Linear Frequency Response Function Research Articles

Assessment of hybrid machine learning models for non-linear system identification of fatigue test rigs

The prediction of system responses for a given fatigue test bench drive signal is a challenging problem, since highly dynamic loads from measurement campaigns must be reproduced accurately. Linear frequency response function models are commonly used for this system identification task, but energy intensive experimental iterations are required to account for system non-linearities. Two novel hybrid modeling strategies are suggested, which augment existing approaches using non-linear Long Short-Term Memory networks. These are trained and deployed on short subsequences of measurement data and recombined using a windowing technique, which enables their application to measurement data with high sampling rates. In addition to fatigue test bench commissioning, these hybrid models can also be employed in the field of virtual sensing. The approach is tested using non-linear experimental data from a servo-hydraulic test rig and this dataset is made publicly available. A variety of metrics in time and frequency domains, as well as fatigue strength under variable amplitudes, are employed in the evaluation. It is shown, that hybrid models can successfully use frequency response function models as a linear baseline estimate, which is further improved by Long Short-Term Memory networks to enable non-linear predictions.

Vibration and force properties of an actuator formed from a piezoelectric stack within a frame structure

This paper examines the vibration and force properties of an actuator formed from a piezoelectric stack within a frame structure. Such configuration avoids the use of the traditional “earth-connected” support and allows the convenient attachment of the actuator to structures, such as power transformer tanks. The steady-state sensitivity of the actuator is analyzed, which comprised a linear frequency response function of the actuator system and a steady-state line frequency spectra of the transducer force at the frequency of the input voltage and its odd harmonics. The sensitivity is first analyzed by examining the forces transmitted to a rigid base structure through the stack and two legs of the frame when the actuator is excited by a single-frequency voltage input. The magnitude of the steady-state sensitivity at the frequency of the input voltage is approximately 15 dB higher than that at the other frequencies. The force transmitted through the piezoelectric stack is approximately twice as large as that through the two frame legs and with opposite phase. Next, the properties of transmitted forces when the rigid base structure is replaced by a beam and a plate of infinite size are also investigated. The interaction between the actuator and the elastic base structure has little effect on the magnitudes of the transmitted forces except at the resonace frequency of the actuator structure. However, the induced structural response is greatly affected and should be considered when describing the excitation capacity of the actuator. This capacity could be illustrated by the power flow between the actuator and the base structure and by the spatial distribution of the structural vibration induced by the actuator.

Fast evaluation of generalized associated linear equations (GALEs) for nonlinear systems characterization and compensation

Nonlinear Output Frequency Response Functions (NOFRFs), which are a one-dimensional extension of the linear Frequency Response Function (FRF) to nonlinear cases, have been introduced for nonlinear system characterization and compensation. The evaluation of NOFRFs is accomplished by solving a series of linear difference or differential equations, known as Generalized Associated Linear Equations (GALEs). However, the derivation of GALEs often includes numerous recursive and symbolic calculations, posing significant challenges in both practical applications and computer programming. In the present study, a fast-computing algorithm, based on combinatorial calculations as opposed to recursive and symbolic calculations, is proposed for fast evaluation of GALEs. This allows for the evaluation of NOFRFs in a more streamlined and straightforward manner. The applications of the proposed fast-computing algorithm for nonlinear system characterization and compensation are demonstrated through several case studies including a High Power Amplifier (HPA) and a Duffing system. The results of the case studies show the effectiveness of applying the GALEs and NOFRFs for system design and safety control across a wide range of engineering applications.

A two-stage model updating method for the linear parts of structures with local nonlinearities

Finite element model updating provides an important supplement for finite element modelling. However, some studies have shown that if the tested structure involves local nonlinearities due to damages, material properties and large deformation et al., it is difficult to achieve an accurate modified model using conventional model updating methods that are based on the assumption of linear structures. To address this issue, a two-stage model updating method separating the effects of local nonlinearities is proposed in this paper. Firstly, the underlying linear frequency response function is obtained by using the conditioned reverse path method. Then, combined with the Sherman-Morrison-Woodbury formula and the model updating objective function established by the frequency response function similarity metric, then structural model updating and damage detection are carried out as the second stage. Three numerical examples are given to illustrate the effectiveness of the proposed method. This method can not only accurately identify the location and quantify the extent of structural damages, but also has the advantages of not based on sensitivity, not depending on the selection of frequency points, not repeatedly calling the initial model et al. The proposed method has high computational efficiency and avoids the numerical problems often encountered by conventional frequency response function-based model updating methods.

