Elastic Modes Research Articles

Full-field Modal Analysis of a Tensegrity Column Using a Three-dimensional Scanning Laser Doppler Vibrometer with a Mirror

Abstract Tensegrity structures become important components of various engineering structures due to their high stiffness, light weight, and deployable capability. Existing studies on their dynamic analyses mainly focus on responses of their nodal points while overlook deformations of their cable and strut members. This study proposes a non-contact approach for experimental modal analysis of a tensegrity structure to identify its three-dimensional (3D) natural frequencies and full-field mode shapes, which include modes with deformations of its cable and strut members. A 3D scanning laser Doppler vibrometer is used with a mirror for extending its field of view to measure full-field vibration of a novel three-strut metal tensegrity column with free boundaries. Tensions and axial stiffnesses of its cable members are determined using natural frequencies of their transverse and longitudinal modes, respectively, to build its theoretical model for dynamic analysis and model validation purposes. Modal assurance criterion (MAC) values between experimental and theoretical mode shapes are used to identify their paired modes. Modal parameters of the first 15 elastic modes of the tensegrity column identified from the experiment, including those of the overall structure and its cable members, can be classified into five mode groups depending on their types. Modes paired between experimental and theoretical results have MAC values larger than 78%. Differences between natural frequencies of paired modes of the tensegrity column are less than 15%. The proposed non-contact 3D vibration measurement approach allows accurate estimation of 3D full-field modal parameters of the tensegrity column.

Effectiveness of palatal anchorage device (PAD) for maxillary anterior retraction with clear aligners using finite element study.

The palatal anchorage device (PAD) is commonly used in fixed appliance for anchorage, but the biomechanics of PAD in patients treated with clear aligners(CAs)remains poorly understood, especially in its elastic mode and force magnitude. This study aimed to assess the biomechanical effects of using PAD for retraction during clear aligner treatment (CAT) with the extraction of two maxillary first premolars. Four finite element models were created: (1) Incisors retraction (IR) with active contraction of the clear aligner only; (2) IR with PAD using different forces (50,75,100g) or not; (3) PAD without IR (forces of 50,75,100g), and (4) Different elastic directions (D1, D2, D3) from aligner to PAD without IR. IR caused lingual tipping and extrusion of the central incisors, as well as mesial and buccal tipping of the first molar. PAD during IR provided strong anchorage for the first molar, with elastics slightly increasing the lingual tipping and extrusion of the central incisors. Elastics applied to the U-shaped arch led to central incisor intrusion and reduced lingual retraction. The highest stress was observed on the mesial attachment of the canine. PAD effectively reinforced posterior anchorage in CAT, while anterior positioning facilitated anterior teeth intrusion. The biomechanical importance of the canine's mesial attachments in maxillary teeth treatment planning was highlighted.

Localization of elastic surface waves based on defect modes in non-Bragg structures

Abstract The non-Bragg defect mode (NBDM) of elastic surface waves is experimentally investigated by inserting a defect in the middle of an antisymmetric periodic corrugated aluminum plate, which has been known as the non-Bragg structures since the observed band gaps are different from the traditional Bragg ones. Generally, the non-Bragg band gaps, existing away from the Bragg ones in a perfectly periodic waveguide, are created by the resonances of different transverse guided modes. The transmission spectra of elastic surface waves in antisymmetric structures with defects reveal the presence of defect modes within the non-Bragg gaps. Notably, the NBDM exhibits significant distribution characteristics in comparison to the traditional Bragg defect mode, including more complex elastic wave higher-order modes and localized wave energy near the defect. Consequently, the NBDM observed in the antisymmetric periodic waveguide with defects holds potential for utilization in other elastic wave functional devices, including filters and wave intensifiers.

Suppression of the Effect of Parametric Instability in the LIGO Voyager Gravitational Wave Detector

The number of unstable combinations of elastic and Stokes optical modes of the LIGO Voyager gravitational wave detector, frequencies and spatial distributions of displacement vectors of elastic modes of mirrors were calculated. The values of overlap factors for unstable modes up to ninth-order optical modes are calculated, taking into account the azimuthal condition of parametric instability. An analysis of the influence of the temperature dependence of the Young’s modulus of the mirror material on the number of unstable modes in the Fabry-Perot resonator was performed.

