Nonlinear Force-displacement Relationship Research Articles

Theoretical nonlinear force-displacement constitutive model of triangular-plate steel damper

Utilizing the inelastic deformation of a steel damper to dissipate seismic energy and control structural damage is a prevalent design strategy in a passive control field. Such a design strategy requires an explicit force-displacement constitutive model of the steel damper in both the elastic and plastic deformation phases. Currently, the nonlinear force-displacement relationship of the steel damper is mainly determined by quasi-static tests or numerical simulations which are generally time-consuming, and there is a lack of a theoretical and straightforward approach to determine the nonlinear force-displacement constitutive model of the steel damper. This study thus theoretically derives the nonlinear force-displacement constitutive model for a commonly used triangular-plate steel damper. The theoretical formula is derived based on the equal curvature assumption commonly applied to the triangular plate and is further modified based on the curvature shape function proposed in this study for the first time. After that, the force-displacement curves obtained from the theoretical formula are compared with those obtained from a large number of numerical simulations and several sets of quasi-static tests. The result shows that the theoretical formula derived based on the curvature shape function can better predict the nonlinear force-displacement curve of the triangular-plate steel damper, which significantly outperforms the theoretical formula derived based on the equal curvature assumption especially as the triangular-plate steel damper has a large plastic deformation. Furthermore, the influence factors of the strain-hardening phenomenon existing in the force-displacement curve of the triangular-plate steel damper are analyzed based on the modified theoretical formula. The result implies that the thickness-to-height ratio of the plate is the most significant factor for the strain-hardening behavior of the triangular-plate steel damper.

Nonlinear Reduced Order Modeling of Structures Near Buckling and Application to an Energy Harvester

Abstract The focus of this investigation is first on assessing the validity to structures under in-plane forces, in particular near buckling, of the reduced order modeling approach for nonlinear geometric response that has been extensively developed in the last two decades. This focus is motivated by a class of piezoelectric energy harvesters that rely on strongly nonlinear behavior, such as large amplitude responses, to achieve broadband energy harvesting. A simple, two-rigid bars linkage that approximates a buckling beam is first considered to discover the features of the nonlinear force–displacement relationship induced by an in-plane loading. It is observed that the corresponding form of this relationship is not consistent with the one derived from a reduced order model (ROM) but can be closely approximated by it over a large displacement range. This analysis emphasizes in particular the role of a group of ROM coefficients that are usually considered unimportant. A similar study is performed next for the buckled harvester modeled within nastran and it is again found that a close match of the force–displacement relationship can be achieved. Based on that positive outlook, a six basis functions ROM of this beam harvester that includes piezoelectric effects is built and identified. It is found to provide a close match of nastran nonlinear predictions over a broad range of transverse and in-plane loadings in static and dynamic conditions. The ROM usefulness in predicting the open-circuit voltage is demonstrated.

Analytical Modeling of Crack Widths and Cracking Loads in Structural RC Members

Crack width is a major performance criterion in reinforced-concrete structures, in general, and is of utmost importance in ensuring bridge performance, in particular. A reliable theory-based method is required to assess crack widths and gain insight into their dependence on material, geometry, and loading parameters. A new, exact analytical method is proposed for a one-dimensional reinforced concrete element based on equilibrium, constitutive, and kinematic relationships, accounting for the geometrical and material behavior of the concrete and reinforcement. A linear interfacial bond stress slip is assumed to represents the small slips associated with the limited allowed crack width. Closed-form expressions have been developed and a wealth of information can be calculated immediately, such as the cracking load levels, the crack width dependence on the load level, the expected number of cracks, and the cracks spacing. The entire nonlinear force-displacement relationship of a cracked reinforced-concrete element may be depicted, demonstrating the tension-stiffening behavior that depends on the variations in the crack width throughout the loading history. Comparisons of the model with experimental data demonstrate very good agreement.

