Hysteretic Elements Research Articles

Topology optimization of structural frames considering material nonlinearity and time-varying excitation

An approach for the topology optimization of structures composed of nonlinear beam elements under time-varying excitation is presented. Central to this approach is a hysteretic beam finite element model that accounts for distributed plasticity and axial-moment interaction through appropriate hysteretic interpolation functions and yield/capacity function, respectively. Nonlinearity is represented via the hysteretic variables for curvature and axial deformations that evolve according to first order nonlinear ordinary differential equations (ODEs), referred to as evolution equations, and the yield function. Hence, the governing dynamic equilibrium equations and hysteretic evolution equations can thus be concisely presented as a system of first-order nonlinear ODEs that can be solved using general ODE solvers without the need for linearization. The approach is applied for the design of frame structures with an objective to minimize the total volume in the domain, such that the maximum displacement at specified node(s) satisfies a specified constraint (i.e., drift limit) for the given excitation. The maximum displacement is approximated using the p-norm and thus permits the completion of the analytical sensitivities required for gradient-based updating. Several numerical examples are presented to demonstrate the approach for the design of structural frames subjected to pulse, harmonic, and seismic base excitation. Topologies obtained using the suggested, nonlinear approach are compared to solutions obtained from topology optimization problems assuming linear-elastic material behavior. These comparisons show that although similarities between the designs exist, in general the nonlinear designs differ in composition and, importantly, outperform the linear designs when assessed by nonlinear dynamic analysis.

A 2D hysteretic DEM model for arbitrarily shaped polygonal particles

A 2D hysteretic Discrete Element Method (DEM) model is developed for simulating the flow of food particles, specifically with multi-material 3D food printing processes in mind. Particles are modeled as arbitrarily shaped polygons due to the diverse nature of food powders, which can be highly irregular in shape. The developed hysteretic force model is applicable to both convex and concave polygonal particles. It is adjusted to use a proportional weighted maximum intersection area upon splitting of contact areas. This results in a continuous force trajectory, which would otherwise not be guaranteed. The model is validated with in literature reported packing ratios for ellipses. Simulated deposition of sugar-shaped particles shows that for dense packings the fraction of splitting interactions occurring can be up to 7%. Furthermore, the influence of particle shape on the coordination number and packing density is shown with a simulation of the deposition of sugar-like material.

Topology optimization of structural systems based on a nonlinear beam finite element model

A method for the topology optimization of structures composed of nonlinear beam elements based on a hysteretic finite element modeling approach is presented. In the context of the optimal design of structures composed of truss or beam elements, studies reported in the literature have mostly considered linear elastic material behavior. However, certain applications require consideration of the nonlinear response of the structural system to the given external forces. Particular to the methodology suggested in this paper is a hysteretic beam finite element model in which the inelastic deformations are governed by principles of mechanics in conjunction with first-order nonlinear ordinary differential equations. The nonlinear ordinary differential equations determine the onset of inelastic deformations and the approximation of the signum function in the differential equation with the hyperbolic tangent function permits the derivation of analytical sensitivities. The objective of the optimization problem is to minimize the volume of the system such that a system-level displacement satisfies a specified constraint value. This design problem is analogous to that of seismic design where inelastic deformations are permitted, yet sufficient stiffness is required to limit the overall displacement of the system to a specified threshold. The utility of the method is demonstrated through numerical examples for the design of two structural frames and a comparison with the solution from the topology optimization assuming linear elastic material behavior. The comparisons show that the nonlinear design is either comprised of a lower volume for a given level of performance, or offers better performance for a given volume in comparison with optimized linear design.