Tool Condition Monitoring Based on Nonlinear Output Frequency Response Functions and Multivariate Control Chart

Tool condition monitoring (TCM) is a key technology for intelligent manufacturing. The objective is to monitor the tool operation status and detect tool breakage so that the tool can be changed in time to avoid significant damage to workpieces and reduce manufacturing costs. Recently, an innovative TCM approach based on sensor data modelling and model frequency analysis has been proposed. Different from traditional signal feature-based monitoring, the data from sensors are utilized to build a dynamic process model. Then the nonlinear output frequency response functions, a concept which extends the linear system frequency response function to the nonlinear case, over the frequency range of the tooth passing frequency of the machining process are extracted to reveal tool health conditions. In order to extend the novel sensor data modelling and model frequency analysis to unsupervised condition monitoring of cutting tools, in the present study, a multivariate control chart is proposed for TCM based on the frequency-domain properties of machining processes derived from the innovative sensor data modelling and model frequency analysis. The feature dimension is reduced by principal component analysis first. Then the moving average strategy is exploited to generate monitoring variables and overcome the effects of noises. The milling experiments of titanium alloys are conducted to verify the effectiveness of the proposed approach in detecting excessive flank wears of solid carbide end mills. The results demonstrate the advantages of the new approach over conventional TCM techniques and its potential in industrial applications.

Locating structural nonlinearities using linear frequency response functions and nonlinear orthogonal projections

This paper addresses the identification of the spatial location of nonlinearities based on classical modal testing. Interpreting nonlinear systems as linear systems with a nonlinear feedback loop shows that the nonlinear contribution to the response and the nonlinear restoring forces are related through the linear frequency response function (FRF) matrix. Specifically, if the nonlinearity is localized, the response’s nonlinear part is intimately linked with the linear FRF at the nonlinearity location. Thus, processing input–output data measured under pseudorandom excitation, the nonlinearity location can be inferred from the comparison of the nonlinear part of the response with the different measured FRFs. A second approach, which provides a direct estimation of the nonlinear restoring forces at the different degrees of freedom, also utilizes the same relations. Both procedures are demonstrated numerically and experimentally.

Global correlation analysis of strongly nonlinear frequency responses using the arclength-based separation and the Correlation-Map

Global correlation analysis is an important technique to quantify both the shape and amplitude differences between two response vectors. In linear dynamic systems, differences between two Frequency Response Functions (FRFs) are quantified as scalar number curves of the Global Shape Criterion (GSC) and the Global Amplitude Criterion (GAC), to represent FRF similarities at different frequencies. From linear to nonlinear, responses are usually obtained at different frequencies to form the Frequency Response Curve (FRC), replacing the FRF for dynamic analysis. Extending the concept of global correlation analysis from linear FRFs to nonlinear FRCs could quantify shape and amplitude similarities between nonlinear models. However, global correlation analysis for multivalued FRCs with a strong nonlinearity is hard to conduct, as strongly nonlinear correlation functions have complex multivalued phenomena with real/fake characteristics. In this paper, the Global Shape Curve Criterion (GSCC) and Global Amplitude Curve Criterion (GACC) are proposed for the correlation analysis of strongly nonlinear FRCs, which can quantify the similarity between two FRCs with different and complex multivalued phenomena. Through the arclength-based separation, multivalued FRCs are separated to single-valued response branches, in order to compute single-valued correlation functions that form the multivalued correlation function. The computed correlations contain the GSCC and GACC, which separately represent shape and amplitude differences between two FRCs at each frequency. The multivalued correlation function is represented as a Correlation-Map (C-MAP) to extract real correlation characteristics, for accurate correlation analysis. The multivalued correlation analysis is first verified on a 3 DOF model with a strong nonlinearity. Differences between the reference and initial multivalued FRCs are successfully quantified as scalar curves and the GACC may be more sensitive than the GSCC on models with a local nonlinearity. Then, the proposed method is further validated on an experimental 3 DOF model. Very complex 15-valued correlation functions between FRCs with different multivalued phenomena are established. Even so, the real correlations are still successfully extracted by the C-MAP. These show the validity and superiority of the proposed method.

