Single Degree Of Freedom System Research Articles

Structural and non‐structural numerical blind prediction of shaking table experimental tests on fixed‐base and base‐isolated hospitals

AbstractBase‐isolated hospitals are frequently preferred to fixed‐base ones because of their improved seismic structural performance. Despite this, the question remains open on the advisability of using this modern seismic protection technology in preference to other conventional solutions, on the grounds of a holistic approach based on limiting non‐structural damage as well as continuity of service to the community in the aftermath of an earthquake. Two full‐scale four‐storey (fixed‐base) and three‐storey (base‐isolated) hospital buildings have been recently built and subjected to three‐dimensional shaking table tests at the National Research Institute for Earth Science and Disaster Prevention (Japan), with particular attention to evaluating and classifying functionality of non‐structural components and vital medical equipment. A two‐phase experimental campaign was carried out considering two earthquakes scaled at different intensity levels and applied along the horizontal and vertical directions. The current study aims to provide results of a numerical structural and non‐structural blind prediction of these hospital settings. A homemade numerical code is developed to account for lumped plasticity modelling of steel frame members and variability of the friction coefficient of spherical sliding bearings. Moreover, three non‐structural components are modelled in the fixed‐base structure: that is, elastic single degree of freedom systems representing two tanks filled with sand at the top floor; elastic beam elements for piping at the third floor; five‐element macro‐model for the in‐plane‐out‐of‐plane nonlinear response of partition walls at the first floor. The identification of predominant vibration periods of the fixed‐base structure is carried out using a homemade numerical code based on the continuous wavelet transforms in combination with the complex Morlet wavelet. Finally, the sliding and rocking motion of three items of medical equipment (i.e., incubator at third floor, dialysis machine at second floor and surgical bed at first floor) are analysed by means of a homemade numerical code, considering acceleration time histories of selected structural nodes of the fixed‐base structure.

Tracking superharmonic resonances for nonlinear vibration of conservative and hysteretic single degree of freedom systems

Many modern engineering structures exhibit nonlinear vibration. Characterizing such vibrations efficiently is critical to optimizing designs for reliability and performance. For linear systems, steady-state vibration occurs only at the forcing frequencies. However, nonlinearities (e.g., contact, friction, large deformation, etc.) can result in nonlinear vibration behavior including superharmonics — responses at integer multiples of the forcing frequency. When the forcing frequency is near an integer fraction of the natural frequency, superharmonic resonance occurs, and the magnitude of the superharmonics can exceed that of the fundamental harmonic that is externally forced. Characterizing such superharmonic resonances is critical to improving engineering designs. The present work extends the concept of phase resonance nonlinear modes (PRNM) to be applicable to general nonlinearities, and is demonstrated for eight different nonlinear forces. The considered forces include stiffening, softening, contact, damping, and frictional nonlinearities that have not been previously considered with PRNM. The proposed variable phase resonance nonlinear modes (VPRNM) method can accurately track superharmonic resonances for hysteretic nonlinearities that exhibit amplitude dependent phase resonance conditions that cannot be captured by PRNM. The proposed method allows for characterization of superharmonic resonances without constructing a full frequency response curve at every force level with the harmonic balance method. Thus, the present method allows for analysis of potential failures due to large amplitudes near the superharmonic resonance with reduced computational cost. The consideration of single degree of freedom systems in the present paper provides insights into superharmonic resonances and a basis for understanding internal resonances for multiple degree of freedom systems.

Analysis of fractional differential equation to compare the dynamic behavior of a single degree of freedom system models using a shaking table

AbstractA mathematical model to describe the dynamic behavior of a single degree of freedom system, SDFS, model in springs and damps assembles in series and parallel models in a laboratory, is presented. Kelvin–Voigt fractional derivative rheological models, KVFRdm, was used to solve the fractional differential equations of motion by numerical methods.

