self-weight and good durability. However, when designing GFRP structures, the high deformability and the susceptibility to instability phenomena seldom allow the full exploitation of the GFRP material. In order to overcome these limitations, several hybrid structural systems have been proposed, namely GFRP-concrete hybrid systems. In this context, the evaluation of the dynamic characteristics of GFRP-concrete hybrid structural systems is very important, especially for footbridge applications, whose design is generally governed by pedestrian comfort criteria. This paper first presents modal identification experimental tests on a 6.0m long and 2.0m wide GFRP-concrete hybrid footbridge prototype, made of two I-shaped GFRP main girders and a thin steel fibre reinforced self-compacting concrete (SFRSCC) deck. The tests consisted of applying an excitation to the deck with an impact hammer and measuring both the applied excitation and the structure’s response. The natural modes of vibration, frequencies and damping were determined using (i) a conventional input–output modal identification technique, based on the Rational Fraction Polynomial method, and (ii) a simple output-only method directly based on the Fast Fourier Transform (FFT) of the response (mode frequencies only). In the second part of the paper, experimental data are compared with analytical and numerical simulations in order to assess the accuracy of such design tools in predicting the dynamic behaviour of GFRP-SFRSCC hybrid structures, in terms of mode shapes and frequencies.

Output-only modal parameter identification for force-embedded acceleration data in the presence of harmonic and white noise excitations

Output-only modal parameter identification is based on the assumption that external forces on a linear structure are white noise. However, harmonic excitations are also often present in real structural vibrations. In particular, it has been realized that the use of forced acceleration responses without knowledge of external forces can pose a problem in the modal parameter identification, because an external force is imparted to its impulse acceleration response function. This paper provides a three-stage identification procedure as a solution to the problem of harmonic and white noise excitations in the acceleration responses of a linear dynamic system. This procedure combines the uses of the mode indicator function, the complex mode indication function, the enhanced frequency response function, an iterative rational fraction polynomial method and mode shape inspection for the correlation-related functions of the force-embedded acceleration responses. The procedure is verified via numerical simulation of a five-floor shear building and a two-dimensional frame and also applied to ambient vibration data of a large-span roof structure. Results show that the modal parameters of these dynamic systems can be satisfactorily identified under the requirement of wide separation between vibration modes and harmonic excitations.

A simple method for the determination of the complex modulus of resilient materials using a longitudinally vibrating three-layer specimen

This paper proposes a new hybrid analytical-experimental methodology for the determination of the complex modulus of resilient materials as for e.g., composition cork-like materials. It is based on a three-layer specimen test and on a simple equation from the analytical model of the specimen. The analytical formulation solves the problem based on the elementary theory of longitudinal vibrations and without taking the transversal motions into account. This methodology is developed to avoid geometric constraints and boundary condition difficulties found on other tests as well as to minimize the computational time and complexity of the coding. The experimental setup and the deduction of the simple equation are described. Metallic layers are used at the extremities of the specimen, for which the Young modulus and mass density are known, while its loss factor may be neglected. For the intermediate resilient layer, which is assumed to present a linear viscoelastic behavior, the mass density is known. Thus, the only unknowns are the storage modulus and loss factor of that intermediate layer. The modal loss factor and the first resonant frequency of the specimen are identified from the experimental receptance frequency response function (FRF) curve using the rational fraction polynomial method (RFP). By using the first resonance frequency obtained experimentally, the analytical response becomes only a function of the storage modulus, resulting in a simple scalar nonlinear equation. Then, a numerical routine is used to determine the storage modulus by calculating the first zero of the nonlinear equation. To validate the proposed methodology, the frequency response assurance criterion (FRAC) and the frequency amplitude assurance criterion (FAAC) are used. For the five composition cork materials tested, the complex modulus shown low frequency dependence in the 40Hz to 1600Hz range. Correlation values between the analytical and the experimental curves are greater than 95%, thus validating the obtained results and the proposed methodology.

Rational Fraction Polynomial Method and Random Decrement Technique for Force-Excited Acceleration Responses

This paper addresses a parameter identification problem of a linear dynamic system resulting from forced acceleration responses when input forces are unknown. The problem is attributed to an input force imposed on its impulse acceleration response function. An identification procedure is provided as a solution to reduce the error caused by the identification problem. The procedure applies the mode indicator function, the complex mode indication function, and the rational fraction polynomial method to identify the modal parameters of the dynamic system from the frequency response functions of the random decrement signatures. Applicability of this procedure is investigated through numerical simulations of a three degree-of-freedom dynamic system loaded by Gaussian white and color noise forces. For the purpose of comparison, the autoregressive model is also used to analyze the same example. Results show that the proposed procedure can improve the identification accuracy of the modal parameters from the acceleration-based random decrement signatures, when comparing with the use of the autoregressive model.