Nonlinear Identification through eXtended Outputs (NIXO) with numerical and experimental validation using geometrically nonlinear structures

This work presents a novel technique for nonlinear system identification that operates in the frequency domain and fits a model to measured spectra to estimate the parameters in a modal domain nonlinear equation of motion (EOM). Nonlinear terms are added to the linear EOM in the form of polynomials, and the proposed algorithm estimates the polynomial coefficients as well as the underlying linear Frequency Response Function (FRF). This method is an extension to a popular nonlinear system identification algorithm called NIFO, from Nonlinear Identification through Feedback of the Outputs. However, NIFO identifies the nonlinear parameters as complex numbers that may be different at each frequency line, even though the mechanical system is expected to be governed by an EOM in which the nonlinear parameters are real and constant with frequency. This might be problematic, because any variation in the identified nonlinear parameters will distort the linear FRFs estimated by NIFO, and those linear FRFs are important to tell the user whether all of the significant nonlinearity has been extracted from the system. The proposed algorithm, here dubbed Nonlinear Identification through eXtended Outputs (NIXO), estimates the nonlinear parameters as frequency-independent and real. Additionally, it is demonstrated that for the systems studied here that the algorithm works when random and swept-sine inputs are used to excite the tested structure, while NIFO only worked well when random inputs were used. The method is first evaluated numerically using benchmark case studies, starting with the SDOF equation and then reduced models of a clamped-clamped flat beam, and the results are compared to those obtained with NIFO. Then the algorithm is applied to swept-sine measurements from a 3D-printed flat beam and the results are validated by computing the primary nonlinear normal mode of the identified model and comparing it with measurements.

Duffing-type digitally programmable nonlinear synthetic inductance for piezoelectric structures

Piezoelectric shunt damping techniques using linear circuits (e.g. resistive-inductive) and switching circuits (e.g. synchronized switch on inductor) have been extremely well studied for suppressing resonant vibrations in flexible structures. Both analog circuits and synthetic impedance circuits with digital control have been explored for linear concepts. In a parallel body of work, from the domain of mechanical methods of vibration attenuation, it is also known that leveraging nonlinearities (e.g. stiffness nonlinearity) can enhance the frequency bandwidth and offer amplitude-dependent suppression over a range of frequencies. However, the existing piezoelectric shunt damping techniques have been mostly limited to linear or switching nonlinear circuits, with the exception of a few nonlinear capacitance efforts. This work aims to introduce cubic inductance to emulate Duffing-type hardening nonlinearity in the shunt circuit with precise digital programming and tuning capability. Experiments are performed on a piezoelectric bimorph cantilever under base excitation for concept demonstration and model validation. First, linear frequency response functions of the cantilever are obtained for the short- and open-circuit conditions, and for linear resistive-inductive synthetic shunt damping, to confirm the standard linear behavior and electromechanical model parameters. Then, cubic inductance is introduced to the circuit and nonlinear experiments (up- and down-frequency sweep) are conducted. Cantilever tip to base motion transmissibility frequency response is measured along with piezoelectric voltage to base motion counterpart for a range of base excitation amplitudes. The distortion of the frequency response curves with increased base acceleration levels is observed. The nonlinear cubic coefficient is then varied to alter the manifestation of nonlinear frequency response at a given base excitation level, demonstrating the ease of tuning and triggering the nonlinear behavior on demand by means of the digitally-controlled synthetic impedance shunt. Nonlinear electromechanical model simulations are also validated against the experiments, yielding a very good agreement.