A novel unsupervised machine learning algorithm for automatic Alfvénic activity detection in the TJ-II stellarator

Abstract A novel sparse encoding algorithm is developed to detect and study plasma instabilities automatically. This algorithm, called Elastic Random Mode Decomposition, is applied to the Mirnov coil signals of a dataset of 1291 discharges of the TJ-II stellarator, enabling the identification of the Alfvénic activity. In the presented approach, each signal is encoded as a collection of basic waveforms called atoms, drawn from a signal dictionary. Then the modes are identified using clustering and correlations with other plasma signals. The performance of the proposed algorithm is dramatically increased by using elastic net regularization and taking advantage of GPU architectures. Therefore the signal size and the number of dictionary elements are no longer limiting factors for encoding complex signals. Once the modes are retrieved from the shots, standard clustering and dimensionality reduction techniques are applied to obtain a 2D map featuring of the physical mode characteristics of this subset of TJ-II shots. The clustering features consider the relationship with the plasma current Ip, the diamagnetic energy W, and inverse square root of electronic density 1/√n, profiling different subtypes of Alfvénic activity. The proposed algorithm can potentially create large databases of labeled modes with unprecedented detail.

Low frequency, broadband and thin acoustic metamaterials for acoustic insulation

Elastic and locally resonant metamaterials can be used to improve the acoustic insulation properties of panels, particularly at low frequencies. Among the possible architectures, we are interested in the case of a very soft panel consisting of a porous material in which a periodic distribution of cylindrical inclusions is inserted. It has recently been established (JSV 571 (2024) 118005) that this configuration gives rise to a large and low-frequency transmission dip, explained by the coexistence of an elastic band gap and the destructive interference resulting from the interaction between an elastic resonance mode of the skeleton and a rigid body mode of the inclusions. The paper aims to describe the essential physical mechanisms involved in using a minimal lumped element model, the characteristics of which are extracted from the cell modes. The relevance of the minimal model is validated utilizing a comparison with reference results obtained using the finite element method. This approach focuses on the mechanisms involved and is not constrained by the choice of a particular geometric configuration (exact shape and size of the inclusion), allowing it to evolve.

Frequency domain solution method for electromagnetic influence analysis on torsional vibrations

In electric motor-driven machines, mechanical torsional dynamics are nonlinearly coupled with the electrical system through the electromagnetic torque and the counter-electromotive force. In this paper, an approach based on small-signal linearization is proposed for modeling the steady-state torsional dynamics of the coupled system. Using the linearized model, resonance interference diagrams, torsional response, and stability can be evaluated rapidly across numerous operating points, while accurately accounting for the electromagnetic effects. The model is validated with conventional time-stepping simulations of two induction machines. An example analysis of a 3 MW motor-driven compressor train displays the modification of torsional properties due to the electromagnetic coupling effect and explains the mechanism of torsional destabilization. Finally, previously published measurements of a 37 kW motor test bench are considered to validate the instability of the elastic torsional mode predicted by the model.

Nonlinear elasticity tailoring and failure mode manipulation of functionally graded honeycombs under large deformation

Design of lattice metamaterials with tailored mechanical properties at a relatively higher (macro) length scale by architecting lower (micro) length scale geometric and material configurations leads to achieving unprecedented mechanical properties for fulfilling advanced multi-functional structural demands. In the design space of innovative microstructural configurations, we propose a novel class of lattice metamaterials with cell walls made of optimally designed functionally graded intrinsic materials. Under different modes of remotely applied mechanical stresses, two different intuitive architectures of functional gradations for the intrinsic cell wall materials are proposed. The large-deformation nonlinear lattices result in broadband modulation of effective stiffness, and an unprecedented manipulation capability of failure modes and corresponding strengths covering ductile and brittle types depending on architected material gradation. For estimating the nonlinear elasticity and microstructural stresses as a measure of failure mode for the proposed functionally graded lattices undergoing large deformation, a multi-scale mechanics-based semi-analytical framework is developed. Geometrically nonlinear functionally graded beams with generalized material gradation, integrated with unit cell architectures, are analyzed through iterative variational energy principle-based Ritz approach. Based on the developed physically insightful computational framework, effective nonlinear elastic properties and failure modes of the functionally graded honeycomb lattices are tailored as a function of the intrinsic material gradation at the lower length scale. The proposed novel class of lattices with optimally designed functionally graded intrinsic materials and coupled unit cell architectures would open up innovative avenues for designing advanced multi-functional engineering structures and mechanical systems.