Numerical and experimental study of a suspension bridge with combined viscous-steel damping system under train and seismic loads

The longitudinal dynamic performance of long-span bridges with passive vibration control devices when subjected to vehicle and seismic loads has raised extensive attention of the civil engineering community. This study aims to provide numerical and experimental assessments of the dynamic performance of a railway suspension bridge with the Combined Viscous-Steel Damping System (CVSDS) under various excitations. The deck is simplified to a single-degree-of-freedom (SDOF) system according to the longitudinal vibration characteristics of the suspension bridge. The Dynamic Equation Method (DEM) and Fourier Expansion Method (FEM) are suggested for the equivalence of the spatiotemporal variable train loads to horizontal excitations. A simplified two-stage analytical model is proposed to simulate the restoring force characteristics and fuse-lock function of the CVSDS. Shake table tests are conducted to further illustrate the feasibility of the simplified model and evaluate the effectiveness of the CVSDS. The results indicate that the SDOF system is reasonable and satisfactory in simulating the deck displacement. The two-stage working mechanism of the CVSDS is verified by both the simulation and experiment, i.e. the viscous fluid damper works under train loads and the steel yielding damper operates under earthquakes. The vibration mitigation effectiveness of the CVSDS is remarkable.

Relationships between ground motion parameters and energy demands for regular low-rise RC frame buildings

AbstractThe present study aims to analyze the correlation between ground motion parameters and energy demands of low-rise RC buildings without shear walls. Two regular 4- and 7-story residential buildings were seismically designed to represent low-rise RC buildings. In order to establish the requirements of single degree of freedom (SDOF) systems as well as multi degree of freedom (MDOF) systems, the dynamic features of “equivalent” SDOF systems were defined by using MDOF systems. The correlation of 20 ground motion parameters (GMPs) of 44 records with the energy demands obtained from a total of 176 nonlinear time history analyses was examined for the SDOF and MDOF systems within the scope of this study. The ground motion parameters (GMPs) were taken as intensity measures (IMs). In contrast, maximum input energy was used as a demand measure (DM) and these energy values were normalized with the masses of the buildings for cases, where the buildings were evaluated together. Parameters related to acceleration and velocity are generally found to yield better results in comparison to the ones related to frequency and displacement. Velocity Spectrum Intensity (VSI) and Arias Intensity (Ia) were obtained to have the highest correlation values as a single parameter. This study suggested new equations by combining multiple ground motion parameters for SDOF and MDOF systems to better reflect damage potential than a single parameter. The use of multiple parameters in combination results in better correlation coefficients.

Ductility demands for stiffness-degrading SDOF systems under pulse-like ground motions of the 2023 Pazarcık (Kahramanmaraş) earthquake

This paper investigates the inelastic displacement ratios (IDRs) and displacement ductility demands of a wide range of single-degree-of-freedom (SDOF) systems subjected to pulse-like ground motions (GMs) of the 2023 Pazarcık (Kahramanmaraş) earthquake. A set of twenty-seven GMs characterized as pulse-like are utilized in the study. As-recorded velocity time histories of horizontal components are rotated over 90° at a step of 1° to attain the waveform with the largest peak ground velocity (PGV) over all horizontal orientations. Inelastic displacement ratio and displacement ductility spectra are computed through nonlinear response history analysis (RHA). Local amplifications of both spectra are observed at some periods. The results of this study show that large inelastic displacement and ductility demands are imposed on certain reinforced concrete (RC) structures. Finally, predictive models of the mean inelastic displacement ratio and mean ductility demand spectra (DDS) are developed based on the Gauss–Newton algorithm (GNA). The model provides a strong correlation between the computed and the estimated data, and sufficient convergence criteria. The results of this study collectively emphasize the necessity of integrating pulse-like GMs into future revisions of earthquake codes.

Truly optimal semi-active damping to control free vibration of a single degree of freedom system

This paper studies a single degree of freedom system under free vibration and controlled by a general semi-active damping. A general integral of squared error is considered as the performance index. A one-time switching damping controller is proposed and optimized. The Pontryagin Maximum Principle is used to prove that no other form of semi-active damping can provide the better performance than the proposed one-time switching damping.