Energy Harvesting from Human Walking Motion using Pendulum-based Electromagnetic Generators

Wearable devices find many applications in health monitoring, communications, and entertainment devices; however, battery life is still a major limitation to their functionality. The kinetic energy from the walking motion can be employed as an energy source for wearable devices with the use of vibration-based energy harvesters. This paper investigates energy harvesting from a pendulum-based system with an electromagnetic generator when placed on different human body joints. The coupled electromechanical equations of motion for the pendulum-based system with an electromagnetic generator energy harvester under combined in-plane movements and the body segment rotation are derived from the Lagrange formulation. The linearized coupled electromechanical equations for small amplitude oscillations are applied for the system parameter identification. The system's nonlinear response depends on identified parameters from linear frequency response functions, while the system's behavior is explored as a function of several critical parameters: excitation frequency, excitation amplitude, resistance load, and placement on the body. The system jumps from the chaotic non-rotary to the periodic rotary solution at the limit point bifurcation (i.e., saddle-node appears on the limit cycle). After this bifurcation point, the harvested energy increases multiple times. To estimate the output power from the walking motion, the trajectories generated by the walking simulation program are substituted into the coupled electromechanical equations of motion. The system's response is always an oscillatory type with the same period as the walking motion, and the output power can be maximized by varying the external load resistance at each specific relative walking velocity. Moreover, a higher walking velocity results in higher output power from the generator. The output voltage and power from wearable energy harvesters can be predicted accurately by using more precise walking trajectories.

Exploration of the Nonlinear Effect of Pendulum Tuned Mass Dampers on Vibration Control

Understanding the nonlinearity of pendulum tuned mass dampers (PTMDs) under large-amplitude vibration is important to enhance control performance. Here, the nonlinear effects of PTMDs on vibration control of a single-degree-of-freedom (SDOF) system under harmonic excitation are explored. The nonlinear governing equations of the linear SDOF structure coupled with the PTMD are formulated. The results obtained via the method of harmonic balance under various excitation amplitudes are compared with those obtained employing the linearized system. It is shown that the effects of PTMD nonlinearity can be ignored when the PTMD rotation angle is below about 9°. Upon increasing the excitation amplitude, remarkable differences are observed between the nonlinear and linear frequency response functions (FRFs). Large mass ratios of PTMD are found to lead to reduced PTMD motion and attenuated nonlinear effects. A parametric study is performed to find the optimal parameters of the nonlinear PTMD under various excitation amplitudes. The results indicate that the optimal tuning frequency ratio increases significantly with excitation amplitudes, whereas the optimal damping is less sensitive to excitation amplitudes. Finally, a 61-m-high steel chimney structure subjected to harmonic and random wind loadings is investigated to verify the proposed PTMD design method.

Measuring the frequency response of the honeybee thorax

Insects with asynchronous flight muscles are believed to flap at the effective fundamental frequency of their thorax–wing system. Flapping in this manner leverages the natural elasticity of the thorax to reduce the energetic requirements of flight. However, to the best of our knowledge, the fundamental frequency of the insect wing–muscle–thorax system has not been measured. Here, we measure the linear frequency response function (FRF) of honeybee Apis mellifera thoraxes about their equilibrium state in order to determine their fundamental frequencies. FRFs relate the input force to output acceleration at the insect tergum and are acquired via a mechanical vibration shaker assembly. When compressed 50 μm, the thorax fundamental frequency averaged across all subjects was about 50% higher than reported wingbeat frequencies. We suspect that the measured fundamental frequencies are higher in the experiment than during flight due to boundary conditions and posthumous muscle stiffening. Next, we compress the thorax between 100–300 μm in 50 μm intervals to assess the sensitivity of the fundamental frequency to geometric modifications. For all specimens considered, the thorax fundamental frequency increased nearly monotonically with respect to level of compression. This implies that the thorax behaves as a nonlinear hardening spring when subject to large displacements, which we confirmed via static force–displacement testing. While there is little evidence that insects utilize this non-linearity during flight, the hardening characteristic may be emulated by small resonant-type flapping wing micro air vehicles to increase flapping frequency bandwidth. Overall, methods established through this work provide a foundation for further dynamical studies on insect thoraxes moving forward.

