Mass-spring Model Research Articles

Inversion study of vehicle frontal collision and front bumper collision

<abstract> <p>The collision of a vehicle is often a process of strong nonlinearity and large deformation, and the finite element method is time-consuming. For this reason, a method of collision inversion research is proposed. Through the definition of the forward and inverse problems, the forward problem is inversely solved from the perspective of the inverse problem, and the collision process can be predicted quickly while the accuracy is ensured. In this paper, the idea of inversion is introduced into the collision test of the front bumper of the vehicle. First, a finite element model is established based on the geometric model of the bumper. The accuracy of the finite element model is verified by comparing the results of the drop weight test with the simulation results of the finite element model. Then, using the built simulation model, spring mass model and drop weight test, the inversion research of vehicle collision and the front bumper of a vehicle is carried out. The inversion research of the bumper first inverts the collision course in the inversion algorithm derived from the vehicle collision, and then compares the collision course derived from the inversion algorithm formula with the simulation test results. The research results show that the change trends of the time-velocity curve and the time-deformation curve obtained by the collision inversion algorithm are basically consistent with the simulation test results, indicating that the collision inversion algorithm can realize the rapid prediction of the collision course and improve derivation efficiency significantly, and it provides a new idea. Finally, under the constant energy condition of the drop weight test E = mgh, through the relationship between energy and deformation, it is concluded that the depth deformation of low-speed collision of the front bumper is greater than that of high-speed collision.</p> </abstract>

Mode conversions for elastic waves transmitted and reflected by ultrathin elastic metamaterial plates with anisotropic resonances

In this work, we study the conversions between longitudinal and transverse modes by one layer of oblique anisotropic dipolar resonators, whose size can be orders smaller than the wavelength. Mode conversions are found for both transmitted and reflected waves by such small resonators with a normal incidence. A mass-spring model is proposed to depict analytically the mode conversion, giving simplified analytical expressions for the conversion rates. The reflected conversion will gradually increase to the maximum while the solid material behind the resonators becomes soft, or vice versa. This work may provide theoretical guidance for practical applications.

Nonlinear Ultrasonic Detection Enhanced by 3-D Printed Phononic Crystals

Nonlinear ultrasonic methods are a novel non-destructive testing and evaluation technology due to their high sensitivity to micro-defects. However, the inherent nonlinearity of ultrasonic equipment hinders the feature extraction with micro-defects information and limits the full-scale use of nonlinear ultrasonic NDT techniques. In this work, we propose a material-enabled filter using phononic crystals (PCs) as functional units to eliminate the unapplicable inherent nonlinearity component. The PCs composed of mass blocks and connective layers symmetrical about the host plate, are simultaneously processed by melting deposition of PLA. The finite element method with Bloch–Floquet boundary condition is implemented to obtain the bandgap at 100 kHz. Based on the mass-spring model, the mechanism of bandgap formation is proved to be attributed to the negative effective mass density band induced by the local resonance of mass block. The PCs successfully opens a propagation wave transmission gap and eliminates the inherent nonlinearity to improve the nonlinear detection of micro-defects in the aluminum plate. This work provides a useful methodology for the wide application of PC-based nonlinear detection methods in the non-destructive testing field.

Extreme springing response statistics of a tethered platform by deconvolution

The research examines the motion response and hydrodynamic wave loads of a deep-water Tension Leg Platform (TLP), emphasising the impacts of the wave sum frequency on the restrained modes of heave, roll, and pitch. The stochastic TLP structural reaction in a random sea state was precisely computed using a Volterra series representation of the TLP corner vertical displacement, which was selected as a response process. The wave loading was evaluated using the second-order diffraction code WAMIT and applied to a linear damped mass-spring model representing the dynamic system. Then, platform displacement response at the design low probability level has been determined using a novel deconvolution approach. Since the Volterra series represented the analytical solution, the exact Volterra and the approximated predictions have been compared in this study. The latter provided an accurate way to validate the effectiveness and precision of the proposed novel deconvolution method. Compared to existing engineering techniques, the most attractive advantage of the proposed deconvolution method is that it does not rely on any pre-assumed asymptotic probability distribution class. The latter may be an attractive point for practical engineering design. Thus the primary objective of this work was to validate a novel deconvolution approach using exact quasi-analytical solutions. This work also highlights the limitations of mean up-crossing rate-based extrapolation methodologies for the situation of narrowband effects, including clustering, which are often included in the springing type of response.

