Equivalent Linearization Research Articles

Evaluation of Modified Frequency Dependent Equivalent Linear Method for Deconvolution Site Response Analysis

ABSTRACT Frequency-dependent equivalent linear (FD-EQL) site response analysis methods, developed as potential substitutes for the EQL procedure to better simulate the nonlinear soil response, have been reported to overpredict the high-frequency wave propagation. Modified procedures proposed to overcome this limitation have demonstrated to improve the fits with the nonlinear analysis results. The method has not yet been applied to perform a deconvolution analysis, where a conventional EQL analysis often fails to converge or provide reliable estimate. The conventional EQL procedure and one of the modified FD-EQL method, which uses an empirical factor f to interpolate the strain spectrum between the EQL and FD-EQL outputs, are used to perform a series of deconvolution analyses using an idealized 1000 m profile and twelve Kik-net downhole arrays. The residuals of recorded and deconvoluted motions are shown to increase with strain for the EQL method even when using the recommended frequency cut-off. For the FD-EQL method, the range of f recommended for convolution analyses is shown to provide unrealistic responses in performing deconvolution analyses. A new range that produces unbiased residuals for all strain amplitudes is proposed. Comparisons highlight that the modified FD-EQL yields reliable predictions of the within motion for all profiles and motion intensities, automatically suppressing the amplification of high-frequency noise typically accompanied in a deconvolution analysis.

Modeling and analysis of a four-parameter vibration isolator with frequency-dependent damping and its implementation based on GERF

To improve the vibration isolation performance, a conventional passive two-parameter vibration isolator (VI) usually adopts the method of adding constant damping for energy dissipation. A compromise is made between the suppression of resonance peak and the rapid attenuation of transmissibility in the high-frequency band. A four-parameter-VI with frequency-dependent damping is a potential solution for this problem. The theoretical models of system transmissibility and equivalent damping are established by a normalization method, and the vibration isolation performance is compared with that of the two-parameter-VI and the three-parameter-VI. A novel four-parameter-VI with damping generated by a giant electrorheological fluid (GERF) damper is designed. The mechanical properties of the GERF damper are tested, and its damping effect is evaluated using an equivalent linearization method. Then, the vibration isolation performance of the four-parameter-VI is tested. The results show that: the vibration isolation performance of the four-parameter-VI can be affected by the damping ratio and the mass ratio. The four-parameter-VI exhibits the frequency-dependent damping characteristic of large damping in the low-frequency band and small damping in the high-frequency band by appropriately setting parameters. The vibration isolation ratio of the four-parameter-VI to white noise reaches 91.1% in the time domain with appropriate parameters, and the peak of the power spectral density curves decreases from 1.2 × 10−3 g2 Hz−1 to 0.16 × 10−3 g2 Hz−1 with a reduction of 1.04 × 10−3 g2 Hz−1 .

Optimization-oriented exponential-polynomial-closure approach for analyzing nonlinear stochastic oscillators

A novel method named optimization-oriented exponential-polynomial-closure approach is proposed in this article. The main idea of this attempt is to extend the original exponential-polynomial-closure solution procedure methodologically by minimizing the resulted residual error square of the governing equation, which is achieved after an exponential polynomial is adopted as the approximate solution. The objective function for computing the parameters in the approximate solutions of nonlinear random oscillators is then formulated. The probabilistic solutions of the oscillators obtained by the presented approach are verified by the exact solutions in some special cases or by Monte Carlo simulation. Numerical examples indicate that the solutions attained by the presented approach match with the exact or Monte Carlo simulation solutions. The advantage of the presented solution procedure is that it can provide a much more accurate solution than the Equivalent Linearization approach and it is much more efficient than Monte Carlo simulation as demonstrated by the numerical examples.