Block-oriented system identification for nonlinear modeling of all-solid-state Li-ion battery technology

The high energy density characteristic of the all-solid-state battery technology makes it one of the promising competitors of current liquid electrolyte-based lithium-ion batteries. However, due to the low ionic conductivity of solid materials and existing issues in active material-solid electrolyte interfaces, all-solid-state batteries show strong nonlinear behavior when the amplitude of input current is relatively high. In this paper, we have developed a methodology based on block-oriented nonlinear systems that can detect, quantize, and model the battery behavior, accurately. For this purpose, two solid-state coin cells, which have different ionic conductivity at the cathode interface are manufactured and excited with multisine input current. The Best Linear Approximation (BLA) method has been used to separate linear frequency response function (FRF) from nonlinear distortion. Then, a Hammerstein-Wiener system has been developed and parametrized to model the nonlinear distortions caused by ionic conductivity variation of the solid electrolyte at the cathode layer. Furthermore, the developed model has been utilized to detect possible faults in the electrode-electrolyte interface based on the nonlinearity level of the measured voltage.

Vortex-induced vibration of bridge decks: Describing function-based model

A describing function (DF)-based model is introduced for the simulation of vortex-induced vibration (VIV) of bridge decks. Similar to the linear frequency response function, the DF is the complex ratio of the first-order component of the nonlinear output to the harmonic input. The DF can be either identified using the forced vibration technique or based on the VIV nonlinear response time history. An iterative procedure is accordingly developed to predict the VIV response with the DF-based model, and an equivalent-damping-ratio-based simplified method is further proposed to efficiently obtain the limit cycle oscillation (LCO) amplitude of VIV. It is demonstrated that the conventional van der Pol-type model is equivalent to a special case of the DF-based model for VIV. Three case studies involving various cross-sections are utilized to validate the simulation accuracy and efficiency of the proposed DF-based model for typical features at VIV lock-in such as LCO and hysteresis phenomena. The vertical VIVs can be well captured by the DF-based model, while its capability of simulating the torsional VIVs requires further improvement. Furthermore, the predictive capability of DF-based model for vertical VIVs of bridge decks within a wide range of mass-damping conditions is highlighted.

Non-Linear Forced Vibration of Fully Clamped FGM Skew Plates using Homogenization Technique

The present work concerns the geometrically non-linear forced vibration of fully clamped functionally graded skew plates (FGSP). The theoretical model based on Hamilton’s principle and spectral analysis previously applied to obtain the non-linear mode shapes of thin straight structures is used. A homogenization technique is developed to reduce the FGSP problem under consideration to that of an equivalent isotropic homogeneous skew plate. Results are given for the linear and non-linear fundamental frequency of fully clamped FGSP, considering different parameters, such as the skew angle, the excitation force level. The results show a non linearity of the hardening type with a shift to the right of the bent non linear frequency response function, in the neighbourhood of the fundamental mode.

Experimental Verification of the Dynamic Model of Turbine Blades Coupled by a Sealing Strip

This paper presents the experimental-numerical comparison of a bladed system with flexible strip dampers. The experimental results constitute, to the authors’ knowledge, the first published experimental evidence on strip dampers. They explore the in-phase and out-of-phase mode of vibration of two blades coupled by a strip damper. The great influence the mode of vibration has on the strip behaviour and consequent effect on the blades’ response is highlighted, analyzed and successfully simulated numerically. Experiments are made possible by a purposely developed loading system based on compressed air. This non-contact system enables the experimenter to apply a realistic value of contact pressure on the strip without adding spurious stiffness to the system or modifying the contact conditions. The availability of experimental data obtained by imposing realistic contact conditions constitutes a true added value. It is here shown how the full stick linear FRFs (Frequency Response Functions), typically used to predict the response in case of limited excitation on blades coupled by rigid dampers, do not offer significant results in the case of flexible strip dampers.

