Single Degree Of Freedom System Research Articles

Seismic response of monopile-supported wind turbine systems: combination laws for kinematic and inertial interaction

The dynamic soil structure interaction imposes both kinematic and inertial interactions on monopile-supported wind turbines. The two effects can be calculated individually, but there is no clear guidance in the current design standard for combining them. A theoretical study using the finite element method to model a simplified single degree of freedom system has been conducted to analyse the phase relationship between the two effects and obtain the combination laws through parametric studies using the enhanced hysteresis control soil constitutive model. The results show that when the fundamental frequency of the soil is smaller than the superstructure, an in-phase relationship is observed. Monopile’s bending moments from the two loadings have different signums. Considering the inertial loading alone with an additional 5% modification is sufficient to capture the maximum bending moment and the profile. When the natural frequency of the soil is larger than the superstructure, an out-of-phase relationship is observed. The kinematic loading has a significant domination on the lower part of the pile. The worst-case scenario can be obtained when both loadings are combined using the simple addition rule. These findings enable engineers to combine two loadings in a simple manner, avoiding performing the lengthy dynamic time-history numerical analysis and facilitating an efficient design.

Dynamic response of SDOF systems with inerter or clutching inerter damper under harmonic excitations

In this study, the dynamic response of single degree-of-freedom (SDOF) systems equipped with an inerter or a clutching inerter damper (CID) under harmonic force or base excitation is investigated. Under harmonic force excitation, an inerter must work with an appropriate damping element to prevent the kinetic energy it absorbs from being returned to the primary system. However, the addition of the damping element introduces extra damping force, which can deteriorate the force transmissibility within certain frequency ranges. In comparison, the piecewise nature of the CID effectively overcomes this limitation. Since this piecewiseness is a typical nonlinear feature, the averaging method is employed to solve the system response. The obtained approximate analytical solution is compared with the solution derived from the equivalent linearization method (ELM). It is found that the resonant frequency calculated by the ELM is significantly higher than the numerical solution for inertance [Formula: see text]. In contrast, the results indicate that the averaging method provides higher accuracy and avoids overestimating the resonant frequency. Under harmonic base excitation, an increase in the inertance of the inerter reduces both the system damping ratio and the excitation amplitude. The reduction in the damping ratio theoretically increases response sensitivity, leading to higher resonance amplitudes. However, this potential increase in resonance amplitude is outweighed by the substantial decrease in excitation amplitude. As a result, the dominant effect of reduced excitation amplitude leads to an overall decrease in the resonance amplitude. In addition, due to the low-pass filtering characteristic of the inerter, its vibration suppression effectiveness rapidly deteriorates in the high-frequency range. Similarly, the CID experiences a decline in vibration suppression at high frequencies for the same reason. Nevertheless, it outperforms the inerter in the resonant region, owing to its distinctive energy-dissipating properties.

The contribution of higher order modes to shock response spectra

Abstract The shock response spectrum is a widely used design tool that engineers use when assessing the severity of transient excitation on a structure. The shock response spectrum is typically defined as the peak response of a series of single degree-of-freedom (DOF) oscillators each with a different natural frequency. For structures representable by a single DOF system, such is the case when only a single mode dominates the response, the shock response spectrum can be used to evaluate this response directly and a brief review of the existing accepted metrics is presented. In general, the response of many structures may comprise contributions from higher order modes leading to inadequate predictions from the shock response spectrum, especially for acceleration responses. In such cases a form of shock response spectrum can be obtained that accounts for all modes of vibration but as a function of shock duration. However, a multi-modal model of the structure is then required and its shock response obtained numerically over a range of shock durations which can be computationally expensive and less amenable to physical interpretation. This paper adopts a previously proposed modal approach to obtaining the shock response of systems with contributions from higher order modes, in which the transient response is calculated for each mode in turn which are then superposed. A key question that arises is the number of modes that need to be retained in the summation. Two numerical case studies are presented which explore modal convergence for displacement, velocity and acceleration shock response spectra. Results suggest that a priori mode selection is not generally possible although inspection of the mode shapes can offer guidance. However, the benefit of the modal approach to interpretability of the results is illustrated.

