Frequency Response Of Structures Research Articles

Open-loop swept frequency response of nonlinear structures subjected to weak coupling

AbstractThe present study demonstrates a common behaviour of a forced nonlinear structure with smooth nonlinearity, while coupled dynamics are apparent, originating from the attached electrodynamic shaker. This appears as a variation in the transmitted forcing amplitude and is often subjected to a hysteretic (multi-state) behaviour for up and down open-loop sweeping. This situation differs from the ideal constant amplitude harmonic excitation, on which parameter extraction and engineering comprehension are based on. Untreated or ignored, this can lead to the misinterpretation of the underlying dynamics through the measured nonlinear frequency response curves and their force-normalised version, often called quasi-frequency response function. In this paper, a post-processing solution is introduced for the correct interpretation of frequency response curves at constant forcing amplitudes through the open-loop construction and resectioning of the so-called frequency response surface. The phenomenon and the proposed methodology are demonstrated using a two-degrees-of-freedom model on a shaker-nonlinear beam structure. First, open-loop frequency sweeps are executed on the mechanical system to create the nonlinear frequency response surface, where their actual amplitudes and hysteresis widths are significantly different from the ideal constant forcing amplitude case. The response surface is then sectioned at the assumed constant forcing values by using an appropriate interpolation law. These resectioned curves represent the forced nonlinear standalone structure under ideal constant harmonic excitation. The frequency response surfaces are characterised and resectioned on a nonlinear structure with stiffening and softening cases. Furthermore, an improvement in the operational resonance decay (ORD) method in its filtering and automation is shown to extract the backbone curves (BBCs). The BBC and the resectioned surface provide a complete picture and cross-validation of the underlying dynamics. Finally, the BBC and its distortion are also shown in the response surfaces in relation with the excitation normalization.

Numerical Simulation Study on Vibration Characteristics and Influencing Factors of Coal Containing Geological Structure

Accurately determining the natural frequency of coal-containing geological structures is crucial for preventing mine dynamic disasters and utilizing vibration waves to break coal and enhance its permeability. Based on the modal theory of rock, vibration models of coal-containing geological structures, including layering and fractures are established. By analysis, the undamped vibration equation and its characteristic equation for both the layered coal system and the fractured coal system are derived. Subsequently, the Lanczos method is employed to solve the system’s vibration modes using ABAQUS. The effects of the layering position, layering thickness, layering physical properties, crack width, and crack length on the natural frequency and vibration response of coal-containing geological structures are investigated. The results indicate that when a single influencing factor is altered, the displacement response distribution of the coal body vibration system with geological structures remains essentially the same, and these single influencing factors have a minimal impact on the vibration displacement of the coal-containing geological structure. The natural frequency of the system decreases exponentially as the distance between the layering and the geometric center of the coal system with geological structures increases. The presence of layering in the coal system with geological structures significantly reduces the system’s natural frequency. The natural frequency of the coal system with geological structures increases in a power function manner as the layering elastic modulus increases. Conversely, the natural frequency decreases with an increase in crack length. When the change ranges of crack width and bedding thickness are the same, the natural frequency of the fractured coal body system exhibits more significant changes. The natural frequency of the coal system with geological structures initially decreases and then increases as bedding thickness and crack width increase. The trend in the natural frequency changes and the position of the extreme point are related to the ratio of the elastic modulus and density of the geological structure.

Frequency data driven damage detection of polymeric composite structural health using machine learning models

AbstractFirstly, the effect of damages (crack and delamination) on frequency responses of the polymeric composite structures is predicted numerically in this research. The responses are computed numerically using the finite element technique associated with a higher‐order deformation kinematic model. The model accuracy has been verified by comparing the published frequency responses and in‐house experimental data. The verified model is extended to generate the desired data (frequencies) utilizing various input parameters related to the geometrical forms and damage types (shapes, sizes, and positions). Further, different machine learning models (MLMs) are developed using Python algorithms for the identification of structural health. In this regard, the extracted data sets are initially used to train the MLM, detect the damages, and identify types of damage and damage‐related data of polymeric structures. Out of all kinds of MLMs, it is understood that the Random Forest Classifier provides the best result, which had an accuracy of 94.66% with the optimal parameters. The precision accomplished is 97% for intact and 94% for damaged structures. The proposed algorithm is also capable of identifying the damage‐related parameters (shape, size, type, and position) and predicting the defects early to prevent unintended mishaps.