Experimental Evaluation of the Effectiveness of the Printed Circuit Board Strain-Based Methodology in Space-Borne Electronics with Vertically Mounted Printed Circuit Boards

The Oh-Park methodology was proposed to overcome the limitations of Steinberg’s theory for evaluating the structural safety of space-borne electronics and has been experimentally verified at the printed circuit board (PCB) specimen level for various types of electronic packages, such as ball grid arrays (BGAs), column grid arrays (CGAs), and small-outline packages (SOPs). However, it is necessary to validate the design methodology because the PCB mounted on the housing is affected by the elastic mode of the mechanical housing. In addition, although the validity of the existing theory based on critical strain has been verified for horizontally mounted structures, there are cases where PCBs are mounted vertically. Therefore, it is essential to consider the dynamic influence of the boundary conditions of mounted electronics. In this study, electronics specimens with corresponding boundary conditions were fabricated, and a fatigue-life test was performed. In addition, a structural analysis using Steinberg’s theory and the Oh-Park methodology was performed, and the results were compared with those of the fatigue-life test. The results showed that the analysis using the Oh-Park methodology accurately represented the test results, and the validity of the Oh-Park methodology for vertical electronics was verified experimentally.

Chirality, anisotropic viscosity and elastic anisotropy in three-dimensional active nematic turbulence

Various active materials exhibit strong spatio-temporal variability of their orientational order known as active turbulence, characterised by irregular and chaotic motion of topological defects, including colloidal suspensions, biofilaments, and bacterial colonies.In particular in three dimensions, it has not yet been explored how active turbulence responds to changes in material parameters and chirality.Here, we present a numerical study of three-dimensional (3D) active nematic turbulence, examining the influence of main material constants: (i) the flow-alignment viscosity, (ii) the magnitude and anisotropy of elastic deformation modes (elastic constants), and (iii) the chirality. Specifically, this main parameter space covers contractile or extensile, flow-aligning or flow tumbling, chiral or achiral elastically anisotropic active nematic fluids. The results are presented using time- and space-averaged fields of defect density and mean square velocity. The results also discuss defect density and mean square velocity as possible effective order parameters in chiral active nematics, distinguishing two chiral nematic states—active nematic blue phase and chiral active turbulence. This research contributes to the understanding of active turbulence, providing a numerical main phase space parameter sweep to help guide future experimental design and use of active materials.

Surrogate models for seismic response analysis of flexible rocking structures

AbstractSeismic design tools are based on surrogate models of the designed structure and the responses of those surrogates to earthquake ground motions. To design symmetric flexible rocking structures, a surrogate model that includes rocking and flexure is needed. In this paper, we derive the equation of motion of a flexible multi‐degree‐of‐freedom (MDOF) structure rocking on its base in modal coordinates. Then, we introduce a set of two‐degree‐of‐freedom (2DOF) surrogate models that accounts only for the first elastic vibration mode of the multimass structure and its rotation about the base pivot points. We investigate the surrogates' ability to represent the dynamics of an elastic MDOF structure that uplifts and rocks and the interaction between rocking and flexure. Therein, we detail the simplifications for the equations of motion of the 2DOF surrogate models and the adopted rocking impact model, and develop and check the sliding initiation condition. We show that the simplified 2DOF surrogate model responses compare well to experimental results. Then we assess the 2DOF surrogate model accuracy in representing the earthquake response of the MDOF model using Cloud Analysis and the coefficient of determination approach. We find that simplified 2DOF surrogate of the MDOF model is quite accurate () in estimating its maximum relative top displacement and acceptably accurate () in estimating its maximum base rotation based on a thousand randomly generated flexible rocking structure earthquake response analyses. Lastly, we discuss using the simplified 2DOF surrogate model of symmetric flexible rocking structures in preliminary seismic design, and give examples featuring a continuous elastic hollow semi‐conical chimney and an inelastic flexible MDOF structure, both with a base that may uplift.

Uniformly Loaded Logarithmic Beam Mode with Spatially Varying Flexural Rigidity

AbstractThis analysis explores natural leading modes represented by logarithmic functions, achieved by imposing four boundary constraints at the ends of an elastic inhomogeneous beam. The beam possessing constant material inertia, is assumed to be uniformly loaded, and is composed of material with variable stiffness. It is sought analytical expressions for beam deflections in terms of logarithmic functions. Our findings demonstrate that such formulae can be derived for a beam under axially uniform load and with spatially distributed flexural rigidity. Subsequently, the beam shapes and material properties for four specific scenarios are identified: free-free logarithmic beam, cantilevered logarithmic beam, simply-supported logarithmic beam, and simply-supported sliding logarithmic beam. Explicit logarithmic beam responses, governed by a limited number of shape parameters, are illustrated graphically using normalized deflections with respect to the maximum deflection. Highly deflected elastic logarithmic modes emerge as a consequence of high flexural rigidity influenced by the uniformly applied transverse load. These elucidated logarithmic beam modes offer potential practical applications in the structural design of functionally graded materials. They also serve as valuable testing platforms for numerical techniques employed in the analysis of more complex beam problems.