Displacement-Based Seismic Design of Multi-Story Resistance Capacitance-Coupled Shear Wall Buildings with Energy-Dissipation Dampers

This research aims to apply the displacement-based design method (DBDM) for the seismic design of reinforced concrete-coupled shear wall buildings equipped with energy dissipation dampers. The DBDM offers design simplicity by focusing on structural design based on a target design displacement, where the building converts into a single degree of freedom (SDOF) system. The implementation of dampers aims to reduce repair costs and downtime for buildings following significant seismic events. Two types of dampers are utilized in this study: metallic damper and viscoelastic damper. The DBDM procedure begins with determining the target displacement, which corresponds to the specific story drift ratio of the structural system, using a nonlinear static pushover analysis. For the structural wall system considered in this study, a target drift ratio of 1/250 is selected due to the inherent rigidity of the structure. The effective damping factor is then determined from the average energy absorption, which is based on the ductility factor of each structural member. Additionally, the effective period of the building is obtained from the displacement spectrum of the design-level earthquakes. Finally, the required damper shear capacity for the SDOF system is calculated based on the target deformation and effective stiffness. The design earthquakes are generated from the acceleration response spectrum for Level 2 earthquakes, as specified in the Japanese seismic code, utilizing three different sets of phase information: Kobe, El Centro, and random phase records. The effectiveness of the DBDM is scrutinized through a comparison with results obtained from time history analysis. The results obtained for 6-, 12-, and 18-story RC-coupled shear walls with energy dissipation dampers indicate that the proposed design methodology effectively meets the specified design objectives.

Transformer winding state diagnosis method based on short-circuit impulse response and multi-feature joint

Abstract To investigate the relationship between the physical characteristics of transformer short-circuit impulse signals and winding deformation, this paper proposes a transformer winding state diagnosis method based on the combination of short-circuit impulse response and multiple features. Firstly, based on the relative displacement model of a single degree of freedom system, a winding short-circuit impulse response spectrum is constructed, and the energy entropy characteristics of different frequency bands are extracted using the Hilbert energy spectrum. Secondly, the energy spectrum is converted into a time-frequency matrix, and its energy density distribution and short-term energy attenuation changes are quantified. Then, we perform support vector regression (SVD) decomposition and calculate the proportion of singular value vectors. Finally, multiple feature quantities are combined to construct a feature set, which is then input into the SVD model optimized by the particle swarm algorithm (PSO) for diagnosis. The experimental results show that the proposed method can accurately extract the physical characteristics of short-circuit impulse signals, and the error between the obtained transformer winding deformation diagnosis value and the actual state value is only 3.6%, which can provide guidance for online monitoring of the transformer winding state.

Comparison of the Dynamic Characteristics of Tuned Liquid Column Dampers with Different Elbow Forms

This study aims to examine experimentally the damping performance of Tuned Liquid Column Dampers (TLCD) by considering different elbow forms. Experiments carried out within the scope of the study point out the dynamic characteristics of TLCD, which has 45 (open) and 90 (closed) degree-elbow forms with theoretical angular frequencies that are the same. Experiments consist of ±5 and ±10 mm harmonic excitation amplitude to observe the relationship between the damping ratio and head loss coefficient with the mass ratio of the TLCD. These experiments with an incremental mass ratio of 0.05% from 0.80% to 1.00% and with an incremental mass ratio of 0.10% from 1.00% to 1.20% had been carried out in detail. Mass and stiffness Modification Factors (MF) are suggested to minimize the difference between numerical and experimental results. Determined MFs are used to examine the earthquake performance of TLCD on a Single-Degree-of-Freedom (SDOF) system. It is shown that the open elbow has an advantage of approximately 25% over the close elbow under earthquake performance on the SDOF system.

Vibrations Control of a SODF Structure Using TMD and MR Damper with Fuzzy Logic Algorithm Based on Velocity

In this study, the effects of using a passive tuned mass damper (TMD) with optimal parameters and a semi-active magnetorheological damper (MR damper) have been evaluated separately and simultaneously to control seismic vibrations of a linear single degree of freedom system, which was a simple "mass-spring-damper" model. OpenSEES software was used to model the single degree of freedom system and TMD, and MATLAB software was used to model MR damper. Fuzzy logic algorithm was used by MATLAB to determine the appropriate voltage for MR damper. By solving the dynamic equation of the damper, its force was calculated and this force was applied to the single degree of freedom system by OpenSEES. In the following, incremental dynamic analysis (IDA) has been performed using 7 earthquake records with maximum acceleration of 0.1g to1.0g with incremental step of 0.1g. Earthquakes are applied to four models without dampers, equipped with optimized TMD, equipped with MR damper, and equipped with both TMD and MR damper. The results showed that the simultaneous use of TMD and MR damper had the greatest effect on controlling the vibration of the system and reduced the maximum displacement by 22.36% and the maximum base shear by 21.85% compared to the case without dampers. Also, using only MR damper had a greater effect than using only TMD on reducing the seismic responses of the single degree of freedom system.

