Equivalent Linearization Research Articles

Non-linear dynamic response of a cable system with a tuned mass damper to stochastic base excitation via equivalent linearization technique

Non-linear dynamic model of a cable–mass system with a transverse tuned mass damper is considered. The system is moving in a vertical host structure therefore the cable length varies slowly over time. Under the time-dependent external loads the sway of host structure with low frequencies and high amplitudes can be observed. That yields the base excitation which in turn results in the excitation of a cable system. The original model is governed by a system of non-linear partial differential equations with corresponding boundary conditions defined in a slowly time-variant space domain. To discretise the continuous model the Galerkin method is used. The assumption of the analysis is that the lateral displacements of the cable are coupled with its longitudinal elastic stretching. This brings the quadratic couplings between the longitudinal and transverse modes and cubic nonlinear terms due to the couplings between the transverse modes. To mitigate the dynamic response of the cable in the resonance region the tuned mass damper is applied. The stochastic base excitation, assumed as a narrow-band process mean-square equivalent to the harmonic process, is idealized with the aid of two linear filters: one second-order and one first-order. To determine the stochastic response the equivalent linearization technique is used. Mean values and variances of particular random state variable have been calculated numerically under various operational conditions. The stochastic results have been compared with the deterministic response to a harmonic process base excitation.

Characterization of challenges in asymmetric nonlinear vibration energy harvesters subjected to realistic excitation

With an explosion of the internet of things (IoT), vibration energy harvesting provides an environmentally friendly solution to replace consumable batteries in powering IoT wireless sensors. Yet, when implemented in practice, ambient vibration input energy is much less periodic than assumptions adopted in previous studies. This becomes especially important given that asymmetries are inevitable in nonlinear device platforms. This research sheds light on these complex challenges of practical vibration energy harvesting by developing and exploring an analytical model based on equivalent linearization. The modeling approach provides an opportunity to understand influences of asymmetry, nonlinearity, and combined excitation response on the DC power delivery of energy harvesters. In the analytical model, a weighted Gaussian joint distribution is utilized to approximate the influences caused by the random excitation. Combined with numerical and experimental validation, the analysis indicates that with the increase of stochastic base acceleration, two outcomes are possible. A first outcome involves an enhancement of DC power by way of triggering large amplitude nonlinear oscillations. A second outcome corresponds to a loss of high power delivery since the noise interferes with the attainment of the snap-through dynamic. Either reducing asymmetry or increasing harmonic excitation component is found to be favorable to induce the power-enhancing dynamics and inhibit the occurrence of the second case. Although with the simplified Gaussian distribution, the analytical framework cannot reproduce exact details of the dynamic responses in every case, the results show that the statistical trends of the analysis are overall borne out in simulation and experiment. This indicates the new modeling of this research may help guide attention to design and deployment techniques for nonlinear vibration energy harvesters in practical combined excitation environments, where limitations on precise manufacture or placement may introduce structural asymmetry.

Stochastic response control of particle dampers under random seismic excitation

In this paper, stochastic response control of particle damper system that is subjected to random seismic excitation is studied and the corresponding damping mechanism is further discussed. Based on the principle of multi-particle damper equivalent to a single particle damper, the nonlinear particle damper is equivalent to a linear system by the equivalent linearization method, and the mathematical expression of stochastic response of the particle damper system is obtained. The non-stationary response and stationary response of the particle damper system are subsequently compared, and the parametric study is conducted based on the stationary response of the particle damper system. It is shown that for a given particle damper system under random seismic excitation, there is an optimal gap clearance to minimize the structural response and this is the key parameter; the mass ratio is closely related to the gap clearance and in practice, the appropriate mass ratio should be properly selected; particle damper can also suppress the vibration response of the primary structure well over a wide frequency range, showing its strong robustness. Certain design suggestions are herein proposed. Finally, a numerical analysis and an experimental study are carried out to verify the viability and reliability of the stochastic vibration analysis of the particle damper system.

Non-linear stochastic dynamics of a cable-mass system with finite bending stiffness via an equivalent linearization technique

The non-linear stochastic dynamic behaviour of a high-rise vertical transportation system modelled as a concentrated mass and a cable with finite bending stiffness is considered. The slow time scale is defined and lateral cable displacements coupled with transverse motions are expanded in terms of approximating functions. The excitation of the high-rise building is assumed in the form of a narrow-band mean-square process equivalent to the harmonic process. The equivalent linearization technique is used to replace the original non-linear system with a linear approximation whose coefficients are determined from minimization of the mean-square equation difference between both systems.

