Nonlinear Spring Research Articles

Computational Modelling of Intra-Module Connections and Their Influence on the Robustness of a Steel Corner-Supported Volumetric Module

This paper investigates the robustness of a single 3D volumetric corner-supported module made of square hollow-section (SHS) columns. Typically, the moment–rotation (M-θ) behaviour of connections within the module (intra-module) is assumed to be fully rigid rather than semi-rigid, resulting in inaccurate assessment (i.e., overestimated vertical stiffness) during extreme loading events, such as progressive collapse. The intra-module connections are not capable of rigidly transferring the moment from the beams to the SHS columns. In this paper, a computationally intensive shell element model (SEM) of the module frame is created. The M-θ relationship of the intra-module connections in the SEM is firstly validated against test results by others and then replicated in a new simplified phenomenological beam element model (BEM), using nonlinear spring elements to capture the M-θ relationship. Comparing the structural behaviour of the SEM and BEM, under notional support removal, shows that the proposed BEM with semi-rigid connections (SR-BEM) agrees well with the validated SEM and requires substantially lower modelling time (98.7% lower) and computational effort (97.4% less RAM). When compared to a BEM with the typically modelled fully rigid intra-module connections (FR-BEM), the vertical displacement in the SR-BEM is at least 16% higher. The results demonstrate the importance of an accurate assessment of framing rotational stiffness and the benefits of a computationally efficient model.

A Hybrid Discrete-Finite Element method for continuous and discontinuous beam-like members including nonlinear geometric and material effects

This paper introduces a novel formulation, called Hybrid Discrete-Finite Element (HybriDFEM) method, for modelling one-directional continuous and discontinuous planar beam-like members, including nonlinear geometric and material effects. In this method, the structure is modelled as a series of distinct rigid blocks, connected to each other through contact pairs distributed along the interfaces. Each of those contact pairs are composed of two nonlinear multidirectional springs in series, which can represent either the deformation of the blocks themselves, or the deformation of their interface. Unlike the Applied Element Method, in which contact pairs are composed of one single spring, the current approach allows capturing phenomena such as sectional deformations or relative deformations between two blocks composed of different materials. This method shares similarities with the Discrete Element Methods in its ability to model contact interfaces between rigid or deformable units, but does not require a numerical time-domain integration scheme. More importantly, its formulation resembles that of the classical Finite Elements Method, allowing one to easily couple the latter with HybriDFEM. Following the presentation of its formulation, the method is benchmarked against analytical solutions selected from the literature, ranging from the linear-elastic response of a cantilever beam to the buckling and rocking response of continuous flexible columns, and rigid block stackings. One final example showcases the coupling of a HybriDFEM element with a linear beam finite element.

A multi-scale model for longitudinal seismic design of prefabricated utility tunnel based on multi-point constraints

A multi-scale model for the longitudinal seismic analysis of underground prefabricated utility tunnel was proposed. In this multi-scale model, beam elements were used to model the utility tunnel segments, shell elements were used to model the weak parts, e.g., tunnel intersections, and node frames with multiple springs were used to simulated the behavior of the tunnel joint. The node frames, whose form are the same as that of the prefabricated joint, were established on a basis of the cross-sectional size and the joint form of the tunnel. By arranging multiple sets of nonlinear springs to connect adjacent node frames, the refined model of the utility tunnel joint was established. Based on the multi-point constraint method, the multi-point constraint equations between the structure model and the refined joint model were given. Finally, a specific straight utility tunnel as well as a specific cross-type utility tunnel were taken as the examples to verify the accuracy and applicability of the proposed model. The results indicate that the proposed model is suitable for the longitudinal seismic analysis of the utility tunnels with asymmetric cross-section and intersections, and may provide reference for the longitudinal seismic design of large-scale utility tunnels.

