Single Degree Of Freedom Oscillator Research Articles

Correlation between seismic intensity measures and engineering demand parameters of reinforced concrete frame buildings through nonlinear time history analysis

In this paper, the correlation between different seismic intensity measures (IM) and engineering demand parameters (EDP) of reinforced concrete (RC) buildings was evaluated by means of nonlinear dynamic analyses (NLDA). The Costa Rican ground motion database and four RC buildings (4-, 6-, 8- and 10-stories) were considered in this research. Estimations of conventional IMs were derived directly from the ground motion records as well as from the response of a single degree of freedom (SDOF) linear oscillator, with a period equal to that of the fundamental period of the buildings. NLDAs were performed using unscaled and scaled records to account for different intensity levels, and the performance of the buildings was characterized in terms of three EDPs: the average inter-story drift ratio, the maximum inter-story drift ratio, and the Park & Ang damage index. Results demonstrate a significant improvement in the correlation between IMs and EDPs when the dynamic characteristics of the structure are accounted for through the response of the SDOF oscillator. On the other hand, the peak ground velocity turned out to be an effective IM, independent of the dynamic characteristics of the buildings. This is very useful when assessing potential seismic damage both for immediate decision-making and for the characterization of the seismic hazard of a region.

Vertical floor acceleration spectra for regular steel frame structures

The scope of this contribution is the assessment of the vertical peak component acceleration of single-degree-of-freedom (SDOF) oscillators attached to elastic moment-resisting steel frames subjected to recorded ground motions. These SDOF oscillators represent nonstructural components (NSCs) flexibly attached to the load-bearing structure whose acceleration demands correlate with their maximum strength demands. The results in terms of floor acceleration spectra show that the interaction between a vertically oscillating NSC and the structure is much more pronounced than for horizontally oscillating NSC of the same mass and natural frequency. This means that even for vertical NSCs with very small mass (in the case of the investigated eight-story frame 0.01% of the frame mass depending on the number of stories) a coupled analysis is required for a realistic prediction of the NSC acceleration demand. In a parametric study of vertical NSCs attached to steel frame structures of different number of stories, a median vertical peak component acceleration was observed up to 5.5 times larger than the vertical peak floor acceleration of the attachment node of the load-bearing structure and up to 20 times larger than the vertical peak ground acceleration. The amplification of the vertical acceleration response is thus of the same order of magnitude as for horizontal NSCs. This means that the vertical dynamic response of flexible NSCs cannot be neglected and should therefore be studied in more detail in the future.

A Wiener path integral technique for determining the stochastic response of nonlinear oscillators with fractional derivative elements: A constrained variational formulation with free boundaries

A technique based on the concept of Wiener path integral (WPI) is developed for determining approximately the joint response probability density function (PDF) of nonlinear oscillators endowed with fractional derivative elements. Specifically, first, the dependence of the state of the system on its history due to the fractional derivative terms is accounted for, alternatively, by augmenting the response vector and by considering additional auxiliary state variables and equations. In this regard, the original single-degree-of-freedom (SDOF) nonlinear system with fractional derivative terms is cast, equivalently, into a multi-degree-of-freedom (MDOF) nonlinear system involving integer-order derivatives only. From a mathematics perspective, the equations of motion referring to the latter can be interpreted as constrained. Second, to circumvent the challenge of increased dimensionality of the problem due to the augmentation of the response vector, a WPI formulation with mixed fixed/free boundary conditions is developed for determining directly any lower-dimensional joint PDF corresponding to a subset only of the response vector components. This can be construed as an approximation-free dimension reduction approach that renders the associated computational cost independent of the total number of stochastic dimensions of the problem. Thus, the original SDOF oscillator joint PDF corresponding to the response displacement and velocity is determined efficiently, while circumventing the computationally challenging task of treating directly equations of motion involving fractional derivatives. Two illustrative numerical examples are considered for demonstrating the reliability of the developed technique. These pertain to a nonlinear Duffing and a nonlinear vibro-impact oscillators with fractional derivative elements subjected to combined stochastic and deterministic periodic loading. Note that alternative standard approximate techniques, such as statistical linearization, need to be significantly modified and extended to account for such cases of combined loading. Remarkably, it is shown herein that the WPI technique exhibits the additional advantage of treating such types of excitation in a straightforward manner without the need for any ad hoc modifications. Comparisons with pertinent Monte Carlo simulation data are included as well.