Constant damage inelastic permanent period shift ratios spectra

The phenomenon of elongation of the structural period after earthquake excitation has been widely observed, and elongated period obtained after earthquake excitations can be used for post-earthquake assessment. The inelastic permanent period shift ratio, an indicator representing the degree of stiffness degradation and closely relating to the damage state, is defined as the ratio of the elongated period Tin of single-degree-of-freedom (SDOF) systems after earthquake excitations to the initial period Tel. The objective of this paper is to investigate the inelastic permanent period shift ratios under constant damage performances through nonlinear time-history analysis of SDOF systems, and the modified Park-Ang damage index is employed to characterize the structural damage. Inelastic permanent period shift ratio spectra are statistically constructed considering both the ground motion characteristics (such as site conditions, velocity pulse, magnitude, closet distance, directivity effects, significant duration and frequency contents) and structural parameters (including damage states, hysteretic models, ultimate deformation capacity, energy dissipation factor and postyield ratio). The results showed that the permanent period shift ratio is a reliable damage indicator for assessment of global damage states. The influence caused by ground motions on the estimation of the mean of the permanent period elongation ratio under constant damage performances is limited, while the effects caused by different structural damage and deformation capacity can reach up to approximately 70% and 33%, respectively. An expression for the relationship between the damage index and the permanent period shift ratio is established. Ignoring the differences of structural performances, using the period shift ratios to directly assess the global damage states will lead to erroneous results.

Output Feedback Dynamical Compensation via EigenStructure Assignment for Active Magnetic Bearings-Rigid Rotor Systems

A rotor system supported by active magnetic bearings (AMBs) is actually a second-order dynamical system, whose dynamical characteristics should be considered in the design process of the controller. The relationships between the dynamical parameters and the control parameters can be established by the eigenstructure assignment. Output feedback dynamical compensators (OFDCs) are proposed based on the eigenstructure assignment for an AMBs-rigid rotor system in this paper. First, the model of the four-degree-of-freedom (4-DOF) AMBs-rigid rotor system was established, and it was decoupled into two single-degree-of-freedom (S-DOF) translational sub-systems and two S-DOF rotational sub- systems through the modal decoupling method. Aiming to the control of the four S-DOF systems, the relationships among the parameters of the OFDCs, the eigenvalues, and the eigenvectors were derived based on the eigenstructure assignment. Then, a multi-objective optimization problem was constructed in which the system stability, the robustness to the bias current perturbation, and the maximum amplitudes of the unbalance displacement and control current were considered. Several Pareto solutions were obtained through NSGA-II, and different solutions were selected to compare the performance of the corresponding systems. The simulation and experimental results verify the effectiveness of the proposed OFDCs and the rationality of the proposed design process.

A two-level performance-based plastic design method for multi-story steel frames with double-yielding systems

The performance-based seismic design (PBSD) is usually conducted under design basis earthquakes (DBE), and then, the performance of the designed structures is examined under maximum considered earthquakes (MCE). As a result, it is challenging to simultaneously satisfy the performance targets associated with the DBE and MCE levels. To this end, this study proposes a two-level performance-based plastic design (PBPD) method. The fused steel frames exhibiting trilinear capacity curves are selected as the example structures. The peak interstory drift ratios under the DBE and MCE levels are defined as the performance targets and they are explicitly utilized at the beginning of the design procedure. The expressions of two energy modification factors are first obtained by establishing energy balance equations for single-degree-of-freedom (SDOF) systems under the DBE and MCE levels, respectively. Nonlinear time history analyses (NLTHA) for SDOF systems with various trilinear capacity curves are conducted to derive necessary design parameters. The versatility and flexibility of this design method are achieved by enabling both the primary and secondary yielding systems to yield under the DBE level. To demonstrate and validate the proposed design method, a six-story benchmark steel frame equipped with idealized metallic yielding damping braces is selected and designed with two sets of performance targets. The nonlinear static and NLTHA results proved that the designed structures can simultaneously meet the prescribed performance targets under both DBE and MCE levels.