A unified dynamic analysis method for kilometre-scale ultra large spacecraft based on dynamic stiffness method

Ultra-kilometre and super-flexible spacecraft are the important development trends in the future. Classical multi-body dynamics and structural dynamics analysis theory have some limitations in dynamic modelling and solution of large deformation structures. In this paper, an improved dynamic stiffness method (DSM) is proposed to deal with the dynamical modelling and solution of ultra-kilometre spacecraft. The method retains the high accuracy and efficiency of the original DSM in dealing with continuous dynamical systems and breaks through the limitations of the original method in dealing with large deformation structures by introducing equivalent linearization techniques. Results show that the modal frequencies and mode shapes in this paper agree well with finite element solutions, and the maximum deviation is less than 1%; Besides, with the increase of spacecraft length, the structure has entered the nonlinear stage from the linear stage, the fundamental frequency of the structure will be lower than 0.001Hz, and the influence of structure length, density, and the mass of additional cabins on natural frequencies is less significant than that in linear stage. The proposed dynamic analysis framework can be further employed in the dynamic response and vibration control problems of super large spacecraft.

Discussion on Adjustment Method of the Characteristic Period of Site Response Spectrum with Soft Soil Layer

Twelve site models were established based on the analysis of the influence of site conditions on earthquake damage and the influence of the soft soil layer on-site seismic response. The equivalent linearization site seismic response analysis is carried out at different input ground motion levels to discuss the influence of soft soil layer thickness and buried depth. The results show that the characteristic period of the response spectrum exhibits a gradual increase as the buried depth or thickness of the soft soil layer increases. Furthermore, the characteristic period of the response spectrum also increases with the rise in the input ground motion peak. Moreover, according to the influence characteristics of soft soil thickness, buried depth, and input ground motion intensity on the characteristic period of the site acceleration response spectrum, a method for adjusting the characteristic period of the site acceleration response spectrum with a soft soil layer is put forward.

Response prediction and fatigue life estimation of thin-walled structures under random excitation using the modified equivalent linearization method

This paper proposes a fast estimation method for the high-cycle fatigue of geometrically nonlinear thin-walled structures under random loading. The core idea of this method is utilizing a modified equivalent linearization method to obtain the statistical properties of the modal displacement response of the thin-walled structures, which is used to generate samples and then transform them into stress samples to determine the peak probability density function of the von Mises stress for fatigue life estimation. There are three critical issues addressed in implementing the method: (1) Using the implicit condensation technique to introduce the membrane softening effect in the regulated equivalent linearization method to improve the computational accuracy of the displacement statistical properties and convert the higher order moments into easily computable second order moments by separating the displacement mean values, so the regulated equations of motion for non-zero mean systems are derived, and fix the problem that the regulated equivalent linearization method does not apply to the prediction of the dynamic response of general thin-walled structures. (2) A nonlinear relationship between modal displacement and stress is derived with the implicit condensation and expansion method to transform the generated displacement samples to stress samples, which are used to determine probability density functions of stress and their derivatives. (3) Simplifying the probability density function of the three-dimensional joint distribution to the probability density function of the two-dimensional joint distribution by taking advantage of the Bayes rule, thus deriving a fast formula for the probability density function of the peak Mises stress.

Transient PDF Solution of Nonlinear Stochastic Oscillator Subjected to Modulated Gaussian White Noise

In this paper, an extended exponential-polynomial-closure (EPC) procedure is presented to investigate the transient probability density function (PDF) of stochastic oscillator under the stimulation being modulated white noise based on the conventional EPC approach. By the presented EPC procedure, an evolutionary exponential–polynomial function is defined as the PDF solution and a set of evolutionary independent base functions are introduced to span a [Formula: see text] space. By integrating the residual error with those evolutionary functions spanning the [Formula: see text] space, a group of ordinary differential equations can then be formulated. Three nonlinear oscillators are analyzed numerically and the results comparisons of various techniques validate the presented EPC procedure in solution efficiency and accuracy, particularly in the PDF solution tails which play a crucial role in reliability analysis. The transient PDF solutions acquired by the presented EPC procedure match well with Monte Carlo simulation even in the PDF solution tails, whereas the solutions obtained by the equivalent linearization are deemed to be unacceptable for the strongly nonlinear oscillators subjected to modulated stimulation. Notably, the presented EPC procedure can highly increase the computational efficiency by about 1[Formula: see text]000 times in all the presented numerical examples in comparison to Monte Carlo simulation.