Sensitivity of 1-D ground motion predictions to analysis codes and material models using KiK-net vertical arrays

One-dimensional (1-D) ground response analysis (GRA) predictions are highly sensitive to the analysis methods and dynamic soil models employed, contributing to significant scatter in analysis outputs. In this paper, a parametric study of 1-D GRA is conducted to evaluate the performance of various models using 60 ground motion records of varying intensity from five Kiban-Kyoshin (KiK-net) downhole array stations in Japan. Four GRA codes are considered, including the equivalent-linear (EL) method, a modified EL method using frequency-dependent soil parameters (FDEL), and two nonlinear (NL) models computed in OpenSees and DEEPSOIL. The soil constitutive models are constrained using a set of five material curves representing a wide range of nonlinear soil behaviour. It was found that all models are sensitive to the material curves at high frequencies, and the NL models are highly sensitive to the parametrization of Rayleigh damping. As the levels of soil nonlinearity increase, all GRA methods tend to over-damp the higher frequency content of ground motions, with substantial discrepancies across the model predictions. Interestingly, the FDEL method displays a lower sensitivity to material curves and reduces the biases at high frequencies. The results of this study suggest that GRA codes and material curves contribute evenly to the overall dispersion when simulating strong ground motions. To quantify the variability of GRA models, along with the epistemic uncertainties, a linear regression relationship in bi-logarithmic space is established across all records between the total standard deviation and the shear strains developed within the soil profiles.

Proposal of an equivalent linearization method to predict seismic hysteretic energy demand considering stiffness degradation effects

SummaryTo overcome complexities and shortcomings of previous studies, a new method is proposed to derive an equivalent linear model for predicting seismic hysteretic energy demand of bilinear single degree of freedom (SDOF) models. A new displacement spectrum is defined, which represents hysteretic energy. It is found that by increasing initial period and damping of a nonlinear system in the correct proportion and defining a linear model with these characteristics, the new developed displacement can be achieved. Error minimization is applied through an algorithm to find the optimum equivalent period corresponding to an equivalent damping utilizing two sets of far‐field and near‐field earthquakes. To analyze the effects of stiffness degradation, the proposed algorithm has been implemented on modified Clough hysteretic model as well. Comparing the results, effects of stiffness degradation on the ratio of equivalent to initial period is evident in the short period range, while with increasing initial period, the effect can almost be neglected at higher values of ductility. Nonlinear regression analysis is carried out to provide the equations for predicting equivalent linear parameters as a function of ductility. Despite the previous predictive equations, the proposed model is independent of earthquake characteristics and response‐related parameters, which has increased efficiency as well as simplicity.

Effect of earthquake statistically correlated vertical component on inelastic demand to regular reinforced-concrete frames

Post-earthquake survey and structural analysis prove that the vertical seismic action can damage buildings and bridges, even those with regular configuration. Moreover, there is evidence that one principal axis of ground motion may be not vertical. The inelastic demand to regular building structures is assessed here considering both uncorrelated and cross-correlated random seismic components, concurrent to deterministic weight. The method is conventional statistical equivalent linearization. Planar reinforced-concrete frames are analyzed parametrically varying the number of stories, the design ductility class, or the foundation ground type, as well as variance and cross-covariance of the vertical seismic component. Following a lumped plasticity approach, the bending moment-rotation relationship is a Bouc-Wen equation extended for asymmetry and interaction with axial force. Based on mean values, variances, and percentiles of response, the effect of uncorrelated vertical excitation is appreciable only on stress resultants of little practical consequence. Much more important is the effect of cross-correlated vertical excitation. As extreme results, the interstory drift demand increases up to four times. The rotation ductility demand to columns, less than 2 under uncorrelated excitations, rises to 5, meaning very likely yielding of columns designed as strong. The probability of axial force in the columns being tensile grows from negligible value to 8%. The beams may yield also around mid-span. The frame plastic mechanism worsens towards soft story, the peak deformation ductility being demanded more than the cyclic ductility. Not only variance of the vertical ground motion, but also cross-covariance with the horizontal one should be considered in the near field.