Using statistical parametric mapping to assess the association of duty factor and step frequency on running kinetic.

Duty factor (DF) and step frequency (SF) were previously defined as the key running pattern determinants. Hence, this study aimed to investigate the association of DF and SF on 1) the vertical and fore-aft ground reaction force signals using statistical parametric mapping; 2) the force related variables (peaks, loading rates, impulses); and 3) the spring-mass characteristics of the lower limb, assessed by computing the force-length relationship and leg stiffness, for treadmill runs at several endurance running speeds. One hundred and fifteen runners ran at 9, 11, and 13km/h. Force data (1000Hz) and whole-body three-dimensional kinematics (200Hz) were acquired by an instrumented treadmill and optoelectronic system, respectively. Both lower DF and SF led to larger vertical and fore-aft ground reaction force fluctuations, but to a lower extent for SF than for DF. Besides, the linearity of the force-length relationship during the leg compression decreased with increasing DF or with decreasing SF but did not change during the leg decompression. These findings showed that the lower the DF and the higher the SF, the more the runner relies on the optimization of the spring-mass model, whereas the higher the DF and the lower the SF, the more the runner promotes forward propulsion.

Bandgaps and topological interfaces of metabeams with periodic acoustic black holes

Acoustic black hole (ABH) with extraordinary properties of wave manipulation and energy focalization has been drawing growing attention in mechanical metamaterial, and the development of topological phenomena provides a novel way for investigation of ABH characteristics. This paper explores the physical mechanism of dispersion curves of three metabeams with embedded periodic ABHs and studies the topological phenomena of the compound metabeam for energy localization. First, the mass-spring models are employed according to lattice dynamics method to clarify the formation mechanism of bandgaps and the effect of the parameters on dispersion relation. Meanwhile, the diatomic mass-spring model provides an analytical framework to investigate the band inversion and the displacement distribution of topological states. Furthermore, numerical analyses are carried out to investigate the band structures, eigenmodes, transmission, and the achievement of topological interface state by proposed compound metabeam. The adjustment of structural parameters, changing the mass and stiffness of the mass-spring model, can broaden the bandgap in low frequency for better vibration attenuation performance. Besides, the modal formation in low frequency is explained by masses vibration distribution. The correctness of the method is verified by comparing the transmission numerically with previous experiments. This work enriches the existing knowledge of ABH on wave manipulating and energy harvesting.

Optimization of Mass-spring Model Parameters by ICA for Assessing Compressional Behavior of Warp-knitted Spacer Fabrics Reinforced Polyurethane Cast Elastomer Composites

Optimization of Mass-spring Model Parameters by ICA for Assessing Compressional Behavior of Warp-knitted Spacer Fabrics Reinforced Polyurethane Cast Elastomer Composites

Computational aerodynamics of insect flight using volume penalization

The state-of-the-art of insect flight research using advanced computational fluid dynamics techniques on supercomputers is reviewed, focusing mostly on the work of the present authors. We present a brief historical overview, discuss numerical challenges and introduce the governing model equations. Two open source codes, one based on Fourier, the other based on wavelet representation, are succinctly presented and a mass-spring flexible wing model is described. Various illustrations of numerical simulations of flapping insects at low, intermediate and high Reynolds numbers are presented. The role of flexible wings, data-driven modeling and fluid–structure interaction issues are likewise discussed.

Loading measurement for intermediate strain rate material test based on dynamic oscillation