Damage Spectra of SDOF Structures Under Tsunami Actions Considering the Nonlinear Dynamic Analysis Method

This paper studies the damage spectra of single-degree-of-freedom (SDOF) systems for tsunami loading. The 2011 Tohoku tsunami wave traces are computed and obtained by adopting the Cornell Multi-grid Coupled Tsunami (COMCOT) model and are further verified by the Tohoku Tsunami Joint Survey (TTJS) data. The tsunami response analyses of SDOF systems are conducted taking into account the dynamic analysis method. The classical ductility damage is considered as the damage performance. The damage spectra are computed for five force reduction factors and 791 tsunami loadings. The effects of inundation depths, hysteretic model, post-yield stiffness ratio, [Formula: see text], viscous damping ratio, [Formula: see text], are explored and interpreted scientifically and statistically. The results demonstrate that the effect of the hysteretic model, force reduction factor, R, post-yield stiffness ratio, [Formula: see text], viscous damping ratio, [Formula: see text], on the damage of the structure is evident. Comparison with the static method and influence analysis of structural damping indicate the necessity for the application of the dynamic method. A simplified predictive formulation is developed for the evaluation of structural damage for tsunami loadings.

A simplified method for the dynamic analysis of structures under column loss

The progressive collapse of buildings as a dynamic phenomenon can be fully understood only within a transient dynamic analysis context considering for damping and inertia effects. The response of a typical beam-column substructure is usually dominated by a single mode of deformation, thus the single-degree-of-freedom (SDOF) modelling assumption provides an accurate representation of the problem. In this study, the dynamic response of an inelastic SDOF system with damping was investigated analytically. Closed form analytical equations of the model were derived that describe the complete dynamic response in the time domain. Moreover, simplified formulae to estimate peak responses and Dynamic Increase Factors (DIFs) were provided. Numerical dynamic analyses in OpenSees were performed to verify the accuracy of the analytical model. The results indicate that the analytical model can trace the full dynamic response accurately while peak displacements can be accurately estimated with the proposed formulas. The estimated DIFs were in agreement with available from the literature results and demonstrate the significant effect of damping and hardening. Application of the analytical methodology on the progressive collapse analysis of a 10-storey steel frame demonstrated the effectiveness and potential of the proposed SDOF system.

Numerical Studies on Seismic Response of Structural Systems Using a Response-Spectra Based Intensity Measure

This paper proposes a novel intensity measure (IM) based on the geometric mean of acceleration response spectral ordinates to assess the probabilistic performance of structures subjected to seismic loading. Instead of relying solely on the fundamental period, the proposed IM is evaluated across a fixed period range for all structural systems, which allows consideration of higher mode effects and period changes due to nonlinearity. The proposed seismic IM is evaluated using two established indices sufficiency and efficiency. Sufficiency quantifies the independence of an engineering demand parameter at a specific intensity level with ground motion characteristics such as seismic magnitude [Formula: see text] and distance from site to fault plane [Formula: see text]. It is calculated by linear regression analysis, or using gradient-based relative sufficiency measures. On the other hand, efficiency is measured as dispersion across ground motions at a given intensity level for any physical response quantity. It helps to reduce computational demand for failure probability assessment by considering a smaller number of records compared to an inefficient IM for similar confidence levels. The effectiveness of the proposed IM along with 10 other IMs is demonstrated on single degree of freedom systems with various fundamental periods by performing nonlinear time history analysis using a far-field ground motion record set. The study is also extended to five degree of freedom lumped mass stick models, 2D models (4-, 8-, and 12-story archetype steel frames), and 3D reinforced concrete shear wall building model. The results indicate that the proposed IM limits dispersion to within 10% for long-time period structures, and demonstrates improved sufficiency across different structural systems. For example, gradient of the proposed IM with respect to magnitude [Formula: see text] and site-to-source distance [Formula: see text] for a 12-story steel frame is reduced by 42.9% and 94%, respectively, compared to spectral acceleration at fundamental time period. Potential application of this research lies in efficiently conducting seismic reliability assessment and design for structural systems.