A COMPARATIVE OF CURVE-FITTING ALGORITHMS FOR THE EXTRACTION OF MODAL PARAMETERS FROM RESPONSE MEASUREMENTS

The main objective of this paper is to perform a comparison of several curve-fitting methods for extraction of the modal parameters from response vibration measurements, and in particular the best damping estimates. Measurements were carried out on a steel beam to which a constrained layer had been added to make the damping more similar to that of vehicle structural components. Two shakers with different excitation signals, a periodic impulsive and a random signal, respectively, excited the structure, but after separation, only the random part was analysed for the results of this paper. This study compares a number of common curve fitting methods, viz: The Rational Fraction Polynomial Method, the Complex Exponential Method, the Complex Cepstrum Method, the Hilbert Envelope Method and the Ibrahim Time Domain method. The most accurate results for detection of the damping and natural frequencies were obtained by using the Ibrahim Time Domain Method, with the Rational Fraction Polynomial method very similar. The Hilbert Envelope method gave comparable damping estimates. The Cepstrum and Complex Exponential methods gave reasonable results for the frequencies, but not for the damping.

Experimental identification of damping

Four different procedures for identification of damping in mechanical systems are utilized. The procedures are based on results of computational simulations of mathematical models obtained through the finite element method––FEM and experimental procedures based on modal analysis. The rational fraction polynomial method (RFPM) is used to modal analysis. The validation of the numeric models is done through confrontations of computational results and experimental data obtained by accelerometers or proximity sensors. Procedures for reduction of the numeric models are used due to the fact that only few points are used to obtain the experimental values. Three different mechanical systems are analyzed: system of two degrees-of-freedom (two bars in balancing), thin plate clamped in an extremity and free in the other, and electric transmission line cable submitted to an axial load.

A comparison of the rational fraction polynomial method and the algorithm of mode isolation for fitting noisy data

A multitude of methods exist for fitting linear, modal models to mechanical system response data. Many of these methods involve rearranging the modal equations of motion of the system so that a linear-least-squares solution is possible. Some methods require simplifying assumptions, such as light damping, while most popular methods are exact for clean data. Because the overall process is nonlinear, two different exact, linear-least-squares methods, can give different results for noisy data. This work compares two methods which are exact for linear, viscous, state-space (or nonproportionally damped) systems, the well known Rational Fraction Polynomial Method (RFP) and the Algorithm of Mode Isolation (AMI). It is shown that while RFP performs exceptionally for clean data, it is much less robust than AMI for noisy data. The performance of both algorithms is compared when applied to noise contaminated analytical data for a multi-degree of freedom frame structure. The frame can be tuned so that high damping ratios and heavy modal coupling are present. [Work supported under a National Science Foundation Graduate Research Fellowship.]

Bias in modal parameters using the direct and iterative RFP identification procedures

In the rational fraction polynomial method [1] the identification of modal parameters is obtained through a direct linear least-square optimization technique but a particular form of fitting error is minimized. An iterative algorithm has been recently developed which uses the true fitting error [2]. In this paper a statistical analysis is developed to estimate the bias effects on the identified parameters when the data are polluted with noise. Both the direct and iterative procedures are considered. Numerical simulations are used to validate the results predicted by the theoretical analysis, which shows that the iterative approach is by far more efficient than the direct method.

An iterative rational fraction polynomial technique for modal identification

The rational fraction polynomial (RFP) modal identification procedure is a well known frequency domain fitting technique. To deal with a linear problem, the RFP procedure does not directly minimize the fitting error, i.e. the difference between the experimental and the analytical frequency response function, but a frequency weighted function of it: this causes bias in the modal parameter estimates. In this paper an iteration procedure is proposed which uses the output of the RFP technique as a starting estimate, and minimizes the true fitting error, expressed as a first order Taylor expansion of the identified parameters. Results are quite satisfactory: the fitting error is notably reduced after few iterations. Moreover, less computational modes with respect to the original RFP method are needed to obtain a good fit in a given frequency band.

A new approach for the modal identification of lightly damped structures

A new approach for the modal identification of lightly damped structures is proposed, being derived from, and a particular case of, the Rational Fraction Polynomial Method. The novelty of the new method lies in the fact that the results are obtained with the minimum of intervention and experience of the user: the method is automatic and can be made “intelligent”, in that it makes judgements and takes decisions. Some examples are presented and discussed in order to illustrate the method.