Effects of P-delta and spectral shape of ground motion records on strength reduction factor and inelastic displacement ratio of SDOF systems with deterioration

Force-based design, due to its usefulness and simplicity with reasonable accuracy, is the main core of current seismic design codes. In this method, the behavior factor and deflection amplification factor are major elements that play a key role in linking linear and nonlinear behavior. This article examines the strength reduction factor due to ductility (Rμ) and inelastic displacement ratio (Cμ), which are important parts of the codes coefficients, considering the spectral shape of ground motion records and the P-delta effect. The nonlinear response history analysis (NL-RHA) is conducted for an extensive range of single-degree-of-freedom (SDOF) systems with different periods by categorizing the ground motion records based on ε, SaRatio, and Np. The most important emphasis of the paper, to which attention has not been paid previously, is to study the effect of spectral shape of ground motion records as an input excitation indicator and the P-delta effect as a structural parameter on strength reduction factor and inelastic displacement ratio of SDOF systems. In addition, the effects of some other structural parameters, including the ductility factor, deterioration, post-capping stiffness ratio on strength reduction factor and inelastic displacement ratio are evaluated. The results demonstrate that the values of Rμ and Cμ change by changing ε in the range of short periods than long periods. However, they vary more clearly with the change in SaRatio and Np. Furthermore, with the increase in θ-αs as an index of the P-delta effect, the system requires more design strength, and Cμ increases as well.

Vibration suppression in SDOF systems coupled to a nonlinear energy sink under colored noise

This study analyzes the stochastic vibration suppression and optimization of the single-degree-of-freedom (SDOF) system equipped with the cubic stiffness nonlinear energy sink (NES) under colored noise excitation. Two theoretical methods are proposed: an integrated method (denoted as EL-ELM) that combines evolutionary Lyapunov theory with the equivalent linearization method, and the other is an empirical formula. Using EL-ELM, the coupled nonlinear system is simplified to an equivalent linear stochastic system, allowing for a theoretical analysis of the impact of NES structural parameters on suppression performance and the precise determination of optimal parameter configurations for the best suppression effects. Subsequently, based on the results from the EL-ELM and the response data, an empirical formula has been developed that clearly describes the comprehensive laws governing the optimal NES parameters as they vary with vibration system parameters and stochastic excitation. Through error analysis and comparison of the two methods, it is found that the empirical formula significantly outperforms EL-ELM in terms of accuracy and computational cost, but it is contingent on solid prior knowledge. This study explores the influence of NES structural parameters on the system’s dynamic response and energy, further validating the effectiveness of the proposed methods in identifying optimal structural parameters. The phenomenon of targeted energy transfer (TET) under different NES structural parameters is also explained. The methodologies introduced in this study have strengthened the theory of vibration suppression. Specifically, the empirical formula excels in accuracy and computational efficiency by effectively using prior knowledge. The EL-ELM method, owing to its theoretical insights, is vital for analyzing complex stochastic nonlinear models. Combining these approaches offers guidance for advancing vibration control in theoretical and practical domains.

Occurrence phase of peak responses to symmetric pulse loads

This study fills a challenging gap in the field of structural dynamics. A potential rule is theoretically proved: The peak displacements of undamped single-degree-of-freedom (SDOF) systems subjected to nonnegative but symmetric pulse loads necessarily occur within the pulse loading duration if the frequency ratio β<1, and after the pulse loading duration if the frequency ratio β>1. As a special case, the first peak displacements accurately take place at the end of the pulse loading when β=1. Also, the occurrence time of the first peak displacements has a theoretic value of tϕ=tp/2+T/4 in the case of β>1. Although this potential rule can be easily verified in certain cases, it has not been theoretically and systematically proved so far. A rigorous and complete proof is presented and featured by the proposed analysis based on Duhamel's integral. The analyzation circumvents the difficulties in analytically solving dynamic responses to different pulse loads in different shapes, but still reaches theoretical conclusions and yields a general law of structural dynamics. The proved law can be used to predict the occurrence phase of the first peak displacements when undamped SDOF systems subjected to nonnegative but symmetric pulse loads.

Performance Based Seismic Design of Reinforced Concrete Building

A novel approach is devised for the displacement-based design of diverse buildings, enabling them to withstand the seismic pressures encountered. The suggested approach necessitates figuring out the structure's yield and final displacements. These characteristics are derived from rough empirical relationships for the preliminary design. The inelastic demand spectrum corresponding to the estimated yield strength and the ductility capacity is then used to calculate the necessary strength of the structure. The structural strength required is then determined from the inelastic demand spectrum that coincides with the yield strength estimate and the ductility capacity. By converting it into an analogous single degree of freedom system, the method can be used to systems with many degrees of freedom. A modal analysis is performed on a model of the structure based on its preliminary design in order to determine the final design. For forces distributed according to the first mode, a pushover analysis of the structure now provides improved estimations of the yield and final displacements. The needed strength is then more precisely calculated using these improved estimates. To achieve convergence between the estimated and calculated values of the design displacements, it could be necessary to perform iterations. Lastly, to take into consideration the impact of higher modes on high-rise moment-resistant frames and shear wall systems.