Vibration Mitigation of Rail Noise Barriers by Hysteretic Absorbers

A strategy is proposed to mitigate the noise barrier vibrations due to the train passage in high speed lines employing a hysteretic vibration absorber. The barrier is modelled as a generalized single degree of freedom system; the absorber consists of a light mass attached to the main structure by a hysteretic element whose restoring force is described by the Bouc-Wen model. The resulting two degrees of freedom system is studied, and it is shown that, for control purposes, beneficial conditions are obtained when the two oscillators are close to the resonance conditions (1:1). A procedure for a preliminary design of the absorber is highlighted; a parametric analysis varying the absorber characteristics is carried out and the optimal values are obtained by maximizing the barrier response performance. The absorber is then realized exploiting high damping rubber elements whose constitutive parameters have been identified through experimental tests. The effectiveness of the realized absorber is assessed by performing dynamic analysis of the two degrees of freedom system under the train excitation at a reference speed and comparing its performances with those of the designed one, observing a similar reduction of the barrier response. Finally, a sensitivity analysis of the performances varying the train speed shows that, even if the stiffness and damping of the absorber are amplitude dependent, its efficiency is confirmed in the speed range of high speed trains.

State transition graph of the Preisach model and the role of return-point memory.

The Preisach model has been useful as a null model for understanding memory formation in periodically driven disordered systems. In amorphous solids, for example, the athermal response to shear is due to localized plastic events (soft spots). As shown recently by Mungan etal. [Phys. Rev. Lett. 123, 178002 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.178002], the plastic response to applied shear can be rigorously described in terms of a directed network whose transitions correspond to one or more soft spots changing states. The topology of this graph depends on the interactions between soft spots and when such interactions are negligible, the resulting description becomes that of the Preisach model. A first step in linking transition graph topology with the underlying soft-spot interactions is therefore to determine the structure of such graphs in the absence of interactions. Here we perform a detailed analysis of the transition graph of the Preisach model. We highlight the important role played by return-point memory in organizing thegraphinto a hierarchy of loops and subloops. Our analysis reveals that the topology of a large portion ofthisgraph is actually not governed by the values of the switching fields that describe the hysteretic behavior of the individual elements but by a coarser parameter, a permutation ρ which prescribes the sequence in which the individual hysteretic elements change their states as the main hysteresis loop is traversed. This in turn allows us to derive combinatorial properties, such as the number of major loops in the transition graph as well as the number of states |R| constituting the main hysteresis loop and its nested subloops. We find that |R| is equal to the number of increasing subsequences contained in the permutation ρ.

The van der Pol oscillator under hysteretic control: regular and chaotic dynamics

In this work we provide the novel hysteretic approach to control the chaotic dynamics of the van der Pol oscillator under periodic excitation. Using the phenomenological Bouc-Wen model, as well as in the frame of the small parameter approach we investigate the influence of the hysteretic control to the dynamical characteristics of this system. Based on the analysis of numerical results in the form of bifurcation diagrams and Lyapunov exponents, the efficiency of the stabilizing role of the hysteretic control element is established. In addition, we investigate the synchronization of oscillations in the system of coupled van der Pol oscillators (with “simple” linear coupling, as well as under hysteretic control). The stabilizing role of the hysteretic control in this case is also demonstrated.

A consistent Timoshenko hysteretic beam finite element model

A parametrized Timoshenko hysteretic beam finite element model is developed, using consistent two-node elements, to efficiently simulate the nonlinear behavior of structures. New displacement interpolation functions are derived that satisfy both the exact equilibrium and kinematic conditions. Therefore, nodal strain field continuity is achieved, resulting in a consistent formulation without shear locking effects. Multiaxial interactions are considered through yield/capacity functions and distributed plasticity is accounted for by appropriate hysteretic interpolation functions. The suggested interpolation functions yield constant element matrices that do not require updating throughout the analysis, and the nonlinearities are captured in terms of hysteretic curvatures, axial and shear deformations, which are set to evolve through ordinary differential equations. A computationally efficient solution scheme is also suggested, without requiring any linearization, to straightforwardly solve the resulting system of equations for quasi-static problems. The consistency, efficiency, versatility and validity of the suggested model is explained in detail and demonstrated through several numerical examples and comparisons with experimental data from available tests.