Loading measurement is one of the critical components of mechanical tests. Different technologies are applied for loading measurement at different spans of time and space. In the past century, the quasi-static hypothesis has been widely applied for low-speed loading conditions (strain rate < 1/s), and the stress wave theory has been applied for high-speed loading conditions (strain rate > 1000/s). However, so far, the feasibility of the quasi-static and stress wave approaches for loading measurement is questionable under intermediate strain rate loading conditions (1 000/s > strain rate > 1/s).For quasi-static approach, the signal ringing (caused by dynamic oscillation) is the main problem that hinders the accuracy of the loading measurement with the increment of strain rate. In this present study, in order to tackle the ringing signal obtained from the intermediate strain rate material test, a signal recognition algorithm was introduced, named the semi-period perturbation algorithm (SPPA). As SPPA is based on the first-order spring-mass oscillatory model, the corresponding test method for intermediate strain rate test was called single-mode method (SMM). To satisfy the basic hypothesis of SPPA, the corresponding load cell and material specimen was designed. Through modal analysis and dynamic simulations, SPPA and the load cell-specimen assembly were proved effective in restoring the loading signal from the ringing signal. Based on the high-speed tests of 6061 aluminum alloy and Q235 steel, and compared with the preponderant material test approach based on SEP 1230/ISO 26,203–2 sample, the SMM was proved effective in the loading measurement of intermediate strain rate tests. What's more, based on the accessibility of the test design and the simplicity of the data processing procedure of SMM, this method exhibited great application potential for similar mechanical scenarios of impact tests.

Reducing cost of transport in asymmetrical gaits: lessons from unilateral skipping.

Unilateral skipping is an asymmetrical gait only exceptionally used by humans, due to high energetic demands. In skipping, the cost of transport decreases as speed increases, and the spring-mass model coexists with the vaulting pendular one. However, the mechanisms of energy transfers and recovery between the vaulting and the bouncing steps are still unclear in this gait. The objective of this work is to study how spatiotemporal and spring-mass asymmetries impact on metabolic cost, lowering it despite speed augmentation. Kinematics and metabolic rates of healthy subjects were measured during running and skipping on a treadmill at controlled speeds. Metabolic power in skipping and running increased with similar slope but different intercepts. This fact determined the different behaviour of the cost of transport: constant in running, decreasing in skipping. In skipping the step time asymmetry remained constant, while the step length asymmetry decreased with speed, almost disappearing at 2.5m/s-1. Leg stiffness in trailing limb increased with higher slope than in leading limb and running; however, the relative leg stiffness asymmetry remained constant. Slow skipping presents short bouncing steps, even shorter than the vaulting, impacting the stride mechanics and the metabolic cost. Faster speeds were achieved by taking longer bouncing steps and a stiffer trailing limb, allowing to improve the effectiveness of the spring-mass mechanism. The step asymmetries' trends with respect to speed in skipping open the possibility to use this gait as an experimental paradigm to study mechanisms of metabolic cost reduction in locomotion.

Does the Extracellular Matrix Support Cell-Cell Communication by Elastic Wave Packets?

The extracellular matrix (ECM) is a fibrous network supporting biological cells and provides them a medium for interaction. Cells modify the ECM by applying traction forces, and these forces can propagate to long ranges and establish a mechanism of mechanical communication between neighboring cells. Previous studies have mainly focused on analysis of static force transmission across the ECM. In this study, we explore the plausibility of dynamic mechanical interaction, expressed as vibrations or abrupt fluctuations, giving rise to elastic waves propagating along ECM fibers. We use a numerical mass-spring model to simulate the longitudinal and transversal waves propagating along a single ECM fiber and across a 2D random fiber network. The elastic waves are induced by an active contracting cell (signaler) and received by a passive neighboring cell (receiver). We show that dynamic wave propagation may amplify the signal at the receiver end and support up to an order of magnitude stronger mechanical cues and longer-ranged communication relative to static transmission. Also, we report an optimal impulse duration corresponding to the most effective transmission, as well as extreme fast impulses, in which the waves are encaged around the active cell and do not reach the neighboring cell, possibly due to the Anderson localization effect. Finally, we also demonstrate that extracellular fluid viscosity reduces, but still allows, dynamic propagation along embedded ECM fibers. Our results motivate future biological experiments in mechanobiology to investigate, on the one hand, the mechanosensitivity of cells to dynamic forces traveling and guided by the ECM and, on the other hand, the impact of ECM architecture and remodeling on dynamic force transmission and its spectral filtering, dispersion, and decay.