Ultra-low wide bandwidth vibrational energy harvesting using a statically balanced compliant mechanism

Structure design of ultra-low wide bandwidth vibrational energy harvesters remains an open issue. In this research, a statically balanced compliant mechanism (SBCM) is proposed, dynamically modelled, and experimentally demonstrated to address this need. This SBCM is designed based on the concept of stiffness compensation between a linear component with positive-stiffness (two double parallelograms in parallel) and a nonlinear component with negative-stiffness (two sets of post-buckled fixed-guided compliant beams in parallel). A design guideline of the SBCM starting from using rapid-design stiffness compensation equation is provided for reasonably approximate and accurate results. Subsequently, a dynamic analytical model of the displacement response of the SBCM to harmonic base excitation has been derived based on the averaging method. The nonlinear force-displacement relationship of the system was obtained through nonlinear finite element analysis (FEA) simulations in this regard and a 5th order polynomial fit was chosen taking only odd terms into account. The accuracy of this analytical model is confirmed by numerical analysis. Next, a prototype of the SBCM was fabricated and its applicability to piezoelectric vibrational energy harvesters (PVEHs) was demonstrated by integrating piezoelectric capacitors made of PVDF films with compliant beams of the SBCM to generate electric outputs in response to the bending of the beams. Static balancing is achieved over a continuous displacement range of 2 mm around the origin. Bi-stability and mono-stability were observed with the same prototype by adjusting the length of the positive-stiffness beams. Under static-balancing condition, the SBCM is able to respond to a wide range of ultra-low frequencies with low accelerations, which has been established experimentally, analytically and numerically. In dynamic experiments, the steady-state relative displacement amplitude H is 2.9 mm under an excitation condition of 6 Hz and 0.1g (1g=9.8 m/s2), and it reaches a maximum of 9.4 mm under 12.75 Hz and 0.25 g. A 30%-Hmax frequency bandwidth is introduced to assess the nonlinear dynamic performance of the SBCM from a mechanical perspective. The maximum experimental 30%-Hmax bandwidth of the structure is 6.75 Hz (from 6 Hz to 12.75 Hz) at 0.25 g and the corresponding theoretical value is 8.73 Hz (from 3.34 Hz to 12.07 Hz). Under the same excitation condition, it is shown that the maximum relative displacement amplitude deceases as the jump-down frequency increases. In addition, super-harmonic and sub-harmonic oscillations are observed in the dynamic experiments. The SBCM can provide a practical structural solution for harvesting energy from a range of vibrational scenarios (e.g. ocean waves, human motions and bridge vibrations) with ultra-low wide bandwidth frequencies and weak accelerations. The concept of SBCM is suitable to be combined with various energy conversion principles and is not limited to piezoelectric sources alone. When miniaturized, the SBCM concept can be of particular interest to researchers active in energy harvesting for microelectromechanical systems (MEMS) technology.

A procedure to model and design elastomeric‐based isolation systems for the seismic protection of rocking art objects

AbstractThis paper investigates the rocking behavior of rigid bodies seismically protected by means of lead rubber bearings and high damping rubber bearings, which are the most popular kinds of elastomeric isolators in the market. The complex nonlinear force–displacement relationship displayed by such devices is predicted by a phenomenological model based on a small set of parameters having a clear mechanical meaning. The algebraic nature of the proposed hysteretic model makes it suitable for a design procedure, using an energy‐based approach, that allows one to obtain the hysteretic model parameters on the basis of the mass and the isolation period. Overturning spectra are evaluated and discussed to illustrate the effect of the isolation devices on the rigid bodies' rocking behavior. Furthermore, nonlinear time history analyses associated with six real earthquakes are carried out on six of Michelangelo's sculptures, located in the Galleria dei Prigioni at the Accademia Gallery of Florence, in order to examine their actual behavior under real strong ground motions.

An inertial macroelement for bridge abutments

In recent years, the designers of long girder bridges in seismic areas have frequently opted for a continuous deck. One implication of this choice is that in many instances bridge abutments are called upon to carry large seismic forces, engaging the dynamic response of the soil–abutment system. To deal with this problem, this paper describes the formulation of a novel one-dimensional, inertial macroelement for simulating the dynamic behaviour of bridge abutments. The non-linear force–displacement relationship is characterised by a multi-surface plasticity model using a rigorous thermodynamic approach. The plastic response of the model is bounded by the ultimate capacity of the soil–abutment system that includes dissymmetry of the soil response in active and passive loading directions, while inertial effects transferred by the near-field approach embankment are simulated through appropriate participating masses in the macroelement formulation. The paper describes a straightforward calibration procedure of the proposed macroelement for horizontal, longitudinal loading of the abutment. The macroelement has been incorporated into a simplified, global, finite-element model of a multi-span girder bridge and validated through comparisons with results from a full three-dimensional (3D) dynamic time domain analysis under seismic loading. The inertial macroelement predictions of abutment deformations, axial deck loads and pier reaction forces are in very good agreement with the 3D soil–structure interaction model, and are achieved at much lower computational costs. The proposed inertial macroelement represents a significant improvement over existing simplified models based on linear response of the soil–abutment system.