Dynamic performance analysis of heavy-haul locomotives running through turnout branches with electric brake

This paper develops a dynamic model of heavy-haul train to investigate the dynamic behaviour of heavy-haul locomotives passing through turnout branch lines with electric brakes. A heavy-haul train dynamic model including 2 HX-type locomotives and 4 sets of coupler and buffer systems has been built up by SIMPACK software. The developed coupler-tail arc surface contact friction element, improved nonlinear hysteretic buffer element and flattened pin rotation stop element are all taken into consideration in the coupler and buffer subsystem. The turnout branch line is simplified as an S-shape curve without considering the profile change of turnout switch and frog rails. This dynamic model is validated by comparing its calculation results with those of the real turnout model and test data respectively. On this basis, the simulations of heavy-haul locomotives passing through Chinese 12# turnout branch lines with electric brakes have been carried out, and the influences of electric brake forces, locomotive and coupler structural parameters have been investigated deeply. Results indicate that both the locomotive electric brake force and the coupler-tail friction coefficient have significant influences, and the locomotive secondary lateral stop also has a great effect. Controlling the locomotive electric brake force and optimising these locomotive and coupler structural parameters could improve the dynamic performance of heavy-haul locomotives passing through turnout branches with electric brakes.

Hysteretic Shell Finite Element

AbstractA hysteretic shell finite element for the nonlinear, static, and dynamic analysis of structures is presented, formulated on the basis of classical theory of plasticity and finite deformatio...

The surrogate system hypothesis for joint mechanics

Abstract The main challenges in the design and analysis of jointed structures deal with the prediction of dissipation across bolted interfaces. Since friction is thought to be one of the main mechanisms for this, the current study investigates the utility of frictional parameter identification applied across different structures. Based on recent discoveries, a Surrogate System Hypothesis has been formulated to systematize this: the hypothesis states in brief, that the physical properties of a joint are independent of its structural context. The current work seeks to obtain a better understanding of the underlying systems by evaluating a confidence metric for the hypothesis for a relatively simple set of systems—physically perturbed configurations of the Brake-Reus Beam. Interfacial friction is modeled using whole jointed patches with distributed hysteretic (Iwan) elements and simulations are conducted using the Quasi-Static Modal Analysis (QSMA) approach to estimate the modal characteristics of the system response (amplitude dependence of natural frequency and dissipation). Posing the estimation as a Multi-Objective Optimization Problem (MOOP) is shown to reveal important features of both the employed constitutive model as well as the structure that is modeled. Consequently, the approach is used to evaluate the epistemic uncertainty inherent in three different friction models. The studies reveal that a confidence metric for the hypothesis can be formulated in such a way that it is nearly independent of the friction constitutive model that is employed.

Conditions for the Existence of Two Limit Cycles in a System with Hysteresis Nonlinearity

This work deals with a two-dimensional automatic control system containing a single nonlinear hysteretic element in the general form. The conditions sufficient for the existence of at least two limit cycles in the system are presented. To prove the existence of cycles, three closed contours embedded into each other are constructed on the phase manifold by “sewing” together pieces of the level lines of various Lyapunov functions. System trajectories cross the inner contour “from outside inwards” and the middle contour “from inside outwards.” The outer contour is crossed by system trajectories “from outside inwards.” The existence of these contours proves the presence of at least two limit cycles in the system. This paper is a continuation of our earlier published work “Conditions for the Global Stability of a Single System with Hysteresis Nonlinearity,” in which the conditions of global stability in this system are formulated.