Performance Evaluation of the Structure of the Klaten DPRD Building When Affected by a 7.0 Magnitude Earthquake

Evaluation of structural performance in Indonesia has been carried out for several high-rise buildings as part of the owner's demands for safety guarantees to determine the level of security that the building has. A simple but accurate nonlinear analysis is needed to determine the structure's performance under earthquake loads. The capacity spectrum method (pushover analysis) documented in ATC 40 is a form of equivalent linearization based on two basic assumptions: secant period and equivalent damping. The performance point estimates where the capacity of the structure can withstand a given (demand) load; In terms of performance, the level of damage to the structure category based on lateral displacement can be seen. It is simulated that an earthquake will shake the Klaten DPRD building with a magnitude of 7.0 and 20 km from the epicenter. The amount of PGA(g) will be calculated based on the equation of Cornell et al. Building structure modeling was carried out with the help of ETABS v9.7.4.

Analysis of nonlinear vibrations of two-degree-of-freedom systems

In this paper, the motions of two-mass systems with two-degree-of-freedom are investigated by using an analytical approach. The masses are connected by linear and nonlinear springs. The motions of systems are described by systems of two coupled strong nonlinear differential equations. Nonlinear differential equations are transferred into a single equation by using some intermediate variables. An analytical method, the equivalent linearization method, is employed to analyze the free nonlinear vibration of systems. The oscillation systems with different values of the parameters are investigated in this paper. In order to verify the accuracy of the obtained results, the present solutions are compared with those achieved by the Hamiltonian approach and the exact solutions. The comparison results show that the obtained solutions are more accurate than those obtained by the Hamiltonian approach.

Reliability analysis for product package via probability density function of acceleration random response

Analytical probability density function of acceleration random response for cubic nonlinear product package has seldomly been investigated due to the difficulty of solving nonlinear random problem and less attention paid to the acceleration response. In this paper, the analytical solution for the probability density function of linear package acceleration response under random vibration is obtained, and on this basis, the equivalent linearization method is introduced to further explore the approximate analytical solution of acceleration response probability density function for cubic nonlinear product package under random vibration. Some suggestions on packaging optimal design are given by analyzing the sensitivity of acceleration random response to the external excitation and system parameters, and the first-passage failure probability for the acceleration response of cubic nonlinear package under random vibration is analyzed. The results show that the equivalent linearization method can precisely predict the acceleration response probability density function for the cubic nonlinear package under certain conditions, and the applicable range of the method is given by the equivalent stiffness formula. Nonlinear characteristic parameter γ of the cubic package activates acceleration response a completely different trend from the quasi-linear system. Properly increasing the system stiffness and decreasing the damping ratio can effectively reduce the acceleration response strength of the product package. As the vibration time increases and the product fragility decreases, the first-passage failure probability of the product package increases. The analysis provides theoretical foundation and guidance for packaging optimization design.

Performance-based model-free adaptive finite-time control for shape memory alloy actuated swing platform

Shape memory alloy (SMA), a kind of smart material, can be used as an actuator in many fields; however, its strong nonlinearity and parameter uncertainty hinders its application in high-tracking accuracy tasks. This paper addresses the tracking control problem of the SMA actuated swing platform suffering from completely unknown nonlinear model information and prescribed finite-time error constraints. First, the equivalent dynamic linearization model of the swing platform is established, and the unknown disturbance is estimated and compensated by the extended state observer. Meanwhile, a novel discrete-time performance function is proposed, and the prescribed finite-time tracking error constraints are transformed into a new equivalent unconstrained task. Second, the model-free adaptive sliding mode controller is designed using the unconstrained error, and the complete control law, including sliding mode control law and equivalent control law, is derived. Third, the stability of the closed-up swing platform is guaranteed by employing the Lyapunov method. Finally, experiments reveal that the proposed method is preferable.