Identification of mechanical properties of adhesive joints using ANN

PurposeThis paper aims to present a new approach to the fast determination of the effective, dynamic, mechanical properties of an adhesive for linear and nonlinear regions of the adhesive response, for both healthy and damaged states of the bond.Design/methodology/approachThe proposed approach is based on the measurement of the linear and nonlinear frequency response function (FRF) of adhesive-bonded structure and using artificial neural network identification technique. For this purpose, linear and nonlinear FRFs are measured for several single-lap joint specimens that are fabricated in healthy and damaged configurations of the bond. The measured FRFs of healthy and damaged specimens are then used to identify the natural frequencies of the specimens. The experimental natural frequencies, in turn, would be used to train artificial neural network (ANN) which would be able to predict the effective Young’s and shear moduli and damping of adhesive in healthy and damaged specimens, for any given excitation level and frequency, within the training domain.FindingsSimultaneous identification of the effective mechanical properties of adhesive for linear and nonlinear response regions, as well as healthy and damages states of the adhesive bond.Practical implicationsThe introduced method is effective to model the assembled structures with the viscoelastic adhesive joints, for linear and nonlinear regions.Originality/valueA fast methodology, using ANN, for identification the effective mechanical properties of adhesives, compared to other methods for both linear and nonlinear regions.

Modification and evaluation of a FRF-based model updating method for identification of viscoelastic constitutive models for a nonlinear polyurethane adhesive in a bonded joint

In this study, a Frequency Response Function (FRF) -based model updating method, was modified for the purpose of the identification of viscoelastic constitutive models. A steel beam, bonded to a heavy rigid steel block by a layer of Sikaflex-252 polyurethane adhesive, was employed as the test setup. Using the concept of Optimum Equivalent Linear FRF (OELF), accelerance FRFs were measured at different random excitation levels which demonstrated the nonlinear behavior of the adhesive. Using a finite element model, the sensitivity analysis showed that the selected FRFs are more sensitive to the storage and loss moduli of the adhesive near the resonances. Therefore, firstly, both of the storage and loss moduli were identified near each resonance separately and the results have been compared with the results based on Inverse Eigen-sensitivity Method (IEM). In continuation, five viscoelastic constitutive models were utilized and identified to characterize the dynamic mechanical properties of the adhesive at different excitation levels. Applying the identified models, the correlation between the FRFs of the FE models and the experimental ones were tested. The results show that amongst the identified models, The Standard Linear Solid (SLS) model in parallel with a viscous or constant structural damper (stiffness proportional) results in better correlation with experiments. Increasing the excitation level, the storage modulus of the adhesive decreases, whereas the loss modulus increases, especially at high frequencies.

A novel method for the identification of a model for the nonlinear characteristic of a bolted lap-joint

Joints are the main source of nonlinearity and energy dissipation in large assembled structures which could be otherwise considered as linear. Consequently, modeling and parameter identification of joints play a significant role in any successful design and finite element (FE) modeling of structures. In the present research, an identification procedure is proposed for the modeling of the nonlinear behavior of a bolted joint. The main emphasis are placed on the simplicity of the experimental procedures involved as well as ease of incorporation of the identified model in the FE model of the structure. Using the concept of the optimum equivalent linear frequency response function, structure was excited by two levels of random force, at two bolt preload levels, and then the eigen values of the nonlinear structure and the inverse eigen-sensitivity identification technique are used, in order to identify the nonlinear properties of bolted joints. The results of implementing the method are promising and indicative of the fact that, in contrast to static Iwan’s model of a bolted joint, the equivalent dynamic characteristics of a bolted joint may be frequency dependent, as the different modes will affect the interface zone of the jointed structures in a different manner.

Identification of the effect of debonding on the linear and nonlinear effective damping of an adhesive joint

In this paper, inverse-eigen sensitivity identification technique is used to investigate the effect of damage on the effective damping of a viscoelastic adhesive, in both linear and nonlinear regions of adhesive behavior. The modal parameters derived from the measured linear and nonlinear Frequency Response Functions (FRF) of the bonded structure are used in identification process. Experimental FRFs are employed to identify linear and nonlinear bending and shear modes which are then used for identifying of damping coefficient at different frequencies. The nonlinear FRFs are measured using the Optimum Equivalent Linear System (OELS) concept. The results indicate that the damping of the adhesive decreases with frequency. Also, debonding damage has a decreasing effect on damping of adhesive, both in linear and nonlinear regions, and with increasing the damage percentage the reduction becomes more significant. Results also show that the linear and nonlinear effective damping are different and hence depend on both amplitude of response and the mode shapes involved.