Geometric Effect on the Nonlinear Force-Displacement Relationship of Awl-Shaped Serpentine Microsprings for In-Plane Deformation.

Even when made by brittle materials, awl-shaped serpentine microsprings (ASSMs) were found to have a nonlinear displacement–force relationship similar to springs made by ductile material. It is found that the nonlinear displacement–force relationship is due to the geometry and dimensions of the ASSMs. The geometric effect of the nonlinear force–displacement relationship of ASSMs for in-plane motion was investigated. A theoretical solution was derived to analyze this nonlinearity. By successfully fabricating and measuring an ASSM, the theoretical results agreed well with the experimental results. The results indicated that ASSMs have a nonlinear force–displacement relationship, which is similar to that of hardening springs. The taper angle has a significant effect on the nonlinear displacement of ASSMs. When the taper angle was small, no obvious effect appeared on the nonlinearity of the microsprings with different numbers of turns. When the beam length increased, the critical force for nonlinear deflection decreased.

An experimental investigation on the influence of an assembly preload on the hysteresis, the drag torque, the lift-off speed and the thermal behavior of three-pad air foil journal bearings

Abstract Rotors supported by air foil journal bearings often show distinctive subharmonic oscillations arising from the nonlinear force-displacement relationship of the elastic foil structure. Also, air film induced instabilities may be observed in such bearings. A straightforward possibility to decrease the amplitude of subharmonic vibrations and to increase the onset speed of rotor instability is to introduce a mechanical assembly preload to the bearing. The resultant rise in stiffness and damping and the improved rotor dynamic behavior goes along with a larger drag torque. This leads to an increase in the bearing power loss and temperature with the risk of a thermal runaway, which may cause bearing failure. In this manuscript, the influence of an assembly preload on the bearing hysteresis, the startup and shutdown torque and the bearing temperature at different bearing loads is investigated experimentally. The experiments are accomplished on an automated high-speed test rig which can run at speeds up to 60,000 rpm . Specifying the external bearing load and the rotor speed, this test rig allows for a measurement of the bearing drag torque, the horizontal and vertical displacement of the bearing and of different temperatures of the rotor and the bearing. It is observed that the lift-off speed changes with the bearing load. The experimental results also show that the shutdown torque increases with the bearing load and the assembly preload, whereas the bearing coefficient of friction and the bearing temperature does not significantly change with the assembly preload. The measured coefficient of friction is compared with results from literature which were obtained from pin-on-disk tests at different speeds and loads. It is shown that the discrepancies between the measured coefficient of friction and corresponding values from pin-on-disk tests reported in literature can be traced back to the high dependency of the friction coefficient on the sliding speed.

Disentangling geometrical, viscoelastic and hyperelastic effects in force-displacement relationships of folded biological tissues

.Drosophila wing discs show a remarkable variability when subject to mechanical perturbation. We developed a stretching bench that allows accurate measurements of instantaneous and time-dependent material behaviour of the disc as a whole, while determining the exact three-dimensional structure of the disc during stretching. Our experiments reveal force relaxation dynamics on timescales that are significant for development, along with a surprisingly nonlinear force-displacement relationship. Concurrently our imaging indicates that the disc is a highly heterogeneous tissue with a complex geometry. Using image-based 3D finite element modelling we are able to identify the contributions of size, shape and materials parameters to the measured force-displacement relations. In particular, we find that simulating the stretching of a disc with stiffness patterns in the extra-cellular matrix (ECM) recapitulates the experimentally found stretched geometries. In our simulations, linear hyperelasticity explains the measured nonlinearity to a surprising extent. To fully match the experimental force-displacement curves, we use an exponentially elastic material, which, when coupled to material relaxation also explains time-dependent experiments. Our simulations predict that as the disc develops, two counteracting effects, namely the discs foldedness and the hardening of the ECM lead to force-relative displacement curves that are nearly conserved during development.Graphical abstract

Neural network modelling of soft tissue deformation for surgical simulation.