Optimal design of tuned inerter eddy current damper

The tuned inerter eddy current damper (TIECD) is a vibration absorber for seismic protection of structures. This paper studies the dynamic characteristics of the TIECD and investigates its optimal design considering the nonlinear behavior of the eddy current damping (ECD). The relationship between the nonlinear parameters of the ECD and its equivalent damping coefficient is revealed by utilizing the energy equivalent linearization method. The optimal design formulae for a TIECD are derived. The comparisons indicate that the proposed solutions offer an ideal estimation for the TIECD optimal design parameters of a lightly damped structure. In contrast with the linear tuned viscous mass damper (TVMD), the nonlinear TIECD achieves better control performances. However, the fluctuation in dynamic loads for the TIECD-structure system could lead to some special nonstationary motions. For the natural ground motions, a practical iterative design procedure for the TIECD is proposed. The results show that the control performance of the TIECD can reach the level of the TVMD with lower maximum damper forces.

Impact of Site-Response Characterization on Probabilistic Seismic Hazard in the Po Plain (Italy)

ABSTRACT We present a probabilistic seismic hazard analysis (PSHA) for the entire Po Plain sedimentary basin (Italy)—one of the widest Quaternary alluvial basins of Europe, to evaluate the impact of site-response characterization on hazard estimates. A large-scale application of approach 3 of the U.S. Nuclear Regulatory Commission (NRC) to include seismic amplification in the hazard is presented. Both 1D amplification related to stratigraphic conditions and 3D amplification due to basin effects are considered with the associated uncertainties, and their impact on the hazard is analyzed through a sensitivity analysis. Whereas 3D basin effects are considered through the application of an empirical, spatial invariant correction term, 1D amplification was estimated throughout the study area by means of dynamic (equivalent linear) ground-response analysis. To separate aleatory variabilities and epistemic uncertainties related to site response, a partially nonergodic approach is used. The results provide a finer picture of the actual seismic hazard, highlighting those areas where the ground motion is affected by amplification effects due to local or regional geological features. We found that, for a return period of 475 yr, neglecting basin effects produces a 30% underestimation of the seismic hazard in the long-period (>1 s) range. Moreover, with reference to the hazard model adopted, such effects are responsible for most of the epistemic uncertainty (up to 80%) in the results. Therefore, such effects deserve special attention in future research related to PSHA in the Po Plain sedimentary basin.

Simplified estimation formula for frequency response function of roll motion of ship in waves

This study proposes a simplified formula for estimating the frequency response function of ship roll motion expressed by the main hull-form parameters of a ship (length L, breadth B, draft d, block coefficient Cb, waterplane area coefficient Cw, height of center of gravity KG‾, and metacenter height GM‾). By taking a theoretical approach, starting from the equation for roll motion and approximating each fluid force component, the authors have developed a formula that is physically clear and simple, yet engineering accurate and extendible. For nonlinear effects, this study assumes the case where decay coefficients or Bertin's N-coefficient are given, and it considers the nonlinearities in the quadratic damping force by equivalently linearizing the force. For equivalent linearization, a convergence calculation is usually required, but in this study, it is replaced by an explicit formula. To validate the proposed equation, numerical calculations using the linear 3D-Green's function method program were performed for 154 actual merchant ships with a wide parameter range, and it was confirmed that the equation is applicable regardless of ship type and size.