Dynamic analysis of track nonlinear energy sinks subjected to simple and stochastice excitations

SummaryTrack nonlinear energy sinks (track NESs) have been shown to be an effective and applicable strategy to mitigate structural response in recent years. However, previous studies on track NESs has mainly focused on demonstrating the benefits of track NESs through numerical simulations and experiments, with relatively little attention paid to the analytical understanding of the unique dynamics of track NESs. This study analyzes the responses of a track NES when subjected to impulsive and harmonic excitations by the harmonic balance method. Special attention is given to the cause and effect of the peaking behavior that is a prominent characteristic of the track NES's restoring force–displacement relationship. Analytical results reveal that the special energy–frequency characteristics of track NESs can be, at times, utilized to enhance the energy robustness that is absent in the conventional cubic NESs. Based on the analytical response expression, an equivalent linearization method (ELM) for the track NESs is developed for stochastic analysis. This ELM is numerically validated on the systems with strong nonlinearities. Stochastic optimization built on the ELM is performed to obtain design parameters of the track NES that can lead to minimum response variances of the primary structure. In particular, the proposed optimization procedure can be applied to seismic optimum design in which the seismic excitations are modeled as filtered white‐noise ground motions. The analytical techniques provided in this study lay the groundwork for the practical implementation of track NESs as a robust and effective control strategy for engineering structures.

Optimal design for a novel inerter-based clutching tuned mass damper system

In this article, a clutching inerter damper is introduced into the conventional tuned mass damper to replace the typical damping element. Regarding the limitation of the typical damping element, the reformed clutching tuned mass damper system is more flexible in parameter design than the optimal tuned mass damper, which may be constrained by the manufacturing process to realize the too small or too large damping coefficient. To investigate the effectiveness of the clutching tuned mass damper, some fundamental analyses are first conducted on the clutching tuned mass damper, and results show that the clutching tuned mass damper system can achieve a similar control effect to the optimal tuned mass damper design. Considering the inherent nonlinearity of the clutching tuned mass damper, the equivalent linearization is performed based on the equivalent linearization parameters drawn from the single-degree-of-freedom system with clutching inerter damper. The equivalent linear system of the clutching tuned mass damper system has been proved to be quite accurate to approximate the nonlinear clutching tuned mass damper system. Based on the equivalent linear system, the performance evaluation and optimal design of the clutching tuned mass damper system are carried out by numerical analysis and analytical solution. Results have shown that there is an optimum inertance for the clutching tuned mass damper to achieve the optimal performance, and the optimum inertance is related to the structural damping ratio and the tuned mass ratio. Finally, the effectiveness of the clutching tuned mass damper system and its equivalent linear system in a multi-degree-of-freedom structure is verified by a numerical study.

Equivalent Linear and Nonlinear Site-Specific Ground Response Analysis of Pashto Cultural Museum Peshawar, Pakistan

Site-specific ground response analysis is needed for seismic hazard estimation, and its mitigation is required for seismically active regions of Pakistan. The work presented in the following includes response analysis of a specific site at Peshawar Khyber Pakhtunkhwa using equivalent linear (EL) and also nonlinear (NL) analyses. The site model was developed using the geotechnical investigation data from field investigation and subsequent laboratory tests on representative soil samples. The earthquake records compatible to the seismic activities in the target area are selected and matched to the target response spectra in order to obtain input excitation force for the base ground model. Several issues that are related to site-specific ground response analysis, i.e., shear strain, mobilized shear strength and ground acceleration along depth of soil model, are studied and compared for both EL and NL analyses. The analysis results are also further studied and compared for ground response spectra and amplification factor using both EL and NL analyses.

Robust model‐free adaptive iterative learning formation for unknown heterogeneous non‐linear multi‐agent systems

The formation problem for a class of non-linear multi-agent systems (MASs) is considered under bounded disturbance and repetitive environment. The agent dynamics here is unknown and heterogeneous, and the disturbances are both time-varying and iteration-varying. Based on a novel disturbance-related dynamic linearisation technique, the agent dynamics is first transformed into an equivalent dynamic linearisation data model along the iteration axis. Then, distributed robust model-free adaptive iterative learning protocols are designed to drive the agents to the desired formation pattern. By theoretic analysis, the formation tracking error of the MASs is ultimately bounded and the result is further extended to an iteration-varying topology and pattern case. As a data-driven method, the protocol design has the feature that only the input and output data of neighbouring agents are used. Finally, numerical examples are presented to show the effectiveness and robustness of this method.