A metastructure conceptual model of cantilever beam for transverse vibration attenuation

Minimum weight along with low vibration level of structures are main desires of engineers in various industries such as aerospace, automobile, and electronics. The metastructures are a kind of nontraditional structures with distributed vibration absorbers without any change in their general shape. In this article, transverse vibration suppression of a proposed metastructure cantilever beam (MSCB) was considered by a lumped mass-spring model. A prismatic cantilever beam with hollow square cross-section, carrying five uniformly distributed absorbers was investigated. Effects of combination of the absorbers’ first natural frequency on vibration suppression of the whole structure were examined. Results were validated by finite element method (FEM). Frequency response functions of the MSCB were compared by those of simple prismatic cantilever beams with the same mass. The results showed that the transverse vibration does not always decrease by increasing the number of absorbers. Also, linear combination of the first natural frequencies of absorbers leads to more vibration suppression for the MSCB.

Frictional Energy Dissipation due to Phonon Resonance in Two-Layer Graphene System

The frictional energy dissipation mechanism of a supported two-layer graphene film under the excitation of the model washboard frequency is investigated by molecular dynamics simulations. The results show that two local maxima in the energy dissipation rate occur at special frequencies as the excitation frequency increases from 0.1 to 0.6 THz. By extracting the vibrational density of states of the graphene, it is found that large numbers of phonons with frequencies equal to the excitation frequency are produced. A two-degree of freedom mass-spring model is proposed to explain the molecular dynamics results. Since the washboard frequency for atomically surfaces in wearless dry friction can be analogous to the excitation frequency in the molecular dynamics simulations, our study indicates that the phonon resonance would occur once the washboard frequency is close to the natural frequency of the frictional system, leading to remarkable local maxima in energy dissipation.Graphical Abstract

Further study on the dynamic loading transmission in cellular solids based on one-dimensional mass-spring model

The characteristics of dynamic loading transmission in cellular solids are studied in the present paper using a one-dimensional (1D) deformation-contact mass-spring model. From the perspective of deformation process, the collisions among adjacent cell walls happen due to the shock-induced localized large deformation and the cellular structure of the material. Two springs representing deformation stress and contact stress, respectively, are introduced to the mass-spring model to represent different deformation and loading transmission mechanisms. Combined with shock-wave model, the contact stress in the densification stage is determined. Both quasi-static and dynamic compression of cellular solids can be described by the 1D mass-spring model, in which the critical impact velocity of the compaction shock is used to distinguish two different compression states. Based on the 1D deformation-contact mass-spring model, several important issues in the dynamic loading transmission in cellular solids, i.e. the deformation-contact process, unloading and reversed loading issue, deformation modes under different loading conditions, stress effectiveness and micro-inertia effect, are examined. The present paper clarifies previous concerns on the mass-spring model and demonstrates its enhanced capability to simulate the dynamic loading transmission in cellular solids.

Wave propagation in two-dimensional elastic metastructures with triangular configuration

A triangular configuration spring–mass model is proposed in this paper to investigate the dependence on structural parameters of elastic wave propagation in two-dimensional (2D) elastic metamaterials/metastructures (EMs). Based on the lattice dynamics, the dispersion relations of the multiple degrees of freedom (DOF) unit cell and the vibration propagation of elastic waves are derived. Moreover, the effects of the configuration angle, spring stiffness, and mass distribution on the frequency and width of bandgaps are explored. The bandgap can be widened by 31.5% or as low as 32 Hz by reasonably adjusting the stiffness and mass distribution. Finally, the asymmetric structure (Model II-1) with the most pronounced full-bandgap difference produced by scanning along two irreducible Brillouin zones (IBZs) of Γ-X-M-Γ and Γ-Y-M-Γ was selected to explore the directional bandgap properties of EM. The dispersion curves obtained by the eigenmode division method are more consistent with the attenuation ranges of the vibration transmittance and can be used to easily and comprehensively identify the bandgap properties. This research provides important clues and theoretical guidance for the design of vibration isolators, beams, plates and other renewed devices.