Dynamic analysis of the three-phase magneto-electro-elastic (MEE) structures with the finite element method enriched by proper enrichment functions

In this study, a carefully designed enriched finite element method (EFEM) is presented to improve the solution accuracy of the conventional FEM by analyzing the dynamic behavior of the magnetic-electric-elastic (MEE) composite structures, which are frequently used in designing various smart and intelligent devices. By formulating the proper EFEM with ideal numerical performance, different enrichment functions are considered and the corresponding solution quality of different versions of the EFEM is compared and examined in great detail. When the Lagrange polynomial basis functions together with the harmonic trigonometric functions are used as enrichment functions, the obtained EFEM shows extremely powerful and ideal numerical performance, which is obviously better than the other versions of EFEM and the conventional FEM, in studying the free vibration and harmonic frequency responses of the MEE structures. Nearly exact numerical solutions for three-phase physical fields of MEE structures can be generated by the proposed EFEM even if very coarse mesh patterns are used. Intensive numerical studies are conducted to confirm and verify the excellent properties of the proposed EFEM in performing dynamic analysis of the MEE structures.

Study of the dynamic process in a nonlinear mathematical model of the transverse oscillations of a moving beam under perturbed boundary conditions

The study of transverse oscillations of systems moving along their axis is a very difficult, but at the same time a very important task. Mathematical models of nonlinear transverse oscillations of a beam moving along its axis are analyzed in this paper work, both for non-resonant and resonant cases. The task becomes even more complicated if we additionally take into account the method of fastening the ends of the beam or the perturbation at its ends. We have obtained dependencies that can be used in construction, transport, industry, mechanical engineering and other domains of technology, ensuring the stability and safety of the operation of such mechanical systems. Mathematical models have been obtained for structural engineers to determine the amplitude–frequency response of relevant structures. These mathematical models are key to researching the dynamics of moving media. The obtained results allow considering not only the influence of kinematic and physical-mechanical parameters on the amplitude–amplitude frequency response of the medium, but also the fastening method. In addition, the correlations obtained in the paper make it possible to study not only the influence of the moving medium parameters on the nature of changes in the frequency and amplitude of oscillations, but also to consider the movement at the points of support of the medium. Namely, even at the stage of designing a pipeline for a liquid flowing at a certain speed, it is possible to consider the influence of the oscillation of the supports or their fastening method on the dynamics of the oscillatory process. The resulting dependencies allow designers to consider the influence of the characteristics given in the paper with a high level of accuracy and predict dynamic phenomena in them. In engineering calculations of various mechanical systems, the resulting dependencies can be used to optimize parameters to avoid negative destructive phenomena during operation.

Indirect measurement method for high frequency response of complex structure based on statistical energy analysis

The harsh working environments often make it difficult to measure high frequency responses of complex structures in the fields such as aerospace and aeronautic, which severely hinders the evaluation of structural performance and safety. An indirect measurement method is proposed to estimate the high frequency responses of non-measurable subsystems using a statistical energy analysis (SEA) framework. Whether or not the non-measurable subsystem includes input load, two indirect measurement approaches are developed by constructing an overdetermined problem and a well-posed problem respectively. Both utilize only the power flow balance equations of unexcited subsystems to solve the problems and thus do not require solving for input loads. The approaches are validated numerically and experimentally through case study of a rocket engine structure. Furthermore, the interval error analyses indicate that the approaches are robust against SEA modeling errors and response measuring errors.