This paper presents a new neural network methodology for modelling of soft tissue deformation for surgical simulation. The proposed methodology formulates soft tissue deformation and its dynamics as the neural propagation and dynamics of cellular neural networks for real-time, realistic, and stable simulation of soft tissue deformation. It develops two cellular neural network models; based on the bioelectric propagation of biological tissues and principles of continuum mechanics, one cellular neural network model is developed for propagation and distribution of mechanical load in soft tissues; based on non-rigid mechanics of motion in continuum mechanics, the other cellular neural network model is developed for governing model dynamics of soft tissue deformation. The proposed methodology not only has computational advantage due to the collective and simultaneous activities of neural cells to satisfy the real-time computational requirement of surgical simulation, but also it achieves physical realism of soft tissue deformation according to the bioelectric propagation manner of mechanical load via dynamic neural activities. Furthermore, the proposed methodology also provides stable model dynamics for soft tissue deformation via the nonlinear property of the cellular neural network. Interactive soft tissue deformation with haptic feedback is achieved via a haptic device. Simulations and experimental results show the proposed methodology exhibits the nonlinear force-displacement relationship and associated nonlinear deformation of soft tissues. Furthermore, not only isotropic and homogeneous but also anisotropic and heterogeneous materials can be modelled via a simple modification of electrical conductivity values of mass points.

Soft tissue deformation modelling through neural dynamics-based reaction-diffusion mechanics.

Soft tissue deformation modelling forms the basis of development of surgical simulation, surgical planning and robotic-assisted minimally invasive surgery. This paper presents a new methodology for modelling of soft tissue deformation based on reaction-diffusion mechanics via neural dynamics. The potential energy stored in soft tissues due to a mechanical load to deform tissues away from their rest state is treated as the equivalent transmembrane potential energy, and it is distributed in the tissue masses in the manner of reaction-diffusion propagation of nonlinear electrical waves. The reaction-diffusion propagation of mechanical potential energy and nonrigid mechanics of motion are combined to model soft tissue deformation and its dynamics, both of which are further formulated as the dynamics of cellular neural networks to achieve real-time computational performance. The proposed methodology is implemented with a haptic device for interactive soft tissue deformation with force feedback. Experimental results demonstrate that the proposed methodology exhibits nonlinear force-displacement relationship for nonlinear soft tissue deformation. Homogeneous, anisotropic and heterogeneous soft tissue material properties can be modelled through the inherent physical properties of mass points. Graphical abstract Soft tissue deformation modelling with haptic feedback via neural dynamics-based reaction-diffusion mechanics.

Experimental Nonlinear Model Identification of a Highly Nonlinear Resonator

In this work, two model identification methods are used to estimate the nonlinear large deformation behavior of a nonlinear resonator in the time and frequency domains. A doubly clamped beam with a slender geometry carrying a central intraspan mass when subject to a transverse excitation is used as the highly nonlinear resonator. A nonlinear Duffing equation has been used to represent the system for which the main source of nonlinearity arises from large midplane stretching. The first model identification technique uses the free vibration of the system and the Hilbert transform (HT) to identify a nonlinear force–displacement relationship in the large deformation region. The second method uses the frequency response of the system at various base accelerations to relate the maximum resonance frequency to the nonlinear parameter arising from the centerline extensibility. Experiments were conducted using the doubly clamped slender beam and an electrodynamic shaker to identify the model parameters of the system using both of the identification techniques. It was found that both methods produced near identical model parameters; an excellent agreement between theory and experiments was obtained using either of the identification techniques. This follows that two different model identification techniques in the time and frequency domains can be employed to accurately predict the nonlinear response of a highly nonlinear resonator.

Abstract 20905: The Force-Displacement Relationship in Commonly Used Resuscitation Manikins: Not Very Human