Physical Model of Shallow Foundation under Dynamic Loads on Sands

Structures built on sands worldwide, with shallow foundations, have experienced damage and collapse during and after earthquakes. Two phenomena triggered the collapse: the liquefaction phenomenon and the P-Δ effects. However, current research and practice do not fully understand granular soil behavior during liquefaction and P-Δ effects, as proven by the sum of investigations on physical models, constitutive models, and laboratory testing proposals about these topics. A question appears at this point: what is the relationship between excitation frequency, displacement amplitude, and the triggering of overturning? To cope with this issue, the authors propose to create a physical 1-g model composed of a single-degree-of-freedom oscillator (SDOFO) capable of transmitting cyclic loadings to the soil in rocking vibration mode. The measurement methodology was based on computer vision using OpenCV by Python, which allowed the “free movement” of the SDOFO. The authors use computer vision as a suitable way to obtain displacements and times without sensors placed directly in the physical model. According to the results, it was possible to define an inversely non-linear relationship between frequency, displacement amplitude, and the total cycles required to reach overturning for different effective grain-size (D10).

Experimental study on ice-structure interaction phenomena of vertically sided structures

This study analyses the results from basin tests with a vertically sided cylindrical pile loaded by ice failing in crushing. Tests were performed with a ‘rigid’ structure and with structural models representing a series of single-degree-of-freedom (SDOF) oscillators covering a wide range of mass, frequency, and damping values. The structural models were represented by a real-time hybrid test setup, which combined physical and numerical components to measure real ice forces, apply the forces to a numerical structural model, and simulate the dynamics of the tested structural models in real-time. The test results are analysed and simple numerical simulations are performed to assess the relevance of several ice force characteristics observed from the ‘rigid’ structure tests to the ice-induced vibrations in the SDOF oscillators. The results from the rigid structure tests show that the median ice forcing frequency is linearly related to the ice drift speed. The mean and standard deviation of the ice forces on the rigid structure show a negative force-velocity gradient at low ice drift speeds and indications of a positive force-velocity gradient at higher ice drift speeds. The comparison of experimental results to the simulations of the single-degree-of-freedom oscillator tests shows that the positive force-velocity gradient at higher ice drift speeds allows to best capture the dynamics during continuous brittle crushing as observed in the experiments. Furthermore, the comparison shows that frequency lock-in initiation is primarily driven by the velocity-independent spatial frequency spectrum of the ice force signal. The added damping caused by the positive force-velocity gradient must be considered to capture the frequency lock-in initiation speeds measured in the constant deceleration experiments. The consideration of the negative force-velocity gradient at low relative velocities is not needed to capture these frequency lock-in initiation speeds as observed in the experiments. However, once frequency lock-in is initiated, the negative gradient is needed to correctly capture the dynamics during frequency lock-in. Analysis of the results shows that the peak forces during intermittent crushing at the end of the load build-up phase have a dependence on relative velocity equal to the load dependence on velocity of rigid structures at low speed. This indicates that intermittent crushing is not a purely brittle type of interaction.

Preliminary results from a time domain model of a clarinet-like system

A time domain model of a clarinet-like system has been built using the Simscape multidomain modeling system. This model largely replicates physical features that have been included in previous models by others. The air column has constant inside diameter. The tube has thirteen tone holes. The first three holes are spaced to give approximate chromatic tone spacing. The remaining holes have the same spacing as the last chromatic pair, and all the holes are sized to give a constant tone hole lattice cuttoff frequency of 1500 Hz. The reed is modeled as a single degree of freedom oscillator that terminates the “closed” end of the tube. Blowing pressure is provided by a mouth cavity of fixed volume. This model allows calculation transient behavior that becomes steady state if the input pressure remains constant. We present results of startup transients, slow crescendo and decrescendo playing, and oscillation behavior in response to opening and closing the tone holes. The current model does not accurately match the design of any particular instrument, but it does demonstrate an ability to calculate the behavior of some interesting playing behaviors of an instrument.