Study on two-level-yielding steel coupling beams for seismic-resistance of shear wall systems

This paper proposes an innovative type of two-level yielding steel coupling beams (TYSCB) to improve the seismic performance of coupled shear wall systems. The proposed TYSCB consists of two key components: shear-yielding beam and bending-yielding beam. For a minor earthquake, the former is expected to yield to dissipate energy while the latter still stays in elastic to provide the stiffness demand. Under a severe earthquake, both the shear and bending-yielding beams are designed to yield to dissipate seismic energy. Experiments are carried out on four full-scale TYSCB specimens to investigate their seismic behavior and yielding mechanism. These specimens vary in lengths, heat treatments of welding joints, web forms and strengthening of end joints. The experimental results show that TYSCBs have fully developed hysteresis behavior and the preferred yielding sequence. The equivalent damping coefficients of all the tested specimens reach up to 45%. The energy dissipation ability increase by 94% with only 27.3% reduction in stiffness compared with traditional complete steel coupling beams under minor earthquakes. On the other hand, its stiffness can increase by 37.5% with only 44.1% reduction in energy dissipation capacity compared with fuse steel coupling beams under minor earthquakes. A trilinear hysteretic model is proposed to reasonably simulate the cyclic behavior of TYSCBs. A nonlinear finite element model is also developed and validated against the test results. The hysteretic and finite element models are to facilitate the application of TYSCBs in the seismic design of shear wall systems.

Hysteretic beam element with degrading smooth models

This work presents the development of a hysteretic beam element in the context of the finite element method, that is suitable for the inelastic dynamic analysis of framed structures. The formulation proposed is able to capture the main characteristics of hysteresis in structural systems and mainly accounts for stiffness degradation, strength deterioration and pinching phenomena, as well as for non-symmetrical yielding that often characterizes their behavior. The proposed formulation is based on the decoupling of deformations into elastic and hysteretic parts by considering additional hysteretic degrees of freedom, i.e., the hysteretic curvatures and hysteretic axial deformations. The direct stiffness method is employed to establish global matrices and determine the mass and viscous damping, as well as the elastic stiffness and the hysteretic matrix of the structure that corresponds to the newly added hysteretic degrees of freedom. All the governing equations of the structure, namely the linear global equations of motion and the nonlinear evolution equations at elemental level that account for degradations and pinching, are solved simultaneously. This is accomplished by converting the system of equations into state space form and implementing a variable-order solver based on numerical differentiation formulas (NDFs) to determine the solution. Furthermore, hysteretic loops and degradation phenomena are easily controlled by modifying the model parameters at the element level enabling simulations of a more realistic response. Numerical results are presented and compared against experimental results and other finite element codes to validate the proposed formulation and verify its ability to simulate complex hysteretic behavior exhibiting cyclic degradations.

Analytical parametric study of bi-linear hysteretic model of dry friction under harmonic, impulse and random excitations

Hysteretic behavior due to some nonlinear sources is a common phenomenon in many dynamical systems. One of the sources of this behavior in mechanical systems is dry friction. Dry friction in bolted or riveted joints of mechanical structures makes their dynamic behavior hysteretic. Bi-linear hysteresis is one of the models that can be used to study these systems which is used in this paper. A SDOF system containing a bi-linear hysteretic element called Jenkins element under harmonic, impulse and random excitations is considered. For all three types of excitations, the effects of system and excitation parameters on the defined equivalent system parameters and the response specifications are studied. Harmonic balance method is employed for harmonic excitation studies, and optimum friction threshold for minimizing response amplitude is obtained versus other system parameters and response amplitude. Energy balance method is used for impulse excitation through which the desired decaying ratio can be achieved by tuning the friction threshold, depending on stiffness ratio. System under random excitation is investigated by equivalent linearization technique in two steps. At the first step, equivalent properties are obtained versus instantaneous amplitude of response. In this step, the paper contains the parametric study of system in which the variations of equivalent parameters are described when physical parameters of system or input intensity vary. Overall variance of system response is determined in the second step, and optimum sliding threshold is obtained to have minimum overall variance of system response.