Research on the interfacial bonding performance of novel composite L‐shaped concrete‐filled steel tubes

SummaryThis study aims to investigate the bonding performance of novel composite specially shaped concrete‐filled steel tubes (CFSTs). Seven novel composite L‐shaped CFST specimens were designed utilizing variations in steel tube wall thickness, steel tube length, and concrete strength as the primary parameters. Their failure modes, load–slip relationships, and longitudinal strain distribution patterns were examined through push‐out tests. Additionally, finite element models of the members were established using nonlinear spring elements based on the experimental data and subjected to numerical analysis. The research findings indicate that the ultimate bond strength of the composite L‐shaped CFSTs is positively correlated with steel tube wall thickness, steel tube length, and concrete strength. The strain distributions on the concave and convex faces of the L‐shaped steel tubes are identical. The results obtained from the finite element analysis closely match the experimental findings.

Topology optimization for rigid and compliant hybrid mechanisms

This paper proposes a topology optimization method integrating the variable trajectory constraints to design rigid and compliant hybrid mechanisms. The variable trajectory constraints are distributed in the design domain and are uniformly modeled by nonlinear spring model. Each of the variable trajectory constraints has an active or inactive state and different states map various mechanical properties of the nonlinear springs. The nonlinear relation between force and displacement of the spring is formulated. A new group of design variables denoting the states of the springs is introduced into the traditional density-based topology optimization model. The design variables representing the element density of the continuum structure and the states of the variable trajectory constraints are uniformly penalized to avoid intermediate values in order to obtain a physically realizable mechanism. Sensitivity analysis is performed by the adjoint equation method. To demonstrate the versatility and the generality of the proposed method, the typical numerical examples including linear, spline and circular trajectory constraints were implemented. The numerical comparisons between the nonlinear spring model and the actual trajectory constraints verify the accuracy. The proposed method expands the application of topology optimization and can be extensively employed to design the mechanisms integrating both the compliant members and rigid parts.

A semi-empirical method for shear response modelling of masonry infilled frame structures

This paper presents a novel experimental and theoretical methodology for the fragility assessment of masonry infilled frame structures subjected to seismic loads. The method uses a Hamiltonian Monte Carlo Bayesian Neural Network trained with laboratory tests, to obtain the constitutive parameters of a non-linear spring model that represents the masonry shear behaviour. The resulting model accounts for several types of masonry units, structural steel and reinforced concrete frames along with the effects of windows and/or doors openings. The results show that the use of deterministic models lead to poor estimations about the in-plane behaviour of the system, whereas the application of the proposed semi-empirical method results in more robust predictions according to the measured data. Also, the proposed approach is tested against two extra data-sets to evaluate its extrapolation capabilities, with satisfactory results. Moreover, the proposed method has been applied to an engineering case study which demonstrates that it can be efficiently applied to robustly assess the safety against collapse of MIF buildings. Finally, a discussion between the proposed method and the current structural standards is provided within the context of the case study.

A novel time-domain approach for identifying nonlinear structural dynamical system with explicit model based on observer/Kalman filter identification method

In this paper, a novel time-domain method for identifying nonlinear structural dynamical system with explicit model is proposed. By exploiting the observer/Kalman filter identification (OKID) and nonlinear feedback interpretation, the nonlinear parameters of the system are estimated accurately. The effectiveness of the proposed nonlinear OKID (NOKID) method is demonstrated through three numerical examples, namely a five degree-of-freedom system with one nonlinear spring, a three degree-of-freedom system with three nonlinear springs and a nonlinear rectangular membrane structure with all edges clamped. Comparing with the time-domain nonlinear subspace identification (TNSI) method, the proposed NOKID method can identify the nonlinear parameters without system order determination, and the identification results obtained by the proposed method are real numbers and have no dependency on frequency.