A new look at the Equivalent Linearization Technique to predict LCO in aeroelastic systems with discrete nonlinearities

Control surfaces nonlinearities can lead to limit cycle oscillations (LCO). Several methods have been proposed to predict LCO, such as Harmonic Balance-based methods (HB). Describing function (DF) is the HB with a single harmonic motion assumed, and the approach can be combined with a classic eigenvalue stability analysis via the Equivalent Linearization Technique (ELT) to predict LCO. On the other hand, High-Order Harmonic Balance (HOHB) methods consider higher number of harmonics, but they lead to a more complex nonlinear algebraic system of equations. This paper introduces a new look at the ELT combining both eigenvalue analysis and describing functions. Two applications are considered. The first one is a new DF written in a matrix form for the ELT to consider both first and third harmonics in LCOs due to freeplay. The second application is an iterative procedure to combine the eigenvalue analysis with a describing function to predict LCOs in systems with both freeplay and friction nonlinearities. Numerical simulations are performed for the aeroelastic typical section airfoil. The results show that the new DF improves the classic DF by providing a more accurate prediction of LCOs that accounts for third harmonic. Also, the iterative-ELT is shown as an excellent predictor of LCO for systems with both freeplay and friction.

3D limit analysis of rock slopes based on equivalent linear failure criterion with tension cut-off

Hoek-Brown (HB) failure criterion is widely used to predict the strength of intact or heavily jointed rock mass. For stability analysis of rock slopes governed by the HB failure criterion, the equivalent linearity to Mohr-Coulomb (MC) criterion is often adopted, leading to the well-known equivalent Mohr-Coulomb method (EMCM). Existing studies on EMCM analysis mainly consider the shear strength of rock material, while consideration of the tensile strength is rare. This contradicts the fact that the underlying tensile strength of rock mass has considerable impact on the rock slope stability in real world. In this regard, this paper proposes a limit analysis-based approach that can account for tension in the three-dimensional (3D) stability analysis of HB rock slope. This approach is established on the equivalent linearity of the HB criterion with consideration of tensile strength, known as the equivalent tension cut-off MC method (ETMCM), and using a horn-like 3D mechanism of limit analysis. The safety factor solutions given by the proposed approach are validated by previous studies and numerical results. Parametric studies are conducted to investigate the effect of rock tensile strength on slope stability. Results show that the consideration of tension leads to a more conservative safety factor and a sharper curvature of the failure surface, and these impacts tend to be more obvious with the increases in slope inclination and slope width. Finally, the stability of the HB rock slope under seepage conditions is studied using the proposed approach. The results indicate that the effect of tensile strength is highly remarkable in seepage circumstances.

Seismic demand estimation of isolated buildings with nonlinear bearings subjected to linear response spectra

Response spectrum analysis is a commonly used method for the seismic design of linear structures with fixed bases since the seismic input is normally provided in terms of the response spectrum in many codes. This study proposes a more generalized response spectrum method for the seismic design of three-dimensional isolated structures. Firstly, the theory basis of this method is derived in the framework of the random vibration theory. The equivalent parameters of each bearing are expressed in terms of the statistics of the displacement amplitude. Then, the complex mode theory is introduced into this method to consider the effect of the nonclassical damping resulting from the isolation layer. Finally, an iterative analysis procedure is presented, which combines the equivalent linearization with the complex mode theory. Moreover, the convergence and applicability of this method are discussed in detail. The parameter analyses show that the relative error of the base shear of the superstructure estimated by using this method is within 10% compared with that of the nonlinear time history analysis. In addition, a benchmark base-isolated structure is used as a numerical example to examine the effectiveness of the proposed method. The results show that the iteration is convergent and computationally efficient. The results also show that the maximum relative errors of the predicted isolation layer displacement, the inter-story drift of the bottom floor and the absolute acceleration of the top floor are 9.5%, 3.9% and 18.9%, respectively, by comparing them with the nonlinear time history analysis, while these responses are underestimated by 9.8%, 25.9% and 40.2%, respectively, when using the classical complete quadratic combination (CQC) method that neglects the nonclassical damping effect. Although in this study the proposed approach mainly focuses on isolated structures, in essence, this approach can also be applied to inelastic structures or structures with nonlinear dampers.