Analytical method for net deployment dynamics modeling and its experimental validation

Tethered-net capturing method is one of the most popular methods for future space debris removal missions. Current modeling methods of net deployment dynamics usually rely on finite element analysis. However, the computational cost of those simulations is dramatically increased when the net is in large-scale and has many degrees of freedom. In this paper, we simplified the configuration of the net and established the dynamic models for the net deployment, including the two-degree-of-freedom model and the three-degree-of-freedom model. In addition, a net shooting device with adjustable ejection angle is designed to study the deployment characteristics of the tethered-net at different ejection angles and velocities. The established dynamic models of the tethered-net is therefore experimentally validated. The comparison between analytical and experimental results shows that the three-degree-of-freedom model is more accurate than the two-degree-of-freedom model in describing the net deployment process. Furthermore, comparison of simulation time between the three-degree-of-freedom model and the widely used mass–spring model demonstrates the computational efficiency of the proposed method. It is concluded that the proposed formula allows to support the preliminary design of a net capturing system and to achieve a real-time simulation.

Modeling of counter-bore and counter-sink screw lap joints

This paper presents an analytical approach to predict the mechanical behavior of different types of screw lap shear joints using a spring-mass model. The spring-mass model captures the axial, bending, bearing, and shear stiffnesses of the components and their masses at the joint. The springs in the model represent the stiffness arising from each of these modes of deformation. We derive the expressions of the individual stiffnesses analytically and use the spring-mass model to obtain the overall joint stiffness. We show that this joint stiffness accurately predicts the experimentally observed mechanics of counter-bore and countersink screw lap joints. The experimental validation is carried out in nine different joint configurations. The analytical model is also validated with the aid of linear finite element analysis. Furthermore, the influence of individual stiffness components on the overall joint behavior is studied by varying the thickness and width of the plates.

Two-dimensional microtwist modeling of topological polarization in hinged Kagome lattices and its experimental validation

Kagome lattices have recently attracted a great attention because of the unique mechanical properties including their topological polarization and localized zero modes at certain edges, which challenge the standard effective continuum theories. The previous study of these systems has been predominantly focused on the ideal Kagome lattice with the spring–mass models. In this study, we stretch this paradigm by exploring the hinged Kagome lattices towards practical application to understand the topological polarization under the framework of the microtwist continuum. The hinges are modeled by ligaments capable of supporting stretching, shear and bending deformations. The microtwist elasticity is then formulated thanks to leading order two-scale asymptotics and its constitutive and balance equations are derived. Performance of the proposed theory is validated by the exact solution for predicting dispersion relations and periodic zero modes. We further demonstrate the effectiveness of this theory through numerical simulations as well as experimental testing. Finally, nonuniform deformation under complex loadings and parity asymmetric surface waves in microtwist media are explored. Our study provides a great potential of using the microtwist medium to design, control and program hinge-based metamaterials.

A locally resonant phononic crystal structure with low-frequency broad bandgap

The vibration induced by low-frequency elastic wave reduces the working accuracy of precision instruments. The locally resonant phononic crystal provides a feasible measure for low-frequency vibration isolation because of its bandgap characteristics. In this paper, a locally resonant phononic crystal structure with a broad bandgap is proposed, which consists of a periodic phononic crystal plate with double-sided steel stubs coated by rubber layers. Through finite element analysis, it can be found that the proposed structure can produce a bandgap with 355 Hz bandwidth within the range of 0–500 Hz and isolate vibration in the range of bandgap. By analyzing the local resonance modes of the proposed structure, we further propose the equivalent mass–spring model to predict its bandgap range. The predicted bandgap results from equivalent models of the proposed structure agree well with the results from the finite element analysis. These equivalent models provide an effective and simple method for bandgap optimization of the proposed phononic crystal structure.

Distributed magnetic attitude control for large space structures

The utility of distributed magnetic torque rods for the attitude control of a large space structure is investigated. Distributed arrays of actuators offer advantages such as distributing structural loads, increasing fault tolerance, allowing structures to be designed modularly, and additionally the actuators may be integrated with on-orbit fabrication strategies. First, distributed torques are shown to effectively rotate highly flexible structures. This is compared with torques applied to the centre-of-mass of the structure, which cause large surface deformations and can fail to enact a rotation. This is demonstrated using a spring–mass model of a planar structure with embedded actuators. A distributed torque algorithm is then developed to control an individually addressable array of actuators. Attitude control simulations are performed, using the array to control a large space structure, again modelled as a spring–mass system. The attitude control system is demonstrated to effectively detumble a representative 75 × 75 m flexible structure, and perform slew manoeuvres, in the presence of both gravity-gradient torques and a realistic magnetic field model.