A time and ensemble equivalent linearization method for nonlinear systems under combined harmonic and random excitation

An Equivalent Linearization technique, termed an Equivalent Linearization Time and Ensemble Expectation (EL-TEE) approach, is used to develop an alternative method for estimating the response of a nonlinear oscillator to a combination of deterministic harmonic and random white noise excitation. The approach is based on applying equivalent linearization and averaging over the time period of one harmonic excitation cycle. This gives a set of coupled nonlinear equations that can be solved for the response averaged over time and across the ensemble. The primary advantages of the proposed method are its computational speed, ability to return physically meaningful linearization matrices and that it can be applied to a wide variety of nonlinearities. The method is applied to three example test systems: the well-known single degree of freedom Duffing oscillator; a single degree of freedom system with a displacement constraint imposing a discontinuous nonlinearity; and a multi degree of freedom oscillator with a localized polynomial nonlinearity that has also been examined experimentally. It is shown that the response predicted matches well with Monte Carlo results from direct time integration at a fraction of the computational cost, and the method is capable of reproducing key results observed experimentally.

A Study on the Relationship between Response Spectrum and Seismic Fragility Using Single Degree of Freedom System

A Study on the Relationship between Response Spectrum and Seismic Fragility Using Single Degree of Freedom System

Evaluation of equivalent linearization methods of MDOF isolated structures in response spectrum analysis

In the seismic design of isolated structures, equivalent linearization of the hysteretic isolators is normally considered. There exist many equivalent linearization methods and evaluation of these methods has been widely studied. However, most of the previous studies concern a single degree of freedom system only, and the used analysis method is time history analysis. In such a case, the evaluation heavily depends on the ground motions selected. Note that the response spectrum analysis is a commonly used method for seismic design of structures. Therefore, it is more valuable to evaluate these equivalent linearization methods based on the response spectrum analysis. In this study, a multiple degree of freedom model is adopted due to the contribution of higher modes of the superstructure to the seismic responses, and the procedure of the response spectrum analysis in conjunction with the equivalent linearization method is presented in detail. Then, considering a series of parameters of isolators, a statistical study is performed to evaluate the accuracy of these equivalent linearization methods by comparing with the nonlinear time history analysis. Moreover, the effect of the damping reduction factor for the response spectrum on the accuracy of the equivalent linearization methods is discussed in terms of probability in a desirable error range. The analytical results show that the accuracy of the equivalent linearization methods is associated with the earthquake intensity. Meanwhile, for different response quantities the accuracy of each equivalent linearization method varies significantly. In addition, to estimate the seismic responses with a better accuracy in the response spectrum analysis, a suitable damping reduction factor should be considered.

On Time-Domain Solutions to the Free Vibration of SDOF Systems with Linear Hysteretic Damping

This article focuses on the free vibration of single-degree-of-freedom (SDOF) systems with linear hysteretic damping and investigates the consistency and convergency of the solution to the free vibration in time domain. The criteria associated with the unit-impulse response function at [Formula: see text] are suggested to check the consistency of possible solutions. Exhaustive and systematic derivations are presented to clarify the confusing results in the literature. Subsequently, the integration in the Hilbert term proves to be nonconvergent when an exponential term ([Formula: see text]) is assumed to be the general form of displacement solutions. Hence, it can be concluded that the exact solution to the free vibration of SDOF systems with linear hysteretic damping is still an open question. At last, a numerical verification method is proposed to assess the accuracy of possible solutions.