A simplified non-linear modelling strategy to generate fragility curves for old masonry buildings

In the context of historical seismology, studying the behaviour of historic masonry buildings is of great importance, as they are witnesses of past events. While interesting methods can be found in the literature to assess the seismic vulnerability of masonry structures subject to strong earthquakes, the topic of moderate seismicity, as encountered in many European countries, is still to be investigated. The present work proposes a global methodology to build fragility curves for existing masonry buildings. An efficient computational method to address the non-linear response of masonry structures is presented. The method is based on the modal decomposition of the structural response. An equivalent non-linear single degree of freedom oscillator is identified for the main modes. The modelling strategy enables to carry out numerous computations with a low computational effort, allowing for a probabilistic approach. To build fragility curves, a damage indicator based on the frequency shift is chosen. To validate the proposed approach, the simplified model of a real masonry building is compared to a full non-linear time history analysis. The method is eventually illustrated by the derivation of fragility curves for industrial masonry buildings.

New equivalent damping for the ATC nonlinear static procedure

The Nonlinear Static Procedure, also known in the literature as Pushover Analysis procedure or Capacity Spectrum Method (CSM), is mainly developed in the report of Applied Technology Council entitled “Seismic evaluation and retrofit of concrete buildings” ATC (1996). This is a widely practical engineering method to calculate the seismically induced ductility demand of a System, Structure or Component (SSC) without performing nonlinear transient analyses. Many authors have improved this method by introducing different equivalent linear concepts. However, discrepancy between predictive results by CSM and references obtained by transient nonlinear analyses remains quite important. In this paper, we present a new equivalent linear concept, in which the conventional equivalent frequency corresponding to the secant stiffness is adopted, and new equivalent damping value is calibrated so that the maximum displacement of equivalent linear oscillator is the same as the one of nonlinear oscillator. The new proposed damping values are derived from systematic analysis of the response of a series of nonlinear single degree of freedom (SDOF) oscillators to seismic type input motions. Oscillators have a bilinear elastic–plastic constitutive relationship with a hardening slope equal to 10% or 20% of the initial stiffness. Their natural frequency (f0) is such that f0/fc varies between 0.1 and 2.0, where fc is the central frequency of the input signal and their damping in elastic regime remained at 5%. Input motion is a set of one thousand synthetic filtered white noise signals. The novelty of the proposed approach is that new equivalent damping values exhibit strong frequency dependence on the f0/fc ratio, which has not been discussed so far. In order to validate the proposed equivalent linear concept, predictive results for nonlinear SDOF oscillators undergoing one hundred natural signals are performed by CSM with the new equivalent damping values and compared to reference results obtained from nonlinear transient analyses. It is concluded that the CSM using the new proposed equivalent damping values can determine the seismically induced ductility demand of SDOF oscillators with a better accuracy than methods used at the moment as a routine.

Analytical investigation of the seismic response of elastic oscillators placed on the top of rocking storey

In the present study the seismic response of elastic single degree of freedom oscillators placed on the top of a rocking storey is examined. A storey comprised of columns that may uplift under ground motion excitations can be regarded as an innovative and cost effective base isolation technique. Regarding the podium structural systems, apart from studying the rocking response in order to avoid the rocking overturn, it is important to examine the seismic demand on the superstructure so as to assess the rocking storey as base isolation system. From this point of view, structural as well as ground motion parameters that affect the seismic demands of the superstructure are examined. For this purpose, the superstructure’s response spectra, of various structural configurations, under free rocking response, analytical pulses and natural ground motions are presented and discussed. Finally, the effect of different assumptions regarding energy dissipation during impact on the response of the superstructure is investigated. The study of the podium structures results in useful remarks concerning the performance of the rocking storey as a base isolation system.