Temperature and strain rate effects on ultra‐high‐molecular‐weight‐polyethylene compression: An experimental and modeling approach

AbstractThis study aims to predict the compression behavior of ultra‐high molecular weight polyethylene (UHMWPE) at various temperatures (25, 40, and 55°C) and strain rates (~10−4/s and ~10−2/s) using a single set of three‐network (TN) viscoplastic model parameters. The TN model is made up of three parallel networks: networks A and B use nonlinear springs and dashpots to control the responses of the crystalline phase and confined amorphous phases, respectively, while network C captures the macromolecular amorphous networks using a single nonlinear spring. A single set of TN model parameters captures the yield and post‐yield hardening responses of the stress–strain curves at experimental temperatures and strain rates. When these TN model calibrated parameters are used as material property in a commercial finite element tool to simulate UHMWPE compression, the predicted and simulated results match well, showing the model's fidelity. Additionally, the model predicts experiments conducted at 40 and 70°C with loading rates of ~10−3/s and ~10−2/s, respectively. The study also correlates the deformations of UHMWPE's crystalline structure and macromolecular amorphous networks with its global stress versus strain response by extracting stresses from individual networks of the TN model at different strain rates and temperatures.Highlights UHMWPE's mechanical behavior predicted by a single set of TN model parameters TN model predicts mechanical response at various strain rates and temperatures. TN model explains microstructural crystalline and amorphous phase deformations.

The Post-Buckling Behavior of a Beam Constrained by Nonlinear Springy Walls

Abstract The post-buckling behavior of a beam that is subjected to lateral constraints is of relevance to a range of medical and engineering applications, such as endoscopic examination of internal organs, the insertion of a guidewire into an artery in-stent procedures, root growth, deep-drilling, and more. In this paper, we address a disconnect between the existing literature and the reality of these systems, in which the lateral constraints are flexible and experience nonlinear deformations. As a step towards bridging this gap, we consider a beam undergoing planar deformations that is laterally constrained by a nonlinear springy wall, i.e., a wall that is laterally pushed by the beam against a nonlinear spring. Based on a simplified mathematical model, we obtain closed-form analytical solutions, which provide valuable insights and intuition. For example, we show that important features of the behavior, such as the transition from point contact to line contact and switching to the next mode, are dictated solely by a non-dimensional force, regardless of all other parameters of the system, and that the full description of the behavior is possible by means of two non-dimensional quantities that describe the relative stiffness of the nonlinear spring compared to that of the beam. The results also highlight the fundamental differences between the behavior with a stiffening spring or with a softening spring, such as the number of attainable modes and the monotonicity of the overall force–displacement relation. These results are then validated by experiments.

Nonlinear elasticity and short-range mechanical coupling govern the rate and symmetry of mouth opening in Hydra.

Hydra has a tubular bilayered epithelial body column with a dome-shaped head on one end and a foot on the other. Hydra lacks a permanent mouth: its head epithelium is sealed. Upon neuronal activation, a mouth opens at the apex of the head which can exceed the body column diameter in seconds, allowing Hydra to ingest prey larger than itself. While the kinematics of mouth opening are well characterized, the underlying mechanism is unknown. We show that Hydra mouth opening is generated by independent local contractions that require tissue-level coordination. We model the head epithelium as an active viscoelastic nonlinear spring network. The model reproduces the size, timescale and symmetry of mouth opening. It shows that radial contractions, travelling inwards from the outer boundary of the head, pull the mouth open. Nonlinear elasticity makes mouth opening larger and faster, contrary to expectations. The model correctly predicts changes in mouth shape in response to external forces. By generating innervated : nerve-free chimera in experiments and simulations, we show that nearest-neighbour mechanical signalling suffices to coordinate mouth opening. Hydra mouth opening shows that in the absence of long-range chemical or neuronal signals, short-range mechanical coupling is sufficient to produce long-range order in tissue deformations.