Story drift and damage level estimation of buildings using relative acceleration responses with multi‐target deep learning models under seismic excitation

AbstractDamage detection is one of the primary purposes of structural health monitoring to inform catastrophic risks of structures right after extreme loadings such as earthquakes and hurricanes. In structural design codes, story drifts are considered as an indicator to estimate the damage states of structures. For instance, when the story drift ratios achieve 0.2‐0.4%, light damage may be present in a building. In addition, the remaining stiffness ratios can also reveal the damage levels of a structure. Previous studies have shown that structural stiffness changes can affect the frequency responses of structures, for example, changing the locations of poles in frequency response functions. In this research, two multi‐target neural network models are developed to concurrently estimate story drifts and remaining stiffness ratios using floor accelerations under seismic excitation. One of the multi‐target neural network models focuses on developing a physics‐guided loss function with a parallel model combination. Meanwhile, the other neural network model sequentially integrates two deep learning approaches by transfer learning. For example, the long short‐term memory units estimate story drift responses from floor accelerations. Then, the short‐time Fourier transform layers of floor accelerations yield the remaining stiffness ratio estimation. The proposed models are numerically investigated and experimentally verified. As a result, both models can estimate story drift and remaining stiffness ratio using the proposed neural network models.

Free vibration of porous graphene oxide powder nano-composites assembled paraboloidal-cylindrical shells

The vibration behavior of porous nano-composite Assembled Paraboloidal-Cylindrical Shell (APCS) structures is evaluated in the present study. In detail, Graphene Oxide Powder (GOP) nanomaterials are served to improve the mechanical features correlated with porous polymer and metal matrices in order to build a Hybrid Polymer Matrix (HPM) and Hybrid Metal Matrix (HMM). For this purpose, the Halpin-Tsai and Rule of Mixture methods are employed to determine the mechanical features associated with the HMM and HPM. In addition, the First Shear Deformation Hypothesis (FSDH) is combined with Hamilton’s principle to determine the governing equations of APCSs, which are then discretized and solved according to the Generalized Differential Quadrature (GDQ) approach. Next, the eigenvalue strategy is implemented to determine the frequency responses of porous nanocomposite APCS structures, whose results are successfully compared to predictions from classical finite elements. Some novel examples are organized and solved to examine the impact of the geometrical and mechanical characteristics on the natural frequencies of porous nanocomposite APCS structures. As beneficial results: FGX and FGO distributions of GOPs throughout the thickness direction for porous HMM and HPM nanocomposite APCS yield the highest and lowest values related to the minimum frequency regarding C-C BC, while these values were determined to couple with the FGV and FGA related to the C-F and F-C BCs. An increased porosity level reduced the overall stiffness of the APCS and frequency response. Subsequently, frequencies related to a non-porous HPM nanocomposite APCS were significantly higher than those from porous nanocomposite APCS. The model of the porosity and values related to the porosity models did not cause to change in the CWN that the minimum frequency happened in.

Single-number values versus subjective judgment of airborne sound insulation in dwellings

There is a long history of adapting airborne sound insulation in buildings to the actual needs. A first pan-European work was published with COST Action TU0901: Integrating and Harmonizing Sound Insulation Aspects in Sustainable Urban Housing Constructions in which the differences between the sound insulation measure and the standard sound level difference were documented. Furthermore, ISO 16283-1: Field measurement of sound insulation in buildings and of building elements—Part 1: Airborne sound insulation states: Compared to DnT, R’ has a weaker connection to the subjective impression of airborne sound insulation. To investigate this relationship a simplified listening test and a computer-simulated variation studies of a synthesized airborne sound insulation were carried out. A test sound was generated from a speech, a music, and a noise signal, each of which was filtered by six frequency responses of solid structures. In a paired comparison task, 16 participants judged the loudness of the resulting 18 sounds. The results of the listening test show that perceived loudness is significantly correlated to the single number quantity DnT,w but less to the single number quantity R’w. This finding was confirmed across all signal types. Thus, the results confirm the statement in ISO 16283-1, that the single number quantity DnT,w has a stronger connection to the subjective impression of airborne sound insulation as the quantity R’w.