A multi-modal-analysis-based simplified seismic design method for high-rise frame-steel plate shear wall dual structures

In this paper, a simplified seismic design method of high-rise frame-steel plate shear wall (SPSW) system was proposed based on the multi-modal-analysis under the framework of performance-based seismic design. The seismic response of high-rise SPSW system was simplified to first and multiple-modal equivalent single degree-of-freedom (SDOF) oscillators. The dual frame-SPSW structure was decomposed into a frame system and SPSW system by controlling the relative contribution of SPSW system, and they were correspondingly simplified to a series of F-SDOF oscillators and S-SDOF oscillators. The analytical models of F-SDOF and S-SDOF oscillators were developed using the modal pushover analysis. By assuming the system responding linearly elastic for higher modes, the equivalent SDOF oscillator (D-SDOF) for the frame-SPSW system was developed by combining the F-SDOF and S-SDOF oscillators in parallel for each mode of vibration. The design procedure was developed based on the comparison of displacement thresholds against the displacement demands derived using the SRSS combination. A 15-story frame-SPSW system was adopted to verify the feasibility and demonstrate the design process of the simplified method. The result also shows the seismic demands derived by the equivalent dual SDOF oscillators have good consistency with that by frame-SPSW structure.

A simplified seismic design method for low-rise dual frame-steel plate shear wall structures

In this paper, a simplified seismic design method for low-rise dual frame-steel plate shear wall (SPSW) structures is proposed in the framework of performance-based seismic design. The dynamic response of a low-rise structure is mainly dominated by the first-mode and the structural system can be simplified to an equivalent single degree-of-freedom (SDOF) oscillator. The dual frame-SPSW structure was decomposed into a frame system and a SPSW system and they were simplified to an equivalent F-SDOF (SDOF for frame) oscillator and an equivalent S-SDOF (SDOF for SPSW) oscillator, respectively. The analytical models of F-SDOF and S-SDOF oscillators were constructed based on the OpenSees platform. The equivalent SDOF oscillator (D-SDOF, dual SDOF) for the frame-SPSW system was developed by combining the F-SDOF and S-SDOF oscillators in parallel. By employing the lateral force resistance coefficients and seismic demands of D-SDOF oscillator, the design approach of SPSW systems was developed. A 7-story frame-SPSW system was adopted to verify the feasibility and demonstrate the design process of the simplified method. The results also show the seismic demands derived by the equivalent dual SDOF oscillator have a good consistence with that by the frame-SPSW structure.

Sommerfeld effect in a single-DOF system with base excitation from motor driven mechanism

Sommerfeld effect is a nonlinear phenomenon observed in rotating machinery driven by a non-ideal source. It is characterised by capture at resonance for a finite range of drive power followed by a sudden release. This paper investigates the dynamics of a direct current (DC) motor driven mechanism which excites the base of a vibrating structure. A slotted cam follower mechanism and a scotch yoke mechanism are chosen for the base excitations of a single degree of freedom oscillator. It is shown that the Sommerfeld effect leads to capture-and-escape (jump) through multiple resonance zones, and as a result, some intermediate speed ranges are missed. The critical power/voltage input to escape through all the resonances may exceed that required to escape the primary resonance. Moreover, while increased structural damping makes it easier to escape resonance capture, it reduces the system efficiency at super-critical operating speeds. These are important design considerations for sizing the motor drives of these mechanisms. Analytical predictions for jumps are obtained from a steady-state power balance whereas the transient analysis is performed with the help of Bond Graph (BG) models and MSC-ADAMS software. The transient responses from the numerical models are used to verify the analytical results.

On discontinuous dynamics of a class of friction-influenced oscillators with nonlinear damping under bilateral rigid constraints

This article is concerned with the discontinuous dynamical behaviors of a class of single degree of freedom oscillators with linear and nonlinear springs and viscous dampers in the co-existence of bilateral rigid impact and friction through the theory of flow switchability and flow barrier in discontinuous dynamical systems, where considering the inequality of static and kinetic friction coefficients. By the analysis of vector fields and G-functions on the corresponding discontinuous boundaries for such an oscillator, the analytical conditions of all possible motions are established, which are in the form of a set of inequalities or equalities and can be used for the selection of control parameters in this kind of system. Additionally, based on mapping dynamics, the two-dimensional basic mappings are defined, the analytical predictions and stability analysis of different periodic motions are accomplished. Finally, our theoretical results are further demonstrated by numerical simulations by means of dimensional and dimensionless parameters with graphical illustrations of the time histories of displacement, velocity, G-function and the trajectories in phase space for the passable motion, stuck motion, impact motion, grazing motion and periodic motion etc., and stick and grazing bifurcation scenarios varying with excitation frequency and amplitude are also given.