Introduction: Manikins are widely used for CPR training and designed to simulate a human in cardiac arrest. Previous studies show a non-linear force-displacement relationship in the human chest. This may not be the case for resuscitation manikins. The aim of this study was to investigate the force-displacement relationship in commonly used resuscitation manikins. Methods: Commonly used infant and adult manikins for resuscitation training were included in the study. Manikins were tested by placing them in a material testing machine (ProLine Z050, Zwick/Roell, Ulm, Germany). A piston was placed on lower half of the sternum of the manikin. The testing machine applied a constant growing force downwards on the manikin chest while recording the force-displacement relationship. The test was stopped when either the manikin was nearly fully compressed or the displacement exceeded 7 cm. On manikins with adjustable compliance of the thorax, the medium setting was used. The force needed to achieve the recommended compression depth i.e. 3 cm and 5 cm were compared for infant and adult manikins respectively. Results: Seven different resuscitation manikins were included in the study; two infant manikins: Resusci Infant (Laerdal) and CPR Anytime® Infant (inflatable; American Heart Association) and five adult manikins: Mini Anne (inflatable), Little Anne®, Resusci Anne, Resusci Anne Advanced(Laerdal) and Ambu® Man (Ambu). Infant manikins required a force of 57 N and 34 N to compress the chest 3 cm. The force required to compress the adult manikins to a depth of 5 cm ranged from 195 N to 543 N. The force-displacement curve is shown in the Figure. Conclusions: Four out of five adult resuscitation manikins and one infant manikin had a nearly linear force-displacement curve different from the human chest. The Mini Anne and Resusci Infant showed a non-linear curve more like the human chest. The force needed to compress the chest to the recommended chest compression depth varied between manikins.

Design and verification of a hybrid nonlinear MRE vibration absorber for controllable broadband performance

In this work, a hybrid nonlinear magnetorheological elastomer (MRE) vibration absorber has been designed, theoretically investigated and experimentally verified. The proposed nonlinear MRE absorber has the dual advantages of a nonlinear force–displacement relationship and variable stiffness technology; the purpose for coupling these two technologies is to achieve a large broadband vibration absorber with controllable capability. To achieve a nonlinear stiffness in the device, two pairs of magnets move at a rotary angle against each other, and the theoretical nonlinear force–displacement relationship has been theoretically calculated. For the experimental investigation, the effects of base excitation, variable currents applied to the device (i.e. variable stiffness of the MRE) and semi-active control have been conducted to determine the enhanced broadband performance of the designed device. It was observed the device was able to change resonance frequency with the applied current; moreover, the hybrid nonlinear MRE absorber displayed a softening-type nonlinear response with clear discontinuous bifurcations observed. Furthermore, the performance of the device under a semi-active control algorithm displayed the optimal performance in attenuating the vibration from a primary system to the absorber over a large frequency bandwidth from 4 to 12 Hz. By coupling nonlinear stiffness attributes with variable stiffness MRE technology, the performance of a vibration absorber is substantially improved.

Neural dynamics-based Poisson propagation for deformable modelling

This paper presents a new methodology from the standpoint of energy propagation for real-time and nonlinear modelling of deformable objects. It formulates the deformation process of a soft object as a process of energy propagation, in which the mechanical load applied to the object to cause deformation is viewed as the equivalent potential energy based on the law of conservation of energy and is further propagated among masses of the object based on the nonlinear Poisson propagation. Poisson propagation of mechanical load in conjunction with non-rigid mechanics of motion is developed to govern the dynamics of soft object deformation. Further, these two governing processes are modelled with cellular neural networks to achieve real-time computational performance. A prototype simulation system with a haptic device is implemented for real-time simulation of deformable objects with haptic feedback. Simulations, experiments as well as comparisons demonstrate that the proposed methodology exhibits nonlinear force–displacement relationship, capable of modelling large-range deformation. It can also accommodate homogeneous, anisotropic and heterogeneous materials by simply changing the constitutive coefficient value of mass points.

Seismic retrofit of structures using steel honeycomb dampers

The purpose of this paper is to investigate the seismic performance of a honeycomb shaped steel hysteretic damper applied to seismic retrofit or strengthening of a structure. The formulas for the initial stiffness and yield strength of a damper unit were derived based on the cell wall bending model, and the results were compared with those obtained from finite element analysis. Bilinear model of the honeycomb damper was developed based on the nonlinear force-displacement relationship obtained from finite element analysis. The honeycomb dampers were applied for seismic retrofit of a 15-story apartment building designed without considering seismic load and for seismic design of a 3-story moment frame designed with reduced seismic load. The analysis results showed that the honeycomb dampers were effective in the enhancement of seismic-load resisting capacity of the model structures.