Slow sound mode prediction and band structure calculation in 1D phononic crystal nanobeams using an artificial neural network.

Phononic crystals, which are artificial crystals formed by the periodic arrangement of materials with different elastic coefficients in space, can display modulated sound waves propagating within them. Similar to the natural crystals used in semiconductor research with electronic bandgaps, phononic crystals exhibit the characteristics of phononic bandgaps. A gap design can be utilized to create various resonant cavities, confining specific resonance modes within the defects of the structure. In studies on phononic crystals, phononic band structure diagrams are often used to investigate the variations in phononic bandgaps and elastic resonance modes. As the phononic band frequencies vary nonlinearly with the structural parameters, numerous calculations are required to analyze the gap or mode frequency shifts in phononic band structure diagrams. However, traditional calculation methods are time-consuming. Therefore, this study proposes the use of neural networks to replace the time-consuming calculation processes of traditional methods. Numerous band structure diagrams are initially obtained through the finite-element method and serve as the raw dataset, and a certain proportion of the data is randomly extracted from the dataset for neural network training. By treating each mode point in the band structure diagram as an independent data point, the training dataset for neural networks can be expanded from a small number to a large number of band structure diagrams. This study also introduces another network that effectively improves mode prediction accuracy by training neural networks to focus on specific modes. The proposed method effectively reduces the cost of repetitive calculations.

A method for computing the pressure distribution in the elastic mode of single-well formation development

A method for computing the pressure distribution in the elastic mode of single-well formation development

Finite element model updating of non-proportional and exponential non-viscous damping systems using complex eigenvalues and eigenvectors

This paper proposes a finite element model updating method for non-proportional viscous and exponential non-viscous damping systems using complex eigenvalues and eigenvectors. The exponential non-viscous damping provides complex eigenvalues/vectors for elastic modes and real-valued eigenvalues/vectors for non-viscous modes. This study first highlights the challenges in accurately identifying real-valued eigenvalues/vectors for non-viscous modes with a commonly used system identification method. In contrast, complex eigenvalues/vectors for elastic modes are reliably identified and, therefore, utilized in the proposed model updating approach. Consequently, an optimization formulation is proposed to minimize the difference between the simulated and experimental complex eigenvalues/vectors. The method applies large structures that can contain multiple substructures with different damping properties. For brevity, damping properties are assumed to be uniform over each substructure. In addition, mass-proportional viscous damping and stiffness-proportional non-viscous damping are adopted for each substructure, which results in overall damping being non-proportional. To improve computational efficiency, the analytical gradient of the proposed optimization formulation is derived and implemented. For validation, the model updating of a full-scale steel pedestrian bridge is performed. The method is first validated in simulation and further validated by experimental data considering the statistical properties of system identification results. The proposed method successfully identifies stiffness and damping parameters using only a limited number of measured DOFs and modes.

Mechanical response and failure modes of three-dimensional auxetic re-entrant LPBF-manufactured steel truss lattice materials

Abstract Auxetic architected materials present a novel class of damage-tolerant materials with tunable mechanical characteristics and high energy absorption due to their unique ability to laterally contract and densify when subjected to axial compressive loading. The current state of research on negative Poisson's ratio materials mainly focuses on 2D geometries and a few families of 3D geometries with limited experimental comparisons between different architectures and various geometrical features. Furthermore, when manufactured via laser powder bed fusion, the influence of as-built deviations of geometrical and material properties inherently present due to the melt pool solidification process for thin features is relatively unexplored in the case of metal architected materials. The authors aim to study the elastic properties, peak characteristics, and failure modes of steel auxetic truss lattices subjected to axial compression while also addressing the uncertainties inherent to the metal laser powder bed fusion additive manufacturing of architected materials. This work presents an experimental and computational exploration and comparison of two promising three-dimensional auxetic truss lattice families of low relative densities. A comprehensive investigation of metal negative Poisson's ratio mechanical metamaterials is presented, including the selection of the architectures, modeling, laser powder bed fusion additive manufacturing, as-built part characterization, material testing, and mechanical testing under axial compression. The study of such architectures can unlock their potential in making them readily adaptable to a wide variety of engineering applications.