Performance Evaluation of Microfluidically Tunable Microwave Filters

This paper presents a contribution to evaluating the performances of tunable devices devoted to RF applications. It is based on reconfiguration by fluids of a capacitor/inductor associated in a monolithic substrate. Indeed, the association of two microfluidic passive devices on the same wafer allows us to increase the total frequency response of microwaves structures. The study evokes the presence and displacement of different conductive and dielectric liquids in the structure microchannels. The theoretical analysis concerns the association of microfluidic devices, a capacitor and inductor, in parallel topology. The obtained results show a good agreement between electrical parameters and the microwave response. Furthermore, a significant frequency variation from 370 MHz to 1720 MHz is achieved, with a tuning range that reaches 364.8%. The experimental part exhibits the fabrication and characterization of two structures in order to evaluate the response of microfluidic actuation for two architectures: a pass-band filter (presented in prior work) and a stop-band filter. The obtained results are in good agreement with the modeled behavior and demonstrate a large tuning range for the stop-band filter.

Numerical and experimental study on optimization of distributed material-orientation for frequency response of CFRP shell structures

In this paper, we present a non-parametric material-orientation optimization method for frequency response, a dynamic design problem of a laminated shell structure composed of orthotropic carbon-fiber-reinforced plastic (CFRP) materials. The squared vibration displacements in the arbitrarily prescribed evaluation area are minimized at a given frequency or in a given frequency band. This optimal design problem to determine the optimal material-orientation distribution is formulated as a distributed-parameter system optimization problem based on the variational method, and the sensitivity function with respect to the variation of the material-orientation is theoretically derived. The optimal material-orientation distribution is determined by the H1 gradient method with Poisson’s equation. With this method, the objective function is minimized while controlling the smoothness and the curvature distribution of the material-orientation. Numerical results show the effectiveness of the proposed method for obtaining the optimal material-orientation distribution of each layer that minimizes the squared vibration displacements at the given frequency or in the given frequency band. Also, the prototypes for the initial and the optimized material-orientation distribution are fabricated by the tailored fiber placement technique (TFP), and the static stiffness and the frequency response are experimentally evaluated. The results are compared with the calculated ones and discussed.

Generalized de-homogenization via sawtooth-function-based mapping and its demonstration on data-driven frequency response optimization

De-homogenization is becoming an effective method to significantly expedite the design of high-resolution multiscale structures, but existing methods have thus far been confined to simple static compliance minimization problems. There are two critical issues in accommodating general design cases: enabling the design of unit-cell orientation and using free-form microstructures. In this paper, we propose a generalized de-homogenization method to address these two issues, significantly increasing its applicability in various multiscale design cases. Instead of using conventional square cells with rectangular holes, we devise a parameterized microstructure composed of bars in different directions to provide more diversity in stiffness while retaining geometrical simplicity. The microstructural geometry-property relationship is then surrogated by a multi-layer neural network to avoid costly homogenization analysis during optimization. A Cartesian representation of the rotation angle is incorporated into homogenization-based optimization to design the unit-cell orientation. Corresponding high-resolution multiscale structures are obtained from the homogenization-based designs through a conformal mapping constructed with sawtooth function fields. This allows us to morph complex microstructures into an oriented and compatible tiling pattern, while preserving the local homogenized properties. To demonstrate our method with a specific application, we optimize the frequency response of structures under harmonic excitations within a given frequency range. It is the first time that the de-homogenization framework, enhanced by the sawtooth function, is applied for complex design scenarios beyond static compliance minimization. The examples illustrate that high-resolution multiscale structures can be generated with high efficiency and much better dynamic performance compared with the macroscale-only optimization. Beyond frequency response design, our proposed framework can be applied to general static and dynamic problems.