Global vibration control of offshore wind turbines with a flexible monopile foundation using a pendulum-tuned mass damper: Risk mitigation and performance incrementation

This study investigates an optimized design approach for a passive-adaptive pendulum-tuned mass damper (PTMD). The PTMD is used to mitigate structural vibrations in offshore wind turbines (OWTs) with flexible monopile foundations considering Pile-Soil Interaction (PSI). The model OWT is based on the 5-MW design proposed by the National Renewable Energy Laboratory (NREL). The PSI incorporates a pile that is modeled as beam–column elements supported by nonlinear springs and accounts for lateral loads (p-y curves) and axial loads (t-z and Q-z curves) at nodal points. Wind and wave spectra estimation, as well as hydrodynamic and aerodynamic load analysis, are performed using a bespoke MATLAB® program operated in conjunction with an ANSYS® 3-D finite element global model. The resultant peak response at the OWT hub was evaluated through power spectral density (PSD) analysis. Optimal design choices for PTMD parameters are obtained via parametric analysis, which assesses the dynamic response in the along and across directions at the hub and stress at the tower base. A systematic representation considers the uncertainties stemming from the geometric and mechanical properties of the tower, environmental loads, and rotating blade dynamics. To address these uncertainties, a Monte Carlo simulation is employed to assess the structural risk via the Performance-Based Wind Engineering (PBWE) framework. Consequently, the probabilistic evaluation of structural response, foundation stresses, and power production assists in the optimal design process of PTMDs that mitigate the global structural vibrations of OWTs.

Seismic Performance Analysis of Semi-rigid Steel Frame Based on Panel Zone Mechanical Characteristics of the Joint Experiment Study

The panel zone, as an important part of the semi-rigid connection (SRC), can effectively affect the force transfer mechanism of the connection. This study aims to evaluate the panel zone mechanical characteristics through test on end-plate connection (EPC) and T-stub connection (TSC) under cyclic loading. Based on failure criteria, the mechanical properties of the panel zone are discussed, including the shear force, plastic energy dissipation, and failure modes. In light of the test studies, an effective and accurate numerical method in OpenSees platform for SRC with a behavior monitored by nonlinear rational springs is proposed. A 4 × 6 plane steel frame with SRC under the loading stimulation of 12 ground motions is conducted to testify the reliability of the model and analyze the panel zone behavior on the frame. The panel zones exhibit stable hysteresis characteristics and excellent ductility, and the analysis results of the effect of stiffness ratio between bolted connections and panel zone on frame energy consumption provide suggestions for structural energy dissipation design. The model conducts an accurate prediction of semi-rigid connection steel frames (SRCSFs) seismic behaviors considering the panel zone mechanical properties, and it can dramatically improve computing efficiency and is conducive to seismic design.

Investigating nonlinear dynamic properties of a swing arm rubber joint and their effects on the dynamic behavior of high-speed trains

The rubber joints mounted on swing arms provide the main part of the yaw stiffness for wheelsets and thus have a significant influence on the dynamic behaviour of railway vehicles. However, their nonlinear dynamic properties concerning the frequency, amplitude, and temperature were not fully considered in previous studies, most of which adopt simple models such as the Kelvin–Voigt model to represent rubber components. This study aims to investigate the radial nonlinear properties of swing arm rubber joints under various conditions and assess their effects on the dynamic performance of high-speed trains. A nonlinear rubber spring model which combines the fractional derivative Zener model with Berg’s friction model is established. To achieve a better fit to the measurements of a swing arm rubber joint, an optimisation-based method is employed to identify the model parameters. The viscous and frictional effects of the rubber joint are compared at different frequencies, amplitudes, and temperatures, by determining the predominant element of the nonlinear model. Additionally, the nonlinear rubber spring model is integrated into a high-speed vehicle dynamic model to investigate the effects of the nonlinear properties of swing arm rubber joints on the vehicle dynamic behaviour through the MATLAB/SIMPACK co-simulation method.