The Characteristics of Seismic Response on Hard Interlayer Sites

Based on the engineering geological data of a nuclear power plant site, nine engineering geological profiles were created with hard interlayers of different thicknesses. The equivalent linearization method of seismic motion segment‐input used for one‐dimensional nonlinear seismic response analysis was applied to study the effect of the interlayer thickness on the peak acceleration and the acceleration response spectra of the site seismic response. The results showed that there was an obvious influence of hard interlayer thickness on site seismic responses. With the increase of hard interlayer thickness, the site nonlinear effect on seismic responses decreased. Under the same thickness of the hard interlayer, the nonlinear effect of the site was strengthened with the higher input peak acceleration. In addition, the short‐period acceleration response spectrum was found to be significantly influenced by the hard interlayer and showed that the longer the period, the less influence of the hard interlayer on the acceleration response spectrum coordinates. Moreover, the influenced frequency band was wider with the increase in the thickness of hard interlayer.

Seismic Response Validation of Simulated Soil Models to Vertical Array Record During A Strong Earthquake

Several soil models, such as linear elastic, equivalent linear, and non-linear models, are employed in seismic ground response analysis. The aim of this study was to validate the seismic responses at ground surface of several soil models with the vertical array record of the Kobe earthquake. One-dimensional seismic response analyses were performed at Port Island using several soil models. The responses at ground surface from the simulated soil models were validated with the vertical array record of the Kobe earthquake. The results showed that the extended hyperbolic model yielded the most appropriate response according to the Kobe earthquake's recorded motion. This means that this model can be considered a suitable soil model to predict the response of strong earthquakes. In general, the results support the recommendation to select the most appropriate soil model for seismic ground response analysis.

An iterative linear procedure using frequency-dependent soil parameters for site response analyses

Local site amplification predictions that derive from the traditional equivalent linear (EL) procedure are deemed to be reliable when weak shaking is considered, while it is well recognized that significant discrepancies can occur for stronger earthquakes where soil nonlinearity dominates the site response. To better capture nonlinearity effects, frequency-dependent equivalent linear methods, commonly referred to as FDEL methods, were proposed by modifying the EL procedure with frequency-dependent soil parameters. However, these methods have never been widely used in earthquake engineering practice. Existing FDEL methods rely on generic spectral shape fitting parameters to approximate the variation of shear strain amplitudes in the frequency domain, which can lead to substantial inconsistencies in the model prediction when applied to a particular seismic event, with a specific signature. This study investigates a range of spectral seismic intensities to identify a proxy that could be used directly to update the shear strain spectra in accordance with the simulated ground motions. A new FDEL algorithm to compute effective shear strain as a function of frequency is proposed and tested on various scenarios. When simulating strong ground motions that exhibit higher levels of soil deformation, the frequency content of ground motions processed within the FDEL procedure is substantially improved as compared to the EL method, with results more consistent with those from NL analyses.

An Approximate Solution for a Nonlinear Duffing – Harmonic Oscillator

The equivalent linearization method introduced by Caughey is a powerful tool for analyzing random oscillations. The method is also easy to apply for deterministic oscillations. However, with strong nonlinear systems, the error of this method is usually quite large and even not acceptable. In conjunction with a weighted averaging, the equivalent linearization method has shown more accuracy than the classical one in which the conventional averaging value is used. Combining advantages of the classical equivalent linearization method and accuracy of the weighted averaging, the proposed method has shown that it is a useful tool for analyzing nonlinear oscillations including strong nonlinear systems. In this paper, the proposed method is applied to analyze a nonlinear Duffing – harmonic oscillator. The present results are compared with the results obtained by using other analytical methods, exact results and numerical results.