A multi-band elastic metamaterial for low-frequency multi-polarization vibration absorption

The vibration of engineering structures in actual practice occurs across numerous frequency ranges and includes diverse polarization modes such as bending, torsion, and expansion. Nevertheless, most reported elastic metamaterials are designed for a single frequency range or a single elastic wave mode, thereby making it challenging to simultaneously suppress the propagation of vibrational energy across multiple frequency ranges and polarization modes. To address this limitation, we propose a two-degree-of-freedom elastic metamaterial exhibiting a multi-frequency range elastic band gap and multi-polarization mode elastic wave band gap. By incorporating periodic vibration-absorbing units, gradient parameter vibration-absorbing units, and different polarization mode vibration-absorbing units to the actual equipment support structure, we achieve respective vibration attenuation enhancement, broadband vibration absorption, and multi-polarization mode vibration absorption. We systematically verify the effectiveness of our design through kinetic theoretical analysis, full-wave field simulations, and experimental tests. This metamaterial vibration absorption device can effectively meet the application requirements of complex vibration reduction involving multi-frequency ranges and modes in practical engineering, providing novel insights into equipment vibration and noise reduction design that holds significant reference value for engineering applications.

Effects of design and fabrication of segmented ship models on test results of wave-induced loads

The design and fabrication of segmented ship models have great effects on wave-induced loads in wave basins. Three different segmented ship models of a 20,000 TEU (twenty-feet equivalent unit) ultra-large container ship are designed and fabricated with different support frames and backbones. The shell and wooden frames of Models A and B were fabricated with the same method, however, the type of backbone and scale are different. Model C is a steel framing model, the dimensions of the frame members are validated by finite element method and the frame members are connected with welding technique. Suitable parameters are selected to design and fabricate Model C with the improved fabrication method. The calculated mode results of Model C are compared with the experimental data to validate the reasonability and advantage of the improved design and fabrication method. The dry and wet modes of the three ship models are measured and compared. The three different ship models are tested under typical conditions in different wave basins, and the corresponding test results are compared to reveal the effects of the fabrication method on structural elastic modes and wave-induced loads. Finally, the uncertainty analysis of the three model tests under typical conditions are conducted.

Impact resistance of assembled plate-lattice auxetic structures

Auxetic materials are attracting increasing interest due to their extraordinary or even abnormal mechanical properties. Different from the traditional truss-lattice structures, this paper proposes a series of novel auxetic plate-lattice structures with an excellent auxetic effect. Utilizing a dual-materials glue-free assembly design involving hyperelastic and plastic materials, these plate-lattices exhibit numerous advantages such as disassembly, assembly, replacement, recyclability, reusability, excellent energy absorption, high specific stiffness, and impact resistance performances. Quasi-static/dynamic tests and simulations are conducted to assess mechanical properties, including elastic constants, deformation modes, and mechanical responses. Simultaneously, the effect of the principal geometric parameter, concave angle, on the structural response was analyzed with various impact velocities. Findings suggest that the concave angle influences the structural elastic constants under compressive loading. The stress–strain response exhibits a distinct dual-plateau phenomenon. Taking the A65 (concave angle is 65°) configuration as an example, the stress on the second plateau is approximately 2.7 times that of the first plateau, significantly enhancing the structure's energy absorption capability. Under dynamic impact loadings, different configurations and varying impact velocities affect the structure's energy absorption performance. This paper introduces a series of assemblable plate-lattice structures with an auxetic effect, offering design insights and guidance tailored to the convenient transportation, unconventional structures, and rapid assembly needs of lightweight, impact-resistant mechanical metamaterials.

Axial compressive stability design of longitudinal asymmetrical shuttle-shaped columns

Longitudinal asymmetrical shuttle-shaped column (LA-SSC) has found its wide application when subjected to combined axial compression and horizontal point load in practical engineering. This paper focuses on elastic and elasto-plastic buckling behaviors of the LA-SSC only under axial compression and the resulting outcome could provide an indispensable basis for further establishing buckling strength prediction of LA-SSC under axial compression and horizontal point load. At first, the elastic buckling behavior of tapered columns (TC) and longitudinal symmetrical shuttle-shaped columns (LS-SSC) are derived in theory and verified by finite element numerical results, and corresponding elastic buckling modes are discussed. Subsequently the elastic buckling load prediction of LA-SSCs is established first and the corresponding normalized slender ratio, as a key design parameter, is generalized. Finally, the stability capacity of LA-SSCs is involved in theoretical analysis and finite element (FE) numerical calculation under axial pressure. Through investigating the axial load-lateral displacement curves and axial stress distributions the buckled LA-SSCs, the effects of different parameters (such as overall length, plate thickness, and steel strength) on the axial compression stability capacity of LS-SSCs are explored. As a result, the complete design formulas for axial compression stability capacity of LA-SSCs are presented first and compared with current major design codes in various countries. It is observed that the proposed strength design formulas are in good correspondence with the FE numerical results. The proposed formulas could provide references and guidance for the axial compression buckling strength of LA-SSCs under axial compression and they further supply fundamentals for establishing the strength design recommendation of LA-SSCs under axial compression and bending moments.