A discrete adjoint gradient approach for equality and inequality constraints in dynamics

The optimization of multibody systems requires accurate and efficient methods for sensitivity analysis. The adjoint method is probably the most efficient way to analyze sensitivities, especially for optimization problems with numerous optimization variables. This paper discusses sensitivity analysis for dynamic systems in gradient-based optimization problems. A discrete adjoint gradient approach is presented to compute sensitivities of equality and inequality constraints in dynamic simulations. The constraints are combined with the dynamic system equations, and the sensitivities are computed straightforwardly by solving discrete adjoint algebraic equations. The computation of these discrete adjoint gradients can be easily adapted to deal with different time integrators. This paper demonstrates discrete adjoint gradients for two different time-integration schemes and highlights efficiency and easy applicability. The proposed approach is particularly suitable for problems involving large-scale models or high-dimensional optimization spaces, where the computational effort of computing gradients by finite differences can be enormous. Three examples are investigated to validate the proposed discrete adjoint gradient approach. The sensitivity analysis of an academic example discusses the role of discrete adjoint variables. The energy optimal control problem of a nonlinear spring pendulum is analyzed to discuss the efficiency of the proposed approach. In addition, a flexible multibody system is investigated in a combined optimal control and design optimization problem. The combined optimization provides the best possible mechanical structure regarding an optimal control problem within one optimization.

Brownian dynamics simulations of bottlebrush polymers in dilute solution under simple shear and uniaxial extensional flows.

We study the dynamics of bottlebrush polymer molecules in dilute solutions subjected to shear and uniaxial extensional flows using Brownian dynamics simulations with hydrodynamic interaction (HI). Bottlebrush polymers are modeled using a coarse-grained representation, consisting of a set of beads interacting pairwise via a purely repulsive potential and connected by finitely extensible nonlinear springs. We present the results for molecular stretching, stress, and solution viscosity during the startup of flow as well as under steady state as a function of side chain length while keeping the backbone length fixed. In extensional flow, the backbone fractional extension and the first normal stress difference decrease with an increase in side chain length at a fixed Weissenberg number (Wi). Using simulation results both in the presence of and in the absence of HI, we show that this is primarily a consequence of steric interaction resulting from the dense grafting of side chains. In shear flow, we observe a shear-thinning behavior in all cases, although it becomes less pronounced with increasing side chain length. Furthermore, nonmonotonicity in the backbone fractional extension is observed under shear, particularly at high Wi. We contextualize our simulation results for bottlebrush polymers with respect to existing studies in the literature for linear polymers and show that the unique dynamical features characterizing bottlebrush polymers arise on account of their additional molecular thickness due to the presence of densely grafted side chains.

The effect of a nonlinear energy sink on the gust response of a wing

In this paper, the potential effectiveness of a nonlinear energy sink (NES) to absorb the energy from a wing that is vibrating as a result of flying in a gusty environment is investigated. The structural dynamics of the wing is simulated using a rigid airfoil mounted on two linear/nonlinear springs to represent the bending and torsional stiffness of the wing. The wing is subjected to a combination of gust and aerodynamic loads. The unsteady aerodynamic lift and moment are modelled using Wagner's theory. Furthermore, the gust loads are obtained by assuming two different gust profiles, e.g. sharp-edged and 1-cosine gust profiles. A nonlinear energy sink, which comprises of a concentrated mass, damper and a nonlinear spring, is attached to the wing, and its effectiveness to absorb the gust energy is investigated. The coupled nonlinear aeroelastic equations are integrated numerically to determine the response of the wing. To verify the developed aeroelastic model, the obtained results are compared with the available results in the literature and an excellent agreement is observed. The results highlight that adding the NES to the wing is capable of reducing the gust oscillation amplitude of the wing significantly when the NES parameters are chosen accordingly.