Seismic Behavior Factors of RC Staggered Wall Buildings

In this study seismic performance of reinforced concrete staggered wall system structures were investigated and their behavior factors such as overstrength factors, ductility factors, and the response modification factors were evaluated from the overstrength and ductility factors. To this end, 5, 9, 15, and 25-story staggered wall system (SWS) structures were designed and were analyzed by nonlinear static and dynamic analyses to obtain their nonlinear force–displacement relationships. The response modification factors were computed based on the overstrength and the ductility capacities obtained from capacity envelopes. The analysis results showed that the 5- and 9-story SWS structures failed due to yielding of columns and walls located in the lower stories, whereas in the 15- and 25-story structures plastic hinges were more widely distributed throughout the stories. The computed response modification factors increased as the number of stories decreased, and the mean value turned out to be larger than the value specified in the design code.

Pipe–soil interaction and pipeline performance under strike–slip fault movements

The performance of pipelines subjected to permanent strike–slip fault movement is investigated by combining detailed numerical simulations and closed-form solutions. First a closed-form solution for the force–displacement relationship of a buried pipeline subjected to tension is presented for pipelines of finite and infinite lengths. Subsequently the solution is used in the form of nonlinear springs at the two ends of the pipeline in a refined finite element model, allowing an efficient nonlinear analysis of the pipe–soil system at large strike–slip fault movements. The analysis accounts for large strains, inelastic material behavior of the pipeline and the surrounding soil, as well as contact and friction conditions on the soil–pipe interface. The numerical models consider infinite and finite length of the pipeline corresponding to various angles β between the pipeline axis and the normal to the fault plane. Using the proposed closed-form nonlinear force–displacement relationship for buried pipelines of finite and infinite length, axial strains are in excellent agreement with results obtained from detailed finite element models that employ beam elements and distributed springs along the pipeline length. Appropriate performance criteria of the steel pipeline are adopted and monitored throughout the analysis. It is shown that the end conditions of the pipeline have a significant influence on pipeline performance. For a strike–slip fault normal to the pipeline axis, local buckling occurs at relatively small fault displacements. As the angle between the fault normal and the pipeline axis increases, local buckling can be avoided due to longitudinal stretching, but the pipeline may fail due to excessive axial tensile strains or cross sectional flattening. Finally a simplified analytical model introduced elsewhere, is enhanced to account for end effects and illustrates the formation of local buckling for relative small values of crossing angle.

Bipolar Piezoelectric Buckling Actuators

A nonlinear piezoelectric amplification mechanism utilizing structural buckling is presented, and its static and dynamic properties are measured and analyzed. Buckling is a pronounced nonlinear effect that occurs at a structurally singular point. A small piezoelectric displacement on the order of 10 μm results in a large buckling displacement on the order of millimeter. Furthermore, the usable stroke is doubled if both sides of the singular point can be reached resulting in bipolar motion. Despite the large gain, buckling is an erratic, singular phenomenon; the side on which deflection will occur is unpredictable. In this paper, multiple design concepts are presented for regulating the buckling direction as well as for extending its usable stroke to bipolar motion. Nonlinear force-displacement relationships are modeled and measured. Nonlinear dynamic analysis using phase planes reveals that the buckling actuator can generate bipolar motion above a specific amplitude. Below this amplitude, it generates only monopolar oscillation. The proposed design concepts are implemented on monolithic flexure mechanisms, and prototype buckling actuators are tested to verify the concepts. Experiments show promising results: 20 N of peak-to-peak output force, and 6.2 mm of bipolar displacement generated by piezoelectric actuators with free displacement of 42 μm.

A polynomial analytical model of rubber bearings based on series of tests

Rubber bearings are among the most frequently applied devices in seismic isolation. Although the behavior of rubber bearings under strong earthquakes is nonlinear, it is often interpreted through the bilinear constitutive law. The investigations presented in this paper represent a successful attempt to simulate nonlinear force–displacement relationships. The first part of the paper covers production and testing of rubber bearings and results obtained. The second part deals with development of a simple nonlinear mathematical model of a rubber bearing involving a polynomial function and eight additional parameters obtained from biaxial tests. The polynomial model can simulate the behavior of natural rubber bearings in case of small and large deformations. The model is capable of covering the strengthening of the rubber in conditions of large deformations, including the loading history effect. Based on comparison between the analytical and experimental results, it is concluded that the proposed polynomial model is capable enough to simulate the force–displacement relationship of rubber bearings.