On Model-Free Adaptive Control and Its Stability Analysis

In this paper, the main issues of model-based control methods are first reviewed, followed by the motivations and the state of the art of the model-free adaptive control (MFAC). MFAC is a novel data-driven control method for a class of unknown nonaffine nonlinear discrete-time systems since neither explicit physical model nor Lyapunov stability theory or key technical lemma is used in the controller design and theoretical analysis except only for the input/output (I/O) measurement data. The basis of MFAC is the dynamic linearization data modeling method at each operating point of the closed-loop system. The established dynamic linearization data model is a virtual equivalent data relationship in the I/O sense to the original nonlinear plant by means of a novel concept called pseudo-partial derivative (PPD) or pseudo-gradient (PG) vector. Based on this virtual equivalent dynamic linearization data model and the time-varying PPD or PG estimation algorithm designed merely using the I/O measurements of a controlled plant, the MFAC system is constructed. The main contribution of this paper is that the theoretical analysis of the bounded-input bounded-output stability, the monotonic convergence of the tracking error dynamics, and the internal stability of the full form dynamic linearization based MFAC scheme are rigorously presented by the contraction mapping principle; the well known PID control and the traditional adaptive control for linear time-invariant systems are explicitly shown as the special cases of this MFAC. The simulation results verify the effectiveness of the proposed approach.

Seismic Vulnerability Assessment of a Reinforced Concrete Building Located in India

Over the recent years the natural disaster especially due to the earthquake effect on buildings increases which causes loss of life and property in many places all over the world. The latest development leads to finding the direct losses and damage states of the buildings for various intensities of earthquake ground motions. In the present study, seismic vulnerability assessment was done for a medium rise building (G+5). The design peak ground acceleration of 0.16g and 0.36g were considered for the risk assessment. The nonlinear static pushover analysis was done to fine the performance point, spectral acceleration and corresponding spectral acceleration by Equivalent Linearization (EL) method given by Federal Emergency Management Agency (FEMA-440). The four damage states such as slight, moderate, extreme and collapse has been considered as per HAZUS-MR4. The seismic vulnerability in terms of fragility curves was developed to evaluate the damage probabilities based on HAZUS methodology. The discrete and cumulative damage probability was found for all the damage states of the building which shows the building at 0.16g experience slight damage whereas at 0.36g the moderate damage state equally becomes predominant.

A novel shape memory alloy damping inerter for vibration mitigation

Various shape memory alloy (SMA) dampers have been developed to reduce structural vibration responses. However, the application of SMA dampers has been restricted by the high material costs of SMAs. Therefore, this study developed and tested an innovative SMA damping inerter (SDI) in which an SMA element and an inerter element were arranged in parallel and deployed in series with a supporting spring element. A single-degree-of-freedom structure with an SDI was employed to analyze the effect of vibration mitigation. A theoretical analysis was conducted via an equivalent linearization method, and parameter studies were then used to evaluate the performance of the SDI-fitted structure from perspectives of structural displacement, acceleration, and energy dissipation as well as the most efficient frequency tuning bandwidth. The performances of the SDI and a conventional SMA damper were compared by using a time-domain analysis with recorded and simulated ground motions, revealing clearly superior results for the SDI with respect to structural response mitigation, and seismic energy dissipation. The system proposed here took advantage of the full potential of combining SMA and inerter elements, making the SDI a robust system for mitigating structural vibration with more economical SMA material costs.

Reliability-based constant-damage ductility demand spectra of mid-rise RC building structures using modified equivalent linearization method

In the displacement-based seismic design and evaluation, the equivalent linearization method is one of the approximation methods to estimate the maximum displacement response of a single-degree-of-freedom (SDOF) system. Based on the past research conducted by Okano & Miyamoto and the seismic design code in Taiwan, this work proposes modified equivalent linearization method (MELM) of a SDOF system for a mid-rise reinforced concrete (RC) building structure. By considering a modification factor suggested by the equivalent linear method proposed by Okano and Miyamoto and performing the nonlinear regression analysis, the accuracy of the proposed method can be improved. A total of 348 ground motions recorded on the various earthquakes are used to calculate error statistics of MELM. Additionally, a closed-form formula for the constant-damage ductility demand spectrum (CDDDS) that corresponds to a specified reliability is suggested in this work.

Nonlinear vibration of microbeams based on the nonlinear elastic foundation using the equivalent linearization method with a weighted averaging

We investigate the nonlinear vibration of microbeams based on the nonlinear elastic foundation through the modified couple stress theory. The equivalent linearization method with a weighted averaging is used to solve approximately the ordinary differential equation that describes the equation of motion of the microbeam. The effects of length scale parameter, the flexural rigidity ratio, the slenderness ratio, the Winkler parameter, the Pasternak parameter and the nonlinear foundation parameter on the nonlinear vibration of the microbeam are studied and discussed.