Inverse Design Framework With Invertible Neural Networks for Passive Vibration Suppression in Phononic Structures

Abstract Automated inverse design methods are critical to the development of metamaterial systems that exhibit special user-demanded properties. While machine learning approaches represent an emerging paradigm in the design of metamaterial structures, the ability to retrieve inverse designs on-demand remains lacking. Such an ability can be useful in accelerating optimization-based inverse design processes. This paper develops an inverse design framework that provides this capability through the novel usage of invertible neural networks (INNs). We exploit an INN architecture that can be trained to perform forward prediction over a set of high-fidelity samples and automatically learns the reverse mapping with guaranteed invertibility. We apply this INN for modeling the frequency response of periodic and aperiodic phononic structures, with the performance demonstrated on vibration suppression of drill pipes. Training and testing samples are generated by employing a transfer matrix method. The INN models provide competitive forward and inverse prediction performance compared to typical deep neural networks (DNNs). These INN models are used to retrieve approximate inverse designs for a queried non-resonant frequency range; the inverse designs are then used to initialize a constrained gradient-based optimization process to find a more accurate inverse design that also minimizes mass. The INN-initialized optimizations are found to be generally superior in terms of the queried property and mass compared to randomly initialized and inverse DNN-initialized optimizations. Particle swarm optimization with INN-derived initial points is then found to provide even better solutions, especially for the higher-dimensional aperiodic structures.

Frequency-dependent analysis of ultrasound apparent absorption coefficient in multiple scattering porous media: application to cortical bone

The effect of matrix viscoelastic absorption on frequency-dependent attenuation in porous structures mimicking simplified cortical bone is addressed in this numerical study. An apparent absorption is defined to quantify the difference between total attenuation (resulting from both absorption and scattering) and attenuation exclusively due to scattering. A power-law model is then used to describe the frequency-dependent apparent absorption as a function of pore diameter and density. The frequency response of the porous structures to a Gaussian pulse is studied to determine the frequency range over which the system can be considered linear. The results show that for low scattering regimes (normalized frequency 0.80), the system and its apparent absorption can be considered linear. Hence, the total attenuation coefficient results from the summation of scattering and absorption coefficients. However, for highly scattering regimes, the system can no longer be considered linear, as the apparent absorption vs. frequency deviates from a linear trend. As the pore density increases, the apparent absorption coefficient increases as well.

A novel frequency-labeled adaptive sparse grid collocation method for uncertainty quantification of the frequency response of general viscoelastic damping structures

In this paper, a frequency-labeled adaptive sparse grid collocation (FL-ASGC) method is proposed to study the uncertainty quantification (UQ) of the frequency response of general viscoelastic damping structures. The advantage of the proposed method is that, on the one hand, it has a good approximation ability for both smooth and non-smooth responses so that it can robustly predict the variability of frequency responses under various damping levels. On the other hand, by evaluating the contribution of each interpolation node to the response at all frequencies, it can adaptively select samples and their accompanying analysis frequency domains, thereby minimizing the computational costs of probabilistic surrogate modeling. To obtain the deterministic solution of each sample, the layer-wise finite element method combined with the direct frequency response (DFR) method is employed. Two numerical examples with different damping materials are provided to test the proposed method. Uncertainties that obey normal distribution in material and geometry are considered. The results show the proposed method can accurately, efficiently, and robustly predict the variability of the frequency response of general viscoelastic damping structures.

Influence of the perforation configuration on dynamic behaviors of multilayered beam structure

The novelty of this paper is to investigates the free vibration, frequency response and modal participation factors of the perforated multilayer microbeam structures using finite element analysis with different hole’s geometry, filling ratio and array of hole. The three squared, rectangular, and circular configurations of perforation are considered through analysis. Elasto-dynamic finite element formulation is implemented. The natural frequencies for the three different perforation configurations at different number of holes are investigated. For better understanding of the dynamic behavior of multilayered perforated beam structure, both modal and harmonic responses are presented and discussed. To specify the most suitable vibrational mode for a certain degree of freedom for base excitation, both participation factor and the associated modal effective mass are presented and analyzed. The obtained results show the remarkable effect of perforation configuration and the number of holes throughout the cross-sectional area on the dynamic behavior of multilayered perforated beam structures. Different boundary conditions are considered through analysis. The obtained results are supportive for the design of perforated beam structures in both macro and micro-scales.