Wind design spectra for generalisation

Previous research has shown that wind acceleration components produce a signal that can vibrate single-degree of-freedom oscillators, whose dynamic responses enable to configure design spectra for structures subject to wind. These wind design spectra present an alternative method for evaluating the dynamic response of structures and are a suitable tool for running modal analyses. Here, a generalised method for producing wind design spectra is proposed. The method consists of scaling existing spectra to adjust to a wider range of building properties and terrain conditions. The modelling technique is tested on a benchmark building to prove that its results are consistent with experimental evidence reported in the past.

System identification of lumped parameter models for weakly nonlinear systems

Experimental system identification through modal analysis is often conducted by assuming linear constitutive laws and kinematic relationships. Significant prediction errors from these identified models can occur when the amplitude dependent contribution of nonlinearities is ignored. In this paper, the impact of such nonlinearities on the parameter identification and accuracy of a linear model is shown to depend significantly on the experimental procedures. Using modal analysis techniques and focusing on vibration near the fundamental mode, a cantilevered beam is modeled as a single degree of freedom oscillator with parameters to be determined through experimental system identification. Various testing procedures are described in detail and used to demonstrate the impact of nonlinearities on the outcome of different system identification processes. The signal control methods compared are (1) open loop, (2) constant input, and (3) constant response excitation. Forcing types considered are single point and distributed. Nonlinearities in experimental response are shown for all procedures, and it is demonstrated that the observed character and degree of the nonlinearities vary depending on the testing procedure. The paper makes various recommendations for experimental system identification procedures using linear structural models, which include developing empirical models for parameter shifts due to nonlinearities and using constant response tests to minimize nonlinear effects in the measured system response. By implementing these identification methods, parameter variations can be included in the linear model to significantly reduce predicted response and parameter errors for weakly nonlinear second-order systems.

Seismic response assessment of rocking systems using single degree‐of‐freedom oscillators

SummaryA new modeling for the seismic response assessment of free‐standing, rigid or flexible, pure rocking systems is presented. The proposed modeling is based on equivalent single degree‐of‐freedom (SDOF) oscillators that can be implemented with common engineering software or user‐made structural analysis codes. The SDOF models adopted use beam elements that are connected to a nonlinear rotational spring with negative stiffness that describes the self‐centering capacity of the rocking member. The loss of energy at impact is treated with an “event‐based” approach consistent with Housner's theory. Different variations pertinent to rigid blocks are first presented, and then the concept is extended to the flexible case. The implementation of the method requires some minor programming skills, while thanks to the versatility of the finite element method, it is capable to handle a variety of rocking problems. This is demonstrated with two applications: (a) a vertically restrained block equipped with an elastic tendon and (b) a rigid block coupled with an elastic SDOF oscillator. The accuracy and the efficiency of the proposed modeling is demonstrated using simple wavelets and historical ground motion records.

Experimental Research on the Impact Environment of Multiplate Cabinet

As an installation and protection device for electrical and electronic components, a shipboard cabinet is a typical multiplate structure. In order to study the impact environment distribution laws of such structures, impact testing was carried out on a shipboard cabinet under four working conditions in this paper. In addition, the impact response characteristics of such a multiplate structure were determined by numerical simulation and theoretical analysis. The impact environments of some pivotal points in cabinet were measured and some laws of dynamic response were found. The impact environment of central position was more severe on a single plate because of the first vibration modal. For different plates, the responses were usually similar at low-frequency band and a little different at high-frequency band. The theoretical analysis of the single degree of freedom oscillator was carried out, and the sensitivity of the response to the different characteristic frequencies was discussed based on the shock spectrum theory. A new method of calculating the response at a special frequency was proposed and verified.