A novel response-amplified shape memory alloy-based damper: Theory, experiment and numerical simulation

The energy dissipation capacity of shape memory alloy (SMA) dampers can be effectively improved owing to the success of response amplification technology. However, conventional response-amplified SMA dampers for building structures still face a risk of failure during extremely rare earthquake events. In this study, a novel response-amplified SMA damper (RASD) is proposed with the advantages of fully exploiting the energy dissipation characteristics of SMA materials and avoiding material failure. First, the components, configuration, and working principles of the proposed damper were described. Subsequently, a restoring-force model of the RASD was established, and its accuracy was verified through cyclic loading tests. Finally, seismic performance mitigation of a single-degree-of-freedom (SDOF) system under different hazard levels was conducted to evaluate the effectiveness and advantages of the proposed RASD quantitatively. The results reveal that the RASD has a good force amplification effect and excellent damping enhancement effect and thus conspicuously mitigates the seismic response of the SDOF system. In particular, the proposed RASD can effectively reduce the force and displacement demand of SMA materials without the risk of SMA failure, while significantly mitigating the structural response.

Numerical steady-state and transient responses of a SDOF system constrained by two optimally designed bumpers

Seismically isolated structures can be subjected to large horizontal displacements relative to the ground, especially in Near-Fault earthquakes, which are characterized by one or more intense pulses of velocity and displacement of long period. One strategy to mitigate the problem of large displacements, which occurs in linearly isolated structures, is the use of deformable and dissipative devices (bumpers). The impact between the structure and the bumpers, if the bumpers are appropriately designed, can produce beneficial effects on the dynamic response of the system, both on displacements and accelerations. In this paper the response obtained from a numerical model of isolated single-degree-of-freedom (SDOF) systems constrained by two bumpers, arranged symmetrically on both sides of the mass of the system with an initial gap, subjected to base harmonic excitation, is studied. This model is called Vibro-Impact Isolation System (V-IIS). The objective of this work is to define a methodology for choosing the design parameter defining the V-IIS (mechanical characteristics of the bumpers, gap and isolation frequency of the system) by observing both steady-state and transient responses of both the system and the bumpers. The study of the transient response is compared with that obtained in the steady-state to assess how representative the latter is of the V-IIS transient response. From the definition of the methodology for choosing the parameters of the V-IIS, through optimal design, the only design parameters are the gap and the isolation frequency of the system. Therefore, an appropriate choice of the gap makes it possible to bring frequency-selective viscous damping in V-IISs, introducing two advantages over linear systems: the reduction of the peak intensity of the responses in the resonance range (both displacement and acceleration) and the reduction of the static displacement of the system, but keeping the dynamic response with which the system is designed unchanged.

Seismic response analysis of single-degree-of-freedom systems with superelastic monocrystalline CuAlBe SMAs

Although the potential of shape memory alloys (SMAs) in seismic applications has been well recognized, the majority of the past research was directed to NiTi SMAs. This study aims to investigate the feasibility of using a novel type of SMA, termed superelastic monocrystalline CuAlBe SMA, in seismic vibration control. According to prior loading tests, CuAlBe SMAs had an excellent superelasticity behavior with the largest recoverable strain over 25%. Within the superelastic range, CuAlBe SMAs shown unloading behavior which was dependent on the strain amplitude in the present cycle, which could not be captured by current constitutive models. Hence, a new constitutive model was firstly proposed to simulate the hysteretic behavior of CuAlBe SMAs, and a good agreement is found between the experimental data and numerical simulations. And then, to explore the characteristics and advantages in terms of seismic applications, case studies on single-degree-of-freedom (SDOF) systems with NiTi and CuAlBe SMAs were carried out. Finally, two design methods were used to size the NiTi and CuAlBe SMAs, and the corresponding SDOF systems were subjected to two ensembles of earthquakes associated with frequently occurred and design basis seismic hazard levels. The simulation results indicated that, compared with NiTi SMAs, CuAlBe SMAs could more effectively control the displacement and acceleration demands, if their ductility capacity was fully utilized in the design procedure.