Computing eigenvalues, eigenvectors and frequency responses of structures with non-proportional damping

This paper presents algorithms to compute eigenvalues, eigenvectors and frequency response matrix of a linear structure with non-proportional damping. The method is based on derivatives of repeated or unrepeated complex eigenvalues and complex eigenvectors . Since eigenvectors are not unique for a repeated eigenvalue, a special algorithm is developed to select the correct eigenvector for a given non-proportional damping. The advantage of the proposed method is that it only requires the knowledge of natural frequencies and mode shapes of the undamped structure. The validity of this method is established by comparing numerical results from this algorithm to exact answers.

Uncertainty propagation of frequency response of viscoelastic damping structures using a modified high-dimensional adaptive sparse grid collocation method

Uncertainty propagation (UP) of the frequency response is essential for the robust design of viscoelastic damping structures. One challenge in solving this problem is enormous computation cost from the UP analysis. This paper aims to develop a novel computational effective method based on the combination of the adaptive sparse grid collocation (ASGC) method and the high dimensional model representation (HDMR) technique to evaluate the variability of the frequency response of viscoelastic damping structures. First, a well-validated layer-wise finite element method is employed to model the viscoelastic damping structures. The direct frequency response (DFR) method is utilized to calculate the response. Then, a modified adaptive strategy using expectation increments as the sampling indicator is proposed for cost-effective sparse grid construction. Lastly, high dimensional model representation (HDMR) technique is introduced to address the difficulty in moderate and high random dimensional situations. Two numerical examples are provided to assess the performances of the proposed method. Variations in constitutive parameters of viscoelastic material and thicknesses of the viscoelastic layer are considered. Numerical results show that, compared to the original high-dimensional ASGC method and Monte Carlo Simulation (MCS), the proposed method accurately predicts the variability of frequency responses and significantly improves the computational efficiency.

Uncertainty quantification of tunable elastic metamaterials using polynomial chaos

Owing to their periodic and intricate configurations, metamaterials engineered for acoustic and elastic wave control inevitably suffer from manufacturing anomalies and deviate from theoretical dispersion predictions. This work exploits the Polynomial Chaos Theory to quantify the magnitude and extent of these deviations and assess their impact on the desired behavior. It is shown that uncertainties stemming from surface roughness, tolerances, and other inconsistencies in a metamaterial’s unit-cell parameters alter the targeted bandgap width, frequency range, and the confidence level with which it is guaranteed. The effect of uncertainties is projected from a Bloch-wave dispersion analysis of three distinct phononic and resonant cellular configurations and is further confirmed in the frequency response of the finite structures. The analysis concludes with a unique algorithm intended to guide the design of metamaterials in the presence of system uncertainties.

On the frequency response computation of geometrically nonlinear flat structures using reduced-order finite element models

This paper presents a general methodology to compute nonlinear frequency responses of flat structures subjected to large amplitude transverse vibrations, within a finite element context. A reduced-order model (ROM)is obtained by an expansion onto the eigenmode basis of the associated linearized problem, including transverse and in-plane modes. The coefficients of the nonlinear terms of the ROM are computed thanks to a non-intrusive method, using any existing nonlinear finite element code. The direct comparison to analytical models of beams and plates proves that a lot of coefficients can be neglected and that the in-plane motion can be condensed to the transverse motion, thus giving generic rules to simplify theROM. Then, a continuation technique, based on an asymptotic numerical method and the harmonic balance method, is used to compute the frequency response in free (nonlinear mode computation) or harmonically forced vibrations. The whole procedure is tested on a straight beam, a clamped circular plate and a free perforated plate for which some nonlinear modes are computed, including internal resonances. The convergence with harmonic numbers and oscillators is investigated. It shows that keeping a few of them is sufficient in a range of displacements corresponding to the order of the structure’s thickness, with a complexity of the simulated nonlinear phenomena that increase very fast with the number of harmonics and oscillators.