Frame buildings are not an answer for earthquakes: The case of the February 2023 earthquakes in Türkiye

The February 2023 earthquakes in Türkiye caused widespread devastation and fatalities, highlighting the critical contrast in the seismic performance of reinforced concrete (RC) buildings with moment-resistant frames and those with structural walls. This study employs analyses of nonlinear single-degree-of-freedom (SDOF) systems using selected accelerograms from the earthquakes to evaluate the behavior of these structural systems. Three SDOF systems representing flexible and stiffer frame buildings, alongside RC wall buildings, were examined. The results highlighted the susceptibility of frame buildings to severe damage and collapse compared with the excellent performance of RC wall buildings. Moreover, the study emphasizes a shift of design focus from life safety to functional recovery. It also identifies potential scenarios regarding consecutive earthquake effects. Overall, the findings advocate adequately designed RC wall buildings for enhanced seismic performance and immediate occupation following major earthquakes.

Parametric Roll Motion Reduction of a Ship with a Passive Anti-Roll Tank

In this study, the influence of a U-tube passive anti-roll tank on parametric roll motion has been investigated as a mechanical absorber of a dynamic system. Specifically, this paper concentrates on how the initial conditions of a coupled roll motion and fluid motion in the tank act on the performance of an anti-roll tank. Parametrically excited roll motion was modeled as a single degree of freedom system incorporating heave and pitch effects by means of a time-varying restoring moment. Fluid motion in the tank was modeled as a single degree of freedom system. Coupled equations of motion were solved numerically and the results were presented in the time domain. Furthermore, the results were comparatively depicted as maximum roll amplitudes vs. initial values for a ship with and without an anti-roll U-tube tank.

Development of Modification Coefficient for Nonlinear Single Degree of Freedom System Considering Plasticity Range for Structures Subjected to Blast Loads

Development of Modification Coefficient for Nonlinear Single Degree of Freedom System Considering Plasticity Range for Structures Subjected to Blast Loads

Assessment of a Simplified Methodology for Preliminary Blast-Resistant Design of Insulated Precast Concrete Wall Panels

Abstract A Blast loading history can be represented by the peak pressure value and the impulse. Combinations of pressure and impulse of various blast loading scenarios can be collected to draw a pressure-impulse (P-I) curve that corresponds to a level of damage of a blast-resistant structural component. The P-I curve is typically used for a preliminary design or assessment of blast resistance structural components. Generating a P I curve for a given structural component and level of protection requires repeatedly ru nning a single degree of freedom system (SDOF) model, which can be tedious for early design stages. Hence, a simplified approach (i.e., the normalization approach) was utilized to promptly generate the P- I curve for insulated precast concrete wall panels, which relies on shifting a control P I curve using normalization factors. The generated P-I curves using the normalization approach are validated with the traditional SDOF approach curves; 95% of the examined insulted panel configurations have an error between +/-7%. As a result of the reached low error percentages, a spreadsheet- design tool was developed using this approach. The tool can further enable rapid evaluation of the performance of insulated precast concrete panels under blast loading during the preliminary design phase.

Fluid Structure Interaction Analyses of Elevated Storage Tank under Seismic Load

Elevated storage tanks are used to store fresh and waste water, cereal, oil and petrochemical materials. Therefore they serve in a wide area of utilization in this day and age. Analysis of a half full with water elevated storage tank is performed in the scope of this study. The structure is composed of a steel circular storage tank and frame typed steel stager. The structure is under the effect of earthquake loads as well as operational ones. Numerical model is generated by Finite Elements Analysis (FEA) software including fluid and structure parts. The interaction of the parts is provided by Coupled Eulerian Lagrangian (CEL) technique. While the structure constitutes the Lagrangian part, the fluid constitutes the Eulerian part. Interaction of Eulerian and Lagrangian parts are provided by general contact algorithms. Free surface movement of the water, maximum displacements and natural frequency values with related mode shapes of the structure are obtained after numerical analysis. Numerical results are verified by semi-analytical model, where the structure is modelled as single degree of freedom system. The accordance between results is numerically and visually obtained. The turbulent movement of water under the influence of an earthquake is modelled by utilizing the CEL technique. So, the effect of the dynamic pressure induced by the turbulence on the tank and the structural system has been taken into consideration.

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

A 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.