Equivalent linearization parameters of soil-MDOF structure systems subjected to pulse-like earthquakes

In seismic design codes and documents, the period and damping ratio of a linear fixed–base single degree-of-freedom (SDOF) system are tuned to return the dynamic response of a given nonlinear fixed–base SDOF structure through a so-called equivalent linearization method. The period and damping ratio of the equivalent linear system are referred to as equivalent or effective linear parameters (ELPs). These ELPs may vary due to higher modes of the superstructure, the soil-structure interaction (SSI), or the pulse effects of the earthquake ground motions. Hence, this paper focuses on an extension of the equivalent linearization concept to nonlinear soil-multi-story structure systems under pulse-like earthquakes where the whole system is substituted with an equivalent fixed-base SDOF oscillator. To achieve this goal, the underlying soil is assumed as a homogeneous half-space medium and the superstructure is modeled as a nonlinear multiple degrees-of-freedom (MDOF) shear building assuming bilinear behavior for each story. Non-dimensional frequency and aspect ratio parameters are used to enforce the SSI effects in the system. The seismic responses are determined under an extensive ensemble of 91 pulse-like earthquakes for two different ranges of fixed-base structure-to-pulse period ratios, and the ELPs are determined for a range of soil-structure systems throughout a statistical framework. It is found that higher modes have decreasing effects on the ELPs of soil-MDOF structure systems, while the effects of the strain-hardening ratio of each story are insignificant. In addition, the impulsive nature of the ground motions strongly affects the ELPs, and this influence is far more pronounced for structure-to-pulse period ratios between 0.8 and 1.2, i.e., very close to the resonance of the first mode of the superstructure.

Computational Fluid Dynamics Reproduction of Nonlinear Loads on a Vertical Column During Extreme Irregular Wave Events

Recently, a method for numerical reproduction of measured irregular wave events has been developed. The measured motion of the wave maker flaps defines the wave kinematics at the boundary of the numerical simulation in order to generate the waves. When such data are not available, the control signal of the wave maker can, instead, be generated from a given free surface elevation following the same procedure as in model tests. This procedure is applied to a model test case with extreme irregular wave events and resulting nonlinear global wave loads on a vertical cylinder, focusing on higher-order ringing excitation. The purpose of the investigation is twofold: (1) to validate the wave reconstruction procedure and (2) to validate the resulting computational fluid dynamics (CFD) ringing loads with the given waves. In order to better understand the frequency content in the CFD-generated loads, wavelet analysis as well as the response of a single degree-of-freedom (SDOF) oscillator is examined and compared with the corresponding results for the third-order wave forcing based on the MacCamy–Fuchs (MF) and Faltinsen, Newman, Vinje (FNV) formulations. The results show generally good agreement between CFD and experiment both in the waves and in the loads; discrepancies found in the loads mainly originate from corresponding uncertainties in the wave reconstruction. Wave breaking may be one source of uncertainty. The MF + FNV formulation showed reasonable prediction of the maximum responses of an SDOF oscillator, but could not capture the loads well at all of the important frequencies.

A Dimensionless Parameter Analysis of a Cylindrical Tube Electromagnetic Vibration Energy Harvester and Its Oscillator Nonlinearity Effect

This paper investigates a single degree of freedom oscillator in a cylindrical tube vibration energy harvester which is applicable in a low-frequency range. A duffing-type nonlinear dynamic differential equation of the oscillator was developed for the nonlinear analysis and solved by using the harmonic balance method in order to widen energy harvesting frequency bandwidth. The exploitation of nonlinear spring force interactions of the oscillator to enhance the generator’s performance is also presented in this paper. The dimensionless harvested power formula of the nonlinear mass-spring-damper system was derived, and a parameter study was conducted for the design optimization of the harvester. The main contribution of this paper is to establish a dimensionless performance analysis method of a nonlinear electromagnetic vibration energy harvester system and to disclose the effects of the parameters and nonlinearity of the system on the harvesting performance.