One dimensional time-domain non-linear site seismic response analysis program integrating two hysteresis models of soil

In the numerical simulation of site seismic responses, traditional equivalent linearization methods typically realized in the frequency domain cannot satisfactorily analyze the high-degree non-linearity of soil under strong input motions. Therefore, the “true” non-linear numerical methods performed in the time domain are often utilized in such cases. However, a crucial element of the time-domain non-linear method, which is the hysteresis model of soil that describes the rule controlling the loading–unloading behavior of soil, has no established guidelines for earthquake engineering. Different researchers presented different models, revealing the epistemic uncertainty related to the dynamic properties of soil. Thus, the time-domain non-linear method should consider this uncertainty in practice. Therefore, in this study, a one-dimensional (1D) time-domain non-linear site seismic response analysis program was developed. The developed program was coded using Fortran95 and integrates two kinds of soil hysteresis models (i.e., extended Masing model and dynamic skeleton curve model). In both models, the damping correction was introduced to calibrate the hysteresis loop area toward the damping ratio measured in the dynamic triaxial test or resonant column test. Moreover, the temporospatial finite difference algorithm was used to resolve the 1D non-linear wave equation, and its precision was demonstrated in comparison with the results of the frequency-domain program for the linear case. Finally, the non-linear seismic response of a specific site was calculated by the proposed program. The findings of the fitting were compared to those of the two popular time-domain non-linear programs DEEPSOIL (Hashash, V6.1) and CHARSOIL (Streeter et al., CHARSOIL, Characteristics Method Applied to Soils, 1974 March 25). Simultaneously, the Japanese KIK-net strong motion observation station data were applied to validate the reliability of this program.

Seismic performance of a clutched inerter for structures with curved surface sliders

The performance of the clutched inerter in limiting the seismic response of structures isolated by curved surface slider (CSS) is investigated. The chosen structure is a flexible multi-floor building with an idealized linear shear-type behaviour. The equations governing the motion of an isolated structure with a clutched inerter under earthquake excitation have been derived. Initially, using the equivalent linearization technique, the isolated building's stationary floor absolute acceleration, isolator displacement, and device forces are evaluated. The stochastic earthquake excitation is considered to be a filtered white-noise Kanai-Tajimi model. The performance of the clutched inerter in mitigating the response of the isolated structure is studied by varying system parameters. The important parameters included are the inertance ratio of the clutched inerter, the period and friction coefficient of the CSS, and the flexibility of the superstructure. Further, there exists an optimum inertance of the clutched inerter for which the structure's highest floor absolute acceleration reaches its minimum level. The influence of important system parameters on the optimum inertance of the clutched inerter is also investigated. Finally, the performance of the clutched inerter as a supplemental device for the base-isolated structure is investigated when subjected to recorded earthquake motions consisting of near-fault as well as far-field types. The trends of results of isolated structures with clutched inerter under the recorded earthquake motions were in good agreement with that under stochastic excitation. It is demonstrated that the supplemental clutched inerter is effective in reducing the displacement response of base-isolated structures.

Equivalent linear method with complex frequency for response analyses of the ground subject to fling-step displacements

The equivalent linear (EL) method is one of the most commonly used methods for seismic ground response analyses in engineering practice. However, it cannot be used to directly calculate the displacement response of the ground subject to fling-step displacements because these are characterized by non-zero final values and ordinary Fourier transform with real frequency breaks down. This article shows that this difficulty can be overcome by introducing a complex frequency in the equivalent linear method. The technique of using complex frequency was used in existing studies for analyses of undamped structures and is known as the exponential window method (EWM). In essence, the same approach is effective for the analysis of fling-step displacement. Herein, why this works for fling-step displacements is described and an application to a vertical array record which captured near-fault fling-step displacements is shown. The reasonable agreement between the observed and synthetic time histories, both for acceleration and displacement, illustrates the applicability of the method to the response analyses of the ground subject to fling-step displacements.