Stability and analysis of the vibrating motion of a four degrees-of-freedom dynamical system near resonance

This article investigates the impact of three harmonically external moments on the motion of a four-degrees-of-freedom (DOF) nonlinear dynamical system composed of a double rigid pendulum connected to a nonlinear spring with linear damping. In light of the system’s generalized coordinates, the governing system (GS) of motion is derived using Lagrange’s equations. With the use of the multiple-scales method (MSM), the approximate solutions (AS) of the equations of this system are obtained up to a higher order of approximation, maybe the third order. Within the framework of the absence of secular terms, the conditions of solvability are obtained. Accordingly, the different resonance cases are categorized, and three of them are investigated simultaneously. Thus, these conditions are updated in preparation for achieving the modulation equations (ME) for the examined system. The numerical solutions (NS) of the GS are achieved using the algorithms of fourth-order Runge–Kutta (4RK), which are compared with the AS, which displays their high degree of consistency and demonstrates the precision of the MSM. The motion’s time history, steady-state solutions, and resonance curves are graphed to demonstrate the influence of various system physical parameters. All relevant fixed points at steady-state situations are identified and graphed in accordance with the Routh–Hurwitz criteria (RHC). Therefore, the zones of stability/instability of are checked and analyzed. Numerous real-world applications in disciplines like engineering and physics attest to the significance of the nonlinear dynamical system under study such as in shipbuilding, automotive devices, structure vibration, developing robots, and analysis of human walking.

A Physics‐Informed Neural Network for the Nonlinear Damage Identification in a Reinforced Concrete Bridge Pier Using Seismic Responses

The condition assessment of reinforced concrete (RC) bridge piers after an earthquake using measured responses is important for ensuring the safety of road and railway users. The problem is nonlinear, and the locations and extents of damages are various. However, previous research works focused on linear structural identification or model updating assuming a limited number of nonlinear materials for reasonable estimates. Leveraging the ability of deep learning (DL) for robustly estimating a large number of unknown parameters, this study proposes an ALL nonlinear spring multi‐degree‐of‐freedom (MDOF) damage identification algorithm based on a physics‐informed neural network (PINN). The algorithm is applied to a stacked bilinear rotational spring and damper model of a pier. The number of unknown parameters reaches about 50. The errors of estimated elastic stiffnesses, damping coefficients, and ductility factors (DFs) using simulated responses added with noises are 0.4%, 0.6%, and 3.1%, respectively. Using full‐scale RC bridge pier shaking table experiments, the algorithm revealed the distributions of elastic stiffnesses and DFs along the pier height and their deteriorations. The effects of different types of local damages are quantitatively evaluated and visualized on the distributions.

Analysis of energy dissipation in non-linear viscoelastic beam in skew bending

This paper deals with the problem of analyzing the energy dissipation in a non-linear viscoelastic beam structural component made of a functionally graded material. The beam is in skew bending. The viscoelastic model used for dealing with the beam mechanical behaviour is set-up by combining of linear as well as non-linear springs and dashpots. The viscoelastic model is under strains which change in time on sinusoidal law. The material properties of the viscoelastic model are distributed smoothly along the thickness of the beam structure. The energy dissipation in the beam is due to dashpots which are parts of the viscoelastic model. Therefore, the strain energy dissipation density is derived by analyzing the stresses and strains in the dashpots. Then the strain energy dissipation density is integrated in the volume of the beam in order to obtain solution of the dissipated energy. The solution is applied for investigating the change of the dissipated energy due to variation of the properties of the model and the loading conditions of the beam.

General analysis of the dissipation of strain energy in circular columns

This theoretical paper is concerned with general analysis of the strain energy dissipation in columns of circular cross-section. The columns are under longitudinal displacements that vary continuously with time. The columns exhibit non-linear viscoelastic behavior that is studied by mechanical models constructed by using non-linear springs and dashpots. Besides, the columns are functionally graded along the radius of the cross-sections. A simple expression for the dissipated strain energy in the columns under consideration is derived. The expression holds for columns having portions with different radius of the cross-section. Also, the expression derived is applicable for columns built-up by concentric layers. Results indicating how the dissipated strain energy is influenced by various factors (distribution of material properties, geometry of the columns and loading) are presented. These results are found by using a non-linear viscoelastic model with one spring and one dashpot. A check-up is performed by determining the dissipated strain energy through subtracting the strain energy in the spring of viscoelastic model from the whole strain energy in the column.