Benchmarking of hysteretic elements in topological transformer model

Transformer core modeling is of importance for some transient studies like inrush currents, ferroresonance and geomagnetically induced current. This paper compares a transformer model with different magnetization representations to actual measurements. Piecewise nonlinear (Type 98) or hysteretic inductors (Type 96) both in parallel to a constant resistance, Jiles–Atherton hysteretic inductance and a newly developed inverse dynamic hysteresis model (DHM) are tested for open circuit response, residual flux after switching out, and inrush currents when energizing the transformer. The models have all problems of reproducing the magnetization current details and there are substantial differences between the models in residual flux estimation resulting in quite different inrush patterns. The DHM model is the easiest to use as few parameters are required and the model gives fairly well agreement with measurements.

Cauer Circuit Representation of the Homogenized Eddy-Current Field Based on the Legendre Expansion for a Magnetic Sheet

Using the Legendre expansion of the magnetic field distribution, we derive the standard Cauer circuit representation of the frequency-dependent properties of magnetic sheets and discuss its physical meaning. The representation of non-linear inductors is derived so as to apply the Cauer circuit in the dynamic hysteresis modeling of silicon steel. This circuit accurately reconstructs the hysteretic property under pulsewidth modulation excitation using only one or two hysteretic elements.

A hysteretic multiscale formulation for validating computational models of heterogeneous structures

A framework for the development of accurate yet computationally efficient numerical models is proposed in this work, within the context of computational model validation. The accelerated computation achieved herein relies on the implementation of a recently derived multiscale finite element formulation, able to alternate between scales of different complexity. In such a scheme, the micro-scale is modeled using a hysteretic finite element formulation. In the micro-level, nonlinearity is captured via a set of additional hysteretic degrees of freedom compactly described by an appropriate hysteric law, which gravely simplifies the dynamic analysis task. The computational efficiency of the scheme is rooted in the interaction between the micro- and a macro-mesh level, defined through suitable interpolation fields that map the finer mesh displacement field to the coarser mesh displacement field. Furthermore, damage-related phenomena that are manifested at the micro-level are accounted for, using a set of additional evolution equations corresponding to the stiffness degradation and strength deterioration of the underlying material. The developed modeling approach is utilized for the purpose of model validation; first, in the context of reliability analysis, and second, within an inverse problem formulation where the identification of constitutive parameters via availability of acceleration response data is sought.

Hysteretic Plate Finite Element

AbstractA hysteretic plate finite element for inelastic, static and dynamic analysis is presented and its performance is compared with currently used methods. A smooth, 3D hysteretic rate-independent model is utilized generalizing the uniaxial Bouc–Wen model. This is expressed in tensorial form, which incorporates the yield criterion and hardening law. The elastic mixed interpolation of tensorial components with four nodes (MITC4) element is extended by considering as additional hysteretic degrees of freedom the plastic strains at the Gauss points of each layer interface, which evolve following Bouc–Wen equations. Incorporating hysteretic relations directly into the element’s formulation is proved computationally advantageous and enables a better identification of the involved parameters in cyclic loading. The solution advances by establishing the equilibrium of external and internal forces on the basis of the initially computed stiffness and hysteretic structural matrices, thereby avoiding Gauss integrat...

Structural hysteresis model of transmitting mechanical systems

We present a structural hysteresis model which describes the dynamic behavior of transmitting mechanical systems with a hysteretic spring and damped bedstop element, both connected in series. From the application point view this approach can be used for predicting the transmitted mechanical force based only on the known kinematic excitation. Using the case study of an elastic gear transmission we show and identify a hysteresis response which multivariate behavior depends on an internal state of the bedstop motion.

Accounting for the Influence of the Tank Walls in the Zero-Sequence Topological Model of a Three-Phase, Three-Limb Transformer

A method is proposed to account for the influence of the tank walls in the topological transient model of a three-phase, three-limb core-type transformer. The influence of the transformer tank walls as a distributed-parameter hysteretic element is reproduced by solving a diffusion equation that describes the penetration of the plane electromagnetic wave into the depth of the wall. The reliability of the model is validated by comparing its zero-sequence impedances to those measured on a 25-MVA transformer, in the presence and absence of a tertiary stabilizing winding. An Electromagnetic Transients Program-Alternate Transients Program implementation of the model is outlined.